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+Weird integrals
+
+$$ \pi \cot(\pi x) = \frac{1}{x} - 2 \int_{0}^{\infty} \frac{\sinh(xt)}{e^t - 1} dt \quad \text{for } |x| < 1 $$
+
+$$ \pi^2 \csc^2(\pi x) = \frac{1}{x^2} + 2 \int_0^\infty \frac{t \cosh(xt)}{e^t - 1} dt \quad \text{for } |x| < 1 $$
+
+Obviously the second one is the result of differentiating and negating the first.
+
+Tested both numerically. Both work well as long as $|x|$ isn't too close to 1.
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+yield another Cantor set, the only remaining case to consider
+is $X = \mu_0 \times \prod_{i=1}^k X_i$ where each $X_i$ is homeomorphic to $\mu_n$ for
+some $n \ge 1$. The path components of $X$ are of the form $\{p\} \times
+\prod_{i=1}^k X_i$ where $p \in \mu_0$. Since self homeomorphisms of $X$ can not
+take two points in a single path component to points in distinct
+path components, it follows that $X$ is not 2-homogeneous. $\square$
+
+The preceding corollary provides a partial negative answer
+to the question in [KKT] as to whether any finite or countable
+product of $\mu_1$ with itself is 2-homogeneous. Also, $\mu_n \times \mu_n$
+provides an example of a homogeneous space that is (2n)-dimensional, that is $LC^{n-1}$, and that is not 2-homogeneous.
+We now proceed to produce an example of a homogeneous $LC^{n-1}$space of dimension ($n+2$) that is not 2-homogeneous.
+
+**Theorem 4.4.** Let $X = \mu_n \times Y$ where $n \ge 1$, $Y$ is compact, path connected, homogenous and locally homologically Čech $n$-connected. Then $X$ is not 2-homogeneous.
+
+*Proof:* We proceed as in the proof of Theorem 4.1. Fix distinct points $x$ and $y$ in $\mu_n$. Choose distinct points $r$ and $s$ in $Y$. We will show that there is no homeomorphism $h : X \to X$ such that $h(\{(x,r), (x,s)\}) = \{(x,r), (y,s)\})$. Assume to the contrary that there is such a homeomorphism $h$.
+
+Let $p$ be projection of $X$ onto $\mu_n$,and let $q$ be projection
+of $X$ onto $Y$. Choose a neighborhood $U$ of $x$ in $\mu_n$ such that
+$p(h(U \times \{r\})) \cap p(h(U \times \{s\})) = \emptyset$, such that $q(h(U \times \{r\})) \cap
+q(h(U \times \{s\})) = \emptyset$, and such that the inclusion induced ho-
+momorphisms from the $n$-th Čech homology of $q(h(U \times \{r\}))$
+and from the $n$-th Čech homology of $q(h(U \times \{s\}))$ to the $n$-th
+Čech homology of $Y$ are trivial.
+
+As in Theorem 4.1, choose an embedding $e: S^n \to U$ and
+let $f_1: S^n \to X$ be the map given by $f_1(a) = (e(a), \{r\})$ and
+let $f_2: S^n \to X$ be the map given by $f_2(a) = (e(a), \{s\})$.
+Since $Y$ is path connected, the maps $f_1$ and $f_2$ are homotopic.
+As before, the map $e$ is essential with respect to $n$-th Čech
+homology. It follows that the maps $f_1$ and $f_2$ are also essential
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diff --git a/samples/texts/4807413/page_11.md b/samples/texts/4807413/page_11.md
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+with respect to *n*-th Čech homology, and so the maps $h \circ f_1$ and $h \circ f_2$ are essential with respect to *n*-th Čech homology.
+
+Since $p \circ h \circ f_1$ and $p \circ h \circ f_2$ are homotopic and have disjoint images, Lemma 3.2 again implies that these maps are not essential with respect to *n*-th Čech homology.
+
+So both $q \circ h \circ f_1$ and $q \circ h \circ f_2$ are essential with respect to *n*-th Čech homology. But this contradicts the fact that the homeomorphisms from the *n*-th Čech homology of $q(h(U \times \{r\})$ and from the *n*-th Čech homology of $q(h(U \times \{s\}))$ to the *n*-th Čech homology of $Y$ are trivial. So there is no homeomorphism $h$ as assumed, and it follows that $X$ is not 2-homogeneous.
+
+**Corollary 4.5.** The product of $\mu_n$, $n \ge 1$ with any ANR or with any manifold is not 2-homogeneous.
+
+**Corollary 4.6.** For each positive integer $n$, there is an $(n+1)$-dimensional homogeneous compact metric space that is $(n-1)$-connected and is not 2-homogeneous.
+
+*Proof:* That the space $\mu_n \times S^1$ satisfies the conditions in the corollary follows directly from the previous theorem. $\square$
+
+Note that the techniques used in proving Theorems 4.2 and 4.4 can be used to prove that any finite product of Menger spaces and manifolds, or any finite product of Menger spaces and spaces that are locally homologically Čech $n$-connected for sufficiently many values of $n$, are not 2-homogeneous. These techniques can also be used to prove that such products are not $n$-homogeneous for values of $n$ greater than 2. One can also use the techniques in the above theorems to analyze the type of self homeomorphisms of finite products of Menger spaces as was done in [KKT] for the product $\mu_1 \times \mu_1$.
+
+## 5. QUESTIONS
+
+The following questions remain open.
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+**Question 1.** Is there a compact metric space of dimension less than ($n + 2$) that is homogeneous, locally $n$-connected, and not 2-homogeneous?
+
+**Question 2.** If a homogeneous compact metric space is locally $n$-connected for all $n$, is the space necessarily 2-homogeneous?
+
+## REFERENCES
+
+[An1] R. D. Anderson, *A characterization of the universal curve and a proof of its homogeneity*, Ann. of Math., **68** (1958), 313-324.
+
+[An2] —, *1-dimensional continuous curves and a homogeneity theorem*, Ann. of Math., **68** (1958), 1-16.
+
+[Be] M. Bestvina, *Characterizing k-dimensional universal Menger compacta*, Memoirs Amer. Math. Soc., **71** (1988), no.380.
+
+[CF] M. L. Curtis and M. K. Fort, *The fundamental group of one-dimensional spaces*, Proc. Amer. Math. Soc., **10** (1959), 140-148.
+
+[ES] S. Eilenberg and N. Steenrod, *Foundations of Algebraic Topology*, Princeton University Press, Princeton, New Jersey, 1952.
+
+[Ga] D. J. Garity, *On multiple homogeneity of products of Menger spaces*, Proc. of the Ninth Annual Workshop in Geometric Topology (J. Henderson, F. Tinsley ed.), Colorado College, Colorado Springs, Colorado, 1992, 18-24.
+
+[Kel] J. Kennedy Phelps, *Homomorphisms of products of universal curves*, Houston J. Math., **6** (1980), 127-143.
+
+[Ke2] —, *A condition under which 2-homogeneity and representability are the same in continua*, Fund. Math., **121** (1984), 89-98.
+
+[Ku] K. Kuperberg, *On the bihomogeneity problem of Knaster*, Trans. Amer. Math. Soc., **321** (1990), 129-143.
+
+[KKT] K. Kuperberg, W. Kuperberg, and W. R. R. Transue, *On the 2-homogeneity of Cartesian products*, Fund. Math. CX (1980), 131-134.
+
+[KKT2] —, *Homology separation and 2-homogeneity*, Continua with the Houston problem book (H. Cook, W. T. Ingram, K. T. Kuperberg, A. Lelek, P. Minc, ed.), vol. 170, Marcel Decker, New Your, 1995.
+
+[Mg] K. Menger, *Kurventheorie*, Teubner, Berlin - Leipzig, 1932.
+
+[Mu] J. Munkres, *Elements of Algebraic Topology*, Benjamin/Cummings, Menlo Park, California, 1984.
+
+[Un] G. S. Ungar, *On all kinds of homogeneous spaces*, Trans. Amer. Math. Soc., **212** (1975), 393-400.
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@@ -0,0 +1,3 @@
+Oregon State University
+Corvallis, OR 97331
+e-mail: garity@math.orst.edu
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diff --git a/samples/texts/4807413/page_8.md b/samples/texts/4807413/page_8.md
new file mode 100644
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@@ -0,0 +1,23 @@
+phism $\lambda$ to $\theta$ can be constructed that takes elements of the
+form $[\alpha_j \otimes \beta_{n-j}]$ to $[\alpha_j] \otimes [\beta_{n-j}]$ in $H_j(M_n^i) \otimes H_{n-j}(X_i)$ when
+$\alpha_j$ and $\beta_{n-j}$ are cycles, and takes $[\alpha_j \otimes \beta_{n-j}]$ to 0 otherwise.
+Since $H_j(M_n^i)$ is nonzero only for $j=0$ or $j=k$, it follows
+that
+
+$$
+\lambda \circ u([c]) = [p_{i\#}(c)] \otimes [q_{i\#}(cB_0)] + [p_{i\#}(cF_0)] \otimes [q_{i\#}(c)]
+$$
+
+Since $\lambda$ and $u$ are isomorphisms, if $[p_{i\#}(c)]$ and $[q_{i\#}(c)]$ are both 0, then $[c] = 0$.
+
+Let $\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n, \ldots)$ be a nontrivial element of the k-th Čech homology of Y that is given by the hypotheses. That is, each $\alpha_i = f_{i*}(\gamma)$ where $\gamma$ is a generator of $H_k(S^k)$ and where $f_i$ is the composition of $f$ with inclusion from Y to $Y_i$. Then there is a positive integer $k$ such that for all $i \ge k$, $\alpha_i \ne 0$ in $H_k(Y_i)$. By the previous paragraph, either $p_{i*}(\alpha_i)$ or $q_{i*}(\alpha_i)$ is nontrivial for all $i \ge k$. The result now follows by considering the definition of the induced homomorphism on Čech homology groups.
+
+If in fact $k < n$, then each $p_{i*}(\alpha_i)$ is trivial, so that $p_*(\alpha)$ is trivial, and thus $q \circ f$ is essential with respect to $k$-th Čech homology. $\square$
+
+4. THE MAIN RESULTS
+
+**Definition 4.1.** A space X is locally homologically Čech *n*-connected if for each point *p* ∈ X and for each neighborhood U of *p*, there is a neighborhood V of *p* such that the inclusion induced homomorphism from the *n*-th Čech homology of V to the *n*-th Čech homology of U is trivial.
+
+**Theorem 4.2.** Let $X = \mu_n \times \prod_{i=1}^k Y_i$ where $n \ge 1$, and where each $Y_i$ is homeomorphic to $\mu_j$ for some $j \ge n$. Then $X$ is not 2-homogeneous.
+
+*Proof:* Fix distinct points x and y in μₙ. Choose points r and s in Πk[i=1 Yᵢ so that for each i, 1 ≤ i ≤ k, qᵢ ≠ rᵢ. We will show that there is no homeomorphism h : X → X such that
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@@ -0,0 +1,11 @@
+$h(\{(x, r), (x, s)\} = \{(x, r), (y, s)\}). Assume to the contrary that there is such a homeomorphism $h$.
+
+Let $p$ be projection of $X$ onto $\mu_n$, let $q$ be projection of $X$ onto $\prod_{i=1}^k Y_i$ and let $q_i$ be projection of $X$ onto $Y_i$. Choose a neighborhood $U$ of $x$ in $\mu_n$ such that $p(h(U \times \{r\}) \cap p(h(U \times \{s\})) = \emptyset$, and such that for each $i$, $q_i(h(U \times \{r\}) \cap q_i(h(U \times \{s\})) = \emptyset$. Choose an embedding $e: S^n \to U$. This is possible by Lemma 3.2. Let $f_1: S^n \to X$ be the map given by $f_1(a) = (e(a), \{r\})$ and let $f_2: S^n \to X$ be the map given by $f_2(a) = (e(a), \{s\})$. Since $\prod_{i=1}^k X_i$ is path connected, the maps $f_1$ and $f_2$ are homotopic. By Lemma 3.1, the map $e$ is essential with respect to $n$-th Čech homology. It follows that the maps $f_1$ and $f_2$ are also essential with respect to $n$-th Čech homology, and so the maps $h \circ f_1$ and $h \circ f_2$ are essential with respect to $n$-th Čech homology.
+
+Lemma 3.3 implies that for each $i$, either $p \circ h \circ f_i$ or $q \circ h \circ f_i$ is essential with respect to $n$-th Čech homology. Since $f_1$ and $f_2$ are homotopic, this implies that either both $p \circ h \circ f_1$ and $p \circ h \circ f_2$ are essential with respect to $n$-th Čech homology, or both $q \circ h \circ f_1$ and $q \circ h \circ f_2$ are essential with respect to $n$-th Čech homology. Since $p \circ h \circ f_1$ and $p \circ h \circ f_2$ are homotopic and have disjoint images, Lemma 3.2 implies that these maps are not essential with respect to $n$-th Čech homology.
+
+So both $q \circ h \circ f_1$ and $q \circ h \circ f_2$ are essential with respect to $n$-th Čech homology. Repeated application of Lemma 3.2 shows that for each $i$, both $q_i \circ h \circ f_1$ and $q_i \circ h \circ f_2$ are not essential with respect to $n$-th Čech homology. This leads to a contradiction. So there is no homeomorphism $h$ as assumed, and it follows that $X$ is not 2-homogeneous. $\square$
+
+**Corollary 4.3.** Any finite product of two or more Menger spaces, where at least one of the Menger spaces is not $\mu_0$, is not 2-homogeneous. In particular, $\mu_m \times \mu_n$, where $\max\{m, n\} \ge 1$ is not 2-homogeneous.
+
+*Proof:* The case where each factor is not $\mu_0$ follows directly from the previous theorem. Since finite products of Cantor sets
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+7th October
+
+Foundation Plus 5-a-day
+
+Work out
+
+$$3\frac{2}{3} - 2\frac{3}{4}$$
+
+A spinner has two sections, yellow and blue.
+
+The probability of the spinner landing on yellow is $\frac{3}{5}$
+
+1st Spin
+
+2nd Spin
+
+Yellow
+$\frac{3}{5}$
+Blue
+Yellow
+Blue
+
+The spinner is spun twice.
+
+Find the probability that the spinner lands on blue twice.
+
+Find the probability that the spinner lands on different colours.
+
+Expand and simplify
+
+$$(x - 10)(x + 8)$$
+
+Rachel buys a DVD for £18.50.
+
+A year later she sells it for £15.91
+
+What is the percentage decrease in value of the DVD?
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+# Partial angular coherence and the angular Schmidt spectrum of entangled two-photon fields
+
+Anand Kumar Jha,¹ Girish S. Agarwal,² and Robert W. Boyd¹
+
+¹The Institute of Optics, University of Rochester, Rochester, New York 14627, USA
+
+²Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
+
+(Received 13 September 2011; published 27 December 2011)
+
+We study partially coherent fields that have a coherent-mode representation in the orbital-angular-momentum-mode basis. For such fields, we introduce the concepts of the angular coherence function and the coherence angle. Such fields are naturally produced by the process of parametric down-conversion—a second-order nonlinear optical process in which a pump photon breaks up into two entangled photons, known as the signal and idler photons. We show that the angular coherence functions of the signal and idler fields are directly related to the angular Schmidt (spiral) spectrum of the down-converted two-photon field and thus that the angular Schmidt spectrum can be measured directly by measuring the angular coherence function of either the signal or the idler field, without requiring coincidence detection.
+
+DOI: 10.1103/PhysRevA.84.063847
+
+PACS number(s): 42.50.Ar, 42.65.Lm, 03.65.Ud, 42.25.Kb
+
+## I. INTRODUCTION
+
+Classical coherence theory is a well-established subject. Its modern interpretation is largely due to Wolf and co-workers [1,2]. The study of coherence in the context of entangled fields has revealed a much deeper understanding of entanglement itself [3–6]. One of the central concepts of classical coherence theory is the cross-spectral-density function, which quantifies the field correlations in the space-frequency domain. A cross-spectral-density function always has a unique coherent-mode representation, which is a way to represent a partially coherent field as an incoherent sum of a finite number of completely coherent fields. In this paper, we study the partially coherent fields that have a coherent-mode representation in the orbital-angular-momentum basis. We show that, for such fields, it is very useful to introduce the concepts of the angular coherence function and the coherence angle. In fact, such fields are naturally produced by the process of parametric down-conversion (PDC), owing to the conservation of orbital angular momentum (OAM) in parametric down-conversion [7,8].
+
+The OAM entanglement of PDC photons [8] is a great resource for quantum-information-based protocols due to the fact that the OAM basis provides a discrete but infinite-dimensional Hilbert space, as opposed to the polarization basis, which provides only a two-dimensional Hilbert space [9,10]. For this reason, an accurate measurement of the dimensionality of the OAM-entangled photons is very important. There are two generic ways in which the dimensionality can be measured. The first is by directly measuring the two-photon intensity in coincidence at different values of the OAM mode indices [11,12] and the second is by using the Hong-Ou-Mandel interference technique [13]. In this paper, we propose an alternative way [14] of measuring the dimensionality, by measuring the angular coherence function of either of the down-converted photons. This scheme is different from the existing schemes in that it requires only singles detection, as opposed to the coincidence detection required in the other schemes.
+
+## II. PARTIALLY COHERENT FIELDS
+
+### A. General representation
+
+In this section, we briefly describe the general representation of partially coherent fields. Let {$V(\mathbf{r},t)$} be an
+
+ensemble representing the statistical properties of a partially coherent field that is both stationary, at least in the wide sense, and ergodic. One way to characterize the statistical correlations of such fields is through the mutual coherence function $\Gamma(\mathbf{r}_1,\mathbf{r}_2,\tau)$, which quantifies the field correlation between the space-time points $(\mathbf{r}_1,t)$ and $(\mathbf{r}_2,t+\tau)$, and is defined as $\Gamma(\mathbf{r}_1,\mathbf{r}_2,\tau) = \langle V^*(\mathbf{r}_1,t)V(\mathbf{r}_2,t+\tau) \rangle$, where $\langle \cdots \rangle_e$ represents the ensemble average. Another way, which is more convenient, to characterize the field correlations is through the cross-spectral-density function $W(\mathbf{r}_1,\mathbf{r}_2,\omega)$, which quantifies the field correlations in the space-frequency domain and is defined as
+
+$$W(\mathbf{r}_1, \mathbf{r}_2, \omega) \equiv \frac{1}{2\pi} \int_{-\infty}^{\infty} \Gamma(\mathbf{r}_1, \mathbf{r}_2, \tau) e^{i\omega\tau} d\tau. \quad (1)$$
+
+For conceptual clarity, we suppress from now on the frequency argument in the definition of the cross-spectral-density function. We also assume that the cross-spectral-density function is a continuous function of $\mathbf{r}_1$ and $\mathbf{r}_2$ within the domain $D$ of interest. The cross-spectral-density function is a bounded function, in the sense that
+
+$$\int_D \int_D |W(\mathbf{r}_1, \mathbf{r}_2)|^2 d\mathbf{r}_1 d\mathbf{r}_2 < \infty. \quad (2)$$
+
+Further, it is a Hermitian function, that is,
+
+$$W^*(\mathbf{r}_1, \mathbf{r}_2) = W(\mathbf{r}_2, \mathbf{r}_1). \quad (3)$$
+
+And most importantly, it is a non-negative definite function, that is,
+
+$$\int_D \int_D W(\mathbf{r}_1, \mathbf{r}_2) f^*(\mathbf{r}_1) f(\mathbf{r}_2) d\mathbf{r}_1 d\mathbf{r}_2 \ge 0, \quad (4)$$
+
+where $f(\mathbf{r})$ is any square integrable function. The physical interpretation of the non-negative-definiteness condition is that the intensity distribution produced by the field, with an aperture function $f(\mathbf{r})$ in domain $D$, on a screen is always non-negative. The above conditions, along with the multidimensional version of the Mercer theorem, imply that the cross-spectral-density function $W(\mathbf{r}_1,\mathbf{r}_2)$ is a Hilbert-Schmidt kernel and that it has a coherent-mode representation of the form [2]
+
+$$W(\mathbf{r}_1, \mathbf{r}_2) = \sum_n \alpha_n \psi_n^*(\mathbf{r}_1) \psi_n(\mathbf{r}_2). \quad (5)$$
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+The functions $\psi_n^*(\mathbf{r})$ are the eigenfunctions and the coefficients $\alpha_n$ are the eigenvalues of the integral equation $\int W(\mathbf{r}_1, \mathbf{r}_2)\psi_n(\mathbf{r}_1)d\mathbf{r}_1 = \alpha_n\psi_n(\mathbf{r}_2)$. The Hermiticity and the non-negative definiteness of $W(\mathbf{r}_1, \mathbf{r}_2)$ ensure that the integral equation has at least one nonzero eigenvalue and that all the eigenvalues are real and non-negative, i.e., $\alpha_n \ge 0$. The above equation can be rewritten as $W(\mathbf{r}_1, \mathbf{r}_2) = \sum_n \alpha_n W^{(n)}(\mathbf{r}_1, \mathbf{r}_2)$, where $W^{(n)}(\mathbf{r}_1, \mathbf{r}_2) \equiv \psi_n^*(\mathbf{r}_1)\psi_n(\mathbf{r}_2)$. This representation implies that for any partially coherent field there exists at least one basis in which the cross-spectral-density function can be represented as a superposition of modes that are completely coherent in the space-frequency domain.
+
+**B. Partially coherent field in the Laguerre-Gaussian basis**
+
+Every type of partially coherent field is characterized by its unique coherent-mode representation. In this paper, we are investigating partially coherent fields that have a coherent-mode representation in the Laguerre-Gaussian (LG) basis. A coherent mode in the LG basis is referred to as an LG mode or an LG beam; these are the exact solutions of the paraxial Helmholtz equation. The normalized field amplitude of these modes at $z=0$ in the cylindrical coordinate system is given by
+
+$$
+\begin{align}
+[\mathrm{LG}_p^l(\rho, \phi)] &\equiv [\mathrm{LG}_p^l(\rho)]e^{il\phi} = \sqrt{\frac{2p!}{\pi(|l|+p)!}} \nonumber \\
+&\quad \times \frac{1}{w_0}\left(\frac{\sqrt{2}\rho}{w_0}\right)^{|l|} L_p^l\left(\frac{2\rho^2}{w_0^2}\right) \exp\left(-\frac{\rho^2}{w_0^2}\right)e^{il\phi}, \tag{6}
+\end{align}
+$$
+
+where $w_0$ is the beam waist radius at $z=0$ and $l$ is the azimuthal mode index. Due to the azimuthal phase dependence of $e^{il\phi}$, these modes carry an orbital angular momentum of $lẖ$ per photon [15]. These modes have been extensively studied in the last few decades. Such fields are very important as they hold promise for many new fascinating applications, especially in quantum-information science. The type of partially coherent fields that we consider in this paper has the following coherent-mode representation:
+
+$$
+\begin{align}
+W(\mathbf{r}_1, \mathbf{r}_2) &\rightarrow W(\rho_1, \phi_1; \rho_2, \phi_2) \nonumber \\
+&= \langle V^*(\rho_1, \phi_1)V(\rho_2, \phi_2) \rangle_e \nonumber \\
+&= \sum_{l,p,p'} \alpha_{lp'} [\mathrm{LG}_p^{*l}(\rho_1, \phi_1)][\mathrm{LG}_{p'}^{l}(\rho_2, \phi_2)], \tag{7}
+\end{align}
+$$
+
+where $V(\rho,\phi)$ is a single realization of the field at location $(\rho,\phi)$. We note that the field is a coherent superposition of modes carrying different values of the orbital-angular-momentum mode index $l$. The specific question that we now ask is the following: “For a field that has the above form for the cross-spectral-density function, what is the correlation between the fields at two different angular positions, after the correlations have been integrated over the radial dimensions?” In order to answer this question, we consider the situation as shown in Fig. 1. Consider a partially coherent field passing through a screen in the form of a double angular slit. The two slits are centered at angular positions $\phi_1$ and $\phi_2$, respectively.
+
+FIG. 1. (Color online) A scheme for studying the angular coherence properties of a partially coherent beam.
+
+The separation between the slits is $\Delta\phi = \phi_1 - \phi_2$. The field $\Psi(\rho,\phi)$ immediately after the aperture is given by
+
+$$
+\Psi(\rho, \phi) = V(\rho, \phi)\Phi(\phi), \quad (8)
+$$
+
+where $V(\rho,\phi)$ is the incoming field and $\Phi(\phi)$ is the amplitude transmission function of the aperture. We decompose the above field in the LG basis as
+
+$$
+\Psi(\rho, \phi) = \sum_{l,p} A_{lp} [\mathrm{LG}_p^l(\rho, \phi)], \quad (9)
+$$
+
+where
+
+$$
+A_{lp} = \iint \rho d\rho d\phi [\mathrm{LG}_p^{kl}(\rho, \phi)] V(\rho, \phi) \Phi(\phi) \quad (10)
+$$
+
+is the probability amplitude for the field to be found in mode $\mathrm{LG}_p^l(\rho, \phi)$. Since we are interested in field correlations at different angular positions, we sum over all the $p$ modes and obtain the intensity $I_l$ of the field for a given value of $l$ as
+
+$$
+I_l = \sum_{p=0}^{\infty} I_{lp} = \sum_{p=0}^{\infty} \langle A_{lp}^* A_{lp} \rangle_e . \qquad (11)
+$$
+
+Using Eqs. (7) and (10), we write $I_l$ as
+
+$$
+\begin{equation}
+\begin{split}
+I_l = {}& \sum_{l', p', p''} \alpha_{l'p'p''} \sum_p \iint \rho \rho' d\rho d\rho' [\mathrm{LG}_p^l(\rho)] \\
+         & \times [\mathrm{LG}_p^{*l}(\rho')] [\mathrm{LG}_{p'}^{*l'}(\rho)] [\mathrm{LG}_{p''}^{l''}(\rho')] \\
+         & \times \iint d\phi d\phi' e^{i(l-l')(\phi-\phi')} \Phi^*(\phi) \Phi(\phi'),
+\end{split}
+\tag{12}
+\end{equation}
+$$
+
+where we have substituted for $\langle V^*(\rho_1, \phi_1)V(\rho_2, \phi_2) \rangle_e$ from Eq. (7). The summation over $p$ can be evaluated by using the identity $\sum_p [\mathrm{LG}_p^l(\rho)][\mathrm{LG}_{p'}^{*l'}(\rho')] = (1/\pi)\delta(\rho^2 - \rho'^2)$ (see Appendix B for the derivation), which gives
+
+$$
+I_l = \sum_{l', p', p''} \alpha_{l'p'p''} \frac{1}{2\pi} \int d\rho d\rho' [\mathrm{LG}_{p'}^{*l'}(\rho)][\mathrm{LG}_{p''}^{l''}(\rho)] \\
+\times \iint d\phi d\phi' e^{i(l-l')(\phi-\phi')} \Phi^*(\phi)\Phi(\phi'). \quad (13)
+$$
+
+The radial integral is evaluated by noting that the radial LG modes with a fixed value for the angular-momentum-mode index form a complete basis, that is, $\int \rho d\rho [\mathrm{LG}_{p'}^{*l'}(\rho)][\mathrm{LG}_{p''}^{l''}(\rho)] = \delta_{p'p''}/2\pi$. Using this formula, we obtain
+
+$$
+I_l = \sum_{l'} \frac{C_{l'}}{2\pi} \iint d\phi d\phi' e^{i(l-l')(\phi-\phi')} \Phi^*(\phi)\Phi(\phi'), \quad (14)
+$$
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+where $C_l' = (1/2\pi) \sum_{p'} \alpha_{l'p'p'}$. Next, we substitute the expression for the aperture function $\Phi(\phi) = k_1\delta(\phi - \phi_1) + k_2\delta(\phi - \phi_2)$. The intensity $I_l$ then assumes the following form:
+
+$$ I_l = \frac{k_1^2}{2\pi} \sum_{l'=-\infty}^{\infty} C_{l'} + \frac{k_2^2}{2\pi} \sum_{l'=-\infty}^{\infty} C_{l'} + \frac{k_1 k_2}{2\pi} \sum_{l'=-\infty}^{\infty} C_{l'} e^{-il'\Delta\phi} e^{il'\Delta\phi} + \text{c.c.,} \quad (15) $$
+
+Equation (15) can be seen to be the angular interference law, since it quantifies the interference between the fields coming from two separate angular positions.
+
+The function
+
+$$ W(\phi_1, \phi_2) = \sum_{l'=-\infty}^{\infty} C_{l'} e^{-il'\Delta\phi} \quad (16) $$
+
+represents the correlation that exists between the fields at $\phi_1$ and $\phi_2$. We refer to $W(\phi_1, \phi_2)$ as the angular coherence function. We note that in Ref. [16] Paterson introduced the “rotational coherence function,” which describes correlation between two field points with the same radial but different angular positions. The angular correlation function constructed above is integrated over the radial dimensions and thus describes only the correlation between field points with different angular positions, without any reference to their radial positions. The field represented by $W(\phi_1, \phi_2)$ is completely coherent if there is only one term in the above expansion. However, when there is more than one term in the expansion, the field is only partially coherent, and, as a consequence, two field points are mutually coherent over only a finite range of angular separation $\Delta\phi$. In order to quantify this thought, we rearrange the above equation to write it as
+
+$$ I_l = \frac{1}{2\pi} \sum_{l'=-\infty}^{\infty} C_{l'} [k_1^2 + k_2^2 + 2k_1 k_2 \lambda(\Delta\phi) \cos(l\Delta\phi + \theta)], \quad (17) $$
+
+where
+
+$$ \lambda(\Delta\phi) = \frac{|W(\phi_1, \phi_2)|}{\sum_{l'=-\infty}^{\infty} C_{l'}} \quad (18) $$
+
+is the degree of angular coherence and $\theta$ the argument of $W(\phi_1, \phi_2)$. For a completely coherent field, the degree of coherence $\lambda(\Delta\phi)$ is equal to unity. The width of $\lambda(\Delta\phi)$ is a measure of the angular separation over which the field remains coherent. We note that $\lambda(\Delta\phi)$ involves a discrete Fourier transform and that therefore it is a periodic function of the argument $\Delta\phi$. For this reason, one has to be careful in defining the width of $\lambda(\Delta\phi)$. However, when $C_{l'}$ has a broad distribution in $l'$ such that the spread of $\lambda(\Delta\phi)$ as a function of $\Delta\phi$ is well within the range $[0, 2\pi]$, the width of $\lambda(\Delta\phi)$ can be defined unambiguously, and this width can, to a very good approximation, be taken as the coherence angle of the field. As shown in Appendix A, when $C_{l'}$ has a broad Gaussian distribution in $l'$ with $\sigma$ being the standard deviation of the distribution, $\lambda(\Delta\phi)$ assumes, to within a very good approximation, the following functional form:
+
+$$ \lambda(\Delta\phi) = \exp\left(-\frac{\sigma^2 \Delta\phi^2}{2}\right). \quad (19) $$
+
+FIG. 2. (Color online) Intensity $I_l$, with $l = 30$ and $k_1 = k_2$, as a function of the angular separation $\Delta\phi$ for three different values of $\sigma$: (a) $\sigma = 1$, (b) $\sigma = 4$, and (c) $\sigma = 8$.
+
+We note that $1/\sigma$ is a measure of the angular width over which the fields at the two angular positions remain mutually coherent. Therefore, $\phi_{coh} \equiv 1/\sigma$ can be defined as the coherence angle of the beam. Figure 2 shows plots of the detection probability $I_l$ as a function of the angular separation $\Delta\phi$ for three different values of $\sigma$. We see that as the width $\sigma$ of the OAM-mode distribution increases, the coherence angle decreases. The visibility of angular interference is given by
+
+$$ V(\Delta\phi) = \frac{2k_1 k_2}{k_1^2 + k_2^2} \lambda(\Delta\phi), \quad (20) $$
+
+and when $|k_1| = |k_2|$, we get $V(\Delta\phi) = \lambda(\Delta\phi)$.
+
+### III. ANGULAR COHERENCE AND OAM ENTANGLEMENT
+
+In this section, we study a process known as parametric down-conversion that produces fields of the type considered in the previous section. We also show how the concept of angular
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+coherence can be useful for characterizing OAM entanglement of the PDC photons [8].
+
+A. Field produced by parametric down-conversion
+
+Parametric down-conversion is a nonlinear optical process in which a pump photon is broken up into two entangled photons known as the signal photon and the idler photon. When the pump field is of the form of a Gaussian beam, that is, an LG beam with $l = 0$ and $p = 0$, the state $|\psi_2\rangle$ of the down-converted two-photon field is given by [17–21]
+
+$$ |\psi_2\rangle = \sum_{l_s, p_s, p_i} \int d\omega_s \chi_{l_s p_s p_i}(\omega_s) \\ \times |l_s, p_s, \omega_s\rangle_s |-l_s, p_i, \omega_0 - \omega_s\rangle_i, \quad (21) $$
+
+where $\chi_{l_s p_s p_i}(\omega_s)$ is the probability amplitude that the signal and idler photons are in the LG modes characterized by indices $(l, p_s)$ and $(-l, p_i)$, respectively. We note that, due to the conservation of OAM in PDC, the signal and idler photons have equal but opposite OAMs. In writing the above state, we have assumed that the pump field is monochromatic with frequency $\omega_0$. We have also assumed perfect frequency phase matching such that $\omega_s + \omega_i = \omega_0$, where $\omega_s$ and $\omega_i$ denote the frequencies of the signal and idler photons. The density operator corresponding to the above two-photon state is $\hat{\rho}_2 = |\psi_2\rangle\langle\psi_2|$. The density operator $\hat{\rho}_s$ corresponding to the signal field can be calculated by taking a partial trace over the idler modes, which gives
+
+$$ \hat{\rho}_s = \text{tr}_i \hat{\rho}_2 = \sum_{l_s, p_s, p'_s} \int d\omega_s C_{l_s p_s p'_s}(\omega_s) |l_s, p_s, \omega_s\rangle_s \langle l_s, p'_s, \omega_s|, \quad (22) $$
+
+where
+
+$$ C_{l_s p_s p'_s}(\omega_s) = \sum_{p''_i} \chi_{l_s p_s p''_i}(\omega_s) \chi^*_{l'_s p'_s p''_i}(\omega_s). \quad (23) $$
+
+Next, by using Glauber's method [22], we calculate the classical correlation function $G_s(\rho_1, \phi_1; \rho_2, \phi_2; \tau)$ corresponding to the density matrix for the signal photon:
+
+$$ G_s(\rho_1, \phi_1; \rho_2, \phi_2; \tau) = \text{tr}[\rho_s \hat{E}^{(-)}(\rho_1, \phi_1, t) \hat{E}^{(+)}(\rho_2, \phi_2, t + \tau)], \quad (24) $$
+
+where
+
+$$ \hat{E}^{(-)}(\rho_1, \phi_1, t) = \sum_{l_1, p_1} \int d\omega_s \hat{s}_{l_1 p_1}^{\dagger}(\omega) [\mathrm{LG}_{p_1}^{*l_1}(\rho_1, \phi_1)] e^{i\omega t}, $$
+
+etc., and where $\hat{s}_{l_1 p_1}^{\dagger}(\omega)$ is the signal-photon annihilation operator for mode $[\mathrm{LG}_{p_1}^{*l_1}(\rho_1, \phi_1)]e^{i\omega t}$. Carrying out the above trace, we obtain
+
+$$ G_s(\rho_1, \phi_1; \rho_2, \phi_2; \tau) = \sum_{l_s, p_s, p'_s} \int d\omega_s C_{l_s p_s p'_s}(\omega_s) [\mathrm{LG}_{p'_s}^{*l'_s}(\rho_1, \phi_1)] \\ \times [\mathrm{LG}_{p'_s}^{l'_s}(\rho_2, \phi_2)] e^{-i\omega_s \tau}. \quad (25) $$
+
+Finally, by taking the Fourier transforms of both sides of the above equation and using the definition in Eq. (1), we obtain the frequency-domain correlation function $W_s(\rho_1, \phi_1; \rho_2, \phi_2; \omega_s)$ for the signal photon:
+
+$$ W_s(\rho_1, \phi_1; \rho_2, \phi_2; \omega) = \sum_{l_s, p_s, p'_s} C_{l_s p_s p'_s}(\omega) [\mathrm{LG}_{p'_s}^{*l'_s}(\rho_1, \phi_1)] \\ \times [\mathrm{LG}_{p'_s}^{l'_s}(\rho_2, \phi_2)]. \quad (26) $$
+
+We see at once that the correlation function for the signal photon has the same functional form as the cross-spectral-density function considered in Eq. (7). Therefore, it follows that, with respect to a detection system that is sensitive only to the azimuthal mode index, the angular coherence function for the signal photon has the same functional form as that of $W(\phi_1, \phi_2)$ in Eq. (16). From now on, we suppress the frequency argument in writing the correlation functions for the signal photon. Starting from Eq. (26) and using the procedure of Sec. II B, one can show that the angular coherence function $W_s(\phi_1, \phi_2)$ corresponding to the signal photon is
+
+$$ W_s(\phi_1, \phi_2) = \sum_{l_s = -\infty}^{\infty} C_{l_s} e^{-il_s \Delta\phi}, \quad (27) $$
+
+where $C_{l_s} = (1/2\pi) \sum_{p_s} C_{l_s p_s p'_s}$. We note that the signal photon field is an incoherent superposition of modes carrying different OAMs. Also, from our discussions in the previous section, we find that if $C_{l_s}$ has a broad distribution in $l_s$, its width can be measured directly by measuring the coherence angle of the signal field. This fact has one very important implication which we discuss in the next section.
+
+B. The angular Schmidt spectrum and the coherence angle
+
+A complete characterization of OAM entanglement of the two-photon state shown in Eq. (21) can be performed through Schmidt decomposition, which yields the Schmidt modes, a natural set of biorthogonal mode pairs that constitute the two-photon state [23]. We note that the two-photon state of Eq. (21) is not in the Schmidt-decomposed form since therein we have summation over three different indices. However, in many quantum-information protocols, such as those based on OAM entanglement, one is concerned with only the OAM-mode index of the photons. In such cases, the detection system is sensitive only to the OAM-mode index and therefore the two-photon state of Eq. (21) can be written in a Schmidt decomposed form that is only one dimensional:
+
+$$ |\psi_2\rangle = \sum_{l=-\infty}^{\infty} \sqrt{C_l} |l\rangle_s |-l\rangle_i. \quad (28) $$
+
+Here $s$ and $i$ stand for signal and idler photons, respectively, and $|l\rangle$ represents an eigenmode of order $l$, corresponding to an azimuthal phase $e^{il\phi}$. $C_l$ is the angular Schmidt coefficient, which is the probability that the signal and idler photons are generated in modes of order $l$ and $-l$, respectively. The distribution of this mode probability is referred to as the angular Schmidt spectrum or the spiral spectrum of the PDC photons [13,24]. For the two-photon state in Eq. (28),
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+the corresponding angular coherence function of the signal
+photon is still given by Eq. (27). The angular Schmidt
+spectrum is directly related to the entanglement of the two-
+photon field through an entanglement measure known as the
+Schmidt number K, which for the normalized angular Schmidt
+spectrum is defined as [23,25,26]:
+
+$$
+K \equiv \frac{1}{\sum_{l=-\infty}^{\infty} C_l^2}. \qquad (29)
+$$
+
+There are two generic ways in which the angular Schmidt
+spectrum of the two-photon field can be measured. First is by
+directly measuring the two-photon intensity in coincidence
+at different values of the OAM-mode index [11,12], and
+the second is by using the Hong-Ou-Mandel interference
+technique [13]. However, comparing Eqs. (28) and (27), we
+find that the OAM-mode spectrum of the signal photon is
+identically equal to the angular Schmidt spectrum of the
+two-photon field. Therefore, it follows that by measuring the
+angular coherence function, as shown in the previous section,
+of the signal field, one can construct the angular Schmidt
+spectrum of the two-photon field. We note that in this method
+one calculates the angular Schmidt (spiral) spectrum without
+doing coincidence measurements, in contrast to the above-
+mentioned methods, which require coincidence detection. In
+situations in which $C_l$ has a broad, Gaussian distribution, it can
+be shown that $K \approx 2\sqrt{\pi}\sigma$, where $\sigma$ is the standard deviation
+of the distribution. This approximate equality becomes an
+exact equality in the limit in which the distribution becomes
+infinitely broad. Now, for the Gaussian distribution, $\sigma$ is
+equal to $1/\phi_{\text{coh}}$, where $\phi_{\text{coh}}$ is the coherence angle of the
+beam. Thus the Schmidt number in this case is inversely
+proportional to the coherence angle of the signal or the idler
+field: $K \approx 2\sqrt{\pi}/\phi_{\text{coh}}$. We thus find that as the entanglement
+of the two-photon field increases the coherence angle of the
+signal and idler fields decreases. We note that the above method
+of estimating the entanglement of the two-photon field by
+performing measurements on the one-photon signal or idler
+field is applicable only when the two-photon field can be
+assumed to be in the Schmidt decomposed form of Eq. (28) and
+fortunately the field produced by the down-converter does have
+this form. In situations in which this assumption is not valid,
+the entanglement has to be estimated by doing measurements
+on the entire two-photon field.
+
+IV. CONCLUSIONS
+
+In conclusion, we have studied partially coherent fields
+that have coherent-mode representations in the OAM-mode
+basis. We have introduced the concepts of the angular
+correlation function and the coherence angle, and by utiliz-
+ing the concept of partial angular coherence we have also
+proposed a method to measure the angular Schmidt spectrum
+of the entangled two-photon field produced by parametric
+down-conversion. This proposed method may have important
+implications as it requires only singles measurement, as
+opposed to the other methods which are based on coincidence
+measurements.
+
+ACKNOWLEDGMENTS
+
+We gratefully acknowledge financial support through a
+MURI grant from the US Army Research Office and by the
+DARPA InPho program through the US Army Research Office
+Award No. W911NF-10-1-0395.
+
+APPENDIX A: ANGULAR COHERENCE OF FIELDS WITH A BROAD DISTRIBUTION FOR C<sub>l</sub>
+
+In this Appendix, we consider partially coherent fields with a broad distribution for $C_l$. For such fields, we can calculate the exact functional form of the angular correlation function $W(\phi_1, \phi_2)$. First, we write the angular correlation function in the following form:
+
+$$
+\begin{align}
+W(\phi_1, \phi_2) &= \frac{1}{2\pi} \sum_{l=-\infty}^{\infty} C_l e^{-il\Delta\phi} \\
+&= \frac{1}{2\pi} \int_{-\infty}^{\infty} C(l) \mathrm{comb}(l) e^{-il\Delta\phi} dl, \quad (\text{A1})
+\end{align}
+$$
+
+where $C(l)$ is a continuous function of $l$. The comb function is defined as $\text{comb}(l) = \sum_{n=-\infty}^{\infty} \delta(l-n)$. In the above equation, the angular correlation function $W(\phi_1, \phi_2)$ is, up to a constant, the Fourier transform of the product of $C(l)$ and $\text{comb}(l)$. We can therefore write it as the convolution of the Fourier transforms of $C(l)$ and $\text{comb}(l)$, that is,
+
+$$
+\begin{align*}
+W(\phi_1, \phi_2) &= \frac{1}{\sqrt{2\pi}} \mathcal{F}[C(l)] \otimes \mathcal{F}[\text{comb}(l)] \\
+&= \frac{1}{\sqrt{2\pi}} C(\Delta\phi) \otimes \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \delta(l-n) e^{-il\Delta\phi} dl \\
+&= \frac{1}{2\pi} C(\Delta\phi) \otimes \sum_{n=-\infty}^{\infty} e^{-i\Delta\phi n}, \tag{A2}
+\end{align*}
+$$
+
+where $\otimes$ represents the convolution and $\mathcal{F}[\cdots]$ the Fourier transformation; $C(\Delta\phi)$ is the Fourier transform of the OAM-mode distribution $C(l)$. Using the formula $\sum_n e^{-i\Delta\phi n} = 2\pi \sum_k \delta(\Delta\phi - 2\pi k)$, we write $W(\phi_1, \phi_2)$ as
+
+$$
+\begin{align}
+W(\phi_1, \phi_2) &= C(\Delta\phi) \otimes \sum_{k=-\infty}^{\infty} \delta(\Delta\phi - 2\pi k) \nonumber \\
+&= \sum_{k=-\infty}^{\infty} C(\Delta\phi - 2\pi k). \tag{A3}
+\end{align}
+$$
+
+Now we restrict the values of Δφ to be between 0 and 2π and assume that the width of C(Δφ) is much smaller than π. This is justified as we have already assumed that C_l has a broad distribution. We thus find that the only significant contribution to W(φ₁, φ₂) comes from the k = 0 term, and thus we obtain W(φ₁, φ₂) = C(Δφ), that is, the angular correlation function is the Fourier transform of the OAM-mode distribution. In the case in which C(l) is Gaussian, that is, C(l) = 1/((√2πσ) exp[-l²/(2σ²)]), where σ is the standard deviation of the distribution, the degree of coherence is given by λ(Δφ) = exp(-σ²Δφ²/2).
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+# APPENDIX B: EVALUATION OF THE SUMMATION IN Eq. (12)
+
+Rearranging the equation
+
+$$ \sum_{p} [\mathrm{LG}_{p}^{l}(\rho)][\mathrm{LG}_{p}^{*l}(\rho')] = \sum_{p} \frac{2p!}{\pi(|l|+p)!} \frac{1}{w_{0}^{2}} \left(\frac{2\rho\rho'}{w_{0}^{2}}\right)^{|l|} \exp\left(-\frac{\rho^{2} + \rho'^{2}}{w_{0}^{2}}\right) L_{p}^{|l|}\left(\frac{2\rho^{2}}{w_{0}^{2}}\right) L_{p}^{|l|}\left(\frac{2\rho'^{2}}{w_{0}^{2}}\right), \quad (B1) $$
+
+we can write the above equation as
+
+$$ \sum_p [\mathrm{LG}_p^l(\rho)][\mathrm{LG}_p^{*l}(\rho')] = \frac{2}{\pi w_0^2} \left(\frac{2\rho\rho'}{w_0^2}\right)^{|l|} \exp\left(-\frac{\rho^2 + \rho'^2}{w_o^2}\right) \sum_p \frac{\Gamma(p+1)}{\Gamma(|l|+p+1)} L_p^{|l|}\left(\frac{2\rho^2}{w_0^2}\right) L_p^{|l|}\left(\frac{2\rho'^2}{w_0^2}\right). \quad (B2) $$
+
+The summation on the right-hand side is a standard result for Laguerre polynomials, using which we get
+
+$$ \sum_{p} [\mathrm{LG}_{p}^{l}(\rho)][\mathrm{LG}_{p}^{*l}(\rho')] = \frac{2}{\pi w_{0}^{2}} \left(\frac{2\rho\rho'}{w_{0}^{2}}\right)^{|l|} \exp\left(-\frac{\rho^{2}+\rho'^{2}}{w_{o}^{2}}\right) \left(\frac{2\rho^{2} 2\rho'^{2}}{w_{0}^{2} w_{0}^{2}}\right)^{-|l|/2} \exp\left(\frac{\rho^{2}+\rho'^{2}}{w_{0}^{2}}\right) \delta\left(\frac{2\rho^{2}}{w_{0}^{2}} - \frac{2\rho'^{2}}{w_{0}^{2}}\right). \quad (B3) $$
+
+Finally after rearranging, we get the desired result
+
+$$ \sum_p [\mathrm{LG}_p^l(\rho)][\mathrm{LG}_p^{*l}(\rho')] = \frac{1}{\pi} \delta(\rho^2 - \rho'^2). \qquad (B4) $$
+
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+[13] H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, *Phys. Rev. Lett.* **104**, 020505 (2010).
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+[14] In a paper by Pires et al., [H. Di Lorenzo Pires, C. H. Monken, and M. P. van Exter, Phys. Rev. A **80**, 022307 (2009)] a similar idea is introduced to measure the entanglement among the transverse modes of a pure two-photon state. In the present paper we concentrate on angular coherence and entanglement in the states of orbital angular momentum.
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+[15] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, *Phys. Rev. A* **45**, 8185 (1992).
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+[17] S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, *Phys. Rev. A* **69**, 023811 (2004).
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+[19] H. H. Arnaut and G. A. Barbosa, *Phys. Rev. Lett.* **85**, 286 (2000).
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+[20] S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, *Phys. Rev. A* **65**, 033823 (2002).
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+[21] A. K. Jha, G. S. Agarwal, and R. W. Boyd, *Phys. Rev. A* **83**, 053829 (2011).
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+[22] R. J. Glauber, *Phys. Rev.* **130**, 2529 (1963).
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+# Error Analysis of the Square Root Operation for the Purpose of Precision Tuning: a Case Study on K-means
+
+Oumaima Matoussi, Yves Durand, Olivier Sentieys, Anca Molnos
+
+► To cite this version:
+
+Oumaima Matoussi, Yves Durand, Olivier Sentieys, Anca Molnos. Error Analysis of the Square Root Operation for the Purpose of Precision Tuning: a Case Study on K-means. ASAP 2019 - 30th IEEE International Conference on Application-specific Systems, Architectures and Processors, Jul 2019, New York, United States. pp.1-8. [hal-02183945](https://hal.inria.fr/hal-02183945)
+
+HAL Id: hal-02183945
+
+https://hal.inria.fr/hal-02183945
+
+Submitted on 15 Jul 2019
+
+**HAL** is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
+
+L'archive ouverte pluridisciplinaire **HAL**, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
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+# Error Analysis of the Square Root Operation for the Purpose of Precision Tuning: a Case Study on K-means
+
+Oumaima Matoussi, Yves Durand
+
+CEA, LETI, Univ. Grenoble Alpes, France
+name.sumame@cea.fr
+
+Olivier Sentieys
+
+Inria, Univ. Rennes, France
+olivier.sentieys@inria.fr
+
+Anca Molnos
+
+CEA, LETI, Univ. Grenoble Alpes, France
+anca.molnos@cea.fr
+
+**Abstract**—In this paper, we propose an analytical approach to study the impact of floating point (FLP) precision variation on the square root operation, in terms of computational accuracy and performance gain. We estimate the round-off error resulting from reduced precision. We also inspect the Newton Raphson algorithm used to approximate the square root in order to bound the error caused by algorithmic deviation. Consequently, the implementation of the square root can be optimized by fittingly adjusting its number of iterations with respect to any given FLP precision specification, without the need for long simulation times. We evaluate our error analysis of the square root operation as part of approximating a classic data clustering algorithm known as K-means, for the purpose of reducing its energy footprint. We compare the resulting inexact K-means to its exact counterpart, in the context of color quantization, in terms of energy gain and quality of the output. The experimental results show that energy savings could be achieved without penalizing the quality of the output (e.g., up to 41.87% of energy gain for an output quality, measured using structural similarity, within a range of [0.95,1]).
+
+**Index Terms**—approximate computing, error analysis, round-off error, algorithmic deviation, square root, Newton Raphson method, precision tuning, k-means, clustering, floating point
+
+## I. INTRODUCTION
+
+The ever increasing volume, diversity and high dimensional- ity of data goes hand in hand with the rapid growth of energy consumption of computer systems. Thus, improving the energy efficiency of computer systems that try to keep pace with the constant growth of information is a critical concern.
+
+In many application domains that deal with huge amounts of data such as multimedia processing (images, audio, video, etc.), data mining and machine learning, computations can be tolerant to some degree of error without critical degradation in the quality of the output. For example, a small image quality loss due to some modification of the color of a group of pixels can be hardly noticed by the user due to the limited capabilities of human perception. In such classes of applications, output accuracy could be traded for energy reduction.
+
+Performance gains can be achieved at the application level using software techniques, thanks to a relatively new comput- ing paradigm called *approximate computing* [1]. Introducing inexactness in computations may lead to energy scaling with little to no loss in accuracy. A way to introduce inexactness in an application is by precision reduction of FLP variables
+
+and computations [2], [3]. Finding the sweet spot between the numerical accuracy and the performance of the application (e.g. energy consumption) by refining the bit-width of the FLP variables is known as *precision tuning*.
+
+A loss in computational accuracy is inevitable due to reduced precision. However, to be able to determine the optimal precision (i.e. bit-width) of FLP variables that minimizes energy cost with the least impact on computational accuracy, two approaches can be considered: FLP simulation and analytical techniques to track round-off error (i.e. error introduced due to limited precision). FLP simulation consists in using libraries (e.g. MPFR) that allow the definition of FLP variables with adjustable bit-widths and provide the appropriate arithmetic operations. Nonetheless, the search space of the optimal bit- width can be very large, in that every change in the format of a FLP variable requires a new simulation, which can be very time consuming.
+
+Analytical approaches, on the other hand, try to determine a mathematical formula that models the impact of quantization error on the accuracy of the output [4], [5]. Once this formula is established, it can be applied to any application with different FLP formats. Unfortunately, this only works with smooth operations (arithmetic operations like addition, subtraction, multiplication, division, etc.). So, analytical approaches help save the time spent on tuning FLP precision but fail to estimate the computational error in the presence of non-smooth operations. In this case, simulation becomes inescapable. Thus, we advocate through our work the combination of simulation and analytical techniques.
+
+A plethora of work using simulation-based approaches for the purpose of FLP precision tuning was propounded [3], [6], [7], etc. As for analytical approaches, efforts were made in the context of fixed-point word-length optimization [4], [8] and mainly focused on common arithmetic operations like addition and multiplication. Square root operation is significantly more costly, in terms of energy consumption, than addition or multiplication and requires several iterative cycles to complete. Moreover, unlike simple arithmetic operations, the square root is usually implemented using the *Newton Raphson* method, which is an approximation of the operation itself, and adds another type of error referred to as *algorithmic deviation* besides the round-off error. Consequently, error analysis of a square
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+root operation is not a straightforward task and entails a two-fold method.
+
+The main contribution of this paper consists in conducting static error analysis of the square root operation and estimating a bound on the errors caused by limited precision and algorithmic deviation. Based on the formally derived error bound, the number of iterations of the square root operation is automatically adjusted for each precision. Therefore, the implementation of the square root (in terms of number of iterations) can be optimized with respect to any given FLP precision specification without the need for long simulation times.
+
+In addition to the stand-alone analysis of the square root operation and for a more complete evaluation of our proposition, the second contribution of this paper consists in evaluating our analytical approach by studying the square root operation as part of a whole application. Our choice fell on a data clustering algorithm called K-means a.k.a. Lloyd's algorithm [9]. It is a type of unsupervised learning algorithm used to cluster a set of unlabeled data into $k$ clusters based on data feature similarity [10]. The similarity is typically determined using the Euclidean distance measure, which involves square root operations. The distance function is the mainstay of the majority of clustering and classification algorithms, which will help us highlight the impact of square root optimization on the performance of the application as a whole.
+
+It should be noted that the proposed analytical approach for the square root operation is independent of the application itself. Any clustering algorithm or any other application (e.g. digital signal and image processing, 3D graphics, spectrum analysis, wireless communications, etc.), for that matter, where square root computations can be found, would also benefit from our error analysis method. The importance of our contribution lies within the integration of our analytical approach, which is aimed for a smooth operation, namely the square root, in an application that also contains non-smooth operations. Hence, the combined effort of simulation and analytical results is needed for precision tuning. We quantify the efficiency of the proposed approach by measuring the quality degradation of K-means' output as well as its energy gain.
+
+The rest of this paper is organized as follows. After overviewing contributions dealing with approximate computing in Section II, we detail our error analysis approach of the square root operation in Section III. We explain the process of sensitivity analysis of K-means, based on our square root analysis results and variable-precision FLP simulation in Section IV. We validate the efficiency of approximate K-means by measuring both SSIM and energy gain and we also discuss the experimental results in Section V, before concluding the paper in Section VI.
+
+## II. RELATED WORK
+
+Precision tuning is a research direction that has received significant attention and is motivated by the fact that FLP operations contribute to the energy footprint of an application.
+
+In [6], a framework called ASAC using statistical methods was proposed. This framework helps discover approximable and non-approximable program parts through sensitivity analysis. The main idea is to perturb program variables and check the resultant output against a correct output, i.e. one fulfilling an acceptable QoS threshold.
+
+Precimonious [7], a dynamic program analysis tool, was proposed to assist developers in choosing the lowest precision that satisfies accuracy and performance constraints. Given a set of FLP variables and their possible types, a search based on the delta-debugging algorithm is performed. The search outputs a type configuration that maps each variable to a type. However, Precimonious requires a representative set of program inputs provided by the user. If the same type configuration is applied on a much worse conditioned input then no guarantees can be made.
+
+The method proposed in [2] is not limited to determining the best mix of single or double precision in a program as in [7], but it computes the precision of FLP variables down to the bit level of the mantissa. A heuristic precision tuning algorithm based on a binary search is performed in order to find the smallest precision possible for each variable while keeping the output error within some user-given bound. The results show that the speedup gained by tuning some programs may be diminished or even eliminated because of the overhead due to data type conversions performed by the compiler.
+
+In [3], hardware implementations of fixed-point and FLP arithmetic operators are compared in terms of area, delay and energy. A custom FLP library called `ct_float` was devised to vary the bit width of the variables. The authors concluded that FLP operators provide a better energy/accuracy tradeoff for small bitwidths but important area, delay and energy overhead for larger bitwidths, compared to fixed-point operators.
+
+The common denominator in the majority of FLP precision tuning approaches is that they rely on varying the precision of the different FLP variables in an application (usually with the help of a multiple precision FLP library) and testing different combinations before deciding on the optimal precision. This process is prohibitively expensive, time wise. Furthermore, when big applications with a huge set of input is under optimization, the search space for optimal precision can be enormous and it might be impossible to cover fully.
+
+Analytical approaches are faster than simulation because once a mathematical expression that models (or bounds) the round-off error is determined, it can be applied to different precisions. So, it is a one-time effort whose results can be applied to multiple FLP formats. Efforts have been made to find analytical models in the context of accuracy evaluation of fixed-point systems. A series of propositions of analytical methods to measure the output noise of signal processing systems were presented in [4], which dealt with smooth operations, then in [5], which catered for decision operators and recently in [8], where a hybrid approach based on both simulation and analytical results was highlighted.
+
+A parametric error analysis of a version of Goldschmidt's square root algorithm that uses directed rounding implemen-
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+tations, was proposed in [11]. The error analysis is based on
+relative errors of intermediate computations. The proposed
+error formulae were intended to help determine the optimal
+hardware implementation (i.e. multiplier dimensions). However,
+their analysis was not demonstrated on a real architecture.
+
+In our work, we study analytically the square root operation,
+which is the backbone of many algorithms such as clustering,
+data mining and signal processing algorithms, with respect to
+FLP precision tuning. Moreover, we quantify the efficiency
+of the proposed error bound in the context of K-means by
+comparing the QoS of the inexact version of K-means to its
+exact counterpart. We also measure relative energy gains using
+energy costs obtained from [3], [12] and profiling information.
+
+### III. ANALYSIS OF THE SQUARE ROOT OPERATION
+
+First, to eliminate any confusion, it should be noted that the
+terms accuracy and precision are not used interchangeably in
+this paper. Nearly all processors and programming languages
+support FLP numbers, which are defined in the IEEE-754
+normalization. A FLP number *f* is represented by an exponent
+*e*, a mantissa *m* and a sign bit *s*: $f = (-1)^s \times m \times 2^e$. We
+designate by precision the number of bits used to represent the
+mantissa of a FLP variable, whereas we refer by accuracy to
+the quality of the result, i.e., the degree of error in the output,
+compared to a reference or a *golden* output.
+
+The square root operation is usually approximated with the *Newton Raphson* method. To compute $y = \sqrt{a}$, $a > 0$, the Newton Raphson method starts with an initial guess (i.e. initial seed value) $y_0 > 0$. The initial guess is then refined by iterating over:
+
+$$y_{n+1} = \frac{1}{2} \left( y_n + \frac{a}{y_n} \right)$$
+
+However, this conventional iteration for the square root computation is not frequently used because it entails a division at each step. Since division is generally much slower than multiplication, the square root reciprocal is usually advocated. The square root reciprocal converges to $\frac{1}{\sqrt{a}}$ and iterates over:
+
+$$x_{n+1} = \frac{x_n}{2} \times (3 - a \times x_n^2).$$
+
+In order to get $\sqrt{a}$, the result is multiplied by *a*.
+
+To optimize the implementation of the Newton Raphson method, we aim at adjusting the number of its iterations according to the selected precision, without jeopardizing the accuracy of the result. To do so, we perform error analysis to statically determine at which iteration it is preferable to stop the computations, for a specific precision *p*, without causing further error. The two main causes of error that we investigated are:
+
+• the round-off error caused by FLP representation and FLP operations,
+
+• and the systematic error (a.k.a. unavoidable error or algorithmic deviation), which results from the Newton Raphson approximation itself.
+
+#### A. Bounding the Round-off Error
+
+Let $fl(x+y) = (x+y)(1+\epsilon_{add})$ and $fl(x\times y) = (x\times y)(1+\epsilon_{mul})$, be approximations of the exact mathematical operations
+
++ and × respectively, $\forall x, y \in \mathbb{R}$, such that $|\epsilon_{add,mul}| \le \epsilon_m$.
+$\epsilon_m$, called machine epsilon or the unit roundoff, is defined as
+the smallest number such that $1 + \epsilon_m > 1$ [13]. $\epsilon_m = \beta^{-p}$,
+where $\beta$ is the base and $p$ is the number of bits used for the
+magnitude of the mantissa of a FLP number represented as
+$r_0.r_1r_2...r_{p-1} \times \beta^e$. We assume that $\epsilon_{add} = \epsilon_{mul} = \epsilon$ and that
+multiplication by $\frac{1}{2}$ does not incur round-off error.
+$\forall x_i \in \mathbb{R}$, let $\hat{x}_i = fl(x_i)$ be the FLP representation of $x_i$;
+$i = 1..n+1$. Machine epsilon is an upper bound on the relative
+error in representing a FLP number: $\frac{|\hat{x}_i-x_i|}{|x_i|} \le \epsilon_m$.
+We would like to bound the round-off error of the Newton
+Raphson reciprocal. To do so, we aim at expressing $\hat{x}_{n+1}$ as:
+$\hat{x}_{n+1} \le x_{n+1} \times (1 + \delta)$, where $\delta$ is the round-off error.
+
+$$ 
+\begin{gather*}
+\hat{x}_{n+1} = fl(x_{n+1}) = fl\left(\frac{\hat{x}_n}{2} \times fl\left(3 - fl(a \times fl(\hat{x}_n^2))\right)\right) \\
+\hat{x}_{n+1} = \frac{\hat{x}_n}{2} \times \left(3 - a \times \hat{x}_n^2(1+\epsilon)^2\right)(1+\epsilon)^2 \quad (1)
+\end{gather*}
+$$
+
+Let $e(\hat{x}_n) = 3 - a \times \hat{x}_n^2 (1 + \epsilon)^2$.
+
+$$|e(\hat{x}_n)| \leq |3 - a \times \hat{x}_n^2| + |2a \times \hat{x}_n^2 \times \epsilon| \quad (2)$$
+
+We have:
+
+$$\sqrt{a} \times x_n < 1 \quad (3)$$
+
+*Proof.* $x_{n+1}\sqrt{a}-1 = -(1+x_n\frac{1}{2}\sqrt{a})(x_n\sqrt{a}-1)^2 < 0$ ☐
+
+Based on Equation 3 we can conclude that
+
+$$\epsilon < \frac{(3 - a \times x_n^2) \times \epsilon}{2}. \quad (4)$$
+
+Then, combining Equations 2 and 4 gives:
+
+$$|e(x_n)| < (3 - a \times x_n^2) \times (1 + \epsilon), \quad (5)$$
+
+and combining Equations 1 and 5 gives
+
+$$\hat{x}_{n+1} < \frac{\hat{x}_n}{2} \times (3 - a \times \hat{x}_n^2) \times (1 + \epsilon)^3.$$
+
+Disregarding $O(\epsilon^n)$ terms, $n > 1$, in the previous inequal-
+ity [13], we obtain
+
+$$\hat{x}_{n+1} < \frac{\hat{x}_n}{2} \times (3 - a \times \hat{x}_n^2) \times (1 + 3\epsilon), \quad (6)$$
+
+from which we can conclude that the round-off error is $3\epsilon$ per
+iteration. Note that $\hat{x}_{n+1}$ converges towards $\frac{1}{\sqrt{a}}$ and that we
+want to bound the error of $\hat{y}_{n+1}$, which converges towards $\sqrt{a}$.
+So, only the last iteration should be multiplied by $a$:
+
+$$ 
+\begin{aligned}
+x_{n+1} \times a &= y_{n+1} &\implies fl(x_{n+1} \times a) &= fl(y_{n+1}) \\
+& & &\implies \hat{x}_{n+1} \times a \times (1 + \epsilon) &= \hat{y}_{n+1}.
+\end{aligned}
+$$
+
+Consequently, the overall round-off error is $\delta \le n \times 3\epsilon + \epsilon$.
+
+#### B. Bounding the Systematic Error
+
+The absolute systematic error in computing $\frac{1}{\sqrt{a}}$ is
+
+$$|x_i - \frac{1}{\sqrt{a}}|.$$
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+Fig. 1: Systematic and round-off errors for different precision values and iteration values of Newton Raphson
+
+The relative systematic error in computing $\frac{1}{\sqrt{a}}$ is
+
+$$ \frac{|x_i - \frac{1}{\sqrt{a}}|}{\frac{1}{\sqrt{a}}} = |\sqrt{a}x_i - 1|. $$
+
+To find an upper bound for the systematic error, we start with a reasonable initial value defined as
+
+$$ \frac{1}{2} \times \frac{1}{\sqrt{a}} \le x_0 \le \frac{3}{2} \times \frac{1}{\sqrt{a}} \Rightarrow 1 + 0.5 \times \sqrt{a} \times x_0 \le \frac{7}{4} \text{ and } \sqrt{a} \times x_0 - 1 \le \frac{1}{2}. $$
+
+So, according to the proof of Equation 3:
+
+$$ |\sqrt{a} \times x_1 - 1| \le \frac{4}{7} \times \left| \frac{7}{4} \times (\sqrt{a} \times x_0 - 1) \right|^2. $$
+
+Since $\frac{7}{4} \times (\sqrt{a} \times x_0 - 1) \le \frac{7}{8}$ then, $|\sqrt{a} \times x_1 - 1| \le \frac{1}{2} \times \frac{7}{8}$.
+
+It is safe to conclude that the systematic error in computing $\frac{1}{\sqrt{a}}$ at iteration $i=0..n$ is
+
+$$ |\sqrt{a} \times x_i - 1| \le \frac{1}{2} \times (\frac{7}{8})^{2^i-1}. \quad (7) $$
+
+The relative systematic error for $\frac{1}{\sqrt{a}}$ is the same as the one for $\sqrt{a}:$
+
+$$ \frac{|y_i - \sqrt{a}|}{\sqrt{a}} = \frac{|ax_i - \sqrt{a}|}{\sqrt{a}} = |\sqrt{a}x_i - 1|. $$
+
+Both the systematic and round-off errors are plotted for different precision values ($p$) in Fig. 1. For space reason, only two of the resulting graphs are illustrated. The straight line corresponds to the round-off error, which increases proportionally to the number of iterations of the *Newton Raphson* method, whereas the curve designates the systematic error, which declines as the number of iterations grows.
+
+The rise of the round-off error is remarkably rapid at lower
+
+precision values (e.g. Fig. 1(a)) but it slows down at higher precision (e.g. Fig. 1(b)).
+
+The systematic error, on the other hand, is independent of the precision. It stabilizes at zero around the sixth iteration. This means that, disregarding the round-off error (round-off error equal to 0), six *Newton Raphson* iterations are sufficient in producing accurate (i.e. error=0) square root approximation. However, when the round-off error is in the picture, which is usually the case, the higher the number of iterations, the higher the round-off error is. Although at six sqrt iterations the systematic error is 0, the round-off error is 0.6 for $p = 4$ and 0.05 for $p = 9$.
+
+The round-off error accumulates as the number of iterations rises, which counteracts the decline of the unavoidable error. That is why, the round-off error should not exceed the systematic error. Thus, the intersection between the two lines is indicative of the *optimal* number of iterations at which we have a *good enough* result.
+
+## IV. APPLICATION TO K-MEANS
+
+So far, we studied the square root operation individually, in the context of error analysis for the purpose of precision tuning. It would be interesting to apply our study of the square root in a fullblown application that encompasses both smooth and non-smooth operations. We chose a clustering algorithm called K-means that uses the square root operation in the computation of the Euclidean distance function.
+
+In this section, we start by describing K-means and its use in color quantization. Then, we present precision tuning of K-means that involves both FLP simulation and our analytical results.
+
+### A. Color Quantization Using K-Means
+
+The basic principle of the clustering problem is as follows. A data set $X = (x_1, x_2, ..., x_n)$ is composed of data elements, where each data element $x_i$ is a $d$-dimensional vector. This data set is partitioned into $k$ clusters of similar data points [14].
+
+In the particular case of color quantization, one of K-means' applications, an image represented as an array of size $N \times d$ where $N$ is the number of pixels in the image and $d$ is the color space (referred to as data dimension or data features), is represented with a smaller number of colors. Most commonly, the space is 3-dimensional ($d=3$) and the coordinates encode the color. For example, RGB is a color space usually encoded as a 3-tuple of 8 bits each. The value of each dimension is within the range of [0, 255].
+
+The flowchart in Fig. 2 gives an overview of the different steps of the clustering process in the context of color quantization with K-means.
+
+The goal of the K-means clustering algorithm is to generate a compressed image out of the original image. To this end, a *palette*, i.e. a set of centroids $C = (c_1, c_2, ..., c_k)$, is firstly chosen by selecting data elements (i.e. colors or pixels) that best represent the original image, for each cluster. In other words, the number of centroids is the number of colors that the palette is made of. Then, each data element is mapped to the *closest*
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+Fig. 2: Flowchart of the K-means algorithm
+
+color in the palette based on a distance measure $||x||_m = (\sum_{i=1}^n |x_i|^m)^{1/m}$ (e.g. $m=1$ for the Manhattan distance, $m=2$ for the Euclidean distance) [15]. This results in a preliminary classification. Following this initial classification, $k$ new centroids are re-computed as barycenters of the previously generated clusters and data elements are re-assigned to the new centroids based on the chosen distance measure. The algorithm is based on an iterative refinement technique and it iterates until centroids do not move anymore (i.e., no data elements change clusters), a maximum number of iterations is reached or the distance measure is minimized, as illustrated in Fig. 2. To quantify the difference between two pixels, a simpler distance function known as the Manhattan distance (sum of absolute differences) is usually used. However the optimal distance, in terms of minimizing within-class variance and producing higher-quality clusters, and more complicated, in terms of computational complexity, is the Euclidean distance (sum of the squared distances) [16]. Unlike other studies on approximate K-means, we use the optimal distance metric:
+
+$$ dist(p1, p2) = \sqrt{(R1 - R2)^2 + (G1 - G2)^2 + (B1 - B2)^2}. $$
+
+## B. Precision Tuning Using a Multiple-Precision Floating Point Library
+
+We focus our precision analysis on the Euclidean distance function, which is the kernel of the K-means algorithm as it is in charge of computing the distance between a given data point and a cluster centroid. These distance values are pivotal in the correct assignment of pixels to their closest cluster and thus in providing a correct classification. We are going to include approximation in the FLP computations of the Euclidean distance function and leverage the results from our analysis in Section III to account for the impact of FLP precision variation on the square root operation.
+
+Introducing inexactness by reducing the number of bits of the mantissa can sometimes provide results of the same accuracy but with better energy efficiency than the exact version of the program. To this aim, we studied the sensitivity of K-means by arbitrarily varying the number of bits of the mantissa (2-23 bits) and observing the repercussions on the output.
+
+As was established in our analytical study of the square root operation (Section III), the number of bits of the mantissa has an impact on the number of square root iterations. So, to study the sensitivity of K-means, we re-wrote K-means' Euclidean distance function using the MPFR library, which is a smooth extension of the IEEE-754 standard where any FLP number can have its own precision [17], and we assigned to each precision its corresponding number of square root iterations according to the analysis in Section III. Our analytical results come in handy in the precision tuning of K-means in that, we do not have to vary the number of square root iterations for each precision simulation in order to determine the optimal pair $(p, sqrt)$. The optimal number of square root iterations is statically determined for each precision (Section III). For example, for a precision $p=4$, the FLP variables in K-means' Euclidean distance function are transformed into MPFR variables ($mpfr_t var$), their precision is set to $4$ ($mpfr_init2(var, 4)$) and the number of iterations in the Newton Raphson method implementation is set directly to $sqrt = 2$.
+
+In addition to the FLP variables, we tracked the operations that use these variables and changed them into MPFR operations. The Euclidean distance function encompasses multiplication, addition, subtraction and square root operations.
+
+### Listing 1: MPFR operations
+
+```c
+mpfr_sub(r, r, r1, MPFR_RNDN); /* r=r-r1; */
+mpfrMul(r, r, r, MPFR_RNDN);   /* r=r*r; */
+mpfr_add(r, r, r2, MPFR_RNDN);  /* r=r+r2; */
+mpfr_Sqrt(r_tmp, r, MPFR_RNDN); /* r_tmp=sqrt(r) */
+```
+
+Listing 1 showcases the transformation of different operations into MPFR operations. We implemented two inexact versions of the Euclidean distance function. We called these versions fused and unfused. In the first, the *sub*, *mul* and *add* operations are computed with full precision and the result is rounded once to *N* significant bits and then passed to the reduced-precision square root function. In the latter, all the operations are performed with reduced-precision, which means that the value is rounded four times before yielding the final result. We examine both fused and unfused versions in Section V to determine which version is more beneficial to implement.
+
+The quality of the result as well as the energy consumption depend not only on the precision $p$ of the FLP variables but also on other parameters such as the number of clusters $k$ and the number of iterations $n$ needed for the clustering process to converge. More clusters means that the compressed image will contain more colors, which is in favor of a better quality. With more iterations, there is a better chance for K-means to converge, which also contributes to the quality of the output. So, we investigated precision variation and its impact on the
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+square root iterations over a range of different values of (k, n) pairs. Thus, we also varied the (k, n) parameters along with the precision and quantified the impact on the output in order to discern the best configuration that yields the desired energy-QoS tradeoff.
+
+V. EXPERIMENTAL RESULTS
+
+To determine whether a configuration (k, n) for a given
+precision p is favorable, we transform the original source code
+of K-means [18] using the configuration in question, compile it
+to an ARM binary, execute it while using energy and instruction
+counters, and check for two criteria: energy consumption and
+QoS.
+
+A. Experimental Setup
+
+In order to quantify the impact of approximation on energy
+consumption, we make use of an energy model based on energy
+measurements of ARM Cortex-A7 instructions at a frequency
+of 500 MHz and without dependencies between instructions
+[12]. We also make use of profile information, i.e., instruction
+types (e.g., add, mul, div) and the number of execution of
+each type. ARM's Cortex-A7 core is a dual issue in-order core
+that consists of one load/store, one multiply, one FLP and two
+integer units. A thorough characterization of the ARM Cortex-
+A7 instruction set with energy metrics for every instruction
+type could be found in [12].
+
+The energy values of integer and FLP instructions are
+normalized with reference to integer multiplication, i.e., integer
+multiplication is used as a unit of measure throughout the
+experiments. Normalized energy values of 32-bit integer and
+FLP instructions are listed in Table I. These normalized energy
+values are used to compute the energy consumption of exact
+K-means, which is considered as a touchstone. As for the
+energy values of variable precision (2-16 bits) operations,
+namely addition, multiplication and division, they are computed
+according to [3] and shown in Fig. 3. It is noticeable from
+Fig. 3 that energy consumption is influenced by the number
+of bits of the mantissa; the higher the precision the higher the
+energy cost.
+
+The overall energy consumption is computed by combining
+the profile information and the energy values
+
+$$E_{total} = \sum_{i=1}^{\#types} op_i \times e_i,$$
+
+where *op*<sub>*i*</sub> is the number of operations of type *i* and *e*<sub>*i*</sub> is the energy consumed per operation of type *i* (normalized to *multi*). We estimate the energy savings by comparing the energy consumption obtained by executing the exact version of K-means (i.e. with the highest precision *p* = 23) to the approximated version.
+
+The quality of the result is measured using the SSIM index.
+SSIM is a perception-based metric that compares two images by
+incorporating a number of terms including contrast, luminance
+and structural information. The degradation of the image quality,
+due to compression with inexact K-means (reduced precision
+
+TABLE I: Energy values (normalized to *muli*) of a sample of integer and 32-bit FLP instructions
+
+<table>
+   <thead>
+    <tr>
+     <td>
+      Instr
+     </td>
+     <td>
+      addi
+     </td>
+     <td>
+      muli
+     </td>
+     <td>
+      si
+     </td>
+     <td>
+      li
+     </td>
+     <td>
+      addf
+     </td>
+     <td>
+      mulf
+     </td>
+     <td>
+      sf
+     </td>
+     <td>
+      lf
+     </td>
+     <td>
+      divf
+     </td>
+    </tr>
+   </thead>
+   <tbody>
+    <tr>
+     <td>
+      Energy
+     </td>
+     <td>
+      1.02
+     </td>
+     <td>
+      1
+     </td>
+     <td>
+      2.41
+     </td>
+     <td>
+      1.79
+     </td>
+     <td>
+      1.16
+     </td>
+     <td>
+      1.16
+     </td>
+     <td>
+      2.34
+     </td>
+     <td>
+      1.92
+     </td>
+     <td>
+      7.80
+     </td>
+    </tr>
+   </tbody>
+  </table>
+
+instr:i:integer instruction, instr:f:FLP instruction
+
+Fig. 3: Normalized energy values for different precisions
+
+and/or parameter variation), is determined with respect to the
+compressed image with the highest precision.
+
+B. Results
+
+In this section, we study the impact of the precision and
+square root iterations (*p*, sqrt) on the quality of the output
+for various *n* (number of iterations needed for K-means to
+converge) and *k* (number of clusters) configurations. To do so,
+we start by jointly varying the precision bits *p* and the number
+of K-means iterations *n*, while fixing the number of clusters
+to *k* = 50, for both unfused and fused operations.
+
+Then, we vary the number of clusters *k* and the precision *p* while fixing the number of K-means iterations to *n* = 10. The SSIM values are averaged over 10 different RGB images chosen from [19]. An SSIM value is within the interval [0, 1], 1 designating the best image quality.
+
+The first set of experiments that we conducted, i.e., SSIM(*p*, *n*) shows that the number of iterations *n* has little to no impact on SSIM for different precisions and for both the fused and unfused cases. This is true for other *k* values as well (10, 100, 500, 1000). Accordingly, the number of iterations *n* is fixed to 10 for the rest of the experiments.
+
+However, by varying the precision with the number of clusters while fixing *n* to 10, a noticeable change in SSIM is observed and the results are reported in Fig. 4. For instance, in case of unfused operations (Fig. 4(a)), SSIM increased from 0.791 with *k* = 10 to 0.969 with *k* = 1000, for *p* = 10. Regarding the fused operations’ case (Fig. 4(b)), the results are almost identical to the unfused case except for *k* = 10, where the unfused version yielded better SSIM values for the different tested precisions, and for *p* = 2 and *k* > 10, where the fused version yielded better SSIM values.
+
+Based on our experiments (SSIM(*p*, *k*) in Fig. 4 and SSIM(*p*, *n*)), fused and unfused operations generate similar
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+Fig. 4: Impact of precision variation and number of clusters on SSIM: for (a) unfused operations and (b) fused operations
+
+SSIM values for the majority of the configurations. Taking into consideration the additional requirements that come with fused operations (i.e., dedicated hardware), which is not always supported by all architectures, as well as a compiler that allows the program to make use of the fused operations, we deem it too big an effort for an insignificant quality enhancement. Consequently, we advocate the use of unfused operations in the context of reduced-precision K-means.
+
+It is also worth mentioning that we conducted the experiments from $p = 2$ to $p = 23$, but no noticeable change was detected starting from $p = 16$, as can be observed in Fig. 4. Thus, the rest of the graphs presented in this paper stop at $p = 16$.
+
+To conclude whether approximating K-means is worthwhile, we also measured the energy gains. Fig. 5 displays the number of executed instructions of the Euclidean distance function per instruction type: integer arithmetic, FLP arithmetic, load/store (both FLP and integer) and other instructions (e.g. branch,
+
+Fig. 5: Executed instructions breakdown for exact K-means ($p = 23$) with different $(k, sqrt)$ configurations
+
+Fig. 6: Energy consumption breakdown for exact K-means ($p = 23$) with different $(k, sqrt)$ configurations
+
+comparison, conversion). The values are generated by applying exact K-means ($p = 23$) while varying the number of clusters and the square root iterations. For the different configurations of $(k, sqrt)$, the FLP arithmetic operations contribute with around 19% of the total number of executed instructions compared to integer arithmetic operations, which represent only 2% when $sqrt = 6$ and almost 3% when $sqrt = 3$.
+
+Fig. 6 shows the energy consumption breakdown for exact K-means with different $(k, sqrt)$ configurations. It is clear that the number of clusters $k$ as well as the number of square root iterations $sqrt$ have a significant impact on the energy footprint of the program. For instance, for the same number of clusters $k = 50$, energy consumption increases by approximately $6 \times 10^9$ from $sqrt = 3$ to $sqrt = 6$.
+
+The percentages of energy savings are plotted against $p$ for different $(k, sqrt)$ configurations in Fig. 7. Each precision is attributed its optimal number of square root iterations according to our analytical study (Section III). For example, for $p = 4$, $sqrt = 2$. The graph indicates that there are energy gains for the different $(p, k, sqrt)$ configurations but with different magnitudes. The energy gain of each configuration $(p, k, sqrt)$ is computed with respect to the reference configuration ($p = 23, k = 100, sqrt = 6$). The smallest gain is observed at $p = 16$ and ($k = 100, sqrt = 6$) with a value of 14.09%. The biggest gain is at $p = 2$ and ($k = 10, sqrt = 1$) with a value of 96.57%. To be able to determine the best energy-QoS tradeoff
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+Fig. 7: Percentage of energy gain of different precisions *p* and with different (*k*, sqrt) configurations
+
+Fig. 8: SSIM of different precisions *p* and with different (*k*, sqrt) configurations
+
+for each precision, SSIM is plotted against *p* in Fig. 8 with the same (*k*, sqrt) configurations as in Fig. 7. The same reference (*p* = 23, *k* = 100, sqrt = 6) also serves as a baseline. The best quality is reached at *p* = 16 and (*k* = 100, sqrt = 6) with an SSIM value of 1. The worst quality is noticed at *p* = 2 and (*k* = 10, sqrt = 1) with an SSIM value of 0.67.
+
+Assuming that an output image with an SSIM value lying within the range of [0.95, 1] is considered a good quality image, then configuring K-means with a 6-bit mantissa, 100 clusters and 4 square root iterations is sufficient in producing such an image while saving 41.87% of energy.
+
+## VI. CONCLUSION
+
+This paper focused on optimizing the implementation of the square root operation. An analytical examination of the Newton Raphson approximation of the square root showed that the number of square root iterations is influenced by the precision bits. Consequently, we associated to each precision its optimal
+
+number of Newton Raphson iterations. For a well-rounded evaluation of our proposition, we aimed at finding opportunities to reduce energy consumption of a classic clustering algorithm called K-means by varying the precision of its FLP variables and adjusting the number of square root iterations of its distance function.
+
+The various approximated versions of K-means were compared to the exact version in terms of QoS, measured with SSIM, and relative energy gain. The obtained results can serve as a guideline in choosing the best configuration of precision bits, number of clusters and square root iterations (*p*, *k*, sqrt), i.e., one that yields the most energy gains, for a desired QoS (e.g., for an SSIM within [0.95, 1], an energy gain of 41.87% is achieved with a (6, 100, 4) configuration).
+
+## REFERENCES
+
+[1] Q. Xu, T. Mytkowicz, and N. S. Kim, "Approximate computing: A survey," *IEEE Design Test*, Feb 2016.
+
+[2] N. M. Ho, E. Manogaran, W. F. Wong, and A. Anoosheh, "Efficient floating point precision tuning for approximate computing," in *ASP-DAC*, Jan 2017.
+
+[3] B. Barrois and O. Sentieys, "Customizing Fixed-Point and Floating-Point Arithmetic - A Case Study in K-Means Clustering," in *IEEE International Workshop on Signal Processing Systems*, Oct. 2017.
+
+[4] R. Rocher, D. Menard, P. Scalart, and O. Sentieys, "Analytical approach for numerical accuracy estimation of fixed-point systems based on smooth operations," *IEEE Transactions on Circuits and Systems*, pp. 2326-2339, Oct 2012.
+
+[5] K. Parashar, R. Rocher, D. Menard, and O. Sentieys, "Analytical approach for analyzing quantization noise effects on decision operators," in *IEEE International Conference on Acoustics, Speech and Signal Processing*, March 2010, pp. 1554-1557.
+
+[6] P. Roy, R. Ray, C. Wang, and W. F. Wong, "Asac: Automatic sensitivity analysis for approximate computing," in *SIGPLAN/SIGBED LCTES*. New York, NY, USA: ACM, 2014.
+
+[7] C. Rubio-González, C. Nguyen, H. D. Nguyen, J. Demmel, W. Kahan, K. Sen, D. H. Bailey, C. Iancu, and D. Hough, "Precimonious: Tuning assistant for floating-point precision," in *SC*, Nov 2013.
+
+[8] K. N. Parashar, D. Menard, and O. Sentieys, "Accelerated performance evaluation of fixed-point systems with un-smooth operations," *Transactions on Computer-Aided Design of Integrated Circuits and Systems*, pp. 599-612, April 2014.
+
+[9] G. Hamerly and J. Drake, *Accelerating Lloyd's Algorithm for k-Means Clustering*. Springer International Publishing, 2015.
+
+[10] J. Silva, E. Faria, R. Barros, E. Hruschka, A. de Carvalho, and J. Gama, "Data stream clustering: A survey," 2014.
+
+[11] P. Seidel, "A parametric error analysis of goldschmidt's square-root algorithm," in *ACSSC*, Nov 2015, pp. 727-731.
+
+[12] E. Vasilakis, "An instruction level energy characterization of arm processors," https://www.ics.forth.gr/carv/greenvm/files/tr450.pdf, CARV Laboratory ICS FORTH, Tech. Rep., 2015, 7/8/2018.
+
+[13] N. J. Higham, *Accuracy and Stability of Numerical Algorithms*. Society for Industrial and Applied Mathematics, 2002.
+
+[14] A. K. Jain, M. N. Murty, and P. J. Flynn, "Data clustering: A review," *ACM Comput. Surv.*, Sep. 1999.
+
+[15] P. Sinha and R. Russell, "A perceptually based comparison of image similarity metrics," *Perception*, 2011.
+
+[16] M. Leeser, J. Theiler, M. Estlick, and J. J. Szymanski, "Design tradeoffs in a hardware implementation of the k-means clustering algorithm," in *IEEE SAM SP Workshop*, 2000.
+
+[17] L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann, "Mpfr: A multiple-precision binary floating-point library with correct rounding," *ACM Trans. Math. Softw.*, no. 2, Jun. 2007.
+
+[18] A. Yazdanbakhsh, D. Mahajan, P. Lotfi-Kamran, and H. Esmaeilzadeh, "Axbench: A benchmark suite for approximate computing across the system stack," 2016.
+
+[19] "Uncompressed rgb images," https://bitbucket.org/act-lab/axbench/src, accessed: 06/08/2018.
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+ON EXISTENCE AND UNIQUENESS OF ASYMPTOTIC N-SOLITON-LIKE
+SOLUTIONS OF THE NONLINEAR KLEIN-GORDON EQUATION
+
+XAVIER FRIEDERICH
+
+**ABSTRACT.** We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in $\mathbb{R}^{1+d}$, $d \ge 1$, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schrödinger equations, we obtain an *N*-parameter family of solutions of (NLKG) which converges exponentially fast to a sum of *N* given (unstable) solitons. For *N* = 1, this family completely describes the set of solutions converging to the soliton considered; for *N* ≥ 2, we prove uniqueness in a class with explicit algebraic rate of convergence.
+
+# 1. INTRODUCTION
+
+**1.1. Setting of the problem.** We consider the following nonlinear Klein-Gordon equation
+
+$$
+\text{(NLKG)} \qquad \partial_t^2 u = \Delta u - u + f(u),
+$$
+
+where $u$ is a real-valued function of $(t,x) \in \mathbb{R} \times \mathbb{R}^d$ and $f$ is a $\mathscr{C}^1$ real-valued function on $\mathbb{R}$. This
+equation classically rewrites as the following first order system in time:
+
+$$
+\text{(NLKG')} \qquad \partial_t U = \begin{pmatrix} 0 & I_d \\ \Delta - I_d & 0 \end{pmatrix} U + \begin{pmatrix} 0 \\ f(u) \end{pmatrix},
+$$
+
+where $U$ is the two-vector $\begin{pmatrix} u \\ \partial_t u \end{pmatrix}$.
+
+Let us denote by $F$ the unique primitive of $f$ on $\mathbb{R}$ which vanishes in $0$. We make the following assumptions:
+
+• if $d=1$,
+
+(H1) $f$ is odd and $f'(0) = 0$.
+
+(H2) There exists $r > 0$ such that $F(r) > \frac{1}{2}r^2$.
+
+• if $d \ge 2$,
+
+(H'1) $f$ is a pure $H^1$-subcritical nonlinearity $r \mapsto \lambda|r|^{p-1}r$, with $\lambda > 0$ and $p > 1$ if
+$d = 2$ and $p \in (1, \frac{d+2}{d-2})$ if $d \ge 3$.
+
+Assumption (H1) for $d=1$ or assumption (H'1) for $d \ge 2$ on the nonlinearity $f$ ensures that
+the Cauchy problem is locally well-posed in the energy space $H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$ [13, 24]. It is even
+globally well-posed if one assumes further sufficient smallness on the initial condition.
+
+Recall also that the following quantities are conserved for $H^1 \times L^2$-solutions $(u, \partial_t u)$ of (NLKG'):
+
+• the energy $\frac{1}{2} \int_{\mathbb{R}^d} \{((\partial_t u)^2 + |\nabla u|^2 + u^2 - 2F(u))\} (t,x) dx$
+
+• the momentum $\int_{\mathbb{R}^d} \{\partial_t u \nabla u\} (t,x) dx.$
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+Similarly there exists $C' > 0$ such that for $t$ large
+
+$$ (3.2) \quad \left\| \Phi_{A'_1, \dots, A'_N}(t) - \Phi_{A'_1, \dots, A'_{i_0-1}}(t) - \sum_{j=i_0}^{N} A'_j e^{-e_j t} Y_{+,j}(t) \right\|_{H^1 \times L^2} \le C' \sum_{j=i_0}^{N} e^{-(e_j+\sigma)t}. $$
+
+Using that $\Phi_{A_1, \dots, A_N}(t) = \Phi_{A'_1, \dots, A'_N}(t)$ and $\Phi_{A_1, \dots, A_{i_0-1}}(t) = \Phi_{A'_1, \dots, A'_{i_0-1}}(t)$, we deduce from (3.1) and (3.2) that for all $t$ sufficiently large
+
+$$ e^{-e_{i_0}t} |A_{i_0} - A'_{i_0}| \le C e^{-(e_{i_0}+\sigma)t}. $$
+
+Hence, letting $t \to +\infty$, we obtain $A_{i_0} - A'_{i_0} = 0$, which leads to a contradiction. This ends the proof of Theorem 1.2.
+
+**3.1. Compactness argument assuming uniform estimate.** The goal of this subsection is to explain how to prove Proposition 3.1; for this, we follow the strategy of Combet [4] and Côte and Muñoz [9], both inspired from pioneering work by Martel [19] and Côte, Martel and Merle [8]. One key ingredient in the construction is the obtainment of uniform estimates satisfied by a sequence of approximating solutions of (NLKG).
+
+We fix $j \in \{1, \dots, N\}$ and $A_j \in \mathbb{R}$. Let $(S_n)_n$ be an increasing sequence of time such that $S_n \to +\infty$. Let us consider $\mathbf{b}_n = (b_{n,k})_{j<k \le N} \in \mathbb{R}^{N-j}$ the generic term of a sequence of parameters to be determined, and let $u_n$ be the maximal solution of (NLKG) such that
+
+$$ (3.3) \qquad U_n(S_n) = \Phi(S_n) + A_j e^{-e_j S_n} Y_{+,j}(S_n) + \sum_{k>j} b_{n,k} Y_{+,k}(S_n), $$
+
+where $U_n := \begin{pmatrix} u_n \\ \partial_t u_n \end{pmatrix}$.
+
+Concerning $u_n$, we claim:
+
+**Proposition 3.2.** There exist $n_0 \ge 0$ and $t_0 > 0$ (independent of $n$) such that for each $n \ge n_0$, there exists $\mathbf{b}_n \in \mathbb{R}^{N-j}$ with $|\mathbf{b}_n| \le 2e^{-(e_j+2\sigma)t}$ and such that $U_n$ is defined on $[t_0, S_n]$ and satisfies
+
+$$ (3.4) \quad \forall t \in [t_0, S_n], \quad \|U_n(t) - \Phi(t) - A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \le C e^{-(e_j+\sigma)t}. $$
+
+The $b_n$ take the role of modulation parameters and are to be determined (if indeed possible) so that $U_n$ fulfills (3.4), thus is a natural candidate in order to "approximate" the desired solution $U$ which is the object of Proposition 3.2.
+
+We postpone the proof of the previous statement at the next subsection; for the time being, let us assume that Proposition 3.2 is satisfied and let us show how it implies Proposition 3.1. In fact, the existence of $U$ is due to the continuity of the flow of (NLKG) for the weak $H^1 \times L^2$ topology. We explicit the construction of $U$ below, following the same strategy as [8, paragraph 2.2, step 2] or [9, section 4].
+
+*Proof of Proposition 3.1.* We observe that the sequence $(\|U_n(t_0)\|_{H^1\times L^2})_{n\in\mathbb{N}}$ is bounded; thus there exist a subsequence of $(U_n(t_0))_{n\in\mathbb{N}}$, say $(U_{n_k}(t_0))_{k\in\mathbb{N}}$, and $U_0 \in H^1(\mathbb{R})\times L^2(\mathbb{R})$ such that $(U_{n_k}(t_0))_{k\in\mathbb{N}}$ converges to $U_0$ in the sense of the weak topology in $H^1(\mathbb{R}) \times L^2(\mathbb{R})$. Let us consider $U$, defined as the maximal solution of (NLKG) such that $U(t_0) = U_0$.
+
+Let $t \ge t_0$. For $k$ sufficiently large, $S_{n_k} \ge t$ and thus $U_{n_k}$ is defined on $[t_0, t]$. By a standard result (we refer to [9, Lemma 10] and [28, Theorem 1.2]), $U$ is defined on $[t_0, t]$ and $(U_{n_k}(t))_{k}$ converges weakly to $U(t)$ in $H^1(\mathbb{R}) \times L^2(\mathbb{R})$.
+
+Moreover, by property of the weak limit,
+
+$$ 
+\begin{align*}
+& \|U(t) - \Phi(t) - A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \le \liminf_{k \to +\infty} \|U_{n_k}(t) - \Phi(t) - A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \\
+& \le C_0 e^{-(e_j + \sigma)t}.
+\end{align*}
+\hspace*{\fill} □
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_11.md b/samples/texts/7376768/page_11.md
new file mode 100644
index 0000000000000000000000000000000000000000..784f47ebc53e08920655fcfc16250ef5aed086dd
--- /dev/null
+++ b/samples/texts/7376768/page_11.md
@@ -0,0 +1,50 @@
+Now, the remainder of Section 3 is devoted to the proof of Proposition 3.2.
+
+**3.2.** **Proof of Proposition 3.2.** For ease of reading, we will drop the index $n$ for the rest of this subsection (except for $S_n$), that is, we will write $U$ for $U_n$, $\mathfrak{b}$ for $\mathfrak{b}_n$, etc.
+
+Let us introduce the following variable (which depends on $n$)
+
+$$W(t) := U(t) - \Phi(t) - A_j e^{-e_j t} Y_{+,j}(t)$$
+
+and for all $k \in \{1, \dots, N\}$,
+
+$$\alpha_{\pm,k}(t) := \langle W(t), Z_{\pm,k}(t) \rangle$$
+
+(which depends on $\mathfrak{b}$ in particular by definition of $U = U_n$ (3.3)). We denote also $\alpha_-(t) := (\alpha_{-,k}(t))_{j<k \le N}$.
+
+**3.2.1. Modulated final data and strategy of the proof of Proposition 3.2.** We make the first step in order to determine the appropriate modulation parameter $\mathfrak{b}$. We obtain $\mathfrak{b}$ as the solution of a well-chosen equation; this is the object of the following
+
+**Lemma 3.3.** There exists $n_0 \ge 0$ such that for all $n \ge n_0$ and for all $\mathfrak{a} \in \mathbb{R}^{N-j}$, there exists a unique $\mathfrak{b} \in \mathbb{R}^{N-j}$ such that $\|\mathfrak{b}\| \le 2\|\mathfrak{a}\|$ and $\alpha_-(S_n) = \mathfrak{a}$.
+
+*Proof.* Let us consider the linear application
+
+$$\Psi: \quad \mathbb{R}^{N-j} \to \mathbb{R}^{N-j} \\ 
+\mathfrak{b} = (b_l)_{j<l \le N} \mapsto \left( \sum_{l>j} b_l \langle Y_{+,l}(S_n), Z_{-,l}(S_n) \rangle \right)_{j<k \le N}.$$
+
+Its matrix in the canonical basis of $\mathbb{R}^{N-j}$ has generic entry $\psi_{k,l} := \langle Y_{+,j+l}(S_n), Z_{-,j+k}(S_n) \rangle$ where $(k,l) \in \{1, \dots, N\}^2$.
+
+Since $\psi_{k,l} = 1$ if $k=l$ and $|\psi_{k,l}| \le C_0 e^{-\sigma S_n}$ for $k \ne l$, with $C_0 > 0$ independent of $n$, we have $\Psi = Id + M$ with $\|M\| \le \frac{1}{2}$ for large values of $n$. Thus $\Psi$ is invertible (for $n$ large) and $\|\Psi^{-1}\| \le 2$. We deduce the content of Lemma 3.3 by taking $n_0$ large enough and by considering, for a given $\mathfrak{a} \in \mathbb{R}^{N-j}$, the element $\mathfrak{b} := \Psi^{-1}(\mathfrak{a})$. $\square$
+
+Roughly speaking, Lemma 3.3 reflects that estimate (3.4) is to be proven by choosing a relevant vector $\mathfrak{a} = a_-(S_n)$.
+
+The reason why we determine $\mathfrak{b}$ according to the value of $\alpha_-(S_n)$ essentially comes from the directions $Z_{-,k}$, which yield "instability" in some sense (given Claim 3.7 below), and also from definition (3.5) below.
+
+At this stage, we notice that we already have:
+
+**Claim 3.4.** *We have:*
+
+(1) $\forall k \in \{1, \dots, N\}, \quad |\alpha_{+,k}(S_n)| \le C|\mathfrak{b}|e^{-2\sigma S_n}$.
+
+(2) $\forall k \in \{1, \dots, j\}, \quad |\alpha_{-,k}(S_n)| \le C|\mathfrak{b}|e^{-2\sigma S_n}$.
+
+(3) $\|W(S_n)\|_{H^1 \times L^2} \le C|\mathfrak{b}|.$
+
+Let $t_0 > 0$ independent of $n$ to be chosen later and $\mathfrak{a} \in B_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n})$ to be determined. We consider the associated data $\mathfrak{b}$ given by Lemma 3.3 and $U$ defined in (3.3).
+
+Let us define
+
+$$T(\mathfrak{a}) := \inf\{T \ge t_0 | \forall t \in [T, S_n], \|W(t)\|_{H^1 \times L^2} \le e^{-(e_j+\sigma)t} \text{ and } e^{(e_j+2\sigma)t} \alpha_-(t) \in B_{\mathbb{R}^{N-j}}(1)\}.$$
+
+We observe that Proposition 3.2 holds if for all $n$, we can find $\mathfrak{a}$ such that $T(\mathfrak{a}) = t_0$. In the rest of the proof, our goal is thus to prove the existence of such an element $\mathfrak{a}$.
+
+To this end, we will first of all improve the estimate on $\|W(t)\|_{H^1 \times L^2}$ which falls within the definition of $T(\mathfrak{a})$. This is the object of the following subsection. Then, we will only need to care about the second condition, which implies a control of $\alpha_-(t)$; this is done in subsection 3.2.3.
\ No newline at end of file
diff --git a/samples/texts/7376768/page_12.md b/samples/texts/7376768/page_12.md
new file mode 100644
index 0000000000000000000000000000000000000000..ee5e56c9e5e63484024582cbaa907e1857c936cf
--- /dev/null
+++ b/samples/texts/7376768/page_12.md
@@ -0,0 +1,49 @@
+**3.2.2. Improvement of the estimate on $||W||_{H^1 \times L^2}$**. For notation purposes and ease of reading, we sometimes omit the index $n$ and also write $O(G(t))$ in order to refer to a function $g$ which a priori depends on $n$ and such that there exists $C \ge 0$ such that for all $n$ large and for all $t \in [t_n^*, S_n]$, $|g(t)| \le C|G(t)|$.
+
+**Lemma 3.5.** There exists $K_0 > 0$ such that for all $t \in [T(a), S_n]$,
+
+$$ ||W(t)||_{H^1 \times L^2} \le \frac{K_0}{t^{1/4}} e^{-(e_j + \sigma)t}. $$
+
+The whole subsection consists of the proof of this lemma.
+
+*Step 1: Estimates on $\alpha_{\pm,k}$. Let us begin with the computation of the time derivative of W.*
+
+**Claim 3.6.** We have for all $k \in \{1, \dots, N\}$,
+
+$$ (3.6) \quad \partial_t W = \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(\varphi) & 0 \end{pmatrix} W + A_j e^{-e_j t} \left[ \begin{pmatrix} \beta_j \partial_x & Id \\ \partial_x^2 - Id + f'(Q_k) & \beta_j \partial_x \end{pmatrix} Y_{+,j} + e_j Y_{+,j} \right] \\ \qquad + A_j e^{-e_j t} \begin{pmatrix} 0 & 0 \\ f'(\phi) - f'(Q_k) & 0 \end{pmatrix} Y_{+,j} + \begin{pmatrix} 0 \\ g \end{pmatrix}, $$
+
+where $g := f(u) - f(\varphi) - f'(\varphi)(u - \varphi)$ satisfies
+
+$$ ||g(t)||_{L^{\infty}} = O(||u - \varphi||_{H^1}^2). $$
+
+*Proof.* Claim 3.6 follows from the fact that both $U$ and $\Phi$ satisfy (NLKG') and is also a consequence of the following Taylor inequality ($f$ is $\mathscr{C}^2$)
+
+$$ |f(u) - f(\varphi) - f'(\varphi)(u - \varphi)|(t) \le C ||u(t) - \varphi(t)||_{L^\infty}^2 $$
+
+and the Sobolev embedding $H^1(\mathbb{R}) \hookrightarrow L^\infty(\mathbb{R})$. $\square$
+
+Now, we are in a position to prove the following estimate on $\alpha_{\pm,k}$.
+
+**Claim 3.7.** For all $k \in \{1, \dots, N\}$ and for all $t \in [T(a), S_n]$, we have
+
+$$ (3.7) \quad \left| \frac{d}{dt} \alpha_{\pm,k}(t) \mp e_k \alpha_{\pm,k}(t) \right| \le C \left( e^{-4\sigma t} ||W(t)||_{H^1 \times L^2} + ||W(t)||_{H^1 \times L^2}^2 + e^{-(e_j+4\sigma)t} \right). $$
+
+*Proof.* Let $k \in \{1, \dots, N\}$. By means of (3.6) and since $\partial_t Z_{\pm,k} = -\beta_k \partial_x Z_{\pm,k}$, we compute
+
+$$ 
+\begin{aligned}
+\frac{d}{dt} \alpha_{\pm,k}(t) &= \langle \partial_t W, Z_{\pm,k} \rangle + \langle W, \partial_t Z_{\pm,k} \rangle \\
+&= \left\langle W, \begin{pmatrix} -\beta_k \partial_x & \partial_x^2 - Id + f'(\varphi) \\ Id & -\beta_k \partial_x \end{pmatrix} Z_{\pm,k} \right\rangle \\
+&\quad + A_j e^{-e_j t} \left\langle Y_{+,j}, \begin{pmatrix} -\beta_j \partial_x & \partial_x^2 - Id + f'(Q_k) \\ Id & -\beta_j \partial_x \end{pmatrix} Z_{\pm,k} \right\rangle \\
+&\quad + A_j e^{-e_j t} \left[ \begin{pmatrix} 0 & f'(Q_k) - f'(\varphi) \\ 0 & 0 \end{pmatrix} Z_{\pm,k} + e_j \langle Y_{+,j}, Z_{\pm,k} \rangle \right] \\
+&\quad + O(||U - \Phi||_{H^1 \times L^2}^2).
+\end{aligned}
+$$
+
+Let us notice first that
+
+$$ \left\langle W, \begin{pmatrix} -\beta_k \partial_x & \partial_x^2 - Id + f'(\varphi) \\ Id & -\beta_k \partial_x \end{pmatrix} Z_{\pm,k} \right\rangle = \left\langle W, \mathcal{H}_k Z_{\pm,k} \right\rangle + \left\langle W, \begin{pmatrix} 0 & f'(\varphi) - f'(Q_k) \\ 0 & 0 \end{pmatrix} Z_{\pm,k} \right\rangle. $$
+
+We have
+
+$$ \langle W, \mathcal{H}_k Z_{\pm,k} \rangle = \langle W, \pm e_k Z_{\pm,k} \rangle = \pm e_k \alpha_{\pm,k}. $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_13.md b/samples/texts/7376768/page_13.md
new file mode 100644
index 0000000000000000000000000000000000000000..e95fcc5a778cb0843b7ebab9df8110722ac7c08c
--- /dev/null
+++ b/samples/texts/7376768/page_13.md
@@ -0,0 +1,84 @@
+and
+
+$$
+\begin{align*}
+& \left| \left\langle W, \begin{pmatrix} 0 & f'(\varphi) - f'(Q_k) \\ 0 & 0 \end{pmatrix} Z_{\pm,k} \right\rangle \right| \\
+& \le \|W\|_{H^1 \times L^2} \| (f'(\varphi) - f'(Q_k)) Z_{\pm,k} \|_{H^1 \times L^2} \\
+& \le C \left\| \varphi - \sum_{i=1}^N Q_i \right\|_{L^\infty} \|W\|_{H^1 \times L^2} + C \|W\|_{H^1 \times L^2} \sum_{i \ne k} \|Q_i Z_{\pm,k}\|_{H^1 \times L^2} \\
+& \le C e^{-4\sigma t} \|W\|_{H^1 \times L^2}.
+\end{align*}
+$$
+
+Similarly, we have
+
+$$
+\left| \left\langle Y_{+,j}, \begin{pmatrix} 0 & f'(Q_k) - f'(\varphi) \\ 0 & 0 \end{pmatrix} Z_{\pm,k} \right\rangle \right| \le C e^{-4\sigma t}
+$$
+
+and
+
+$$
+\begin{align*}
+\left\langle Y_{+,j}, \begin{pmatrix} -\beta_j \partial_x & \partial_x^2 - Id + f'(Q_k) \\ Id & -\beta_j \partial_x \end{pmatrix} Z_{\pm,k} \right\rangle &= \langle Y_{+,j}, \mathcal{H}_k Z_{\pm,k} \rangle + (\beta_k - \beta_j) \langle Y_{+,j}, \partial_x Z_{\pm,k} \rangle \\
+&= \pm \langle Y_{+,j}, e_k Z_{\pm,k} \rangle + O(e^{-4\sigma t}).
+\end{align*}
+$$
+
+Indeed, we notice that
+
+$$
+(\beta_k - \beta_j) \langle Y_{+,j}, \partial_x Z_{\pm,k} \rangle = \begin{cases} 0 & \text{if } k = j \\ O(e^{-4\sigma t}) & \text{if } k \neq j. \end{cases}
+$$
+
+Hence, we obtain
+
+$$
+(3.8) \quad \frac{d}{dt} \alpha_{\pm,k}(t) = \pm e_k \alpha_{\pm,k} + O(e^{-4\sigma t} \|W\|_{H^1 \times L^2}) \\
+\phantom{(3.8)\quad} + A_j e^{-e_j t} [\pm \langle Y_{+,j}, e_k Z_{\pm,k} \rangle + e_j \langle Y_{+,j}, Z_{\pm,k} + O(e^{-4\sigma t} + \|U-\Phi\|^2_{H^1\times L^2}) ].
+$$
+
+Now, we observe that
+
+$$
+\pm e_k \langle Y_{+,j}, Z_{\pm,k} \rangle + e_j \langle Y_{+,j}, Z_{\pm,k} \rangle = O(e^{-4\sigma t}).
+$$
+
+This is clear if $k \neq j$ and for $k = j$, we have
+
+$$
+\pm e_j \langle Y_{+,j}, Z_{\pm,j} \rangle + e_j \langle Y_{+,j}, Z_{\pm,j} \rangle = \begin{cases} 0+0=0 & \text{if } \pm = + \\ -e_j + e_j = 0 & \text{if } \pm = -; \end{cases}
+$$
+
+indeed, we recall from Proposition 2.1
+
+$$
+(Y_{+,j}, Z_{+,j}) = 0 \quad \text{and} \quad (Y_{+,j}, Z_{-,j}) = 1.
+$$
+
+In addition, we have by the well-known inequality $(a+b)^2 \le 2(a^2+b^2)$,
+
+$$
+\|U - \Phi\|_{H^1 \times L^2}^2 \le C (\|W\|_{H^1 \times L^2}^2 + e^{-2e_j t}).
+$$
+
+Considering that $2e_j \ge e_j + 4\sigma$, we have thus finished the proof of the claim.
+
+**Step 2: Control of the stable directions.**
+
+**Claim 3.8.** We have for all $k \in \{1, \dots, N\}$, for all $t \in [T(a), S_n]$,
+
+$$
+(3.9) \qquad |α_{+,k}(t)| ≤ Ce^{-(e_j+4σ)t}.
+$$
+
+*Proof.* Due to Claim 3.7 and (3.5), we obtain
+
+$$
+\left| \frac{d}{dt} \alpha_{+,k}(t) - e_k \alpha_{+,k}(t) \right| \le C e^{-(e_j+4\sigma)t},
+$$
+
+that is, for all $t \in [T(a), S_n]$,
+
+$$
+|(e^{-e_k t} \alpha_{+,k}(t))'| \le C e^{-(e_j+e_k+4\sigma)t}.
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_14.md b/samples/texts/7376768/page_14.md
new file mode 100644
index 0000000000000000000000000000000000000000..992dc06573df40644c62deda7779df6fb7079e71
--- /dev/null
+++ b/samples/texts/7376768/page_14.md
@@ -0,0 +1,72 @@
+Integrating, we deduce that for all $t \in [T(\mathfrak{a}), S_n]$,
+
+$$|e^{-e_k S_n} \alpha_{+,k}(S_n) - e^{-e_k t} \alpha_{+,k}(t)| \le Ce^{-(e_j+e_k+4\sigma)t}.$$
+
+Thus,
+
+$$|\alpha_{+,k}(t)| \le |\alpha_{+,k}(S_n)| + C e^{-(e_j+4\sigma)t}.$$
+
+From Claim 3.4 and Lemma 3.3, we have
+
+$$
+\begin{align*}
+|\alpha_{+,k}(S_n)| &\le Ce^{-2\sigma S_n} |b| \\
+&\le Ce^{-2\sigma S_n} e^{-(e_j+2\sigma)S_n} \\
+&\le Ce^{-(e_j+4\sigma)t}.
+\end{align*}
+$$
+
+Consequently, Claim 3.8 indeed holds.
+
+**Step 3: Control of the unstable directions for $k \le j$.**
+
+**Claim 3.9.** We have for all $k \in \{1, \dots, j\}$, for all $t \in [T(\mathfrak{a}), S_n]$,
+
+$$ (3.10) \qquad |\alpha_{-,k}(t)| \le Ce^{-(e_j+4\sigma)t}. $$
+
+*Proof.* As in the preceding step, we have for all $k \in \{1, \dots, N\}$ and $t \in [T(\mathfrak{a}), S_n]$,
+
+$$ (3.11) \qquad \left| \frac{d}{dt} \alpha_{-,k}(t) + e_k \alpha_{-,k}(t) \right| \le Ce^{-(e_j+4\sigma)t}, $$
+
+which writes also
+
+$$ |(e^{e_k t} \alpha_{-,k}(t))'| \le Ce^{(e_k - e_j + 4\sigma)t}. $$
+
+For $k \le j$, we have $e_k \le e_j$, and so by integration, we obtain
+
+$$ |\alpha_{-,k}(t)| \le e^{e_k(S_n-t)} |\alpha_{-,k}(S_n)| + Ce^{-(e_j+4\sigma)t}. $$
+
+But again from Claim 3.4 and Lemma 3.3, we infer
+
+$$
+\begin{align*}
+e^{e_k(S_n-t)} |\alpha_{-,k}(S_n)| &\le Ce^{e_k(S_n-t)} e^{-2\gamma S_n} e^{-(e_j+2\sigma)S_n} \\
+&\le Ce^{(S_n-t)(e_k-e_j)} e^{-e_j t} e^{-4\sigma S_n} \\
+&\le Ce^{-(e_j+4\sigma)t}.
+\end{align*}
+$$
+
+Thus
+
+$$ \forall k \in \{1, \dots, j\}, \quad \forall t \in [T(\mathfrak{a}), S_n], \qquad |\alpha_{-,k}(t)| \le Ce^{-(e_j+4\sigma)t}. $$
+
+**Step 4: Control of a Lyapunov functional satisfying a coercivity property.** Let us consider
+
+$$
+\psi : \mathbb{R} \to \mathbb{R} \\
+x \mapsto \frac{2}{\pi} \operatorname{Arctan}(e^{-x}).
+$$
+
+We define for all $k = 1, \dots, N-1$,
+
+$$ \psi_k(t, x) := \psi\left(\frac{1}{\sqrt{t}}\left(x - \frac{\beta_{\eta(k)} + \beta_{\eta(k+1)}}{2}t - \frac{x_{\eta(k)} + x_{\eta(k+1)}}{2}\right)\right), $$
+
+and then
+
+$$
+\phi_1(t) = \psi_1(t) \\
+\phi_k(t) = \psi_k(t) - \psi_{k-1}(t) \text{ for all } k = 2, \dots, N-1, \\
+\phi_N(t) = 1 - \psi_{N-1}(t).
+$$
+
+Recall that the permutation $\eta$ has been chosen so that $-1 < \beta_{\eta(1)} < \dots < \beta_{\eta(N)} < 1$.
\ No newline at end of file
diff --git a/samples/texts/7376768/page_15.md b/samples/texts/7376768/page_15.md
new file mode 100644
index 0000000000000000000000000000000000000000..d4aa0bbdeeacc784939a6ab001272edf17cf9672
--- /dev/null
+++ b/samples/texts/7376768/page_15.md
@@ -0,0 +1,60 @@
+Now, let us introduce for all $k \in \{1, \dots, N\}$
+
+$$ \mathcal{F}_{W,k}(t) = \int_{\mathbb{R}} (w_1^2 + (\partial_x w_1)^2 + w_2^2 - f'(Q_{\eta(k)})w_1^2 + 2\beta_{\eta(k)}\partial_x w_1 w_2) \phi_k \, dx, $$
+
+and
+
+$$ \mathcal{F}_W(t) := \sum_{k=1}^{N} \mathcal{F}_{W,k}(t). $$
+
+By means of Proposition 2.2 and a usual localization argument [21, 22], we obtain that $\mathcal{F}_W$ is coercive on a subspace of $H^1 \times L^2$ of finite codimension. More precisely, there exists $\mu > 0$ such that
+
+$$ (3.12) \quad \mathcal{F}_W(t) \geq \mu \|W(t)\|_{H^1 \times L^2}^2 - \frac{1}{\mu} \sum_{k=1}^{N} \left( \langle W, \partial_x R_k \rangle^2 + \langle W, Z_{+,k} \rangle^2 + \langle W, Z_{-,k} \rangle^2 \right). $$
+
+We state the following control about the derivative of $\mathcal{F}_W$:
+
+**Claim 3.10.** For $t_0$ large and for all $t \in [T(a), S_n]$,
+
+$$ (3.13) \qquad \left| \frac{d}{dt} \mathcal{F}_W(t) \right| \le \frac{C}{\sqrt{t}} \|W\|_{H^1 \times L^2}^2. $$
+
+*Proof.* Let us rewrite $\mathcal{F}_{W,k}$ differently, using the notations developed in the introduction. Relying on integrations by parts, our computations lead to:
+
+$$ \langle (H_{\eta(k)} W) \phi_k, W \rangle = \mathcal{F}_{W,k}(t) - \frac{1}{2} \int_{\mathbb{R}} w_1^2 \partial_x^2 \phi_k \, dx + \beta_{\eta(k)} \int_{\mathbb{R}} w_1 w_2 \partial_x \phi_k \, dx. $$
+
+Thus
+
+$$ (3.14) \qquad \mathcal{F}_{W,k}(t) = \langle (H_{\eta(k)} W)\phi_k, W\rangle + O\left(\frac{1}{\sqrt{t}}\|W\|_{H^1\times L^2}^2\right). $$
+
+We immediately have
+
+$$ \frac{d}{dt} \langle (H_{\eta(k)} W) \phi_k, W \rangle = \langle (H_{\eta(k)} W) \phi_k, \partial_t W \rangle + \langle \partial_t (H_{\eta(k)} W) \phi_k, W \rangle + \langle (H_{\eta(k)} W) \partial_t \phi_k, W \rangle. $$
+
+Besides
+
+$$ 
+\begin{align*}
+& \langle \partial_t (H_{\eta(k)} W) \phi_k, W \rangle + \langle (H_{\eta(k)} W) \partial_t \phi_k, W \rangle \\
+&= \langle (H_{\eta(k)} \partial_t W) \phi_k, W \rangle + \beta_{\eta(k)} \int_{\mathbb{R}} \partial_x Q_{\eta(k)} f''(Q_{\eta(k)}) w_1^2 \phi_k \, dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1 \times L^2}^2\right) \\
+&= \langle H_{\eta(k)} (\partial_t W), W \phi_k \rangle + \beta_{\eta(k)} \int_{\mathbb{R}} \partial_x Q_{\eta(k)} f''(Q_{\eta(k)}) w_1^2 \phi_k \, dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1 \times L^2}^2\right).
+\end{align*}
+$$
+
+Since $H_{\eta(k)}$ is a self-adjoint operator, we have
+
+$$ \langle H_{\eta(k)} (\partial_t W), W \phi_k \rangle = \langle \partial_t W, H_{\eta(k)} (W \phi_k) \rangle. $$
+
+By a straightforward calculation, we have moreover
+
+$$ \langle H_{\eta(k)}(W\phi_k), \partial_t W \rangle = (\langle H_{\eta(k)}(W)\phi_k, \partial_t W \rangle + O(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2})). $$
+
+At this stage, we thus obtain
+
+$$ 
+\begin{align*}
+& \frac{d}{dt} \langle (H_{\eta(k)} W) \phi_k, W \rangle = 2\langle (H_{\eta(k)} W) \phi_k, \partial_t W \rangle \\
+& \phantom{{}= 2\langle (H_{\eta(k)} W) } + \beta_{\eta(k)} \int_{\mathbb{R}} \partial_x Q_{\eta(k)} f''(Q_{\eta(k)}) w_1^2 \phi_k dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}\right).
+\end{align*}
+$$
+
+Now, by (3.6), we write
+
+$$ (\langle H_{\eta(k)} W )\phi_k , \partial_t W\rangle = I_1 + I_2 + I_3 $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_16.md b/samples/texts/7376768/page_16.md
new file mode 100644
index 0000000000000000000000000000000000000000..5853abc9ed64a1ca6cbed804e83abf7f1b4da92c
--- /dev/null
+++ b/samples/texts/7376768/page_16.md
@@ -0,0 +1,54 @@
+where
+
+$$
+I_1 := \left\langle \begin{pmatrix} T_{\eta(k)} & 0 \\ 0 & Id \end{pmatrix} W \phi_k, \begin{pmatrix} 0 & 0 \\ \partial_x^2 - Id + f'(\varphi) & 0 \end{pmatrix} \right\rangle W
+$$
+
+by denoting $T_i = -\partial_x^2 + Id - f'(Q_i)$ for all $i = 1, \dots, N$,
+
+$$
+I_2 := \beta_{\eta(k)} \left\langle \begin{pmatrix} 0 & -\partial_x \\ \partial_x & 0 \end{pmatrix} W \phi_k, \begin{pmatrix} 0 & Id \\ -T_{\eta(k)} + f'(\varphi) - f'(Q_{\eta(k)}) & 0 \end{pmatrix} \right\rangle_W
+$$
+
+and
+
+$$
+I_3 := A_j e^{-e_j t} \left\langle (H_{\eta(k)} W) \phi_k, \begin{pmatrix} \beta_j \partial_x & Id \\ \partial_x^2 - Id + f'(Q_j) & \beta_j \partial_x \end{pmatrix} Y_{+,j} \right\rangle \\
+\qquad + \left\langle (H_{\eta(k)} W) \phi_k, e_j Y_{+,j} + \begin{pmatrix} 0 & 0 \\ f'(\varphi) - f'(Q_j) & 0 \end{pmatrix} Y_{+,j} \right\rangle.
+$$
+
+Let us deal with $I_1$: we observe that
+
+$$
+\begin{align*}
+I_1 &= \left\langle \begin{pmatrix} T_{\eta(k)} & 0 \\ 0 & Id \end{pmatrix} W \phi_k, \begin{pmatrix} 0 & 0 \\ -T_{\eta(k)} + f'(\varphi) - f'(Q_{\eta(k)}) & Id \end{pmatrix} W \right\rangle \\
+&= \left\langle W, \begin{pmatrix} T_{\eta(k)} & 0 \\ 0 & Id \end{pmatrix} \begin{pmatrix} 0 & Id \\ -T_{\eta(k)} & 0 \end{pmatrix} W \phi_k \right\rangle + O(e^{-4\sigma t} \|W\|_{H^1\times L^2}^2) \\
+&= \left\langle W, \begin{pmatrix} 0 & T_{\eta(k)} \\ -T_{\eta(k)} & 0 \end{pmatrix} W \phi_k \right\rangle + O(e^{-4\sigma t} \|W\|_{H^1\times L^2}^2) \\
+&= -\int_{\mathbb{R}} w_1 \partial_x^2 w_2 \phi_k dx + \int_{\mathbb{R}} w_2 \partial_x^2 w_1 \phi_k dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2\right) \\
+&= O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2\right).
+\end{align*}
+$$
+
+We have
+
+$$
+I_2 = \beta_{\eta(k)} \left\langle \begin{pmatrix} 0 & -\partial_x \\ \partial_x & 0 \end{pmatrix} W \phi_k, \begin{pmatrix} 0 & Id \\ -T_{\eta(k)} & 0 \end{pmatrix} W \right\rangle + O\left(e^{-(e_j+4\sigma)t} \|W\|_{H^1\times L^2}\right) \\
+= -\beta_{\eta(k)} \int_{\mathbb{R}} \partial_x w_2 w_2 \phi_k dx + \beta_{\eta(k)} \int_{\mathbb{R}} \partial_x w_1 (\partial_x^2 w_1 - w_1 + f'(Q_{\eta(k)})w_1) \phi_k dx \\
++ O\left(e^{-(e_j+4\sigma)t} \|W\|_{H^1\times L^2}\right) \\
+= -\frac{\beta_{\eta(k)}}{2} \int_{\mathbb{R}} w_1^2 \partial_x Q_{\beta_{\eta(k)}} f''(Q_{\beta_{\eta(k)}}) \phi_k dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2 + e^{-(e_j+4\sigma)t} \|W\|_{H^1\times L^2}\right).
+$$
+
+In addition, we have
+
+$$
+I_3 = A_j e^{-e_j t} \langle (H_{\eta(k)} W) \phi_k, JZ_{+,j} + e_j Y_{+,j} \rangle + O \left( e^{-(4\sigma+e_j)t} \|W\|_{H^1\times L^2} \right) \\
+= A_j e^{-e_j t} \left( \langle W\phi_k, -\mathcal{H}_{\eta(k)} Z_{+,j} \rangle + e_j \langle W\phi_k, H_{\eta(k)} Y_{+,k} \rangle \right) \\
++ O \left( \frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2 + e^{-(4\sigma+e_j)t} \|W\|_{H^1\times L^2} \right) \\
+= O \left( \frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2 + e^{-(4\sigma+e_j)t} \|W\|_{H^1\times L^2} \right).
+$$
+
+Note that the last line of the previous equality is a consequence of the following observation: if $\eta(k) = j$, we have $\mathcal{H}_j Z_{+,j} = e_j Z_{+,j}$ and $H_j Y_{+,j} = Z_{+,j}$ so that
+
+$$
+A_j e^{-e_j t} (\langle W \phi_k, -\mathcal{H}_{\eta(k)} Z_{+,j} \rangle + e_j \langle W \phi_k, H_{\eta(k)} Y_{+,j} \rangle) = 0.
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_17.md b/samples/texts/7376768/page_17.md
new file mode 100644
index 0000000000000000000000000000000000000000..c01adc912aae4bd5ee063d3b1a015e986a5c4a2e
--- /dev/null
+++ b/samples/texts/7376768/page_17.md
@@ -0,0 +1,49 @@
+If $\eta(k) \neq j$, we have
+
+$$A_j e^{-e_j t} (\langle W\phi_k, -\mathcal{H}_{\eta(k)}Z_{+,j} \rangle + e_j \langle W\phi_k, H_{\eta(k)}Y_{+,j} \rangle) = O(e^{-(e_j+4\sigma)t} \|W\|_{H^{1\times L^2}}).$$
+
+Gathering the preceding computations yields
+
+$$\left|\frac{d}{dt}\mathcal{F}_{W,k}(t)\right| \le C \left(\frac{1}{\sqrt{t}}\|\mathcal{W}\|_{H^{1\times L^2}}^2 + e^{-(e_j+4\sigma)t}\|\mathcal{W}\|_{H^{1\times L^2}}^2\right),$$
+
+hence the expected claim, by summing on $k$. $\square$
+
+**Step 5: Control of the directions $\partial_x R_k$.** To obtain a control of the scalar products $\langle W, \partial_x R_k \rangle$ which is more precise than the a priori control by $\|\tilde{W}\|_{H^{1\times L^2}}$, let us introduce the following modulated variable $\tilde{W}$:
+
+$$ (3.15) \qquad \tilde{W}(t) = W(t) + \sum_{k=1}^{N} a_k(t) \partial_x R_k(t), $$
+
+where $a_k(t) \in \mathbb{R}, k = 1, \dots, N$ are chosen so that for all $l = 1, \dots, N$, $\langle \tilde{W}(t), \partial_x R_l(t) \rangle = 0$. Existence and uniqueness of the family $(a_k(t))_{k \in \{1, \dots, N\}}$ are justified by the fact that the (interaction) $N \times N$-matrix with generic entry $\langle \partial_x R_k(t), \partial_x R_l(t) \rangle$ is invertible for $t$ large enough.
+
+Notice that
+
+$$ (3.16) \qquad |a_k(t)| \le C \|W(t)\|_{H^{1\times L^2}} \le Ce^{-(e_j+\sigma)t} $$
+
+The functional $\mathcal{F}_{\tilde{W}}(t)$, defined as $\mathcal{F}_W(t)$ by changing $W$ in $\tilde{W}$, satisfies the following coercivity property:
+
+$$ (3.17) \qquad \|\tilde{W}\|_{H^{1\times L^2}}^2 \le C \left( \mathcal{F}_{\tilde{W}}(t) + \sum_{k=1}^{N} \left( \langle \tilde{W}, Z_{+,k} \rangle^2 + \langle \tilde{W}, Z_{-,k} \rangle^2 \right) \right). $$
+
+We have
+
+$$ \mathcal{F}_{\tilde{W}}(t) \le \mathcal{F}_W(t) + O\left(e^{-4\sigma t} \|W\|_{H^{1\times L^2}}^2\right) $$
+
+and we have moreover by Proposition 2.3 and (3.15).
+
+$$ \langle \tilde{W}, Z_{\pm,k} \rangle^2 \le \alpha_{\pm,k}^2 + e^{-2(e_j+5\sigma)t}. $$
+
+**Claim 3.11 (Estimate on $\tilde{W}$).** We have
+
+$$ \forall t \in [T(a), S_n], \qquad \|\tilde{W}(t)\|_{H^{1\times L^2}}^2 \le \frac{1}{\sqrt{t}} e^{-(2e_j+2\sigma)t}. $$
+
+*Proof.* Let $t$ belong to $[T(a), S_n]$. We obtain by (3.17) and by integration of (3.13) on $[t, +\infty)$ (which is indeed possible by definition of $T(a)$) that
+
+$$ \|\tilde{W}(t)\|_{H^{1\times L^2}}^2 \le \frac{C}{\sqrt{t}} e^{-2(e_j+\sigma)t} + C \sum_{\pm,k} \alpha_{\pm,k}^2 + C e^{-2(e_j+4\sigma)t}. $$
+
+Using the estimate on $\alpha_{\pm,k}$ provided by the definition of $T(a)$ and Claim 3.8, we then infer:
+
+$$ \|\tilde{W}(t)\|_{H^{1\times L^2}}^2 \le \frac{C}{\sqrt{t}} e^{-2(e_j+\sigma)t} + Ce^{-(2e_j+4\sigma)t}. $$
+
+This concludes the proof of the claim. $\square$
+
+**Claim 3.12 (Control of the modulation parameters).** We have for all $k = 1, \dots, N$,
+
+$$ \forall t \in [T(a), S_n], \quad |a_k(t)| \le \frac{C}{t^{4/3}} e^{-(e_j+\sigma)t}. $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_18.md b/samples/texts/7376768/page_18.md
new file mode 100644
index 0000000000000000000000000000000000000000..22a8f1b1284d604ffa402f05bd4a40d1d83ecf6e
--- /dev/null
+++ b/samples/texts/7376768/page_18.md
@@ -0,0 +1,80 @@
+*Proof.* By definition of the modulation parameters $a_k$, we have $\langle \tilde{W}, \partial_x R_k \rangle = 0$. Thus, we have by differentiation with respect to $t$:
+
+$$
+\langle \partial_t \tilde{W}, \partial_x R_k \rangle + \langle \tilde{W}, \partial_t \partial_x R_k \rangle = 0.
+$$
+
+By Proposition 2.3, we have for $l \neq k$,
+
+$$
+\langle \partial_x R_l, \partial_x R_k \rangle = O(e^{-4\sigma t})
+$$
+
+and for all $l$,
+
+$$
+\langle \partial_t \partial_x R_l, \partial_x R_k \rangle = O(e^{-4\sigma t}).
+$$
+
+We deduce that
+
+$$
+a'_{k}(t) \langle \partial_{x} R_{k}, \partial_{x} R_{k} \rangle + \langle \partial_{t} W, \partial_{x} R_{k} \rangle + \langle \tilde{W}, \partial_{t} \partial_{x} R_{k} \rangle = O \left( e^{-4\sigma t} \|W(t)\|_{H^{1} \times L^2} \right).
+$$
+
+We have in addition
+
+$$
+\left\langle W, \begin{pmatrix} 0 \\ Id \end{pmatrix} \partial_x^2 - Id + f'(\varphi) \begin{pmatrix} 0 \\ 0 \end{pmatrix} \partial_x R_k \right\rangle + \langle \tilde{W}, \partial_{t,x} R_k \rangle = O (\|\tilde{W}\|_{H^{1} \times L^2}) .
+$$
+
+What is more,
+
+$$
+|A_j e^{-e_j t} e_j \langle Y_{+,j}, \partial_x R_k \rangle| \le C e^{-(e_j+4\sigma)t},
+$$
+
+again by Proposition 2.3.
+
+Hence,
+
+$$
+\begin{align*}
+|a'_k(t)| &\le C \| \tilde{W}(t) \|_{H^{1} \times L^2} + C e^{-(e_j+4\sigma)t} \\
+&\le \frac{C}{t^{1/4}} e^{-(e_j+\sigma)t} + e^{-(e_j+3\sigma)t}.
+\end{align*}
+$$
+
+$\square$
+
+Now, gathering (3.15), Claim 3.11, and Claim 3.12, we immediately deduce the expected estimate of $\|W\|_{H^1\times L^2}$, which ends the proof of Lemma 3.5.
+
+**3.2.3. Control of the unstable directions for $k > j$ and end of the proof.** To control $\alpha_- = (\alpha_{-,k})_{j<k\le N}$ and eventually obtain the following statement, we resort to a classical topological argument, already set up in [4] and initially developed by Côte, Martel and Merle [8].
+
+**Lemma 3.13.** For $t_0$ large enough, there exists $\mathfrak{a} \in B_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n})$ such that $T(\mathfrak{a}) = t_0$.
+
+The proof follows that of Combet [4]. We write it below for the sake of completeness.
+
+*Proof.* We first choose $t_0$ sufficiently large such that $\frac{K_0}{\sqrt{t_0}} \le \frac{1}{2}$. Then, we have by Lemma 3.5
+
+$$
+\forall t \in [T(\mathbf{a}), S_n], \quad \|W(t)\|_{H^1 \times L^2} \le \frac{1}{2} e^{-(e_j+\sigma)t}.
+$$
+
+Assume, for the sake of contradiction, that for all $\mathfrak{a} \in B_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n})$, $T(\mathfrak{a}) > t_0$. As $\|W(T(\mathfrak{a}))\|_{H^1\times L^2} \le \frac{1}{2}e^{-(e_j+\sigma)T(\mathfrak{a})}$, by definition of $T(\mathfrak{a})$ and continuity of the flow, we have:
+
+$$
+|\alpha_-(T(\mathbf{a}))| = 1.
+$$
+
+(We recall that $\alpha_-(t) = (\alpha_{-,k}(t))_{j<k\le N}$.) In other words, the map
+
+$$
+\begin{array}{r@{\quad}c@{\quad}l@{\quad}\rightrightarrows}
+\mathcal{M}: & B_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n}) & S_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n}) \\
+& \mathfrak{a} & e^{-(e_j+2\sigma)(S_n-T(\mathfrak{a}))}\alpha_-(T(\mathfrak{a}))
+\end{array}
+$$
+
+is well-defined. Now, we aim at showing that $\mathcal{M}$ is continuous and that its restriction to $S_{\mathbb{R}^{N-j}}(e^{-(e_j+2\gamma)S_n})$
+is the identity.
\ No newline at end of file
diff --git a/samples/texts/7376768/page_19.md b/samples/texts/7376768/page_19.md
new file mode 100644
index 0000000000000000000000000000000000000000..2270e2642042b69514f3648bd7bae9ade7f52947
--- /dev/null
+++ b/samples/texts/7376768/page_19.md
@@ -0,0 +1,57 @@
+Let $T \in [T_0, T(\mathbf{a})]$ be such that $W$ is defined on $[T, S_n]$ and, by continuity,
+
+$$\forall t \in [T, S_n], \quad \|W(t)\|_{H^{1} \times L^2} \le 1.$$
+
+We consider, for all $t \in [T, S_n]$:
+
+$$\mathcal{N}(t) := \mathcal{N}(\alpha_{-}(t)) = \|e^{(e_j+2\sigma)t}\alpha_{-}(t)\|^2.$$
+
+**Claim 3.14.** For $t_0$ large enough, and for all $t \in [T, S_n]$ such that $\mathcal{N}(t) = 1$, we have:
+
+$$\mathcal{N}'(t) \le -(e_{j+1} - e_j - 2\sigma).$$
+
+*Proof of Claim 3.14.* Let us start from estimate (3.11): for all $k \in \{j+1, \dots, N\}$, for all $t \in [T, S_n]$,
+
+$$\left|\frac{d}{dt}\alpha_{-,k} + e_k\alpha_{-,k}\right| \le Ce^{-(e_j+4\sigma)t}.$$
+
+Thus we obtain for all $k \in \{j+1, \dots, N\}$,
+
+$$\alpha_{-,k} \frac{d}{dt}\alpha_{-,k} + e_{j+1}\alpha_{-,k}^2 \le \alpha_{-,k} \frac{d}{dt}\alpha_{-,k} + e_k\alpha_{-,k}^2 \le Ce^{-(e_j+4\sigma)t}|\alpha_{-,k}|.$$
+
+Then, summing on $k \in \{j+1, \dots, N\}$ leads to
+
+$$(|\alpha_{-}(t)|^2)' + 2e_{j+1}|\alpha_{-}(t)|^2 \le Ce^{-(e_j+4\sigma)t}|\alpha_{-}(t)|.$$
+
+Therefore we can estimate:
+
+$$\begin{aligned} \mathcal{N}'(t) &= e^{2(e_j+2\sigma)t} \left[ 2(e_j + 2\sigma)|\alpha_{-}(t)|^2 + (|\alpha_{-}(t)|^2)' \right] \\ &\le e^{2(e_j+2\sigma)t} \left[ 2(e_j + 2\sigma)|\alpha_{-}(t)|^2 - 2e_{j+1}|\alpha_{-}(t)|^2 + Ce^{-(e_j+4\sigma)t}|\alpha_{-}(t)| \right]. \end{aligned}$$
+
+Hence we have for all $t \in [T, S_n]$,
+
+$$\mathcal{N}'(t) \le -\theta\mathcal{N}(t) + Ce^{e_j t}|\alpha_{-}(t)|,$$
+
+where $\theta = 2(e_{j+1} - e_j - 2\sigma) > 0$ by definition of $\sigma$. In particular, for all $\tau \in [T, S_n]$ satisfying $\mathcal{N}(\tau) = 1$, we have:
+
+$$\mathcal{N}'(\tau) \le -\theta + Ce^{e_j \tau} |\alpha_{-}(\tau)| \le -\theta + Ce^{e_j \tau} e^{-(e_j+2\sigma)\tau} \le -\theta + Ce^{-2\sigma t_0}.$$
+
+Now, we fix $t_0$ large enough such that $Ce^{-2\sigma t_0} \le \frac{\theta}{2}$. Thus for all $\tau \in [T, S_n]$ such that $\mathcal{N}(\tau) = 1$, we have
+
+$$\mathcal{N}'(\tau) \le -\frac{\theta}{2}.$$
+
+□
+
+Finally, we claim that $\mathbf{a} \mapsto T(\mathbf{a})$ is continuous. Indeed, let $\varepsilon > 0$. By definition of $T(\mathbf{a})$ and by Claim 3.14, there exists $\delta > 0$ such that for all $t \in [T(\mathbf{a}) + \varepsilon, S_n]$, $\mathcal{N}(t) < 1 - \delta$, and such that $\mathcal{N}(T(\mathbf{a}) - \varepsilon) > 1 + \delta$. But from continuity of the flow, there exists $\eta > 0$ such that for all $\tilde{\mathbf{a}}$ satisfying $\|\tilde{\mathbf{a}} - \mathbf{a}\| \le \eta$, we have
+
+$$\forall t \in [T(\mathbf{a}) - \varepsilon, S_n], \quad |\mathcal{N}(\tilde{\mathbf{a}}) - \mathcal{N}(\mathbf{a})| \le \frac{\delta}{2}.$$
+
+We finally deduce that
+
+$$T(\mathbf{a}) - \varepsilon \le T(\tilde{\mathbf{a}}) \le T(\mathbf{a}) + \varepsilon.$$
+
+Hence, $\mathbf{a} \mapsto T(\mathbf{a})$ is continuous.
+
+We then obtain that the map $\mathcal{M}$ is continuous. What is more, for $\mathbf{a} \in S_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n})$, as $\mathcal{N}'(S_n) \le -(e_{j+1} - e_j - 2\sigma) < 0$, we then deduce by definition of $T(\mathbf{a})$ that $T(\mathbf{a}) = S_n$, and thus, $\mathcal{M}(\mathbf{a}) = \mathbf{a}$.
+
+The existence of such a map $\mathcal{M}$ contradicts Brouwer's fixed point theorem. Thus, we have finished proving Lemma 3.13.
+
+□
\ No newline at end of file
diff --git a/samples/texts/7376768/page_2.md b/samples/texts/7376768/page_2.md
new file mode 100644
index 0000000000000000000000000000000000000000..dde046a3dcf7fb94d128021085a39ee8ccc5eaaf
--- /dev/null
+++ b/samples/texts/7376768/page_2.md
@@ -0,0 +1,39 @@
+Moreover, the structure of the equation is left invariant under the action of $\mathbb{R} \times \mathbb{R}^d$ by (time and space) translation, and under the action of the Lorentz group $O(1, d)$ which consists of the linear automorphisms of $\mathbb{R}^{1+d}$ that preserve the quadratic form $(t, x_1, \dots, x_d) \mapsto t^2 - \sum_{i=1}^d x_i^2$. In other words, precising this latter action, for all $\beta \in \mathbb{R}^d$ with $\mathbb{R}^d$-euclidean norm $|\beta| < 1$ and $\gamma := \frac{1}{\sqrt{1-|\beta|^2}}$, $u$ is a solution of (NLKG) if and only if
+
+$$ (t, x) \mapsto u(\Lambda_{\beta}(t, x)) $$
+
+is still a solution to (NLKG), where $\Lambda_{\beta}$ is the linear transformation with matrix
+
+$$ \begin{pmatrix} \gamma & -\gamma\beta \\ -\gamma\beta^{\top} & I_d + \frac{\gamma-1}{|\beta|^2}\beta^{\top}\beta \end{pmatrix} $$
+
+in the canonical basis of $\mathbb{R}^{1+d}$. We observe in particular that
+
+$$ \Lambda_{\beta}(t,x) = (\gamma(t-\beta x), \gamma(x-\beta t)) $$
+
+if $d=1$. We refer to [9] for further details concerning the Lorentz transformations in all dimensions.
+
+It is well-known that (NLKG) admits a family of solitons indexed by two parameters: the velocity parameter $\beta \in \mathbb{R}^d$ with $|\beta| < 1$ and the translation parameter $x_0 \in \mathbb{R}^d$. Let $Q$ denote the unique (up to translation) positive $H^1$ solution of the following stationary elliptic problem, associated with (NLKG):
+
+$$ (1.1) \qquad \Delta Q - Q + f(Q) = 0 $$
+
+which we take as radial; for the record, existence of $Q$ follows from a standard result of Berestycki and Lions [2] due to (H2) or (H'1) and uniqueness has been proved in Kwong [17] (in the case where $f(u) = |u|^{p-1}u$ is the particular power nonlinearity) and in Serrin and Tang [27]. We recall that $Q$ and its partial derivatives up to order 3 decay exponentially. Then for all $\beta \in \mathbb{R}^d$ such that $|\beta| < 1$, for all $x_0 \in \mathbb{R}^d$, the boosted ground state
+
+$$ Q_{\beta,x_0} : (t,x) \mapsto Q(p r \circ \Lambda_{\beta}(t, x - x_0)), $$
+
+where $\gamma := \frac{1}{\sqrt{1-|\beta|^2}}$ and $pr$ is the canonical projection $\mathbb{R}^{1+d} \to \mathbb{R}^d$ on the last $d$ coordinates, is a solution of (NLKG) known as *soliton*. In the one-dimensional case, this soliton rewrites
+
+$$ Q_{\beta,x_0} : (t,x) \mapsto Q(x - \beta t - x_0). $$
+
+Soliton theory concerning (NLKG) has extensively been studied in many articles. One major result is linked to the classification of the solutions with energy near that of the ground state. Dynamics of the solutions $u$ of (NLKG) on the threshold energy $E(u) = E(Q)$ has been investigated in Duyckaerts and Merle [10]. More generally, classification of the solutions with energy less than a quantity slightly larger than the energy of the ground state has been done by Nakanishi and Schlag [25] and by Krieger, Nakanishi and Schlag [16].
+
+Let us also mention that solitons of (NLKG) are known to be orbitally unstable in $H^1(\mathbb{R}^d)$ by a general property by Grillakis, Shatah and Strauss [14].
+
+We further develop soliton analysis by exploring solutions which behave as a soliton or a sum of solitons as time goes to infinity.
+
+For all $\beta \in \mathbb{R}^d$ such that $|\beta| < 1$ and $x_0 \in \mathbb{R}^d$, let us denote
+
+$$ R_{\beta,x_0}(t,x) := \begin{pmatrix} Q_{\beta,x_0}(t,x) \\ \partial_t Q_{\beta,x_0}(t,x) \end{pmatrix} = \begin{pmatrix} Q_{\beta,x_0}(t,x) \\ -\beta \cdot \nabla Q_{\beta,x_0}(t,x) \end{pmatrix}. $$
+
+When $x_0 = 0$, we will write $R_{\beta}$ instead of $R_{\beta,0}$ for the sake of simplification.
+
+Drawing on the work by Grillakis, Shatah and Strauss [14, 15], Côte and Muñoz [9] have developed and proved spectral results adapted to the unstable dynamic around the (vector) soliton $R_{\beta}$. Essential
\ No newline at end of file
diff --git a/samples/texts/7376768/page_20.md b/samples/texts/7376768/page_20.md
new file mode 100644
index 0000000000000000000000000000000000000000..62562da9eedff4db5b90c7533e5a0cbd4a8de07f
--- /dev/null
+++ b/samples/texts/7376768/page_20.md
@@ -0,0 +1,70 @@
+**4. CLASSIFICATION UNDER CONDITION OF THE MULTI-SOLITONS OF (NLKG)**
+
+Let $N \ge 2$ and $x_1, \dots, x_N, \beta_1, \dots, \beta_N$ be $2N$ parameters as in Theorem 1.2. Let $U$ be a solution of (NLKG) such that
+
+$$
+(4.1) \qquad \left\| U(t) - \sum_{i=1}^{N} R_{\beta_i}(t) \right\|_{H^1 \times L^2} = O \left( \frac{1}{t^\alpha} \right) \quad \text{as } t \to +\infty
+$$
+
+for some $\alpha > 3$.
+
+The goal of this section is to prove the existence of $A_1, \dots, A_N \in \mathbb{R}$ such that
+
+$$
+U = \Phi_{A_1, \dots, A_N}.
+$$
+
+Here again, we make the proof for $d = 1$.
+
+We denote by $\varphi$ a multi-soliton solution associated with these parameters, satisfying (2.2) and
+$\Phi := \begin{pmatrix} \varphi \\ \partial_t \varphi \end{pmatrix}$. Let us consider $Z := U - \Phi = \begin{pmatrix} z \\ \partial_t z \end{pmatrix}$. Obviously,
+
+$$
+\|Z(t)\|_{H^1 \times L^2} = O\left(\frac{1}{t^\alpha}\right), \quad \text{as } t \to +\infty.
+$$
+
+Our first objective is to improve this comparison, and namely to pass from the polynomial decay
+to an exponential one.
+
+**4.1. Exponential convergence to 0 at speed $e_1$ of $\|Z(t)\|_{H^1 \times L^2}$**
+
+**4.1.1. Introduction of a new variable by modulation.** In a standard way, we modulate the vari-
+able Z in order to obtain suitable orthogonality properties, making it possible to obtain crucial esti-
+mates when we apply the spectral theory available for (NLKG).
+
+**Lemma 4.1.** There exists $t_0 > 0$ and $\mathscr{C}^1$ functions $a_i : [t_0, +\infty) \to \mathbb{R}$ and $b_i : [t_0, +\infty) \to \mathbb{R}$ for all $i = 1, \dots, N$ such that, defining
+
+$$
+E := Z - \sum_{i=1}^{N} a_i \partial_x R_i - \sum_{i=1}^{N} b_i Y_{+,i},
+$$
+
+we have for all $i = 1, \dots, N$ and for all $t \ge t_0$:
+
+$$
+(4.2) \qquad \langle E(t), \partial_x R_i(t) \rangle = 0
+$$
+
+$$
+(4.3) \qquad \langle E(t), Z_{-,i}(t) \rangle = 0.
+$$
+
+Moreover, we have for all *i* = 1, ..., *N*:
+
+$$
+(4.4) \qquad a_i(t) = \frac{1}{\|\partial_x R_i\|^2} \langle Z(t), \partial_x R_i(t) \rangle + O(e^{-4\sigma t} \|Z\|_{H^1\times L^2})
+$$
+
+$$
+(4.5) \qquad b_i(t) = \langle Z(t), Z_{-,i}(t) \rangle + O\left(e^{-4\sigma t} \|Z\|_{H^1\times L^2}\right).
+$$
+
+Proof. This lemma follows from the consideration of the system with unknown variables $a_i$ and $b_i$ which is obtained by replacing $E$ by its definition in (4.2) and (4.3). See also [6] for similar considerations in the case of modulation for the nonlinear Schrödinger equations. □
+
+**4.1.2. Control of the Z<sub>+,i</sub> and Z<sub>-,i</sub> directions.** Define α<sub>±,i</sub> := ⟨Z, Z<sub>±,i</sub>⟩ for all i = 1, ..., N. We
+claim:
+
+**Lemma 4.2.** The following bounds hold: for all *i* = 1, . . . , *N*, for all *t* ≥ *t*₀,
+
+$$
+|\alpha'_{\pm,i}(t) \mp e_i \alpha_{\pm,i}(t)| \le C \left( e^{-4\sigma t} \|Z\|_{H^1\times L^2} + \|Z\|_{H^1\times L^2}^2 \right).
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_21.md b/samples/texts/7376768/page_21.md
new file mode 100644
index 0000000000000000000000000000000000000000..cd14fd0df0cdf57d0e31cfaa6363d3b4926d430d
--- /dev/null
+++ b/samples/texts/7376768/page_21.md
@@ -0,0 +1,67 @@
+*Proof.* The proof is in a similar fashion as that of Claim 3.7. We note that
+
+$$
+\partial_t Z = \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(\varphi) & 0 \end{pmatrix} Z + \begin{pmatrix} 0 \\ f(u) - f(\varphi) - (u-\varphi)f'(\varphi) \end{pmatrix}
+$$
+
+and that $|f(u) - f(\varphi) - (u-\varphi)f'(\varphi)| \le C|u-\varphi|^2 \le C\|Z\|_{H^{1\times L^2}}^2$. Thus for $i=1, \dots, N$, we have
+
+$$
+\begin{align*}
+a'_{\pm,i} &= \langle \partial_t Z, Z_{\pm,i} \rangle + \langle Z, \partial_t Z_{\pm,i} \rangle \\
+&= \left( \left( \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(\varphi) & 0 \end{pmatrix} Z, Z_{\pm,i} \right) - \beta_i \langle Z, \partial_x Z_{\pm,i} \rangle + O(\|Z\|_{H^{1\times L^2}}^2) \right) \\
+&= \left( Z, \begin{pmatrix} -\beta_i \partial_x & \partial_x^2 - Id + f'(Q_i) \\ Id & -\beta_i \partial_x \end{pmatrix} Z_{\pm,i} \right) + \left( Z, \begin{pmatrix} 0 & f'(\varphi) - f'(Q_i) \\ 0 & 0 \end{pmatrix} Z_{\pm,i} \right) \\
+&\quad + O(\|Z\|_{H^{1\times L^2}}^2) \\
+&= \langle Z, \pm e_i Z_{\pm,i} \rangle + O(e^{-4\sigma t} \|Z\|_{H^{1\times L^2}} + \|Z\|_{H^{1\times L^2}}^2).
+\end{align*}
+$$
+
+**4.1.3. Control of the remaining modulation parameters.**
+
+**Lemma 4.3.** For all *i* = 1, ..., *N*, we have
+
+$$
+(4.6) \qquad |a'_i| \le C (\|E\|_{H^{1\times L^2}} + \|Z\|_{H^{1\times L^2}}^2).
+$$
+
+*Proof.* We do not detail the proof of this lemma which is similar to Claim 3.12. It suffices to start by differentiating the orthogonality relation $\langle E, (R_i)_x \rangle = 0$ with respect to $t$ and then to control terms by means of $\|E\|_{H^{1\times L^2}}$.
+
+**4.1.4. Study of a Lyapunov functional.** Taking some inspiration in [20, 29, 30], we consider for all $t \ge t_0$:
+
+$$
+F_z(t) := \int_{\mathbb{R}} \{\partial_x z^2 + \partial_t z^2 + z^2 - f'(\varphi)z^2\} dx + 2 \int_{\mathbb{R}} \chi \partial_x z \partial_t z dx,
+$$
+
+where $\chi$ is defined as follows.
+
+To begin with, recall that the parameters are ordered in such a way: $-1 < \beta_{\eta(1)} < \cdots < \beta_{\eta(N)} < 1$; let us denote, for some small $\delta > 0$ which will be determined later:
+
+$$
+\bar{l}_i := \beta_{\eta(i)} + \delta (\beta_{\eta(i+1)} - \beta_{\eta(i)})
+$$
+
+$$
+\underline{l}_i := \beta_{\eta(i)} - \delta (\beta_{\eta(i+1)} - \beta_{\eta(i)}).
+$$
+
+We then define for all $t \ge t_0$ and for all $x \in \mathbb{R}$:
+
+$$
+\chi(t,x) := \begin{cases}
+\beta_{\eta(1)} & \text{if } x \in (-\infty, \bar{l}_1 t] \\
+\beta_{\eta(i)} & \text{if } x \in [\underline{l}_i t, \bar{l}_i t] \\
+\beta_{\eta(N)} & \text{if } x \in [\bar{l}_N t, +\infty) \\
+\dfrac{x}{(1-2\delta)t} - \dfrac{\delta}{1-2\delta}(\beta_{\eta(i)} + \beta_{\eta(i+1)}) & \text{if } x \in [\bar{l}_i t, \underline{l}_{i+1} t], i \in \{1, \dots, N-1\}.
+\end{cases}
+$$
+
+For all $t \ge t_0$, $\chi(t)$ is a piecewise $\mathscr{C}^1$ function.
+
+Set $\Omega(t) := \bigcup_{i=1}^{N} (\bar{l}_i t, \underline{l}_{i+1} t)$. It follows from the definition of $\chi$ that
+
+$$
+\begin{gather*}
+\partial_t \chi(t,x) = \partial_x \chi(t,x) = 0 && \text{if } x \in \Omega(t)^\mathcal{C} \\
+\partial_x \chi(t,x) = \frac{1}{(1-2\delta)t}, && \partial_t \chi(t,x) = -\frac{x}{(1-2\delta)t^2} && \text{if } x \in \Omega(t).
+\end{gather*}
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_22.md b/samples/texts/7376768/page_22.md
new file mode 100644
index 0000000000000000000000000000000000000000..283085f74aa88abc03e72693427e3ed0f0c05880
--- /dev/null
+++ b/samples/texts/7376768/page_22.md
@@ -0,0 +1,62 @@
+**Lemma 4.4.** There exists $\gamma > 0$ such that
+
+$$
+\begin{equation}
+\begin{aligned}
+\mathcal{F}_z'(t) ={}& 2 \int_{\Omega(t)} \partial_x z \partial_t z \partial_t \chi \, dx - \int_{\Omega(t)} \left\{ (\partial_t z)^2 + (\partial_x z)^2 - z^2 + f'(\varphi) z^2 \right\} \partial_x \chi \, dx \\
+& + O \left( e^{-\gamma t} \|Z\|_{H^{1} \times L^2}^2 + \|Z\|_{H^{1} \times L^2}^3 \right).
+\end{aligned}
+\end{equation}
+$$
+
+*Proof.* We essentially have to use the identity $\partial_t^2 z = \partial_x^2 z - z + f(u) - f(\varphi)$ in the expression of $\mathcal{F}_z'(t)$. We compute
+
+$$
+\begin{align}
+\mathcal{F}_z'(t) &= 2 \int_{\mathbb{R}} \left\{ z_t x z_x + z_{tt} z_t + z_t z - f'(\varphi) z_t z - \frac{1}{2} \varphi_t f''(\varphi) z^2 \right\} dx \nonumber \\
+&\quad + 2 \int_{\mathbb{R}} \chi_t z_x z_t \, dx + 2 \int_{\mathbb{R}} \chi z_x z_{tt} \, dx + 2 \int_{\mathbb{R}} \chi z_{xt} z_t \, dx \tag{4.8} \\
+&= 2 \int_{\mathbb{R}} z_t (-z_{xx} + z_{tt} + z - f'(\varphi)z) \, dx + 2 \int_{\mathbb{R}} \chi_t z_x z_t \, dx \nonumber \\
+&\quad + 2 \int_{\mathbb{R}} \chi z_x (z_{xx} - z + f(u) - f(\varphi)) \, dx - \int_{\mathbb{R}} z_t^2 \chi_x \, dx - \int_{\mathbb{R}} \varphi_t f''(\varphi) z^2 \, dx \nonumber
+\end{align}
+$$
+
+Notice that
+
+$$
+(4.9) \quad \int_{\mathbb{R}} z_t(-z_{xx} + z_{tt} + z - f'(\varphi)z) \, dx = \int_{\mathbb{R}} z_t(f(u) - f(\varphi) - f'(\varphi)z) \, dx = O(\|Z\|_{H^1 \times L^2}^3)
+$$
+
+and
+
+$$
+(4.10) \quad \int_{\mathbb{R}} \chi z_x (f(u) - f(\varphi)) \, dx = \int_{\mathbb{R}} \chi z_x f'(\varphi) z \, dx + O(\|Z\|_{H^1 \times L^2}^3).
+$$
+
+Hence, collecting (4.8), (4.9), and (4.10),
+
+$$
+\begin{align}
+\mathcal{F}_z'(t) &= 2 \int_{\mathbb{R}} \chi_t z_x z_t \, dx - \int_{\mathbb{R}} \chi_x (z_x^2 + z_t^2 - z^2) \, dx \notag \\
+&\quad - \int_{\mathbb{R}} z^2 (\chi_x f'(\varphi) + \chi_\varphi x f''(\varphi)) \, dx - \int_{\mathbb{R}} \varphi_t f''(\varphi) z^2 \, dx + O(\|Z\|_{H^{1}\times L^2}^3) \tag{4.11} \\
+&= 2 \int_{\mathbb{R}} z_x z_t \chi_t \, dx - \int_{\mathbb{R}} \{z_x^2 + z_t^2 - z^2 + f'(\varphi)z^2\} \chi_x \, dx \notag \\
+&\quad - \int_{\mathbb{R}} z^2 f''(\varphi)(\varphi_t + \chi_\varphi x) \, dx + O(\|Z\|_{H^{1}\times L^2}^3). \notag
+\end{align}
+$$
+
+Lastly, observe that
+
+$$
+\begin{align*}
+& \int_{\mathbb{R}} z^2 f''(\varphi)(\varphi_t + \chi_\varphi x) \, dx = \sum_{i=1}^{N} \int_{\mathbb{R}} z^2 f''(\varphi)((R_k)_t + \chi(R_k)_x) \, dx + O(\|Z\|_{H^{1}\times L^2}^3) \\
+& = I + J + O(\|Z\|_{H^{1}\times L^2}^3),
+\end{align*}
+$$
+
+where
+
+$$
+\begin{cases}
+I = \sum_{k=1}^{N} \int_{\Omega(t)} z^2 f''(\varphi) ((R_k)_t + \chi(R_k)_x) dx \\
+J = \sum_{k=1}^{N} \sum_{i=1}^{N} \int_{L_i(t)} l_i^2 z^2 f''(\varphi) ((R_k)_t + \chi(R_k)_x) dx.
+\end{cases}
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_23.md b/samples/texts/7376768/page_23.md
new file mode 100644
index 0000000000000000000000000000000000000000..97d04c8b7a43905c2b49cd2fa5970cfde8a0238c
--- /dev/null
+++ b/samples/texts/7376768/page_23.md
@@ -0,0 +1,76 @@
+On the one hand,
+
+$$
+\begin{align*}
+J &= \sum_{i=1}^{N} \sum_{k=1}^{N} \int_{L_i(t)}^{\bar{l}_i t} z^2 f''(\varphi) ((R_k)_t + \chi(R_k)_x) \, dx \\
+&= \sum_{i=1}^{N} \int_{L_i(t)}^{\bar{l}_i t} z^2 f''(\varphi) ((R_{\eta(i)})_t + \chi(R_{\eta(i)})_x) \, dx + O\left(e^{-4\sigma t} \|Z\|_{H^1 \times L^2}^2\right) \\
+&= O\left(e^{-4\sigma t} \|Z\|_{H^1 \times L^2}^2\right).
+\end{align*}
+$$
+
+Indeed, for all $x \in [\bar{l}_i(t), \bar{l}_i t]$, we have $(R_{\eta(i)})_t(t,x) + \chi(t,x)(R_{\eta(i)})_x(t,x) = 0$.
+
+On the other, for $x \in \Omega(t)$, there exists $k' \in \{1, \dots, N\}$ such that $\bar{l}_{k'}t \le x \le \underline{l}_{k'+1}t$.
+
+Then
+
+$$
+(\bar{l}_{k'} - \beta_k)t \leq x - \beta_k t \leq (\underline{l}_{k'+1} - \beta_k)t
+$$
+
+and thus
+
+$$
+|x - \beta_k t| \geq \delta \min_{j=1,\dots,N-1} \{\beta_{\eta(j+1)} - \beta_{\eta(j)}\} t > 0.
+$$
+
+As a consequence, we have for all $x \in \Omega(t)$
+
+$$
+\begin{equation}
+\begin{aligned}
+|(R_k)_x(t,x)| &\le Ce^{-\sigma|x-\beta_k t|} \\
+&\le Ce^{-\gamma t},
+\end{aligned}
+\tag{4.13}
+\end{equation}
+$$
+
+where $0 < \gamma < \sigma\delta \min_{j=1,...,N-1}\{\beta_{\eta(j+1)} - \beta_{\eta(j)}\}$.
+
+Hence
+
+$$
+(4.14) \qquad |I| \le C e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2.
+$$
+
+Finally we gather (4.11), (4.12), and (4.14) in order to obtain (4.7). $\square$
+
+Let us introduce the components $\epsilon$ and $\epsilon_2$ of the vector
+
+$$
+E = \begin{pmatrix} \epsilon := z - \sum_i \{a_i \partial_x Q_i + b_i (Y_{+,i})_1\} \\ \epsilon_2 := \partial_t z - \sum_i \{a_i (-\beta_t \partial_{xx} Q_i) + b_t (Y_{+,i})_2\} \end{pmatrix}.
+$$
+
+**Corollary 4.5.** We have
+
+$$
+(4.15) \quad \mathcal{F}_z'(t) = 2 \int_{\Omega(t)} \epsilon_x \epsilon_2 \chi_t \, dx - \int_{\Omega(t)} \{ \epsilon_2^2 + \epsilon_x^2 - \epsilon^2 + f'(\varphi) \epsilon^2 \} \chi_x \, dx \\
+\phantom{(4.15) \quad \mathcal{F}_z'(t) = 2 \int_{\Omega(t)} } + O\left( e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2 + \|Z\|_{H^1 \times L^2}^3 \right).
+$$
+
+*Proof.* The corollary immediately follows from (4.7) and bounds for the derivatives of $R_k$ and $Y_{+,k}$ which are analogous to (4.13). $\square$
+
+In the spirit of [20, Proposition 4.2], we will state an almost monotonicity property satisfied by $\mathcal{F}_z$. Let us define
+
+$$
+\mathcal{F}_{\varepsilon, \Omega(t)}(t) := \int_{\Omega(t)} \{\varepsilon_x^2 + \varepsilon_2^2 + 2\chi\varepsilon_x\varepsilon_2\} dx.
+$$
+
+Let $\lambda \in (1, \alpha - 1)$. The choice of $\lambda$ is linked with the integrability of particular quantities and will appear naturally later.
+
+**Proposition 4.6.** There exists $\delta > 0$ and $t_0 > 0$ such that for all $t \ge t_0$,
+
+$$
+(4.16) \qquad -\mathcal{F}'_z(t) \leq \frac{\lambda}{t} \mathcal{F}_{\varepsilon, \Omega(t)}(t) + O\left(e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2 + \|Z\|_{H^1 \times L^2}^3\right).
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_24.md b/samples/texts/7376768/page_24.md
new file mode 100644
index 0000000000000000000000000000000000000000..1bcfb2a44939e30180e65062b667e84d76b0c6c2
--- /dev/null
+++ b/samples/texts/7376768/page_24.md
@@ -0,0 +1,76 @@
+*Proof.* Since $f'(0) = 0$ and $f'$ is continuous, there exists $r_0 > 0$ such that for all $r \in [0, r_0]$, $|f'(r)| \le 1$. There exists $K > 0$ (independent of $t$) such that for all $t \ge t_0$ and for all $x \in \Omega(t)$, $\lvert\varphi(t,x)\rvert \le Ke^{-\gamma t}$. Even if it means increasing $t_0$, we can assume that $Ke^{-\gamma t} \le r_0$ for all $t \ge t_0$.
+
+In addition $\chi_x \ge 0$. Thus, for $t \ge t_0$,
+
+$$ 
+\begin{aligned}
+& -2 \int_{\Omega(t)} \varepsilon_x \varepsilon_2 \chi_t \, dx + \int_{\Omega(t)} \{\varepsilon_x^2 + \varepsilon_2^2 - \varepsilon^2 + f'(\varphi)\varepsilon^2\} \chi_x \, dx \\
+& \le -2 \int_{\Omega(t)} \varepsilon_x \varepsilon_2 \chi_t \, dx + \int_{\Omega(t)} \{\varepsilon_x^2 + \varepsilon_2^2\} \chi_x \, dx.
+\end{aligned}
+$$
+
+Moreover,
+
+$$ 
+\begin{aligned}
+-2 \int_{\Omega(t)} \varepsilon_x \varepsilon_2 \chi_t \, dx + \int_{\Omega(t)} \{\varepsilon_x^2 + \varepsilon_2^2\} \chi_x \, dx &= \frac{1}{(1-2\delta)t} \int_{\Omega(t)} \{2\varepsilon_x\varepsilon_2 \frac{x}{t} + \varepsilon_x^2 + \varepsilon_2^2\} \, dx \\
+&= \frac{1}{(1-2\delta)t} \left( \mathcal{F}_{\varepsilon, \Omega(t)} + 2 \int_{\Omega(t)} \left(\frac{x}{t} - \chi\right) \varepsilon_x \varepsilon_2 \, dx \right).
+\end{aligned}
+$$
+
+Now, for $x \in \Omega(t)$, we have
+
+$$ 
+\begin{aligned}
+\left|\frac{x}{t} - \chi(t,x)\right| &\leq \left|\frac{x}{t}\left(1-\frac{1}{1-2\delta}\right)\right| + \frac{\delta}{1-2\delta} \times 2 \max_i |\beta_i| \\
+&\leq 2\delta \left(\left|\frac{x}{t}\right| + C\right) \\
+&\leq C\delta.
+\end{aligned}
+$$
+
+Thus,
+
+$$ 
+\frac{2}{(1-2\delta)t} \int_{\Omega(t)} \left(\frac{x}{t} - \chi\right) \varepsilon_x \varepsilon_2 \, dx = O\left(\delta \|E\|_{H^{1}\times L^2}\right).
+$$
+
+Noticing moreover that
+
+$$ 
+\begin{aligned}
+\mathcal{F}_{\varepsilon, \Omega(t)} &\geq \int_{\Omega(t)} (\varepsilon_x^2 + \varepsilon_2^2) \, dx - 2\|\chi(t)\|_{\infty} \int_{\Omega(t)} \varepsilon_x \varepsilon_2 \, dx \\
+&\geq \int_{\Omega(t)} (\varepsilon_x^2 + \varepsilon_2^2) \, dx - \|(\chi(t))'\|_{\infty} \int_{\Omega(t)} (\varepsilon_x^2 + \varepsilon_2^2) \, dx \\
+&\geq (1 - \|(\chi(t))'\|_{\infty}) \|E\|_{H^{1}\times L^2(\Omega(t))}^2,
+\end{aligned}
+$$
+
+we obtain
+
+$$ 
+\frac{2}{(1-2\delta)t} \int_{\Omega(t)} \left(\frac{x}{t} - \chi\right) \epsilon_x \epsilon_2 dx = O(\delta F_{\epsilon, \Omega(t)}).
+$$
+
+Finally,
+
+$$ 
+\begin{aligned}
+-F'_z(t) &\leq \frac{1}{(1-2\delta)t}(1+\mathcal{O}(\delta))F_{\epsilon,\Omega(t)} + O(e^{-\gamma t} \|Z\|_{H^{1}\times L^2}^2 + \|Z\|_{H^{1}\times L^2}^3) \\
+&\leq \frac{\lambda}{t} F_{\epsilon,\Omega(t)} + O(e^{-\gamma t} \|Z\|_{H^{1}\times L^2}^2 + \|Z\|_{H^{1}\times L^2}^3),
+\end{aligned}
+$$
+
+provided $\delta$ is chosen small enough. $\square$
+
+We now introduce the functional
+
+$$ 
+F_\epsilon(t) := \int_{\mathbb{R}} (\epsilon_x^2 + \epsilon_2^2 + \epsilon^2 - f'(\varphi)\epsilon^2) dx + 2 \int_{\mathbb{R}} \chi \epsilon_x \epsilon_2 dx
+$$
+
+and compare it with $F_z$.
+
+**Lemma 4.7.** We have
+
+$$ 
+F_\epsilon(t) = F_z(t) - 2 \sum_{i=1}^{N} \alpha_{-,i}(t) \alpha_{+,i}(t) + G(t),
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_25.md b/samples/texts/7376768/page_25.md
new file mode 100644
index 0000000000000000000000000000000000000000..a1b9be69e2d1f36c1be6f577e26c7fb358138b0e
--- /dev/null
+++ b/samples/texts/7376768/page_25.md
@@ -0,0 +1,39 @@
+where $\mathcal{G}(t) = O(e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2)$ and $\mathcal{G}'(t) = O(e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2)$.
+
+*Proof.* Let $M$ be the matrix $\begin{pmatrix} -\partial_x^2 + Id - f'(\varphi) & 0 \\ 0 & Id \end{pmatrix}$. For all $i=1, \dots, N$, we have the decomposition
+
+$$ (4.17) \qquad M = H_i + \begin{pmatrix} 0 & \beta_i \partial_x \\ -\beta_i \partial_x & 0 \end{pmatrix} + \begin{pmatrix} f'(Q_i) - f'(\varphi) & 0 \\ 0 & 0 \end{pmatrix}. $$
+
+Then we infer
+
+$$ 
+\begin{align*}
+\mathcal{F}_\varepsilon(t) &= \langle ME, E \rangle + 2 \int_{\mathbb{R}} \chi \varepsilon_x \varepsilon_2 \, dx \\
+&= \langle MZ, Z \rangle - 2 \left\langle M \left( \sum_{i=1}^N \{a_i(R_i)_x + b_i Y_{+,i}\} \right), Z \right\rangle \\
+&\quad + \left\langle M \left( \sum_{i=1}^N \{a_i(R_i)_x + b_i Y_{+,i}\} \right), \sum_{j=1}^N \{a_j(R_j)_x + b_j Y_{+,j}\} \right\rangle \\
+&\quad + 2 \int_{\mathbb{R}} \chi z_x z_t \, dx + 2 \int_{\mathbb{R}} \chi \sum_{i=1}^N \{a_i(Q_i)_{xx} + b_i(Y_{+,i})_{1,x}\} \sum_{j=1}^N \{a_j(Q_j)_{xt} + b_j(Y_{+,j})_2\} \, dx \\
+&\quad - 2 \int_{\mathbb{R}} \chi z_x \sum_{i=1}^N \{a_i(Q_i)_{xt} + b_i(Y_{+,i})_2\} \, dx - 2 \int_{\mathbb{R}} \chi z_t \sum_{i=1}^N \{a_i(Q_i)_{xx} + b_i(Y_{+,i})_{1,x}\} \, dx \\
+&= \mathcal{F}_z(t) - 2 \sum_{i=1}^N b_i \langle Z_{+,i}, Z \rangle + \tilde{\mathcal{G}}(t) \\
+&= \mathcal{F}_z(t) - 2 \sum_{i=1}^N \alpha_{-,i}(t) \alpha_{+,i}(t) + \mathcal{G}(t),
+\end{align*}
+$$
+
+with
+
+$$ 
+\begin{align*}
+MY_{+,i} &= Z_{+,i} + \beta_i \begin{pmatrix} (Y_{+,i})_{2,x} \\ -(Y_{+,i})_{1,x} \end{pmatrix} + \begin{pmatrix} (f'(Q_i) - f'(\varphi))(Y_{+,i})_1 \\ 0 \end{pmatrix} \\
+M(R_i)_x &= \begin{pmatrix} -\beta_i^2 (Q_i)_{xxx} \\ -\beta_i (Q_i)_{xx} \end{pmatrix} + \begin{pmatrix} (f'(Q_i) - f'(\varphi))(Q_i)_x \\ 0 \end{pmatrix}.
+\end{align*}
+$$
+
+and
+
+$$ 
+\begin{align*}
+\tilde{\mathcal{G}}(t) ={}& -2 \left\langle \sum_{i=1}^{N} \left\{ a_i \begin{pmatrix} -\beta_i^2 (Q_i)_{xxx} \\ \beta_i (Q_i)_{xx} \end{pmatrix} + b_i \beta_i \begin{pmatrix} (Y_{+,i})_{2,x} \\ -(Y_{+,i})_{1,x} \end{pmatrix} \right\}, Z \right\rangle + O(e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2) \\
+&+ \sum_{i=1}^{N} \left\langle a_i \begin{pmatrix} -\beta_i^2 (Q_i)_{xxx} \\ \beta_i (Q_i)_{xx} \end{pmatrix} + b_i \beta_i \begin{pmatrix} (Y_{+,i})_{2,x} \\ -(Y_{+,i})_{1,x} \end{pmatrix} + b_i Z_{+,i}, a_i (R_i)_x + b_i Y_{+,i} \right\rangle \\
+&+ 2 \sum_{i=1}^{N} \int_{\mathbb{R}} \chi (a_i (Q_i)_{xx} + b_i (Y_{+,i})_{1,x}) (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx \\
+&- 2 \sum_{i=1}^{N} \int_{\mathbb{R}} \chi z_x (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx - 2 \sum_{i=1}^{N} \int_{\mathbb{R}} \chi z_t (a_i (Q_i)_{xx} + b_i (Y_{+,i})_{1,x}) \, dx.
+\end{align*}
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_26.md b/samples/texts/7376768/page_26.md
new file mode 100644
index 0000000000000000000000000000000000000000..009f11144061137181f8e7bf0ac84834f2e07202
--- /dev/null
+++ b/samples/texts/7376768/page_26.md
@@ -0,0 +1,56 @@
+Integrating by parts, we obtain
+
+$$
+\begin{align*}
+& \int_{\mathbb{R}} \chi z_x (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx \\
+&= - \int_{\mathbb{R}} \chi z (-\beta_i a_i (Q_i)_{xxx} + b_i (Y_{+,i})_{2,x}) \, dx - \int_{\mathbb{R}} \chi_x z (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx.
+\end{align*}
+$$
+
+Consequently,
+
+$$
+\begin{align*}
+\tilde{\mathcal{G}}(t) &= -2 \sum_{i=1}^{N} a_i \int_{\mathbb{R}} (\chi - \beta_i) (-\beta_i z(Q_i)_{xxx} + z_t(Q_i)_{xx}) \, dx \\
+&\quad - 2 \sum_{i=1}^{N} b_i \int_{\mathbb{R}} (\chi - \beta_i) (-z(Y_{+,i})_{2,x} + z_t(Y_{+,i})_{1,x}) \, dx \\
+&\quad + 2 \sum_{i=1}^{N} a_i^2 \beta_i \int_{\mathbb{R}} (\beta_i - \chi)(Q_i)_{xx}^2 \, dx + 2 \sum_{i=1}^{N} b_i^2 \int_{\mathbb{R}} (\beta_i - \chi)(Y_{+,i})_1 (Y_{+,i})_{2,x} \\
+&\quad + 2 \sum_{i=1}^{N} \int_{\Omega(t)} \chi_x z (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx \\
+&\quad + 2 \sum_{i=1}^{N} a_i b_i \int_{\mathbb{R}} (\chi - \beta_i) (-\beta_i (Y_{+,i})_{1,x}(Q_i)_{xx} + (Q_i)_{xx}(Y_{+,i})_2) \, dx + O(e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2) \\
+&= O(e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2).
+\end{align*}
+$$
+
+□
+
+We deduce from Proposition 4.6 and Lemma 4.7 the following "weak" monotonicity property.
+
+**Corollary 4.8.** We have for all $t \geq t_0$,
+
+$$
+(4.18) \quad - \mathcal{F}_{\varepsilon}'(t) \leq \frac{\lambda}{t} \mathcal{F}_{\varepsilon}(t) + \frac{C}{t} \sum_{i=1}^{N} \alpha_{+,i}^2 + O \left( e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2 + \|Z\|_{H^{1\times L^2}}^3 \right).
+$$
+
+Proof. From Lemma 4.2, we obtain: for all $i = 1, \dots, N$, for all $t \ge t_0$,
+
+$$
+(4.19) \quad |(\alpha_{-,i} \alpha_{+,i})'| \le C (e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2 + \|Z\|_{H^{1\times L^2}}^3).
+$$
+
+Thus, we have
+
+$$
+- \mathcal{F}_{\epsilon}'(t) = - \mathcal{F}_z'(t) + 2 \sum_{i=1}^{N} (\alpha_{-,i} \alpha_{+,i})' + O(e^{-\gamma t} \|Z\|^2) \\
+\leq \frac{\lambda}{t} \mathcal{F}_{\epsilon, \Omega(t)} + O(e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2 + \|Z\|_{H^{1\times L^2}}^3).
+$$
+
+We now make use of the following property satisfied by $\mathcal{F}_\epsilon$ which is a consequence of a localized version of Proposition 2.1 (we refer to [20, proof of (4.12) and (4.21)] for similar considerations in the case of the energy-critical wave equation):
+
+$$
+(4.20) \qquad \mathcal{F}_{\epsilon, \Omega(t)} \leq \mathcal{F}_{\epsilon}(t) + C \sum_{i=1}^{N} \langle \epsilon, Z_{+,i} \rangle^2,
+$$
+
+to deduce that
+
+$$
+- \mathcal{F}_{\varepsilon}'(t) \leq \frac{\lambda}{t} \mathcal{F}_{\varepsilon}(t) + \frac{C}{t} \sum_{i=1}^{N} a_{+,i}^2 + O\left(e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2 + \|Z\|_{H^{1\times L^2}}^3\right). \quad \square
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_27.md b/samples/texts/7376768/page_27.md
new file mode 100644
index 0000000000000000000000000000000000000000..bfecad1029b43a81f8a54885265b0c48e7b09713
--- /dev/null
+++ b/samples/texts/7376768/page_27.md
@@ -0,0 +1,66 @@
+We are now in a position to prove
+
+**Proposition 4.9.** Even if it means taking a larger $t_0$, we have for all $t \ge t_0$
+
+$$
+(4.21) \qquad \|Z(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{i=1}^{N} |\alpha_{-,i}(t)|.
+$$
+
+Proof. Multiplying the estimate obtained in Corollary 4.8 by $t^\lambda$, we have for all $t \ge t_0$,
+
+$$
+-(t^\lambda \mathcal{F}_E)'(t) \le C \left( \sum_{i=1}^{N} t^{\lambda-1} \alpha_{+,i}^2 + t^\lambda e^{-\gamma t} \|Z\|_{H^{1}\times L^2}^2 + t^\lambda \|Z\|_{H^{1}\times L^2}^3 \right).
+$$
+
+Since, $t^{\lambda-1}\alpha_{+,i}^2, t^\lambda \|Z\|_{H^1 \times L^2}^3$ are integrable functions of $t$ on $[t_0, +\infty)$ and $t^\lambda \mathcal{F}_E(t) \to 0$ as $t \to +\infty$, we infer that
+
+$$
+(4.22) \qquad \mathcal{F}_E(t) \le C \left( \frac{1}{t^\lambda} \sum_{i=1}^N \int_t^{+\infty} t^{\lambda-1} \alpha_{+,i}^2(t') dt' + \frac{C}{t^\lambda} \int_t^{+\infty} t^{\lambda} \|Z(t')\|_{H^1 \times L^2}^3 dt' + e^{-\gamma t} \sup_{t' \ge t} \|Z(t')\|^2 \right).
+$$
+
+By the coercivity property satisfied by $\mathcal{F}_E$, we thus obtain
+
+$$
+\begin{align*}
+\|E(t)\|_{H^1 \times L^2}^2 &\le C \left( \mathcal{F}_E(t) + \sum_{i=1}^N \alpha_{+,i}^2(t) + e^{-2\gamma t} \|Z(t)\|_{H^1 \times L^2}^2 \right) \\
+&\le \frac{C}{t^\lambda} \sum_{i=1}^N \int_t^{+\infty} t^{\lambda-1} \alpha_{+,i}^2(t') dt' + \frac{C}{t^\lambda} \int_t^{+\infty} t^{\lambda'} \|Z(t')\|_{H^1 \times L^2}^3 dt' \\
+&\quad + C e^{-\gamma t} \sup_{t' \ge t} \|Z(t')\|_{H^1 \times L^2}^2 + C \sum_{i=1}^N \alpha_{+,i}^2(t).
+\end{align*}
+$$
+
+In other words, there exists $C \ge 0$ such that for all $t \ge t_0$,
+
+$$
+(4.23) \qquad \|E(t)\|_{H^1 \times L^2} \le \frac{C}{t^{\frac{1}{2}}} \left( \int_t^{+\infty} t^{\lambda-1} \sum_{i=1}^{N} \alpha_{+,i}^2(t') dt' \right)^{\frac{1}{2}} + C \left( \sum_{i=1}^{N} \alpha_{+,i}^2(t) \right)^{\frac{1}{2}} \\
+\qquad\qquad + \frac{C}{t^{\frac{1}{2}}} \left( \int_t^{+\infty} t^{\lambda'} \|Z(t')\|_{H^{1}\times L^2}^3 dt' \right)^{\frac{1}{2}} + C e^{-\frac{\gamma}{2}t} \sup_{t'\ge t} \|Z(t')\|_{H^{1}\times L^2}.
+$$
+
+From Lemma 4.2 we recall the estimate
+
+$$
+|\alpha'_{+,i}(t) - e_i \alpha_{+,i}(t)| \le C (e^{-\gamma t} \|Z(t)\|_{H^1 \times L^2} + \|Z(t)\|_{H^1 \times L^2}^2)
+$$
+
+which is equivalent to
+
+$$
+|(e^{-e_i t} \alpha_{+,i})'(t)| \le C e^{-e_i t} (e^{-\gamma t} \|Z(t)\|_{H^1 \times L^2} + \|Z(t)\|_{H^1 \times L^2}^2).
+$$
+
+Integrating the preceding inequality (which is indeed possible), we deduce:
+
+$$
+(4.24) \qquad |α_{+,i}(t)| \le C \left( e^{-γt} \sup_{t'≥t} ||Z(t')||_{H^1×L^2} + \sup_{t'≥t} ||Z(t')||_{H^1×L^2}^2 \right).
+$$
+
+Now,
+
+$$
+\|Z(t)\|_{H^1 \times L^2} \le \|E(t)\|_{H^1 \times L^2} + C \sum_{i,\pm} |\alpha_{\pm,i}(t)| + C \sum_{i=1}^{N} |a_i(t)| + C e^{-\gamma t} \|Z(t)\|_{H^1 \times L^2}.
+$$
+
+By Lemma 4.3, it follows that
+
+$$
+(4.25) \qquad \|Z(t)\|_{H^1 \times L^2} \le C \left( \|E(t)\|_{H^1 \times L^2} + \sum_{\pm, i} |\alpha_{\pm, i}(t)| + \int_t^{+\infty} (\|E(t')\|_{H^1 \times L^2} + \|Z(t')\|_{H^1 \times L^2}) dt' \right).
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_28.md b/samples/texts/7376768/page_28.md
new file mode 100644
index 0000000000000000000000000000000000000000..ff09450675253b8a75ff69cb3bfeeb343f21e897
--- /dev/null
+++ b/samples/texts/7376768/page_28.md
@@ -0,0 +1,41 @@
+Observing that the quantity $\int_t^{+\infty} \|Z(t')\|_{H^1 \times L^2} dt'$ makes sense and tends to 0 as $t \to +\infty$ (because $\alpha > 1$) and that
+
+$$ \int_t^{+\infty} \|Z(t')\|_{H^1 \times L^2}^2 dt' \leq \sup_{t' \geq t} \|Z(t')\|_{H^1 \times L^2} \int_t^{+\infty} \|Z(t')\|_{H^1 \times L^2} dt, $$
+
+we deduce that for $t$ sufficiently large
+
+$$ (4.26) \quad \|Z(t)\|_{H^1 \times L^2} \le C \left( \sup_{t' \ge t} \|E(t')\|_{H^1 \times L^2} + \int_t^{+\infty} \|E(t')\|_{H^1 \times L^2} dt' + \sup_{t' \ge t} \sum_{\pm, i} |\alpha_{\pm, i}(t')| \right). $$
+
+Now, we replace (4.24) in (4.23) and use (4.26). We notice that the following well-defined quantities tend to 0 as $t \to +\infty$ (because $\alpha > 3$ and by the choice of $\lambda < \alpha - 1$):
+
+$$ \begin{align*} & \int_t^{+\infty} t'^{\lambda-1} \sup_{t'' \ge t'} \|Z(t'')\|_{H^1 \times L^2}^2 dt', && \int_t^{+\infty} \sup_{t'' \ge t'} \|Z(t'')\|_{H^1 \times L^2}^2 dt' \\ & \int_t^{+\infty} t'^{\lambda} \|Z(t')\|_{H^1 \times L^2}^2 dt', && \int_t^{+\infty} \frac{1}{u^{\frac{4}{5}}} \left( \int_u^{+\infty} t'^{\lambda-1} \sup_{t'' \ge t'} \|Z(t'')\|_{H^1 \times L^2}^2 dt' \right)^{\frac{1}{2}} du \\ & \int_t^{+\infty} \frac{1}{u^{\frac{4}{5}}} \left( \int_u^{+\infty} t'^{\lambda} \|Z(t')\|_{H^1 \times L^2}^2 dt' \right)^{\frac{1}{2}} du. \end{align*} $$
+
+We then obtain for $t$ sufficiently large
+
+$$ \|Z(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{i=1}^{N} |\alpha_{-,i}(t')|. $$
+
+□
+
+**Proposition 4.10.** We have for all $t \ge t_0$, for all $i = 1, \dots, N$,
+
+$$ (4.27) \qquad |\alpha_{-,i}(t)| \le Ce^{-e_1 t} $$
+
+*Proof.* From Lemma 4.2 and Proposition 4.9, it results that for all $i = 1, \dots, N$, for all $t \ge t_0$,
+
+$$ (4.28) \qquad |\alpha'_{-,i}(t) + e_i \alpha_{-,i}(t)| \le C \left( e^{-\sigma t} \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')| + \left( \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')| \right)^2 \right). $$
+
+Then, for all $i = 1, \dots, N$,
+
+$$ (4.29) \qquad |\alpha'_{-,i}(t)\alpha_{-,i}(t) + e_i\alpha_{-,i}^2(t)| \le C \left( e^{-\sigma t} \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')||\alpha_{-,i}(t)| + \sup_{t' \ge t} \left( \sum_{j=1}^{N} |\alpha_{-,j}(t')| \right)^2 |\alpha_{-,i}(t)| \right). $$
+
+Let us denote $\mathcal{A} := \sum_{j=1}^N \alpha_{-,j}^2$. Summing on $i = 1, \dots, N$, we have in particular:
+
+$$ |\mathcal{A}'(t) + 2e_1\mathcal{A}(t)| \le C e^{-\sigma t} \left( \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')| \right) \sum_{i=1}^{N} |\alpha_{-,i}(t)| + C \left( \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')| \right)^2 \sum_{i=1}^{N} |\alpha_{-,i}(t)|. $$
+
+Noticing that $(\sum_{j=1}^N |\alpha_{-,j}|)^2 \le C\mathcal{A}$, we obtain the existence of $c > 0$ such that for all $t \ge t_0$,
+
+$$ |\mathcal{A}'(t) + 2e_1\mathcal{A}(t)| \le C(e^{-\sigma t} + (\mathcal{A}(t))^{1/2})\sup_{t' \ge t}\mathcal{A}(t'). $$
+
+Lastly we observe that $\xi: t \mapsto e^{-\sigma t} + \mathcal{A}(t)^{1/2}$ is integrable on $[t_0, +\infty)$ since
+
+$$ \mathcal{A}(t)^{\frac{1}{2}} = O\left(\|Z(t)\|_{H^1 \times L^2}^{\frac{1}{2}}\right) = O\left(\frac{1}{t^{\frac{\alpha}{2}}}\right). $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_29.md b/samples/texts/7376768/page_29.md
new file mode 100644
index 0000000000000000000000000000000000000000..8c109d1f4cbb105a04d8931e5a20311489efb7ea
--- /dev/null
+++ b/samples/texts/7376768/page_29.md
@@ -0,0 +1,68 @@
+By Lemma 7.1 in Appendix, we obtain $\mathcal{A}(t) \le Ce^{-2e_1 t}$ for $t \ge t_0$. Consequently, for all $i = 1, \dots, N$,
+$$|\alpha_{-,i}(t)| \le Ce^{-e_1 t}.$$
+
+Gathering Propositions 4.9 and 4.10, we deduce
+
+**Proposition 4.11.** There exists $C > 0$ such that for $t$ sufficiently large,
+
+$$\|Z(t)\|_{H^1 \times L^2} \le C e^{-e_1 t}.$$
+
+**4.2. Identification of the solution.** Recall that we have constructed in Section 3 a family of multi-solitons $(\phi_{A_1, \dots, A_N})$ such that for all $j=1, \dots, N$, for all $t \ge t_0$,
+
+$$ (4.30) \quad \| \Phi_{A_1, \dots, A_j}(t) - \Phi_{A_1, \dots, A_{j-1}}(t) - A_j e^{-e_j t} Y_{+,j}(t) \|_{H^1 \times L^2} \le C e^{-(e_j+\sigma)t}. $$
+
+(We can always assume that $\sigma < \min\{e_1, \min_{j=2,\dots,N}\{e_j - e_{j-1}\}\}$).
+
+Following the strategy of Combet [4], our goal is to establish
+
+**Proposition 4.12.** For all $j = 1, \dots, N$, there exist $C \ge 0$, $t_0 \ge 0$, and $A_1, \dots, A_j \in \mathbb{R}$ such that, defining
+
+$$E_j := U - \Phi_{A_1, \dots, A_j},$$
+
+we have:
+
+$$ (4.31) \qquad \|E_j(t)\|_{H^1 \times L^2} \le C e^{-e_j t}. $$
+
+Moreover, denoting $\alpha_{\pm,j,k} := \langle E_j, Z_{\pm,k} \rangle$ for all $k = 1, \dots, N$, we have
+
+$$ (4.32) \quad \forall k \in \{1, \dots, j\}, \quad e^{e_k t} \alpha_{-,j,k}(t) \to 0, \text{ as } t \to +\infty. $$
+
+*Proof.* We proceed by induction on $j$. First, we focus on the case where $j = 1$. We have $\|Z(t)\|_{H^1 \times L^2} \le C e^{-e_1 t}$ by Proposition 4.11. Thus, by Lemma 4.2 and given that $\sigma < e_1$,
+
+$$|(e^{e_1 t} \alpha_{-,1})'| \le C e^{-\sigma t}.$$
+
+Since $t \mapsto e^{-\sigma t}$ is integrable in $+\infty$, there exists $A_1 \in \mathbb{R}$ such that
+
+$$e^{e_1 t} \alpha_{-,1}(t) \to A_1, \quad \text{as } t \to +\infty.$$
+
+We then define $E_1 := U - \Phi_{A_1}$. We notice that $E_1 = E + (\Phi - \Phi_{A_1})$ so that
+
+$$
+\begin{align*}
+\|E_1(t)\|_{H^1 \times L^2} &\le \|E(t)\|_{H^1 \times L^2} + \|(\Phi - \Phi_{A_1})(t)\|_{H^1 \times L^2} \\
+&\le C e^{-e_1 t} + \|(\Phi_{A_1} - \Phi)(t) - A_1 e^{-e_1 t} Y_{+,1}(t)\|_{H^1 \times L^2} + \|A_1 e^{-e_1 t} Y_{+,1}(t)\|_{H^1 \times L^2} \\
+&\le C e^{-e_1 t}.
+\end{align*}
+$$
+
+Moreover
+
+$$
+\begin{align*}
+\alpha_{-,1,1} &= \langle E, Z_{-,1} \rangle + (\Phi - \Phi_{A_1} - A_1 e^{-e_1 t} Y_{+,1}, Z_{-,1}) + A_1 e^{-e_1 t} \langle Y_{+,1}, Z_{-,1} \rangle \\
+&= -\alpha_{-,1} + A_1 e^{-e_1 t} + O(e^{-(e_1+\sigma)t}) \\
+&= o(e^{-e_1 t}),
+\end{align*}
+$$
+
+the last line resulting from the definition of $A_1$.
+
+Thus Proposition 4.12 is true for $j = 1$.
+
+We now assume that there exist $A_1, \dots, A_{j-1} \in \mathbb{R}$ such that $\|E_{j-1}(t)\|_{H^1 \times L^2} \le C e^{-e_{j-1}t}$ and for all $k = 1, \dots, j-1$, $e^{e_k t} \alpha_{-,j-1,k}(t) \to 0$ as $t \to +\infty$.
+
+Let us show
+
+**Claim 4.13.** We have
+
+$$\|E_{j-1}(t)\|_{H^1 \times L^2} \le C e^{-e_j t}.$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_3.md b/samples/texts/7376768/page_3.md
new file mode 100644
index 0000000000000000000000000000000000000000..4363017160c38c417eb2e06fceb2a9eb58266212
--- /dev/null
+++ b/samples/texts/7376768/page_3.md
@@ -0,0 +1,52 @@
+properties which are needed in this paper, as well as the introduction of useful notations, are presented
+in the next subsection. Note that a similar spectral theory was firstly considered by Pego and Weinstein
+[26] in the context of the generalized Korteweg-de Vries equations.
+
+Starting from this point of view, we are interested in solutions which converge to a soliton or a
+sum of solitons for large values of $t$; these solutions are classically known as *multi-solitons*.
+
+Let us consider an integer $N \ge 1$ and $2N$ parameters
+
+$x_1, \dots, x_N \in \mathbb{R}^d \text{ and } \beta_1, \dots, \beta_N \in \mathbb{R}^d$
+
+such that
+
+$$
+\forall i = 1, \dots, N, \quad |\beta_i| < 1 \quad \text{and} \quad \forall i \neq j, \quad \beta_i \neq \beta_j.
+$$
+
+We recall the following theorem by Côte and Muñoz which states the existence of at least one multi-soliton.
+
+**Theorem 1.1** ([9]). There exist $\sigma_0$, $t_0 \in \mathbb{R}$ and $C_0 > 0$, only depending on the sets $(\beta_i)_i$, $(x_i)_i$, and a solution $U = \begin{pmatrix} u \\ \partial_t u \end{pmatrix} \in \mathcal{C}([t_0, +\infty), H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d))$ of (NLKG) such that for all $t \ge t_0$,
+
+$$
+\left\| U(t) - \sum_{i=1}^{N} R_{\beta_i, x_i}(t) \right\|_{H^1 \times L^2} \le C_0 e^{-\sigma_0 t}.
+$$
+
+Let us mention that, dealing with complex valued solutions of (NLKG), thus opening the possibi-
+lity of considering stable solitons, Bellazzani, Ghimenti and Le Coz [1] obtained a similar existence
+result for (NLKG) in this particular stable case. We also notice that the previous theorem has been
+extended to solutions describing multi-bound states by Côte and Martel [7], that is to multi-traveling
+waves made of any number *N* of decoupled general (excited) bound states. In the present paper, we
+will however only focus on (real valued) multi-solitary waves in the above sense.
+
+Since solitons are unstable, a solution of (NLKG) which behaves as a soliton in large time is not
+expected to be necessarily a soliton. One of our goals is thus to precise the dynamic of the flow of
+(NLKG) near a soliton. Similarly, the dynamic near a sum of solitons is also supposed to be more
+complex as time goes to infinity.
+
+**1.2. Main results.** Given $N$ distinct velocity parameters, we aim at proving the existence of a whole family of multi-solitons which turns out to be the unique family of multi-solitons in a certain class of solutions. Our first result reads as follows.
+
+**Theorem 1.2.** Assume that $f$ is of class $\mathcal{C}^2$ and $0 < |\beta_N| < \dots < |\beta_1| < 1$. There exist $\sigma > 0$, $0 < e^{\beta_1} < \dots < e^{\beta_N}$, $Y_{+,i} \in \mathcal{C}(\mathbb{R}, H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)) \cap L^\infty(\mathbb{R}, H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d))$ for $i = 1, \dots, N$ and an $N$-parameter family $(\varphi_{A_1, \dots, A_N})_{(A_1, \dots, A_N) \in \mathbb{R}^N}$ of solutions of (NLKG) such that, for all $(A_1, \dots, A_N) \in \mathbb{R}^N$, there exist $t_0 \in \mathbb{R}$ and $C > 0$ such that
+
+$$
+(1.2) \quad \forall t \ge t_0, \quad \left\| \Phi_{A_1, \ldots, A_N}(t) - \sum_{i=1}^{N} R_{\beta_i, x_i}(t) - \sum_{i=1}^{N} A_i e^{-e^{\beta_i} t} Y_{+,i}(t) \right\|_{H^{1} \times L^{2}} \le C e^{-(e^{\beta_N}+\sigma)t},
+$$
+
+where $\Phi_{A_1,...,A_N} := (\varphi_{A_1,...,A_N})_{\partial_t \varphi_{A_1,...,A_N}}$. In addition, if $(A'_1,...,A'_N) \neq (A_1,...,A_N)$, then $\varphi_{A'_1,...,A'_N} \neq \varphi_{A_1,...,A_N}$.
+
+*Remark 1.3.* The parameters $e_{\beta_i}$ and the functions $Y_{+,i}$ ($i = 1, \dots, N$) are defined in Proposition 2.1 and in subsection 2.2.
+
+One can moreover precise the value of $\sigma$ in Theorem 1.2; for this, we refer to (2.1).
+
+Our next result is concerned with the classification of multi-solitons. We aim at proving that any multi-soliton should belong to the family constructed in Theorem 1.2 above. This is indeed the case, if one knows that the multi-soliton converges sufficiently fast to its profile. As it is stated below, a decay in a power of *t* of degree larger than 3 is a sufficient rate.
\ No newline at end of file
diff --git a/samples/texts/7376768/page_30.md b/samples/texts/7376768/page_30.md
new file mode 100644
index 0000000000000000000000000000000000000000..146a1c5ab1e1574fbb3986b25eef38083065d905
--- /dev/null
+++ b/samples/texts/7376768/page_30.md
@@ -0,0 +1,75 @@
+* To prove this claim, we show that, if $\|E_{j-1}(t)\| \le C e^{-\sigma_0 t}$ with $e_{j-1} < \sigma_0 < e_j - \sigma$, then
+
+$$ \|E_{j-1}(t)\|_{H^1 \times L^2} \le C e^{-(\sigma+\sigma_0)t}. $$
+
+As
+
+$$ 
+\begin{aligned}
+|\alpha'_{\pm,j-1,k}(t) \mp e_k \alpha_{\pm,j-1,k}(t)| &\le C (e^{-\sigma t} \|E_{j-1}(t)\|_{H^1 \times L^2} + \|E_{j-1}(t)\|^2_{H^1 \times L^2}) \\
+&\le Ce^{-(\sigma+\sigma_0)t}
+\end{aligned}
+ $$
+
+(by the same calculations and arguments as those developed in the proof of Lemma 4.2), we
+have for all $k=1, \dots, j-1$,
+
+$$ |(e^{e_k t} \alpha_{-,j-1,k})'| \le C e^{-(\sigma+\sigma_0-e_k)t}. $$
+
+Since $t \mapsto e^{-(\sigma+\sigma_0-e_k)t}$ is integrable in the neighborhood of $+\infty$ (since $e_k \le e_{j-1}$), and by assumption, $e^{e_k t} \alpha_{-,j-1,k}(t) \to 0$ as $t \to +\infty$, we have by integration
+
+$$ |\alpha_{-,j-1,k}(t)| \le C e^{-(\sigma+\sigma_0)t}. $$
+
+For all $k = j, \dots, N$, we have $\sigma + \sigma_0 - e_k \le \sigma + \sigma_0 - e_j < 0$, thus by integration on $[t_0, t]$,
+we obtain
+
+$$ |e^{e_k t} \alpha_{-,j-1,k}(t) - e^{e_k t_0} \alpha_{-,j-1,k}(t_0)| \le C e^{(e_k - \sigma_0 - \sigma)t}. $$
+
+Eventually, we obtain (by a "cut-and-paste" of the argument exposed in subsection 4.1)
+
+$$ \|E_{j-1}(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{k=1}^{N} |\alpha_{-,j-1,k}(t')| \le C e^{-(\sigma_0+\sigma)t}, $$
+
+which is what was expected.
+
+* Now, from the preceding induction, there exists $\tilde{\sigma}_0 \in (e_j - \sigma, e_j)$ such that
+
+$$ \|E_{j-1}(t)\|_{H^1 \times L^2} \le C e^{-\tilde{\sigma}_0 t}, $$
+
+from which we deduce
+
+$$ |(e^{e_k t} \alpha_{-,j-1,k})'| \le C e^{-(\sigma+\tilde{\sigma}_0-e_k)t}. $$
+
+Now, for $k \in \{1, \dots, j-1\}$, $e_k - \sigma - \tilde{\sigma}_0 \le e_{j-1} - \sigma - e_j < 0$, we thus have
+
+$$ |\alpha_{-,j-1,k}(t)| \le C e^{-(\sigma_0+\tilde{\sigma})t} \le C e^{-e_j t}. $$
+
+For $k=j$, we have $|(e^{e_j t} \alpha_{-,j-1,j})'| \le C e^{(e_j - \tilde{\sigma}_0 - \sigma)t}$. Thus, there exists $A_j \in \mathbb{R}$ such that
+
+$$ e^{e_j t} \alpha_{-,j-1,j}(t) \to A_j, \quad \text{as } t \to +\infty. $$
+
+For $k \in \{j+1, \dots, N\}$, we have $\sigma + \tilde{\sigma}_0 - e_k < \sigma + e_j - e_k < 0$, thus by integration
+
+$$ 
+\begin{aligned}
+|\alpha_{-,j-1,k}(t)| &\le Ce^{-e_k t} + Ce^{-(\tilde{\sigma}_0+\gamma)t} \\
+&\le Ce^{-e_j t}.
+\end{aligned}
+ $$
+
+Hence,
+
+$$ \|E_{j-1}(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{k=1}^{N} |\alpha_{-,j-1,k}(t')| \le C e^{-e_j t}. $$
+
+Let us conclude the proof of Proposition 4.12. We define at this stage $E_j := U - \Phi_{A_1, \dots, A_j}$. We immediately have
+
+$$ E_j(t) = E_{j-1}(t) + \Phi_{A_1, \dots, A_{j-1}}(t) - \Phi_{A_1, \dots, A_j}(t). $$
+
+Then,
+
+$$ 
+\begin{align*}
+\|E_j(t)\|_{H^1 \times L^2} &\le \|E_{j-1}(t)\|_{H^1 \times L^2} + \|(\Phi_{A_1, \dots, A_j}(t) - \Phi_{A_1, \dots, A_{j-1}}(t)) - A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \\
+&\quad + \|A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \\
+&\le Ce^{-e_j t}.
+\end{align*}
+ $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_31.md b/samples/texts/7376768/page_31.md
new file mode 100644
index 0000000000000000000000000000000000000000..85d866f817b6cd7178a5fcb701f25648e8c92785
--- /dev/null
+++ b/samples/texts/7376768/page_31.md
@@ -0,0 +1,80 @@
+What is more,
+
+$$
+\begin{align*}
+\alpha_{-,j,k}(t) &= \langle E_j(t), Z_{-,k}(t) \rangle \\
+&= \alpha_{-,j-1,k}(t) - A_j e^{-e_j t} \langle Y_{+,j}, Z_{-,k} \rangle + O(e^{-(e_j+\sigma)t}).
+\end{align*}
+$$
+
+For $k = 1, \dots, j-1$, we have:
+
+$$
+\begin{align*}
+e^{e_k t} |\alpha_{-,j,k}(t)| &\le C e^{e_k t} |\alpha_{-,j-1,k}(t)| + O(e^{-(e_j-e_k+\sigma)t}) \\
+&\le C e^{(e_k-e_j)t} \xrightarrow{t \to +\infty} 0.
+\end{align*}
+$$
+
+For $k = j$,
+
+$$
+e^{e_j t} \alpha_{-,j,j}(t) = e^{e_j t} \alpha_{-,j-1,j}(t) - A_j + O(e^{-\sigma t}) \xrightarrow{t \to +\infty} 0.
+$$
+
+This finishes the induction argument.
+
+Finally we obtain that $U = \Phi_{A_1, \dots, A_N}$ by means of
+
+**Corollary 4.14.** For *t* sufficiently large, ||*E*<sub>*N*</sub>(*t*)||<sub>H<sup>1</sup>×L<sup>2</sup></sub> = 0.
+
+*Proof.* As in the preceding proofs, the following bounds hold:
+
+$$
+(4.33) \qquad \|E_N(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{i=1}^N |\alpha_{-,N,i}(t')|
+$$
+
+and
+
+$$
+(4.34) \quad |a'_{-,N,i}(t) + e_i a_{-,N,i}(t)| \le C (e^{-\sigma t} \|E_N(t)\|_{H^1 \times L^2} + \|E_N(t)\|_{H^1 \times L^2}^2).
+$$
+
+We observe that $t \mapsto e^{e_i t} (e^{-\sigma t} \|E_N(t)\|_{H^1 \times L^2} + \|E_N(t)\|_{H^1 \times L^2}^2)$ is integrable on $[t_0, +\infty)$; this is
+due to the fact that $\|E_N(t)\|_{H^1 \times L^2} \le Ce^{-e_N t}$. Since for all $i=1, \dots, N$, $e^{e_i t} \alpha_{-,N,i}(t) \to 0$ as
+$t \to +\infty$, we obtain by integration of (4.34) on $[t, +\infty)$:
+
+$$
+|\alpha_{-,N,i}(t)| \le C \left( e^{-\sigma t} \sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2} + \sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2}^2 \right).
+$$
+
+Then, using (4.33), we obtain
+
+$$
+\sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2} \le C \left( e^{-\sigma t} \sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2} + \sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2}^2 \right).
+$$
+
+This implies that ||*E*<sub>*N*</sub>(*t*)||<sub>*H*<sup>1</sup>×*L*<sup>2</sup></sub> = 0 for *t* sufficiently large.
+
+5. CONSTRUCTION OF A ONE-PARAMETER FAMILY OF SOLUTIONS CONVERGING TO A SOLITON
+
+The goal of this section is to prove the existence part in Theorem 1.6. Once again we restrict our focus to *d* = 1.
+
+**5.1. Outline of the construction.** Let $A \in \mathbb{R}$.
+
+Let $(S_n)_{n \in \mathbb{N}}$ be an increasing sequence of real numbers which tends to $+\infty$ and, for all $n \in \mathbb{N}$,
+define $u_n$ as the maximal solution of (NLKG) such that
+
+$$
+(5.1) \qquad U_n(S_n) = R_\beta(S_n) + A e^{-e_\beta S_n} Y_{+, \beta}(S_n),
+$$
+
+with obvious notations.
+
+We aim at proving the following key proposition:
+
+**Proposition 5.1.** There exist $t_0 \ge 0$ and $C_0 \ge 0$ such that for *n* large,
+
+$$
+(5.2) \quad \forall t \in [t_0, S_n], \quad \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+, \beta}(t)\|_{H^1 \times L^2} \le C_0 e^{-2e_\beta t}.
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_32.md b/samples/texts/7376768/page_32.md
new file mode 100644
index 0000000000000000000000000000000000000000..17c65198ee89c720cb949923bfaf06b6c0ff4682
--- /dev/null
+++ b/samples/texts/7376768/page_32.md
@@ -0,0 +1,33 @@
+To this end, we will set up a bootstrap argument and show
+
+**Proposition 5.2.** There exist $\alpha_0 > 0$, $C_0 > 0$, and $t_0 \ge 0$ such that for $n$ sufficiently large, if there exists $t_n^* \in [t_0, S_n]$ such that for all $t \in [t_n^*, S_n]$,
+
+$$ (5.3) \qquad \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+,\beta}(t)\|_{H^1 \times L^2} \le \alpha_0, $$
+
+then for all $t \in [t_n^*, S_n]$,
+
+$$ (5.4) \qquad \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+,\beta}(t)\|_{H^1 \times L^2} \le C_0 e^{-2e_\beta t}. $$
+
+Let us show how to deduce Proposition 5.1 from Proposition 5.2.
+
+*Proof of Proposition 5.1.* Assume momentarily that Proposition 5.2 holds true. Let us consider $\alpha_0$ and $C_0$ as in Proposition 5.2 and suppose (even if it means enlarging $t_0$) that $C_0 e^{-2e_\beta t_0} \le \frac{\alpha_0}{2}$. We define for all $n$ such that $S_n > t_0$:
+
+$$ t_n^* := \inf \{ t \in [t_0, S_n], \forall \tau \in [t, S_n], \|U_n(\tau) - R_\beta(\tau) - A e^{-e_\beta \tau} Y_{+, \beta}(\tau)\|_{H^1 \times L^2} \le \alpha_0 \}. $$
+
+By (5.1) and by continuity in time of $U_n$, $R_\beta$, and $Y_{+,\beta}$, $t_n^*$ is indeed well-defined and we necessarily have $t_0 \le t_n^* < S_n$. Since (5.3) implies (5.4), for all $t \in [t_n^*, S_n]$,
+
+$$ \begin{align*} \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+, \beta}(t)\|_{H^1 \times L^2} &\le C_0 e^{-2e_\beta t} \\ &\le C_0 e^{-2e_\beta t_0} \\ &\le \frac{\alpha_0}{2}. \end{align*} $$
+
+Let us assume for the sake of contradiction that $t_n^* > t_0$ for some $n$. Then, observing the preceding inequality, we obtain (again by continuity in time of $U_n$, $R_\beta$, and $Y_{+,\beta}$) the existence of $\tau_n > 0$ such that $t_n^* - \tau_n \ge t_0$ and for all $t \in [t_n^* - \tau_n, S_n]$,
+
+$$ \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+, \beta}(t)\|_{H^1 \times L^2} \le \frac{3\alpha_0}{4} < \alpha_0. $$
+
+This contradicts the definition of $t_n^*$ as an infimum. Hence $t_n^* = t_0$ and (5.3) (and thus (5.4)) holds on $[t_0, S_n]$ for all $n$. This achieves the proof of Proposition 5.1. □
+
+The existence of $u^A$ (and $U^A$), as stated in Theorem 1.6 is a consequence of Proposition 5.1 and the continuity of the flow of (NLKG) for the weak $H^1 \times L^2$ topology. We will not detail the construction of $U^A$ considering that it is a sort of cut and paste of what was done in order to prove Proposition 3.1 in the context of multiple solitons.
+
+Similarly, we do not repeat the arguments exposed at the beginning of section 3 which justify that the map $A \mapsto u^A$ is one-to-one. We devote the next subsection to the proof of Proposition 5.2.
+
+**5.2. Proof of Proposition 5.2.** We assume that $U_n(t)$ is defined on some interval $[t_n^*, S_n]$ and satisfies (5.3). We want to show that (5.4) holds, provided that the parameters $\alpha_0$ and $t_0$ are well chosen.
+
+In this subsection again, for notation purposes and ease of reading, we sometimes omit the index $n$ and also write $O(G(t))$ in order to refer to a function $g$ which a priori depends on $n$ and such that there exists $C \ge 0$ (independent of $n$) such that for all $n$ large and for all $t \in [t_n^*, S_n]$, $|g(t)| \le C|G(t)|$.
\ No newline at end of file
diff --git a/samples/texts/7376768/page_33.md b/samples/texts/7376768/page_33.md
new file mode 100644
index 0000000000000000000000000000000000000000..f61b40ebd62ccf6555e9a475c14f3d3a32067531
--- /dev/null
+++ b/samples/texts/7376768/page_33.md
@@ -0,0 +1,84 @@
+**5.2.1. Step 1: Set up of a modulation argument.**
+
+**Lemma 5.3.** For $t_0 \ge 0$ sufficiently large and $\alpha_0 > 0$ sufficiently small, there exists a unique $\mathscr{C}^1$
+function $x : [t_n^*, S_n] \to \mathbb{R}$ such that if we set
+
+$$
+W_n(t) := U_n(t) - \tilde{R}_\beta(t) - Ae^{-e_\beta t} \tilde{Y}_{+\beta}(t),
+$$
+
+with $\tilde{R}_\beta(t) := R_\beta(t, \cdot -x(t))$ and $\tilde{Y}_{+\beta}(t) := Y_{+\beta}(t, \cdot -x(t))$, then for all $t \in [t_n^*, S_n]$,
+
+$$
+(5.5) \qquad \langle W_n(t), \partial_x \tilde{R}_\beta(t) \rangle = 0.
+$$
+
+Moreover there exists $K_1 > 0$ such that for all $t \in [t_n^*, S_n]$,
+
+$$
+(5.6) \qquad \|W_n(t)\|_{H^{1}\times L^2} + |x(t)| \le K_1 \alpha_0,
+$$
+
+$$
+(5.7) \qquad |\dot{x}(t)| \le K_1 (\|W_n(t)\|_{H^1 \times L^2} + e^{-2e_\beta t}).
+$$
+
+*Remark 5.4.* Notice that by uniqueness of the function *x* and by definition of *u*<sub>*n*</sub> (see (5.1)), we have *W*<sub>*n*</sub>(*S*<sub>*n*</sub>) = 0 and *x*(*S*<sub>*n*</sub>) = 0.
+
+*Proof.* The existence of *x* such that (5.5) is granted and the existence of *K*₂ > 0 such that
+
+$$
+\|W_n(t)\|_{H^1 \times L^2} + |x(t)| \leq K_2 \alpha_0
+$$
+
+are standard consequences of the implicit function theorem.
+
+Now, let us prove (5.7). For this, we notice that $W = W_n$ satisfies the following equation:
+
+$$
+(5.8) \quad \partial_t W = \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(Q_\beta) & 0 \end{pmatrix} (W + A e^{-e_\beta t} \tilde{Y}_{+\beta}) + \left( f(u) - f(Q_\beta) - f'(Q_\beta)(u - Q_\beta) \right) \\
+\qquad + \dot{x}(t) \partial_x \tilde{R}_\beta - A e^{-e_\beta t} (\partial_t \tilde{Y}_{+\beta} - \dot{x}(t) \partial_x \tilde{Y}_{+\beta} - e_\beta \tilde{Y}_{+\beta}),
+$$
+
+where $\tilde{Q}_{\beta}(t,x) = Q_{\beta}(t,x-x(t))$.
+
+Since $\frac{d}{dt}\langle W, \partial_x \tilde{R}_\beta \rangle = 0$, we have:
+
+$$
+\langle \partial_t W, \partial_x \tilde{R}_\beta \rangle + \langle W, \partial_{tx} \tilde{R}_\beta - \dot{x} \partial_x^2 \tilde{R}_\beta \rangle = 0.
+$$
+
+Observing moreover that
+
+$$
+f(u) - f(\tilde{Q}_{\beta}) - f'(\tilde{Q}_{\beta})(u - \tilde{Q}_{\beta}) = O\left(\|U - \tilde{R}_{\beta}\|_{H^{1}\times L^{2}}^{2}\right) = O\left(\|W(t)\|_{H^{1}\times L^{2}}^{2} + e^{-2e_{\beta}t}\right)
+$$
+
+by Taylor formula ($f$ is $\mathcal{C}^2$), we have thus:
+
+$$
+\begin{align*}
+0 &= \left\langle \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(\tilde{Q}_\beta) & 0 \end{pmatrix} W, \partial_x \tilde{R}_\beta \right\rangle + O\left(\|W(t)\|_{H^{1}\times L^2}^2 + e^{-2e_\beta t}\right) + \dot{x}(t) \| \partial_x R_\beta \|_{H^{1}\times L^2}^2 \\
+&\quad + A e^{-e_\beta t} \left\langle (\beta + \dot{x}) \partial_x \tilde{Y}_{+\beta} + e_\beta \tilde{Y}_{+\beta}, \partial_x \tilde{R}_\beta \right\rangle - (\beta + \dot{x}) \langle W, \partial_x^2 \tilde{R}_\beta \rangle \\
+&= \dot{x} \left( \| \partial_x R_\beta \|_{H^{1}\times L^2}^2 + A e^{-e_\beta t} \langle \partial_x \tilde{Y}_{+\beta}, \partial_x \tilde{R}_\beta \rangle - \langle W, \partial_x^2 \tilde{R}_\beta \rangle \right) + O\left(\|W(t)\|_{H^{1}\times L^2}^2 + e^{-2e_\beta t}\right) \\
+&\quad + \left\langle \begin{pmatrix} \beta \partial_x & Id \\ \partial_x^2 - Id + f'(\tilde{Q}_\beta) & \beta \partial_x \end{pmatrix} W, \partial_x \tilde{R}_\beta \right\rangle + A e^{-e_\beta t} \left\langle \begin{pmatrix} \beta \partial_x & Id \\ \partial_x^2 - Id + f'(\tilde{Q}_\beta) & \beta \partial_x \end{pmatrix} \tilde{Y}_{+\beta}, \partial_x \tilde{R}_\beta \right\rangle.
+\end{align*}
+$$
+
+Notice that we have used $\langle Y_{+\beta}, \partial_x R_\beta \rangle = 0$ (see Proposition 2.1). We now observe that
+
+$$
+\left( \begin{pmatrix} \beta & 0 \\ 0 & Id \end{pmatrix} - Id + f'(Q_{\beta}) & 0 \\ 0 & Id \end{pmatrix} = -J\tilde{H}_{\beta},
+$$
+
+where $\tilde{H}_{\beta}$ is the matrix operator defined like $H_{\beta}$ by replacing $Q_{\beta}$ by $\tilde{Q}_{\beta}$, that is
+
+$$
+(5.9) \qquad \tilde{H}_{\beta} := \begin{pmatrix} -\partial_x^2 + Id - f'(\tilde{Q}_{\beta}) & -\beta\partial_x \\ 0 & \beta\partial_x \\ 0 & Id \end{pmatrix}.
+$$
+
+We obtain:
+
+$$
+\left\langle\left(\begin{array}{cc} 0 & 0 \\ 0 & Id \\ 0 & 0 \end{array}\right)-Id+f^{\prime}\left(\tilde{Q}_{\beta}\right)\right\rangle=\left\langle\tilde{Y}_{+,\beta}, \tilde{H}_{\beta} J \partial_{x} \tilde{R}_{\beta}\right\rangle=\left\langle\tilde{Y}_{+,\beta}, 0\right\rangle=0.
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_34.md b/samples/texts/7376768/page_34.md
new file mode 100644
index 0000000000000000000000000000000000000000..db46108d9bb5707f5d942a55dc76bc345ebab895
--- /dev/null
+++ b/samples/texts/7376768/page_34.md
@@ -0,0 +1,69 @@
+Finally we can choose $t$ large enough such that
+
+$$A e^{-e\beta t} |\langle \partial_x \tilde{Y}_{+\beta}, \partial_x \tilde{R}_\beta \rangle| \leq \frac{1}{4} \| \partial_x R_\beta \|_{H^1 \times L^2}^2$$
+
+and we can take $\alpha_0 > 0$ sufficiently small such that
+
+$$|\langle W, \partial_x^2 \tilde{R}_\beta \rangle| \le K_2 \alpha_0 \| \partial_x^2 \tilde{R}_\beta \|_{H^1 \times L^2} \le \frac{1}{4} \| \partial_x R_\beta \|_{H^1 \times L^2}^2$$
+
+and such that $K_2\alpha_0 \le 1$ (then $\|W(t)\|_{H^1\times L^2}^2 \le \|W(t)\|_{H^1\times L^2}$). Consequently for $t$ large and $\alpha_0$ small, we have:
+
+$$
+\begin{align*}
+|\dot{x}(t)| &\le \frac{2}{\|\partial_x R_\beta\|_{H^1\times L^2}^2} \left| \left\langle W, \begin{pmatrix} \beta \partial_x & \partial_x^2 - Id + f'(\tilde{Q}_\beta) \\ Id & \beta \partial_x \end{pmatrix} \partial_x \tilde{R}_\beta \right\rangle \right| + O\left( \|W(t)\|_{H^1\times L^2} + e^{-2e_\beta t} \right) \\
+&\le K_3 (\|W(t)\|_{H^1\times L^2} + e^{-2e_\beta t}),
+\end{align*}
+$$
+
+for some constant $K_3 > 0$. Finally, take $K_1 := \max(K_2, K_3)$ to obtain Lemma 5.3. □
+
+**5.2.2. Step 2: Control of particular directions.** Let us denote
+
+$$ (5.10) \qquad \alpha_{\pm,\beta}(t) := \langle W(t), \tilde{Z}_{\pm,\beta}(t) \rangle, $$
+
+where $\tilde{Z}_{\pm,\beta}(t) := Z_{\pm,\beta}(t, \cdot -x(t))$.
+
+**Lemma 5.5.** We have for all $t \in [t_n^*, S_n]$,
+
+$$ \left| \frac{d\alpha_{\pm,\beta}}{dt}(t) - e_{\beta}\alpha_{\pm,\beta}(t) \right| + \left| \frac{d\alpha_{-\beta}}{dt}(t) + e_{\beta}\alpha_{-\beta}(t) \right| = O\left( \|W(t)\|_{H^1\times L^2}^2 + e^{-2e_{\beta}t} \right). $$
+
+*Proof.* Observe that
+
+$$
+\begin{align*}
+\partial_t \tilde{Z}_{\pm, \beta}(t, x) &= -(\beta + \dot{x}(t)) \partial_x Z_{\pm, \beta}(t, x - x(t)) \\
+&= -(\beta + \dot{x}(t)) \partial_x \tilde{Z}_{\pm, \beta}(t, x).
+\end{align*}
+$$
+
+Thus
+
+$$ (5.11) \qquad \frac{d\alpha_{\pm,\beta}}{dt} = (\partial_t W, \tilde{Z}_{\pm,\beta}) + (\langle W, -\beta \partial_x Z_{\pm,\beta} \rangle + \langle W, -\dot{x} \partial_x Z_{\pm,\beta} \rangle). $$
+
+By (5.7), we obtain
+
+$$ (5.12) \qquad \langle W, -\dot{x}\partial_x Z_{\pm,\beta} \rangle = O\left(\|W(t)\|_{H^1\times L^2}^2 + e^{-2e_\beta t}\right). $$
+
+In addition, defining $\mathcal{H}_\beta$ like $\mathcal{H}_\beta$ by replacing $Q_\beta$ by $\tilde{Q}_\beta$, we have by (5.8)
+
+$$
+\begin{align}
+(\partial_t W, \tilde{Z}_{\pm,\beta}) + (\langle W, -\beta \partial_x Z_{\pm,\beta} \rangle) &= (\langle W, \mathcal{H}_\beta \tilde{Z}_{\pm,\beta} \rangle + \dot{x} (\partial_x \tilde{R}_\beta, \tilde{Z}_{\pm,\beta})) \nonumber \\
+&\quad + Ae^{-e_\beta t} (\langle \tilde{Y}_{+\beta}, \mathcal{H}_\beta \tilde{Z}_{\pm,\beta} \rangle + O(\|W(t)\|_{H^1\times L^2}^2 + e^{-2e_\beta t})) \nonumber \\
+&\quad + Ae^{-e_\beta t} (\dot{x} (\partial_x \tilde{Y}_{+\beta}, \tilde{Z}_{\pm,\beta}) + e_\beta (\langle \tilde{Y}_{+\beta}, \tilde{Z}_{\pm,\beta} \rangle)). 
+\end{align}
+$$
+
+Since
+
+$$ (\tilde{Y}_{+\beta}, \tilde{Z}_{\pm,\beta}) = (\langle Y_{+\beta}, Z_{\pm,\beta} \rangle) = \begin{cases} 1 & \text{if we take } \pm = - \\ 0 & \text{if we take } \pm = +, \end{cases} $$
+
+we always obtain
+
+$$ (5.14) \qquad A e^{-e\beta t} (\tilde{Y}_{+\beta}, \mathcal{H}_{\beta} \tilde{Z}_{\pm,\beta}) + A e^{-e\beta t} e_{\beta} (\tilde{Y}_{+\beta}, \tilde{Z}_{\pm,\beta}) = 0. $$
+
+We have in addition
+
+$$ |e^{-e\beta t}\dot{x}| \leq \frac{1}{2}(e^{-2e\beta t} + \dot{x}^2) = O(\|W(t)\|_{H^1\times L^2}^2 + e^{-2e\beta t}) $$
+
+and $\langle \partial_x \tilde{R}_\beta, \tilde{Z}_{+\beta} \rangle$ by Proposition 2.1.
\ No newline at end of file
diff --git a/samples/texts/7376768/page_35.md b/samples/texts/7376768/page_35.md
new file mode 100644
index 0000000000000000000000000000000000000000..2375bbfdd05a65f11e4aec83e4509a772526fb51
--- /dev/null
+++ b/samples/texts/7376768/page_35.md
@@ -0,0 +1,91 @@
+Hence, gathering (5.11), (5.12), (5.13), and (5.14), we infer:
+
+$$
+\frac{d\alpha_{\pm,\beta}}{dt} = \pm e_{\beta}\langle W, \tilde{Z}_{\pm,\beta} \rangle + O\left(\|W(t)\|_{H^{1}\times L^2}^2 + e^{-2e_{\beta}t}\right),
+$$
+
+which proves Lemma 5.5.
+
+□
+
+**5.2.3. Step 3: Exponential control of ||W_n||_{H^1 \times L^2}.** Let us introduce the functional
+
+$$
+(5.15) \qquad \mathcal{F}_W(t) := \langle \hat{H}_\beta W(t), W(t) \rangle.
+$$
+
+(We recall that $\tilde{H}_{\beta}$ is defined in (5.9).)
+
+**Lemma 5.6 (Control of F<sub>W</sub>).*** There exists $C > 0$ such that for $t_0$ sufficiently large, we have for all $n$
+such that $S_n \ge t_0$, for all $t \in [t_n^*, S_n]$:
+
+$$
+|\mathcal{F}_W(t)| \le C \left( \|W(t)\|_{H^1 \times L^2}^3 + e^{-3e_\beta t} + e^{-e_\beta t} |\alpha_{+, \beta}(t)| \right).
+$$
+
+Proof. We have
+
+$$
+(5.16) \qquad \begin{aligned}
+\mathcal{F}_W(t) &= \langle \hat{H}_\beta (U - \tilde{R}_\beta), U - \tilde{R}_\beta \rangle - 2Ae^{-e_\beta t} \langle \hat{H}_\beta \tilde{Y}_{+\beta}, W \rangle \\
+&= \langle \hat{H}_\beta (U - \tilde{R}_\beta), U - \tilde{R}_\beta \rangle - 2Ae^{-e_\beta t} \alpha_{+\beta}.
+\end{aligned}
+$$
+
+Notice that we have used $\tilde{H}_{\beta}\tilde{Y}_{+\beta} = \tilde{Z}_{+\beta}$ and $\langle\tilde{Y}_{+\beta}, \tilde{Z}_{+\beta}\rangle = 0$.
+
+Now let us focus on the quadratic term $\langle \tilde{H}_{\beta}(U - \tilde{R}_{\beta}), U - \tilde{R}_{\beta} \rangle$. This term rewrites
+
+$$
+\langle \tilde{H}_{\beta}(U - \tilde{R}_{\beta}), U - \tilde{R}_{\beta} \rangle = \int_{\mathbb{R}} \{\epsilon_2^2 + (\partial_x \epsilon_1)^2 + \epsilon_1^2 - f'(\tilde{Q}_{\beta}) \epsilon_1^2 + 2\beta \epsilon_2 \partial_x \epsilon_1\} (t,x) dx,
+$$
+
+where $\epsilon_1$ and $\epsilon_2$ are defined as follows:
+
+$$
+\epsilon_1(t,x) := u(t,x) - Q_\beta(t,x-x(t)) \quad \text{and} \quad \epsilon_2(t,x) := \partial_t u(t,x) - \partial_t Q_\beta(t,x-x(t)).
+$$
+
+In a compact manner, we can write:
+
+$$
+\epsilon_1 := u - \tilde{Q}_{\beta} \quad \text{and} \quad \epsilon_2 := \partial_t u + \beta \partial_x \tilde{Q}_{\beta}
+$$
+
+since $\partial_t Q_\beta(t,x) = -\beta \partial_x Q_\beta(t,x)$.
+
+We observe that
+
+$$
+\int_{\mathbb{R}} \left\{ (\partial_t Q_\beta(t, x - x(t)))^2 + (\partial_x Q_\beta(t, x - x(t)))^2 + (Q_\beta(t, x - x(t)))^2 \right\} dx \\
+= \int_{\mathbb{R}} \left\{ (\partial_t Q_\beta(t, x))^2 + (\partial_x Q_\beta(t, x))^2 + (Q_\beta(t, x))^2 \right\} dx
+$$
+
+and
+
+$$
+\int_{\mathbb{R}} \partial_x Q_\beta(t, x - x(t)) \partial_t Q_\beta(t, x - x(t)) dx = \int_{\mathbb{R}} \partial_x Q_\beta(t, x) \partial_t Q_\beta(t, x) dx.
+$$
+
+Considering that $u_n$ and $Q_\beta$ are solutions of (NLKG), the energy and the momentum of these
+solutions as defined in introduction are conserved. Thus, there exists $C_n \in \mathbb{R}$ such that
+
+$$
+\begin{align*}
+& \langle \tilde{H}_{\beta}(U - \tilde{R}_{\beta}), U - \tilde{R}_{\beta} \rangle(t) \\
+&= C_n + 2 \int_{\mathbb{R}} \left\{
+  F(u(t,x)) - F(\tilde{Q}_{\beta}(t,x)) - \frac{1}{2} f'(\tilde{Q}_{\beta}(t,x)) \varepsilon_1(t,x)^2
+\right\} dx \\
+&\quad - 2 \int_{\mathbb{R}} \partial_t u(t,x) (-\beta \partial_x \tilde{Q}_{\beta}(t,x)) dx \\
+&\quad - 2 \int_{\mathbb{R}} \partial_x u(t,x) \partial_x \tilde{Q}_{\beta}(t,x) dx \\
+&\quad - 2 \int_{\mathbb{R}} u(t,x) \tilde{Q}_{\beta}(t,x) dx \\
+&\quad - 2\beta \int_{\mathbb{R}} \partial_t u(t,x) \partial_x Q_{\beta}(t,x-x(t)) dx \\
+&\quad - 2\beta \int_{\mathbb{R}} \partial_x u(t,x) \partial_t Q_{\beta}(t,x-x(t)) dx.
+\end{align*}
+$$
+
+Now, we notice that
+
+$$
+\int_{\mathbb{R}} \partial_x u(t, x) \partial_x \tilde{Q}_\beta(t, x) dx = - \int_{\mathbb{R}} u(t, x) \partial_x^2 \tilde{Q}_\beta(t, x) dx
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_36.md b/samples/texts/7376768/page_36.md
new file mode 100644
index 0000000000000000000000000000000000000000..ac65b1cd7fa7bb5934f7e8cb6602d8f1c640b644
--- /dev/null
+++ b/samples/texts/7376768/page_36.md
@@ -0,0 +1,70 @@
+and
+
+$$
+\partial_x \tilde{Q}_\beta(t, x) = \partial_x Q_\beta(t, x - x(t)).
+$$
+
+Thus
+
+$$
+(5.17) \qquad \begin{aligned}
+\langle \hat{H}_{\beta}(U - \tilde{R}_{\beta}), U - \tilde{R}_{\beta} \rangle (t) \\
+&= C_n + 2 \int_{\mathbb{R}} \left\{ F(u) - F(\tilde{Q}_{\beta}) - f(\tilde{Q}_{\beta})\varepsilon_1 - \frac{1}{2}f'(\tilde{Q}_{\beta})\varepsilon_1^2 \right\} (t,x) \, dx \\
+&\quad + 2 \int_{\mathbb{R}} u(t,x) \left( (1-\beta^2)\partial_x^2 \tilde{Q}_{\beta}(t,x) - \tilde{Q}_{\beta}(t,x) + f(\tilde{Q}_{\beta}(t,x)) \right) dx.
+\end{aligned}
+$$
+
+The last integral is zero by the equation satisfied by $Q_\beta$. By means of Taylor inequality, we claim
+
+$$
+(5.18) \qquad \begin{aligned}[t]
+& \int_{\mathbb{R}} \left\{ F(u(t,x)) - F(\tilde{Q}_{\beta}(t,x)) - f(\tilde{Q}_{\beta}(t,x))\varepsilon_1(t,x) - \frac{1}{2}f'(\tilde{Q}_{\beta}(t,x))\varepsilon_1(t,x)^2 \right\} dx \\
+&= O \left( \|U - \tilde{R}_{\beta}(t)\|_{H^{1}\times L^2}^3 \right) \\
+&= O \left( \|W(t)\|_{H^{1}\times L^2}^3 + e^{-3e_{\beta}t} \right).
+\end{aligned}
+$$
+
+Hence, collecting (5.16), (5.17), and (5.18), we have:
+
+$$
+\mathcal{F}_W(t) = C_n + O \left( \|W(t)\|_{H^1 \times L^2}^3 + e^{-3e_{\beta}t} + e^{-e_{\beta}t} |\alpha_{+,\beta}(t)| \right).
+$$
+
+On the other hand, we immediately have $|\mathcal{F}_W(t)| \le C \|W(t)\|_{H^1\times L^2}^2$. Since $W(S_n) = 0$ and $\alpha_{+\beta}(S_n) = 0$, we deduce that $|C_n| \le Ce^{-3e_\beta S_n} \le Ce^{-3e_\beta t}$ for all $t \in [t_n^*, S_n]$. $\square$
+
+**Corollary 5.7.** There exists $C > 0$ such that for $t_0$ sufficiently large, we have for all $n$ such that $S_n \ge t_0$, for all $t \in [t_n^*, S_n]$:
+
+$$
+|\mathcal{F}_W(t)| \le C \left( \|W(t)\|_{H^1 \times L^2}^3 + e^{-e_{\beta}t} \sup_{t' \in [t, S_n]} \|W(t')\|_{H^1 \times L^2}^2 + e^{-3e_{\beta}t} \right).
+$$
+
+Proof. From Lemma 5.5 we deduce that
+
+$$
+\forall t \in [t_n^*, S_n], \quad \left| \frac{d}{dt} (e^{-e_{\beta}t} \alpha_{+, \beta}(t)) \right| \le C \left( e^{-e_{\beta}t} \sup_{t' \in [t, S_n]} \|W(t')\|_{H^1 \times L^2}^2 + e^{-3e_{\beta}t} \right).
+$$
+
+Given that $e^{-e_\beta S_n} \alpha_{+\beta}(S_n) = 0$, we deduce by integration of the preceding inequality that:
+
+$$
+\forall t \in [t_n^*, S_n], \quad |e^{-e_\beta t} \alpha_{+\beta}(t)| \le C \left( e^{-e_\beta t} \sup_{t' \in [t, S_n]} \|W(t')\|_{H^1 \times L^2}^2 + e^{-3e_\beta t} \right).
+$$
+
+Now the corollary follows from the combination of this last result with Lemma 5.6. $\square$
+
+**Lemma 5.8.** There exists $c > 0$ such that for $\alpha_0 > 0$ sufficiently small and $t_0 \ge 0$ sufficiently large, for all $n$ such that $S_n \ge t_0$, for all $t \in [t_n^*, S_n]$:
+
+$$
+(5.19) \qquad \left\{
+\begin{aligned}
+\left|\frac{d}{dt}\alpha_{+\beta}(t) - e_\beta\alpha_{+\beta}(t)\right| &\le c \min\left\{\left(\sup_{t'\in[t,S_n]}\left(\alpha_{+\beta}^2(t')+a_{-\beta}^2(t')\right)+e^{-2e_\beta t}\right), a_{+\beta}^2(t)+a_{-\beta}^2(t)+e^{-e_\beta t}\right\}, \\
+\left|\frac{d}{dt}\alpha_{-\beta}(t)+e_\beta\alpha_{-\beta}(t)\right| &\le c \min\left\{\left(\sup_{t'\in[t,S_n]}\left(\alpha_{+\beta}^2(t')+a_{-\beta}^2(t')\right)+e^{-2e_\beta t}\right), a_{+\beta}^2(t)+a_{-\beta}^2(t)+e^{-e_\beta t}\right\}.
+\end{aligned}
+\right.
+$$
+
+*Proof.* Due to Proposition 2.2, we have on the one hand
+
+$$
+(5.20) \qquad \|W\|_{H^{1}\times L^{2}}^{2} \leq \frac{1}{\mu} \mathcal{F}_{W}(t) + \frac{1}{\mu^{2}} (\alpha_{+, \beta}^{2} + \alpha_{-, \beta}^{2}) .
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_37.md b/samples/texts/7376768/page_37.md
new file mode 100644
index 0000000000000000000000000000000000000000..17d4bea2fece3dbc5e966562da318300ebbefb21
--- /dev/null
+++ b/samples/texts/7376768/page_37.md
@@ -0,0 +1,82 @@
+On the other,
+
+$$
+(5.21) \quad |\mathcal{F}_W(t)| \le C \left( \|W\|_{H^1 \times L^2}^3 + e^{-e_\beta t} \sup_{t' \in [t, S_n]} \|W(t')\|_{H^1 \times L^2}^2 + e^{-3e_\beta t} \right).
+$$
+
+Now, inserting (5.21) into (5.20), we obtain
+
+$$
+(5.22) \qquad \|W\|_{H^1 \times L^2}^2 \le \frac{2C}{\mu} e^{-3e_\beta t} + \frac{2}{\mu^2} \sup_{t' \in [t, S_n]} (\alpha_{+, \beta}^2(t') + \alpha_{-, \beta}^2(t'))
+$$
+
+provided $\alpha_0$ is chosen small enough such that $\frac{C}{\mu} \sup_{t' \in [t, S_n]} \|W(t')\|_{H^1 \times L^2} \le \frac{1}{4}$ (for all $t$) and $t_0$ is
+chosen large enough such that $\frac{C}{\mu} e^{-e_\beta t_0} \le \frac{1}{4}$.
+
+Then, combining (5.22) and the estimates obtained in Lemma 5.5, we deduce:
+
+$$
+\begin{align*}
+\left| \frac{d\alpha_{\pm,\beta}}{dt} \mp e_\beta \alpha_{\pm,\beta} \right| &\le C \left( \|W\|_{H^1\times L^2}^2 + e^{-2e_\beta t} \right) \\
+&\le C \left( \sup_{t'\in[t,S_n]} \left( \alpha_{+,\beta}^2(t') + \alpha_{-,\beta}^2(t') \right) + e^{-2e_\beta t} \right).
+\end{align*}
+$$
+
+In order to avoid the supremum in front of the expression $\alpha_{+\beta}^2 + \alpha_{-\beta}^2$ and hence to obtain the second way of estimating $\frac{d}{dt}\alpha_{\pm,\beta} \mp e_\beta \alpha_{\pm,\beta}$ which is described by Lemma 5.8, we can rewrite (5.21) more simply as follows
+
+$$
+|\mathcal{F}_W(t)| \le C (\|W\|_{H^1 \times L^2}^3 + e^{-e_\beta t} + e^{-3e_\beta t}).
+$$
+
+Then the analog of (5.22) takes the following form:
+
+$$
+\|W\|_{H^1 \times L^2}^2 \le Ce^{-e_\beta t} + (\alpha_{+, \beta}^2(t) + \alpha_{-, \beta}^2(t));
+$$
+
+the last arguments remain unchanged. However note that, in this case, the obtained estimates are at
+the cost of a worse exponential term.
+$\square$
+
+**Proposition 5.9.** There exists $C > 0$ such that for all $\alpha_0 \ge 0$ sufficiently small, for all $t_0$ sufficiently large, for all $n$ such that $S_n \ge t_0$, for all $t \in [t_n^*, S_n]$, the following inequalities hold:
+
+$$
+(5.23) \qquad |α_{-β}(t)| + |α_{+,β}(t)| ≤ Ce^{-2e_βt}
+$$
+
+and
+
+$$
+(5.24) \qquad \|W(t)\|_{H^1 \times L^2} \le C e^{-\frac{3}{2} e_{\beta} t}.
+$$
+
+*Proof.* Let us start with the system (5.19) obtained in Lemma 5.8. We have in particular
+
+$$
+(5.25) \qquad \left\{
+\begin{aligned}
+\left|\frac{d}{dt}\alpha_{+, \beta} - e_{\beta}\alpha_{+, \beta}\right| &\le c\left(\alpha_{+, \beta}^{2}(t) + \alpha_{-, \beta}^{2}(t) + e^{-e_{\beta}t}\right) \\
+\left|\frac{d}{dt}\alpha_{-, \beta} + e_{\beta}\alpha_{-, \beta}\right| &\le c\left(\alpha_{+, \beta}^{2}(t) + \alpha_{-, \beta}^{2}(t) + e^{-e_{\beta}t}\right).
+\end{aligned}
+\right.
+$$
+
+Taking some inspiration in [3, paragraph 4.4.2], we will first show the existence of $M \ge 0$ such that
+for all $t_0$ large enough, for all $t \in [t_n^*, S_n]$,
+
+$$
+(5.26) \qquad |α_{+,β}(t)| \le M (α_{-,β}^2(t) + e^{-e_β t}) .
+$$
+
+In order to prove (5.26), let us consider, for some positive constants $M$ and $\tilde{M}$, the function
+
+$$
+h : t \mapsto \alpha_{+, \beta}(t) - M\alpha_{-, \beta}^{2}(t) - \tilde{M}e^{-e_{\beta}t}
+$$
+
+and show that it is always negative on $[t_n^*, S_n]$, provided $M$ and $\tilde{M}$ are well chosen.
+We compute
+
+$$
+h'(t) = \alpha'_{+, \beta}(t) - 2M\alpha'_{-, \beta}(t)\alpha_{-, \beta}(t) + \tilde{M}e_{\beta}e^{-e_{\beta}t}.
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_38.md b/samples/texts/7376768/page_38.md
new file mode 100644
index 0000000000000000000000000000000000000000..720d553a2307b258afa5886f2e35d8cde4508ff6
--- /dev/null
+++ b/samples/texts/7376768/page_38.md
@@ -0,0 +1,68 @@
+By (5.25), we obtain
+
+$$
+\begin{align*}
+h' \ge e_{\beta} \alpha_{+, \beta} - c (\alpha_{+, \beta}^2 + \alpha_{-, \beta}^2 + e^{-e_{\beta} t}) \\
+\qquad - 2M (-e_{\beta} \alpha_{-, \beta}^2 + c |\alpha_{-, \beta}| (\alpha_{+, \beta}^2 + \alpha_{-, \beta}^2 + e^{-e_{\beta} t})) + \tilde{M} e_{\beta} e^{-e_{\beta} t}.
+\end{align*}
+$$
+
+Replacing $\alpha_{+, \beta}$ by its expression in terms of $h$, $\alpha_{-, \beta}^2$, and $e^{-e_{\beta}t}$ and using that
+
+$$
+(h + M\alpha_{-, \beta}^2 + \tilde{M}e^{-e_{\beta}t})^2 \leq 2h^2 + 4M^2\alpha_{-, \beta}^4 + 4\tilde{M}^2e^{-2e_{\beta}t}
+$$
+
+lead to the following estimate of $h'$:
+
+$$
+\begin{align*}
+h' \geq & e_\beta h - 2c (1 + 2M|\alpha_{-, \beta}|) h^2 \\
+& + (e_\beta \tilde{M} - c - 2Mc|\alpha_{-, \beta}| - 4\tilde{M}^2 c e^{-e_\beta t} - 8M\tilde{M}^2 c |\alpha_{-, \beta}| e^{-e_\beta t}) e^{-e_\beta t} \\
+& + \alpha_{-, \beta}^2 (3Me_\beta - c - 2Mc|\alpha_{-, \beta}| - 4cM^2\alpha_{-, \beta}^2 - 8M^3c|\alpha_{-, \beta}| \alpha_{-, \beta}^2).
+\end{align*}
+$$
+
+Now we choose $M := \frac{c}{e_\beta}$ and $\tilde{M} := \frac{2c}{e_\beta}$ so that the expressions
+
+$$
+3Me_{\beta} - c - 2Mc|\alpha_{-, \beta}| - 4cM^{2}\alpha_{-, \beta}^{2} - 8M^{3}c|\alpha_{-, \beta}|^{2}
+$$
+
+and
+
+$$
+e_\beta \tilde{M} - c - 2Mc|\alpha_{-, \beta}| - 4\tilde{M}^2 c e^{-e_\beta t} - 8M\tilde{M}^2 c |\alpha_{-, \beta}| e^{-e_\beta t}
+$$
+
+are positive for $\alpha_0$ sufficiently small and $t_0$ large enough. Thus for such values of $\alpha_0$ and $t_0$, there exists $c_M > 0$ such that for all $t \in [t_n^*, S_n]$,
+
+$$
+h'(t) \geq e_{\beta}h(t) - c_{M}h(t)^{2}.
+$$
+
+At this stage, we can deduce that for all $t$ in $[t_n^*, S_n]$, $h(t) \le 0$. Assume for the sake of contradiction that there exists $\tilde{t}_n^* \in [t_n^*, S_n]$ such that $h(\tilde{t}_n^*) > 0$. Then, we can define
+
+$$
+T := \sup\{t \in [\tilde{t}_n^*, S_n], h(t) > 0\}.
+$$
+
+We necessarily have $h(T) = 0$. Indeed, $h(T) < 0$ is excluded by continuity of $h$ in $T$ (on the left side) and by definition of the supremum; if $h(T) > 0$, then $T < S_n$ (since $h(S_n) = -\tilde{M}e^{-e_\beta S_n}$) which leads once more to a contradiction, using the continuity of $h$ in $T$ (on the right side) and the definition of $T$ as a supremum. It follows that $h'(T) \ge 0$; thus $h$ is non-decreasing in the neighborhood of $T$ and in particular, $h(t) \le 0$ on $[T-\eta, T]$ for some $\eta > 0$. This again contradicts the definition of $T$. Hence, for all $t$ in $[\tilde{t}_n^*, S_n]$, $h(t) \le 0$. Given that $-\alpha_{+, \beta}$ satisfies the same differential system as $\alpha_{+, \beta}$, we finally obtain (5.26).
+
+Now, we have for all $t \in [t_n^*, S_n]$,
+
+$$
+\sup_{t' \in [t, S_n]} |\alpha_{+, \beta}(t')| \le M \left( \sup_{t' \in [t, S_n]} \alpha_{-, \beta}^2(t') + e^{-e_{\beta}t} \right).
+$$
+
+By using Lemma 5.8 and even if it means reducing $\alpha_0$ and increasing $t_0$, we obtain that for all $t \in [t_n^*, S_n]$,
+
+$$
+|\alpha'_{-, \beta}(t) + e_{\beta}\alpha_{-, \beta}(t)| \le C \left( \sup_{t' \in [t, S_n]} \alpha^2_{-, \beta}(t') + e^{-2e_{\beta}t} \right) \le \frac{e_{\beta}}{10} |\alpha_{-, \beta}(t)| + C e^{-2e_{\beta}t}.
+$$
+
+We claim that this implies the estimates in Proposition 5.9. The preceding inequality rewrites as follows:
+
+$$
+\left| \frac{d}{dt} (e^{e_\beta t} \alpha_{-, \beta})(t) \right| \leq \frac{e_\beta}{10} e^{e_\beta t} \sup_{t' \in [t, S_n]} |\alpha_{-, \beta}(t')| + C e^{-e_\beta t}.
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_39.md b/samples/texts/7376768/page_39.md
new file mode 100644
index 0000000000000000000000000000000000000000..470c7590b523357b98f4237df8863d9a1431693c
--- /dev/null
+++ b/samples/texts/7376768/page_39.md
@@ -0,0 +1,53 @@
+For $t$ belonging to $[t_n^*, S_n]$, we obtain by integration on $[t, S_n]$:
+
+$$e^{e_{\beta}t} |\alpha_{-, \beta}(t)| \le Ce^{-e_{\beta}t} + \frac{e_{\beta}}{10} \int_t^{S_n} e^{e_{\beta}s} \sup_{t' \in [s, S_n]} |\alpha_{-, \beta}(t')| ds.$$
+
+Passing to the supremum and defining $y(t) := e^{e\beta t} \sup_{t' \in [t, S_n]} |\alpha_{-, \beta}(t')|$ on $[t_n^*, S_n]$, this leads to
+
+$$ (5.27) \qquad y(t) \le Ce^{-e\beta t} + \frac{e_\beta}{10} \int_t^{S_n} y(s) ds. $$
+
+Now, a standard Grönwall argument allows us to see that $y(t) \le Ce^{-e\beta t}$, which precisely provides the expected estimate of the parameter $\alpha_{-, \beta}$ in Proposition 5.9. Then, the similar estimate of the parameter $\alpha_{+, \beta}$ follows from the integration of the inequality
+
+$$ | \alpha'_{+, \beta}(t) - e_{\beta} \alpha_{+, \beta}(t) | \le Ce^{-2e_{\beta}t} $$
+
+and finally (5.24) follows from (5.22) and (5.23).
+
+For the reader's convenience, let us write explicitly the Grönwall argument. The function $\psi(t) := e^{\frac{e_\beta}{10}t} \int_t^{S_n} y(s) ds$ is $\mathscr{C}^1$ on $[t_n^*, S_n]$ and for all $t \in [t_n^*, S_n]$,
+
+$$ \psi'(t) = e^{\frac{e_\beta}{10}t} \left( -y(t) + \frac{e_\beta}{10} \int_t^{S_n} y(s) ds \right) \ge -Ce^{-\frac{9e_\beta}{10}t} $$
+
+by (5.27). Observing that $\psi(S_n) = 0$, it follows that $\psi(t) \le Ce^{-\frac{9e_\beta}{10}t}$. As a consequence,
+
+$$ \int_t^{S_n} y(s) ds \le Ce^{-e\beta t} $$
+
+and thus, by (5.27) again,
+
+$$ \forall t \in [t_n^*, S_n], \qquad y(t) \le Ce^{-e\beta t}. $$
+
+**5.2.4. Step 4: Improvement of the exponential decay rate.** The goal of this paragraph is to optimize the exponential decay rate in the estimate of $\|W\|_{H^1 \times L^2}$. We actually prove
+
+**Proposition 5.10.** *The following estimate holds for all $t \in [t_n^*, S_n]$:*
+
+$$ \|W(t)\|_{H^1 \times L^2} \le Ce^{-2e\beta t}. $$
+
+*Proof.* Let us consider this time the derivative of $\mathcal{F}_W$ (and not the functional $\mathcal{F}_W$ itself). From the definition (5.15) and the symmetric property of $\tilde{H}_\beta$ for $(\cdot, \cdot)$, we immediately obtain:
+
+$$ \frac{d}{dt} \mathcal{F}_W(t) = 2\langle \tilde{H}_\beta W, \partial_t W \rangle + \beta \int_{\mathbb{R}} \partial_x \tilde{Q}_\beta f''(\tilde{Q}_\beta) w_1^2 dx, $$
+
+where $w_1$ (resp. $w_2$) is the first (resp. the second) component of $W$. Replacing $\partial_t W$ by its expression obtained in (5.8) and noticing that
+
+$$ \begin{pmatrix} \beta \partial_x & Id \\ \partial_x^2 - Id + f'(\tilde{Q}_\beta) & \beta \partial_x \end{pmatrix} \tilde{Y}_{+, \beta} = J \tilde{H}_\beta \tilde{Y}_{+, \beta} = J \tilde{Z}_{+, \beta}, $$
+
+we have:
+
+$$ 
+\begin{aligned}
+& \langle \tilde{H}_{\beta} W, \partial_t W \rangle \\
+& = \left\langle \tilde{H}_{\beta} W, \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(\tilde{Q}_{\beta}) & 0 \end{pmatrix} W \right\rangle + \left\langle \tilde{H}_{\beta} W, \begin{pmatrix} 0 \\ f(u) - f(\tilde{Q}_{\beta}) - f'(\tilde{Q}_{\beta})(u - \tilde{Q}_{\beta}) \end{pmatrix} W \right\rangle \\
+& + \langle \tilde{H}_{\beta} W, J\tilde{Z}_{+, \beta} \rangle + e_{\beta} A e^{-e_{\beta}t} \langle W, \tilde{Z}_{+, \beta} \rangle + \dot{x}\langle \tilde{H}_{\beta} W, \partial_x \tilde{R}_{\beta} \rangle + A e^{-e\beta t} \dot{x}\langle W, \tilde{H} \partial_x \tilde{Y}_{+, \beta} \rangle.
+\end{aligned}
+$$
+
+Let us analyze each term appearing in the preceding decomposition. Since $\tilde{H}_\beta$ is self-adjoint, we infer:
+
+$$ (5.28) \qquad \langle \tilde{H}_\beta W, \partial_x \tilde{R}_\beta \rangle = \langle W, \tilde{H}_\beta \partial_x \tilde{R}_\beta \rangle = 0 $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_4.md b/samples/texts/7376768/page_4.md
new file mode 100644
index 0000000000000000000000000000000000000000..5ad2b1236ea368a72b6a350ff238003990111558
--- /dev/null
+++ b/samples/texts/7376768/page_4.md
@@ -0,0 +1,41 @@
+Let us emphasize that this is a much weaker assumption than exponential decay, which is natural in view of the convergence (1.2). We also underline that we obtain for (NLKG) an effective rate. As a comparison, in the context of the nonlinear Schrödinger equations, uniqueness of a multi-soliton (in the stable case) was shown in a class of solutions with convergence faster than any power of $1/t$, see [6]: this is still much weaker than the exponential convergence, but not well behaved as what is proven here for (NLKG). We nevertheless conjecture that the classification should hold in the most general case when no decay rate is assumed.
+
+Here is the precise statement.
+
+**Theorem 1.4.** Under the assumptions of Theorem 1.2 and keeping the same notations, if $u$ is a solution of (NLKG) such that
+
+$$ (1.3) \qquad \left\| U(t) - \sum_{i=1}^{N} R_{\beta_i} x_i(t) \right\|_{H^1 \times L^2} = O\left(\frac{1}{t^\alpha}\right) \quad \text{as } t \to +\infty, $$
+
+where $U = \begin{pmatrix} u \\ \partial_t u \end{pmatrix}$ and where $\alpha > 3$, then there exist $A_1, \dots, A_N \in \mathbb{R}$ and $t_0 \in \mathbb{R}$ such that for all $t \ge t_0$, $U(t) = \Phi_{A_1, \dots, A_N}(t)$.
+
+**Remark 1.5.** Notice that Theorem 1.2 and Theorem 1.4 apply in dimension $d \le 5$ only. Indeed, assuming (**H'1**) and $f$ of class $\mathcal{C}^2$ forces to have $p > 2$, hence $2 < \frac{d+2}{d-2}$ if $d \ge 3$.
+
+In the case where only one soliton is considered, one can moreover improve the preceding Theorem by completely characterizing solutions which converge to a soliton in large time.
+
+**Theorem 1.6.** Let $\beta \in \mathbb{R}^d$, $|\beta| < 1$ and assume that $f$ is of class $\mathcal{C}^2$. There exist $e_\beta > 0$, $Y_{+\beta} \in \mathcal{C}(\mathbb{R}, H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d))$, and a one-parameter family $(u^A)_{A \in \mathbb{R}}$ of solutions of (NLKG) such that for all $A \in \mathbb{R}$, there exists $t_0 = t_0(A) \in \mathbb{R}$ such that for all $t \ge t_0$
+
+$$ (1.4) \qquad \|U^A(t) - R_\beta(t) - Ae^{-e_\beta t} Y_{+\beta}(t)\|_{H^1 \times L^2} \le Ce^{-2e_\beta t}, $$
+
+where $U^A := \begin{pmatrix} u^A \\ \partial_t u^A \end{pmatrix}$. In addition, if $A \neq A'$, then $u^A \neq u^{A'}$.
+
+Moreover, if $u$ is a solution of (NLKG) such that
+
+$$ (1.5) \qquad \|U(t) - R_{\beta}(t)\|_{H^1 \times L^2} \to 0 \quad \text{as } t \to +\infty, $$
+
+where $U = \begin{pmatrix} u \\ \partial_t u \end{pmatrix}$, then there exist $A \in \mathbb{R}$ and $t_0 \in \mathbb{R}$ such that for all $t \ge t_0$, $U(t) = U^A(t)$.
+
+**Remark 1.7.** The parameter $e_\beta$ (which depends on $\beta$) and the function $Y_{+\beta}$ are defined in Proposition 2.1; they are intimately related to the spectral theory dealing with the flow around $R_\beta$.
+
+It is interesting to remark that there are only three special solutions among the elements of the preceding family $(U^A)_{A \in \mathbb{R}}$, up to translations in time and in space. This is the object of the following
+
+**Corollary 1.8.** Consider the family of solutions $(U^A)_{A \in \mathbb{R}}$ defined in Theorem 1.6.
+
+(1) If $A > 0$, there exists $t_A \in \mathbb{R}$ such that for all possible $t$, $U^A(t) = U^1(t+t_A, \cdot +\beta t_A)$.
+
+(2) If $A < 0$, there exists $t_A \in \mathbb{R}$ such that for all possible $t$, $U^A(t) = U^{-1}(t+t_A, \cdot +\beta t_A)$.
+
+(3) For all $t \in \mathbb{R}$, $U^0(t) = R_\beta(t)$.
+
+**Remark 1.9.** Let us observe that Remark 1.5 is valid for Theorem 1.6 and Corollary 1.8 too.
+
+Theorem 1.6 provides the behavior of the solutions converging to solitons at the order $O(e^{-2e_\beta t})$ in $H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$. The instability direction due to the existence of one negative eigenvalue (related to $e_\beta$) for the linearized operator around $Q_\beta$ yields infinitely many possibilities (described by the real line) to perturb the soliton $Q_\beta$ and to retrieve another solution of (NLKG) which however remains exponentially close in time (with decay rate $e_\beta$) to $Q_\beta$. Let us emphasize that this phenomenon appears quite naturally in the study of unstable solitons. In a stable mode such as for the $L^2$-subcritical
\ No newline at end of file
diff --git a/samples/texts/7376768/page_40.md b/samples/texts/7376768/page_40.md
new file mode 100644
index 0000000000000000000000000000000000000000..e922a03a3a5dbde2ab40913043dcd168ddb1cda5
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+++ b/samples/texts/7376768/page_40.md
@@ -0,0 +1,73 @@
+and
+
+$$
+(5.29) \quad \langle \tilde{H}_{\beta} W, J \tilde{Z}_{+\beta} \rangle = -A e^{-e_{\beta} t} \langle W, \mathcal{H}_{\beta} \tilde{Z}_{+\beta} \rangle = -e_{\beta} A e^{-e_{\beta} t} \langle W, \tilde{Z}_{+\beta} \rangle.
+$$
+
+Moreover,
+
+$$
+(5.30) \quad \left| \left\langle \tilde{H}_{\beta} W, \begin{pmatrix} 0 & 0 \\ f(u) - f(\tilde{Q}_{\beta}) & -f'(\tilde{Q}_{\beta})(u - \tilde{Q}_{\beta}) \end{pmatrix} \right\rangle \right| \le C \|W\|_{H^{1}\times L^2} (\|W\|_{H^{1}\times L^2}^2 + e^{-2e_{\beta}t}).
+$$
+
+We notice that (5.7) implies
+
+$$
+(5.31) \quad |A e^{-e_{\beta} t} \dot{x} \langle W, \tilde{H}_{\beta} \partial_x \tilde{Y}_{+\beta} \rangle| \le C e^{-e_{\beta} t} \|W(t)\|_{H^{1}\times L^2}^2 + C e^{-3e_{\beta} t} \|W(t)\|_{H^{1}\times L^2}.
+$$
+
+Defining $\tilde{T} := -\partial_x^2 + Id - f'(\tilde{Q}_\beta)$, it remains us to examine
+
+$$
+\begin{align*}
+& \left\langle \tilde{H}_{\beta} W, \begin{pmatrix} 0 & Id \\ -\tilde{T} & 0 \end{pmatrix} W \right\rangle \\
+&= \left\langle \begin{pmatrix} \tilde{T} & 0 \\ 0 & Id \end{pmatrix} W, \begin{pmatrix} 0 & Id \\ -\tilde{T} & 0 \end{pmatrix} W \right\rangle + \left\langle \begin{pmatrix} 0 & -\beta \partial_x \\ \beta \partial_x & 0 \end{pmatrix} W, \begin{pmatrix} 0 & Id \\ -\tilde{T} & 0 \end{pmatrix} W \right\rangle.
+\end{align*}
+$$
+
+On the one hand, we observe that
+
+$$
+\left\langle \begin{pmatrix} \tilde{T} & 0 \\ 0 & Id \end{pmatrix} W, \begin{pmatrix} 0 & Id \\ -\tilde{T} & 0 \end{pmatrix} W \right\rangle = 0
+$$
+
+and on the other
+
+$$
+\begin{align*}
+\left\langle \begin{pmatrix} 0 & -\beta \partial_x \\ \beta \partial_x & 0 \end{pmatrix} W, \begin{pmatrix} 0 & Id \\ -\tilde{T} & 0 \end{pmatrix} W \right\rangle &= \int_{\mathbb{R}} \{-\beta \partial_x w_2 w_2 + \beta \partial_x w_1 (-\tilde{T} w_1)\} dx \\
+&= \beta \int_{\mathbb{R}} \partial_x w_1 f'(\tilde{Q}_\beta) w_1 \\
+&= -\frac{\beta}{2} \int_{\mathbb{R}} \partial_x \tilde{Q}_\beta f''(\tilde{Q}_\beta) w_1^2 dx.
+\end{align*}
+$$
+
+Gathering the above lines, we deduce
+
+$$
+\left|\frac{dF_W}{dt}\right| \le C \left(\|W\|_{H^1\times L^2}^3 + e^{-2e_\beta t} \|W\|_{H^1\times L^2} + e^{-e_\beta t} \|W\|_{H^1\times L^2}^2\right).
+$$
+
+Using the estimate of $\|W\|_{H^1\times L^2}$ obtained in Proposition 5.9, we infer:
+
+$$
+(5.32) \qquad \left| \frac{d F_W}{dt} \right| \le C \left( e^{-2e_{\beta t}} \|W\|_{H^1 \times L^2} + e^{-e_{\beta t}} \|W\|_{H^1 \times L^2}^2 \right).
+$$
+
+By integration of (5.32), we obtain for all $t \in [t^*, S_n]$:
+
+$$
+(5.33) \quad |\mathcal{F}_W(t)| &\le C \sup_{\tau \in [t, S_n]} \|W(\tau)\|_{H^{1}\times L^2} \int_t^{S_n} e^{-2e_\beta \tau} d\tau + C \sup_{\tau \in [t, S_n]} \|W(\tau)\|_{H^{1}\times L^2}^2 \int_t^{S_n} e^{-e_\beta \tau} d\tau \\
+&\le C \sup_{\tau \in [t, S_n]} \|W(\tau)\|_{H^{1}\times L^2} e^{-2e_\beta \tau} + C \sup_{\tau \in [t, S_n]} \|W(\tau)\|_{H^{1}\times L^2}^2 e^{-e_\beta \tau}.
+$$
+
+At this stage, using once again the coercivity property provided by Proposition 2.2 and the estimates on $\alpha_{\pm,\beta}$ given in Proposition 5.9, and taking $t_0$ large enough so that $e^{-e_\beta t} < \frac{1}{2}$ for all $t \ge t_0$, we have:
+
+$$
+\sup_{\tau \in [t, S_n]} \|W(\tau)\|_{H^{1}\times L^2}^2 \le C \left( \sup_{\tau \in [t, S_n]} \|W(\tau)\|_{H^{1}\times L^2} e^{-2e_\beta t} + e^{-4e_\beta t} \right).
+$$
+
+We now deduce the existence of $C > 0$ such that for all $n$, for all $t \in [t_n^*, S_n]$,
+
+$$
+\|W(t)\|_{H^{1}\times L^2}^2 \le C e^{-2e_{\beta}t}.
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_41.md b/samples/texts/7376768/page_41.md
new file mode 100644
index 0000000000000000000000000000000000000000..08b6679793d1cab003a1a20aae223408d162cca4
--- /dev/null
+++ b/samples/texts/7376768/page_41.md
@@ -0,0 +1,76 @@
+Now, Proposition 5.2 is obtained as a corollary of Proposition 5.10. The triangular inequality implies
+
+$$
+\|U_n(t) - R_\beta(t) - A e^{-e_{\beta} t} Y_{+, \beta}(t)\|_{H^1 \times L^2} \le \|W_n(t)\|_{H^1 \times L^2} + C|x(t)| \\
+\text{and since } x(S_n) = 0, \text{ the result follows from the integration of}
+$$
+
+$$
+|\dot{x}(t)| \leq C (\|W_n(t)\|_{H^1 \times L^2} + e^{-2e_{\beta}t}) \leq Ce^{-2e_{\beta}t}.
+$$
+
+**6. CLASSIFICATION OF THE ASYMPTOTIC SOLITON-LIKE SOLUTIONS**
+
+Let us consider a solution $u$ of (NLKG), denote $U = \left( \frac{u}{\partial_t u} \right)$, and assume that
+
+$$
+\|U(t) - R_{\beta}(t)\|_{H^{1} \times L^{2}} \to 0, \quad \text{as } t \to +\infty.
+$$
+
+We want to show that $U$ equals to $U^A$, for some $A \in \mathbb{R}$. In this section again, we consider the one-dimensional case.
+
+**6.1. Modulation of U and coercivity property.**
+
+**Lemma 6.1.** There exist $T \in \mathbb{R}$, $C > 0$, and $x : [T, +\infty) \to \mathbb{R}$ of class $\mathcal{C}^1$ such that, denoting
+
+$$
+\tilde{R}_{\beta}(t,x) := \begin{pmatrix} Q_{\beta}(t, x-x(t)) \\ \partial_t Q_{\beta}(t, x-x(t)) \end{pmatrix} \quad \text{and} \quad E := U - \tilde{R}_{\beta},
+$$
+
+we have ||*E*(t)||<sub>*H*<sup>1</sup>*×*L*<sup>2</sup></sub> → 0 as *t* → +∞, *x*(*t*) → 0 as *t* → +∞, and for all *t* ≥ *T*,
+
+$$
+(6.1) \qquad \langle E(t), \partial_x \tilde{R}_\beta(t) \rangle = 0
+$$
+
+and
+
+$$
+(6.2) \qquad |\dot{x}(t)| \le C \|E(t)\|_{H^1 \times L^2}.
+$$
+
+*Proof.* Lemma 6.1 is proved similarly as Lemma 5.3. Defining
+
+$$
+\tilde{Q}_{\beta}(t, x) := Q_{\beta}(t, x - x(t)),
+$$
+
+the equation satisfied by $E$ reads:
+
+$$
+(6.3) \qquad \partial_t E = \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(Q_\beta) & 0 \end{pmatrix} E + \dot{x} \partial_x \tilde{R}_\beta \\
+\qquad\qquad + \begin{pmatrix} 0 \\ f(u) - f(Q_\beta) - f'(Q_\beta)(u - Q_\beta) \end{pmatrix}.
+\hfill \square
+$$
+
+Let us denote $\varepsilon_1(t,x) := u(t,x) - Q_\beta(t,x-x(t))$ and $\varepsilon_2(t,x) := \partial_t u(t,x) - \partial_t Q_\beta(t,x-x(t))$ so that $E(t,x) := \begin{pmatrix} \varepsilon_1(t,x) \\ \varepsilon_2(t,x) \end{pmatrix}$.
+
+Then, introduce the functional $\mathcal{F}_E$ defined as follows: for all $t \ge T$,
+
+$$
+(6.4) \qquad \mathcal{F}_E(t) := \int_{\mathbb{R}} \{\epsilon_2^2 + (\partial_x \epsilon_1)^2 + \epsilon_1^2 - f'(\tilde{Q}_\beta)\epsilon_1^2 + 2\beta\epsilon_2\partial_x\epsilon_1\} (t,x) dx.
+$$
+
+With analogous notations as that employed in the previous section, we denote
+
+$$
+\tilde{Z}_{\pm,\beta}(t,x) := Z_{\pm,\beta}(t,x-x(t)).
+$$
+
+**Proposition 6.2.** There exists $\mu > 0$ such that for all $t \ge T$:
+
+$$
+(6.5) \qquad \mathcal{F}_E(t) \geq \mu \|E(t)\|_{H^1 \times L^2}^2 - \frac{1}{\mu} [\langle E(t), \tilde{Z}_{-\beta}(t) \rangle^2 + \langle E(t), \tilde{Z}_{+\beta}(t) \rangle^2].
+$$
+
+*Proof.* We observe that $\mathcal{F}_E(t) = \langle \tilde{H}_\beta E(t), E(t) \rangle$ so that Proposition 6.2 follows from Proposition 2.2 and from (6.1). $\square$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_42.md b/samples/texts/7376768/page_42.md
new file mode 100644
index 0000000000000000000000000000000000000000..4cf48f484b2ceabb851f046035d617a2a4d8c4b7
--- /dev/null
+++ b/samples/texts/7376768/page_42.md
@@ -0,0 +1,51 @@
+**6.2. Exponential control of $\|E(t)\|_{H^1 \times L^2}$**. We improve the control of $\|E(t)\|_{H^1 \times L^2}$, proceeding as in section 5.
+
+**6.2.1. Control of the functional $\mathcal{F}_E$.**
+
+**Proposition 6.3.** *We have*
+
+$$ (6.6) \qquad \mathcal{F}_E(t) = O\left(\|E(t)\|_{H^1\times L^2}^3\right). $$
+
+*Proof.* Let us take again the proof of Lemma 5.6 (replacing $A$ in that proof by $0$ here). We obtain in the same way the existence of $K \in \mathbb{R}$ such that
+
+$$ \mathcal{F}_E(t) = K + 2 \int_{\mathbb{R}} \left\{ F(u) - F(\tilde{Q}_\beta) - f(\tilde{Q}_\beta) \varepsilon_1 - \frac{1}{2} f'(\tilde{Q}_\beta) \varepsilon_1^2 \right\} (t,x) \, dx $$
+
+By Taylor inequality,
+
+$$ \int_{\mathbb{R}} \left\{ F(u) - F(\tilde{Q}_{\beta}) - f(\tilde{Q}_{\beta})\varepsilon_{1} - \frac{1}{2}f'(\tilde{Q}_{\beta})\varepsilon_{1}^{2} \right\} (t,x) \, dx = O\left(\|E(t)\|_{H^1 \times L^2}^3\right). $$
+
+It follows that
+
+$$ \mathcal{F}_E(t) = K + O\left(\|E(t)\|_{H^1 \times L^2}^3\right). $$
+
+On the other hand, $\mathcal{F}_E(t) = O(\|E(t)\|_{H^1 \times L^2}^2)$; thus $K=0$ and Proposition 6.3 is proved. $\square$
+
+**6.2.2. Control of the unstable directions.** Let us denote
+
+$$ \alpha_{\pm,\beta}(t) := \langle E(t), \bar{Z}_{\pm,\beta}(t) \rangle. $$
+
+**Lemma 6.4.** *We have*
+
+$$ \left|\frac{d\alpha_{+\beta}}{dt} - e_\beta \alpha_{+\beta}\right| + \left|\frac{d\alpha_{-\beta}}{dt} + e_\beta \alpha_{-\beta}\right| = O\left(\|E(t)\|_{H^{1}\times L^2}^2\right). $$
+
+*Proof.* The proof follows the same lines as that of Lemma 5.5 (by taking $A=0$). We compute
+
+$$ 
+\begin{align*}
+\frac{d\alpha_{\pm,\beta}}{dt} &= \langle \partial_t E, \tilde{Z}_{\pm,\beta} \rangle + \langle E, \partial_t \tilde{Z}_{\pm,\beta} \rangle - \dot{x} \langle E(t), \partial_x \tilde{Z}_{\pm,\beta} \rangle \\
+&= \left\langle E, \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(Q_\beta) & 0 \end{pmatrix} \tilde{Z}_{\pm,\beta} \right\rangle + O\left(\|E\|_{H^{1}\times L^2}^2\right) \\
+&\quad + \dot{x} \langle \partial_x \tilde{R}_\beta, \tilde{Z}_{\pm,\beta} \rangle + \left\langle E, \begin{pmatrix} -\beta \partial_x & 0 \\ 0 & -\beta \partial_x \end{pmatrix} \tilde{Z}_{\pm,\beta} \right\rangle \\
+&= \langle E, \mathcal{H}_\beta \tilde{Z}_{\pm,\beta} \rangle + O\left(\|E\|_{H^{1}\times L^2}^2\right) \\
+&= \pm e_\beta \alpha_{\pm,\beta} + O\left(\|E\|_{H^{1}\times L^2}^2\right),
+\end{align*}
+$$
+
+which precisely yields Lemma 6.4. $\square$
+
+**Proposition 6.5.** There exists $C \ge 0$ such that for $t$ large enough,
+
+$$ (6.7) \qquad |\alpha_{-\beta}(t)| \le Ce^{-e_\beta t} \quad \text{and} \quad |\alpha_{+,\beta}(t)| \le Ce^{-2e_\beta t}. $$
+
+*Proof.* For $t$ large enough, we obtain as a consequence of Lemma 6.2 and Proposition 6.3:
+
+$$ (6.8) \qquad \|E(t)\|_{H^1 \times L^2}^2 \le C (\alpha_{+,\beta}^2 + \alpha_{-,\beta}^2). $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_43.md b/samples/texts/7376768/page_43.md
new file mode 100644
index 0000000000000000000000000000000000000000..f21d8b5e2781a1f87ad631519acaa6c579980f4f
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@@ -0,0 +1,51 @@
+Now, we deduce from Lemma 6.4 the following differential system, for some constant $C \ge 0$ and for all $t$ large enough:
+
+$$ (6.9) \qquad \left\{ \begin{aligned} \left| \frac{d\alpha_{+,\beta}}{dt} - e_\beta \alpha_{+,\beta} \right| &\le C (\alpha_{+,\beta}^2 + \alpha_{-,\beta}^2) \\ \left| \frac{d\alpha_{-,\beta}}{dt} + e_\beta \alpha_{-,\beta} \right| &\le C (\alpha_{+,\beta}^2 + \alpha_{-,\beta}^2). \end{aligned} \right. $$
+
+Then, the argument exposed in [3, paragraph 4.4.2] shows that
+
+$$ |\alpha_{+, \beta}(t)| \le C \alpha_{-, \beta}^2(t) $$
+
+and allows us to conclude to (6.7). □
+
+**Proposition 6.6.** There exists $C \ge 0$ such that for $t$ large enough,
+
+$$ (6.10) \qquad \|E(t)\|_{H^1 \times L^2} \le Ce^{-e_\beta t} $$
+
+*Proof.* This is an immediate consequence of (6.7) and (6.8). □
+
+### 6.2.3. Identification of U with $U^A$ for some $A \in \mathbb{R}$.
+
+**Lemma 6.7.** There exist $A \in \mathbb{R}$ and $C \ge 0$ such that for $t$ sufficiently large,
+
+$$ (6.11) \qquad |e^{e_\beta t} \alpha_{-, \beta}(t) - A| \le Ce^{-e_\beta t}. $$
+
+*Proof.* We have from (6.9):
+
+$$ (6.12) \qquad \left| \frac{d}{dt} (e^{e_\beta t} \alpha_{-, \beta}(t)) \right| \le Ce^{-e_\beta t}. $$
+
+Thus the derivative of $t \mapsto e^{e_\beta t} \alpha_{-, \beta}(t)$ is integrable in the neighborhood of $+\infty$. Hence, there exists $A \in \mathbb{R}$ such that $e^{e_\beta t} \alpha_{-, \beta}(t) \to A$ as $t \to +\infty$. We finally obtain Lemma 6.7 by integration of (6.12). □
+
+**Lemma 6.8.** There exists $x_\infty \in \mathbb{R}$ such that for $t$ sufficiently large,
+
+$$ (6.13) \qquad |x(t) - x_\infty| \le Ce^{-e_\beta t}. $$
+
+*Proof.* As (6.11) was a consequence of (6.12), (6.13) is obtained by means of the estimate
+
+$$ |x'(t)| \le C \|E(t)\|_{H^1 \times L^2} \le Ce^{-e_\beta t}. $$
+
+□
+
+Now let $V(t,x) := U(t,x) - U^A(t,x-x_\infty)$ where $A$ and $x_\infty$ are defined in Lemma 6.7 and Lemma 6.8 respectively.
+
+**Lemma 6.9.** There exists $C \ge 0$ such that for $t$ large enough, $\|V(t)\|_{H^1 \times L^2} \le Ce^{-e_\beta t}$.
+
+*Proof.* Define $V^A(t) := U^A(t) - R_\beta(t) - Ae^{-e_\beta t}Y_{+,\beta}(t)$ and decompose $V(t,x)$ as follows:
+
+$$ (6.14) \qquad V(t,x) = E(t,x) + \tilde{R}_\beta(t,x) - R_\beta(t, x-x_\infty) - Ae^{-e_\beta t} Y_{+,\beta}(t, x-x_\infty) - V^A(t, x-x_\infty). $$
+
+Now, we have
+
+$$ (6.15) \qquad \tilde{R}_{\beta}(t,x) - R_{\beta}(t,x-x_{\infty}) = (x(t)-x_{\infty})\partial_x R_{\beta}(t,x-x_{\infty}) + O(|x(t)-x_{\infty}|^2). $$
+
+Hence, Lemma 6.9 is obtained as a consequence of Proposition 6.6, Lemma 6.8, and the estimate of $\|V^A(t)\|_{H^1 \times L^2}$ given by (1.4), that is $\|V^A(t)\|_{H^1 \times L^2} = O(e^{-2e_\beta t})$. □
\ No newline at end of file
diff --git a/samples/texts/7376768/page_44.md b/samples/texts/7376768/page_44.md
new file mode 100644
index 0000000000000000000000000000000000000000..c0e829b1fd1f2024ec0c5a223a8251715efc9d95
--- /dev/null
+++ b/samples/texts/7376768/page_44.md
@@ -0,0 +1,76 @@
+Let us decompose $V(t,x)$ as follows:
+
+$$
+(6.16) \quad V(t,x) = \alpha_{+, \beta}^{A}(t)Y_{-, \beta}(t, x - x_{\infty}) + \alpha_{-, \beta}^{A}(t)Y_{+, \beta}(t, x - x_{\infty}) + \lambda(t)\partial_x R_\beta(t, x) + V_{\perp}(t, x),
+$$
+
+where $\alpha_{+, \beta}^A(t) = \langle V(t), Z_{+, \beta}(t, \cdot - x_{\infty}) \rangle$, $\alpha_{-, \beta}^A(t) = \langle V(t), Z_{-, \beta}(t, \cdot - x_{\infty}) \rangle$ and
+
+$$
+\lambda(t) := \frac{1}{\|\partial_x R_\beta\|_{H^1 \times L^2}} \langle V(t) - \alpha_{+, \beta}^A(t) Y_{-, \beta}(t, \cdot -x_{\infty}) - \alpha_{-, \beta}^A(t) Y_{+, \beta}(t, x-x_{\infty}), \partial_x R_\beta(t) \rangle.
+$$
+
+Then, we have
+
+$$
+\langle V_{\perp}(t), \partial_x R_{\beta}(t) \rangle = \langle V_{\perp}(t), Z_{+, \beta}(t, \cdot -x_{\infty}) \rangle = \langle V_{\perp}(t), Z_{-, \beta}(t, \cdot -x_{\infty}) \rangle = 0.
+$$
+
+Thus, by Proposition 2.2, there exists $C \ge 0$ such that
+
+$$
+(6.17) \qquad \|V_{\perp}(t)\|_{H^1 \times L^2}^2 \le C \langle H_{\beta}V_{\perp}(t), V_{\perp}(t) \rangle.
+$$
+
+We claim
+
+**Lemma 6.10.** The following assertions hold:
+
+$$
+(1) \quad \langle H_\beta V(t), V(t) \rangle = \langle H_\beta V_\perp(t), V_\perp(t) \rangle + 2\alpha_{+, \beta}^A(t)\alpha_{-, \beta}^A(t).
+$$
+
+$$
+(2) \quad \frac{d}{dt} \langle H_{\beta}V(t), V(t) \rangle = O\left(e^{-e_{\beta}t} \|V(t)\|_{H^{1} \times L^{2}}^{2}\right).
+$$
+
+(3) $|\alpha_{+, \beta}^A(t)| \le Ce^{-e_{\beta}t} \|V(t)\|_{H^1 \times L^2}$ and $|\alpha_{-, \beta}^A(t)| \le Ce^{-e_{\beta}t} \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt'$
+
+$$
+(4) \quad \lambda'(t) = O\left(e^{-e_{\beta}t} \|V(t)\|_{H^1 \times L^2} + e^{-e_{\beta}t} \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' + \|V_{\perp}(t)\|_{H^1 \times L^2}\right).
+$$
+
+*Proof.* The first assertion in Lemma 6.10 is obtained by the decomposition of *V* in terms of *V*<sub>⊥</sub> (6.16) and by means of the following properties (see Proposition 2.1):
+
+$$
+\langle Z_{\pm, \beta}, Y_{\pm, \beta} \rangle = 0, \quad \langle Z_{\pm, \beta}, Y_{\mp, \beta} \rangle = 1, \quad \text{and} \quad H_{\beta}(\partial_x R_{\beta}) = 0.
+$$
+
+Let us now prove assertion (2). Let $v := u - u^A$; recall that $U = \begin{pmatrix} u \\ \partial_t u \end{pmatrix}$ and $U^A = \begin{pmatrix} u^A \\ \partial_t u^A \end{pmatrix}$.
+
+We observe that
+
+$$
+\langle H_\beta V(t), V(t) \rangle = \int_{\mathbb{R}} \{( (\partial_t v)^2 + (\partial_x v)^2 + v^2 - f'(Q_\beta)v^2 + 2\beta \partial_t v \partial_x v )\} (t,x) dx.
+$$
+
+Using the fact that $u$ and $u^A$ satisfy (NLKG), we obtain
+
+$$
+\begin{align*}
+& \frac{d}{dt} \langle H_\beta V(t), V(t) \rangle \\
+&= -2 \int_{\mathbb{R}} [f(u) - f(u^A) - f'(Q_\beta)v] \partial_t v \, dx \\
+&\quad - 2\beta \int_{\mathbb{R}} \partial_x v (f(u) - f(u^A)) \, dx + \beta \int_{\mathbb{R}} \partial_x Q_\beta f''(Q_\beta) v^2 \, dx \\
+&= O(\|V(t)\|_{H^{1\times L^2}}^3) - 2\beta \int_{\mathbb{R}} (\nu f'(u^A) + O(v^2)) \, dx + \beta \int_{\mathbb{R}} \partial_x Q_\beta f''(Q_\beta) v^2 \, dx \\
+&= O(\|V(t)\|_{H^{1\times L^2}}^3) - 2\beta \int_{\mathbb{R}} (\nu f'(Q_\beta) + O(v^2 + e^{-e_\beta t} v)) \, dx + \beta \int_{\mathbb{R}} \partial_x Q_\beta f''(Q_\beta) v^2 \, dx \\
+&= O(\|V(t)\|_{H^{1\times L^2}}^3 + e^{-e_\beta t} \|V(t)\|_{H^{1\times L^2}}^2).
+\end{align*}
+$$
+
+By Lemma 6.9, assertion (2) is thus proved.
+
+In order to prove (3) and (4), let us write
+
+$$
+(6.18) \qquad \partial_t V = \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(Q_\beta) & 0 \end{pmatrix} V + \begin{pmatrix} 0 \\ f(u) - f(u^A) \end{pmatrix}.
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_45.md b/samples/texts/7376768/page_45.md
new file mode 100644
index 0000000000000000000000000000000000000000..ea806174fa36c521c2012f72fb7ebf4977ab7d9d
--- /dev/null
+++ b/samples/texts/7376768/page_45.md
@@ -0,0 +1,55 @@
+Then,
+
+$$
+\begin{align*}
+\frac{d}{dt} \alpha_{\pm, \beta}^{A}(t) &= \langle \partial_t V, Z_{\pm, \beta}(t, \cdot -x_{\infty}) \rangle + \langle V, \partial_t Z_{\pm, \beta}(t, \cdot -x_{\infty}) \rangle \\
+&= \left\langle V, \begin{pmatrix} -\beta \partial_x & \partial_x^2 - Id + f'(Q_\beta) \\ Id & -\beta \partial_x \end{pmatrix} Z_{\pm, \beta}(t, \cdot -x_{\infty}) \right\rangle + O(e^{-e_\beta t} \|V(t)\|_{H^1 \times L^2}) \\
+&= \pm e_\beta \alpha_{\pm, \beta}^{A}(t) + O(e^{-e_\beta t} \|V(t)\|_{H^1 \times L^2})
+\end{align*}
+$$
+
+Note that we have used that $\partial_t Z_{\pm,\beta}(t, \cdot -x_{\infty}) = -\beta \partial_x Z_{\pm,\beta}(t, \cdot -x_{\infty})$. We then deduce, in a similar way as for Proposition 6.5:
+
+$$|\alpha_{+, \beta}^{A}(t)| \leq C e^{-e_{\beta} t} \|V(t)\|_{H^{1} \times L^{2}}$$
+
+because $t \mapsto e^{-2e_{\beta}t} ||V(t)||_{H^{1}\times L^{2}}$ is integrable in $+\infty$ and
+
+$$|e^{-e_{\beta}t} \alpha_{+, \beta}^{A}(t)| \leq e^{-e_{\beta}t} \|V(t)\|_{H^{1}\times L^{2}} \underset{t \to +\infty}{\longrightarrow} 0.$$
+
+In addition we deduce
+
+$$\left| \frac{d}{dt} (e^{e_{\beta}t} \alpha_{-, \beta}^{A}(t)) \right| \leq C \|V(t)\|_{H^{1} \times L^{2}}.$$
+
+Arguing similarly as for $\alpha_{+, \beta}^A(t)$, we then obtain $|\alpha_{-, \beta}^A(t)| \le Ce^{-e_\beta t} \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt'$. This is due to the fact that $t \mapsto \|V(t)\|_{H^1 \times L^2}$ is integrable in $+\infty$ and the fact that, by (6.14), (6.15), and (6.11),
+
+$$
+\begin{align*}
+\alpha_{-, \beta}^{A}(t) &= \alpha_{-, \beta}(t) - A e^{-e_{\beta} t} \langle Y_{+, \beta}(t, \cdot -x_{\infty}), Z_{-, \beta}(t, \cdot -x_{\infty}) \rangle + O(e^{-2e_{\beta} t}) \\
+&= O(e^{-2e_{\beta} t}),
+\end{align*}
+$$
+
+which explains that $e^{e_\beta t} \alpha_{-, \beta}^A(t) \to 0$ as $t \to +\infty$.
+
+It remains to prove (4). By definition of $\lambda(t)$ and the decomposition (6.16) of $V$,
+
+$$
+\begin{align*}
+\lambda'(t) &= \frac{1}{\|\partial_x R_\beta\|_{H^1\times L^2}^2} \left\langle \partial_t V - \alpha_{+, \beta}^A \partial_t Y_{-, \beta}(t, \cdot -x_\infty) - \alpha_{-, \beta}^A \partial_t Y_{+, \beta}(t, \cdot -x_\infty), \partial_x R_\beta \right\rangle \\
+&\quad - \frac{1}{\|\partial_x R_\beta\|_{H^1\times L^2}^2} \left\langle (\alpha_{+, \beta}^A)' Y_{-, \beta}(t, \cdot -x_\infty) + (\alpha_{-, \beta}^A)' Y_{+, \beta}(t, \cdot -x_\infty), \partial_x R_\beta \right\rangle \\
+&\quad + \frac{1}{\|\partial_x R_\beta\|_{H^1\times L^2}^2} \langle V_\perp, \partial_{x_t} R_\beta \rangle \\
+&= \frac{1}{\|\partial_x R_\beta\|_{H^1\times L^2}^2} \left\langle \partial_t V - \alpha_{+, \beta}^A \partial_t Y_{-, \beta}(t, \cdot -x_\infty) - \alpha_{-, \beta}^A \partial_t Y_{+, \beta}(t, \cdot -x_\infty), \partial_x R_\beta \right\rangle \\
+&\quad + \frac{1}{\|\partial_x R_\beta\|_{H^1\times L^2}^2} \langle V_\perp, \partial_{x_t} R_\beta \rangle.
+\end{align*}
+$$
+
+since $\langle Y_{\pm,\beta}, \partial_x R_\beta \rangle = 0$ (we refer to Proposition 2.1). Hence,
+
+$$
+\begin{align*}
+\lambda'(t) = {}& \left( \left( \begin{array}{cc} 0 & Id \\ \partial_x^2 - Id + f'(Q_\beta) & 0 \end{array} \right) V, \partial_x R_\beta \right) + O(e^{-e_\beta t} \|V\|_{H^1\times L^2}) + \frac{1}{\|\partial_x R_\beta\|_{H^1\times L^2}^2} \langle V_\perp, \partial_{x_t} R_\beta \rangle \\
+& + \frac{1}{\|\partial_x R_\beta\|_{H^1\times L^2}^2} \left\langle \beta \alpha_{+, \beta}^A \partial_x Y_{-, \beta}(t, -x_\infty) + \beta \alpha_{-, \beta}^A \partial_x Y_{+, \beta}(t, -x_\infty), \partial_x R_\beta \right\rangle.
+\end{align*}
+$$
+
+Now, using that $\partial_x R_\beta = JZ_0$ and ${}^t J (\begin{matrix} \beta & Id \\ -Id + f'(Q_\beta) & 0 \end{matrix}) = H_\beta$, we have:
\ No newline at end of file
diff --git a/samples/texts/7376768/page_46.md b/samples/texts/7376768/page_46.md
new file mode 100644
index 0000000000000000000000000000000000000000..72ab67006fc0e743c3824278a50e6a50f19437c9
--- /dev/null
+++ b/samples/texts/7376768/page_46.md
@@ -0,0 +1,62 @@
+$$
+\left\langle \left( \begin{array}{cc} 0 & Id \\ \partial_x^2 - Id + f'(Q_\beta) & 0 \end{array} \right) V, \partial_x R_\beta \right\rangle \\
+= \left\langle \left( \begin{array}{cc} \beta \partial_x & Id \\ \partial_x^2 - Id + f'(Q_\beta) & \beta \partial_x \end{array} \right) V, JZ_0 \right\rangle - \left\langle \left( \begin{array}{cc} \beta \partial_x & 0 \\ 0 & \beta \partial_x \end{array} \right) V, \partial_x R_\beta \right\rangle \\
+= \langle H_\beta V, Z_0 \rangle - \frac{1}{\| \partial_x R_\beta \|_{H^{1\times L^2}}^2} \left\langle \beta \alpha_{+,\beta}^A \partial_x Y_{-,\beta}(t, \cdot -x_\infty) - \beta \alpha_{-,\beta}^A \partial_x Y_{+,\beta}(t, \cdot -x_\infty), \partial_x R_\beta \right\rangle \\
++ \frac{1}{\| \partial_x R_\beta \|_{H^{1\times L^2}}^2} \langle V_\perp, \beta \partial_x^2 R_\beta \rangle.
+$$
+
+Finally, let us notice that
+
+$$
+\begin{align*}
+\langle H_{\beta}V, Z_0 \rangle &= \langle H_{\beta}V_{\perp}, Z_0 \rangle + O(\alpha_{+, \beta}^A + \alpha_{-, \beta}^A) \\
+&= \langle V_{\perp}, H_{\beta}Z_0 \rangle + O(\alpha_{+, \beta}^A + \alpha_{-, \beta}^A) \\
+&= O(\alpha_{+, \beta}^A + \alpha_{-, \beta}^A + \|V_{\perp}\|_{H^{1\times L^2}}).
+\end{align*}
+$$
+
+Hence,
+
+$$
+\lambda'(t) = O\left(e^{-e\beta t} \|V(t)\|_{H^{1\times L^2}} + |\alpha_{+, \beta}^A| + |\alpha_{-, \beta}^A| + \|V_{\perp}\|_{H^{1\times L^2}}\right)
+$$
+
+and assertion (4) indeed holds.
+
+It follows from (6.17) and (1), (2), and (3) in Lemma 6.10 that for $t$ large,
+
+$$
+(6.19) \quad \|V_{\perp}(t)\|_{H^{1\times L^2}}^2 \le C \int_t^{+\infty} e^{-e\beta t'} \|V(t')\|_{H^{1\times L^2}}^2 dt' \\
+\qquad + C e^{-2e\beta t} \|V(t)\|_{H^{1\times L^2}} \|V(t')\|_{H^{1\times L^2}} dt'.
+$$
+
+Since
+
+$$
+\int_t^{+\infty} e^{-e\beta t'} \|V(t')\|_{H^{1\times L^2}}^2 dt' \le e^{-e\beta t} \sup_{t'\ge t} \|V(t')\|_{H^{1\times L^2}} \int_t^{+\infty} \|V(t')\|_{H^{1\times L^2}} dt',
+$$
+
+we deduce that for $t$ large,
+
+$$
+(6.20) \quad \|V_{\perp}(t)\|_{H^{1\times L^2}}^2 \le C e^{-\frac{1}{2}e\beta t} \sup_{t'\ge t} \|V(t')\|_{H^{1\times L^2}}^{\frac{1}{2}} \left( \int_t^{+\infty} \|V(t')\|_{H^{1\times L^2}} dt' \right)^{\frac{1}{2}} .
+$$
+
+**Lemma 6.11** (Estimate of $\|V(t)\|_{H^{1\times L^2}}$ in terms of $\int_t^{+\infty} \|V(t')\|_{H^{1\times L^2}} dt'$ and $\|V_\perp(t)\|_{H^{1\times L^2}}$. We have the existence of $C \ge 0$ such that for $t$ sufficiently large:
+
+$$
+(6.21) \quad \|V(t)\|_{H^{1\times L^2}}^2 \le C e^{-e\beta t} \int_t^{+\infty} \|V(t')\|_{H^{1\times L^2}} dt' \\
+\phantom{(6.21) \quad \|V(t)\|_{H^{1\times L^2}}^2} + C \left( \|V_\perp(t)\|_{H^{1\times L^2}}^2 + \int_t^{+\infty} \|V_\perp(t')\|_{H^{1\times L^2}} dt' \right).
+$$
+
+*Proof.* Using the decomposition (6.16) of *V*, we have:
+
+$$
+\|V(t)\|_{H^{1\times L^2}}^2 \le C(|\alpha_{+,,\beta}^A(t)| + |\alpha_{-,,\beta}^A(t)| + |\lambda(t)| + \|V_\perp(t)\|_{H^{1\times L^2}}).
+$$
+
+By Lemma 6.10 which gives estimates of $|\alpha_{+,,\beta}^A(t)|$, $|\alpha_{-,,\beta}^A(t)|$, and $\lambda'(t)$ in terms of $\|V(t)\|_{H^{1\times L^2}}$ and $\|V_\perp(t)\|_{H^{1\times L^2}}$, we obtain:
+
+$$
+(6.22) \quad \lambda(t) = O\left(e^{-e\beta t} \int_t^{+\infty} \|V(t')\|_{H^{1\times L^2}} dt' + \int_t^{+\infty} \|V_\perp(t')\|_{H^{1\times L^2}} dt'\right).
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_47.md b/samples/texts/7376768/page_47.md
new file mode 100644
index 0000000000000000000000000000000000000000..d067ff6845aecee2c0667af5872fc91bc64045dd
--- /dev/null
+++ b/samples/texts/7376768/page_47.md
@@ -0,0 +1,73 @@
+Thus
+
+$$
+\begin{align*}
+\|V(t)\|_{H^1 \times L^2} \le C \left( e^{-e\beta t} \|V(t)\|_{H^1 \times L^2} + e^{-e\beta t} \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' + \|V_\perp(t)\|_{H^1 \times L^2} \right) \\
+\qquad + C \int_t^{+\infty} \|V_\perp(t')\|_{H^1 \times L^2} dt'.
+\end{align*}
+$$
+
+Hence, even if it means taking larger values of $t$, this finishes proving Lemma 6.11.
+□
+
+**Lemma 6.12.** We have
+
+$$
+(6.23) \qquad \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \le C e^{-e\beta t} \sup_{t'\ge t} \|V(t')\|_{H^1 \times L^2}
+$$
+
+Proof. From (6.20), it follows that
+
+$$
+(6.24) \qquad \int_t^{+\infty} \|V_\perp(t')\|_{H^1 \times L^2} dt' \le C \sup_{t' \ge t} \|V(t')\|_{H^1 \times L^2}^{1/2} \left( \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \right)^{1/2} \int_t^{+\infty} e^{-t/2} e_{\beta t'} dt'
+$$
+
+$$
+\le C e^{-\frac{1}{2}e\beta t} \sup_{t'\ge t} \|V(t')\|_{H^1\times L^2}^{\frac{1}{2}} \left(\int_t^{+\infty} \|V(t')\|_{H^1\times L^2} dt'\right)^{\frac{1}{2}}.
+$$
+
+Gathering Lemma 6.11, (6.20), and (6.24), we obtain:
+
+$$
+\begin{align*}
+\|V(t)\|_{H^1 \times L^2} &\le C e^{-e\beta t} \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \\
+&\qquad + C \left( e^{-\frac{1}{2}e\beta t} \sup_{t' \ge t} \|V(t')\|_{H^1 \times L^2}^{\frac{1}{2}} \left( \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \right)^{\frac{1}{2}} \right).
+\end{align*}
+$$
+
+Now integrating the preceding inequality (which is obviously possible) leads to:
+
+$$
+\begin{align*}
+\int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' &\le C \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt'. \int_t^{+\infty} e^{-e\beta t'} dt' \\
+&\quad + C \sup_{t' \ge t} \|V(t')\|_{H^1 \times L^2}^{\frac{1}{2}} \left( \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \right)^{\frac{1}{2}} \int_t^{+\infty} e^{-\frac{1}{2}e\beta t'} dt'.
+\end{align*}
+$$
+
+Or more simply
+
+$$
+\begin{align*}
+\int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' &\le C e^{-e\beta t} \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \\
+&\quad + C e^{-\frac{1}{2}e\beta t} \sup_{t' \ge t} \|V(t')\|_{H^1 \times L^2}^{\frac{1}{2}} \left( \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \right)^{\frac{1}{2}}.
+\end{align*}
+$$
+
+Even if it means considering larger values of $t$, we obtain:
+
+$$
+\int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \le C e^{-\frac{1}{2}e\beta t} \sup_{t' \ge t} \|V(t')\|_{H^1 \times L^2}^{\frac{1}{2}} \left( \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \right)^{\frac{1}{2}},
+$$
+
+which immediately implies the expected result.
+□
+
+Now, let us show
+
+**Proposition 6.13.** For $t$ sufficiently large, $V(t) = 0$.
+
+*Proof.* Gathering Lemma 6.11 and Lemma 6.12, we infer that for $t$ large:
+
+$$
+\|V(t)\|_{H^1 \times L^2} \le C \left( \sup_{t' \ge t} \|V_\perp(t')\|_{H^1 \times L^2} + \int_t^{+\infty} \|V_\perp(t')\|_{H^1 \times L^2} dt' \right).
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_48.md b/samples/texts/7376768/page_48.md
new file mode 100644
index 0000000000000000000000000000000000000000..83c96a58887c18b2c7aae17da1f1c6e1dce09572
--- /dev/null
+++ b/samples/texts/7376768/page_48.md
@@ -0,0 +1,70 @@
+From (6.20) and (6.24), we deduce
+
+$$
+\|V(t)\|_{H^1 \times L^2} \le C e^{-\frac{1}{2} e \beta t} \sup_{t' \ge t} \|V(t')\|_{H^1 \times L^2}^{\frac{1}{2}} \left( \int_t^{+\infty} \|V(t')\|_{H^1 \times L^2} dt' \right)^{\frac{1}{2}}.
+$$
+
+Now, it results from Lemma 6.12 again that
+
+$$
+\|V(t)\|_{H^{1}\times L^{2}} \leq C e^{-e\beta t} \sup_{t'\geq t} \|V(t')\|_{H^{1}\times L^{2}}.
+$$
+
+Thus we deduce that ||V(t)||<sub>H<sup>1</sup>×L<sup>2</sup></sub> = 0 for large values of t.
+
+Finally let us observe below that $x_{\infty} = 0$ so that $U = U^A$.
+
+**Proposition 6.14.** There exists $t_0 \ge 0$ such that for all $t \ge t_0$, $U(t) = U^A(t)$.
+
+*Proof.* On the one hand, we have $\|U^A(t, \cdot - x_\infty) - R_\beta(t, \cdot - x_\infty)\|_{H^1 \times L^2} \le Ce^{-e\beta t}$. On the other, we have $\|U(t) - R_\beta(t)\|_{H^1 \times L^2} \to 0$ as $t \to +\infty$. Since we have $U(t) = U^A(t, \cdot - x_\infty)$, it follows from the triangular inequality that
+
+$$
+\| R_{\beta}(t) - R_{\beta}(t, \cdot - x_{\infty}) \|_{H^{1} \times L^{2}} \to 0 \quad \text{as } t \to +\infty.
+$$
+
+Hence $x_{\infty} = 0$ by the following claim, which is a consequence of Taylor formula.
+
+**Claim 6.15.** *There exist $h_0 > 0$, $\epsilon_0 > 0$, and $\delta > 0$ such that*
+
+$$
+(1) \text{ if } |h| \le h_0, \text{ then } \delta h^2 \le \|Q(\cdot + h) - Q\|_{H^1}^2 \le 4\delta h^2;
+$$
+
+$$
+(2) \text{ if } |h| > h_0, \text{ then } \|Q(\cdot + h) - Q\|_{H^1}^2 > \epsilon_0.
+$$
+
+7. APPENDIX
+
+**7.1. Extension of the proofs to higher dimensions.** The main parts of the proofs remain ob-
+viously unchanged. Essentially three notable adaptations are to be made, passing from the one-
+dimensional case to higher dimensions.
+
+In a first instance, one has to be careful about how establishing several estimates. Although all
+estimates we have proved in the previous sections (in dimension 1) are identical for general d, the
+way we establish them when d ≥ 2 can be altered.
+
+For example, we point out that it is no longer possible to use the Sobolev embedding $H^1 \hookrightarrow L^\infty$ when $d \ge 2$. Particularly, multi-solitons in dimension $d \ge 2$ do not necessarily take values in $L^\infty(\mathbb{R}^d)$ and in order to estimate a quantity like
+
+$f(u) - f(\varphi) - f'(\varphi)(u - \varphi)$
+
+(as for proving Claim 3.6), we would proceed as follows: by (H’1), we deduce that $|f''(r)| = p(p-1)|r|^{p-2}$, thus applying Taylor formula, we have for fixed time $t \in \mathbb{R}$ and position $x \in \mathbb{R}^d$,
+
+$$
+\begin{align*}
+|f(u) - f(\varphi) - f'(\varphi)(u - \varphi)|(t, x) &\le \frac{|u - \varphi|^2(t, x)}{2} \sup_{r \in [\varphi(t,x), u(t,x)]} |f''(r)| \\
+&\le C|u - \varphi|^2(t, x) (|\varphi|^{p-2}(t,x) + |u|^{p-2}(t,x)).
+\end{align*}
+$$
+
+Now, for all $\psi \in L^\infty(\mathbb{R}^d)$ and for all $z \in H^1(\mathbb{R}^d)$, Hölder inequality yields
+
+$$
+\left| \int_{\mathbb{R}^d} |\psi| |u(t) - \varphi(t)|^2 |z|^{p-2} dx \right| \le C \| \psi \|_{L^\infty} \left( \int_{\mathbb{R}^d} |u(t) - \varphi(t)|^p dx \right)^{\frac{2}{p}} \left( \int_{\mathbb{R}^d} |z|^p dx \right)^{\frac{p-2}{p}} .
+$$
+
+Finally, replacing z by $\varphi(t)$ and $u(t) - \varphi(t)$, we obtain an estimate of
+
+$$
+\int_{\mathbb{R}^d} \psi (f(u) - f(\varphi) - f'(\varphi)(u-\varphi)) dx
+$$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_49.md b/samples/texts/7376768/page_49.md
new file mode 100644
index 0000000000000000000000000000000000000000..6c1301109946ba7c6ca47b5745bb9af56682812e
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+in terms of $\|u - \varphi\|_{H^1}$, $\|u\|_{H^1}$, and $\|\varphi\|_{H^1}$ due to the Sobolev embedding $H^1(\mathbb{R}^d) \hookrightarrow L^p(\mathbb{R}^d)$.
+
+Secondly, in view of Proposition 2.1, one has to take into account, for all $i = 1, \dots, N$ the $d$ directions which generate the kernel of the operator $\mathcal{H}_{\beta_i}$ when we practice modulation in dimension $d$. For instance, in Lemma 4.1, we would define $E$ as follows:
+
+$$E := Z - \sum_{i=1}^{N} a_i \cdot \nabla R_i - \sum_{i=1}^{N} b_i Y_{+,i},$$
+
+with $a_i(t) \in \mathbb{R}^d$ and $b_i(t) \in \mathbb{R}$ such that for all $i = 1, \dots, N$ and for all $j = 1, \dots, d$:
+
+$$ (7.1) \qquad \langle E(t), \partial_{x_j} R_i(t) \rangle = 0 $$
+
+$$ (7.2) \qquad \langle E(t), Z_{-,i}(t) \rangle = 0. $$
+
+But each time this extension does not affect the sequel; in other words, the estimates we are supposed to obtain afterwards and their proofs are the same.
+
+A third change to be done concerns the way we define the different Lyapunov functionals which are studied throughout the article. To deal with dimensions greater or equal than 2, we reduce the problem to the case of a one-dimensional variable. For instance, let us explain how to generalize Step 4 in subsection 3.2.2 to all dimensions. The subset
+
+$$ \mathcal{M} := \bigcup_{i \neq j} \{\ell \in \mathbb{R}^d | \ell \cdot (\beta_j - \beta_i) = 0\} $$
+
+of $\mathbb{R}^d$ is of zero Lebesgue measure. Hence, there exists $\ell \in \mathbb{R}^d$ such that for all $i \neq j$,
+
+$$ \ell \cdot (\beta_j - \beta_i) \neq 0. $$
+
+In particular $\ell \neq 0$ and, even if it means considering $\frac{\ell}{|\ell|}$, we can assume that $|\ell| = 1$, so that $\forall i = 1, \dots, N$, $|\ell \cdot \beta_i| < 1$. Now, defining $\tilde{\beta}_i := \ell \cdot \beta_i$, and even if it means changing the permutation $\eta$, we have
+
+$$ -1 < \tilde{\beta}_{\eta(1)} < \tilde{\beta}_{\eta(2)} < \dots < \tilde{\beta}_{\eta(N)} < 1. $$
+
+Then, the direction described by $\ell$ is to be favored: we consider the following cut-off functions:
+
+$$ \psi_k(t) = \psi\left(\frac{1}{\sqrt{t}}\left(\ell \cdot x - \frac{\tilde{\beta}_{\eta(k)} + \tilde{\beta}_{\eta(k+1)}}{2} - \ell \cdot \frac{x_{\eta(k)} + x_{\eta(k+1)}}{2}\right)\right). $$
+
+At this stage, the definition of the functions $\phi_k$ in terms of the $\psi_k$ is kept unchanged and the corresponding Lyapunov functional is to be written:
+
+$$ F_w(t) = \sum_{k=1}^{K} \int_{\mathbb{R}^d} (w_1^2 + (\partial_x w_1)^2 + w_2^2 - f'(Q_{\eta(k)})w_1^2 + 2\beta_{\eta(k)} \cdot \nabla w_1 w_2) \phi_k dx. $$
+
+**7.2. Proof of Corollary 1.8.** The proof is an immediate adaptation of that of Proposition 4.12 in [3].
+
+Let $A > 0$ and denote $t_A := -\frac{\ln(A)}{e_\beta}$. In the sense of the $H^1 \times L^2$-norm, we have:
+
+$$ 
+\begin{align*}
+U^1(t+t_A, \cdot + \beta t_A) &= R_\beta(t+t_A, \cdot + \beta t_A) + e^{-e_\beta(t+t_A)} Y_{+,\beta}(t+t_A, \cdot + \beta t_A) + O(e^{-2e_\beta t}) \\
+&= R_\beta(t) + A e^{-e_\beta t} Y_{+,\beta}(t) + O(e^{-2e_\beta t}).
+\end{align*}
+$$
+
+Then, $\|U^1(t+t_A, \cdot + \beta t_A) - R_\beta(t)\|_{H^1 \times L^2} \to 0$ so that there exist $\tilde{A} \in \mathbb{R}$ and $t_0 = t_0(\tilde{A}) \in \mathbb{R}$ such that for all $t \ge t_0$,
+
+$$ U^{\tilde{A}}(t) = U^1(t+t_A, \cdot+\beta t_A). $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_5.md b/samples/texts/7376768/page_5.md
new file mode 100644
index 0000000000000000000000000000000000000000..759f51ab1390921ad95187ffa15e5e7c2d0178a3
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+generalized Korteweg-de Vries (gKdV) equation, a solution which converges to a soliton as time goes to infinity is pretty well-known to be exactly the corresponding soliton.
+
+What is more, Theorem 1.6 is built exactly as in [3]. It reminds us of similar classification results, previously obtained for the $L^2$-supercritical gKdV equations by Combet [3, Theorem 1.1] and for the three-dimensional cubic Schrödinger equation by Duyckaerts and Roudenko [12, Proposition 3.1], both inspired from pioneering works of Duyckaerts and Merle [10, 11].
+
+As far as the multi-soliton case is concerned, Theorem 1.2 provides an *N*-parameter family of solutions to (NLKG) which behave as a sum of *N* (differently boosted) solitons in large time. Here again, the existence result looks like that of Combet [4, 5] and reinforces the idea that, in general, non-uniqueness holds for multi-solitons in an unstable context. Furthermore, Theorem 1.4 opens the way to treat the question of the classification of multi-solitons for other models, at least in the restraint class of solutions with algebraic convergence in the sense of (1.3); on this matter and given the main existence results in [5, 1] one can for instance explore the issue for the $L^2$-supercritical non-linear Schrödinger equation or the nonlinear Klein-Gordon equation with complex valued solutions and stable solitons.
+
+Regarding the global approach developed in this article, we shed new light on the construction of the one-parameter family $(U^A)_{A \in \mathbb{R}}$ by a compactness procedure. Note that the special solution $U^{-1}$ in [3] has also been obtained in a first rigorous way by a compactness method (the starting point was the proof of instability of the soliton), but the question of obtaining $U^1$ (as well as the other solutions $U^A$ for $A > 0$) by such method was raised (see Remark 4.15 in [3]) and, to our knowledge, remained unanswered. Obviously, our process can be thought and used as an alternative to prove similar results in the context of other partial differential equations which involve unstable solitons, and for which the spectral theory around the ground states is well understood (and actually analogous to the present case). This is a nice feature of our paper.
+
+A second interesting point to be discussed is about proving uniqueness of multi-solitary waves. Exploring the possibility of obtaining a "weak" monotonicity property like (4.18) in Corollary 4.8 would be a promising direction of research in order to classify multi-solitons of other models.
+
+Besides, a significant issue would be to know to what extent the classification obtained in Theorem 1.6 could be transcribed to the multi-soliton topic. Indeed, contrary to the gKdV setting [3] and especially as for the NLS case [5, 6], proving general uniqueness in the sense of (1.5), where $R_\beta$ is replaced by a sum of several solitons $R_{\beta_i, x_i}$, still remains unclear.
+
+Another question to be addressed could be related to the potential generalizations of the previous theorems to multi-bound states.
+
+**1.3. Outline and organization of the article.** In the spirit of [19] (dealing with the construction of multi-soliton solutions), our approach to constructing the families in Theorem 1.2 and Theorem 1.6 is based on backward uniform $H^1 \times L^2$-estimates satisfied by well-chosen sequences of solutions of (NLKG) which aim to approximate the desired solutions. We entirely exploit a coercivity property available in the present matter (see Proposition 2.2 stated below) in order to establish those estimates and, in fact, to obtain the expected exponential convergences to zero in time. We finally use continuity of the flow of (NLKG) for the weak $H^1 \times L^2$-topology to obtain special solutions which fulfill (1.2) or (1.4).
+
+Based on compactness and energy methods, the proof of Theorem 1.2 follows more precisely the strategy of [4, 5]. The construction of $\Phi_{A_1, \dots, A_N}$ is done by iteration, by means of Proposition 3.1 which roughly asserts that each multi-soliton can be perturbed slightly at the order $e^{-\beta_j t}$ around the soliton $R_{\beta_j, x_j}$. In order to establish this key proposition, we particularly rely on the topological ingredient set up originally by Côte, Martel and Merle [8] for the construction of one given multi-soliton in unstable situations.
+
+For the construction of $U^A$ in the one-soliton case (Theorem 1.6), we also substantially rely on the spectral theory available for (NLKG), built on the linearized operator of the flow around the boosted ground state. In particular, we point out that our approach differs from previous articles [10, 11, 12, 3]
\ No newline at end of file
diff --git a/samples/texts/7376768/page_50.md b/samples/texts/7376768/page_50.md
new file mode 100644
index 0000000000000000000000000000000000000000..3b08373a6a783b2fbb98e2d74bc13d0912ba7270
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@@ -0,0 +1,57 @@
+But on the other hand,
+
+$$U^{\tilde{A}}(t) = R_{\beta}(t) + \tilde{A}e^{-e_{\beta}t}Y_{+\beta}(t) + O(e^{-2e_{\beta}t}).$$
+
+Hence,
+
+$$ (A - \tilde{A})e^{-e_{\beta}t}Y_{+\beta}(t) = O(e^{-2e_{\beta}t}), $$
+
+which implies $A = \tilde{A}$. Consequently, $U^A(t) = U^1(t + t_A, \cdot + \beta t_A)$.
+
+If $A < 0$, we have just to repeat the above argument with $-A$ instead of $A$.
+
+Lastly, let us identify $U^0$. Given that $R_\beta$ is a solution of (NLKG) which satisfies (1.5), Theorem 1.2 provides the existence of $A \in \mathbb{R}$ and of $t_0 \in \mathbb{R}$ such that for all $t \ge t_0$, $U^A(t) = R_\beta(t)$.
+Since $U^A$ satisfies (1.4), we deduce that
+
+$$ \| Ae^{-e_{\beta}t} Y_{+\beta}(t) \|_{H^1 \times L^2} \le C e^{-2e_{\beta}t}. $$
+
+Thus $A=0$ and $U^0 = R_\beta$ is defined for all $t \in \mathbb{R}$.
+
+### 7.3. A result of analytic theory of differential equations.
+
+**Lemma 7.1.** Let $t_0 \in \mathbb{R}$, $\mathcal{A}: [t_0, +\infty) \to \mathbb{R}$ be a $\mathscr{C}^1$ bounded function, and $\xi: [t_0, +\infty) \to \mathbb{R}^+$ be continuous and integrable.
+If, for some $\rho > 0$,
+
+$$ \forall t \ge t_0, \quad |\mathcal{A}'(t) + \rho\mathcal{A}(t)| \le \xi(t) \sup_{t' \ge t} |\mathcal{A}(t')|, $$
+
+then there exists $c > 0$ such that
+
+$$ \forall t \ge t_0, \quad |\mathcal{A}(t)| \le c e^{-\rho t}. $$
+
+*Proof.* Let us assume that
+
+$$ (7.3) \qquad \forall t \ge t_0, \quad |\mathcal{A}'(t) + \rho\mathcal{A}(t)| \le \xi(t) \sup_{t' \ge t} |\mathcal{A}(t')|, $$
+
+for some $\rho > 0$. Then for all $t \ge t_0$,
+
+$$ |(e^{\rho t} \mathcal{A})'(t)| \le \xi(t) e^{\rho t} \sup_{t' \ge t} |\mathcal{A}(t')|. $$
+
+Let us consider $t \ge t_0$. For $t' \ge t$, we obtain by integration
+
+$$ |e^{\rho t'} \mathcal{A}(t') - e^{\rho t} \mathcal{A}(t)| \le \int_t^{t'} \xi(s)e^{\rho s} \sup_{u \ge s} |\mathcal{A}(u)| ds. $$
+
+This implies that, for $t' \ge t$,
+
+$$ e^{\rho t'} |\mathcal{A}(t')| \le e^{\rho t} |\mathcal{A}(t)| + \sup_{u \ge t} |\mathcal{A}(u)| e^{\rho t'} \int_t^{t'} \xi(s) ds. $$
+
+From the preceding line, we deduce that for all $t' \ge t$,
+
+$$ (7.4) \qquad |\mathcal{A}(t')| \le |\mathcal{A}(t)| + \sup_{u \ge t} |\mathcal{A}(u)| \int_t^{+\infty} \xi(s) ds. $$
+
+Now we consider $t_1 \ge t_0$ such that $\int_{t_1}^{+\infty} \xi(s) ds < \frac{1}{2}$ (which is indeed possible given that $\int_{t_1}^{+\infty} \xi(s) ds \to 0$ as $t \to +\infty$). By passing to the supremum on $t'$ in (7.4), we obtain for all $t \ge t_1$,
+
+$$ \sup_{t' \ge t} |\mathcal{A}(t')| \le 2|\mathcal{A}(t)|. $$
+
+Consequently, assumption (7.3) becomes
+
+$$ (7.5) \qquad \forall t \ge t_1, \quad |\mathcal{A}'(t) + \rho\mathcal{A}(t)| \le 2\xi(t)|\mathcal{A}(t)|. $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_51.md b/samples/texts/7376768/page_51.md
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+Let us define $y(t) := e^{\rho t} |\mathcal{A}(t)|$. By integration of (7.5), we obtain
+
+$$ (7.6) \qquad \forall t \ge t_1, \quad y(t) \le y(t_1) + \int_{t_1}^{t} 2\xi(s)y(s) ds. $$
+
+By a standard Grönwall argument, we conclude to the existence of $C > 0$ such that for all $t \ge t_1$, $y(t) \le C$, which implies the desired result. For the sake of completeness, let us explicit this argument.
+
+We define $Y(t) := \exp(-\int_{t_1}^t 2\xi(s) ds) \int_{t_1}^t 2\xi(s)y(s) ds$ for $t \ge t_1$. The function $Y$ is $\mathcal{C}^1$ on $[t_1, +\infty)$ and for all $t \ge t_1$,
+
+$$ 
+\begin{aligned}
+Y'(t) &= 2\xi(t) \exp\left(-\int_{t_1}^{t} 2\xi(s) ds\right) \left[y(t) - \int_{t_1}^{t} 2\xi(s)y(s) ds\right] \\
+&\le 2y(t_1)\xi(t) \exp\left(-\int_{t_1}^{t} 2\xi(s) ds\right)
+\end{aligned}
+ $$
+
+by (7.6). Integrating the preceding inequality and observing that $Y(t_1) = 0$, we have
+
+$$ Y(t) \le \int_{t_1}^{t} 2\xi(s)y(t_1) \exp\left(-\int_{t_1}^{s} 2\xi(u) du\right) ds. $$
+
+We then infer
+
+$$ (7.7) \quad \int_{t_1}^{t} 2\xi(s)y(s) ds = \exp\left(\int_{t_1}^{t} 2\xi(s) ds\right) Y(t) \le 2y(t_1) \int_{t_1}^{t} \xi(s) \exp\left(\int_{s}^{t} 2\xi(u) du\right) ds. $$
+
+Lastly, we denote $v := \int_{t_1}^{+\infty} \xi(s) ds$; gathering (7.6) and (7.7), we obtain
+
+$$ \forall t \ge t_1, \quad y(t) \le y(t_1) + 2y(t_1)e^{2v}v. $$
+
+This achieves the proof of Lemma 7.1. □
+
+## REFERENCES
+
+[1] Jacopo Bellazzani, Marco Ghimenti, and Stefan Le Coz. Multi-solitary waves for the nonlinear Klein-Gordon equation. *Communications in Partial Differential Equations*, 39(8):1479–1522, 2014.
+
+[2] Henri Berestycki and Pierre-Louis Lions. Non linear scalar field equations, I. Existence of a ground state. *Arch. Ration. Mech. Anal.*, 82(4):313–345, 1983.
+
+[3] Vianney Combet. Construction and characterization of solutions converging to solitons for supercritical gKdV equations. *Differential Integral Equations*, 23(5/6):513–568, 05 2010.
+
+[4] Vianney Combet. Multi-soliton solutions for the supercritical gKdV equations. *Communications in Partial Differential Equations*, 36, 02 2010.
+
+[5] Vianney Combet. Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension. *Discrete Contin. Dyn. Syst.*, 34(5):1961–1993, 2014.
+
+[6] Raphaël Côte and Xavier Friederich. On smoothness and uniqueness of multi-solitons of the non-linear Schrödinger equations. *Communications in Partial Differential Equations*, 2021.
+
+[7] Raphaël Côte and Yvan Martel. Multi-travelling waves for the nonlinear klein-gordon equation. *Transactions of the American Mathematical Society*, 2018.
+
+[8] Raphaël Côte, Yvan Martel, and Frank Merle. Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations. *Rev. Mat. Iberoam.*, 27(1):273–302, 2011.
+
+[9] Raphaël Côte and Claudio Muñoz. Multi-solitons for nonlinear Klein-Gordon equations. *Forum of Mathematics, Sigma*, 2:e15, 2014.
+
+[10] Thomas Duyckaerts and Frank Merle. Dynamics of threshold solutions for energy-critical wave equation. *International Mathematics Research Papers*, 2008, 01 2008. rpn002.
+
+[11] Thomas Duyckaerts and Frank Merle. Dynamic of threshold solutions for energy-critical NLS. *Geometric and Functional Analysis*, 18:1787–1840, 2009.
+
+[12] Thomas Duyckaerts and Svetlana Roudenko. Threshold solutions for the focusing 3D cubic Schrödinger equation. *Rev. Mat. Iberoamericana*, 26(1):1–56, 03 2010.
+
+[13] J. Ginibre and G. Velo. The global Cauchy problem for the nonlinear Klein-Gordon equation. *Mathematische Zeitschrift*, 189(4):487–505, 1985.
+
+[14] Manoussos Grillakis, Jalal Shatah, and Walter Strauss. Stability theory of solitary waves in the presence of symmetry, I. *J. Funct. Anal.*, 74:160–197, 1987.
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diff --git a/samples/texts/7376768/page_52.md b/samples/texts/7376768/page_52.md
new file mode 100644
index 0000000000000000000000000000000000000000..e60feb2c76f719c28130b68b06dd99e86cb0cbe0
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+++ b/samples/texts/7376768/page_52.md
@@ -0,0 +1,34 @@
+[15] Manoussos Grillakis, Jalal Shatah, and Walter Strauss. Stability theory of solitary waves in the presence of symmetry, II. *J. Funct. Anal.*, 94:308–348, 1990.
+
+[16] Joachim Krieger, Kenji Nakanishi, and Wilhelm Schlag. Global dynamics above the ground state energy for the one-dimensional NLKG equation. *Mathematische Zeitschrift*, 272, 11 2010.
+
+[17] Kwok-Kun Kwong. Uniqueness of positive solutions of $\Delta u - u + u^p = 0$ in $\mathbb{R}^n$. *Arch. Ration. Mech. Anal.*, 105(3):243-266, 1989.
+
+[18] Mihai Maris. Existence of nonstationary bubbles in higher dimensions. *Journal de Mathématiques Pures et Appliquées*, 81(12):1207–1239, 2002.
+
+[19] Yvan Martel. Asymptotic N-Soliton-like Solutions of the Subcritical and Critical Generalized Korteweg-de Vries Equations. *American Journal of Mathematics*, 127(5):1103–1140, 2005.
+
+[20] Yvan Martel and Frank Merle. Construction of multi-Solitons for the energy-critical wave equation in dimension 5. *Archive for Rational Mechanics and Analysis*, 222(3):1113–1160, 2016.
+
+[21] Yvan Martel, Frank Merle, and Tai-Peng Tsai. Stability and Asymptotic Stability for Subcritical gKdV Equations. *Communications in Mathematical Physics*, 231(2):347–373, 2002.
+
+[22] Yvan Martel, Frank Merle, and Tai-Peng Tsai. Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations. *Duke Math. J.*, 133(3):405–466, 06 2006.
+
+[23] Kevin McLeod. Uniqueness of positive radial solutions of $\Delta u + f(u) = 0$ in $R^n$, II. *Transactions of the American Mathematical Society*, 339(2):495–505, 1993.
+
+[24] Makoto Nakamura and Tohru Ozawa. The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces. *Publ. Res. Inst. Math. Sci.*, 37, 11 2001.
+
+[25] K. Nakanishi and W. Schlag. Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation. *Journal of Differential Equations*, 250(5):2299–2333, 2011.
+
+[26] Robert L. Pego and Michael I. Weinstein. Eigenvalues, and instabilities of solitary waves. 340:47-94, 1992.
+
+[27] James Serrin and Moxun Tang. Uniqueness of ground states for quasilinear elliptic equations. *Indiana Univ. Math. J.*, 49:897-923, 2000.
+
+[28] Terence Tao. Low regularity semi-linear wave equations. *Communications in Partial Differential Equations*, 24(3-4):599-629, 1999.
+
+[29] Xu Yuan. On multi-solitons for the energy-critical wave equation in dimension 5. *Nonlinearity*, 32(12):5017-5048, nov 2019.
+
+[30] Xu Yuan. Construction of excited multi-solitons for the 5D energy-critical wave equation. *J. Hyperbolic Differ. Equ.*, 2021.
+
+INSTITUT DE RECHERCHE MATHÉMATIQUE AVANCÉE UMR 7501, UNIVERSITÉ DE STRASBOURG, STRASBOURG, FRANCE
+*Email address:* friederich@math.unistra.fr
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diff --git a/samples/texts/7376768/page_6.md b/samples/texts/7376768/page_6.md
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index 0000000000000000000000000000000000000000..5fffe5b83293fb547fbabc8d9e02aa9041f7d0ba
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+where the construction centers around the contraction principle.
+
+Regarding the question of classification, from (1.4) and orthogonality properties exposed in the next section, we notice that $A$ corresponds to the limit of $e^{\epsilon\beta t}\langle U^A, Z_{-\cdot,\beta} \rangle$ as $t \to +\infty$ in Theorem 1.6. This is precisely useful for the uniqueness part of this theorem, where the goal is to identify $U$ with an element of the one-parameter family $(U^A)_{A\in\mathbb{R}}$. Actually, to prove the second part of Theorem 1.6, we follow [3] in the first instance (up to the obtainment of an exponential control of $U - R_\beta$). Then, a refined version of the coercivity argument considered in [3], and indeed an elementary but careful analysis of the available estimates, allows us to reach the conclusion, that is to show that $U$ equals some $U^A$ (already constructed in the first part of the theorem), and that without making use of any supplementary tool. We underline once again that we do not need any fixed point argument to conclude.
+
+Yet, we do not exclude the possibility to prove Theorem 1.6 via the contraction principle; if the nonlinearity $f$ is sufficiently regular ($C^{s+1}$ for instance), one could precise by this means the behavior of $U^A$ in large time, and indeed expand the solution at the order $O(e^{-(s+1)\epsilon\beta t})$ in the Sobolev space $H^s$, in line with [10]. For clarity purposes, we will not explore this path further and anyway, our description of the family $(U^A)_{A\in\mathbb{R}}$ is sufficient to characterize solutions verifying (1.5).
+
+Concerning Theorem 1.4, the identification of the solution satisfying (1.3) is done step by step as in [4]. The core of the proof is the obtainment of an almost monotonicity property, inspired by Martel and Merle [20]. By means of a technical lemma of analysis (we refer to Lemma 7.1 in Appendix), this monotonicity property allows us to see that any multi-soliton in the class with polynomial convergence to zero converges in fact exponentially (see subsection 4.1.4), provided one assumes suitable integrability conditions in the neighborhood of $+\infty$ (and indeed $\alpha > 3$). Such a "weak" monotonicity property has a priori been used so far only for the construction of multi-solitons or multi-bound states of the energy-critical wave equation [20, 29, 30]. We also underline that, by Lemma 7.1, we directly obtain the adequate exponential convergence rate which allows us to identify $A_1$; this is in contrast with [4]. The monotonicity property is also used as a key ingredient to identify the other parameters $A_2, \dots, A_N$.
+
+This paper is organized as follows. In Section 2, we introduce essential notations and tools which are used throughout our article. In the following sections, we establish the proofs of our main results; for ease of writing, each proof will be made in dimension 1. Section 3 is devoted to the construction of the family of multi-solitons described in Theorem 1.2. Section 4 deals with the classification of multi-solitons, which is the object of Theorem 1.4. In Section 5, we study the existence part of Theorem 1.6 focusing on one soliton. Section 6 aims at proving the second part of this latter theorem, that is general uniqueness of the one-parameter family previously constructed. In the appendix, we explain how to adapt the proof to all dimensions, we justify Corollary 1.8, and we state and prove the lemma of analytic theory of differential equations used in Section 4.
+
+As usual, $C$ denotes a positive constant which may depend on the soliton parameters and change from one line to the other, but which is always independent of $t$ and $x$.
+
+**1.4. Acknowledgments.** The author would like to thank his supervisor Raphaël Côte for his constant encouragements and for fruitful discussions. The author is also grateful to Rémi Carles for his comments which improved the quality of this paper.
+
+## **2. NOTATIONS, REVIEW OF SPECTRAL THEORY, AND MULTI-SOLITONS**
+
+**2.1. Elements of spectral theory concerning (NLKG).** For all $U = \begin{pmatrix} u_1 \\ u_2 \end{pmatrix}, V = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \in H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$, we define the scalar product:
+
+$$ \langle U, V \rangle := \int_{\mathbb{R}^d} (u_1 v_1 + u_2 v_2) \, dx $$
\ No newline at end of file
diff --git a/samples/texts/7376768/page_7.md b/samples/texts/7376768/page_7.md
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+and the energy norm
+
+$$
+\|U\|_{H^1 \times L^2} := (\|u_1\|_{H^1}^2 + \|u_2\|_{L^2}^2)^{\frac{1}{2}}.
+$$
+
+Under assumption (H'1) (or (H1) and (H2) in the particular one-dimensional case), the operator $L := -\Delta + Id - f'(Q)$ admits a unique simple negative eigenvalue, which we denote by $-\lambda_0$. The kernel of $L$ is spanned by $(\partial_{x_i} Q)_{i=1,\dots,d}$ [18, 23].
+
+Note that for a general nonlinearity $f$ and for $d \ge 2$, the operator $L$ possibly counts several and multiple negative eigenvalues. We refer to Côte and Martel [7] for the detail of the spectral properties in this case.
+
+With a slight abuse of notation, we still denote by $Q_\beta$ the function defined on $\mathbb{R}^d$ by $Q_\beta(x) := Q(\gamma x)$, where $\gamma = \frac{1}{\sqrt{1-|\beta|^2}}$ so that for all $t$, $Q_\beta(x) = Q_\beta(t,x)$. In the sequel, we sometimes omit the variables $x$ and $t$ when there is no ambiguity (we work with functions which either depend on time or not).
+
+For all $\beta \in \mathbb{R}^d$ with $|\beta| < 1$, we consider the matrix operator
+
+$$
+H_{\beta} := \begin{pmatrix} -\Delta + \mathrm{Id} - f'(Q_{\beta}) & -\beta \cdot \nabla \\ \beta \cdot \nabla & \mathrm{Id} \end{pmatrix},
+$$
+
+the matrix $J := \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, and the operator
+
+$$
+\mathcal{H}_{\beta} := -H_{\beta}J = \begin{pmatrix} -\beta \cdot \nabla & \Delta - Id + f'(Q_{\beta}) \\ Id & \beta \cdot \nabla \end{pmatrix}.
+$$
+
+We define for all $j = 1, \dots, d$
+
+$$
+Z_{j,\beta} := \begin{pmatrix} -\beta \cdot \nabla (\partial_{x_j} Q_{\beta}) \\ \partial_{x_j} Q_{\beta} \end{pmatrix}.
+$$
+
+**Proposition 2.1** (Côte and Muñoz [9]). We have $\mathcal{H}_\beta Z_{j,\beta} = 0$ for all $j \in \{1, \dots, d\}$ and there exist two functions $Z_{\pm,\beta}$ whose components decrease exponentially in space and such that
+
+$$
+\mathcal{H}_{\beta} Z_{\pm, \beta} = \pm e_{\beta} Z_{\pm, \beta}
+$$
+
+where $e_\beta := \frac{\sqrt{\lambda_0}}{\gamma}$.
+
+Moreover there exist unique functions $Y_{\pm,\beta}$ (whose components are exponentially decreasing in space)
+such that
+
+$$
+H_{\beta} Y_{\pm, \beta} \in \operatorname{Span}\{Z_{\pm, \beta}\}, \quad \langle J Z_{j, \beta}, Y_{\pm, \beta} \rangle = 0, \quad \text{and} \quad \langle Y_{\pm, \beta}, Z_{\mp, \beta} \rangle = 1.
+$$
+
+In addition, the following orthogonality properties hold:
+
+$$
+\langle Y_{\pm,\beta}, Z_{\pm,\beta} \rangle = 0 \quad \text{and} \quad \langle JZ_{0,\beta}, Z_{\pm,\beta} \rangle = 0.
+$$
+
+The following coercivity property turns out to be a crucial tool in our paper.
+
+**Proposition 2.2** (Almost coercivity of $H_\beta$; Côte and Muñoz [9]). There exists $\mu > 0$ such that for all $V \in H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$,
+
+$$
+\langle H_\beta V, V \rangle \geq \mu \|V\|_{H^1 \times L^2}^2 - \frac{1}{\mu} \left[ \langle V, Z_{+, \beta} \rangle^2 + \langle V, Z_{-, \beta} \rangle^2 + \sum_{j=1}^{d} \langle V, JZ_{j, \beta} \rangle^2 \right].
+$$
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diff --git a/samples/texts/7376768/page_8.md b/samples/texts/7376768/page_8.md
new file mode 100644
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+**2.2. Multi-soliton results.** Let us consider a set of $2N$ parameters as given in Theorem 1.2 and the associated (vector) solitons $R_i = \begin{pmatrix} Q_i \\ \partial_t Q_i \end{pmatrix} := R_{\beta_i, x_i}, i = 1, \dots, N$. We introduce moreover the vectors:
+
+* $Y_{\pm,i}(t,x) := Y_{\pm,\beta_i} (pr \circ \Lambda_{\beta_i}(t, x - x_i))$
+
+* $Z_{\pm,i}(t,x) := Z_{\pm,\beta_i} (pr \circ \Lambda_{\beta_i}(t, x - x_i)),$
+
+where $\gamma_i := \frac{1}{\sqrt{1-|\beta_i|^2}}$. We denote $e_i = e_{\beta_i} := \frac{\sqrt{\lambda_0}}{\gamma_i} = \sqrt{\lambda_0(1 - |\beta_i|^2)}$.
+
+In particular, let us observe that for all $i = 1, \dots, N$, $Y_{\pm,i}$ belongs to $\mathcal{C}(\mathbb{R}, H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)) \cap L^{\infty}(\mathbb{R}, H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d))$.
+
+There exists $\ell \in \mathbb{R}^d$ such that
+
+$$ \forall i \neq j, \quad \ell \cdot \beta_i \neq \ell \cdot \beta_j \quad \text{and} \quad \forall i, \quad |\ell \cdot \beta_i| < 1; $$
+
+we postpone the argument in the appendix. (If $d=1$, one can take obviously $\ell=1$.) Let us consider
+the permutation $\eta$ of $\{1, \dots, N\}$ such that
+
+$$ -1 < \ell \cdot \beta_{\eta(1)} < \cdots < \ell \cdot \beta_{\eta(N)} < 1. $$
+
+We denote also
+
+$$ (2.1) \quad \sigma := \frac{1}{16} \min \{e_1, \gamma_N \min\{\ell \cdot (\beta_{\eta(2)} - \beta_{\eta(1)}), \dots, \ell \cdot (\beta_{\eta(N)} - \beta_{\eta(N-1)})\}\} > 0. $$
+
+We can quantify the interactions between the solitons $R_i$ and the functions $Y_{\pm,i}$ and $Z_{\pm,i}$, for $i = 1, \dots, N$ in terms of the parameter $\sigma$. This is the object of the following
+
+**Proposition 2.3.** We have for all $i \neq j$, for all $k$, $l \in \{0, 1, 2\}$, and for all $t \ge 0$,
+
+$$ 
+\begin{align*}
+\langle \partial_x^k R_i(t), \partial_x^l R_j(t) \rangle &= O(e^{-4\sigma t}) \\
+\langle Y_{\pm,i}(t), Y_{\pm,j}(t) \rangle &= O(e^{-4\sigma t}) \\
+\langle Z_{\pm,i}(t), Z_{\pm,j}(t) \rangle &= O(e^{-4\sigma t}) \\
+\langle Y_{\pm,i}(t), \partial_x^l R_j(t) \rangle &= O(e^{-4\sigma t}) \\
+\langle Z_{\pm,i}(t), \partial_x^l R_j(t) \rangle &= O(e^{-4\sigma t}) \\
+\langle Y_{\pm,i}(t), Z_{\pm,j}(t) \rangle &= O(e^{-4\sigma t})
+\end{align*}
+$$
+
+What is more, due to Theorem 1.1, there exist $t_0 \in \mathbb{R}$ and $C > 0$, only depending on the sets $(\beta_i)_i$, $(x_i)_i$, and a solution $\Phi_0 = \begin{pmatrix} \varphi_0 \\ \partial_t \varphi_0 \end{pmatrix} \in C([t_0, +\infty), H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d))$ of (NLKG) such that for all $t \ge t_0$,
+
+$$ (2.2) \qquad \left\| \Phi_0(t) - \sum_{i=1}^{N} R_i(t) \right\|_{H^1 \times L^2} \le Ce^{-4\sigma t}. $$
+
+When dealing with the multi-soliton case, we will need to consider in the present article the eu-
+clidean space $(\mathbb{R}^k, | \cdot |)$ and euclidean balls and spheres of radius $r > 0$ in $\mathbb{R}^k$, $k = 1, \dots, N$; in
+particular we define:
+
+$$ B_{\mathbb{R}^k}(r) := \{x \in \mathbb{R}^k | |x| \le r\} $$
+
+$$ S_{\mathbb{R}^k}(r) := \{x \in \mathbb{R}^k | |x| = r\}. $$
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diff --git a/samples/texts/7376768/page_9.md b/samples/texts/7376768/page_9.md
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+**3. CONSTRUCTION OF A FAMILY OF MULTI-SOLITONS FOR N ≥ 2**
+
+In this section, we give a detailed proof of Theorem 1.2 in the one-dimensional case.
+
+Let $N \ge 2$ and $x_1, \dots, x_N, \beta_1, \dots, \beta_N$ be $2N$ parameters as in Theorem 1.2. Denote by $\varphi$ a multi-soliton solution associated with these parameters, satisfying (2.2) and consider $\Phi := \begin{pmatrix} \varphi \\ \partial_t \varphi \end{pmatrix}$.
+
+As it was firstly observed in [4], the existence of $(\varphi_{A_1,...,A_N})_{(A_1,...,A_N)\in\mathbb{R}^N}$ verifying (1.2) in Theorem 1.2 is a consequence of the following crucial
+
+**Proposition 3.1.** Let $j \in \{1, \dots, N\}$ and $A_j \in \mathbb{R}$. Then there exist $t_0 > 0$, $C > 0$, and a solution $u$ of (NLKG), defined on $[t_0, +\infty)$ such that
+
+$$
+\begin{gather*}
+\forall t \ge t_0, \quad \|U(t) - \Phi(t) - A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \le Ce^{-(e_j+\sigma)t}, \\
+\text{where } U := \begin{pmatrix} u \\ \partial_t u \end{pmatrix}.
+\end{gather*}
+$$
+
+By (2.2), assuming that the preceding proposition holds, and considering $A_1, \dots, A_N \in \mathbb{R}$, we
+indeed obtain a solution $\phi_{A_1}$ of (NLKG) and $t_1 > 0$ such that
+
+$$
+\forall t \geq t_1, \quad \| \Phi_{A_1}(t) - \Phi_0(t) - A_1 e^{-e_1 t} Y_{+,1}(t) \|_{H^1 \times L^2} \leq C e^{-(e_1+\sigma)t}
+$$
+
+with obvious notations. We notice that $\phi_{A_1}$ is a multi-soliton. Now, assume that we have constructed,
+for some $j \in \{1, \dots, N-1\}$, a family of multi-solitons $\phi_{A_1}, \dots, \phi_{A_1, \dots, A_j}$ such that there exists $t_j > 0$
+such that for all $k = 1, \dots, j$,
+
+$$
+\forall t \geq t_j, \quad \| \Phi_{A_1, \dots, A_k}(t) - \Phi_{A_1, \dots, A_{k-1}}(t) - A_k e^{-e_k t} Y_{+,k}(t) \|_{H^1 \times L^2} \leq C e^{-(e_k+\sigma)t},
+$$
+
+where $\Phi_{A_1,...,A_{k-1}} = \Phi_0$ if $k=1$. Hence we can apply Proposition 3.1 with $\Phi_{A_1,...,A_j}$ instead of $\Phi$
+and there exist $\Phi_{A_1,...,A_{j+1}}$ and $t_{j+1} > 0$ such that
+
+$$
+\forall t \geq t_{j+1}, \quad \| \Phi_{A_1, \dots, A_{j+1}}(t) - \Phi_{A_1, \dots, A_j}(t) - A_{j+1} e^{-e_{j+1}t} Y_{+,j+1}(t) \|_{H^1 \times L^2} \leq C e^{-(e_{j+1}+\sigma)t}.
+$$
+
+Thus, by induction on $j$, we obtain a family of multi-solitons $\phi_{A_1}, \dots, \phi_{A_1, \dots, A_j}$ such that for all
+$t \ge t_0 := \max\{t_j | j = 1, \dots, N\}$,
+
+$$
+\left\| \Phi_{A_1, \dots, A_N}(t) - \Phi_0(t) - \sum_{j=1}^{N} A_j e^{-e_j t} Y_{+,j}(t) \right\|_{H^1 \times L^2} \\
+\leq \sum_{j=1}^{N} \| \Phi_{A_1, \dots, A_j}(t) - \Phi_{A_1, \dots, A_{j-1}}(t) - A_j e^{-e_j t} Y_{+,j}(t) \|_{H^1 \times L^2} \\
+\leq C \sum_{j=1}^{N} e^{-(e_j + \sigma)t}.
+$$
+
+At this stage, we conclude to (1.2) in Theorem 1.2, using once more (2.2) and the triangular inequality.
+
+Let us already justify also that for all $(A_1, \dots, A_N) \neq (A'_1, \dots, A'_N)$, we have $\varphi_{A_1, \dots, A_N} \neq
+\varphi_{A'_1, \dots, A'_N}$.
+
+Assume for the sake of contradiction that $\varphi_{A_1,\ldots,A_N} = \varphi_{A'_1,\ldots,A'_N}$ for some $N$-uples $(A_1,\ldots,A_N) \neq (A'_1,\ldots,A'_N)$. Then, we denote
+
+$$
+i_0 := \min\{i \in \{1, \dots, N\} | A'_i \neq A_i\}.
+$$
+
+From the construction of $\varphi_{A_1,...,A_N}$, there exists $C > 0$ such that for $t$ large
+
+$$
+(3.1) \quad \left\| \Phi_{A_1, \dots, A_N}(t) - \Phi_{A_1, \dots, A_{i_0-1}}(t) - \sum_{j=i_0}^{N} A_j e^{-e_j t} Y_{+,j}(t) \right\|_{H^1 \times L^2} \le C \sum_{j=i_0}^{N} e^{-(e_j+\sigma)t}.
+$$
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diff --git a/samples/texts/7426999/page_1.md b/samples/texts/7426999/page_1.md
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+Low Complexity, Parallel Algorithms, and Scalable
+Architectures for Real Time Coherent Optical OFDM
+Systems
+
+Pramod Udupa
+
+► To cite this version:
+
+Pramod Udupa. Low Complexity, Parallel Algorithms, and Scalable Architectures for Real Time Coherent Optical OFDM Systems. Signal and Image processing. Université de Rennes 1, 2014. English. tel-01099824
+
+HAL Id: tel-01099824
+
+https://hal.inria.fr/tel-01099824
+
+Submitted on 8 Jan 2015
+
+**HAL** is a multi-disciplinary open access
+archive for the deposit and dissemination of sci-
+entific research documents, whether they are pub-
+lished or not. The documents may come from
+teaching and research institutions in France or
+abroad, or from public or private research centers.
+
+L'archive ouverte pluridisciplinaire **HAL**, est
+destinée au dépôt et à la diffusion de documents
+scientifiques de niveau recherche, publiés ou non,
+émanant des établissements d'enseignement et de
+recherche français ou étrangers, des laboratoires
+publics ou privés.
\ No newline at end of file
diff --git a/samples/texts/7426999/page_10.md b/samples/texts/7426999/page_10.md
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+<table>
+  <tbody>
+    <tr>
+      <td>2.2.1</td>
+      <td>Linear Impairments</td>
+      <td>28</td>
+    </tr>
+    <tr>
+      <td>2.2.2</td>
+      <td>Non-Linear Impairments.</td>
+      <td>31</td>
+    </tr>
+    <tr>
+      <td>2.3</td>
+      <td>Differences between Wireless-OFDM and CO-OFDM Systems</td>
+      <td>31</td>
+    </tr>
+    <tr>
+      <td>2.4</td>
+      <td>Typical CO-OFDM System</td>
+      <td>32</td>
+    </tr>
+    <tr>
+      <td>2.4.1</td>
+      <td>Coherent Detection.</td>
+      <td>33</td>
+    </tr>
+    <tr>
+      <td>2.4.2</td>
+      <td>OFDM System</td>
+      <td>34</td>
+    </tr>
+    <tr>
+      <td>2.4.3</td>
+      <td>Digital Transmitter.</td>
+      <td>35</td>
+    </tr>
+    <tr>
+      <td>2.4.4</td>
+      <td>RF-to-Optical Up Converter</td>
+      <td>36</td>
+    </tr>
+    <tr>
+      <td>2.4.5</td>
+      <td>Optical-to-RF Down Converter</td>
+      <td>38</td>
+    </tr>
+    <tr>
+      <td>2.4.6</td>
+      <td>Digital OFDM Receiver</td>
+      <td>38</td>
+    </tr>
+    <tr>
+      <td>2.5</td>
+      <td>Complexity Analysis of the System</td>
+      <td>40</td>
+    </tr>
+    <tr>
+      <td>2.5.1</td>
+      <td>Digital Transmitter.</td>
+      <td>41</td>
+    </tr>
+    <tr>
+      <td>2.5.2</td>
+      <td>Digital Receiver</td>
+      <td>42</td>
+    </tr>
+    <tr>
+      <td>2.5.3</td>
+      <td>Time/Frequency Synchronization</td>
+      <td>43</td>
+    </tr>
+    <tr>
+      <td>2.5.4</td>
+      <td>CFO Compensation</td>
+      <td>44</td>
+    </tr>
+    <tr>
+      <td>2.5.5</td>
+      <td>FFT</td>
+      <td>45</td>
+    </tr>
+    <tr>
+      <td>2.5.6</td>
+      <td>Integer CFO Estimation</td>
+      <td>45</td>
+    </tr>
+    <tr>
+      <td>2.5.7</td>
+      <td>Channel Estimation and Equalization</td>
+      <td>46</td>
+    </tr>
+    <tr>
+      <td>2.5.7.1</td>
+      <td>Least Squares (LS).</td>
+      <td>46</td>
+    </tr>
+    <tr>
+      <td>2.5.7.2</td>
+      <td>Normalized Least Mean Squares (NLMS)</td>
+      <td>47</td>
+    </tr>
+    <tr>
+      <td>2.5.8</td>
+      <td>CPE Estimation and Compensation</td>
+      <td>47</td>
+    </tr>
+    <tr>
+      <td>2.5.9</td>
+      <td>Demapper.</td>
+      <td>49</td>
+    </tr>
+    <tr>
+      <td>2.6</td>
+      <td>Observations</td>
+      <td>49</td>
+    </tr>
+    <tr>
+      <td>2.7</td>
+      <td>Conclusions.</td>
+      <td>49</td>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3 Timing Synchronization in OFDM Systems</th>
+      <th style="font-weight: bold;">50</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.1 Introduction.</th>
+      <th style="font-weight: bold;">50</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.2 Timing Synchronization in Wireless OFDM Systems</th>
+      <th style="font-weight: bold;">50</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.3 Proposed Hierarchical Low-Complexity Synchronizer for Wireless OFDM Systems</th>
+      <th style="font-weight: bold;">52</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.3.1 OFDM System Description.</th>
+      <th style="font-weight: bold;">52</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.3.2 Proposed Hierarchical Method.</th>
+      <th style="font-weight: bold;">53</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.3.3 Carrier Frequency Offset (CFO) Estimation.</th>
+      <th style="font-weight: bold;">56</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.4 Simulation Results.</th>
+      <th style="font-weight: bold;">57</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.4.1 Parameters.</th>
+      <th style="font-weight: bold;">57</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.4.2 Mean Square Error (MSE) of Timing Estimate.</th>
+      <th style="font-weight: bold;">57</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.4.3 Mean Square Error (MSE) of CFO Estimate.</th>
+      <th style="font-weight: bold;">58</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.4.4 Complexity of Calculations.</th>
+      <th style="font-weight: bold;">59</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.5 Hierarchical Synchronizer Proposed for CO-OFDM System.</th>
+      <th style="font-weight: bold;">60</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.6 Simulation Results.</th>
+      <th style="font-weight: bold;">61</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.6.1 Parameters.</th>
+      <th style="font-weight: bold;">61</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.6.2 MSE of Timing Estimate.</th>
+      <th style="font-weight: bold;">61</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.6.3 MSE of CFO Estimate.</th>
+      <th style="font-weight: bold;">63</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.7 Need for Parallel Timing Synchronization Architecture.</th>
+      <th style="font-weight: bold;">63</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.8 Proposed Block Parallel Architecture for Auto-Correlation.</th>
+      <th style="font-weight: bold;">66</th>
+    </tr>
+    <tr style="font-weight: bold;">
+      <th colspan="2">3.9 Partial-Streaming Block-Parallel (PSBP) Architecture.</th>
+      <th style="font-weight: bold;">69</th>
+    </tr>
+  </tbody></table>
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diff --git a/samples/texts/7426999/page_100.md b/samples/texts/7426999/page_100.md
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index 0000000000000000000000000000000000000000..6cec772794a591ace501af544bb3c24942722b50
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+FIGURE 3.17: *R* = 4-Parallel Initial Point Auto-Correlation Computation Block for MBA
+
+FIGURE 3.18: R = 4-Parallel Initial Point Energy Computation Block for MBA
+
+TABLE 3.7: Architectural Complexity of $P_{mb}$ for FSBP Architecture as a function of *R*-parallel input/output
+
+<table><thead><tr><th>Algorithm</th><th>Real Multipliers</th><th>Real Adders</th></tr></thead><tbody><tr><td>P<sub>mb</sub> (Initial Point)</td><td>4R</td><td>4R</td></tr><tr><td>P<sub>mb</sub> (Iterative Point)</td><td>4(R + 3)</td><td>2(3R + 3)</td></tr><tr><td>P<sub>mb</sub> (Total)</td><td>8R + 12</td><td>10R + 6</td></tr><tr><td>R<sub>mb</sub> (Initial Point)</td><td>2R</td><td>4R</td></tr><tr><td>R<sub>mb</sub> (Iterative Point)</td><td>2(R + 3)</td><td>(3R + 3)</td></tr><tr><td>R<sub>mb</sub> (Total)</td><td>4R + 6</td><td>7R + 3</td></tr></tbody></table>
+
+### 3.10.3 Comparison of Architectural Complexity
+
+Comparison of Architectural Complexity of proposed FSBP architecture of SCA and MBA with Kaneda's architecture is done. Figures 3.19 and 3.20 show number of multipliers and adders for FSBP architecture as a function of *R*-parallel input/output respectively. Comparison based on area occupied is not straightforward since the number of multipliers required is high compared to Kaneda's architecture but the number of adders required is significantly smaller.
+
+For comparison, area estimates of 2-stage pipelined multiplier and adder of 90 nm
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index 0000000000000000000000000000000000000000..c39964e51b32d79e4b94474c61449004c4400513
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+technology node (Table 3.8) [50] is calculated. Area for proposed FSBP architecture and Kaneda’s architecture for SCA is calculated in Table 3.9. Area savings from 21 to 74% are observed. Area for proposed FSBP architecture for MBA with Kaneda’s architecture is calculated in Table 3.10. Area savings from 17 to 72% are observed.
+
+TABLE 3.8: Area estimates for 2-Stage Pipelined Adders and Multipliers for 90nm technology node
+
+<table><thead><tr><td>Bit Width</td><td>Adder Area (μm²)</td><td>Multiplier Area (μm²)</td></tr></thead><tbody><tr><td>4</td><td>81</td><td>612</td></tr><tr><td>5</td><td>109</td><td>912</td></tr><tr><td>8</td><td>207</td><td>2020</td></tr><tr><td>10</td><td>278</td><td>2953</td></tr><tr><td>16</td><td>530</td><td>6667</td></tr><tr><td>32</td><td>1235</td><td>24444</td></tr><tr><td>64</td><td>2588</td><td>98519</td></tr></tbody></table>
+
+FIGURE 3.19: Multiplier requirement as a function of R-parallel output for FSBP and Kaneda's architecture, M = 32
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+FIGURE 3.20: Adder requirement as a function of R-parallel output for FSBP and Kaneda's architecture, M = 32
+
+FIGURE 3.21: Parallel conjugate symmetric correlation on $R = 4$ PSPB/FSPB architecture. `iter_flag = 0` for this operation.
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diff --git a/samples/texts/7426999/page_103.md b/samples/texts/7426999/page_103.md
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index 0000000000000000000000000000000000000000..e592b184ec88b4845547b19e0dbdbb8145393af0
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+TABLE 3.9: Area calculation of FSBP (Schmidl-Cox) and Kaneda's architecture at 90nm technology node for 5-bit multiplier, 10-bit adder for R = 16-Parallel input/output for Schmidl-Cox Algorithm
+
+<table><thead><tr><td>Training Symbol ($M_{sc}$)</td><td>Proposed Arch. Area ($\mu m^2$)</td><td>Kaneda Arch. Area ($\mu m^2$)</td><td>Area Savings (%)</td></tr></thead><tbody><tr><td>32</td><td>165420</td><td>209600</td><td>21.08</td></tr><tr><td>64</td><td>165420</td><td>351936</td><td>53</td></tr><tr><td>128</td><td>165420</td><td>636608</td><td>74.02</td></tr></tbody></table>
+
+TABLE 3.10: Area calculation of FSBP (Minn-Bhargava) and Kaneda's architecture at 90 nm technology node for 5-bit multiplier, 10-bit adder for R = 16-Parallel input/output for Schmidl-Cox Algorithm
+
+<table><thead><tr><th>Training<br>Symbol ($M_{mb}$)</th><th>Proposed Arch.<br>Area ($\mu m^2$)</th><th>Kaneda Arch.<br>Area ($\mu m^2$)</th><th>Area<br>Savings (%)</th></tr></thead><tbody><tr><td>32</td><td>173828</td><td>209600</td><td>17.07</td></tr><tr><td>64</td><td>173828</td><td>351936</td><td>50.61</td></tr><tr><td>128</td><td>173828</td><td>636608</td><td>72.69</td></tr></tbody></table>
+
+FIGURE 3.22: Parallel energy calculation on R = 4 PSPB/FSPB architecture. iter_flag = 0
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diff --git a/samples/texts/7426999/page_104.md b/samples/texts/7426999/page_104.md
new file mode 100644
index 0000000000000000000000000000000000000000..1ecb1fef36d5f6f313cfb11c65deeafa1ce23d4d
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+## 3.11 Mapping Conjugate Symmetric Correlation onto Proposed PSPB/FSPB architecture
+
+To support the proposed hierarchical synchronization algorithm, the architecture has to support conjugate symmetric correlation operation (Eq. 3.26, Eq. 3.27). Since proposed hierarchical algorithm uses MBA for the first step, the conjugate symmetric correlation operation is mapped onto the proposed PSPB/FSPB architecture for MBA. Figure 3.21 shows the conjugate symmetric correlation mapping on $R = 4$-Parallel PSPB/FSPB architecture. Figure 3.22 shows the mapping of energy calculation of fine time metric calculation on $R = 4$-Parallel PSPB/FSPB architecture. The mapping achieves a parallelism factor of three on $R = 4$-parallel architecture. Table 3.11 shows the factor of parallelism for conjugate symmetric correlation achieved with PSPB/FSPB architecture for different values of $R$-parallel auto-correlation PSPB/FSPB architecture. Note that the mapping only uses PSPB architecture and does not use initial point architecture of FSPB architecture.
+
+TABLE 3.11: Conjugate Symmetric Correlation Parallelism Factor achieved on PSPB/FSPB architecture
+
+<table><thead><tr><td>R</td><td>Parallelism Factor</td></tr></thead><tbody><tr><td>2</td><td>1</td></tr><tr><td>4</td><td>3</td></tr><tr><td>8</td><td>5</td></tr><tr><td>16</td><td>9</td></tr></tbody></table>
+
+## 3.12 Conclusions
+
+In this chapter, a low complexity time synchronization algorithm is proposed which can work in a highly dispersive channel. For complexity reduction of ≈ 80%, a similar MSE performance comparable to cross-correlation only estimator is observed. This algorithm is adapted to optical channel and performance of the algorithm is compared with other auto-correlation algorithms. Next, two types of block parallel architectures were proposed for synchronization algorithms. PSBP architecture provided partial streaming output and required 82% (Schmidl-Cox)/81% (Minn-Bhargava) lesser adder resources compared to Kaneda's proposal. FSBP architecture supports full streaming output and area gains of 21-72% (Schmidl-Cox) and 17-72% (Minn-Bhargava) were observed. Then, conjugate symmetric correlation operation is accelerated on the proposed PSBP/FSBP architecture for MBA to improve the timing estimation. The proposed architecture is scalable and can be generalized for use with any auto-correlation based algorithms.
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diff --git a/samples/texts/7426999/page_105.md b/samples/texts/7426999/page_105.md
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+# Chapter 4
+
+## End-to-End Parallel Streaming Architecture for CO-OFDM System
+
+### 4.1 Introduction
+
+To reach 100 Gb/s total data rate, multi-band CO-OFDM (MB-CO-OFDM) approach is adopted to reduce the pressure on signal converters (DAC/ADC), which presently form the bottleneck in the signal processing chain. Using 50 GHz bandwidth allocated by International Telecommunication Union (ITU) standard, MB-CO-OFDM divides this total bandwidth into multiple non-overlapping sub-bands. Target data rate of 100 Gb/s requires that the total line rate to be around 117 Gb/s to accommodate for overheads in transmission. Considering this scenario, it results in the requirement of single-band to support data rates of Gb/s. Due to requirement of single-polarization to support data rates of the order of Gb/s, choice of algorithms and efficient realization of algorithms used plays a huge part in realizing the goal. OFDM frame structure choices like position and number of training symbols decides the kind of estimation algorithms. For example, training symbol based synchronization significantly reduces the complexity of detection at the receiver compared to blind synchronization. The number of training symbols used can be traded-off with the complexity of channel estimation algorithm used at the receiver, like whether to use LMS or time-frequency domain averaging techniques to keep the complexity down. Efficient realization of algorithms can be broken into two parts, namely adopting efficient parallel architectures and optimizing on fixed-point precision without incurring too much penalty on BER value.
+
+A high level synthesis (HLS) approach has been used for realization of the CO-OFDM architecture, which is described in Section 4.2. This approach is first of its kind in case of CO-OFDM systems. Section 4.3 describes the frame structure, algorithms used for the transmitter and receiver blocks of single-polarization single-band CO-OFDM system.
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diff --git a/samples/texts/7426999/page_106.md b/samples/texts/7426999/page_106.md
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+Section 4.4 shows parallel architecture of the transmitter and the associated fixed-point analysis. Section 4.6 explains the parallel receiver architecture starting from frame synchronization to demapper block. Section 4.7 explains the fixed-point analysis of the receiver architecture. Section 4.8 concludes the results showing the gains due to parallel transceiver architecture and fixed-point optimizations.
+
+## 4.2 A HLS Approach to Designing CO-OFDM System
+
+CO-OFDM system realization has been done using a high-level synthesis language and the CatapultC synthesis tool [21] which allows design entry using C/C++ language with additional libraries for modeling and synthesis of FIFOs (*ac_channel*), complex data types (*ac_complex*) and fixed-point (*ac_fixed*) data types. The major attractive feature of realization using CatapultC is that same source is used for functional verification with Matlab models and simulation/synthesis of RTL code for downstream ASIC/FPGA tools. Figure 4.1 shows CatapultC synthesis flow from specification to RTL code generation and its integration with Matlab for verification of functionality and testing. The flow starts from specification about CO-OFDM system parameters like FFT size, cyclic prefix, algorithms to be used etc. Matlab is a very efficient tool for designing complex mathematical systems and offers visualization capabilities for debugging. Matlab is first used to realize the system and perform simulation varying parameters of all the blocks. The Matlab model now serves a golden reference for checking the functionality of the C/C++ implementation. Hence, integration with Matlab helps in verification of CatapultC C/C++ code.
+
+Now, C/C++ implementation is done which implements the same blocks as in Matlab. C/C++ test bench written checks the basic functionality of the code. Extensive testing is done using Matlab integration. For Matlab testing, wrappers for interfacing with C/C++ code are generated. The wrapper and C/C++ function are imported into Matlab and compiled using MEX compiler. The generated executable is called from Matlab function and is given the same set of inputs as the golden reference. The outputs generated are compared and decision of C/C++ code satisfying the specifications are taken. Initially all data types in C/C++ code are double-precision variables. After passing initial specification test, the data types are converted to fixed-point data types. Again, the simulation of both golden reference and C/C++ is done and outputs compared to check whether it passes specifications. Here, fixed-point exploration is done to minimize the error criterion of choice. After this iteration process with Matlab, the fixed-point data types are fixed and then architecture exploration is done in CatapultC using options provided like Loop Pipelining, Loop Unrolling, FIFO sizing, mapping of memories to either register or SRAM/DRAM, etc. to meet the hardware specifications of operating frequency. Choice of interface is done based on kind of application at hand, for example, in case of streaming application *ac_channel* is used for input and output. Here the tool offers capabilities to target either ASIC or FPGA and also provides accelerated libraries for FPGA implementation. After successfully satisfying the constraints of clock frequency, the RTL code generated is simulated in
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diff --git a/samples/texts/7426999/page_107.md b/samples/texts/7426999/page_107.md
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+FIGURE 4.1: HLS Block Diagram of CatapultC synthesis flow and Matlab Integration
+
+Modelsim using C/C++ test bench. This helps in verifying the generated RTL code and CatapultC also generates scripts for downstream synthesis tools.
+
+The generated RTL code is imported into ASIC/FPGA tool flows. A RTL testbench is written which verifies the functionality of this code for testing after synthesis step. The major advantages in using CatapultC based HLS flow are
+
+* Fixed-point exploration using C code in Matlab, which is used for RTL generation. Same fixed-point libraries used for both simulation and synthesis.
+
+* Architecture exploration by selection of interface, loop pipelining, loop unrolling which are beneficial for DSP algorithms.
+
+## 4.3 Transceiver Algorithms and Frame Structure
+
+In this section, algorithms and frame structure selected for the realization of CO-OFDM systems are explained. The frame structure consists of training symbols (timing synchronization, fractional CFO estimation, integer CFO estimation and channel estimation) and
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diff --git a/samples/texts/7426999/page_108.md b/samples/texts/7426999/page_108.md
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+data symbols. Data symbols contain data and pilot symbols. Pilot symbols are used for phase offset estimation. Frame structure used for single and dual-polarization CO-OFDM systems are shown in Figures 4.2 and 4.3 respectively.
+
+FIGURE 4.2: OFDM frame format for single polarization (Pol$_{X}$) CO-OFDM system
+
+FIGURE 4.3: OFDM frame format for dual polarization (Pol$_{X}$, Pol$_{Y}$) CO-OFDM system
+
+TS₁ (training symbol) is used for timing synchronization and fractional CFO estimation. TS₂ is used for integer CFO estimation and channel estimation. TS₂ is repeated twice [51] to improve the accuracy of channel estimation. QPSK mapping scheme is used for data and pilot symbols. In subsection 4.3.1, the selection of sizes of IFFT/FFT and cyclic prefix (CP) are discussed for a single band single-polarization CO-OFDM system and data rate achieved using this setup is calculated. Subsections 4.3.2 and 4.3.3 describe the algorithms adopted for transmitter and receiver in a single-polarization single-band CO-OFDM system.
+
+### 4.3.1 Design of OFDM Parameters
+
+According to International Telecommunication Union (ITU) standard, a total of 50 GHz is allocated for each band. Optical channel used is standard single mode fiber (SSMF), with dispersion parameters being $\eta^{CD} = 17 \frac{ps}{nm-km}$ and $\eta^{PMD} = 12 ps$. ADC and DAC available with effective number of bits (ENOB) of 8 bits and voltage range of 0.5 Vp-p are used. The bandwidth ($B_w$) used for single-band is 5 GHz, which is the sampling frequency of both DAC and ADC. This sets the bandwidth available to single-band OFDM. The maximum data rate achievable with a single band CO-OFDM is given by
+
+$$D_b = p \cdot \log_2 M \cdot B_w \quad (4.1)$$
+
+where $D_b$ is the data rate in a single-band, $p$ is the number of polarizations used, $p = 1/2$ for single/dual polarization respectively, $M$ is given by the mapping scheme used, $M = 4$ for QPSK mapping, $B_w$ is the bandwidth of the OFDM system. The length of cyclic prefix
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+is calculated by using maximum dispersion delay of SSMF channel.
+
+$$ \tau_{max} = \tau_{max}^{CD} + \tau_{max}^{PMD} \quad (4.2) $$
+
+$$ \tau_{max}^{CD} = \eta_{CD} \cdot L_f \cdot c \cdot B_w / f_0^2 \quad (4.3) $$
+
+$$ \tau_{max}^{PMD} = 3.5 \cdot \eta_{PMD} \cdot \sqrt{L_f} \quad (4.4) $$
+
+where $L_f$ is the fiber length in km, $\eta^{CD}$ is the chromatic dispersion coefficient, $c$ is the speed of light in m/s, $B_w$ is the bandwidth of the system in Hz, $f_0$ is the LASER frequency used in Hz, $\eta^{PMD}$ is the polarization mode dispersion coefficient. The length of cyclic prefix ($L_{cyp}$) has to be greater than maximum dispersion delay ($\tau_{max}$). The number of samples in cyclic prefix is given by
+
+$$ N_{cyp} = L_{cyp} \cdot F_{DAC} \quad (4.5) $$
+
+where $F_{DAC}$ is the sampling frequency of the DAC. $N_{cyp}$ must be sufficiently small compared to length of IFFT/FFT ($N$). A priori, loss in spectral efficiency is fixed
+
+$$ N = N_{cyp} \cdot \frac{1 - \epsilon_{cyp}}{\epsilon_{cyp}} \quad (4.6) $$
+
+The constraint on maximum length of $N$ is given by phase noise variation of LASER, which fixes the maximum value of $N$ [52]. It limits $N$ to values less than or equal to 256 and requires allocation of 10% of sub-carriers for phase offset tracking. Generally, a value of $\epsilon_{cyp} = 0.2$ is chosen. Table 4.1 shows one such calculation for $N_{cyp}$ and $N$. The total
+
+TABLE 4.1: Calculation of $N_{cyp}$ and $N$ given SMF parameters
+
+<table><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Carrier Frequency (f<sub>0</sub>)</td><td>193.1 THz</td></tr><tr><td>Sampling Frequency (F<sub>s</sub><sup>DAC</sup>)</td><td>5 GHz</td></tr><tr><td>Bandwidth (B<sub>w</sub>)</td><td>5 GHz</td></tr><tr><td>Spectral Efficiency loss assumed (&epsilon;<sub>cyp</sub>)</td><td>20 %</td></tr><tr><td>Fiber Lemgth (L<sub>f</sub>)</td><td>1000 km</td></tr><tr><td>Mapping scheme used (M)</td><td>QPSK</td></tr><tr><td>Maximum Delay (&tau;<sub>max</sub>)</td><td>0.69 ns</td></tr><tr><td>Cylic Prefix size (N<sub>cyp</sub>)</td><td>8</td></tr><tr><td>IFFT/FFT size (N)</td><td>256</td></tr><tr><td>Spectral Efficiency loss achieved</td><td>3.125 %</td></tr><tr><td>OFDM symbol duration (T<sub>sym</sub>)</td><td>52.8 ns</td></tr><tr><td>Sub-carrier spacing (<sup>F<sub>s</sub>DAC</sup>&frasl;<sub>N</sub>)</td><td>19.531 MHz</td></tr></tbody></table>
+
+data rate achieved by the single-band OFDM system considering the parameters calculated and by considering loss of spectral efficiency due to forward error correction (FEC) ($\epsilon_{fec}$), loss of spectral efficiency due to training symbols ($\epsilon_{tr}$), loss of efficiency due to use of null
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diff --git a/samples/texts/7426999/page_11.md b/samples/texts/7426999/page_11.md
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+<table>
+  <tbody>
+    <tr>
+      <td>3.9.1 Proposed PSBP architecture for Schmidl-Cox algorithm (SCA)</td>
+      <td>69</td>
+    </tr>
+    <tr>
+      <td>3.9.2 Proposed PSBP architecture of Minn-Bhargava algorithm (MBA)</td>
+      <td>71</td>
+    </tr>
+    <tr>
+      <td>3.9.3 Comparison of Architectural Complexity</td>
+      <td>73</td>
+    </tr>
+    <tr>
+      <td>3.10 Full-Streaming Block-Parallel (FSBP) Architecture</td>
+      <td>74</td>
+    </tr>
+    <tr>
+      <td>3.10.1 Proposed FSBP architecture for SCA</td>
+      <td>74</td>
+    </tr>
+    <tr>
+      <td>3.10.2 Proposed FSBP architecture for MBA</td>
+      <td>75</td>
+    </tr>
+    <tr>
+      <td>3.10.3 Comparison of Architectural Complexity</td>
+      <td>76</td>
+    </tr>
+    <tr>
+      <td>3.11 Mapping Conjugate Symmetric Correlation onto Proposed PSPB/FSPB architecture</td>
+      <td>80</td>
+    </tr>
+    <tr>
+      <td>3.12 Conclusions</td>
+      <td>80</td>
+    </tr>
+    <tr>
+      <td><b>4 End-to-End Parallel Streaming Architecture for CO-OFDM System</b></td>
+      <td><b>81</b></td>
+    </tr>
+    <tr>
+      <td>4.1 Introduction.</td>
+      <td>81</td>
+    </tr>
+    <tr>
+      <td>4.2 A HLS Approach to Designing CO-OFDM System</td>
+      <td>82</td>
+    </tr>
+    <tr>
+      <td>4.3 Transceiver Algorithms and Frame Structure</td>
+      <td>83</td>
+    </tr>
+    <tr>
+      <td>4.3.1 Design of OFDM Parameters</td>
+      <td>84</td>
+    </tr>
+    <tr>
+      <td>4.3.2 Transmitter Algorithm Design</td>
+      <td>86</td>
+    </tr>
+    <tr>
+      <td>4.3.3 Receiver Algorithm Design</td>
+      <td>87</td>
+    </tr>
+    <tr>
+      <td>4.4 Parallel Transmitter Architecture</td>
+      <td>90</td>
+    </tr>
+    <tr>
+      <td>4.5 Fixed Point Analysis of Transmitter Architecture</td>
+      <td>91</td>
+    </tr>
+    <tr>
+      <td>4.6 Parallel Receiver Architecture</td>
+      <td>93</td>
+    </tr>
+    <tr>
+      <td>4.7 Fixed-point Analysis of Receiver Architecture</td>
+      <td>98</td>
+    </tr>
+    <tr>
+      <td>4.7.1 Analysis &amp; Choice of Fixed-point Precision</td>
+      <td>99</td>
+    </tr>
+    <tr>
+      <td>4.7.2 Area vs. Precision</td>
+      <td>99</td>
+    </tr>
+    <tr>
+      <td>4.8 Conclusions</td>
+      <td>104</td>
+    </tr>
+    <tr>
+      <td><b>5 Experimental Validation of CO-OFDM System</b></td>
+      <td><b>107</b></td>
+    </tr>
+    <tr>
+      <td>5.1 Introduction.</td>
+      <td>107</td>
+    </tr>
+    <tr>
+      <td>5.2 Sampling Clock Offset (SCO) Estimation Algorithm</td>
+      <td>108</td>
+    </tr>
+    <tr>
+      <td>5.3 Electrical Back-to-Back (B2B) Experiment.</td>
+      <td>110</td>
+    </tr>
+    <tr>
+      <td>5.4 Electrical B2B Configuration with RF Amplifier</td>
+      <td>112</td>
+    </tr>
+    <tr>
+      <td>5.5 Optical B2B Configuration with Homodyne Coherent Detection.</td>
+      <td>113</td>
+    </tr>
+    <tr>
+      <td>5.6 Heterodyne Coherent Detection Configuration.</td>
+      <td>116</td>
+    </tr>
+    <tr>
+      <td>5.7 Real-Time FPGA Platform</td>
+      <td>117</td>
+    </tr>
+    <tr>
+      <td>5.8 Performance of the Proposed Timing Synchronization Algorithm on Real-Time FPGA Platform</td>
+      <td>123</td>
+    </tr>
+    <tr>
+      <td>5.9 Future Experiments proposed for Real-Time Platform</td>
+      <td>124</td>
+    </tr>
+    <tr>
+      <td>5.10 Conclusions</td>
+      <td>125</td>
+    </tr>
+    <tr>
+      <td><b>6 Conclusions and Perspectives</b></td>
+      <td><b>126</b></td>
+    </tr>
+    <tr>
+      <td>6.1 Overview</td>
+      <td>126</td>
+    </tr>
+    <tr>
+      <td>6.2 Future Work</td>
+      <td>128</td>
+    </tr>
+    <tr>
+      <td>6.2.1 Real-time FPGA platform experiments</td>
+      <td>128</td>
+    </tr>
+    <tr>
+      <td>6.2.2 Dual-polarization CO-OFDM System</td>
+      <td>128</td>
+    </tr>
+    <tr>
+      <td>6.2.3 Time Domain Sampling Clock Offset (SCO) Algorithm.</td>
+      <td>128</td>
+    </tr>
+    <tr>
+      <td>6.3 Scaling to more than 100 Gb/s with MB-CO-OFDM system.</td>
+      <td>128</td>
+    </tr>
+  </tbody>
+</table>
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+sub-carriers ($\epsilon_{null}$),
+
+$$D_b = (1 - \epsilon_{fec})(1 - \epsilon_{tr})(1 - \epsilon_{null})(1 - \epsilon_{cyp}) \log_2(M) B_w \quad (4.7)$$
+
+Considering $\epsilon_{fec} = 0.0627$, $\epsilon_{tr} = 0.1$, $\epsilon_{null} = 0.1$, we get $D_b = 7.362$ Gb/s for single polarization. For all further calculations, the values of $N_{cyp} = 8$ and $N = 256$ is used for designing both optical experiments and hardware implementation. To attain 100 Gb/data rate, eight sub-bands with dual polarizations are used. The total data rate achievable is
+
+$$D_{b,total} = 2 \cdot D_b \cdot N_{sb} \quad (4.8)$$
+
+$$= 117.792 \text{ Gb/s} \quad (4.9)$$
+
+where $D_{b,total}$ is the total data rate in 50 GHz channel, $N_{sb}$ is the total number of sub-bands used. A guard band of 1 GHz is used for separating the sub-bands.
+
+### 4.3.2 Transmitter Algorithm Design
+
+In transmitter, IFFT block is the major block to design. For the transmitter, the choice of radix used for IFFT can reduce the total number of operations. The algorithmic complexity for $N = 256$ in terms of millions of operations per second (MOPS)/giga operations per second (GOPS)/tera operations per second (TOPS) is calculated for radix-2,4,2² and split-radix algorithms. Table 4.2 shows the number of operations per single-band for supporting 7.3 Gb/s bit rate. Then, total number of operations for supporting $D_{b,total} \ge 100$ Gb/s is shown in last column of Table 4.2.
+
+In case of sub-band design, same IFFT is used across all sub-bands and polarizations. Any gains obtained by reduction of number of operations in single-polarization single-band IFFT is multiplied across all polarizations and sub-bands. Hence, a choice of low-complexity blocks in single-band in MB-CO-OFDM results in savings of area that will be multiplied by number of sub-bands used. From the last column of Table 4.2, it can be seen that total savings of 800 GOPS for radix-4/2² over radix-2 is obtained, while split-radix gains 200 GOPS over radix-4/2². A choice of radix-2² is made over radix-4/split-radix due to lesser complex parallel architecture. GOPS calculation is done as follows: GOPS = Total Operations · $B_w/N$. TOPS = GOPS · 16.
+
+TABLE 4.2: Algorithmic Complexity for calculation of *N* output for IFFT size of 256
+
+<table><thead><tr><th>Radix Used</th><th>Real Multipli-cations</th><th>Real Additions</th><th>Total Operations</th><th>GOPS (D<sub>b</sub>) for 7.3 Gb/s</th><th>TOPS (D<sub>b,total</sub>) for 117 Gb/s</th></tr></thead><tbody><tr><td>Radix-2</td><td>4096</td><td>6144</td><td>10240</td><td>294.4</td><td>4.7</td></tr><tr><td>Radix-4</td><td>3072</td><td>5632</td><td>8704</td><td>248.2</td><td>3.9</td></tr><tr><td>Radix-2<sup>2</sup></td><td>3072</td><td>5632</td><td>8704</td><td>248.2</td><td>3.9</td></tr><tr><td>Split-Radix</td><td>2731</td><td>5462</td><td>8193</td><td>233.6</td><td>3.7</td></tr></tbody></table>
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diff --git a/samples/texts/7426999/page_111.md b/samples/texts/7426999/page_111.md
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+### 4.3.3 Receiver Algorithm Design
+
+The algorithms used for timing synchronization, fractional CFO estimation, FFT, integer CFO estimation, channel estimation and equalization, phase error estimation and compensation are described. Algorithmic complexity for calculation of *N* outputs is shown and the total complexity for 117 Gb/s system is also calculated. Optimizations obtained by using specific data format of training symbols are used to eliminate multiplications. For example, in case of LS channel estimation, multiplication by reference symbol of [±1 ± 1j] can be converted into a look-up table and hence complex multiplications can be avoided. Wherever such optimizations are used, reduction in complexity is calculated.
+
+* Coarse Time Synchronization - The algorithm used is the proposed algorithm due to its superior performance over other auto-correlation algorithms. The algorithmic complexity for calculation of *N* outputs is shown in Table 4.3, for one sub-band and for 117 Gb/s output. Fractional CFO is estimated using auto-correlation value at the index corresponding to start point.
+
+$$ 
+\begin{aligned}
+P_{mb}[n+1] = P_{mb}[n] & - r^*[n] \cdot r[n + M_{mb}] && (4.10) \\
+& - r^*[n + 3M_{mb} \cdot r[n + 4M_{mb}]] \\
+& + 2 \cdot r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}]
+\end{aligned}
+$$
+
+where $P_{mb}$ is the auto-correlation function, $M_{mb}$ is the length of repeating training symbol used [$A A A - A$].
+
+TABLE 4.3: Algorithmic Complexity (auto-correlation function only) for Proposed Syn-chronization Algorithm
+
+<table><thead><tr><th>Algo. Used</th><th>Real Multipli-cations</th><th>Real Additions</th><th>Total Operations</th><th>GOPS ($D_b$) per sub-band</th><th>TOPS ($D_{b,total}$) for 117 Gb/s</th></tr></thead><tbody><tr><td>Minn-Bhargava ($L = 4$)</td><td>3072</td><td>3072</td><td>6144</td><td>175.2</td><td>2.8</td></tr></tbody></table>
+
+* FFT - Radix-2² FFT is chosen because of lower algorithmic complexity compared to radix-2 and its better architectural complexity and scalability which will be shown in Section 4.4.
+
+* Integer CFO Estimation - Cross-correlation with known sequence of training symbol is chosen for integer CFO estimation. Consider a known complex sequence of length $N_{ifo} = N/4$, given by $z[n]$. Since, integer CFO estimation is done in frequency domain, the known sequence can be QPSK sequence with values (±1 ± 1j). The
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+cross-correlation operation to determine the integer CFO can be written as
+
+$$M_{ifo}[n] = |P_{ifo}[n]|^2 \quad (4.11)$$
+
+$$P_{ifo}[n] = \sum_{m=0}^{N_{ifo}-1} r[n+m] \cdot z^*[n+m] \quad (4.12)$$
+
+where $n$ is the search index, $n \in [-W_s, \dots, -2, 0, 2, \dots, W_s]$, where $W_s$ is the maximum search index. Here $W_s = 20$ is chosen as the maximum search window value. The value of $N_{ifo}$ is chosen to be 32. The algorithmic complexity is shown in Table 4.4. Due to use of QPSK constellation ($\pm 1 \pm 1j$), complex multiplications can be completely avoided and complexity reduced. Savings of 39.8 MOPS is obtained by this optimization.
+
+TABLE 4.4: Algorithmic Complexity for integer CFO estimation algorithm
+
+<table><thead><tr><th>Algo.<br>Used</th><th>Real<br>Multipli-<br>cations</th><th>Real<br>Add-<br>itions</th><th>Total<br>Oper-<br>ations</th><th>MOPS (D<sub>b</sub>)<br>per<br>sub-band</th><th>MOPS<br>(D<sub>b,total</sub>) for<br>100Gb/s</th></tr></thead><tbody><tr><td>IFO<br>Estimation<br>Non-Optimized</td><td>15360</td><td>7680</td><td>23040</td><td>3.68</td><td>59</td></tr><tr><td>IFO<br>Estimation<br>Optimized</td><td>82</td><td>2624</td><td>2706</td><td>1.3</td><td>20.8</td></tr></tbody></table>
+
+* Channel Estimation & Equalization - The algorithms of least squares (LS) and normalized mean least squares (NLMS) algorithms have been used for channel estimation. Algorithmic Complexity is given in Table 4.5. Here, LS method's for complexity can be reduced by using multiplication by symbol [$\pm 1 \pm 1j$] and complex multiplications avoided. This optimization works for both Single Polarization and Dual Polarization transmission. Savings of 29.2 GOPS is obtained for LS channel estimation and 21.9 GOPS is obtained for NLMS channel estimation method for single-polarization, single-band by using this optimization.
+
+* CPE Estimation & Compensation - Pilot based CPE estimation [15] is done for estimation of LASER phase noise. Optimization can be done by using [$\pm 1 \pm 1j$] symbol and thus complex multiplications can be avoided. Algorithmic Complexity of CPE compensation is given in Table 4.6.
+
+From the algorithms chosen, it can be seen that savings of 800 GOPS is obtained by choosing radix-2² IFFT/FFT over radix-2. In the case of Integer CFO, lower complexity method cross-correlation was adopted and optimized to save 39.8 MOPS. In case of LS/NLMS channel estimation, savings of 29.2/21.9 GOPS were obtained due to optimization.
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diff --git a/samples/texts/7426999/page_113.md b/samples/texts/7426999/page_113.md
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+TABLE 4.5: Algorithmic Complexity for Channel Estimation algorithms
+
+<table><thead><tr><th>Algo.<br>Used</th><th>Real<br>Multipli-<br>cations</th><th>Real<br>Add-<br>itions</th><th>Total<br>Oper-<br>ations</th><th>GOPS (D<sub>b</sub>)<br>per<br>sub-band</th><th>TOPS<br>(D<sub>b,total</sub>) for<br>117 Gb/s</th></tr></thead><tbody><tr><td>LS<br>Single Pol.<br>Non-Optimized</td><td>2048</td><td>768</td><td>2816</td><td>80.3</td><td>1.28</td></tr><tr><td>LS<br>Single Pol.<br>Optimized</td><td>1024</td><td>768</td><td>1792</td><td>51.1</td><td>0.81</td></tr><tr><td>LS<br>Dual Pol.<br>Non-Optimized</td><td>8192</td><td>3072</td><td>11264</td><td>321.2</td><td>5.1</td></tr><tr><td>LS<br>Dual Pol.<br>Optimized</td><td>4096</td><td>3072</td><td>7168</td><td>204.4</td><td>3.2</td></tr><tr><td>NLMS<br>Single Pol.<br>Non-Optimized</td><td>2816</td><td>2048</td><td>4864</td><td>138.7</td><td>2.2</td></tr><tr><td>NLMS<br>Single Pol.<br>Optimized</td><td>2304</td><td>1792</td><td>4096</td><td>116.8</td><td>1.8</td></tr><tr><td>NLMS<br>Dual Pol.<br>Non-Optimized</td><td>5632</td><td>4096</td><td>9728</td><td>277.4</td><td>4.4</td></tr><tr><td>NLMS<br>Dual Pol.<br>Optimized</td><td>4608</td><td>3584</td><td>8192</td><td>233.6</td><td>3.7</td></tr></tbody></table>
+
+TABLE 4.6: Algorithmic Complexity for CPE Compensation
+
+<table><thead><tr><th>Algo.<br>Used</th><th>Real<br>Multipli-<br>cations</th><th>Real<br>Add-<br>itions</th><th>Total<br>Oper-<br>ations</th><th>GOPS (D<sub>b</sub>)<br>per<br>sub-band</th><th>GOPS<br>(D<sub>b,total</sub>) for<br>100Gb/s</th></tr></thead><tbody><tr><th>CPE<br>Estimation<br>Optimized</th><td>1024</td><td>512</td><td>1536</td><td>43.8</td><td>700.8</td></tr></tbody></table>
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+## 4.4 Parallel Transmitter Architecture
+
+In this section, parallel architecture for CO-OFDM transmitter is explained. Proposed parallel architecture of transmitter consists of a parallel mapper, IFFT and pre-emphasis block where number of parallel output times FPGA clock frequency is equal to DAC clock frequency. The mapper block supports mapping of normalized constellations from BPSK, QPSK, 16-QAM and 64-QAM. It is implemented using look-up table (LUT) where a single table is used to support all the above constellations. Since CO-OFDM Transmitter complexity depends on IFFT size [53], choice of radix plays a big role in deciding the overall complexity. The IFFT is implemented using radix-2$^2$ algorithm [54] which uses the same number of complex multipliers as radix-4 but uses radix-2 butterfly unit as basic block of computation, which is much simpler than radix-4 basic block of computation. There are mainly two types of pipelined architectures [54] for IFFT:
+
+* Feed-forward
+
+1. Multipath Delay Commutator (MDC)
+
+2. Single-path Delay Commutator (SDC)
+
+* Feedback
+
+1. Multipath Delay Feedback (MDF)
+
+2. Single path Delay Feedback (SDF)
+
+Since feedback based architectures do not provide parallel outputs every clock cycle, only feed-forward based architectures are considered. Similar comment holds good for SDC feed-forward architecture. Only MDC feedforward architectures can provide parallel outputs every clock cycle and can be parallelized to provide higher number of parallel outputs.
+The IFFT equation of radix-2$^2$ is given by
+
+$$x(n_1 + 2n_2 + 4n_3) = \sum_{k_3=0}^{\frac{N}{4}-1} \left[ H(n_1, n_2, k_3) \cdot W_N^{k_3(n_1+2n_2)} \right] W_{\frac{N}{4}}^{n_3 k_3} \quad (4.13)$$
+
+$$H(n_1, n_2, k_3) = \left[ X(k_3 + (-1)^{n_1} X(k_3 + \frac{N}{2}) + (-j)^{n_1+2n_2} X(k_3 + \frac{N}{4}) + (-1)^{n_1} X(k_3 + \frac{3N}{4}) \right] \quad (4.14)$$
+
+where $x[n]$ is the IFFT output, $X[k]$ is the input, $N$ is the size of IFFT, $W$ is the twiddle factor multiplication. There are two kinds of architecture based on order of input for radix-2$^2$ MDC IFFT. A novel architecture shown in Figure 4.4 is proposed based on input order supplied with even and odd indices [55] separated. The proposed architecture has more uniform routing architecture, but uses one extra complex multiplier compared to previously proposed architecture shown in Figure 4.5 [5]. In Figure 4.5, the inputs are applied in normal order. The routing architecture is more complicated.
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diff --git a/samples/texts/7426999/page_115.md b/samples/texts/7426999/page_115.md
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+FIGURE 4.4: IFFT/FFT Architecture of 4-Parallel radix-2² for N = 256, when input is given in even and odd index order
+
+FIGURE 4.5: IFFT/FFT Architecture of 4-Parallel radix-2² for N = 256, when input is given in normal order
+
+Since architecture of Figure 4.5 uses one less complex multiplier compared to proposed
+architecture (Figure 4.4), the architecture with normal input order is chosen. Table 4.7
+compares the architectural complexity of radix-2<sup>2</sup> with radix-2/4/8/16 for different parallel
+outputs. It shows the scalability of radix-2<sup>2</sup> for different parallel outputs compared to
+radix-2/4/8/16. The amount of resources required by radix-2<sup>2</sup> is closest to the minimum
+resources for all number of parallel outputs. Comparatively, as number of parallel outputs
+increases, lower radix IFFT consumes more resources.
+
+## 4.5 Fixed Point Analysis of Transmitter Architecture
+
+Fixed-point analysis of the transmitter architecture is done in this Section. A normalized
+IFFT is implemented where $1/\sqrt{N}$ scaling is applied to both IFFT and FFT equations. After
+addition of cyclic prefix, the output signal is then scaled and clipped so as to match the
+input amplitude range of DAC. The DAC takes parallel samples from output of scaling
+block and converts it into analog output. The mapper receives binary data at input and
+maps it onto complex constellations, which is represented by fixed-point data type. The
+mapper outputs complex symbols from a normalized constellation of either BPSK, QPSK,
+16-QAM or 64-QAM. Since 64-QAM has the largest range of output, the mapper's dynamic
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+TABLE 4.7: Architectural Complexity (normal input order) for full streaming outputs for $N = 256$, with input and output in natural order. Resource count is generated by using SPIRAL tool [4] for radix-2/4/8/16 and using [5] for radix-2²
+
+<table><thead><tr><th>Radix</th><th>Multipliers</th><th>Adders</th></tr></thead><tbody><tr><td colspan="3">R = 2-Parallel Output</td></tr><tr><td>2</td><td>28</td><td>46</td></tr><tr><td>2<sup>2</sup></td><td>24</td><td>44</td></tr><tr><td colspan="3">R = 4-Parallel Output</td></tr><tr><td>2</td><td>48</td><td>88</td></tr><tr><td>4</td><td>36</td><td>82</td></tr><tr><td>2<sup>2</sup></td><td>36</td><td>82</td></tr><tr><td colspan="3">R = 8-Parallel Output</td></tr><tr><td>2</td><td>84</td><td>172</td></tr><tr><td>4</td><td>72</td><td>164</td></tr><tr><td>8</td><td>60</td><td>162</td></tr><tr><td>2<sup>2</sup></td><td>72</td><td>164</td></tr><tr><td colspan="3">R = 16-Parallel Output</td></tr><tr><td>2</td><td>156</td><td>340</td></tr><tr><td>4</td><td>120</td><td>320</td></tr><tr><td>8</td><td>120</td><td>324</td></tr><tr><td>16</td><td>108</td><td>318</td></tr><tr><td>2<sup>2</sup></td><td>108</td><td>310</td></tr></tbody></table>
+
+range is fixed by maximum and minimum values of 64-QAM output. Consider the IFFT equation implemented by the transmitter:
+
+$$x[n] = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X[k] \cdot e^{j2\pi nk/N} \quad (4.15)$$
+
+From a fixed-point design perspective of IFFT, the main observation is that IFFT block provides input to DAC, which is precision limited to 6-8 bits. Hence, the output precision of IFFT is precision limited by precision input of DAC. This is opposite of that in receiver, where FFT occurs after ADC block. Hence, fixed-point precision computation at IFFT and FFT needs to be different and this asymmetry can be used for resource optimization in case of IFFT. Computation precision of IFFT which is closer to DAC precision is sufficient and any extra precision used for calculation in IFFT will be discarded at the DAC. Based on this observation, IFFT area optimization is done.
+
+Table 4.8 shows the variation of Root Mean Square Error (RMSE) with resolution of input/output bits ($W_i$) and twiddle factor inputs ($W_t$). RMSE for fixed-point output is evaluated using
+
+$$RMSE = \sqrt{\frac{\sum_{n=0}^{N-1} (S_n - T_n)^2}{N}} \quad (4.16)$$
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+where $S_n$ is the actual fixed-point output of IFFT while $T_n$ is the double-precision floating-point output used as reference. Figure 4.6 shows a semilog plot of mean of RMSE as a function of variation of $W_i$ for different values of $W_t$. It can be observed that minimum value of $W_t \ge 7$ is required to ensure low value of mean of RMSE. The value of $W_i$ chosen depends on the input resolution of DAC. Next, parallel architecture of transmitter with different values of $W_t \ge 7$ and $W_i \ge 6$ is generated with CatapultC HLS tool. It allows hardware exploration in terms of pipelining, loop unrolling, etc. to achieve a high throughput architecture. The gains in area due to usage of lower fixed-point precision value to achieve a particular value of RMSE is explored. Table 4.9 shows the resources usage in terms of LUTs used as a function of different values of $W_i$ and $W_t$. From Table 4.9, it can be seen that resource consumption of IFFT is a strong function of input precision ($W_i$) and a weak function of twiddle factor precision ($W_t$). Percentage increase in area from $W_i = 6$ to 10 bits for $W_t = 8,9,10$ bits is 57%,62%,66% respectively. Fixed-point optimization with respect to $W_i$ does offer huge savings in resource usage.
+
+TABLE 4.8: Mean (μ) and Standard Deviation (σ) of RMSE for variation of Bitwidths of inputs/outputs $W_i$ and Twiddle Factor $W_t$
+
+<table><thead><tr><th rowspan="2">Bitwidth<br>$W_t$</th><th rowspan="2">RMSE</th><th colspan="7">Bitwidth ($W_i$)</th></tr><tr><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th></tr></thead><tbody><tr><td rowspan="2">4</td><td>$\mu$</td><td>5.5x10<sup>-2</sup></td><td>3.2x10<sup>-2</sup></td><td>2.1x10<sup>-2</sup></td><td>1.9x10<sup>-2</sup></td><td>1.8x10<sup>-2</sup></td><td>1.7x10<sup>-2</sup></td><td>1.7x10<sup>-2</sup></td></tr><tr><td>$\sigma$</td><td>1.6x10<sup>-3</sup></td><td>7.7x10<sup>-4</sup></td><td>5.4x10<sup>-4</sup></td><td>5.4x10<sup>-4</sup></td><td>5.5x10<sup>-4</sup></td><td>5.3x10<sup>-4</sup></td><td>5.3x10<sup>-4</sup></td></tr><tr><td rowspan="2">5</td><td>$\mu$</td><td>5.3x10<sup>-2</sup></td><td>2.7x10<sup>-2</sup></td><td>1.5x10<sup>-2</sup></td><td>1.0x10<sup>-2</sup></td><td>8.8x10<sup>-3</sup></td><td>8.4x10<sup>-3</sup></td><td>8.3x10<sup>-3</sup></td></tr><tr><td>$\sigma$</td><td>1.6x10<sup>-3</sup></td><td>8.8x10<sup>-4</sup></td><td>4.3x10<sup>-4</sup></td><td>3.0x10<sup>-4</sup></td><td>2.5x10<sup>-4</sup></td><td>2.7x10<sup>-4</sup></td><td>2.6x10<sup>-4</sup></td></tr><tr><td rowspan="2">6</td><td>$\mu$</td><td>5.2x10<sup>-2</sup></td><td>2.6x10<sup>-2</sup></td><td>1.3x10<sup>-2</sup></td><td>7.4x10<sup>-3</sup></td><td>4.8x10<sup>-3</sup></td><td>3.9x10<sup>-3</sup></td><td>3.7x10<sup>-3</sup></td></tr><tr><td>$\sigma$</td><td>1.5x10<sup>-3</sup></td><td>7.4x10<sup>-4</sup></td><td>3.8x10<sup>-4</sup></td><td>2.2x10<sup>-4</sup></td><td>1.2x10<sup>-4</sup></td><td>1.2x10<sup>-4</sup></td><td>1.1x10<sup>-4</sup></td></tr><tr><td rowspan="2">7</td><td>$\mu$</td><td>5.2x10<sup>-2</sup></td><td>2.6x10<sup>-2</sup></td><td>1.3x10<sup>-2</sup></td><td>7.0x10<sup>-3</sup></td><td>3.9x10<sup>-3</sup></td><td>2.8x10<sup>-3</sup></td><td>2.4x10<sup>-3</sup></td></tr><tr><td>$\sigma$</td><td>1.3x10<sup>-3</sup></td><td>7.5x10<sup>-4</sup></td><td>3.9x10<sup>-4</sup></td><td>1.7x10<sup>-4</sup></td><td>9.5x10<sup>-5</sup></td><td>6.9x10<sup>-5</sup></td><td>6.3x10<sup>-5</sup></td></tr><tr><td rowspan="2">8</td><td>$\mu$</td><td>5.2x10<sup>-2</sup></td><td>2.6x10<sup>-2</sup></td><td>1.3x10<sup>-2</sup></td><td>6.6x10<sup>-3</sup></td><td>3.4x10<sup>-3</sup></td><td>2.0x10<sup>-3</sup></td><td>1.4x10<sup>-3</sup></td></tr><tr><td>$\sigma$</td><td>1.5x10<sup>-3</sup></td><td>6.8x10<sup>-4</sup></td><td>3.9x10<sup>-4</sup></td><td>2.0x10<sup>-4</sup></td><td>1.0x10<sup>-4</sup></td><td>5.7x10<sup>-5</sup></td><td>3.4x10<sup>-5</sup></td></tr><tr><td rowspan="2">9</td><td>$\mu$</td><td>5.2x10<sup>-2</sup></td><td>2.5x10<sup>-2</sup></td><td>1.3x10<sup>-2</sup></td><td>6.5x10<sup>-3</sup></td><td>3.3x10<sup>-3</sup></td><td>1.7x10<sup>-3</sup></td><td>9.7x10<sup>-4</sup></td></tr><tr><td>$\sigma$</td><td>1.4x10<sup>-3</sup></td><td>7.8x10<sup>-4</sup></td><td>3.3x10<sup>-4</sup></td><td>1.8x10<sup>-4</sup></td><td>1.0x10<sup>-4</sup></td><td>5.1x10<sup>-5</sup></td><td>2.8x10<sup>-5</sup></td></tr><tr><td rowspan="2">10</td><td>$\mu$</td><td>5.2x10<sup>-2</sup></td><td>2.6x10<sup>-2</sup></td><td>1.3x10<sup>-2</sup></td><td>6.5x10<sup>-3</sup></td><td>3.2x10<sup>-3</sup></td><td>1.6x10<sup>-3</sup></td><td>8.3x10<sup>-4</sup></td></tr><tr><td>$\sigma$</td><td>1.4x10<sup>-3</sup></td><td>7.6x10<sup>-4</sup></td><td>3.4x10<sup>-4</sup></td><td>1.9x10<sup>-4</sup></td><td>9.9x10<sup>-5</sup></td><td>4.4x10<sup>-5</sup></td><td>2.6x10<sup>-5</sup></td></tr></tbody></table>
+
+## 4.6 Parallel Receiver Architecture
+
+In this section, parallel architecture of the receiver blocks in the chain are explained starting from time synchronization to demapper. Each of the blocks are parallelized and for all illustrations $R = 4$-parallel architecture is shown. Figure 4.7 shows the end-to-end connectivity of the parallel processing blocks starting from ADC to demapper. There are three memory blocks used for data to be stored temporarily while the different estimation blocks perform processing.
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+FIGURE 4.6: Plot of Mean of RMSE output of IFFT as function of $W_t$ and $W_t$.
+
+TABLE 4.9: Area Occupied for variation of Bitwidths of inputs/outputs $W_t$ and Twiddle Factor $W_t$
+
+<table><thead><tr><th rowspan="2">Bitwidth<br>$W_t$</th><th rowspan="2">Area in<br>Xilinx FPGA</th><th colspan="3">Bitwidth ($W_t$)</th></tr><tr><th>6</th><th>8</th><th>10</th></tr></thead><tbody><tr><td rowspan="2">8</td><td>LUTs</td><td>38294</td><td>50187</td><td>60186</td></tr><tr><td>DSP Multipliers</td><td>36</td><td>36</td><td>36</td></tr><tr><td rowspan="2">9</td><td>LUTs</td><td>38922</td><td>52543</td><td>63107</td></tr><tr><td>DSP Multipliers</td><td>36</td><td>36</td><td>36</td></tr><tr><td rowspan="2">10</td><td>LUTs</td><td>40165</td><td>54652</td><td>66763</td></tr><tr><td>DSP Multipliers</td><td>36</td><td>36</td><td>36</td></tr></tbody></table>
+
+1. Time/Frequency Synchronization Memory - It receives data from ADC and gives it to the synchronization block. This memory is organized to read/write $R = 4$ samples every cycle and provides input to $R = 4$-parallel time synchronization block. Along with this memory, there is also a ping-pong memory block which also processes $R = 4$ samples every cycle. These two memory blocks together provide eight samples every cycle required for the time synchronization block. The size of synchronization memory is $6N$ samples, while the ping-pong memory is $N$ samples. Figure 4.8 shows the organization of data in the synchronization memory for $R = 4$-Parallel input. Data organization in the ping-pong memory is the same as in this memory. After start of symbol estimation and CFO estimation, the starting address is given to FFT memory. The size and data organization of FFT memory is exactly the same
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+FIGURE 4.7: Proposed CO-OFDM Receiver Architecture Block Diagram
+
+FIGURE 4.8: Data organization in the Synchronization Memory
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+CONTENTS ix
+
+Publications 130
+
+Bibliography 131
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+TABLE 4.10: Architectural Complexity of Time/Frequency Architecture for $R = 4$-Parallel input/output
+
+<table><thead><tr><th>Algorithm</th><th>Real Multipliers</th><th>Real Adders</th><th>Total Memory Locations</th></tr></thead><tbody><tr><td>P<sub>mb</sub> (Total)</td><td>44</td><td>46</td><td>4096</td></tr></tbody></table>
+
+as synchronization memory.
+
+2. **Time/Frequency Synchronization Block** - It uses Full-Streaming Block-Parallel (FSBP) architecture presented in Chapter 3 to provide real-time low-complexity synchronization. Architectural complexity is calculated in Table 4.10, which also includes memory requirement of synchronization memory.
+
+3. **CFO Compensation** - This block receives data from FFT memory after removal of cyclic prefix. It receives fractional CFO estimate from synchronization block and integer CFO estimate from integer CFO estimation block. It compensates the CFO by using $R = 4$-parallel multipliers. Architectural Complexity is 16 real multipliers and 16 real adders.
+
+4. **FFT** - It receives CFO compensated data and outputs data in frequency domain. It uses radix-2$^2$ $R = 4$-parallel architecture. The architectural complexity is given in Table 4.7.
+
+5. **Integer CFO Estimation** - Figure 4.9 shows the parallel architecture for integer CFO estimation using Equation 4.11. The look-up table implemented for multiplying by complex conjugate of reference input is given in Table 4.11. Hence it requires only two adders for performing multiplication with conjugate of reference input symbol. IFO estimation block uses a $4N$-size memory which stores the input till output is computed. The delay incurred is equal to receiving $3N$ amount of samples. IFO compensation involved at this point is done by reading from starting address which is equal to integer CFO predicted. Architectural complexity is given in Table 4.12.
+
+FIGURE 4.9: Parallel Architecture for IFO Estimation
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+TABLE 4.11: Look-up table implemented for complex multiplication of conjugate of reference symbol with input $r = a + jb$.
+
+<table><thead><tr><th>Reference QPSK Value</th><th>Real Output</th><th>Imaginary Output</th></tr></thead><tbody><tr><td>1 + j</td><td>a + b</td><td>-a + b</td></tr><tr><td>1 - j</td><td>a - b</td><td>a + b</td></tr><tr><td>-1 + j</td><td>-a + b</td><td>-a - b</td></tr><tr><td>-1 - j</td><td>-a - b</td><td>a - b</td></tr></tbody></table>
+
+TABLE 4.12: Architectural Complexity of IFO Estimation Architecture for R = 4-Parallel input
+
+<table><thead><tr><th>Algorithm</th><th>Real Multipliers</th><th>Real Adders</th><th>Memory Size</th></tr></thead><tbody><tr><td>IFO Estimator</td><td>2</td><td>19</td><td>2048</td></tr></tbody></table>
+
+6. De-interleaver - It removes the unused sub-carriers in the OFDM symbol and provides data and pilot sub-carriers to the next block. It also calculates energy of the non-zero sub-carrier samples. Architectural complexity is 8 real multipliers and 512 memory locations.
+
+7. Channel Estimation & Compensation - Figure 4.10 shows the architecture of the channel estimation and equalization block for single sample input. It receives input sample and energy calculated from de-interleaver block. It also gets reference symbol for LS channel estimation or old NLMS channel estimate from memory. The updated value is written back to memory. Channel equalization is done using $H_{k,old}$ and updated $H_k$ value calculated is written to memory for use in next iteration. The multiplexer selects input to be given to LS channel estimator or NLMS estimator. The LUT block is used to calculate the inverse of input energy. Architectural Complexity for R = 4-Parallel Channel Estimator and Equalizer is given in Table 4.13.
+
+TABLE 4.13: Architectural Complexity of Channel Estimator/Equalizer for R = 4- Parallel input/output
+
+<table><thead><tr><th>Algorithm</th><th>Real Multipliers</th><th>Real Adders</th><th>Total Memory Locations</th></tr></thead><tbody><tr><td>Channel Estimator and Equalizer</td><td>36</td><td>40</td><td>512</td></tr></tbody></table>
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+FIGURE 4.10: Channel Estimation and Equalization Architecture which supports both LS and NLMS equalizers
+
+8. Common Phase Estimation & Compensation - Figure 4.11 shows the architecture of CPE estimation. Compensation consists of complex multiplication by using the phase error estimated. Architectural Complexity for CPE Estimation and Compensation is given in Table 4.14.
+
+FIGURE 4.11: CPE Estimation Block
+
+## 4.7 Fixed-point Analysis of Receiver Architecture
+
+The goal of fixed-point analysis is to reduce the bitwidth used for computation in all DSP blocks of the receiver without degrading the performance significantly. The analysis gives details about the bitwidth for each of the DSP blocks in the receiver chain. It helps
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+TABLE 4.14: Architectural Complexity of CPE Estimator and Compensator for R = 4-Parallel input/output
+
+<table><thead><tr><td>Algorithm</td><td>Real Multipliers</td><td>Real Adders</td></tr></thead><tbody><tr><td>CPE Estimator and Compensator</td><td>16</td><td>22</td></tr></tbody></table>
+
+in identification of blocks whose precision affect more BER more significantly compared to others. This helps in aggressive optimization of such blocks. After selection of fixed-point bitwidths of all the blocks, the area vs. bitwidth variation for each of the blocks is calculated. This table helps explore optimizations which can lead to huge area savings in large blocks like FFT, channel estimator, etc with certain loss in BER. Finally, the area occupied by individual blocks after fixed-point optimization is shown in a pie-chart.
+
+### 4.7.1 Analysis & Choice of Fixed-point Precision
+
+Performance of the system using floating-point computation is considered as reference for fixed-point optimization. The conversion of the floating-point system to fixed-point system is done in a step-by-step manner. Starting from the first block of Time/Frequency synchronization, each block is converted first from floating-point computation to higher precision of fixed-point computation (generally 16-bit fixed-point number). Then, performance attained by this higher precision fixed-point computation is noted and then precision of the blocks is lowered in a linear manner. The results of this process is shown in Table 4.16 for different configurations after optimization. The various configurations used are detailed in Table 4.15. Figure 4.12 shows the performance of the fixed-point receiver for various configurations for homodyne setup. Fixed-point optimization for CFO compensation block and integer CFO estimation block is done using BER vs. OSNR curve of heterodyne setup. Table 4.18 shows the different configurations explored for heterodyne setup and Figure 4.13 shows the performance of the fixed-point receiver.
+
+### 4.7.2 Area vs. Precision
+
+Variation of area vs. fixed-point is plotted for all major blocks of the receiver. All the blocks were coded with CatapultC HLS tool. The design input is done using C language with fixed-point library (`ac_fixed`) for modeling and synthesis of fixed-point effects. It supports FIFOs (`ac_channel`) used for data exchange between blocks for streaming applications. The output of the HLS tool is Verilog/VHDL that can be targeted to either ASIC or FPGA. Architecture exploration is done using Loop Pipelining, Loop Unrolling, Mapping of memories to register array or RAMs/ROMs, Finite State Machine (FSM) coding using either gray/binary coding. Loop Pipelining has been done with Initiation Interval (II) of
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+FIGURE 4.12: BER vs. OSNR plot for floating-point and various fixed-point configurations in Homodyne setup
+
+FIGURE 4.13: BER vs. OSNR plot for floating-point and various fixed-point configurations in Heterodyne setup
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+TABLE 4.15: Fixed-point configurations for Homodyne setup
+
+<table><thead><tr><th>Block Name</th><th>Fixed-point config0</th><th>Fixed-point config1</th><th>Fixed-point config2</th><th>Fixed-point config3</th><th>Fixed-point config4</th></tr></thead><tbody><tr><td></td><td colspan="5" style="text-align:center;">Bitwidth W<sub>i</sub></td></tr><tr><td>Time/Frequency Synchronization</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td></tr><tr><td>CFO Compensation</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td></tr><tr><td>FFT</td><td>10</td><td>10</td><td>10</td><td>8</td><td>6</td></tr><tr><td>Integer CFO Estimation</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td></tr><tr><td>De-Interleaver</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Channel Estimation &amp; Equalization</td><td>12</td><td>10</td><td>8</td><td>8</td><td>8</td></tr><tr><td>CPE Estimation &amp; Compensation</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Demapper</td><td>8</td><td>8</td><td>8</td><td>8</td><td>8</td></tr></tbody></table>
+
+TABLE 4.16: BER vs. ONSR for floating-point and various fixed-point configurations in Homodyne setup
+
+<table><thead><tr><th rowspan="2">OSNR (in dB)</th><th colspan="6">BER</th></tr><tr><th>5.3</th><th>6.8</th><th>7.9</th><th>9.16</th><th>10.1</th><th>11.09</th></tr></thead><tbody><tr><th>Floating-point configuration</th><td>3.5x10<sup>-2</sup></td><td>8.0x10<sup>-3</sup></td><td>1.6x10<sup>-3</sup></td><td>4.7x10<sup>-4</sup></td><td>1.9x10<sup>-4</sup></td><td>9.1x10<sup>-5</sup></td></tr><tr><th>Fixed-point config0</th><td>5.9x10<sup>-2</sup></td><td>1.4x10<sup>-2</sup></td><td>2.6x10<sup>-3</sup></td><td>7.8x10<sup>-4</sup></td><td>3.6x10<sup>-4</sup></td><td>1.9x10<sup>-4</sup></td></tr><tr><th>Fixed-point config1</th><td>9.4x10<sup>-2</sup></td><td>2.1x10<sup>-2</sup></td><td>5.6x10<sup>-3</sup></td><td>1.3x10<sup>-3</sup></td><td>4.9x10<sup>-4</sup></td><td>3.2x10<sup>-4</sup></td></tr><tr><th>Fixed-point config2</th><td>1.2x10<sup>-1</sup></td><td>3.1x10<sup>-2</sup></td><td>9.6x10<sup>-3</sup></td><td>1.9x10<sup>-3</sup></td><td>8.3x10<sup>-4</sup></td><td>5.2x10<sup>-4</sup></td></tr><tr><th>Fixed-point config3</th><td>1.7x10<sup>-1</sup></td><td>5.1x10<sup>-2</sup></td><td>1.8x10<sup>-2</sup></td><td>4.9x10<sup>-3</sup></td><td>1.7x10<sup>-3</sup></td><td>1.1x10<sup>-3</sup></td></tr><tr><th>Fixed-point config4</th><td>2.2x10<sup>-1</sup></td><td>6.6x10<sup>-2</sup></td><td>2.4x10<sup>-2</sup></td><td>6.9x10<sup>-3</sup></td><td>2.7x10<sup>-3</sup></td><td>1.7x10<sup>-3</sup></td></tr></tbody></table>
+
+$W_i$ means that the receiver reads and writes data every clock cycle. The generated Verilog/VHDL code is synthesized using Xilinx ISE tool targeted towards Virtex-7 Development Board. The blocks were designed to work at a frequency of 200 MHz. Each of the blocks of the receiver were synthesized individually and resources taken at different input/output precision values are given. Tables 4.19 to 4.25 give area calculations for the blocks in the receiver starting from Time Synchronization to CPE Estimation and Compensation.
+
+The values of precision selected for the blocks using "Fixed-point config0" are given in Table 4.26 and plotted in Figure 4.14. If instead of "Fixed-point config1" was chosen, then a savings of 13.3% in LUTs would have been obtained for a small degradation of BER.
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+TABLE 4.17: Fixed-point configurations for Heterodyne setup
+
+<table><thead><tr><th rowspan="2">Block Name</th><th>Fixed-point config5</th><th>Fixed-point config6</th><th>Fixed-point config7</th></tr><tr><th colspan="3">Bitwidth W<sub>i</sub></th></tr></thead><tbody><tr><td>Time/Frequency Synchronization</td><td>6</td><td>6</td><td>6</td></tr><tr><td>CFO Compensation</td><td>10</td><td>8</td><td>6</td></tr><tr><td>FFT</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Integer CFO Estimation</td><td>6</td><td>6</td><td>6</td></tr><tr><td>De-Interleaver</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Channel Estimation &amp; Equalization</td><td>12</td><td>12</td><td>12</td></tr><tr><td>CPE Estimation &amp; Compensation</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Demapper</td><td>8</td><td>8</td><td>8</td></tr></tbody></table>
+
+TABLE 4.18: BER vs. ONSR for floating-point and various fixed-point configurations in Heterodyne setup
+
+<table><thead><tr><th rowspan="2">OSNR (in dB)</th><th colspan="6">BER</th></tr><tr><th>6.5</th><th>7.4</th><th>8.6</th><th>9.8</th><th>10.9</th><th>11.85</th></tr></thead><tbody><tr><td>Floating-point configuration</td><td>6.5x10<sup>-3</sup></td><td>3.5x10<sup>-3</sup></td><td>1.0x10<sup>-3</sup></td><td>2.9x10<sup>-4</sup></td><td>1.6x10<sup>-4</sup></td><td>6.4x10<sup>-5</sup></td></tr><tr><td>Fixed-point config5</td><td>9.5x10<sup>-3</sup></td><td>5.0x10<sup>-3</sup></td><td>1.3x10<sup>-3</sup></td><td>3.8x10<sup>-4</sup></td><td>2.3x10<sup>-4</sup></td><td>9.4x10<sup>-5</sup></td></tr><tr><td>Fixed-point config6</td><td>1.3x10<sup>-2</sup></td><td>9.0x10<sup>-3</sup></td><td>1.9x10<sup>-3</sup></td><td>5.3x10<sup>-4</sup></td><td>3.5x10<sup>-4</sup></td><td>1.5x10<sup>-4</sup></td></tr><tr><td>Fixed-point config7</td><td>1.7x10<sup>-2</sup></td><td>1.4x10<sup>-2</sup></td><td>2.9x10<sup>-3</sup></td><td>7.3x10<sup>-4</sup></td><td>4.8x10<sup>-4</sup></td><td>2.3x10<sup>-4</sup></td></tr></tbody></table>
+
+TABLE 4.19: Area Occupied vs. Bitwidth for Time/Frequency Synchronization block
+
+<table><thead><tr><th colspan="3">Time/Frequency Synchronization</th></tr><tr><th>Bitwidth<br/>W<sub>i</sub></th><th>Area in<br/>Xilinx FPGA</th><th>Area<br/>Numbers</th></tr></thead><tbody><tr><td rowspan="2">6</td><td>LUTs</td><td>36210</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr><tr><td rowspan="2">8</td><td>LUTs</td><td>44873</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr><tr><td rowspan="2">10</td><td>LUTs</td><td>51374</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr></tbody></table>
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+TABLE 4.20: Area vs. Bitwidth for CFO Compensation block
+
+<table><thead><tr><th colspan="3">CFO Compensation</th></tr><tr><th>Bitwidth<br>W<sub>i</sub></th><th>Area in<br>Xilinx FPGA</th><th>Area<br>Numbers</th></tr></thead><tbody><tr><td rowspan="2">6</td><td>LUTs</td><td>1807</td></tr><tr><td>DSP Multipliers</td><td>15</td></tr><tr><td rowspan="2">8</td><td>LUTs</td><td>1985</td></tr><tr><td>DSP Multipliers</td><td>15</td></tr><tr><td rowspan="2">10</td><td>LUTs</td><td>2437</td></tr><tr><td>DSP Multipliers</td><td>15</td></tr></tbody></table>
+
+TABLE 4.21: Area vs. Bitwidth for FFT block
+
+<table><thead><tr><th colspan="3">FFT</th></tr><tr><th>Bitwidth<br>W<sub>i</sub></th><th>Area in<br>Xilinx FPGA</th><th>Area<br>Numbers</th></tr></thead><tbody><tr><td rowspan="2">8</td><td>LUTs</td><td>73529</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr><tr><td rowspan="2">10</td><td>LUTs</td><td>85653</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr><tr><td rowspan="2">12</td><td>LUTs</td><td>97979</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr></tbody></table>
+
+TABLE 4.22: Area vs. Bitwidth for Integer CFO Estimation block
+
+<table><thead><tr><th colspan="3">Integer CFO Estimation</th></tr><tr><th>Bitwidth<br>W<sub>i</sub></th><th>Area in<br>Xilinx FPGA</th><th>Area<br>Numbers</th></tr></thead><tbody><tr><td rowspan="2">6</td><td>LUTs</td><td>5987</td></tr><tr><td>DSP Multipliers</td><td>2</td></tr><tr><td rowspan="2">8</td><td>LUTs</td><td>8200</td></tr><tr><td>DSP Multipliers</td><td>2</td></tr><tr><td rowspan="2">10</td><td>LUTs</td><td>10843</td></tr><tr><td>DSP Multipliers</td><td>2</td></tr><tr><td rowspan="2">12</td><td>LUTs</td><td>12984</td></tr><tr><td>DSP Multipliers</td><td>2</td></tr></tbody></table>
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+TABLE 4.23: Area vs. Bitwidth for De-interleaver block
+
+<table><thead><tr><th colspan="3">De-Interleaver</th></tr><tr><th>Bitwidth<br>W<sub>i</sub></th><th>Area in<br>Xilinx FPGA</th><th>Area<br>Numbers</th></tr></thead><tbody><tr><td rowspan="2">8</td><td>LUTs</td><td>9622</td></tr><tr><td>DSP Multipliers</td><td>8</td></tr><tr><td rowspan="2">10</td><td>LUTs</td><td>11543</td></tr><tr><td>DSP Multipliers</td><td>8</td></tr><tr><td rowspan="2">12</td><td>LUTs</td><td>13678</td></tr><tr><td>DSP Multipliers</td><td>8</td></tr></tbody></table>
+
+TABLE 4.24: Area vs. Bitwidth for Channel Estimation & Equalization
+
+<table><thead><tr><th colspan="3">Channel Estimation & Equalization</th></tr><tr><th>Bitwidth<br>W<sub>i</sub></th><th>Area in<br>Xilinx FPGA</th><th>Area<br>Numbers</th></tr></thead><tbody><tr><td rowspan="2">8</td><td>LUTs</td><td>34520</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr><tr><td rowspan="2">10</td><td>LUTs</td><td>41975</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr><tr><td rowspan="2">12</td><td>LUTs</td><td>48467</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr></tbody></table>
+
+TABLE 4.25: Area vs. Bitwidth for CPE Estimation & Compensation
+
+<table><thead><tr><th colspan="3">CPE Estimation & Compensation</th></tr><tr><th>Bitwidth<br>W<sub>i</sub></th><th>Area in<br>Xilinx FPGA</th><th>Area<br>Numbers</th></tr></thead><tbody><tr><td rowspan="2">8</td><td>LUTs</td><td>1540</td></tr><tr><td>DSP Multipliers</td><td>16</td></tr><tr><td rowspan="2">10</td><td>LUTs</td><td>1756</td></tr><tr><td>DSP Multipliers</td><td>16</td></tr><tr><td rowspan="2">12</td><td>LUTs</td><td>1943</td></tr><tr><td>DSP Multipliers</td><td>16</td></tr></tbody></table>
+
+## 4.8 Conclusions
+
+In this Chapter, an end-to-end fully streaming parallel architecture for CO-OFDM system was proposed. For the transmitter, low-complexity radix-2<sup>2</sup> IFFT algorithm was used, for which scalable parallel architecture was utilized. Utilizing the idea that IFFT is present before DAC and hence limited by its precision, architecture was designed to use precision closer to DAC precision. This helps in area savings compared to FFT at the receiver which is not precision limited since it receives data from ADC. The frame structure and algorithms chosen is done to reduce long feedback loops. The proposed architecture has
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+TABLE 4.26: Fixed-Point Allocation and Area for all blocks of the R = 4-Parallel Receiver
+
+<table><thead><tr><th>Block Name</th><th>Bitwidth W<sub>i</sub></th><th>Area in Xilinx FPGA</th><th>Area Numbers</th></tr></thead><tbody><tr><td rowspan="2">Time/Frequency Synchronization</td><td rowspan="2">6</td><td>LUTs</td><td>36210</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr><tr><td rowspan="2">CFO Compensation</td><td rowspan="2">10</td><td>LUTs</td><td>2437</td></tr><tr><td>DSP Multipliers</td><td>15</td></tr><tr><td rowspan="2">FFT</td><td rowspan="2">10</td><td>LUTs</td><td>85653</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr><tr><td rowspan="2">Integer CFO Estimation</td><td rowspan="2">6</td><td>LUTs</td><td>5987</td></tr><tr><td>DSP Multipliers</td><td>2</td></tr><tr><td rowspan="2">De-Interleaver</td><td rowspan="2">10</td><td>LUTs</td><td>11543</td></tr><tr><td>DSP Multipliers</td><td>8</td></tr><tr><td rowspan="2">Channel Estimation &amp; Equalization</td><td rowspan="2">12</td><td>LUTs</td><td>41975</td></tr><tr><td>DSP Multipliers</td><td>36</td></tr><tr><td rowspan="2">CPE Estimation &amp; Compensation</td><td rowspan="2">10</td><td>LUTs</td><td>1756</td></tr><tr><td>DSP Multipliers</td><td>16</td></tr><tr><td rowspan="2">Demapper</td><td rowspan="2">8</td><td>LUTs</td><td>640</td></tr><tr><td>DSP Multipliers</td><td>0</td></tr></tbody></table>
+
+FIGURE 4.14: Pie Chart of area Occupation of all blocks of R = 4-Parallel CO-OFDM Receiver (Fixed-point config0)
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+# List of Figures
+
+<table><tr><td>1</td><td>Architecture typique d'un réseau optique. CN - Core Node, EN - Edge Node, AN - Access Node</td><td>1</td></tr><tr><td>2</td><td>Tracé des fonctions de métriques temporelles grossière (a) et fine (b)</td><td>4</td></tr><tr><td>3</td><td>MSE de l'estimation temporelle en fonction du SNR dans un canal ISI</td><td>6</td></tr><tr><td>4</td><td>MSE de l'estimation temporelle en fonction du OSNR pour des canaux SMS et un CFO = 4.75</td><td>8</td></tr><tr><td>5</td><td>Architecture PSPB proposée pour le calcul de P<sub>mb</sub> avec MBA</td><td>10</td></tr><tr><td>6</td><td>Architecture parallèle FSPB proposée pour le calcul de P<sub>mb</sub> avec MBA et R = 4</td><td>11</td></tr><tr><td>7</td><td>Configuration hétérodyne à détection cohérente avec une fibre SSMFde 50 km</td><td>15</td></tr><tr><td>8</td><td>BER vs SNR pour un système CO-OFDM simple bande hétérodyne</td><td>15</td></tr><tr><td>9</td><td>Plateforme FPGA temps réel d'émission</td><td>16</td></tr><tr><td>10</td><td>Plateforme FPGA temps réel de réception</td><td>16</td></tr><tr><td>1.1</td><td>Cisco Visual Networking Index (VNI) Prediction of growth of internet by Application Type (Updated May 2013). The ordinate units is in Eta Bytes (EB). Total traffic is 2017 is predicted to be three times larger than 2012 [1].</td><td>19</td></tr><tr><td>1.2</td><td>Typical Optical Network Architecture, CN - Core Node, EN - Edge Node, AN - Access Node</td><td>20</td></tr><tr><td>1.3</td><td>Power Savings Possible at each stage in Top down VLSI Design Flow</td><td>24</td></tr><tr><td>2.1</td><td>Fiber loss coefficient vs. different wavelengths for a typical low-loss optical fiber (SSMF) and fiber without the water absorption peak (Allwave). [Reproduced from Essiambre et al.[2]]</td><td>29</td></tr><tr><td>2.2</td><td>Tolerance of various phase-amplitude constellations to ASE. Reproduced from [3].</td><td>33</td></tr><tr><td>2.3</td><td>Single band of a single/dual polarization CO-OFDM system</td><td>35</td></tr><tr><td>2.4</td><td>Digital OFDM Transmitter, S/P - Serial-to-Parallel, P/S - Parallel-to-Serial</td><td>35</td></tr><tr><td>2.5</td><td>Single Polarization RF-to-Optical Up Converter. I<sub>X</sub> - Real Part of X-Polarization, Q<sub>X</sub> - Imaginary Part of X-Polarization, DAC - Digital-to-Analog Converter, LPF - Low Pass Filter, RFD - RF Driver, MZM - Mach-Zender Modulator, ECL - External Cavity LASER, VOA - Variable Optical Amplifier.</td><td>36</td></tr></table>
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+only one feedback loop whose output is required once every 80 OFDM symbols. At the receiver, scalable parallel architecture for Time Synchronization was used. Low-Complexity Parallel blocks for Integer CFO Estimation, Channel Estimation and CPE Estimation was proposed. The implementation was done for *R* = 4-Parallel CO-OFDM Transceiver on Xilinx FPGA using CatapultC. Fixed-point exploration was done using *ac\_fixed* models and whole parallel CO-OFDM receiver fits in a single Virtex-7 development board. Area vs. BER trade-off exploration was indicated.
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+# Chapter 5
+
+## Experimental Validation of
+CO-OFDM System
+
+### 5.1 Introduction
+
+In this chapter, offline and real-time experiments conducted for validation of OFDM frame format, transceiver algorithms adopted are explained. In the first half, optical experiments conducted using arbitrary waveform generator (AWG) as transmitter, digital storage oscilloscope (DSO) for sampling received signal and Matlab for generating and decoding data are described. The OFDM frame format and transceiver algorithms used are from Section 4.3. The algorithm used for selection of best sampling point when using oversampling at the receiver is described in Section 5.2. The experiments are done in a step-by-step manner starting from electrical back-to-back (B2B) experiment, which is detailed in Section 5.3. In Section 5.4, the effect of addition of RF driver to the setup is explored. Section 5.5 describes optical B2B configuration using same LASER for both transmitter and receiver. Performance is characterized by plotting BER as a function of optical signal-to-noise ratio (OSNR). In section 5.6, performance of the system is explored in the presence of separate LASER sources for transmitter and receiver, which resembles real-world data transfer using optical communication systems.
+
+The second half details experiments conducted on real-time FPGA platform. Section 5.7 provides details about the transmitter and receiver FPGA prototype platform, OFDM frame structure and algorithms used. It primarily validates the use of real-time full-streaming block-parallel (FSBP) architecture proposed for timing synchronization. The timing synchronization block provides input samples to FFT in a continuous manner. The FPGA transceiver prototype platform was developed as part of FUI 100GFLEX project. Section 5.8 gives the results achieved by the use of proposed architecture in presence of synchronous and asynchronous sampling at the receiver. Section 5.10 concludes the chapter.
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+## 5.2 Sampling Clock Offset (SCO) Estimation Algorithm
+
+The OFDM frame format used is the same as the one proposed for developing parallel transceiver architecture. Figure 5.1 shows the frame structure used for optical experiments. consists of one initial training symbol ($TS_1$) which is used for time synchronization and fractional CFO estimation and of two repeated training symbols ($TS_2$) for integer CFO and channel estimation. A total of 77 data symbols follow these three training symbols. A single frame contains totally 80 OFDM symbols. Each data symbol contains eight pilot sub-carriers for phase estimation.
+
+FIGURE 5.1: OFDM frame format for single polarization (Pol$_{X}$) CO-OFDM system
+
+The algorithms used in receiver are:
+
+* Time Frequency Synchronization - Proposed Hierarchical algorithm.
+
+* Integer CFO Estimation - Cross-correlation with known training symbol.
+
+* Channel Estimation - LS for initial estimation and NLMS Equalizer for tracking.
+
+* Phase Tracking - Common Phase Error (CPE) estimation using pilots in data symbols.
+
+In the optical experiments conducted, the signal was oversampled at the receiver and for decoding, the best sampling point needs to be found. To find the optimal sampling point in the oversampled signal, fine timing estimation algorithm was used. Fine timing estimation algorithm calculates residual timing offset in the OFDM signal after the FFT operation. The output of this operation has integer and fractional parts. The integer part gives residual integer timing offset estimate, while the fractional part gives sampling clock offset (SCO) estimate.
+
+$$ \eta_{\text{fine}} = \eta_i + \eta_{\text{SCO}} \quad (5.1) $$
+
+where $\eta_{\text{fine}}$ is the total fine timing offset estimate, $\eta_i$ is the integer timing offset estimate and $\eta_{\text{SCO}}$ (the fractional part) is the SCO estimate. For the estimation of fine timing offset, the algorithm proposed by Lee et al [56] is used. It uses the phase information present in the sub-carriers to derive estimates of timing offset. Consider the received signal after FFT affected by residual timing offset after coarse timing synchronization
+
+$$ Y_j[k] = X_j[k] \cdot e^{j2\pi k(\eta_i + \eta_{\text{SCO}})/N} + N_j[k] \quad (5.2) $$
+
+where $Y_j[k]$ and $X_j[k]$ is the $k^{th}$ sub-carrier of $j^{th}$ OFDM symbol, $\eta_i$ is the integer timing offset, $\eta_{\text{SCO}}$ is the SCO, $N_j[k]$ is the noise at the sub-carrier. Data symbols are mapped using QPSK scheme. The effect of timing offset in the frequency domain is the rotation
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+of sub-carrier proportional to sub-carrier number. The basic idea is subtracting phase
+difference between neighbouring sub-carriers and taking mean of the phase difference. An
+improved method is averaging phase difference calculated on a group of sub-carriers. The
+algorithm is given as follows
+
+$$
+Y p4[j] = \frac{1}{B} \sum_{k=-B/2}^{B/2-1} Y^4[j+k], |j| < \frac{N}{2} \quad (5.3)
+$$
+
+where *B* is the group of sub-carriers, *Y*p4 is the fourth power of group of sub-carriers of *Y*. Fourth power is taken to neutralize the effect of QPSK modulation on sub-carriers.
+The groups are selected to be continuous set of sub-carriers. The group is selected so as to exclude group of null sub-carriers. Here *B* = 4 is used for group size. The mean complex phase rotation of array of *Y*p4 is given by
+
+$$
+G = \frac{1}{K} \sum_{k=0}^{K-1} Y p4[k] \cdot Y p4^*[k+1] \quad (5.4)
+$$
+
+where *G* is the mean of complex phase rotation, it contains both integer and fractional
+timing offset contribution, *K* is the total number of group of sub-carriers. The fine symbol
+timing estimation is obtained by taking angle of *G*:
+
+$$
+\eta = \frac{N}{2\pi B} \angle G \tag{5.5}
+$$
+
+where η denotes total fine timing offset, η = ηᵢ + ηSCO. In an oversampled signal, this calculation is done on all the sample streams obtained by down sampling. For example, in case of oversampling by a factor of 10, ηSCO is calculated for all 10 sample streams and sample stream with the lowest ηSCO is selected for further demodulation. But, calculation of ηSCO for all streams for every OFDM symbol in each frame results in too much computational complexity. To reduce the computational complexity, average SCO value is calculated using a 20 OFDM data symbols for single downsampled stream. Then, the next downsampled stream is chosen for calculation of SCO over next 20 OFDM symbols. It is done till all the downsampled streams are covered. Stream corresponding to minimum of the average SCO values calculated is selected for further data decoding. For example, if received signal is oversampled by 10, then for each of the 10 streams, SCO is calculated using 20 data OFDM symbols. After this calculation, stream corresponding to minimum SCO value is chosen. This method reduces the computational complexity and assuming ADC sampling clock remains stable over many OFDM frames, this calculation needs to be repeated after many OFDM frames (1000 OFDM Frames).
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+FIGURE 5.2: Configuration of Electrical B2B Experiment. Green blocks indicate analogue blocks.
+
+## 5.3 Electrical Back-to-Back (B2B) Experiment
+
+Figure 5.2 shows the electrical B2B configuration, with Matlab being used for generation and decoding data offline. The configuration consists of arbitrary waveform generator (AWG) directly connected to digital storage oscilloscope (DSO). The trigger signal for DSO is provided by the Marker signal from AWG. The objective is to characterize the Matlab transceiver in presence of DAC/ADC's limited resolution, frequency response and bandwidth of the analogue components in the configuration. Parameters of OFDM system, DAC and ADC are given in Table 5.1.
+
+TABLE 5.1: Parameters of the Electrical B2B Experiment
+
+<table><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>N</td><td>256</td></tr><tr><td>N<sub>cyp</sub></td><td>8</td></tr><tr><td>F<sup>DAC</sup><sub>s</sub></td><td>1 GHz</td></tr><tr><td>DAC Resolution</td><td>8 bits</td></tr><tr><td>DAC Voltage Range</td><td>1 Vp - p</td></tr><tr><td>LASER Wavelength</td><td>1557.07 nm</td></tr><tr><td>F<sup>ADC</sup><sub>s</sub></td><td>10 GHz</td></tr><tr><td>ADC Resolution</td><td>8 bits</td></tr><tr><td>ADC Voltage Range</td><td>0.25 Vp - p</td></tr><tr><td>Number of Used sub-carriers</td><td>176</td></tr><tr><td>Number of Pilot sub-carriers</td><td>8</td></tr><tr><td>Mapping Scheme</td><td>QPSK</td></tr></tbody></table>
+
+The spectrum of OFDM signal after AWG taken using spectrum analyzer is given in Figure 5.3. Due to high-frequency sub-carriers (sub-carriers near N/2) being switched off, aliasing effects are reduced. The selection of best sampling stream is done using the estimated values of η<sub>SCO</sub> (shown in Figure 5.4) and its gradient (Figure 5.5). The minimum value of absolute value of product of η<sub>SCO</sub> and gradient of η<sub>SCO</sub> is used as a metric for deciding the best sampling stream among the possible choices.
+
+Theoretical BER for QPSK modulation is given by
+
+$$BER = 0.5 \ erfc \left( \sqrt{\frac{E_s}{2N_0}} \right) \qquad (5.6)$$
+
+$$erfc(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \qquad (5.7)$$
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+FIGURE 5.3: OFDM Signal Spectrum
+
+FIGURE 5.4: Estimated Values of η$_{SCO}$
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+FIGURE 5.5: Gradient of estimated value of η$_{SCO}$
+
+Figure 5.6 shows BER as a function of E_s/N_0. E_s/N_0 is varied by adding gaussian noise at the receiver. BER is calculated by averaging over 1000 frames in a single acquisition. Each frame consists of 77 OFDM data symbols. It can be seen that experimental BER curve follows theoretical curve very closely and validates the Electrical B2B configuration.
+
+## 5.4 Electrical B2B Configuration with RF Amplifier
+
+To generate an Optical OFDM signal, the output signal level of AWG is not sufficient to drive the optical Mach-Zender Modulator (MZM). To boost the signal, RF driver is used and its impact on the performance of CO-OFDM transceiver is evaluated. Figure 5.7 shows the electrical B2B experiment after the addition of RF driver for amplification. The output of AWG is fed into RF driver introduced. The RF driver now drives the input of DSO. A 20 dB attenuator is connected to the output of RF driver to limit the maximum voltage input to DSO. The RF driver provides a fixed gain over a large bandwidth. The parameter to vary is the peak-to-peak voltage output from AWG to input of RF driver. The range of AWG output voltage is 0.5 Vp-p to 1 Vp-p. The AWG output voltage is varied to find the value which does not saturate the RF amplifier and the value is found to be 0.6 Vp-p. The output of RF amplifier corresponding to 0.6 Vp-p after amplification by RF driver is around 10 dBm. The output spectrum after RF driver is similar to Figure 5.3. Figure 5.8 shows the variation of BER as a function of E_s/N_0. BER is calculated using average over 1000 OFDM frames and plotted in Figure 5.8 along with electrical B2B
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+FIGURE 5.6: BER vs SNR for Electrical B2B experiment (Theoretical and Experimental)
+
+FIGURE 5.7: Configuration of Electrical B2B Experiment with RF Driver. Green blocks indicate analogue blocks.
+
+curves. From Figure 5.8, it can be observed that since BER curve after RF driver is very close to BER curve of Electrical B2B experiment. It shows that the addition of RF driver does not introduce non-linearities in the transmission chain.
+
+## 5.5 Optical B2B Configuration with Homodyne Coherent Detection
+
+After validation of Electrical B2B configuration, optical components are introduced. Figure 5.9 shows the optical B2B configuration after the addition of optical modulator and demodulator. Since a common LASER is used for modulating and demodulating optical signal, it is called homodyne detection. The configuration is described in detail below.
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+FIGURE 5.8: BER vs SNR for Electrical B2B experiment with RF driver (Theoretical, Experimental with and without RF driver)
+
+FIGURE 5.9: Configuration of Homodyne Coherent Detection. DSP processing is done offline in Matlab. Green blocks indicate analogue blocks. Light Blue blocks indicate Optical components.
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+* Electro-Optic Transmitter - The carrier frequency for the transmitter is generated using the external-cavity LASER (ECL) at a wavelength of 1540 nm. It is passed through optical amplifier using a polarization maintaining fiber (PMF) which amplifies it to 15 dBm power level. The 3 dB coupler divides the power input and one output is given to polarization controller. The other output is connected to local oscillator (LO) input of coherent detector. The polarization controller is used to maximize optical power at the output of optical modulator (Mach-Zender Modulator (MZM)) because the optical modulator is sensitive to the signal polarization. It is used to minimize the optical loss and maximize modulation depth in MZM. The output of polarization controller, whose power level is at 11 dBm, serves as the carrier frequency input of MZM block. The transmitted signal generated in Matlab is passed through AWG and RF Driver. The output of RF Driver is connected to modulation input of MZ Modulator, power level being around 10 dBm. The modulated optical signal is connected to optical attenuator.
+
+* Opto-Electronic Receiver - The optical attenuator is used to vary the optical signal-to-noise ratio (OSNR). The attenuated signal is then amplified for transmission and measurement. 10% of the amplified signal is given to optical signal analyzer (OSA) for measuring OSNR of the modulated signal and 90% of the signal is used for transmission. The optical band pass filter (BPF) selects the bandwidth around the carrier frequency and the filtered output is passed through optical attenuator. The bandwidth range of optical BPF is 3 nm. The optical attenuator is used to control the maximum optical power level at the input of coherent detector. The maximum value of signal input of coherent detector is fixed at -5 dBm. The attenuated output is connected to polarization controller which allows to maximize the output level of CoD corresponding to X-polarization or Y-polarization. The LO power level is around 11 dBm and signal after coherent detection is given to digital storage oscilloscope (DSO). The sampled digital data are then transferred to computer running Matlab for offline processing.
+
+The bias of MZM needs to be adjusted for operating it in linear region. The setting of bias is done using the first training symbol ($TS_1$), which has the property that it is PSK signal both in time domain and frequency domain. For adjusting the bias of MZM, $TS_1$ is selected and zoomed in at the DSO. The bias of MZ Modulator is now adjusted to have circular constellation for $TS_1$. Faithful reproduction of circular constellation at the DSO indicates that MZ Modulator is operating in the linear region and this is done on everyday before the start of the experiment, since bias drifts with temperature. During the experiment, using the optical attenuator, the OSNR value is varied from 2 to 13 dB and corresponding BER obtained by Matlab program is tabulated. Figure 5.10 shows the BER obtained for different values of OSNR for the optical B2B experiment.
+
+Figure 5.11 shows the homodyne configuration with the addition of standard single mode optical fiber (SSMF) of length 50 km. Again OSNR value is varied and BER values
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index 0000000000000000000000000000000000000000..b756ef692d2ace0dce71d2bd20f03602bd396635
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+<table>
+  <tbody>
+    <tr>
+      <td>2.6</td>
+      <td>Dual Polarization RF-to-Optical Up Converter. I<sub>X</sub> - Real Part of X-Polarization, Q<sub>X</sub> - Imaginary Part of X-Polarization, I<sub>Y</sub> - Real Part of Y-Polarization, Q<sub>Y</sub> - Imaginary Part of Y-Polarization, DAC - Digital-to-Analog Converter, LPF - Low Pass Filter, RFD - RF Driver, MZM - Mach-Zender Modulator, ECL - External Cavity LASER, PBS - Polarization Beam Splitter, VOA - Variable Optical Amplifier, PBC - Polarization Beam Combiner.</td>
+      <td>36</td>
+    </tr>
+    <tr>
+      <td>2.7</td>
+      <td>Resolution vs. Sampling Rate for fastest DAC available. GSa/s - Giga Samples/second.</td>
+      <td>37</td>
+    </tr>
+    <tr>
+      <td>2.8</td>
+      <td>Optical-to-RF Down Converter. BPF - Band Pass Filter, ECL - External Cavity LASER, LO - Local Oscillator, PBS - Polarization Beam Splitter, ADC - Analog-to-Digital Converter, I<sub>X</sub> - Real Part of X-Polarization, Q<sub>X</sub> - Imaginary Part of X-Polarization, I<sub>Y</sub> - Real Part of Y-Polarization, Q<sub>Y</sub> - Imaginary Part of Y-Polarization.</td>
+      <td>38</td>
+    </tr>
+    <tr>
+      <td>2.9</td>
+      <td>Resolution vs. Sampling Rate for fastest ADC available. GSa/s - Giga Samples/second.</td>
+      <td>39</td>
+    </tr>
+    <tr>
+      <td>2.10</td>
+      <td>Digital Receiver of PDM-CO-OFDM System. TFSYNC - Time Frequency Synchronization, CFO - Carrier Frequency Offset, FCOMP - CFO Compensation, FFT - Fast Fourier Transform, ICFO - Integer CFO Estimation, CEE - Channel Estimation &amp; Equalization, CPE - Common Phase Error, CPEC - CPE Estimation &amp; Compensation, DMAP - Demapper.</td>
+      <td>39</td>
+    </tr>
+    <tr>
+      <td>3.1</td>
+      <td>Plot of Coarse (a) and Fine (b) Timing Metric Functions</td>
+      <td>54</td>
+    </tr>
+    <tr>
+      <td>3.2</td>
+      <td>MSE of Timing Estimation versus SNR in ISI channel</td>
+      <td>58</td>
+    </tr>
+    <tr>
+      <td>3.3</td>
+      <td>MSE of CFO Estimation versus SNR in ISI channel</td>
+      <td>59</td>
+    </tr>
+    <tr>
+      <td>3.4</td>
+      <td>MSE of Timing Estimation vs. OSNR in SSMF channel with CFO = 0.75</td>
+      <td>62</td>
+    </tr>
+    <tr>
+      <td>3.5</td>
+      <td>MSE of Timing Estimation vs. OSNR in SSMF channel with CFO = 4.75</td>
+      <td>63</td>
+    </tr>
+    <tr>
+      <td>3.6</td>
+      <td>MSE of CFO Estimation vs. OSNR in SSMF channel for CFO = 0.75</td>
+      <td>64</td>
+    </tr>
+    <tr>
+      <td>3.7</td>
+      <td>Parallel Architecture proposed by Kaneda et al for Schmidl-Cox Algorithm</td>
+      <td>65</td>
+    </tr>
+    <tr>
+      <td>3.8</td>
+      <td>Parallel Architecture proposed by Chen et al for cross-correlation operation</td>
+      <td>66</td>
+    </tr>
+    <tr>
+      <td>3.9</td>
+      <td>Proposed R = 4-Parallel PSBP Architecture for P<sub>sc</sub> calculation in case of SCA. iter_flag = 0 indicates non-iterative computation mode, while iter_flag = 1 indicates iterative computation mode.</td>
+      <td>70</td>
+    </tr>
+    <tr>
+      <td>3.10</td>
+      <td>Proposed R = 4-Parallel PSBP Architecture for R<sub>sc</sub> calculation in case of SCA. iter_flag = 0 indicates non-iterative computation mode, while iter_flag = 1 indicates iterative computation mode.</td>
+      <td>70</td>
+    </tr>
+    <tr>
+      <td>3.11</td>
+      <td>Proposed PSPB Architecture for calculation of P<sub>mb</sub> in case of MBA. iter_flag = 0 indicates non-iterative computation mode, while iter_flag = 1 indicates iterative computation mode.</td>
+      <td>72</td>
+    </tr>
+    <tr>
+      <td>3.12</td>
+      <td>Proposed R = 4-Parallel PSBP Architecture for R<sub>mb</sub> calculation in case of MBA. iter_flag = 0 indicates non-iterative computation mode, while iter_flag = 1 indicates iterative computation mode.</td>
+      <td>72</td>
+    </tr>
+    <tr>
+      <td>3.13</td>
+      <td>Multiplier requirement as a function of R-parallel output for PSBP and Kaneda's architecture, M = 32</td>
+      <td>73</td>
+    </tr>
+    <tr>
+      <td>3.14</td>
+      <td>Adder requirement as a function of R-parallel output for PSBP and Kaneda's architecture, M = 32</td>
+      <td>74</td>
+    </tr>
+    <tr>
+      <td>3.15</td>
+      <td>R = 4-Parallel Initial Point Auto-Correlation Computation Block for SCA</td>
+      <td>75</td>
+    </tr>
+    <tr>
+      <td>3.16</td>
+      <td>R = 4-Parallel Initial Point Energy Computation Block for SCA</td>
+      <td>75</td>
+    </tr>
+  </tbody>
+</table>
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+FIGURE 5.10: BER vs SNR for single-band Optical Back-to-Back Experiment
+
+are noted. Figure 5.10 shows the BER as a function of OSNR with introduction of SSMF. The introduction of SSMF does not change the performance of the system due to presence of cyclic prefix (CP) which helps in tolerance of chromatic dispersion and is same as optical B2B configuration.
+
+## 5.6 Heterodyne Coherent Detection Configuration
+
+Figure 5.12 shows the heterodyne coherent detection configuration with separate LASER sources used for LO input of MZ Modulator and Coherent Detector. This depicts a real-world scenario where transmitter and receiver have different LASER sources. Again OSNR is varied and BER is calculated using offline Matlab receiver. Due to large variations of frequency of receiver LASER compared to transmitter LASER, it was found that frame size of 80 OFDM symbols was very long and BER obtained was very high. To overcome this problem, frame size was decreased by reducing the number of data symbols in each frame. This results in more frequent estimation of CFO using training symbols. Frame size was reduced in steps of 5 from 80 onwards. BER vs. OSNR curve was calculated at every frame size value. It was found with 45 OFDM symbol frame size, the BER vs. OSNR performance was lower than for higher frame sizes and also it remained the same when frame size was further lowered. Results of BER vs. OSNR are plotted in Figure 5.13, which validates the experimental configuration and also the full Matlab transceiver. From these optical physical layer experiments, the algorithms used were validated in presence of OSNR
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+FIGURE 5.11: Configuration of Homodyne Coherent Detection with SMF of 50 km. DSP processing is done offline in Matlab. Green blocks indicate analogue blocks. Light Blue blocks indicate Optical components.
+
+degradation and the size of frame which can be used in case of heterodyne configuration was found. This value of frame size of around 45 OFDM symbols is used in the real-time FPGA prototype platform. The DSP algorithms are realized in hardware on the real-time FPGA platform and characterization of performance of these algorithms is done in the next Section.
+
+## 5.7 Real-Time FPGA Platform
+
+A real-time FPGA platform was built as part of 100GFLEX project. Ekinops, industrial partner of the project gave us Altera FPGA platform on which to develop the DSP algorithms. It is to contain DSP blocks of transmitter and receiver in a single platform. The second platform of Xilinx was used to only interface DAC and ADC blocks and link between Altera and Xilinx FPGA was done using SFP+ interface. Hence, both Xilinx and Altera FPGA were used for the real-time FPGA prototype. The different building blocks used for real-time transmitter and receiver are explained below.
+
+**Transmitter Platform** - Figure 5.14 shows the different blocks and the connections starting from bit generation to analog output. The OFDM frame format adopted in 100GFLEX project is shown in Figure 5.15. The training symbol $TS_1$ is used for timing
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+FIGURE 5.12: Configuration of Heterodyne Coherent Detection with standard single mode fiber (SSMF) of 50 km. DSP processing is done offline in Matlab. Green blocks indicate analogue blocks. Light Blue blocks indicate Optical components.
+
+FIGURE 5.13: BER vs SNR for single-band CO-OFDM system for Heterodyne Detection
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+FIGURE 5.14: Real-Time FPGA Transmitter Block Diagram. PLL - Phase Locked Loop, SFP+ - Enhanced Small Form-factor Pluggable, SMF Cable - Single Mode Fiber Cable, I/F - Interface, CDR I/F - Clock Data Recovery Interface, DAC I/F - Digital-to-Analog Converter Interface.
+
+synchronization and fractional CFO estimation. Training symbol $TS_2$ is used for integer CFO estimation, least squares channel estimation. Channel and Phase tracking is done using least mean squares (LMS) algorithm and common phase error (CPE) algorithm respectively. Frame contains 47 data symbols and thus totally 49 symbols in each frame. Table 5.2 shows the parameters used for $TS_1$, $TS_2$ and $DS_{1,2,...,47}$.
+
+FIGURE 5.15: 100GFLEX Frame Format
+
+The digital OFDM transceiver is implemented in an Altera Stratix V FPGA development board, which consists of Mapper, IFFT, Cyclic Prefix Addition blocks. The Stratix V FPGA used is 5SGXEA7K2F40C2. The clock for Altera FPGA transmitter is generated by on-board quartz and PLL. It operates at a clock frequency of 180 MHz. Data is generated by four Pseudo Random Binary Sequence (PRBS) generators, which are fed into four IFFTs operating at 180 MHz. The IFFT uses radix-2$^2$ algorithm and uses single delay feedback (SDF) architecture, providing one sample per cycle output. After addition of cyclic prefix, the four outputs are fed into SFP+(enhanced small form-factor pluggable)
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+TABLE 5.2: 100GFLEX Frame Format Parameters
+
+<table><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Sampling Frequency (F<sub>DAC</sub>, F<sub>ADC</sub>)</td><td>720 MHz</td></tr><tr><td>FPGA Clock Frequency (F<sub>clk</sub>)</td><td>180 MHz</td></tr><tr><td>Parallel Factor Used (R)</td><td>4</td></tr><tr><td>Cylic Prefix size (N<sub>cyp</sub>)</td><td>8</td></tr><tr><td>IFFT/FFT size (N)</td><td>256</td></tr><tr><td>Symbol size (N<sub>sym</sub>)</td><td>264</td></tr><tr><td>Training Symbols in each frame</td><td>2</td></tr><tr><td>Data Symbols in each frame</td><td>47</td></tr><tr><td>OFDM Symbols in each frame</td><td>49</td></tr><tr><td>Mapping scheme used (M)</td><td>QPSK</td></tr><tr><td>Sub-carrier spacing (F<sub>DAC</sub>/N)</td><td>2.8125 MHz</td></tr><tr><td>OFDM symbol duration (T<sub>sym</sub>)</td><td>366.6 ns</td></tr><tr><td>Training Symbol TS<sub>1</sub></td><td>Minn-Bhargava Type</td></tr><tr><td>TS<sub>1</sub> Pattern used</td><td>[1 1 -1 1]</td></tr><tr><td>Number of Pilots per OFDM Symbol (N<sub>p</sub>)</td><td>8</td></tr><tr><td>Position of Pilots</td><td>[31 63 95 127 159 191 223 255]</td></tr><tr><td>DAC Voltage range</td><td>1 Vp - p</td></tr><tr><td>ADC Voltage range</td><td>0.5 Vp - p</td></tr></tbody></table>
+
+interface which connects Altera FPGA board and Xilinx FPGA board using single mode fiber (SMF) cable.
+
+The Xilinx Virtex 7 development board (VC707) uses clock data recovery circuit to recover clock from the received data. The clock extracted is 180 *MHz* clock, which is fed into PLL (SILABS SI5338) to generate the 50 *MHz* reference clock for PLL (HITTITE HMC833LP6GE). The PLL (HITTITE HMC833LP6GE) generates twice the sampling frequency of 1440 *MHz* and gives it to a clock divider (HMC394LP4 programmable divider) circuit. Xilinx FPGA board provides four samples per cycle at 180 *MHz* to DAC sampling at 720 *MHz*. The output range of DAC (FMC204) is 1 *Vp - p* and resolution is 10-bits. The output range used by the platform is 0.4 *Vp - p*. The real and imaginary outputs are then fed into two 3 dB attenuators to reduce the peak-to-peak voltage. This reduction is done to make the voltage to fit in the range of ADC (FMC126), which has an input range of 0.25 *Vp - p*. The output of attenuator is fed into ADC, which is equivalent to a electrical back-to-back (B2B) experiment.
+
+**Receiver Platform** - Figure 5.16 shows the receiver platform both in case of synchronized sampling and asynchronous sampling by the use of switch. In case of synchronized sampling, the sampling frequency for the ADC will be given by the PLL (HITTITE HMC833LP6GE) of the transmitter chain. It will give a 1440 *MHz* clock to ADC clock, which has an internal divide by 2 circuit. This mode is indicated by switch value of "0". In asynchronous sampling mode (switch value of "1"), the sampling clock for ADC is generated by another PLL (HITTITE HMC833LP6GE).
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+FIGURE 5.16: Real-Time FPGA Receiver Block Diagram. ADC - Analog-to-Digital Converter, I/F - Interface, SFP+ I/F - Enhanced Small Form-factor Pluggable Interface, SMF Cable - Single Mode Fiber Cable, CDR I/F - Clock Data Recovery Interface, PLL - Phase Locked Loop.
+
+The incoming real and imaginary data are sampled at 720 *MHz* and given to Xilinx FPGA Development Board (VC707). The clock for Xilinx FPGA is given by the clock output of ADC, which is 180 *MHz*. The ADC has a resolution of 10 bits. It gives four parallel samples every cycle to Xilinx FPGA due to FPGA clock frequency being one-fourth that of ADC sampling frequency. The Xilinx FPGA then transfers four parallel real and imaginary samples to Altera FPGA Board by using SFP+ interface on SMF cable. On the Altera FPGA board side, the clock data recovery (CDR) circuit recovers the clock of 180 *MHz* and it is used as the clock for OFDM Receiver. The OFDM receiver consists of the blocks in the following order: Time synchronization block, fractional CFO estimation block, CFO compensation block, FFT block, integer CFO estimation block, Channel estimation and compensation block, CPE estimation block and demapper block. Altera SignalTap block is connected to sample the outputs at various points of the receiver to verify the correctness of the operation of the block.
+
+Time Synchronization block uses the 4-Parallel block-parallel full-streaming architecture proposed in this thesis. It is slightly modified to take care of the different sign pattern used in 100GFLEX TS₁. After detection of starting point of the frame, it gives 4-parallel samples to CFO compensation block along with estimate of fractional CFO estimate. The output of CFO compensation block is given to FFT. Here, four separate FFTs are present which work in a round robin fashion to process the input samples. The FFT uses radix-2² algorithm and single delay feedback (SDF) architecture and produces single output per
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+FIGURE 5.17: Snapshot of Real-Time FPGA Transceiver Platform. The topmost rack shows the power supply for the configuration, the second rack is the EKINOPS Altera FPGA Digital Transceiver, the third rack shows the Xilinx Virtex-7 FPGA interfaced to DAC board, the bottom most rack shows the other Xilinx Virtex-7 FPGA interfaced to ADC. The yellow cables are single mode fiber (SMF) cables to connect using SFP+ interface.
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+cycle. The output of FFT is then given to integer CFO estimation block, which uses cross-correlation with known TS₂ portion to estimate integer CFO. Then TS₂ symbol is passed through Least Squares equalizer to produce an initial channel estimate. During data symbol compensation, it uses LMS equalizer to update the channel coefficients. Finally, CPE is estimated using the pilots embedded in the OFDM symbol and compensated. Here, the architecture is not end-to-end parallel like the one proposed in Chapter 4. A snapshot of fully connected Real-time Transceiver platform is shown in Figure 5.17.
+
+## 5.8 Performance of the Proposed Timing Synchronization Algorithm on Real-Time FPGA Platform
+
+FIGURE 5.18: Altera SignalTap Snapshot of coarse synchronization output. Presence of periodic zeros indicate cyclic prefix removal and bigger gap zeros indicate the removal of the first training symbol in the output fed into FFT block.
+
+In the Electrical B2B experiment, with no presence of CFO, the only unknown to estimate to demodulate OFDM signal is start of frame. The objective is to validate the time synchronization algorithm. Proposed block-parallel architecture for Minn-Bhargava algorithm was used for estimation of the starting point. The Verilog HDL generated using CatapultC was integrated into the setup and performance of the system was examined. To make the performance easier to observe, all the data symbols were coded the same and hence validation of the correct starting point could be done easily by observing SignalTap outputs after synchronization block. Figure 5.18 shows the output captured after timing synchronization block. The bigger zero pulses and corresponding zeros indicate TS₁ which is not passed through the FFT and small zero pulses indicate cyclic prefix removed. The gap between two large zeros indicate the time between two full frames. It can be observed that width of zero pulses remains constant which is indication of correct synchronization estimation. Figure 5.19 shows the zoomed-in version, where values of four parallel inputs
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diff --git a/samples/texts/7426999/page_148.md b/samples/texts/7426999/page_148.md
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+FIGURE 5.19: Altera SignalTap Snapshot of coarse synchronization output of OFDM symbols. Starting from second row, it contains real and imaginary signals alternatively. Correctness of the synchronization is verified by observing that alternate rows have repeating values indicating correct synchronization is achieved.
+
+to FFT are indicated. The values in the four lanes has to be identical, since all data symbols have the same values. It can be again observed that the four data symbols have the same value and it is the correct starting point of the symbol. Thus, proposed synchronization algorithm is validated using real-time FPGA platform experiment. The data captured through SignalTap is then analyzed to see whether synchronization performance remains the same over large number of frames. Similar results were observed even with asynchronous sampling setup. Hence, the real-time platform setup and synchronization algorithm were validated by observing the outputs after synchronization step. BER of the system was calculated using the captured data. The data was captured repeatedly and on every acquisition five OFDM frames were captured. BER was averaged over multiple acquisition and found to be near zero.
+
+## 5.9 Future Experiments proposed for Real-Time Platform
+
+Real-time FPGA platform took a long time in setting up and a working prototype was available very recently. Hence, experiments including optical setup and FPGA platform could not be done. The two experiments remaining with the addition of optical components are:
+
+* Homodyne Optical B2B experiment - The AWG and DSO in the offline configuration (Figure 5.9) will be replaced by FPGA transmitter (Figure 5.14) and FPGA receiver (Figure 5.16) respectively. BER is calculated using data acquired from Signal-Tap at different values of OSNR.
+
+* Heterodyne Optical B2B experiment - Again AWG and DSO in the offline Heterodyne configuration (Figure 5.12 will be replaced by FPGA transmitter and receiver. BER calculation is done at various values of OSNR).
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+## 5.10 Conclusions
+
+In this chapter, optical experiments were done to validate the frame structure and algorithms adopted for demodulating CO-OFDM signal. The experiments with real optical equipments prove the validity of the frame structure and the algorithms used. Value of frame size obtained in heterodyne configuration was used in real-time FPGA prototype platform to validate the hardware implementation of timing synchronization algorithm. Both synchronous sampling and asynchronous sampling yielded similar results. It also showed the architecture's suitability for real-time processing of optical OFDM signals. Further experiments to be performed with real-time FPGA platform is detailed and also experiments using dual-polarization CO-OFDM system to estimate and compensate for polarization effect.
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+<table>
+  <tbody>
+    <tr>
+      <td>3.17</td>
+      <td><i>R</i> = 4-Parallel Initial Point Auto-Correlation Computation Block for MBA</td>
+      <td>76</td>
+    </tr>
+    <tr>
+      <td>3.18</td>
+      <td><i>R</i> = 4-Parallel Initial Point Energy Computation Block for MBA</td>
+      <td>76</td>
+    </tr>
+    <tr>
+      <td>3.19</td>
+      <td>Multiplier requirement as a function of <i>R</i>-parallel output for FSBP and Kaneda's architecture, <i>M</i> = 32</td>
+      <td>77</td>
+    </tr>
+    <tr>
+      <td>3.20</td>
+      <td>Adder requirement as a function of <i>R</i>-parallel output for FSBP and Kaneda's architecture, <i>M</i> = 32</td>
+      <td>78</td>
+    </tr>
+    <tr>
+      <td>3.21</td>
+      <td>Parallel conjugate symmetric correlation on <i>R</i> = 4 PSPB/FSPB architecture. iter_flag = 0 for this operation.</td>
+      <td>78</td>
+    </tr>
+    <tr>
+      <td>3.22</td>
+      <td>Parallel energy calculation on <i>R</i> = 4 PSPB/FSPB architecture. iter_flag = 0</td>
+      <td>79</td>
+    </tr>
+    <tr>
+      <td>4.1</td>
+      <td>HLS Block Diagram of CatapultC synthesis flow and Matlab Integration</td>
+      <td>83</td>
+    </tr>
+    <tr>
+      <td>4.2</td>
+      <td>OFDM frame format for single polarization (Pol<sub>X</sub>) CO-OFDM system</td>
+      <td>84</td>
+    </tr>
+    <tr>
+      <td>4.3</td>
+      <td>OFDM frame format for dual polarization (Pol<sub>X</sub>,Pol<sub>Y</sub>) CO-OFDM system</td>
+      <td>84</td>
+    </tr>
+    <tr>
+      <td>4.4</td>
+      <td>IFFT/FFT Architecture of 4-Parallel radix-2<sup>2</sup> for <i>N</i> = 256, when input is given in even and odd index order</td>
+      <td>91</td>
+    </tr>
+    <tr>
+      <td>4.5</td>
+      <td>IFFT/FFT Architecture of 4-Parallel radix-2<sup>2</sup> for <i>N</i> = 256, when input is given in normal order</td>
+      <td>91</td>
+    </tr>
+    <tr>
+      <td>4.6</td>
+      <td>Plot of Mean of RMSE output of IFFT as function of <i>W</i><sub>i</sub> and <i>W</i><sub>t</sub>.</td>
+      <td>94</td>
+    </tr>
+    <tr>
+      <td>4.7</td>
+      <td>Proposed CO-OFDM Receiver Architecture Block Diagram</td>
+      <td>95</td>
+    </tr>
+    <tr>
+      <td>4.8</td>
+      <td>Data organization in the Synchronization Memory</td>
+      <td>95</td>
+    </tr>
+    <tr>
+      <td>4.9</td>
+      <td>Parallel Architecture for IFO Estimation.</td>
+      <td>96</td>
+    </tr>
+    <tr>
+      <td>4.10</td>
+      <td>Channel Estimation and Equalization Architecture which supports both LS and NLMS equalizers.</td>
+      <td>98</td>
+    </tr>
+    <tr>
+      <td>4.11</td>
+      <td>CPE Estimation Block.</td>
+      <td>98</td>
+    </tr>
+    <tr>
+      <td>4.12</td>
+      <td>BER vs. OSNR plot for floating-point and various fixed-point configurations in Homodyne setup.</td>
+      <td>100</td>
+    </tr>
+    <tr>
+      <td>4.13</td>
+      <td>BER vs. OSNR plot for floating-point and various fixed-point configurations in Heterodyne setup.</td>
+      <td>100</td>
+    </tr>
+    <tr>
+      <td>4.14</td>
+      <td>Pie Chart of area Occupation of all blocks of R = 4-Parallel CO-OFDM Receiver (Fixed-point config0).</td>
+      <td>105</td>
+    </tr>
+    <tr>
+      <td>5.1</td>
+      <td>OFDM frame format for single polarization (Pol<sub>X</sub>) CO-OFDM system.</td>
+      <td>108</td>
+    </tr>
+    <tr>
+      <td>5.2</td>
+      <td>Configuration of Electrical B2B Experiment. Green blocks indicate analogue blocks.</td>
+      <td>110</td>
+    </tr>
+    <tr>
+      <td>5.3</td>
+      <td>OFDM Signal Spectrum.</td>
+      <td>111</td>
+    </tr>
+    <tr>
+      <td>5.4</td>
+      <td>Estimated Values of η<sub>SCO</sub></td>
+      <td>111</td>
+    </tr>
+    <tr>
+      <td>5.5</td>
+      <td>Gradient of estimated value of η<sub>SCO</sub></td>
+      <td>112</td>
+    </tr>
+    <tr>
+      <td>5.6</td>
+      <td>BER vs SNR for Electrical B2B experiment (Theoretical and Experimental)</td>
+      <td>113</td>
+    </tr>
+    <tr>
+      <td>5.7</td>
+      <td>Configuration of Electrical B2B Experiment with RF Driver. Green blocks indicate analogue blocks.</td>
+      <td>113</td>
+    </tr>
+    <tr>
+      <td>5.8</td>
+      <td>BER vs SNR for Electrical B2B experiment with RF driver (Theoretical, Experimental with and without RF driver)</td>
+      <td>114</td>
+    </tr>
+    <tr>
+      <td>5.9</td>
+      <td>Configuration of Homodyne Coherent Detection. DSP processing is done offline in Matlab. Green blocks indicate analogue blocks. Light Blue blocks indicate Optical components.</td>
+      <td>114</td>
+    </tr>
+    <tr>
+      <td>5.10</td>
+      <td>BER vs SNR for single-band Optical Back-to-Back Experiment.</td>
+      <td>116</td>
+    </tr>
+  </tbody>
+</table>
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+# Chapter 6
+
+## Conclusions and Perspectives
+
+### 6.1 Overview
+
+In this thesis, low-complexity algorithms and parallel architectures were explored for efficient realization of the digital signal processing (DSP) blocks of a CO-OFDM transceiver. To achieve the total data rate of 100 Gb/s using present day data converters (DAC and ADC) bandwidth, multi-band CO-OFDM (MB-CO-OFDM) is adopted. MB-CO-OFDM divides total bandwidth of 50 GHz into smaller sub-bands and thus bandwidth requirement of DAC/ADC is reduced significantly. Hence, the total MB-CO-OFDM architecture consists of identical transceiver chains which transmit/decode the data in both polarizations of every sub-band. The major idea is that, since identical DSP architectures are used in each polarization of every sub-band, gains obtained due to resource optimization will be multi-fold. Hence, exploration of low-complexity algorithms and parallel architectures was done for single-polarization, single-band CO-OFDM transceiver. The only block which changes from single-polarization and dual-polarization is the channel estimation block in the receiver, with rest of the DSP blocks replicated. Also, realization of architecture on FPGA platforms makes it necessary to have parallel architecture, since FPGA can reach a maximum of few hundreds of MHz, while DAC/ADC interfaced to it will be in range of GHz. Hence, scalable parallel architectures are required for every DSP block in the CO-OFDM signal processing chain to avoid costly replication to match the input sampling rate. With these set of requirements, the major contributions of this thesis are listed below.
+
+* A novel low-complexity hierarchical time synchronization algorithm is proposed for intersymbol interference (ISI) channel. The mean square error (MSE) performance is comparable to high-efficiency cross-correlation only algorithms and computational complexity comparable to low-complexity auto-correlation algorithms. It also provides fractional carrier frequency offset (CFO) estimate.
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+* The low-complexity hierarchical time synchronization is modified for an optical channel. The modification reduces the computational complexity still further to a two-step process from a three-step process and attains MSE performance superior to auto-correlation algorithms.
+
+* A scalable block-parallel architecture is proposed for efficient parallelization of auto-correlation algorithm. The required amount of resources (complex multipliers) scales linearly with number of parallel outputs. The proposed approach of parallelization is very general and can be used to parallelize any auto-correlation algorithm. Comparison with previous parallel architecture proposals show a 17 to 72% decrease in resource usage for Minn-Bhargava algorithm parallelization.
+
+* The proposed block-parallel architecture is then modified to support the proposed hierarchical synchronization algorithm and parallelization obtained for auto-correlation and cross-correlation algorithms is reported.
+
+* A new multipath delay (MPD) architecture for radix-2² IFFT/FFT algorithm is proposed which uses lesser resources compared to radix-2 architecture and is easily scalable to higher number of parallel outputs.
+
+* An end-to-end parallel architecture for CO-OFDM system is proposed, with each block parallelized to handle multiple samples per cycle inputs. Scalability and reduction in algorithmic/architectural complexity due to use of proposed time synchronization algorithm, MPD architecture of radix-2² for IFFT/FFT, data representation optimization leading to savings of multipliers in case of channel estimation and common phase estimation (CPE) is shown. Fixed-point analysis of the CO-OFDM transceiver for adoption of reduced bitwidth for meeting a particular value of root mean square error (RMSE) in case of transmitter and getting BER curve as close as possible to floating-point computation in case of receiver. Since multi-band CO-OFDM (MB-CO-OFDM) for reaching total data rate of 100 Gbs/ uses the same parallel architecture for all the sub-bands, the optimization of resources in a single sub-band leads to multiplied savings in resources across all the sub-bands.
+
+* The algorithms and the frame structure adopted are validated by experiments performed in the optical laboratory. Offline experiments using Matlab transmitter/receiver are performed to validate the algorithm performance in homodyne and heterodyne coherent detection configurations. From the heterodyne configuration, frame size suitable for use with our optical setup was found. This value of frame size was used in the development of the real-time FPGA platform. The validation of timing synchronization algorithm was done using real-time experiment in a electrical back-to-back (B2B) configuration.
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+## 6.2 Future Work
+
+Following sub-sections indicate future work possible in experimental and algorithm domains.
+
+### 6.2.1 Real-time FPGA platform experiments
+
+As mentioned in Chapter 5, optical experiments need to be conducted using the real-time FPGA platform. Firstly, homodyne configuration is performed to check the performance of the digital receiver only in presence of timing and phase offset. BER vs. OSNR curve will characterize the system performance. Next, in heterodyne configuration the effect of carrier frequency offset (CFO) will be seen on the BER curve. This configuration will exercise all the DSP blocks in the receiver chain. Finally, a very long length of standard single mode fiber (SSMF) (1000 km) will be connected to reach the complete real-world scenario for single-polarization single-band CO-OFDM system.
+
+### 6.2.2 Dual-polarization CO-OFDM System
+
+All the experiments in this Thesis have been conducted using only one polarization. Since coherent detection allows both polarizations to be used and thus higher utilization of the optical channel, transition to dual polarization single-band CO-OFDM system is necessary from throughput perspective. From a signal processing perspective, all the blocks except channel estimation will be replicated for the polarization added and frame structure needs to be changed slightly to accommodate for training symbols for channel estimation for both polarizations. Offline experiments conducted first to validate the channel estimation algorithm and then with real-time platform and long length of SSMF to reach the complete real-world scenario.
+
+### 6.2.3 Time Domain Sampling Clock Offset (SCO) Algorithm
+
+A low-complexity sampling clock offset (SCO) tracking algorithm and architecture in time domain will be very beneficial to control the sampling clock of the ADC and reduce the errors due to SCO offset. Position of SCO estimation in time domain reduces the loop delay for estimation and compensation compared to presently available SCO estimation algorithms [56] in frequency domain which is after FFT block. Due to presence of SCO estimation block after FFT, the delay of the feedback path is long and cannot adapt to sudden changes.
+
+## 6.3 Scaling to more than 100 Gb/s with MB-CO-OFDM system
+
+Scaling to data rate higher than 100 Gb/s like 400 Gb/s or 1 Tb/s can be achieved by increasing the size of FFT/IFFT to achieve higher data rate per polarization of a
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+single-band. Presently, the maximum size of FFT/IFFT is limited by the LASER phase
+noise variation to values less than or equal to 256. Digital CPE estimation algorithms
+assume that phase noise is constant across a single OFDM symbol, which breaks in case of
+larger OFDM symbol. RF-Pilot phase noise estimation scheme [57] has been proposed to
+overcome this. But the method is computationally very complex. A phase noise estimation
+scheme [58] which can use both RF-based pilot scheme and CPE method to handle large
+OFDM symbol size and still be computationally efficient would enable very high rates.
+The next option is when higher resolution DAC/ADC signal converters become available,
+it will be possible to support higher constellations in sub-carriers namely 16-QAM and
+64-QAM.
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+# Publications
+
+## Journal Publications
+
+1. P. Udupa, O. Sentieys and L. Bramerie, "A Scalable Parallel Architecture for Coarse Time Synchronization for Coherent Optical-OFDM Systems," submitted to *IEEE Transaction Briefs on Very Large Scale Integration (VLSI) Systems*, 2014.
+
+## International Conference Publications
+
+1. P. Udupa, O. Sentieys and P. Scalart, "A Novel Hierarchical Low Complexity Syn- chronization Method for OFDM Systems," in *IEEE VTC-Spring 2013*, 1-5, June 2013.
+
+2. P. Udupa, O. Sentieys and P. Scalart, "A Block-Parallel Architecture for Initial and Fine Synchronization in OFDM Systems," in *IEEE ICC 2013*, 4761-4765, June 2013.
+
+## National Conference/Colloquium Publications
+
+1. P. Udupa, O. Sentieys and L. Bramerie, "Design and Implementation of DSP algo- rithms for 100 Gbps Optical OFDM System," in *GRETSI 2013*, September 2013.
+
+2. P. Udupa, O. Sentieys and L. Bramerie, "Design and Real Time FPGA Prototyping of 100Gb/s Optical MB-OFDM System and Beyond," in *GDR SOC/SIP 2012*, June 2012.
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+[31] Inan, B. and Adhikari, S. and Karakaya, O. and Kainzmaier, P. and Mocker, M. and von Kirchbauer, H. and Hanik, N. and Jansen, S.L., "Realization of a real-time 93.8-Gb/s polarization-multiplexed OFDM transmitter with 1024-point IFFT," in *Optical Communication (ECOC), 2011 37th European Conference and Exhibition on*, 2011, pp. 1–3.
+
+[32] Y. M. S Chen, Q Yang and W. Shieh, "Multi-Gigabit Real-Time Coherent Optical OFDM Receiver," in *Proc. OFC/NFOEC, San Diego, CA*, March 2009, pp. 1–3.
+
+[33] T. Schmidl and D. Cox, "Robust Frequency and Timing Synchronization for OFDM," *IEEE Transactions on Communications*, vol. 45, no. 12, pp. 1613–1621, December 1997.
+
+[34] H. Minn, V. Bhargava, and K. Letaief, "A Robust Timing and Frequency Synchronization for OFDM Systems," *IEEE Transactions on Wireless Communications*, vol. 2, no. 4, pp. 822–839, July 2003.
+
+[35] K. Shi and E. Serpedin, "Coarse Frame and Carrier Synchronization of OFDM Systems: A New Metric and Comparison," *IEEE Transactions on Wireless Communications*, vol. 3, no. 4, pp. 1271–1284, July 2004.
+
+[36] B. Park, H. Cheon, C. Kang, and D. Hong, "A Novel Timing Estimation Method for OFDM Systems," *IEEE Communications Letters*, vol. 7, no. 5, pp. 239–241, May 2003.
\ No newline at end of file
diff --git a/samples/texts/7426999/page_158.md b/samples/texts/7426999/page_158.md
new file mode 100644
index 0000000000000000000000000000000000000000..28738936bcf73b2e6074b52d63497ee6c44310f8
--- /dev/null
+++ b/samples/texts/7426999/page_158.md
@@ -0,0 +1,23 @@
+[37] S. D. Choi, J. M. Choi, and J. H. Lee, "An Initial Timing Offset Estimation Method for OFDM Systems in Rayleigh Fading Channel," in *IEEE 64th Vehicular Technology Conference*, September 2006, pp. 1-5.
+
+[38] E. Zhou, X. Hou, Z. Zhang, and H. Kayama, "A Preamble Structure and Synchronization Method Based on Central-Symmetric Sequence for OFDM," in *IEEE Vehicular Technology Conference*, 2008, pp. 1478-1482.
+
+[39] Yun Hee Kim and Young-Kwon Hahm and Hye Jung Jung and Iickho Song, "An Efficient Frequency Offset Estimator for Timing and Frequency Synchronization in OFDM Systems," in *IEEE Pacific Rim Conference on Communications, Computers and Signal Processing*, 1999, pp. 580-583.
+
+[40] Chiueh, Tzi-Dar and Tsai, Pei-Yun and Lai, I-Wei, *Baseband Receiver Design for Wireless MIMO-OFDM Communications*. John Wiley and Sons Singapore Pte. Ltd., 2007, ch. 7, pp. 167-208.
+
+[41] M. Morelli, C.-C. Kuo, and M.-O. Pun, "Synchronization Techniques for Orthogonal Frequency Division Multiple Access (OFDMA): A Tutorial Review," *Proceedings of the IEEE*, vol. 95, no. 7, pp. 1394-1427, July 2007.
+
+[42] M. Speth, F. Classen, and H. Meyr, "Frame Synchronization of OFDM Systems in Frequency Selective Fading Channels," in *IEEE 47th Vehicular Technology Conference*, vol. 3, May 1997, pp. 1807-1811.
+
+[43] T. Pollet, M. Van Bladel, and M. Moeneclaey, "BER Sensitivity of OFDM Systems to Carrier Frequency Offset and Wiener Phase Noise," *IEEE Transactions on Communications*, vol. 43, no. 234, pp. 191-193, 1995.
+
+[44] J. van de Beek, M. Sandell, and P. Borjesson, "ML estimation of time and frequency offset in ofdm systems," *IEEE Transactions on Signal Processing*, vol. 45, no. 7, pp. 1800-1805, July 1997.
+
+[45] M. Morelli and U. Mengali, "An Improved Frequency Offset Estimator for OFDM Applications," in *Communication Theory Mini-Conference*, June 1999, pp. 106-109.
+
+[46] T. Bhatt, V. Sundaramurthy, J. Zhang, and D. McCain, "Initial Synchronization for 802.16e Downlink," in *Fortieth Asilomar Conference on Signals, Systems and Computers*, November 2006, pp. 701-706.
+
+[47] "Third-generation partnership project ts 36.211, physical channels and modulation (release 8) technical specification group," Radio Access Network:Evolved Universal Terrestrial Radio Access (E-UTRAN), Tech. Rep., 2008.
+
+[48] P. Serena, M. Bertolini and A. Vannucci. (2009) Optilux Toolbox. [Online]. Available: http://www.optilux.sourceforge.net
\ No newline at end of file
diff --git a/samples/texts/7426999/page_159.md b/samples/texts/7426999/page_159.md
new file mode 100644
index 0000000000000000000000000000000000000000..1c2c680b4b1ce44450f5665f0a144d1b293f1b8b
--- /dev/null
+++ b/samples/texts/7426999/page_159.md
@@ -0,0 +1,19 @@
+[49] P. Udupa, O. Sentieys and P.Scalart, "A Block-Parallel Architecture for Initial and Fine Synchronization in OFDM Systems," in *IEEE International Conference on Communications (ICC)*, 2013, pp. 4761–4765.
+
+[50] V. C. Kurapati, "Analysis of IP Based Implementations of Adders and Multipliers in Submicron and Deep Submicron Technologies," Master of Science, Oklahoma State University, December 2008.
+
+[51] Sander L. Jansen and Itsuro Morita and Noriyuki Takeda and Hideaki Tanaka, "20-Gb/s OFDM Transmission over 4,160-km SSMF Enabled by RF-Pilot Tone Phase Noise Compensation," in *Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference*. Optical Society of America, 2007.
+
+[52] W. Shieh, Q. Yang, and Y. Ma, "107 Gb/s Coherent Optical OFDM Transmission over 1000-km SSMF Fiber using Orthogonal Band Multiplexing," *Optics express*, vol. 16, no. 9, pp. 6378–6386, 2008.
+
+[53] R. Bouziane, P. Milder, R. Koutsoyannis, Y. Benlachtar, J. Hoe, M. Glick, and R. Killey, "Dependence of Optical OFDM Transceiver ASIC Complexity on FFT size," in *Optical Fiber Communication Conference and Exposition (OFC/NFOEC), 2012 and the National Fiber Optic Engineers Conference*, 2012, pp. 1–3.
+
+[54] Shousheng He and Torkelson, M., "A New Approach to Pipeline FFT Processor," in *The 10th International Parallel Processing Symposium*, 1996, pp. 766–770.
+
+[55] P. Udupa, O. Sentieys and L.Bramerie, "Design and Implementation of DSP algorithms for 100Gbps Optical OFDM System," in *XXIV Colloque GRETSI*, September 2013.
+
+[56] Lee, D. and Kyungwhoon Cheun, "A New Symbol Timing Recovery Algorithm for OFDM Systems," *IEEE Transactions on Consumer Electronics*, vol. 43, no. 3, pp. 767–775, August 1997.
+
+[57] S. Randel, S. Adhikari, and S. Jansen, "Analysis of RF-Pilot-Based Phase Noise Compensation for Coherent Optical OFDM Systems," *Photonics Technology Letters*, *IEEE*, vol. 22, no. 17, pp. 1288–1290, sept.1, 2010.
+
+[58] S.Hussin, K.Puntsri and R.Noe, "Improvement of RF-Pilot Phase Noise Compensation for Coherent Optical OFDM Systems via CPE Equalizer," 2013.
\ No newline at end of file
diff --git a/samples/texts/7426999/page_16.md b/samples/texts/7426999/page_16.md
new file mode 100644
index 0000000000000000000000000000000000000000..cd9c582a3cb5b45e44555e05b13c330d1b3fbf57
--- /dev/null
+++ b/samples/texts/7426999/page_16.md
@@ -0,0 +1,49 @@
+<table>
+  <tbody>
+    <tr>
+      <td>5.11</td>
+      <td>Configuration of Homodyne Coherent Detection with SMF of 50 km. DSP processing is done offline in Matlab. Green blocks indicate analogue blocks. Light Blue blocks indicate Optical components.</td>
+      <td>117</td>
+    </tr>
+    <tr>
+      <td>5.12</td>
+      <td>Configuration of Heterodyne Coherent Detection with standard single mode fiber (SSMF) of 50 km. DSP processing is done offline in Matlab. Green blocks indicate analogue blocks. Light Blue blocks indicate Optical components.</td>
+      <td>118</td>
+    </tr>
+    <tr>
+      <td>5.13</td>
+      <td>BER vs SNR for single-band CO-OFDM system for Heterodyne Detection</td>
+      <td>118</td>
+    </tr>
+    <tr>
+      <td>5.14</td>
+      <td>Real-Time FPGA Transmitter Block Diagram. PLL - Phase Locked Loop, SFP+ - Enhanced Small Form-factor Pluggable, SMF Cable - Single Mode Fiber Cable, I/F - Interface, CDR I/F - Clock Data Recovery Interface, DAC I/F - Digital-to-Analog Converter Interface.</td>
+      <td>119</td>
+    </tr>
+    <tr>
+      <td>5.15</td>
+      <td>100GFLEX Frame Format</td>
+      <td>119</td>
+    </tr>
+    <tr>
+      <td>5.16</td>
+      <td>Real-Time FPGA Receiver Block Diagram. ADC - Analog-to-Digital Converter, I/F - Interface, SFP+ I/F - Enhanced Small Form-factor Pluggable Interface, SMF Cable - Single Mode Fiber Cable, CDR I/F - Clock Data Recovery Interface, PLL - Phase Locked Loop.</td>
+      <td>121</td>
+    </tr>
+    <tr>
+      <td>5.17</td>
+      <td>Snapshot of Real-Time FPGA Transceiver Platform. The topmost rack shows the power supply for the configuration, the second rack is the EKINOPS Altera FPGA Digital Transceiver, the third rack shows the Xilinx Virtex-7 FPGA interfaced to DAC board, the bottom most rack shows the other Xilinx Virtex-7 FPGA interfaced to ADC. The yellow cables are single mode fiber (SMF) cables to connect using SFP+ interface.</td>
+      <td>122</td>
+    </tr>
+    <tr>
+      <td>5.18</td>
+      <td>Altera SignalTap Snapshot of coarse synchronization output. Presence of periodic zeros indicate cyclic prefix removal and bigger gap zeros indicate the removal of the first training symbol in the output fed into FFT block.</td>
+      <td>123</td>
+    </tr>
+    <tr>
+      <td>5.19</td>
+      <td>Altera SignalTap Snapshot of coarse synchronization output of OFDM symbols. Starting from second row, it contains real and imaginary signals alternatively. Correctness of the synchronization is verified by observing that alternate rows have repeating values indicating correct synchronization is achieved.</td>
+      <td>124</td>
+    </tr>
+  </tbody>
+</table>
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diff --git a/samples/texts/7426999/page_160.md b/samples/texts/7426999/page_160.md
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index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391
diff --git a/samples/texts/7426999/page_161.md b/samples/texts/7426999/page_161.md
new file mode 100644
index 0000000000000000000000000000000000000000..4909c444544c4b699e821e08c52629ddfa9365fa
--- /dev/null
+++ b/samples/texts/7426999/page_161.md
@@ -0,0 +1,7 @@
+**Résumé en français** : Optique Cohérente-OFDM (CO-OFDM) a été proposée comme un candidat viable pour 100 Gigabit Ethernet (100 GbE) nœud. CO-OFDM que tout traitement à l'aide de signaux numériques de traitement (DSP) algorithmes pour estimer et compenser tous les non-idéalités de canal et optoélectronique les systèmes d'extrémité avant. Dans cette thèse, les algorithmes de faible complexité, les architectures parallèles évolutives pour grands blocs de calcul complexe de CO-OFDM émetteur-récepteur sont explorées. Un temps faible complexité synchronisation est proposé qui donne de meilleures performances que algorithmes d'auto-corrélation de canal optique. Une architecture parallèle évolutive est proposé pour l'algorithme qui peut prendre en charge plusieurs échantillons parallèles et réduit l'utilisation des ressources de l'ordre de 70% par rapport à la proposition précédente. Un parallèle bout-à-bout CO-OFDM l'architecture d'émetteur-récepteur est proposé qui intègre parallèlement à radix-2² bloc IFFT/FFT, ce qui réduit considérablement la complexité de calcul par rapport à radix-2 architecture et canal bloc d'estimation qui utilise la représentation des données optimisations pour supprimer multiplicateurs, entraînant des gains de 24% de la région. Enfin, les algorithmes et architectures ont été validés par des expériences hors ligne/Matlab et FPGA en temps réel la plate-forme expériences, respectivement.
+
+**Mots clés** : Optique cohérente-OFDM, la synchronisation du temps/fréquence, algorithmes de faible complexité, architectures parallèles, fibre optique
+
+**Résumé en anglais** : Coherent Optical-OFDM (CO-OFDM) has been proposed as a viable candidate for 100 gigabit Ethernet (100GbE) node. CO-OFDM does all processing using digital signal processing (DSP) algorithms to estimate and compensate all non-idealities of channel and opto-electronic front end systems. In this thesis, low-complexity algorithms, scalable parallel architectures for major computationally complex blocks of CO-OFDM transceiver are explored. A low-complexity time synchronization is proposed which gives better performance than auto-correlation algorithms in optical channel. A scalable parallel architecture is proposed for the algorithm which can support multiple parallel samples and reduces resource usage of around 70% compared to previous proposal. An end-to-end parallel CO-OFDM transceiver architecture is proposed which incorporates parallel radix-2<sup>2</sup> IFFT/FFT block, which reduces the computational complexity significantly compared to radix-2 architecture and channel estimation block which uses data representation optimizations to remove multipliers, resulting in area gains of 24%. Finally, the algorithms and architectures were validated using offline/Matlab experiments and real-time FPGA platform experiments respectively.
+
+**Keywords** : Coherent Optical-OFDM, time/frequency synchronization, low-complexity algorithms, parallel architectures, optical fiber
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diff --git a/samples/texts/7426999/page_17.md b/samples/texts/7426999/page_17.md
new file mode 100644
index 0000000000000000000000000000000000000000..ad7dac73304e2a964352afdebb3d41a3b4ac9a62
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@@ -0,0 +1,136 @@
+# List of Tables
+
+<table>
+  <tbody>
+    <tr>
+      <td>1</td>
+      <td>Nombre d'opérations réelles pour le calcul d'un point de la métrique temporelle</td>
+      <td>7</td>
+    </tr>
+    <tr>
+      <td>2</td>
+      <td>Complexité architecturale en fonction de R</td>
+      <td>10</td>
+    </tr>
+    <tr>
+      <td>3</td>
+      <td>Complexité architecturale en fonction de R</td>
+      <td>11</td>
+    </tr>
+    <tr>
+      <td>4</td>
+      <td>Complexité algorithmique pour le calcul de N sorties de la IFFT avec N=256</td>
+      <td>12</td>
+    </tr>
+    <tr>
+      <td>5</td>
+      <td>Complexité algorithmique (auto-corrélation) de notre proposition</td>
+      <td>12</td>
+    </tr>
+    <tr>
+      <td>6</td>
+      <td>Complexité algorithmique de l'estimation CFO entière</td>
+      <td>13</td>
+    </tr>
+    <tr>
+      <td>7</td>
+      <td>Complexité algorithmique de l'estimation de canal</td>
+      <td>14</td>
+    </tr>
+    <tr>
+      <td>8</td>
+      <td>Complexité algorithmique de l'estimation et de la compensation CPE</td>
+      <td>14</td>
+    </tr>
+    <tr>
+      <td>2.1</td>
+      <td>DWDM Band Wavelength Range</td>
+      <td>28</td>
+    </tr>
+    <tr>
+      <td>2.2</td>
+      <td>Specifications of commercially available single mode fibers (Corning Fibers)</td>
+      <td>30</td>
+    </tr>
+    <tr>
+      <td>2.3</td>
+      <td>Cost of Optical Transceiver for CO-OFDM, CO-QPSK and IM-DD Systems</td>
+      <td>40</td>
+    </tr>
+    <tr>
+      <td>2.4</td>
+      <td>Algorithmic Complexity in terms of size of IFFT/FFT N.</td>
+      <td>41</td>
+    </tr>
+    <tr>
+      <td>2.5</td>
+      <td>Architectural Complexity of feedforward pipelined IFFT/FFT for 2/4/8- Parallel Outputs as a function of IFFT/FFT size (N). MDC - Multipath Delay Commutator.</td>
+      <td>42</td>
+    </tr>
+    <tr>
+      <td>2.6</td>
+      <td>Real-time CO-OFDM Transmitter Implementation</td>
+      <td>42</td>
+    </tr>
+    <tr>
+      <td>2.7</td>
+      <td>Computational Complexity for CO-OFDM Transmitter</td>
+      <td>42</td>
+    </tr>
+    <tr>
+      <td>2.8</td>
+      <td>Algorithmic Complexity of Coarse Time Synchronization Algorithms. Calculations count only correlation operation and not the energy calculation.</td>
+      <td>43</td>
+    </tr>
+    <tr>
+      <td>2.9</td>
+      <td>Architectural Complexity of Coarse Time Synchronization Algorithms</td>
+      <td>44</td>
+    </tr>
+    <tr>
+      <td>2.10</td>
+      <td>Algorithmic/Architectural Complexity for integer CFO Estimation. R - number of parallel outputs.</td>
+      <td>46</td>
+    </tr>
+    <tr>
+      <td>2.11</td>
+      <td>Algorithmic/Architectural Complexity for Channel Estimation and Equalization. R - number of parallel outputs.</td>
+      <td>48</td>
+    </tr>
+    <tr>
+      <td>2.12</td>
+      <td>Algorithmic/Architectural Complexity for CPE Estimation and Compensation. R - number of parallel outputs.</td>
+      <td>48</td>
+    </tr>
+    <tr>
+      <td>3.1</td>
+      <td>Simulation Parameters.</td>
+      <td>57</td>
+    </tr>
+    <tr>
+      <td>3.2</td>
+      <td>Number of Real Operations for calculation of a single timing metric point</td>
+      <td>60</td>
+    </tr>
+    <tr>
+      <td>3.3</td>
+      <td>Simulation Parameters for CO-OFDM System Simulation</td>
+      <td>62</td>
+    </tr>
+    <tr>
+      <td>3.4</td>
+      <td>Architectural Complexity calculation as a function of R-parallel input/output for SCA</td>
+      <td>71</td>
+    </tr>
+    <tr>
+      <td>3.5</td>
+      <td>Architectural Complexity calculation as a function of R-parallel input/output for MBA.</td>
+      <td>73</td>
+    </tr>
+    <tr>
+      <td>3.6</td>
+      <td>Architectural Complexity of P<sub>sc</sub> for FSBP Architecture as a function of R-parallel input/output</td>
+      <td>75</td>
+    </tr>
+  </tbody>
+</table>
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diff --git a/samples/texts/7426999/page_18.md b/samples/texts/7426999/page_18.md
new file mode 100644
index 0000000000000000000000000000000000000000..859fe023be8e52de7d035f5c7eb07643ee81e84f
--- /dev/null
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@@ -0,0 +1,134 @@
+<table>
+  <tbody>
+    <tr>
+      <td>3.7</td>
+      <td>Architectural Complexity of P<sub>mb</sub> for FSBP Architecture as a function of R-parallel input/output</td>
+      <td>76</td>
+    </tr>
+    <tr>
+      <td>3.8</td>
+      <td>Area estimates for 2-Stage Pipelined Adders and Multipliers for 90nm technology node</td>
+      <td>77</td>
+    </tr>
+    <tr>
+      <td>3.9</td>
+      <td>Area calculation of FSBP (Schmidl-Cox) and Kaneda's architecture at 90nm technology node for 5-bit multiplier, 10-bit adder for R = 16-Parallel input/output for Schmidl-Cox Algorithm</td>
+      <td>79</td>
+    </tr>
+    <tr>
+      <td>3.10</td>
+      <td>Area calculation of FSBP (Minn-Bhargava) and Kaneda's architecture at 90 nm technology node for 5-bit multiplier, 10-bit adder for R = 16-Parallel input/output for Schmidl-Cox Algorithm</td>
+      <td>79</td>
+    </tr>
+    <tr>
+      <td>3.11</td>
+      <td>Conjugate Symmetric Correlation Parallelism Factor achieved on PSPB/FSPB architecture</td>
+      <td>80</td>
+    </tr>
+    <tr>
+      <td>4.1</td>
+      <td>Calculation of N<sub>cyp</sub> and N given SMF parameters</td>
+      <td>85</td>
+    </tr>
+    <tr>
+      <td>4.2</td>
+      <td>Algorithmic Complexity for calculation of N output for IFFT size of 256</td>
+      <td>86</td>
+    </tr>
+    <tr>
+      <td>4.3</td>
+      <td>Algorithmic Complexity (auto-correlation function only) for Proposed Synchronization Algorithm</td>
+      <td>87</td>
+    </tr>
+    <tr>
+      <td>4.4</td>
+      <td>Algorithmic Complexity for integer CFO estimation algorithm.</td>
+      <td>88</td>
+    </tr>
+    <tr>
+      <td>4.5</td>
+      <td>Algorithmic Complexity for Channel Estimation algorithms</td>
+      <td>89</td>
+    </tr>
+    <tr>
+      <td>4.6</td>
+      <td>Algorithmic Complexity for CPE Compensation</td>
+      <td>89</td>
+    </tr>
+    <tr>
+      <td>4.7</td>
+      <td>Architectural Complexity (normal input order) for full streaming outputs for N = 256, with input and output in natural order. Resource count is generated by using SPIRAL tool [4] for radix-2/4/8/16 and using [5] for radix-2<sup>2</sup></td>
+      <td>92</td>
+    </tr>
+    <tr>
+      <td>4.8</td>
+      <td>Mean (μ) and Standard Deviation (σ) of RMSE for variation of Bitwidths of inputs/outputs W<sub>i</sub> and Twiddle Factor W<sub>t</sub></td>
+      <td>93</td>
+    </tr>
+    <tr>
+      <td>4.9</td>
+      <td>Area Occupied for variation of Bitwidths of inputs/outputs W<sub>i</sub> and Twiddle Factor W<sub>t</sub></td>
+      <td>94</td>
+    </tr>
+    <tr>
+      <td>4.10</td>
+      <td>Architectural Complexity of Time/Frequency Architecture for R = 4-Parallel input/output</td>
+      <td>96</td>
+    </tr>
+    <tr>
+      <td>4.11</td>
+      <td>Look-up table implemented for complex multiplication of conjugate of reference symbol with input r = a + jb.</td>
+      <td>97</td>
+    </tr>
+    <tr>
+      <td>4.12</td>
+      <td>Architectural Complexity of IFO Estimation Architecture for R = 4-Parallel input.</td>
+      <td>97</td>
+    </tr>
+    <tr>
+      <td>4.13</td>
+      <td>Architectural Complexity of Channel Estimator/Equalizer for R = 4-Parallel input/output</td>
+      <td>97</td>
+    </tr>
+    <tr>
+      <td>4.14</td>
+      <td>Architectural Complexity of CPE Estimator and Compensator for R = 4-Parallel input/output</td>
+      <td>99</td>
+    </tr>
+    <tr>
+      <td>4.15</td>
+      <td>Fixed-point configurations for Homodyne setup</td>
+      <td>101</td>
+    </tr>
+    <tr>
+      <td>4.16</td>
+      <td>BER vs. ONSR for floating-point and various fixed-point configurations in Homodyne setup.</td>
+      <td>101</td>
+    </tr>
+    <tr>
+      <td>4.17</td>
+      <td>Fixed-point configurations for Heterodyne setup.</td>
+      <td>102</td>
+    </tr>
+    <tr>
+      <td>4.18</td>
+      <td>BER vs. ONSR for floating-point and various fixed-point configurations in Heterodyne setup.</td>
+      <td>102</td>
+    </tr>
+    <tr>
+      <td>4.19</td>
+      <td>Area Occupied vs. Bitwidth for Time/Frequency Synchronization block.</td>
+      <td>102</td>
+    </tr>
+    <tr>
+      <td>4.20</td>
+      <td>Area vs. Bitwidth for CFO Compensation block.</td>
+      <td>103</td>
+    </tr>
+    <tr>
+      <td>4.21</td>
+      <td>Area vs. Bitwidth for FFT block.</td>
+      <td>103</td>
+    </tr>
+  </tbody>
+</table>
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+<table>
+  <tr>
+    <td>4.22</td>
+    <td>Area vs. Bitwidth for Integer CFO Estimation block</td>
+    <td>103</td>
+  </tr>
+  <tr>
+    <td>4.23</td>
+    <td>Area vs. Bitwidth for De-interleaver block</td>
+    <td>104</td>
+  </tr>
+  <tr>
+    <td>4.24</td>
+    <td>Area vs. Bitwidth for Channel Estimation &amp; Equalization</td>
+    <td>104</td>
+  </tr>
+  <tr>
+    <td>4.25</td>
+    <td>Area vs. Bitwidth for CPE Estimation &amp; Compensation</td>
+    <td>104</td>
+  </tr>
+  <tr>
+    <td>4.26</td>
+    <td>Fixed-Point Allocation and Area for all blocks of the R = 4-Parallel Receiver</td>
+    <td>105</td>
+  </tr>
+  <tr>
+    <td>5.1</td>
+    <td>Parameters of the Electrical B2B Experiment</td>
+    <td>110</td>
+  </tr>
+  <tr>
+    <td>5.2</td>
+    <td>100GFLEX Frame Format Parameters</td>
+    <td>120</td>
+  </tr>
+</table>
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+# THÈSE / UNIVERSITÉ DE RENNES 1
+
+sous le sceau de l'Université Européenne de Bretagne
+
+pour le grade de
+
+DOCTEUR DE L'UNIVERSITÉ DE RENNES 1
+
+Mention : Traitement du Signal et Télécommunications
+
+École doctorale : MATISSE
+
+présentée par
+
+Pramod UDUPA
+
+préparée à l'unité de recherche : IRISA - UMR 6074
+
+Institut de recherche en informatique et systèmes aléatoires - CAIRN
+
+École Nationale Supérieure des Sciences Appliquées et de Technologie
+
+## Composition du jury :
+
+**Lilian BOSSUET**
+Maître de Conférences HDR, Télécom Saint-Etienne
+Université Jean Monnet / Examinateur
+
+**Erwan PINCEMIN**
+Ingénieur, Orange Labs
+Examinateur
+
+**Michel JEZEQUEL**
+Professeur, Electronics Department
+Télécom Bretagne / Examinateur
+
+**Emmanuel BOUTILLON**
+Professeur, Lab-STICC
+Université de Bretagne Sud / Rapporteur
+
+**Christophe JEGO**
+Professeur, IMS, Bordeaux
+Universités - IPB/ENSEIRB-MATMECA / Rappor-
+teur
+
+**Olivier SENTIEYS**
+Directeur de Recherche, IRISA/INRIA
+Université de Rennes 1 / Directeur de thèse
+
+**Laurent BRAMERIE**
+Ingénieur de Recherche, FOTON, ENSSAT
+Université de Rennes 1 / Co-directeur de thèse
+
+Algorithmes parallèles et
+architectures évolutives
+de faible complexité
+pour systémes optiques OFDM
+cohérents temps réel
+
+Low Complexity,
+Parallel Algorithms
+and Scalable Architectures
+for Real Time Coherent Optical
+OFDM Systems
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+# List of Abbreviations
+
+<table><tr><td>100 GbE</td><td>100 Gigabit Ethernet</td></tr><tr><td>ADC</td><td>Analog to Digital Converter</td></tr><tr><td>ASE</td><td>Amplified Spontaneous Noise</td></tr><tr><td>ASIC</td><td>Application Specific Integrated Circuit</td></tr><tr><td>AWG</td><td>Arbitrary Waveform Generator</td></tr><tr><td>AWGN</td><td>Additive White Gaussian Noise</td></tr><tr><td>BER</td><td>Bit Error Rate</td></tr><tr><td>BPF</td><td>Band Pass Filter</td></tr><tr><td>BPSK</td><td>Binary Phase Shift Keying</td></tr><tr><td>CD</td><td>Chromatic Dispersion</td></tr><tr><td>CE</td><td>Channel Estimation</td></tr><tr><td>CFO</td><td>Carrier Frequency Offset</td></tr><tr><td>CMZM</td><td>Complex Mach-Zender Modulator</td></tr><tr><td>CO</td><td>Coherent Optical</td></tr><tr><td>CO-DP-QPSK</td><td>Coherent Optical Dual Polarization QPSK</td></tr><tr><td>CO-OFDM</td><td>Coherent Optical-OFDM</td></tr><tr><td>CoD</td><td>Coherent Detection</td></tr><tr><td>CP</td><td>Cyclic Prefix</td></tr><tr><td>CPE</td><td>Common Phase Error</td></tr><tr><td>DAC</td><td>Digital to Analog Converter</td></tr><tr><td>DCF</td><td>Dispersion Compensated Fiber</td></tr><tr><td>DD</td><td>Direct Detection</td></tr><tr><td>DFT</td><td>Discrete Fourier Transform</td></tr><tr><td>DGD</td><td>Differential Group Delay</td></tr><tr><td>DP-QPSK</td><td>Dual Polarization QPSK</td></tr></table>
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+<table>
+   <tr>
+    <td>DPSK</td>
+    <td>Differential Phase Shift Keying</td>
+   </tr>
+   <tr>
+    <td>DSO</td>
+    <td>Digital Storage Oscilloscope</td>
+   </tr>
+   <tr>
+    <td>DSP</td>
+    <td>Digital Signal Processing</td>
+   </tr>
+   <tr>
+    <td>DWDM</td>
+    <td>Dense Wavelength Division Multiplexing</td>
+   </tr>
+   <tr>
+    <td>EB</td>
+    <td>Eta Bytes (10<sup>18</sup>)</td>
+   </tr>
+   <tr>
+    <td>ECL</td>
+    <td>External Cavity LASER</td>
+   </tr>
+   <tr>
+    <td>EDC</td>
+    <td>Electronic Dispersion Compensation</td>
+   </tr>
+   <tr>
+    <td>EDFA</td>
+    <td>Erbium-Doped Fiber Amplifier</td>
+   </tr>
+   <tr>
+    <td>ENOB</td>
+    <td>Effective Number of Bits</td>
+   </tr>
+   <tr>
+    <td>EVM</td>
+    <td>Error Vector Magnitude</td>
+   </tr>
+   <tr>
+    <td>FEC</td>
+    <td>Forward Error Correction</td>
+   </tr>
+   <tr>
+    <td>FFT</td>
+    <td>Fast Fourier Transform</td>
+   </tr>
+   <tr>
+    <td>FIR</td>
+    <td>Finite Impulse Response</td>
+   </tr>
+   <tr>
+    <td>FPGA</td>
+    <td>Field Programmable Gate Array</td>
+   </tr>
+   <tr>
+    <td>FSF</td>
+    <td>Frequency Selective Fading</td>
+   </tr>
+   <tr>
+    <td>FSM</td>
+    <td>Finite State Machine</td>
+   </tr>
+   <tr>
+    <td>FTTH</td>
+    <td>Fiber-to-the-Home</td>
+   </tr>
+   <tr>
+    <td>GB</td>
+    <td>Giga Bytes (10<sup>9</sup>)</td>
+   </tr>
+   <tr>
+    <td>GOPS</td>
+    <td>Giga Operations per second</td>
+   </tr>
+   <tr>
+    <td>GVD</td>
+    <td>Group Velocity Dispersion</td>
+   </tr>
+   <tr>
+    <td>HLS</td>
+    <td>High Level Synthesis</td>
+   </tr>
+   <tr>
+    <td>ICI</td>
+    <td>Inter Carrier Interference</td>
+   </tr>
+   <tr>
+    <td>IDFT</td>
+    <td>Inverse Discrete Fourier Transform</td>
+   </tr>
+   <tr>
+    <td>IEEE</td>
+    <td>Institute of Electrical and Electronic Engineers</td>
+   </tr>
+   <tr>
+    <td>IFFT</td>
+    <td>Inverse Fast Fourier Transform</td>
+   </tr>
+   <tr>
+    <td>IIR</td>
+    <td>Infinite Impulse Response</td>
+   </tr>
+   <tr>
+    <td>IM</td>
+    <td>Intensity Modulation</td>
+   </tr>
+   <tr>
+    <td>IM-DD</td>
+    <td>Intensity Modulation-Direct Detection</td>
+   </tr>
+   <tr>
+    <td>ISI</td>
+    <td>Inter Symbol Interference</td>
+   </tr>
+   <tr>
+    <td>ITU</td>
+    <td>International Telecommunication Union</td>
+   </tr>
+   <tr>
+    <td>LAN</td>
+    <td>Local Area Network</td>
+   </tr>
+  </table>
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+**LASER** Light Amplification by Stimulated Emission of Radiation
+
+**LD** LASER Diode
+
+**LEAF** Large Effective Area Fiber
+
+**LMS** Least Mean Squares
+
+**LO** Local Oscillator
+
+**LPF** Low Pass Filter
+
+**LS** Least Squares
+
+**LTE** Long Term Evolution
+
+**MAN** Metropolitan Area Network
+
+**MB** Multi-Band
+
+**MB-OFDM** Multi-band OFDM
+
+**MDC** Multipath Delay Commutator
+
+**MDF** Multipath Delay Feedback
+
+**MHz** Mega Hertz
+
+**MIMO** Multi Input Multi Output
+
+**MLSE** Maximum Likelihood Sequence Estimation
+
+**MOPS** Million of Operations per second
+
+**MSE** Mean Square Error
+
+**MZM** Mach-Zender Modulator
+
+**N-WDM** Nyquist Wavelength Division Multiplexing
+
+**NGI CO-OFDM** No-Guard Interval CO-OFDM
+
+**NLMS** Normalized Least Mean Square
+
+**NRZ** Non Return-to-Zero
+
+**OBM-OFDM** Orthogonal Band Multiplexed OFDM
+
+**OBPF** Optical BPF
+
+**OEO** Optical-to-Electrical-to-Optical
+
+**OFDM** Orthogonal Frequency Division Multiplexing
+
+**OOK** On-Off Keying
+
+**OSA** Optical Spectrum Analyzer
+
+**OSNR** Optical Signal to Noise Ratio
+
+**PAPR** Peak to Average Power Ratio
+
+**PB** Peta Bytes (10¹⁵)
+
+**PBC** Polarization Beam Combiner
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+<table>
+  <tr>
+    <td>PBS</td>
+    <td>Polarization Beam Splitter</td>
+  </tr>
+  <tr>
+    <td>PCD</td>
+    <td>Polarization Dependent Chromatic Dispersion</td>
+  </tr>
+  <tr>
+    <td>PD</td>
+    <td>Photo Diode</td>
+  </tr>
+  <tr>
+    <td>PDM</td>
+    <td>Polarization Division Multiplexing</td>
+  </tr>
+  <tr>
+    <td>PDM-CO-OFDM</td>
+    <td>Polarization Division Multiplexing-Coherent Optical-Orthogonal Frequency Division Multiplexing</td>
+  </tr>
+  <tr>
+    <td>PDM-QPSK</td>
+    <td>Polarization Division Multiplexing-Quadrature Phase Shift Keying</td>
+  </tr>
+  <tr>
+    <td>PDR</td>
+    <td>Polarization Diverse Receiver</td>
+  </tr>
+  <tr>
+    <td>PMD</td>
+    <td>Polarization Mode Dispersion</td>
+  </tr>
+  <tr>
+    <td>PRBS</td>
+    <td>Pseudo Random Bit Sequence</td>
+  </tr>
+  <tr>
+    <td>PS</td>
+    <td>Polarization Scrambler</td>
+  </tr>
+  <tr>
+    <td>PSCF</td>
+    <td>Pure Silica Core Fiber</td>
+  </tr>
+  <tr>
+    <td>PSK</td>
+    <td>Phase Shift Keying</td>
+  </tr>
+  <tr>
+    <td>PSP</td>
+    <td>Principle State of Polarization</td>
+  </tr>
+  <tr>
+    <td>PSS</td>
+    <td>Primary Synchronization Signal</td>
+  </tr>
+  <tr>
+    <td>QAM</td>
+    <td>Quadrature Amplitude Modulation</td>
+  </tr>
+  <tr>
+    <td>QoS</td>
+    <td>Quality of Service</td>
+  </tr>
+  <tr>
+    <td>QPSK</td>
+    <td>Quadrature Phase Shift Keying</td>
+  </tr>
+  <tr>
+    <td>RF</td>
+    <td>Radio Frequency</td>
+  </tr>
+  <tr>
+    <td>RGI CO-OFDM</td>
+    <td>Reduced Guard Interval CO-OFDM</td>
+  </tr>
+  <tr>
+    <td>RMSE</td>
+    <td>Root Mean Square Error</td>
+  </tr>
+  <tr>
+    <td>ROADM</td>
+    <td>Reconfigurable Optical Add Drop Multiplexer</td>
+  </tr>
+  <tr>
+    <td>RS</td>
+    <td>Reed-Solomon</td>
+  </tr>
+  <tr>
+    <td>RZ</td>
+    <td>Return-to-Zero</td>
+  </tr>
+  <tr>
+    <td>RZ-DQPSK</td>
+    <td>Return-to-Zero Differential Quadratic Phase Shift Keying</td>
+  </tr>
+  <tr>
+    <td>SCO</td>
+    <td>Sampling Clock Offset</td>
+  </tr>
+  <tr>
+    <td>SDC</td>
+    <td>Single Path Delay Commutator</td>
+  </tr>
+  <tr>
+    <td>SDF</td>
+    <td>Single Path Delay Feedback</td>
+  </tr>
+  <tr>
+    <td>SNR</td>
+    <td>Signal to Noise Ratio</td>
+  </tr>
+  <tr>
+    <td>SOP</td>
+    <td>State of Polarization</td>
+  </tr>
+</table>
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+**SOPMD** Second Order PMD
+
+**SPM** Self Phase Modulation
+
+**SQNR** Signal Quantization to Noise Ratio
+
+**SSMF** Standard Single Mode Fiber
+
+**TB** Tera Bytes (10¹²)
+
+**TS** Training Symbol
+
+**ULH** Ultra Long Haul
+
+**VNI** Visual Networking Index
+
+**VOA** Variable Optical Attenuator
+
+**WAN** Wide Area Network
+
+**WDM** Wavelength Division Multiplexing
+
+**XPM** Cross Phase Modulation
+
+**ZF** Zero Forcing
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+# Chapitre 0
+
+## Résumé étendu
+
+### 0.1 Système de communications optiques OFDM à détection cohérente
+
+L'arrivée massive des équipements connectés tels que les smartphones, tablettes, etc., et l'augmentation de la demande des services basés sur la vidéo, impliquent une augmentation exponentielle de la bande passante des réseaux, ce qui augmente la pression sur tous les nœuds du réseau Internet. Les réseaux de communication optique basés sur des fibres optiques SMF (single-mode optical fiber) sont utilisés pour transporter les données à haut débit sur de longues distances (long-haul), des distances moyennes (metro) ou des distances plus courtes (Fiber-to-the-Home (FTTH)). Les réseaux optiques peuvent donc être divisés en trois partie majeures comme illustrées sur la Figure 1.
+
+FIGURE 1: Architecture typique d'un réseau optique. CN - Core Node, EN - Edge Node, AN - Access Node
+
+Pour transmettre des données sur une fibre optique simple mode (SMF), des techniques de modulation et détection directes IM-DD (intensity modulation-direct detection) sont utilisées pour obtenir des débits de 10 Gb/s sur des réseaux longues distances. Pour supporter l'augmentation de la demande en débit, l'objectif est de supporter des liaisons à 100 Gb/s [6]. Ceci ne peut être atteint avec le schéma IM-DD de façon efficace à cause de phénomènes tels que la dispersion chromatique CD (chromatic dispersion) ou la dispersion du mode de polarisation PMD (polarization mode dispersion) qui apparaissant à de telles
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+vitesses. Pour atteindre de tels débits, la détection cohérente CoD (coherent detection) a été introduite dans les systèmes de communications optiques et rendue possible grâce aux progrès des circuits intégrés électroniques. La détection cohérente [7][8][9] offre des avantages grâce à sa meilleure sensibilité de détection, à un débit symbole plus élevé, à l'utilisation de la double polarisation (dual polarization) et, de façon plus importante, à la conservation de l'information de phase et d'amplitude entre les domaines optiques et électroniques, ce qui ouvre la possibilité d'utiliser de puissants algorithmes de traitement numérique du signal pour la compensation électronique des dispersions EDC (electronic dispersion compensation) à un faible coût et de façon flexible grâce aux DSP. L'utilisation des modulations OFDM (orthogonal frequency division multiplexing) a été proposée pour être utilisée conjointement avec la CoD pour atteindre des débits de 100 Gb/s avec une meilleure flexibilité. La modulation OFDM est immune à la CD grâce à la présence d'un préfix cyclique (cyclic prefix (CP)) et à la réduction de la complexité de l'égaliseur avec l'utilisation de symboles d'apprentissage (training symbols (TS)). De plus, l'OFDM offre un clair avantage en termes de flexibilité pour l'allocation de puissance par sous porteuse (bit-power loading) et la présence de symboles pilotes dans les sous porteuses en fonction des conditions du canal. L'OFDM multi-bande à détection cohérente MB-CO-OFDM (Multiband-Coherent Optical-OFDM) a donc été proposé en se basant sur les technologies récentes de convertisseurs numériques DAC et ADC et la réalisation possible dans les circuits intégrés ASIC ou FPGA.
+
+## 0.2 Contexte du travail
+
+Les systèmes CO-OFDM sont sensibles aux non linéarités présentes dans la chaîne de traitement du signal comme le PAPR (peak-to-average-power-ratio). Il sont également sensibles aux offsets de phase et de fréquence du LASER et aux erreurs de timing. Dans ces systèmes CO-OFDM, l'estimation et la compensation des imperfections sont réalisées dans le domaine numérique à l'aide d'algorithmes de traitement du signal. Avec la contrainte que chaque bande du système CO-OFDM travaille à un débit au Gb/s, les algorithmes de traitement du signal doivent être de faible complexité pour pouvoir supporter de telles fréquences de fonctionnement et d'échantillonnage et être implémentés dans des circuits comme des FPGA. De plus, les fréquences d'horloge des FPGA atteignent quelques centaines de MHz tandis que les fréquences d'échantillonnage des DAC/ADC requises sont de plusieurs GHz, ce qui implique des traitements hyper-parallèles au sein des architectures numériques de traitement. L'objectif de cette thèse est de proposer et d'implémenter des algorithmes parallèles performants et à faible complexité pour un transmetteur (émetteur + récepteur) CO-OFDM simple-bande et simple-polarisation, l'extension à plusieurs bandes et deux polarisations étant simple vu la scalabilité des algorithmes proposés.
+
+L'approche que nous proposons pour réduire la complexité est tout d'abord d'utiliser
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+des algorithmes (e.g. synchronisation) à faible complexité avant de proposer des architectures parallèles efficaces pour leur implémentation matérielle. Les sections suivantes détaillent les résultats obtenus. Premièrement, une architecture et un algorithme à faible complexité sont proposés pour la synchronisation temporelle des trames et symboles OFDM. Deuxièmement, une architecture d'un transmetteur CO-OFDM complet est détaillé. Finalement, les architectures et algorithmes sont validés dans un contexte d'expérimentation offline puis temps réel.
+
+## 0.3 Algorithme de synchronisation temporelle à faible complexité pour les systèmes OFDM
+
+Un algorithme de synchronisation temporelle à faible complexité pour les systèmes OFDM est proposé dans un contexte de canaux sans fil sélectifs en fréquence. La proposition est basée sur une nouvelle séquence de symboles d'apprentissage basée sur les séquences de Chu modifiées (CAZAC) [10] définies par
+
+$$ a_k^{(r)} = \begin{cases} \exp\left(i \frac{2\pi}{N_s} \left\lfloor \frac{rk^2}{2} \right\rfloor\right), & \text{pour } N_s \text{ pair} \\ \exp\left(i \frac{2r\pi k(k+1)}{N_s}\right), & \text{pour } N_s \text{ impair} \end{cases} \tag{1} $$
+
+$$ a_k^{(r)} = \begin{cases} \exp\left(i \frac{2\pi}{N_s} \left\lfloor \frac{rk^2}{2} \right\rfloor\right), & \text{pour } N_s \text{ pair} \\ \exp\left(i \frac{2r\pi k(k+1)}{N_s}\right), & \text{pour } N_s \text{ impair} \end{cases} \tag{2} $$
+
+où $0 \le k < N_s$, $\gcd(r, N_s) = 1$ et $\lfloor a \rfloor$ dénotent la partie entière de $a$. Ici $r = 1$ est utilisé. La taille de l'alphabet est $N_s$ pour la séquence de Chu modifiée en comparaison avec $2N_s$ pour la séquence de Chu. La nouvelle séquence de symboles d'apprentissage proposée dans cette thèse [11] est
+
+$$ [C C C - C], C = [A B], B = A^*[-n] $$
+
+La partie **A** est construite en prenant la IFFT de la séquence de Chu modifiée [12] sur une taille $N_s = \frac{N}{8}$. Ensuite, **B** est construit à partir de **A** par un renversement temporel et une opération de conjugaison. Le motif de signes $[1 1 1 -1]$ est conçu de façon à assurer une transition raide pour l'algorithme d'estimation grossière (coarse). L'algorithme proposé contient trois étapes.
+
+• Auto-corrélation initiale : l'opération d'auto-corrélation basée sur le délai est effectuée en utilisant un motif répétitif dans le symbole d'apprentissage. La métrique temporelle pour l'auto-corrélation initiale est donnée par
+
+$$ TM_{init}[n] = \left( \frac{L}{L-1} \cdot \frac{|P_{init}[n]|}{R_{init}[n]} \right)^2 \quad (3) $$
+
+ou $P_{init}$ est la fonction d'auto-corrélation, $R_{init}$ est la fonction d'énergie, $TM_{init}$ est la métrique temporelle et $L$ est le nombre de répétitions (ici $L = 4$) dans le symbole d'apprentissage proposé. Le terme $\frac{L}{L-1}$ est utilisé pour normaliser la valeur maximale à 1 au point de démarrage correct. Les expressions pour $P_{init}$ et $R_{init}$ sont
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+$$P_{init}[n] = \sum_{k=0}^{L-2} u[k] \sum_{m=0}^{M-1} r^*[n+kM+m] \cdot r[n+(k+1)M+m] \quad (4a)$$
+
+$$R_{init}[n] = \sum_{k=0}^{L-1} \sum_{m=0}^{M-1} |r[n+kM+m]|^2 \quad (4b)$$
+
+où $u[k] = p[k] \cdot p[k+1]$, $p[k]$ contient le motif de signes $[1 1 1 -1]$, $k = 0, 1, ..., (L-1)$ et $M = N/L$. L'index de temps correspondant à la valeur maximale donne l'estimation temporelle initiale.
+
+$$\hat{\eta}_{init} = \arg \max_{n} TM_{init}[n] \quad (5)$$
+
+La figure 2.a trace $TM_{init}[n]$ pour un *Signal to Noise Ratio* (SNR) de 10 dB dans un canal sans fil sélectif en fréquence. L'algorithme d'estimation fine consiste en la correction d'un petit décalage pour trouver le point de démarrage correct.
+
+FIGURE 2: Tracé des fonctions de métriques temporelles grossière (a) et fine (b)
+
+• Corrélation conjuguée symétrique : la métrique d'estimation temporelle fine est donnée par
+
+$$TM_{fine}[n] = |P_{fine}[n]|^2 \quad (6)$$
+
+avec
+
+$$P_{fine}[n] = \sum_{k=0}^{\frac{N}{2}-1} r[n-k-1] \cdot r[n+k] - \sum_{k=\frac{N}{4}}^{\frac{N}{2}-1} r[n-k-1] \cdot r[n+k] \quad (7)$$
+
+où $TM_{fine}$ est la métrique temporelle fine et $P_{fine}$ est l'opération de corrélation conjuguée symétrique. Le signe négatif pour $k \in [\frac{N}{4}, \frac{N}{2}-1]$ est dû au motif de signes $[1 1 1 -1]$, $n \in [-N_{cyp}, N_{cyp}]$. Cette intervalle pour $n$ a été choisi pour que l'estimation initiale ne produise pas de pics en dehors de la longueur maximale du canal multi-trajets. La métrique de temps produit des pics qui sont proportionnels au carré des gains des trajets du canal. $Q[n]$ est calculé en normalisant toutes les
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+valeurs de $TM_{fine}[n]$ par la valeur maximale de $TM_{fine}[n]$ :
+
+$$Q[n] = \frac{TM_{fine}[n]}{\max(TM_{fine}[n])} \qquad (8)$$
+
+La figure 2.b montre le tracé de $Q[n]$ pour un SNR de 10 dB. La figure montre les pics correspondant aux gains des trajets multiples. L'index temporel de la valeur maximale de $Q[n]$
+
+$$\hat{\eta}_{fine} = \arg\max_n(Q[n]) \qquad (9)$$
+
+est utilisé comme point de démarrage pour la méthode des sommes fenêtrées.
+
+• Sommation basée sur le seuil : les valeurs de $Q[n]$ sont limitées par un seuil de valeur $\beta$.
+
+$$Q[n] = \begin{cases} Q[n], & Q[n] > \beta, \\ 0, & \text{sinon}, \end{cases} \qquad (10)$$
+
+$\beta$ est le seuil qui sépare le signal de la composante de bruit dans $Q[n]$. Ce seuil est déterminé en utilisant la distribution de probabilité de la composante de bruit dans $Q[n]$. Les étapes sont les suivantes :
+
+1. La séquence $Q[n]$ est passée dans l'algorithme de quantification de Lloyd-Max [13] utilisant trois niveaux de quantification.
+
+2. Le niveau de quantification le plus bas est considéré comme du bruit. Il est observé que ce "cluster" de points suit une distribution log-normale. La moyenne ($\mu_n$) et la variance ($\sigma_n^2$) du cluster de bruit sont calculées en premier. $\mu$ et $\sigma$ pour une distribution log-normale sont
+
+$$\mu = \log \left( \frac{\mu_n^2}{\sqrt{\sigma_n^2 + \mu_n^2}} \right) \qquad (11a)$$
+
+$$\sigma = \sqrt{\log \left( \frac{\sigma_n^2}{\mu_n^2} + 1 \right)} \qquad (11b)$$
+
+3. Une vitesse constante de fausse alarme (constant false alarm rate (CFAR)) de "$\alpha$" est utilisée pour le calcul du seuil. L'équation est dérivée de l'intégrale de la fonction de distribution des probabilités (probability distribution function) du bruit dans l'intervalle [$\beta$, $\infty$].
+
+$$\beta = e^{(\sqrt{2\cdot\sigma}\cdot\operatorname{erf}^{-1}(1-2\cdot\alpha)+\mu)} \qquad (12)$$
+
+CFAR est utilisé pour toutes les valeurs de SNR. Une somme fenêtrée est calculée après avoir supprimé les valeurs de bruit en dessous du seuil $\beta$ calculé.
+
+$$E_p(n) = \sum_{k=0}^{S_w-1} Q(\hat{\eta}_{fine} - n + k) \qquad (13)$$
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+To,
+**AMMA** and **APPA**
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+où $S_w$ est la longueur de la fenêtre de sommation et $J_m$ est la fenêtre de recherche pour la composante signal. Puis, le premier chemin d'arrivée est donné par
+
+$$ \hat{\eta}_{first} = \arg \max_{n} E_{p}(n) : n = 0, 1, \dots, J_{m} \quad (14) $$
+
+Finalement,
+
+$$ \hat{\eta}_{final} = \hat{\eta}_{init} - \hat{\eta}_{first} \quad (15) $$
+
+indique l'estimation finale de l'index du début du symbole OFDM.
+
+## Résultats de simulation
+
+La figure 3 montre l'erreur quadratique moyenne (MSE) de l'estimation temporelle dans un canal présentant des interférences entre symboles (ISI) pour différentes méthodes de synchronisation. Les méthodes basées sur la corrélation des délais (Schmidl, Minn, Shi) ont un MSE plus grand comparé aux méthodes utilisant des corrélations symétriques conjuguées (Park, Choi). La méthode proposée est meilleure que Park et est comparable à celle de Choi mais avec une complexité de calcul largement plus faible. Le nombre d'opérations sur des nombres réels en fonction de N est décrit dans la Table 1 pour différents algorithmes. Une réduction d'environ 80% de la complexité de calcul est obtenue pour la méthode proposée (pour $N_{sym} = 1126$, $N_{cyp} = 102$) par rapport à celles de Choi, Park et Zhou, tandis que les performances MSE restent très proches de celles de Choi.
+
+FIGURE 3: MSE de l'estimation temporelle en fonction du SNR dans un canal ISI
+
+## 0.4 Synchronisation temporelle hiérarchique à faible complexité pour les systèmes CO-OFDM
+
+La méthode synchronisation proposée dans la section précédente pour les systèmes sans-fil OFDM possède trois étapes pour atteindre de faible valeur de MSE :
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+TABLE 1: Nombre d'opérations réelles pour le calcul d'un point de la métrique temporelle
+
+<table><thead><tr><th>Algorithme</th><th>Multiplication</th><th>Addition</th><th>Division</th></tr></thead><tbody><tr><td>Schmidl-Cox</td><td>15</td><td>13</td><td>1</td></tr><tr><td>Minn(L = 4)</td><td>31</td><td>29</td><td>1</td></tr><tr><td>Shi</td><td>59</td><td>61</td><td>1</td></tr><tr><td>Park</td><td>(2N + 11)</td><td>(2N + 7)</td><td>1</td></tr><tr><td>Choi</td><td>(2N + 7)</td><td>(2N + 3)</td><td>1</td></tr><tr><td>Zhou</td><td>(2N + 22)</td><td>(2N + 16)</td><td>1</td></tr><tr><td>Algorithme proposé (coarse step) (L = 4)</td><td>31</td><td>29</td><td>1</td></tr><tr><td>Algorithme proposé (fine step)</td><td>(2N + 3)</td><td>(2N - 1)</td><td>1</td></tr></tbody></table>
+
+• opération d'auto-corrélation ($TM_{init}$, Eq. 3),
+
+• opération de corrélation conjuguée symétrique ($TM_{fine}$, Eq. 6),
+
+• opération de sommation fenêtrée ($E_p$, Eq. 13).
+
+Dans le cas de canaux optiques SMF, les valeurs de dispersion ne sont pas trop élevées et restent stables en comparaison avec l'effet multi-trajets des canaux sans fil qui peuvent présenter des retards très importants. Par conséquent, l'étape de sommation fenêtrée peut être éliminée et l'algorithme est réduit à ses deux premières étapes. Cet algorithme modifié est utilisé dans le cadre des canaux optiques SMF. Pour les comparaisons de performance, seuls les algorithmes basés sur l'auto-corrélation (Schmidl-Cox, Minn-Bhargava, Shi-Serpedin) sont reportés. En effet, les algorithmes de cross-corrélation (Choi, Park) sont trop complexes dans un contexte optique et ne génèrent pas de sorties à chaque cycle, comme requis dans ce contexte. Les étapes pour calculer le point de départ du symbole dans un système CO-OFDM $\eta_{final} = \eta_{init} - \eta_{fine}$ sont donc :
+
+• opération d'auto-corrélation : l'équation 3 est utilisée sans modification pour le calcul de $\eta_{init}$;
+
+• opération de corrélation conjuguée symétrique : l'équation 6 est modifiée comme ci-dessous pour réduire la complexité et la normalisation est effectuée en utilisant l'énergie du symbole.
+
+$$ \eta_{fine} = \arg\max_n (TM_{fine}^{lc}[n]) \qquad (16) $$
+
+$$ TM_{fine}^{lc}[n] = \frac{|P_{fine}^{lc}[n]|^2}{R_{fine}^2} \qquad (17) $$
+
+$$ P_{fine}^{lc}[n] = \sum_{k=0}^{\frac{N}{4}-1} r[n-k-1] \cdot r[n+k] \qquad (18) $$
+
+$$ R_{fine}[n] = \sum_{k=0}^{\frac{N}{4}-1} |r[n+k]|^2 \qquad (19) $$
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+## Résultats de simulation
+
+La figure 4 trace le MSE de l'estimation temporelle dans un canal SMF avec un CFO de 4.75 pour les sous porteuses. On peut observer que l'algorithme proposé engendre une légère dégradation pour de faibles OSNR à cause de sa faible complexité dans le calcul de $P_{fine}^{lc}$. L'algorithme proposé donne par contre des améliorations significatives pour des OSNR plus élevés. Rappelons que dans tous les cas la complexité de calcul est largement réduite par rapport à l'état de l'art ce qui représente un grand avantage dans un contexte de communications optiques à très haut débit.
+
+FIGURE 4: MSE de l'estimation temporelle en fonction du OSNR pour des canaux SMS et un CFO = 4.75
+
+## 0.5 Architecture parallèle pour l'auto-corrélation
+
+L'objectif de la parallélisation de l'algorithme d'auto-corrélation est d'obtenir une architecture "scalable" pour gérer de façon efficace des échantillons d'entrée multiples à chaque cycle d'horloge et atteindre le débit élevé désiré. Une architecture parallèle et évolutives est proposée dans ce travail et comparée aux travaux existants. L'architecture proposée possède un parallélisme au niveau bloc et utilise à la fois la forme itérative et celle non-itérative du calcul de l'auto-corrélation pour atteindre un niveau de parallélisme suffisant. La forme non-itérative est utilisée pour initialiser les calculs tandis que la forme itérative calcule les point restant. Le choix de la taille des blocs permet de déterminer le partage des ressources. Soit l'algorithme d'auto-corrélation de Minn-Bhargava appliqué au symbole d'apprentissage $[AA^T - A]$ :
+
+$$ 
+\begin{align}
+P_{mb}[n] &= \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+kM_{mb}] \nonumber \\
+&\quad \cdot r[n+m+(k+1)M_{mb}] \tag{20} \\
+P_{mb}[n] &= P_{mb}[n-1] + r^*[n] \cdot r[n+M_{mb}] - r^*[n+3M_{mb}] \cdot r[n+4M_{mb}] \nonumber \\
+&\quad + 2r^*[n+2M_{mb}] \cdot r[n+3M_{mb}] \tag{21}
+\end{align}
+$$
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+où Eq. 20 est la forme non itérative et Eq. 21 est la forme itérative, $P_{mb}$ est la fonction d'auto-corrélation, $M_{mb}$ est la taille de la partie répétitive (A), et $M_{mb} = \frac{N}{4}$. Si les équations sont réécrites pour un niveau de parallélisme $R = 4$ avec une taille de bloc $M_{mb}$, on obtient
+
+$$ 
+\begin{align*}
+P_{mb}[n] &= \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+kM_{mb}] \\
+P_{mb}[n+M_{mb}] &= \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+(k+1)M_{mb}] \\
+P_{mb}[n+2M_{mb}] &= \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+(k+2)M_{mb}] \\
+P_{mb}[n+3M_{mb}] &= \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+(k+3)M_{mb}]
+\end{align*}
+\tag{22}
+$$
+
+$$ 
+\begin{align}
+P_{mb}[n+1] &= P_{mb}[n] - r^*[n] \cdot r[n + M_{mb}] - r^*[n + 3M_{mb}] \cdot r[n + 4M_{mb}] \nonumber \\
+&\quad + 2r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}] \nonumber \\
+P_{mb}[n+1] &= P_{mb}[n] - r^*[n] \cdot r[n + M_{mb}] - r^*[n + 3M_{mb}] \cdot r[n + 4M_{mb}] \nonumber \\
+&\quad + 2r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}] \nonumber \\
+P_{mb}[n+1] &= P_{mb}[n] - r^*[n] \cdot r[n + M_{mb}] - r^*[n + 3M_{mb}] \cdot r[n + 4M_{mb}] \nonumber \\
+&\quad + 2r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}] \nonumber \\
+P_{mb}[n+1] &= P_{mb}[n] - r^*[n] \cdot r[n + M_{mb}] - r^*[n + 3M_{mb}] \cdot r[n + 4M_{mb}] \nonumber \\
+&\quad + 2r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}] \tag{23}
+\end{align}
+$$
+
+### 0.5.1 Architecture parallèle partielle (PSBP)
+
+Dans cette première version d'une architecture parallèle, les ressources de calcul sont partagées entre les calculs itératifs et non itératifs. L'architecture possède donc deux modes d'opération, un mode non-itératif pour l'initialisation du calcul et un second mode itératif pour le reste des calculs. Soit le calcul de $R M_{mb}$ sorties de l'auto-corrélation avec une architecture PSBP incluant $R$ blocs en parallèle, chaque bloc calcule $\frac{R M_{mb}}{R}$ sorties. L'ordre des calculs est le suivant :
+
+• *R* points initiaux sont calculés en mode non itératif ce qui nécessite $M_{mb}$ cycles pour l'algorithme MBA,
+
+• les ($M_{mb} - 1$) points restant sont calculés en mode itératif.
+
+Après le calcul de $R M_{mb}$ sorties, le même processus est répété pour les $R M_{mb}$ sorties suivantes. L'architecture prend ($2M_{mb} - 1$) cycles pour le calcul de $M_{mb}$ points d'auto-corrélation par bloc. L'architecture est appelée "parallèle partielle" car elle ne produit pas de sortie à chaque cycle et elle possède un délai équivalent à la partie initialisation. Cependant le nombre de ressources est plus faible que dans la proposition suivante.
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+FIGURE 5: Architecture PSPB proposée pour le calcul de $P_{mb}$ avec MBA
+
+La table 2 présente la complexité architecturale en fonction de R pour l’algorithme de Minn-Bhargava. Elle est comparée avec la proposition de Kaneda pour l’algorithme de Schmidl-Cox. Notre architecture requiert 12 multiplieurs de plus que celle de Kaneda du fait de la plus grande complexité de l’algorithme considéré. Mais la surface totale reste plus faible en particulier au nombre réduit d’additionneurs.
+
+TABLE 2: Complexité architecturale en fonction de R
+
+Architecture proposée pour Minn-Bhargava
+
+<table><thead><tr><th>Algorithm</th><th>Real Multipliers</th><th>Real Adders</th></tr></thead><tbody><tr><td>P<sub>mb</sub></td><td>4(R + 3)</td><td>2(3R + 3)</td></tr></tbody></table>
+
+Architecture de Kaneda pour Schmidl-Cox
+
+<table><thead><tr><th>Algorithm</th><th>Real Multipliers</th><th>Real Adders</th></tr></thead><tbody><tr><td>P<sub>sc</sub></td><td>4R</td><td>2R(M<sub>sc</sub>/2 + 1)</td></tr></tbody></table>
+
+### 0.5.2 Architecture parallèle complète (FSBP)
+
+L'architecture proposée dans la section 0.5.1 utilise $M_{mb}$ cycles pour initialiser les calculs ce qui induit un délai à chaque nouveau calcul et implique donc des ressources mémoire supplémentaires pour le stockage des symboles OFDM ainsi qu'un non respect des contraintes temps réel. L'architecture est donc modifiée pour éviter ces défauts et obtenir une architecture complètement parallèle capable de produire $M_{mb}$ sortie d'auto-corrélation en $M_{mb}$ cycles. La modification proposée consiste en l'ajout d'un bloc pour le calcul des points initiaux en parallèle de façon àce que l'auto-corrélation produise un calcul à cycle. Une
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+architecture parallèle à $R = 4$ blocs pour MBA est présentée à la figure 6. La table 3 donne la complexité architecturale pour le calcul de $P_{mb}$. Comparée à nouveau à celle de Kaneda [14], des réductions en termes de surface de 17 à 72% sont obtenus en fonction de la taille du symbole.
+
+FIGURE 6: Architecture parallèle FSPB proposée pour le calcul de $P_{mb}$ avec MBA et $R = 4$
+
+TABLE 3: Complexité architecturale en fonction de R
+
+<table><thead><tr><th>Algorithm</th><th>Real Multipliers</th><th>Real Adders</th></tr></thead><tbody><tr><td>P<sub>mb</sub> (Initial Point)</td><td>4R</td><td>4R</td></tr><tr><td>P<sub>mb</sub> (Iterative Point)</td><td>4(R + 3)</td><td>2(3R + 3)</td></tr><tr><td>P<sub>mb</sub> (Total)</td><td>8R + 12</td><td>10R + 6</td></tr></tbody></table>
+
+## 0.6 Architecture parallèle pour les systèmes CO-OFDM
+
+Dans cette section une architecture parallèle d'émetteur-récepteur pour un transmetteur CO-OFDM est proposée.
+
+### 0.6.1 Emetteur
+
+A l'émission, la IFFT est le bloc principal en termes de complexité. Le choix du radix utilisé pour le FFT est donc crucial et peut donc influencer la complexité de l'architecture. Pour $N = 256$ la complexité en millions d'opérations par seconde (MOPS) est calculée pour des FFT radix-2/4/2² et split-radix et reportée dans la table 4 pour supporter un débit de 7.3 Gbit/s. Ensuite, le nombre total d'opérations pour supporter un débit total $D_{b,total} \ge 100$ Gb/s est reporté dans la dernière colonne de la table 4. Ces résultats montrent qu'un gain de 800 GOPS peut être obtenu pour les algorithmes radix-4/2² par rapport au radix-2, tandis que 200 GOPS supplémentaires peuvent être atteints par split-radix. Nous avons retenu le radix-2² car sa complexité architecturale est plus faible.
+
+### 0.6.2 Récepteur
+
+Les algorithmes utilisés pour la synchronisation temps, l'estimation du CFO, la FFT, l'estimation du canal, l'égalisation, l'estimation de l'erreur en phase et la compensation
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+TABLE 4: Complexité algorithmique pour le calcul de N sorties de la IFFT avec N=256
+
+<table><thead><tr><th>Radix Used</th><th>Real Multipli-cations</th><th>Real Additions</th><th>Total Operations</th><th>GOPS (D<sub>b</sub>) for 7.3 Gb/s</th><th>TOPS (D<sub>b,total</sub>) for 117 Gb/s</th></tr></thead><tbody><tr><td>Radix-2</td><td>4096</td><td>6144</td><td>10240</td><td>294.4</td><td>4.7</td></tr><tr><td>Radix-4</td><td>3072</td><td>5632</td><td>8704</td><td>248.2</td><td>3.9</td></tr><tr><td>Radix-2<sup>2</sup></td><td>3072</td><td>5632</td><td>8704</td><td>248.2</td><td>3.9</td></tr><tr><td>Split-Radix</td><td>2731</td><td>5462</td><td>8193</td><td>233.6</td><td>3.7</td></tr></tbody></table>
+
+sont décrits dans cette section. La complexité algorithmique pour le calcul de *N* sorties et pour un système à 117 Gb/s y est aussi présentée. Des optimisations spécifiques sur le format des données sont réalisées pour réduire la complexité.
+
+• Synchronisation temporelle grossière - l'algorithmme utilisé est celui proposé dans cette thèse. Sa complexité pour le calcul de *N* sorties est donnée dans la table 5, pour une sous bande et pour un débit des sorties de 117 Gb/s. Le CFO est estimé par auto-corrélation au point de départ du symbole.
+
+$$ 
+\begin{aligned}
+P_{mb}[n+1] ={}& P_{mb}[n] - r^*[n] \cdot r[n + M_{mb}] && (24) \\
+             & - r^*[n + 3M_{mb} \cdot r[n + 4M_{mb}]] \\
+             & + 2 \cdot r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}]
+\end{aligned}
+$$
+
+où $P_{mb}$ est l'auto-corrélation, $M_{mb}$ est la longueur du symbole d'apprentissage utilisé [$A A A - A$].
+
+TABLE 5: Complexité algorithmique (auto-corrélation) de notre proposition
+
+<table><thead><tr><th>Algo. Used</th><th>Real Multipli-cations</th><th>Real Additions</th><th>Total Operations</th><th>GOPS (D<sub>b</sub>) per sub-band</th><th>TOPS (D<sub>b,total</sub>) for 117 Gb/s</th></tr></thead><tbody><tr><td>Minn-Bhargava (L = 4)</td><td>3072</td><td>3072</td><td>6144</td><td>175.2</td><td>2.8</td></tr></tbody></table>
+
+• FFT - Une FFT Radix-2² est choisie et implémentée.
+
+• Estimation CFO entière - Une cross-corrélation avec une séquence connue de symboles est réalisée pour l'estimation CFO. Soit une séquence connue de longueur $N_{ifo} = N/4$ notée $z[n]$. Comme, l'estimation CFO est faite dans le domaine de fréquences la séquence connue peut être construite à partir de symboles QPSK de valeur ($\pm 1 \pm 1j$). L'opération de cross-corrélation devient
+
+$$ 
+\begin{gathered}
+M_{ifo}[n] = |P_{ifo}[n]|^2 && (25) \\
+P_{ifo}[n] = \sum_{m=0}^{N_{ifo}-1} r[n+m] \cdot z^*[n+m] && (26)
+\end{gathered}
+$$
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+où *n* est l'index de recherche, *n* ∈ [-W<sub>s</sub>, ..., -2, 0, 2, ..., W<sub>s</sub>], où W<sub>s</sub> est l'index maximum. Ici W<sub>s</sub> = 20 est choisi comme valeur maximale. La valeur de N<sub>ifo</sub> est fixée à 32. La complexité algorithmique est reportée dans la table 6. Grâce à l'utilisation d'une constellation QPSK, (±1 ± j), les multiplications complexes peuvent être complètement éliminées et la complexité ainsi réduite. Un gain de 39.8 MOPS est ainsi obtenu.
+
+TABLE 6: Complexité algorithmique de l'estimation CFO entière
+
+<table><thead><tr><th>Algo.<br>Used</th><th>Real<br>Multipli-<br>cations</th><th>Real<br>Add-<br>itions</th><th>Total<br>Oper-<br>ations</th><th>MOPS (<i>D</i><sub>b</sub>)<br>per<br>sub-band</th><th>MOPS<br>(<i>D</i><sub>b,total</sub>) for<br>100Gb/s</th></tr></thead><tbody><tr><td>IFO<br>Estimation<br>Non-Optimized</td><td>15360</td><td>7680</td><td>23040</td><td>3.68</td><td>59</td></tr><tr><td>IFO<br>Estimation<br>Optimized</td><td>82</td><td>2624</td><td>2706</td><td>1.3</td><td>20.8</td></tr></tbody></table>
+
+• Estimation du canal et égalisation - Les algorithmes des moindres-carrés (LS) et LS normalisés sont utilisés pour l'estimation du canal. La complexité algorithmique est reportée dans la table 7. A nouveau, la complexité de la méthode LS peut être réduite en utilisant le symbole (±1 ± j). Cette optimisation reste valable pour les modes à simple et double polarisation. Des gains de 29.2 GOPS pour LS et de 21.9 GOPS pour NLMS sont atteints.
+
+• Estimation CPE et compensation - Une estimation CPE basée sur des symboles pilotes [15] est réalisée pour l'estimation du bruit de phase du LASER. La complexité algorithmique est reportée dans la table 8.
+
+Le bilan final montre un gain en complexité de plus 800 GOPS par rapport à une implémentation non optimisée.
+
+## 0.7 Experimentations
+
+Des expérimentations temps réel et off-line ont finalement été réalisées dans cette thèse pour valider les paramètres du système OFDM dans un contexte de communications optiques. Tout d'abord, des scénarios en temps différé (off-line) ont été conduits à l'aide d'un générateur de signaux AWG comme émetteur et d'un oscilloscope numérique rapide (DSO) comme récepteur. La figure 7 montre la configuration hétérodyne dans laquelle des sources LASER distinctes sont utilisées à l'émission et à la réception.
+
+• Emetteur Electro-Optique - La fréquence porteuse de l'émetteur et générée par un LASER à cavité externe (ECL) àune longueur d'ondes de 1540 nm. Le signal optique est amplifiée avec une fibre PMF (*polarization maintaining fiber*) qui amplifie le
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new file mode 100644
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+TABLE 7: Complexité algorithmique de l'estimation de canal
+
+<table><thead><tr><th>Algo. Used</th><th>Real Multipli-cations</th><th>Real Add-itions</th><th>Total Oper-ations</th><th>GOPS (D<sub>b</sub>) per sub-band</th><th>TOPS (D<sub>b,total</sub>) for 117 Gb/s</th></tr></thead><tbody><tr><td>LS Single Pol. Non-Optimized</td><td>2048</td><td>768</td><td>2816</td><td>80.3</td><td>1.28</td></tr><tr><td>LS Single Pol. Optimized</td><td>1024</td><td>768</td><td>1792</td><td>51.1</td><td>0.81</td></tr><tr><td>LS Dual Pol. Non-Optimized</td><td>8192</td><td>3072</td><td>11264</td><td>321.2</td><td>5.1</td></tr><tr><td>LS Dual Pol. Optimized</td><td>4096</td><td>3072</td><td>7168</td><td>204.4</td><td>3.2</td></tr><tr><td>NLMS Single Pol. Non-Optimized</td><td>2816</td><td>2048</td><td>4864</td><td>138.7</td><td>2.2</td></tr><tr><td>NLMS Single Pol. Optimized</td><td>2304</td><td>1792</td><td>4096</td><td>116.8</td><td>1.8</td></tr><tr><td>NLMS Dual Pol. Non-Optimized</td><td>5632</td><td>4096</td><td>9728</td><td>277.4</td><td>4.4</td></tr><tr><td>NLMS Dual Pol. Optimized</td><td>4608</td><td>3584</td><td>8192</td><td>233.6</td><td>3.7</td></tr></tbody></table>
+
+TABLE 8: Complexité algorithmique de l'estimation et de la compensation CPE
+
+<table><thead><tr><th>Algo.<br>Used</th><th>Real<br>Multipli-<br>cations</th><th>Real<br>Add-<br>itions</th><th>Total<br>Oper-<br>ations</th><th>GOPS (D<sub>b</sub>)<br>per<br>sub-band</th><th>GOPS<br>(D<sub>b,total</sub>) for<br>100Gb/s</th></tr></thead><tbody><tr><th scope="row">CPE<br>Estimation<br>Optimized</th><td>1024</td><td>512</td><td>1536</td><td>43.8</td><td>700.8</td></tr></tbody></table>
+
+signal de 15 dBm. Un coupleur 3 dB divise la puissance pour attaquer le contrôleur de polarisation de façon à maximiser la puissance optique sur le modulateur optique MZM (Mach-Zender Modulator). La sortie du contrôleur avec une puissance de 11 dBm sert de porteuse au bloc MZM. Le signal, généré à partir du logiciel Matlab, est envoyé à l'AWG et au driver RF.
+
+* Récepteur Opto-Electronique - Un atténuateur optique est utilisé pour faire varier le rapport signal à bruit (optical signal-to-noise ratio (OSNR)). Un filtre passe-bande optique (BPF) sélectionne la bande passante autour de la fréquence porteuse dans l'intervalle des 3 nm. L'atténuateur permet de régler la puissance du signal optique à l'entrée du détecteur cohérent et la valeur maximale du signal est fixée à -5 dBm. Le
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+signal après détection cohérente est envoyé à l'oscilloscope pour échantillonnage et stockage. Le signal ainsi échantillonné peut être traité en temps différé sur le logiciel Matlab.
+
+La figure 8 montre les performances en termes de taux d'erreurs binaires (BER) de cette expérimentation ce qui permet de valider les différents algorithmes du transmetteur.
+
+FIGURE 7: Configuration hétérodyne à détection cohérente avec une fibre SSMFde 50 km
+
+FIGURE 8: BER vs SNR pour un système CO-OFDM simple bande hétérodyne
+
+Dans un second temps, des expérimentations "temps-réel" sont réalisées et reportées. Elles utilisent une plateforme temps-réel FPGA développée dans le cadre du projet FUI 100GFLEX. Cette plateforme intègre des FPGA Xilinx et Altera associés à des DAC et ADC rapides. Les différents blocs implémentés sont décrits ci dessous. Les figures 9 et 10 montrent les plateformes d'émission et de réception respectivement.
+
+Une expérimentation *back-to-back* électrique sans CFO permit de valider les algorithmes de synchronisation de trame OFDM. L'architecture proposée pour l'algorithme
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+FIGURE 9: Plateforme FPGA temps réel d'émission
+
+FIGURE 10: Plateforme FPGA temps réel de réception
+
+de Minn-Bhargava est utilisée pour l'estimation du point de départ de la trame. L'architecture est synthétisée à partir du langage C en utilisant l'outil de synthèse de haut niveau HLS CatapultC. Les résultats reportés dans la thèse valident les algorithmes et démontrent l'efficacité de l'architecture proposée en termes de performance BER et de surface.
+
+## 0.8 Conclusion
+
+Les systèmes de communications optiques à très haut débit sont construits à partir des techniques de pointe pour la détection, la modulation et la compensation de dispersion tels que, la détection cohérente, les modulations multi-porteuses orthogonales (OFDM) et la compensation électronique des dispersions (EDC). La réapparition de la détection cohérente dans les systèmes de communication optique a été rendue notamment possible par les progrès dans les circuits numériques dans les technologies avancées. La détection cohérente possède une meilleure sensibilité pour la détection du signal par rapport aux méthodes de détection directe. Elle permet d'utiliser des transmissions à double polarisation et conserve les informations de phase du signal optique et les transfert dans le
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+domaine électrique. L'utilisation de la modulation OFDM fournit une flexibilité significative et l'utilisation efficace de la bande passante allouée. En raison de la disponibilité des informations de phase dans le domaine numérique, les processeurs DSP de faible coût peuvent être utilisés pour la compensation des dispersions dans le domaine numérique qui rend la solution flexible et reconfigurable. Mais, l'introduction du système CO-OFDM (Coherent-Optical OFDM) à la place de système de IM-DD (Intensity Modulation-Direct Detection) augmente significativement le coût du système avec un plus grand nombre de composants optiques et une quantité plus élevée de ressources électroniques requises pour la réception du signal. À l'heure actuelle, cela rend cette solution uniquement justifiable pour des transmissions à longue portée, même si le nombre de ressources par rapport à un système mono-porteuse à détection cohérente et modulation à quatre états (DP-CO-QPSK). Le choix de l'algorithme et l'optimisation de la précision des calculs en virgule fixe de l'architecture peuvent réduire de façon significative les ressources nécessaires pour la réalisation de systèmes CO-OFDM.
+
+Dans cette thèse, des algorithmes à faible complexité et des architectures parallèles et efficaces sont explorés pour les systèmes CO-OFDM. Tout d'abord, des algorithmes de faible complexité pour la synchronisation et l'estimation du décalage en fréquence en présence d'un canal dispersif sont étudiés. Un nouvel algorithme de synchronisation temporelle à faible complexité qui peut résister à grande quantité de retard dispersif est proposé et comparé par rapport aux propositions antérieures. Ensuite, le problème de la réalisation d'une architecture parallèle à faible coût est étudié et une architecture parallèle générique et évolutives qui peut être utilisée pour réaliser tout type d'algorithme d'auto-corrélation est proposé. Cette architecture est ensuite étendue pour gérer plusieurs échantillons issus du convertisseur analogique/numérique (ADC) en parallèle et fournir une sortie qui suive la fréquence des ADC. L'évolutivité de l'architecture pour un nombre plus élevé de sorties en parallèle et les différents types d'algorithmes d'auto-corrélation sont explorés.
+
+Une approche d'adéquation algorithme-architecture est ensuite appliquée à l'ensemble de la chaîne de l'émetteur-récepteur CO-OFDM. Du côté de l'émetteur, un algorithme IFFT à radix-2² est choisi pour et une architecture parallèle Multipath Delay Commutator (MDC) Feed-forward (FF) est choisie car elle consomme moins de ressources par rapport aux architectures MDC-FF en radix-2/4. Au niveau du récepteur, un algorithme efficace pour l'estimation du Integer CFO est adopté et implémenté de façon optimisée sans l'utilisation de multiplicateurs complexes. Une réduction de la complexité matérielle est obtenue grâce à la conception d'architectures efficaces pour la synchronisation temporelle, la FFT et l'estimation du CFO. Une exploration du compromis entre la précision des calculs en virgule fixe et la complexité du matériel est réalisée pour la chaîne complète de l'émetteur-récepteur, de façon à trouver des points de fonctionnement qui n'affectent pas le taux d'erreur binaire (TEB) de manière significative. Les algorithmes proposés sont validés à l'aide d'une part d'expériences off-line en utilisant un générateur AWG (arbitrary waveform generator) à l'émetteur et un oscilloscope numérique à mémoire (DSO) en sortie de la détection cohérente au récepteur, et d'autre part un émetteur-récepteur temps-réel
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+basé sur des plateformes FPGA et des convertisseurs numériques. Le TEB est utilisé pour montrer la validité du système intégré et en donner les performances.
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+# Chapter 1
+
+# Introduction
+
+In today's world, the demand of Internet bandwidth is increasing at an exponential rate, compared to increase in demand in the previous decade. One of the reasons being the arrival of always connected devices like smartphones, tablets, and the emergence of video based applications like YouTube, the Web moving towards richer interactive applications. With more and more people connected to the Internet, it creates huge burden on all the nodes of the communication network. Figure 1.1 shows the predicted increase in monthly traffic for the next five years by application type. It can be seen that video-based applications will continue to increase in number and put enormous pressure on the underlying network.
+
+FIGURE 1.1: Cisco Visual Networking Index (VNI) Prediction of growth of internet by Application Type (Updated May 2013). The ordinate units is in Eta Bytes (EB). Total traffic is 2017 is predicted to be three times larger than 2012 [1].
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+The Internet is built upon many communication standards, which use different types of physical medium to communicate data bits around the world. One of the most important physical medium which forms the backbone of the network is Optical Communication system. Optical Communication system carries data presently over very long distances (Submarine networks, Long-haul networks), medium distances (Metro networks, Access Networks). With the introduction of Fiber-to-the-Home (FTTH), optical fiber communication system is also serving end users directly. Submarine networks are undersea networks which have links supporting distances of more than 2000 km. The terrestrial optical communication network can be divided into three major types. Figure 1.2 shows these three types of network which are classified as a function of distance. Core Network (CN)
+
+FIGURE 1.2: Typical Optical Network Architecture, CN - Core Node, EN - Edge Node, AN - Access Node
+
+is a long-haul interconnection network that covers hundreds/thousands of kilometers connecting large cities, countries and even continents. It uses mesh topology. Optical Edge Network (EN) connects smaller geographical areas, covering distances of tens of kilometers, which is commonly known as Metropolitan Area Network (MAN). The Access Network (AN) is the peripheral part of the optical network, which is commonly known as Local Area Network (LAN). It uses star topology. Presently, the optical network uses 10/40 Gb/s single carrier modulation scheme in a single band for the data transmission.
+
+Optical communication system uses optical fiber as the primary medium for transmission. Optical fiber has low attenuation coefficient ($\alpha_{dB} \le 0.35$ dB/km) in the wavelength range of 1300 nm to 1700 nm. It offers huge amount of bandwidth for transmission. Coupled with Erbium-Doped Fiber Amplifier (EDFA) technology, which amplifies optical signals in the wavelength range of 1530 nm - 1600 nm, optical communication system can reach thousands of kilometers. Total data rate is increased in optical communication systems by the use of Wavelength Division Multiplexing (WDM) techniques and Dense Wavelength Division Multiplexing (DWDM). The bandwidth (1530 nm - 1565 nm) used
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+Presently Ethernet is used at a line rate of 10 Gb/s. Due to large presence of Ethernet, future increases in line rate will want to use Ethernet standard and change only the technology to support higher line rates. The next upgrade step is 100 Gb/s Ethernet. The jump from 10 Gb/s Ethernet to 100 Gb/s is necessary because router-to-router trunk connectivity has already reached 100 Gb/s [20] and also achieving line rate of 100 Gb/s compared to 10 lines of 10 Gb/s results in cost reduction per Gb/s. This makes achieving 100 Gigabit Ethernet (100GbE) a very important milestone to support the present day demands. Along with the present goal of 100 GbE and towards a future goal of 1 Tb/s Ethernet (1TbE), the solutions adopted should have these desirable properties, which can make the solution future proof.
+
+* They should be compatible with the present optical infrastructure which comprises of single mode fiber having varying range of CD, Dispersion Compensation Fiber (DCF) and other types of fiber.
+
+* They should be scalable to higher speeds easily and can support reconfigurable networks which manages bandwidth at a higher software level.
+
+## 1.1 Context of the Work
+
+With traffic on the Internet growing exponentially and management of quality-of-service (QoS) requiring flexibility at all levels of hierarchy of the optical communication system. The presence of WDM and Reconfigurable Optical Add Drop Multiplexer (ROADM) gives flexibility to cope up with dynamic changes in bandwidth requirement in different parts of network. But, with required bandwidth moving towards 100 Gb/s between routers, it has become necessary for the network to be dynamically reconfigurable and software controlled. Also, the physical layer has to become transparent to different types of network it is traversing through. As the speed is increased to 100 Gb/s, the signal loses its immunity to CD, PMD and ROADM filtering effects. With the presence of different types of optical fibers having large range of CD and PMD values, it is very difficult to maintain CD and PMD compensation using lossy, bulky optical compensators. With these set of flexibility and inherent immunity to dispersion requirements of optical networks, it is important to choose a solution, which offers these features without significant costs.
+
+CO-OFDM has inherent advantages of being immune to dispersion mechanisms and also is extremely flexible, e.g. using different data mapping schemes on sub-carriers (bit-loading), pilot insertion, etc. Also, with the use of Multiband-OFDM (MB-OFDM), the pressure on DAC/ADC is reduced and it allows the present day solution of multiple sub-banded OFDM systems to reach the total data rate of 100 Gb/s. But, CO-OFDM has some disadvantages, which put the brakes on scaling the system efficiently. CO-OFDM system is sensitive to non-linearities in the optical communication chain and also it has a high Peak to Average Power Ratio (PAPR). CO-OFDM system is sensitive to frequency and phase offsets of the LASER in the system and also to timing offset. Also, all the
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+estimation and compensation of non-idealities are done in digital domain, which makes the algorithms complex and has to be adapted to optical system taking from Wireless OFDM domain. These algorithms now have to operate at much higher speeds compared to Wireless OFDM implementation and have to be easily scalable as well.
+
+With FPGA maximum clock frequencies much lower than DAC/ADC frequencies, it forces every DSP block to be parallelized or simply replicated to match the input data rate. Simply replicating the blocks results in huge amount of area and is not feasible in the long term. In this thesis, the goal is to have a fully scalable parallel CO-OFDM system that can support a very-high data rate. At the transmitter, Inverse Fast Fourier Transform (IFFT) is the major component, which needs to be parallelized and it needs to be parallelized efficiently. The choice of radix and number of parallel outputs is explored to get the best area efficient design. CO-OFDM transmitter is a feed-forward system, while CO-OFDM receiver is a system with a feedback loop. Also, it consists of time, frequency, phase offset estimation and compensation blocks which needs to be efficiently parallelized along with Fast Fourier Transform (FFT) which forms the major block of the receiver. Fixed-point analysis of the complete OFDM chain is done which helps in reduction of the area. All the analysis is done for a single polarization CO-OFDM system. It can be extended to dual polarization CO-OFDM system by the inclusion of Multiple Input Multiple Output (MIMO) block which can separate the two polarization components and feed into corresponding chains.
+
+## 1.2 Contributions
+
+The contributions of this work are given as follows.
+
+1. A new low-complexity hierarchical coarse time synchronization algorithm is proposed. For an OFDM signal in the presence of a multi-path channel, initial time synchronization is estimated by using hierarchical approach of auto-correlation and cross-correlation. It also gives fractional Carrier Frequency Offset (CFO) estimate.
+
+2. A scalable and parallel architecture for initial time synchronization algorithm is proposed. The proposed architecture is required to perform real-time synchronization when the ADC sampling frequency is much higher than FPGA operating frequency.
+
+3. A complete parallel architecture for all the blocks of the CO-OFDM transceiver is proposed. The scalability towards 100 Gb/s is detailed. The scalability of individual algorithms is explored in detail. A complete fixed-point analysis of the proposed parallel CO-OFDM system is done. Area reduction due to the analysis is reported.
+
+4. A High-Level Synthesis (HLS) approach to designing of the CO-OFDM blocks is taken, which reduces design time and helps in Design Space Exploration of the DSP blocks.
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+are outlined. Finally perspectives on future work are given.
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+# Chapter 2
+
+## CO-OFDM Transceiver System
+
+### 2.1 Introduction
+
+An introduction to the different blocks of the CO-OFDM transceiver is presented in this Chapter. A CO-OFDM system combines coherent detection and orthogonal multi-carrier modulation to reach higher data rates greater than which is possible by IM-DD systems. CO-OFDM system combines the use of coherent detection, OFDM multi-carrier modulation and electronic dispersion compensation to extract more data rate out of optical fiber channel. Since, the context is optical, the characteristics of single-mode optical fiber are detailed in Section 2.2. Major linear and non-linear phenomena in the optical channel which impair high-speed transmission are explained. Dispersion values of different types of single-mode optical fiber used in core, metro, and access networks are given. Section 2.3 gives the major differences between Wireless and CO-OFDM systems, which helps in understanding the unique challenges posed by the optical fiber and analogue/optical front-ends of the system.
+
+Section 2.4 explains a complete end-to-end CO-OFDM system, giving details separately about digital, analogue/RF, and digital blocks in the subsections that follow. CO-OFDM system is an expensive system compared to IM-DD systems, in terms of optical/analogue/digital components required for its realization. Section 2.5 calculates the resource increase for CO-OFDM system in optical/analogue/digital domains, with detailed analysis done on DSP algorithms used and their complexities. A survey of offline and real-time CO-OFDM experiments is done and then the algorithmic/architectural complexity of those systems is calculated. Section 2.6 lists the observations done by this survey and Section 2.7 concludes the chapter.
+
+### 2.2 Single-Mode Optical Fiber (SMF)
+
+Single-mode optical fiber is an optical fiber which is designed to carry a single ray of light (mode). The mode defines how the light wave is distributed in space. A typical SMF has core diameter between 8 and 10.5 µm and cladding diameter of 125 µm. SMF allows
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+a single mode to propagate and is better at retaining the fidelity of light pulse over longer distances. It has lower attenuation and much higher bandwidth than multi-mode optical fibers (MMF). When light pulse travels in SMF, it undergoes pulse width broadening and attenuation along the fiber. Present day attenuation values of SMF fibers are 0.2 dB/km, which requires optical amplifiers (EDFA) only at distances of 50 km apart from each other. EDFA is the most deployed optical amplifier because its amplification window coincides with the band of lowest attenuation (C-band and L-band) in SMF. Different transmission windows used in SMF are listed in Table 2.1. Phenomena which contribute to degradation of signal as it travels through SMF can be grouped into linear and non-linear phenomena. Description about these impairments are given in the following subsections.
+
+TABLE 2.1: DWDM Band Wavelength Range
+
+<table><thead><tr><td>Band Name</td><td>Wavelengths (in nm)</td></tr></thead><tbody><tr><td>O-Band</td><td>1260 - 1360</td></tr><tr><td>E-Band</td><td>1360 - 1460</td></tr><tr><td>S-Band</td><td>1460 - 1530</td></tr><tr><td>C-Band</td><td>1530 - 1565</td></tr><tr><td>L-Band</td><td>1565 - 1625</td></tr><tr><td>U-Band</td><td>1625 - 1675</td></tr></tbody></table>
+
+### 2.2.1 Linear Impairments
+
+Major linear impairments for signal traversing through SMF are fiber attenuation, chromatic dispersion (CD) and polarization mode dispersion (PMD). Each of the phenomenon is explained below.
+
+* Fiber Attenuation: Signal travelling the optical fiber experiences constant attenuation ($\alpha_{dB}$) as a function of length ($L_F$), given by
+
+$$ \alpha_{dB} = \frac{10}{L_F} \log_{10} \frac{P_0}{P} \qquad (2.1) $$
+
+where $P_0$ is the injected power, $P$ is the received power, and $L_F$ is the length of optical fiber. The attenuation can be classified into intrinsic and extrinsic losses. The intrinsic loss mechanisms are:
+
+1. Rayleigh scattering - caused by density fluctuations within a fiber.
+
+2. OH⁻ Absorption Loss - OH⁻ is the major impurity responsible for this. It causes attenuation peaks at 1380 nm, 1250 nm, and 950 nm. The peak in the transmission window at 1380 nm is shown in Figure 2.1, which increases the losses to nearly 0.4 dB/km. This peak has been removed by improved manufacturing techniques.
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+3. Silica Absorption Loss - Pure silica causes absorption loss in two regions above 2000 nm.
+
+The extrinsic loss mechanisms are due to bending loss and connection between two fiber pieces. Figure 2.1 [reproduced from [2]] shows the variation of Fiber attenuation as a function of wavelength. It shows a region of low attenuation in the C-Band and L-Band.
+
+FIGURE 2.1: Fiber loss coefficient vs. different wavelengths for a typical low-loss optical fiber (SSMF) and fiber without the water absorption peak (Allwave). [Reproduced from Essiambre et al.[2]]
+
+* Chromatic Dispersion (CD): Different frequency components of the optical pulse travel with different velocities inside the optical fiber. This leads to pulse broadening and causes interference among neighbouring symbols leading to ISI. It is also called intramodal dispersion. The chromatic dispersion can be expressed as a sum of two components
+
+$$ 
+\begin{align} 
+CD &= \frac{d \left( -\frac{\lambda^2}{2\pi c} \frac{d\beta}{d\lambda} \right)}{d\lambda} \nonumber \\
+&= -\frac{1}{2\pi c} \left( 2\lambda \frac{d\beta}{d\lambda} + \lambda^2 \frac{d^2\beta}{d\lambda^2} \right) \nonumber \\
+&= D_M + D_W \tag{2.2}
+\end{align} 
+$$
+
+where $D_M$ - material dispersion, $D_W$ - waveguide dispersion, $\lambda$ - wavelength of optical signal, $\beta$ - propagation constant, $c$ - speed of light in vacuum. The units of
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+CD is $ps/(nm-km)$. The material dispersion is due to refractive index variation in fiber core material which makes different wavelength components travel with unequal speeds. The waveguide dispersion is due to $\beta$ (propagation constant) being a function of fiber parameters and also wavelength. CD is the major limiting factor for achieving higher single-band data rates using IM-DD systems. As pulse width becomes smaller, ISI increases and complexity of time-domain equalizer in the DD receiver becomes very high.
+
+* Polarization Mode Dispersion (PMD): The State of Polarization (SoP) of the electric field changes as the signal traverses through the optical fiber. The changes in SoP is random because of fluctuating birefringence. Geometric birefringence and anisotropic stress are the major sources of variation of birefringence. Variation in birefringence means variation of refractive index, which leads to variation in propagation constant ($\beta$). PMD is statistical in nature and is given by the following equation:
+
+$$D_p = \frac{\langle (\Delta T)^2 \rangle^{0.5}}{\sqrt{L_F}} \quad (2.3)$$
+
+where $D_p$ - PMD [$\frac{ps}{\sqrt{km}}$], $\Delta T$ - mean square Differential Group Delay (DGD) value, which is a Maxwellian distributed random variable, $L_F$ - length of optical fiber. In case of IM-DD systems, dual polarization is not used. But for systems using dual polarization, it changes channel coefficients and equalizer coefficients have to be updated regularly to accommodate this.
+
+Different SMF types are used based on distances involved in transmission. In undersea network (submarine), distance involved is more than 2000 km. Terrestrial communication networks is divided into core, metro and access networks. Core network covers distances upto few hundreds to thousands of kilometres connecting cities or countries. Metro network connects core and access network, covering several tens of kilometres. Access network provides connectivity to the end users. Typical values of fiber used in all these types of networks with values for fiber attenuation, CD and PMD are given in Table 2.2.
+
+TABLE 2.2: Specifications of commercially available single mode fibers (Corning Fibers)
+
+<table><thead><tr><th>Fiber Name</th><th>ITU-T Naming</th><th>CD @1550 nm (ps/nm - km)</th><th>PMD ps/√km</th><th>α<sub>dB</sub> @1550 nm dB/km</th><th>Network Usage</th></tr></thead><tbody><tr><td>PSCF</td><td>G.654</td><td>20.2</td><td>≤ 0.05</td><td>0.158</td><td>Submarine</td></tr><tr><td>SSMF</td><td>G.652.D</td><td>18</td><td>0.1</td><td>0.21</td><td>Backbone</td></tr><tr><td>LEAF</td><td>G.655</td><td>4.4</td><td>≤ 0.04</td><td>0.19</td><td>Metro</td></tr><tr><td>SMF-28</td><td>G.652</td><td>18</td><td>≤ 0.04</td><td>0.18</td><td>Access</td></tr></tbody></table>
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+## 2.2.2 Non-Linear Impairments
+
+If launched power into the optical fiber exceeds several milliWatts in single channel system, then non-linear behaviour of optical fiber becomes significant [22]. In modern WDM technology, high-power semiconductor LASERs and optical amplifiers are used which can exceed several milliWatts. Fiber non-linearities can be classified into two major groups.
+
+1. Kerr Nonlinearities caused by the dependence on the index of refraction on light intensity. It causes pulse distortion due to power variation.
+
+* Self-phase Modulation (SPM) - Changes in refractive index caused by power variation within the channel leading to pulse distortion.
+
+* Cross-phase Modulation (XPM) - Pulse distortion caused by variations of power of other wavelength channels in addition to its own channel.
+
+* Four-wave Mixing (FWM) - New channels are created due to interaction of several wavelength channels. FWM effect depends on chromatic dispersion and powers of interacting channels.
+
+2. Simulated Scattering caused by parametric interaction materials of the fiber and optical light.
+
+* Simulated Raman Scattering (SRS) - In this scattering, interaction occurs between light and material through vibrations leading to energy transfer from short wavelength channels to long wavelength channels. This causes to crosstalk between channels.
+
+* Stimulated Brillouin Scattering (SBS) - Interaction occurs between light and material through acoustic waves, leading to coupling with backward propagating waves, which limits the available power per channel.
+
+Non-linear impairments are directly proportional to transmission length ($L_F$) and inversely proportional to cross-sectional area of the optical fiber. Since non-linear impairments are caused due to higher power signals, the non-linear effects are reduced for attenuated signal. For longer fiber lengths and smaller cross-sectional areas, non-linear interaction is stronger.
+
+## 2.3 Differences between Wireless-OFDM and CO-OFDM Systems
+
+Differences between wireless and optical OFDM systems is tabulated to contrast the kind of algorithms, architectures which will be specifically required for CO-OFDM systems and which algorithms can be borrowed from Wireless OFDM systems. The main differences are due to channel used, optical/analogue front-end, and data rates involved.
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+* Wireless channel can have deep spectral nulls in the bandwidth depending on external environment of operation, resulting in frequency selective fading of the signal. The CO-OFDM system uses optical fiber channel which has no spectral nulls in the region of operation.
+
+* Wireless channel varies much faster compared to optical channel whose time constants of variation are of the order of ms. Optical channel is an engineered channel with variations in channel parameters caused by temperature, fiber bending, etc.
+
+* Wireless OFDM systems converts signal from RF to baseband signal using RF-to-analog down converter, while CO-OFDM system converts from optical to RF and then RF to baseband signal using LASER as local oscillator. Because of linewidth of LASER, it results in integer carrier frequency offset (CFO) and rapid phase variations. Rapid phase variations limit the length of symbol size which can be used for OFDM when using digital common phase error (CPE) estimation technique.
+
+* Due to large bandwidth involved for CO-OFDM systems, the data converters (DAC and ADC) become the bottleneck of the system since effective number of bits (ENOB) available at such high bandwidth is also constrained. This imposes resolution constraints on data transmission at DAC and on reception at ADC.
+
+* Data rates of Wireless OFDM systems are in the range of Mb/s, while CO-OFDM systems must support data rates of the order of Gb/s. This difference in data rate makes it necessary for each block to support multiple parallel input/output.
+
+Based on observation of differences between Wireless-OFDM and CO-OFDM systems, the following points are important for the choice of algorithms/architectures for realization of CO-OFDM systems:
+
+* Due to the absence of spectral nulls and channel variations, channel estimation algorithm can be simplified and update rate of the coefficients can be reduced.
+
+* Integer CFO estimation block is mandatory compared to Wireless OFDM systems because of large variation of LASER frequency and efficient phase estimation techniques are required. Continuous monitoring of both frequency and phase is required.
+
+* Due to high data rates involved, highly parallel and scalable architecture are required for all the blocks in the transceiver processing chain. Also, it is necessary to avoid long feedback loops with large delay in critical path of computation.
+
+## 2.4 Typical CO-OFDM System
+
+CO-OFDM system replaces direct detection in intensity modulation systems with Coherent Detection and single-carrier modulation with OFDM modulation scheme. The use of
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+coherent detection supports dual polarization in the optical fiber, thus essentially doubling the data rate of a single polarization CO-OFDM system. A brief introduction of Coherent Detection and OFDM modulation is given in subsections below. Next, all the blocks (digital, analogue and optical) of the single polarization and dual polarization CO-OFDM transmitter and receiver are explained.
+
+**2.4.1 Coherent Detection**
+
+Design of optical transmission system is done by budgeting for different effects in the optical signal processing chain that degrade the signal. The metric used to measure goodness of transmission is done at the receiver by calculation of SNR. SNR can be improved by improvement in noise tolerance at the receiver i.e. better receiver sensitivity. Receiver sensitivity can be defined by the minimum received optical power to keep SNR at the specified level. Receiver sensitivity of CoD and DD are contrasted below.
+
+CoD uses a local oscillator at the receiver which beats at the same frequency as the one at the transmitter to down convert the incoming optical signal. Figure 2.2 [3] shows the noise resilience of CoD and DD methods with different types of symbol mapping. CoD BPSK provides a 4.3 dB improvement in amplified spontaneous noise (ASE) noise tolerance over DD scheme at 40 GBaud signalling rate. Similar tolerance is obtained for dual polarization QPSK at 10 GBaud. This makes CoD a better candidate to work at higher speeds.
+
+FIGURE 2.2: Tolerance of various phase-amplitude constellations to ASE. Reproduced from [3].
+
+CoD was researched heavily in the 1980s in the quest for providing improved receiver sensitivity by detecting low signal powers caused by fiber loss. The invention of EDFA resulted in low cost optical amplifiers that compensate for fiber loss. Due to its low cost, IM-DD systems gained importance and CoD scheme was neglected. But in optical communication systems operating in excess of 20Gb/s data rates, CD and PMD effects became
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+very computationally complex to compensate for in DD scheme. CoD started gaining im-
+portance because of its ability to give access to optical electrical field. While DD scheme
+only detects incoming intensity of the optical signal, CoD scheme detects both amplitude
+and phase of optical signal. This enables the use of Electronic Dispersion Compensa-
+tion (EDC), which uses DSP techniques for estimation and compensation of these linear
+dispersion effects of the channel. With the use of DSP, cost of the system can be brought
+down and flexibility of the system increases significantly. CoD scheme enables higher QAM
+mapping schemes like QPSK, 16-QAM which increase the bits per symbol. It enables dual
+polarization schemes which doubles the data rate per band and requires MIMO processing
+at the receiver to separate the two polarizations. So, with optical community significantly
+adopting DSP based solutions for very high data rate systems, CoD scheme has seen a
+revival recently.
+
+2.4.2 OFDM System
+
+OFDM is a special class of multi-carrier modulation system, in which all sub-carriers are orthogonal to each other. The modulation is realized using IDFT at the transmitter and DFT at the receiver. The complexity of implementation of transmitter and receiver is reduced by the usage of FFT.
+
+The major advantages offered by OFDM communication system is as follows:
+
+*   **Cyclic Prefix (CP)** - A portion of end of OFDM symbol is pre-appended to the symbol. This is called CP, whose length is more than maximum delay of the multi-path signal. Hence, ISI effect due to Chromatic Dispersion is eliminated.
+
+*   **Resistant to FSF** - By division of bandwidth into narrow band flat fading channels, it is more resistant to FSF effects of the channel. Frequency nulls can be avoided or bit-power loading can be employed.
+
+*   **Spectral Efficiency** - Efficient usage of bandwidth by spectrum overlap by using orthogonal sub-carriers.
+
+*   **One-tap Equalization** - Due to addition of CP, linear convolution with the channel is converted to circular convolution and hence a single-tap equalizer per sub-carrier is sufficient.
+
+The disadvantages are:
+
+* PAPR - Due to summation of *N* sub-carriers at the transmitter by IFFT, dynamic range of peak value to mean value varies by a large value [23]. This causes problems for the blocks in the transmission chain to handle such large dynamic range. Clipping at the DAC and introduction of non-linearities at the RF amplifier are some of the effects due to PAPR.
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+Résumé
+
+Les systèmes de communications optiques à très haut débit sont construits à partir des techniques de pointe pour la détection, la modulation et la compensation de dispersion tels que, la détection cohérente, les modulations multi-porteuses orthogonales (OFDM) et la compensation électronique des dispersions (EDC). La réapparition de la détection cohérente dans les systèmes de communication optique a été rendue notamment possible par les progrès dans les circuits numériques dans les technologies avancées. La détection cohérente possède une meilleure sensibilité pour la détection du signal par rapport aux méthodes de détection directe. Elle permet d'utiliser des transmissions à double polarisation et conserve les informations de phase du signal optique et les transfert dans le domaine électrique. L'utilisation de la modulation OFDM fournit une flexibilité significative et l'utilisation efficace de la bande passante allouée. En raison de la disponibilité des informations de phase dans le domaine numérique, les processeurs DSP de faible coût peuvent être utilisés pour la compensation des dispersions dans le domaine numérique qui rend la solution flexible et reconfigurable. Mais, l'introduction du système CO-OFDM (Coherent-Optical OFDM) à la place de système de IM-DD (Intensity Modulation-Direct Detection) augmente significativement le coût du système avec un plus grand nombre de composants optiques et une quantité plus élevée de ressources électroniques requises pour la réception du signal. À l'heure actuelle, cela rend cette solution uniquement justifiable pour des transmissions à longue portée, même si le nombre de ressources par rapport à un système mono-porteuse à détection cohérente et modulation à quatre états (DP-CO-QPSK). Le choix de l'algorithme et l'optimisation de la précision des calculs en virgule fixe de l'architecture peuvent réduire de façon significative les ressources nécessaires pour la réalisation de systèmes CO-OFDM.
+
+Dans cette thèse, des algorithmes à faible complexité et des architectures parallèles et efficaces sont explorés pour les systèmes CO-OFDM. Tout d'abord, des algorithmes de faible complexité pour la synchronisation et l'estimation du décalage en fréquence en présence d'un canal dispersif sont étudiés. Un nouvel algorithme de synchronisation temporelle à faible complexité qui peut résister à grande quantité de retard dispersif est proposé et comparé par rapport aux propositions antérieures. Ensuite, le problème de la réalisation d'une architecture parallèle à faible coût est étudié et une architecture parallèle générique et évolutive qui peut être utilisée pour réaliser tout type d'algorithme d'auto-corrélation est proposé. Cette architecture est ensuite étendue pour gérer plusieurs échantillons issus du convertisseur analogique/numérique (ADC) en parallèle et fournir une sortie qui suive la fréquence des ADC. L'évolutivité de l'architecture pour un nombre plus élevé de sorties en parallèle et les différents types d'algorithmes d'auto-corrélation sont explorés.
+
+Une approche d'adéquation algorithme-architecture est ensuite appliquée à l'ensemble de la chaîne de l'émetteur-récepteur CO-OFDM. Du côté de l'émetteur, un algorithme IFFT à radix-2² est choisi pour et une architecture parallèle Multipath Delay Commutator (MDC) Feed-forward (FF) est choisie car elle consomme moins de ressources par rapport aux architectures MDC-FF en radix-2/4. Au niveau du récepteur, un algorithme efficace pour l'estimation du Integer CFO est adopté et implémenté de façon optimisée sans l'utilisation de multiplicateurs complexes. Une rÃduction de la complexité matérielle est obtenue grâce à la conception d'architectures efficaces pour la synchronisation temporelle, la FFT et l'estimation du CFO. Une exploration du compromis entre la précision des calculs en virgule fixe et la complexité du matériel est réalisée pour la chaîne complète de l'émetteur-récepteur, de façon à trouver des points de fonctionnement qui n'affectent pas le taux
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+* Add CP - It adds portion of last part of OFDM symbol to the front. It avoids ISI due to multipath channel when the length of CP is greater than maximum dispersion delay of the channel. It provides immunity against CD of optical fiber.
+
+* Scale, Clip - Output of IFFT is scaled and clipped to fit in the input voltage range of the DAC. Clipping value must be chosen to minimize clipping distortion as well as quantization noise.
+
+### 2.4.4 RF-to-Optical Up Converter
+
+Figure 2.5 shows the RF-to-Optical Converter for a single polarization CO-OFDM system, while Figure 2.6 corresponds to dual polarization CO-OFDM system.
+
+FIGURE 2.5: Single Polarization RF-to-Optical Up Converter. $I_X$ - Real Part of X-Polarization, $Q_X$ - Imaginary Part of X-Polarization, $DAC$ - Digital-to-Analog Converter, $LPF$ - Low Pass Filter, $RFD$ - RF Driver, $MZM$ - Mach-Zender Modulator, $ECL$ - External Cavity LASER, $VOA$ - Variable Optical Amplifier.
+
+FIGURE 2.6: Dual Polarization RF-to-Optical Up Converter. $I_X$ - Real Part of X-Polarization, $Q_X$ - Imaginary Part of X-Polarization, $I_Y$ - Real Part of Y-Polarization, $Q_Y$ - Imaginary Part of Y-Polarization, $DAC$ - Digital-to-Analog Converter, $LPF$ - Low Pass Filter, $RFD$ - RF Driver, $MZM$ - Mach-Zender Modulator, $ECL$ - External Cavity LASER, $PBS$ - Polarization Beam Splitter, $VOA$ - Variable Optical Amplifier, $PBC$ - Polarization Beam Combiner.
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+FIGURE 2.9: Resolution vs. Sampling Rate for fastest ADC available. GSa/s - Giga Samples/second.
+
+FIGURE 2.10: Digital Receiver of PDM-CO-OFDM System. **TFSYNC** - Time Frequency Synchronization, **CFO** - Carrier Frequency Offset, **FCOMP** - CFO Compensation, **FFT** - Fast Fourier Transform, **ICFO** - Integer CFO Estimation, **CEE** - Channel Estimation & Equalization, **CPE** - Common Phase Error, **CPEC** - CPE Estimation & Compensation, **DMAP** - Demapper.
+
+*   **FFT** - It converts input time domain samples to frequency domain output samples. It is the most complex block in the receiver chain.
+
+$$X[k] = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} x[n] e^{j2\pi kn/N} \quad (2.5)$$
+
+*   **Integer Frequency Estimation** - In case of CO-OFDM systems, Integer CFO is present due to variations of LASER frequency by large amount. Integer CFO is estimated by use of training symbol. The estimated value is fed back to CFO compensator block.
+
+*   **Channel Estimation & Equalization** - Channel Estimation is done using training symbols and then tracking is done either using decision-directed equalizers like LMS Equalizers or averaging techniques in time/ frequency domain.
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@@ -0,0 +1,181 @@
+used in transmitter and receiver is calculated. Algorithmic complexity gives the total number of real multiplications and additions required for computation of single sample of output. For example, total number of complex multiplications and additions required for one output of IFFT is expressed in terms of size of IFFT (N). Then, the architectural complexity of the algorithms is calculated for throughput of one output every clock cycle. Throughput of one output clock is necessary to support high data rates and to avoid large buffer memory. Architectural complexity involves calculation of number of real multipliers and adders required for realization of the algorithm.
+
+**2.5.1 Digital Transmitter**
+
+The major computational block in the digital transmitter is IFFT. The algorithmic com-
+plexity of radix-2/4/2² or split radix IFFT block is given in Table 2.4, which gives total
+number of multiplications and additions as a function of IFFT size (N). Architectural com-
+plexity of Feedforward pipelined architectures using radix-2/4/8 in terms of total number
+of multipliers, adders and memory requirement is given in Table 2.5.
+
+TABLE 2.4: Algorithmic Complexity in terms of size of IFFT/FFT N.
+
+<table>
+   <thead>
+    <tr>
+     <td>
+      Radix
+     </td>
+     <td>
+      Real Multiplications
+     </td>
+     <td>
+      Real Additions
+     </td>
+    </tr>
+   </thead>
+   <tbody>
+    <tr>
+     <td>
+      Radix-2
+     </td>
+     <td>
+      2
+      <i>
+       N
+      </i>
+      &middot;
+      <i>
+       log
+      </i>
+      <sub>
+       2
+      </sub>
+      <i>
+       N
+      </i>
+     </td>
+     <td>
+      3
+      <i>
+       N
+      </i>
+      &middot;
+      <i>
+       log
+      </i>
+      <sub>
+       2
+      </sub>
+      <i>
+       N
+      </i>
+     </td>
+    </tr>
+    <tr>
+     <td>
+      Radix-4
+     </td>
+     <td>
+      3⁄2
+      <i>
+       N
+      </i>
+      &middot;
+      <i>
+       log
+      </i>
+      <sub>
+       2
+      </sub>
+      <i>
+       N
+      </i>
+     </td>
+     <td>
+      5⁄4
+      <i>
+       N
+      </i>
+      &middot;
+      <i>
+       log
+      </i>
+      <sub>
+       2
+      </sub>
+      <i>
+       N
+      </i>
+     </td>
+    </tr>
+    <tr>
+     <td>
+      Radix-2<sup>2</sup>
+     </td>
+     <td>
+      3⁄2
+      <i>
+       N
+      </i>
+      &middot;
+      <i>
+       log
+      </i>
+      <sub>
+       2
+      </sub>
+      <i>
+       N
+      </i>
+     </td>
+     <td>
+      5⁄4
+      <i>
+       N
+      </i>
+      &middot;
+      <i>
+       log
+      </i>
+      <sub>
+       2
+      </sub>
+      <i>
+       N
+      </i>
+     </td>
+    </tr>
+    <tr>
+     <td>
+      Split-Radix
+     </td>
+     <td>
+      4⁄3
+      <i>
+       N
+      </i>
+      &middot;
+      <i>
+       log
+      </i>
+      <sub>
+       2
+      </sub>
+      <i>
+       N
+      </i>
+     </td>
+     <td>
+      8⁄3
+      <i>
+       N
+      </i>
+      &middot;
+      <i>
+       log
+      </i>
+      <sub>
+       2
+      </sub>
+      <i>
+       N
+      </i>
+     </td>
+    </tr>
+   </tbody>
+  </table>
+
+Survey of previously reported real-time transmitter with offline receiver is done. The objective is to calculate transmitter’s architectural complexity, which inherently comes down to calculation of IFFT complexity. Table 2.6 lists real-time CO-OFDM experiments on standard single mode fiber (SSMF) which have achieved gigabit per second using real-time implementation on an FPGA.
+
+The computational complexity of the proposed real-time solutions is given in Table 2.7. Proposal by Inan et al. [31] uses radix-2 IFFT for larger parallel factor of 64, which is inefficient considering higher radix can be used at such high parallel output. Proposal by Schmogrow et al. [29] does not use multipliers, but huge number of adders and LUTs. Since, all multiplier combinations are stored in memory, it is limited to small size (N) of IFFT of 64. This approach is not scalable to higher speeds.
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diff --git a/samples/texts/7426999/page_66.md b/samples/texts/7426999/page_66.md
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+TABLE 2.5: Architectural Complexity of feedforward pipelined IFFT/FFT for 2/4/8-Parallel Outputs as a function of IFFT/FFT size (N). MDC - Multipath Delay Commutator.
+
+<table><thead><tr><th>Radix</th><th>Architecture Type</th><th>Real Multipliers</th><th>Real Adders</th><th>Total Memory</th></tr></thead><tbody><tr><td colspan="5">2-PARALLEL INPUT/OUTPUT</td></tr><tr><td>Radix-2 [26]</td><td>MDC</td><td>8(log<sub>4</sub> N &minus; 1)</td><td>12 log<sub>4</sub> N &minus; 4</td><td>2N</td></tr><tr><td colspan="5">4-PARALLEL INPUT/OUTPUT</td></tr><tr><td>Radix-2</td><td>MDC</td><td>16(log<sub>4</sub> N &minus; 1)</td><td>24 log<sub>4</sub> N &minus; 8</td><td>2N</td></tr><tr><td>Radix-4 [27]</td><td>MDC</td><td>12(log<sub>4</sub> N &minus; 1)</td><td>22 log<sub>4</sub> N &minus; 6</td><td>2N</td></tr><tr><td colspan="5">8-PARALLEL INPUT/OUTPUT</td></tr><tr><td>Radix-2 [28]</td><td>MDC</td><td>32(log<sub>4</sub> N &minus; 1)</td><td>48 log<sub>4</sub> N &minus; 16</td><td>2N</td></tr><tr><td>Radix-4</td><td>MDC</td><td>24(log<sub>4</sub> N &minus; 1)</td><td>44 log<sub>4</sub> N &minus; 12</td><td>2N</td></tr><tr><td>Radix-8 [27]</td><td>MDC</td><td>24(log<sub>4</sub> N &minus; 7)</td><td>44 log<sub>4</sub> N &minus; 14</td><td>(32N)/7</td></tr></tbody></table>
+
+TABLE 2.6: Real-time CO-OFDM Transmitter Implementation
+
+<table><thead><tr><th>Reference</th><th>Bandwidth (GHz)</th><th>Data Rate (Gb/s)</th><th>IFFT size (N)</th><th>Cyclic Prefix</th><th>Year</th></tr></thead><tbody><tr><td>Schmogrow et al. [29]</td><td>25.4</td><td>101.5</td><td>64</td><td>0</td><td>2011</td></tr><tr><td>Inan et al. [30]</td><td>11.9</td><td>23.9</td><td>1024</td><td>64</td><td>2011</td></tr><tr><td>Inan et al. [31]</td><td>23.4</td><td>93.8</td><td>1024</td><td>64</td><td>2011</td></tr></tbody></table>
+
+TABLE 2.7: Computational Complexity for CO-OFDM Transmitter
+
+<table><thead><tr><th>Reference</th><th>Real Multipliers</th><th>Real Adders</th><th>IFFT size (N)</th><th>Radix used</th></tr></thead><tbody><tr><td>Schmogrow et al. [29]</td><td>0 (uses LUTs to store values)</td><td>3776</td><td>64</td><td>Radix-2/4</td></tr><tr><td>Inan et al. [31]</td><td>844</td><td>1732</td><td>1024</td><td>Radix-2 (64-Parallel)</td></tr></tbody></table>
+
+### 2.5.2 Digital Receiver
+
+For all blocks in the receiver, algorithmic and architectural complexity is calculated for single/multiple parallel output. The major blocks of the receiver are: Time/Frequency Synchronization, CFO Compensation, FFT, Integer CFO Estimation, Channel Estimation & Equalization, CPE Estimation & Compensation, and Demapper. For architectural complexity comparison, only two papers are available which have implemented real-time
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diff --git a/samples/texts/7426999/page_7.md b/samples/texts/7426999/page_7.md
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+d'erreur binaire (TEB) de manière significative. Les algorithmes proposés sont validés à l'aide d'une part d'expériences off-line en utilisant un générateur AWG (arbitrary wave-form generator) à l'émetteur et un oscilloscope numérique à mémoire (DSO) en sortie de la détection cohérente au récepteur, et d'autre part un émetteur-récepteur temps-réel basé sur des plateformes FPGA et des convertisseurs numériques. Le TEB est utilisé pour montrer la validité du système intégré et en donner les performances.
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diff --git a/samples/texts/7426999/page_70.md b/samples/texts/7426999/page_70.md
new file mode 100644
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+TABLE 2.10: Algorithmic/Architectural Complexity for integer CFO Estimation. R - number of parallel outputs.
+
+<table><thead><tr><th>Algorithm</th><th>Real Multiplications</th><th>Real Additions</th><th>Real Multipliers</th><th>Real Adders</th></tr></thead><tbody><tr><td>Method 1</td><td>11NW/2</td><td>2NW</td><td>10R</td><td>7R</td></tr><tr><td>Method 2</td><td>3NW</td><td>3NW/2</td><td>6R</td><td>5R</td></tr></tbody></table>
+
+### 2.5.7 Channel Estimation and Equalization
+
+#### 2.5.7.1 Least Squares (LS)
+
+Channel estimation using LS approach is given for both single/dual polarizations.
+
+1. Single polarization - Consider the received signal,
+
+$$R_{km} = H_k \cdot c_{km} + N_{km} \quad (2.12)$$
+
+where $R_{km}$ is the $k^{th}$ sub-carrier of $m^{th}$ received OFDM symbol, $H_k$ is the channel response for $k^{th}$ sub-carrier, $c_{km}$ is the $k^{th}$ sub-carrier of $m^{th}$ transmitted OFDM symbol, $N_{km}$ is the additive noise. Using the training symbol, channel frequency response can be estimated according to LS criterion using optimization criterion [40],
+
+$$\hat{H}_k = \arg_{min} ||R_k - H_k c_k||^2. \quad (2.13)$$
+
+$\hat{H}_k$ can be calculated using
+
+$$\hat{H}_k = \frac{R_k^* c_k}{|R_k|^2} \quad (2.14)$$
+
+where $k \in [0, 1, ..., N_{usc}]$, $N_{usc}$ is the used sub-carrier number. Since optical channel does not have frequency nulls in the band of operation, the problem of noise enhancement is avoided. Algorithmic/Architectural Complexity for LS channel estimation is given in Table 2.11. The equalization is done using $\hat{H}_k$ given by
+
+$$C_{km} = R_{km} \cdot \hat{H}_k. \quad (2.15)$$
+
+where $C_{km}$ is the equalized signal.
+
+2. Dual Polarization - In dual polarization system, the received signal can be written as
+
+$$R_{km,x} = H_k^{xx} \cdot c_{km,x} + H_k^{xy} \cdot c_{km,y} \quad (2.16)$$
+
+$$R_{km,y} = H_k^{yx} \cdot c_{km,x} + H_k^{yy} \cdot c_{km,y} \quad (2.17)$$
+
+There are four coefficients to be calculated, which can be simplified by transmitting training symbols in only one polarization at a time. The system of equations using
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diff --git a/samples/texts/7426999/page_71.md b/samples/texts/7426999/page_71.md
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index 0000000000000000000000000000000000000000..c9d9e06a02320418889078c0de882e126c8233e7
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+this scheme can be written as
+
+$$
+\begin{bmatrix} R_{km,x} & R_{k(m+1),x} \\ R_{km,y} & R_{k(m+1),y} \end{bmatrix} = \begin{bmatrix} H_k^{xx} & H_k^{xy} \\ H_k^{yx} & H_k^{yy} \end{bmatrix} \cdot \begin{bmatrix} c_{km,x} & 0 \\ 0 & c_{k(m+1),y} \end{bmatrix} \quad (2.18)
+$$
+
+Estimation of the coefficients is given by
+
+$$
+\hat{H}_k^{xx} = \frac{R_{km,x}}{c_{km,x}}, \quad \hat{H}_k^{xy} = \frac{R_{k(m+1),x}}{c_{k(m+1),y}} \tag{2.19}
+$$
+
+$$
+\hat{H}_{k}^{yx} = \frac{R_{km,y}}{c_{km,x}}, \quad \hat{H}_{k}^{yy} = \frac{R_{k(m+1),y}}{c_{k(m+1),y}} \qquad (2.20)
+$$
+
+The equalization can be done using $\hat{H}_k$
+
+$$
+C_{km} = \hat{H}_{k}^{-1} \cdot R_{km} \tag{2.21}
+$$
+
+where $C_{km}$ is the equalized signal, $\hat{H}_k^{-1}$ is the inverse of 2x2 $\hat{H}_k$ matrix. Since the $\hat{H}_k$ is a unitary matrix, the inverse calculation can be done by using a Hermitian transpose. Algorithmic/Architectural complexity is given in Table 2.11.
+
+2.5.7.2 Normalized Least Mean Squares (NLMS)
+
+After initial estimation using training symbols by LS estimation, NLMS method can be
+used to track the channel. The equations used for single polarization are
+
+$$
+e_k = \hat{R}_k - R_k^{\text{ideal}} \tag{2.22}
+$$
+
+$$
+|R|_k^2 = k_1 \cdot |R|_{k,old}^2 + (1-k_1) \cdot |R|_k^2 \quad (2.23)
+$$
+
+$$
+\hat{H}_k = \hat{H}_{k,old} + step \cdot e_k \cdot \frac{R_k^*}{|R|^2_k} \quad (2.24)
+$$
+
+where $e_k$ is the error between received symbol ($R_k$) and ideal constellation symbol ($R_k^{ideal}$),
+$k_1$ is the coefficient for updating energy, $|R|_{k,old}^2$ is the old value of energy, step is the co-
+efficient for updating equalizer coefficients, $\hat{H}_{k,old}$ is the old value of equalizer coefficient.
+Algorithmic Complexity for NLMS Estimation is given in Table 2.11. Equalization in-
+volves complex multiplications with channel estimated and its algorithmic/architectural
+complexity is given in Table 2.11. Kaneda et al. simplified the channel estimation signif-
+icantly by using look-up table implementation. No multipliers were required for channel
+estimation.
+
+**2.5.8 CPE Estimation and Compensation**
+
+CPE Estimation [14] is done by comparing and averaging received pilot phases with refer-
+ence phase to calculate phase noise. Compensation is done by rotating the symbols using
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diff --git a/samples/texts/7426999/page_74.md b/samples/texts/7426999/page_74.md
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+# Chapter 3
+
+# Timing Synchronization in OFDM Systems
+
+## 3.1 Introduction
+
+OFDM systems are sensitive to timing, carrier frequency offset (CFO) [41][42] and phase offset [43]. Loss of timing synchronization causes inter-carrier interference (ICI) and inter-symbol interference (ISI). It also leads to reduced accuracy in carrier frequency offset estimation and causes sub-carrier dependent phase rotation after FFT. Uncompensated CFO also causes rotation of sub-carriers proportional to frequency offset. Thus, loss of timing and frequency synchronization reduces the advantages provided by single-carrier OFDM over single-carrier systems.
+
+In Section 3.2, a survey of timing synchronization algorithms proposed for Wireless OFDM systems is done. Survey was done to look at possible improvements possible in timing estimation. This led to a novel proposal of hierarchical low-complexity synchronizer for Wireless OFDM systems which is given in Section 3.3. Performance of the proposal is evaluated in Section 3.4 in an ISI channel. In Section 3.5, the proposal is adapted to optical channel with modifications to reduce complexity. The performance of adapted proposal is evaluated using single mode optical fiber channel in Section 3.6. In Sections 3.9 and 3.10, the proposed streaming parallel architectures for synchronization algorithm is explained. Architectural complexity of the proposed architectures is calculated to show the scalability of the architecture and compared with previous proposals. Section 3.12 concludes the chapter.
+
+## 3.2 Timing Synchronization in Wireless OFDM Systems
+
+In the OFDM transmitter, input bits are mapped to QAM constellation by mapper block. These mapped symbols are considered to be in frequency domain. The mapped symbols are passed through IFFT and converted to time domain. Every OFDM symbol (training, data) is passed through IFFT and converted to time domain before transmission. Since
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diff --git a/samples/texts/7426999/page_75.md b/samples/texts/7426999/page_75.md
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+timing synchronization operation is done before IFFT in the receiver, it is performed on time domain data. All the methods (auto-correlation/cross-correlation) proposed for timing synchronization work on time domain data to obtain the starting point estimation of OFDM symbol. Some of the methods also give a coarse estimate of carrier frequency offset (CFO) which is used to compensate the received OFDM symbol before passing it through the FFT.
+
+Many methods have been proposed for initial timing synchronization previously. The synchronization methods have either been based on cyclic prefix [44] or known preamble symbols [33][34][35][36][37]. Cyclic prefix (CP) based methods utilize the cyclic prefix length for synchronization which constitutes only a small part of the OFDM symbol. In a multipath channel, cyclic prefix samples are corrupted due to ISI. This leads to reduction in the number of uncorrupted samples for synchronization which leads to reduction in the accuracy of estimation.
+
+Preamble-based synchronization methods utilize specially designed training symbol for achieving synchronization. Preamble symbols are designed with specific structure for maximizing the detection of start point of the OFDM symbol. Schmidl et al. [33] proposed a training symbol (TS) which consists of two repetitive parts. The correlation peak of the timing metric has a plateau equal to the length of cyclic prefix in AWGN channel. The length of plateau is equal to the length of the uncorrupted portion of the cyclic prefix in frequency selective (ISI) channels. The training symbol also allows a CFO estimation of ± 1 subcarrier spacing. Minn et al. [34] proposed an algorithm consisting of multiple repetitive parts with specific sign pattern for timing synchronization. The algorithm has steeper timing roll-off compared to Schmidl's timing metric roll-off. CFO estimation was done using algorithm by Morelli [45]. The symbol has a CFO estimation range of ± L/2 sub-carrier spacings, where L is the number of repetitive parts in the TS. Shi and Serpedin [35] proposed a four part TS which is similar to Minn's, but the algorithm used for timing estimation is more generalized and uses all repetitive parts of the TS. The CFO estimation range is ± 2 sub-carrier spacing.
+
+Park et al. [36] proposed a new training symbol which utilizes conjugate symmetry property for timing synchronization. The timing metric has a steeper roll-off compared to Minn's method. Choi et al. [37] proposed a training symbol for multipath channels which also utilizes conjugate symmetry for correlation operation. The symbol is based on CAZAC (Zadoff-Chu) sequence, which has constant amplitude and has much better correlation properties than m-sequences. The timing metric has impulse like roll-off at the correct starting point. Although these methods [37][36] have very steep timing roll-off metric, they are computationally intensive in number of operations required per point of timing metric.
+
+Zhou [38][46] proposed a hybrid method for synchronization using preamble symbols. The method uses both kinds of correlation (auto and cross-correlation) operations with TS and multiple thresholds for detection of first path in multipath channel. But it uses both correlation operations together simultaneously, thus using a lot of computations per
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diff --git a/samples/texts/7426999/page_78.md b/samples/texts/7426999/page_78.md
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+estimate.
+
+$$ \hat{\eta}_{init} = \arg \max_{n} TM_{init}[n] \quad (3.8) $$
+
+Figure 3.1a shows the plot of $TM_{init}[n]$ for a Signal-to-Noise Ratio (SNR) of 10 dB in a frequency selective channel. The maximum peak is not exactly at zero index which is the actual start of the OFDM symbol, but slightly shifted to the right due to multipath effect. The fine estimation algorithm consists of correcting this unknown shift and finding the correct starting point. The fine estimation algorithm does not assume a dominant first path and can work with non-dominant first path in multipath channel.
+
+FIGURE 3.1: Plot of Coarse (a) and Fine (b) Timing Metric Functions
+
+The fine estimation algorithm uses the conjugate symmetry present in the training
+symbol to estimate the correct starting point. The fine time estimation algorithm starts
+from the point $\hat{\eta}_{center} = \hat{\eta}_{init} + \frac{N}{2}$. Since the length of **A** or **B** part is greater than the
+maximum delay spread of the multipath signal, all the four parts of the training symbol are
+exposed to similar multipath channel environment. A search distance of [-$N_{cyp}$, $N_{cyp}$] is
+covered from $\hat{\eta}_{center}$ for finding all paths of multipath channel. The fine timing estimation
+metric is given by
+
+$$ TM_{fine}[n] = |P_{fine}[n]|^2 \qquad (3.9) $$
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diff --git a/samples/texts/7426999/page_79.md b/samples/texts/7426999/page_79.md
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+with
+
+$$P_{\text{fine}}[n] = \sum_{k=0}^{\frac{N}{4}-1} r[n-k-1] \cdot r[n+k] - \sum_{k=\frac{N}{4}}^{\frac{N}{2}-1} r[n-k-1] \cdot r[n+k] \quad (3.10)$$
+
+where $TM_{fine}$ is the fine timing estimation metric and $P_{fine}$ is the conjugate symmetric correlation operation. The negative sign for $k \in [\frac{N}{4}, \frac{N}{2} - 1]$ is because of the sign pattern $[1 \ 1 \ 1 \ -1]$, $n \in [-N_{cyp}, N_{cyp}]$. This range for $n$ was chosen since the initial estimate does not produce peaks outside the maximum length of multipath channel. The timing metric produces peaks which are proportional to individual squared channel path gains. The expansion of $P_{fine}[n]$ in terms of channel coefficients is given by Equation 3.11.
+
+$$ 
+\begin{align}
+P_{\text{fine}}[n] &= \sum_{k=0}^{M-1} r[n-k-1] \cdot r[n+k] \tag{3.11} \\
+&= \sum_{k=0}^{M-1} \left( \sum_{m=0}^{L_h-1} h_m x[n-k-1-\tau_m] + w[n-k-1] \right) \sum_{m'=0}^{L_h-1} h_{m'} x[n+k-\tau_{m'}] + w[n+k] \nonumber \\
+&= \sum_{k=0}^{M-1} \sum_{\substack{m=0, \\ m=m'}}^{L_h-1} h_m^2 |x[n-k-1-\tau_m]|^2 + \sum_{k=0}^{M-1} \left( \sum_{\substack{m=0, \\ m'=0, \\ m \neq m}}^{L_h-1} h_m h_{m'} x[n-k-1-\tau_m] \cdot x[n+k-\tau_{m'}] \right) \nonumber \\
+&\qquad + W[n] \nonumber
+\end{align}
+$$
+
+$$P_{\text{fine}}[n] \approx \sum_{k=0}^{M-1} \sum_{\substack{m=0, \\ m=m'}}^{L_h-1} h_m^2 |x[n-k-1-\tau_m]|^2 \quad (3.12)$$
+
+The term $W[n]$ refers to the correlation terms produced due to noise and signal components. The first term in Equation 3.11 corresponds to peaks produced in the timing metric. $Q[n]$ is calculated by normalizing all values of $TM_{fine}[n]$ by the maximum value of $TM_{fine}[n]$:
+
+$$Q[n] = \frac{TM_{\text{fine}}[n]}{\max(TM_{\text{fine}}[n])} \quad (3.13)$$
+
+Figure 3.1b shows the plot of $Q[n]$ for SNR of 10 dB. The figure shows peaks corresponding to multipath gains. The threshold shown in 3.1b helps in selecting signal only components which are to be used for the windowed summation method to find the first arrival path. The time index of the maximum value of $Q[n]$:
+
+$$\hat{\eta}_{\text{fine}} = \arg\max_n(Q[n]) \quad (3.14)$$
+
+is used as the starting point for windowing summation method which is similar to the one used in Choi's. The values of $Q[n]$ are thresholded by a value $\beta$.
+
+$$Q[n] = \begin{cases} Q[n], & Q[n] > \beta, \\ 0, & \text{otherwise,} \end{cases} \quad (3.15)$$
+
+$\beta$ is the threshold which separates signal and noise components in $Q[n]$. This threshold is
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+FIGURE 3.8: Parallel Architecture proposed by Chen et. al for cross-correlation operation
+
+operation. Further, the proposed architecture is shown to accelerate conjugate symmetric correlation operation. Hence, it could accelerate both auto-correlation and conjugate symmetric correlation operation, thus helping in implementing the proposed hierarchical synchronization algorithm.
+
+## 3.8 Proposed Block Parallel Architecture for Auto-Correlation
+
+The goal of efficient parallelization of auto-correlation algorithm is to have a scalable architecture for efficiently handling multiple parallel inputs and match the input data rate. Block-level parallelism is proposed for achieving a scalable architecture compared to sample-level parallel approach. Proposed block-level architecture uses both non-iterative and iterative forms of equations to achieve parallel computation. Non-iterative equation is used to calculate auto-correlation for the first sample point of the blocks, while iterative equation is used for auto-correlation sample calculation for the remaining points in the block. The choice of block size for parallel computation determines the possible sharing of resources. The proposed block parallel method is explained as follows using Schmidl-Cox algorithm (SCA) and Minn-Bhargava algorithm (MBA) auto-correlation equations. For SCA, with a training symbol $[A \ A]$, the non-iterative and iterative auto-correlation
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