samples/texts/1775930/page_3.md

Before proceeding to the description of the proposed method, it should be emphasized that addressing this problem in the third order statistics rather than the signal domain offers two advantages.

1. Statistical similarity is exploited, which means that valid results are obtained even if $y(\mathbf{n})$ and $\mathbf{x}(\mathbf{n})$ correspond to different realizations of a stochastic process. This feature is of great importance in situations where the reference (high resolution) and the test (low resolution) signals are not acquired simultaneously. This is the case, e.g., with pattern classification applications that use pre-stored data.

2. The well known immunity of the third order statistics to a wide class of additive noises, makes the proposed method appropriate for situations where only noisy data is available.

Proposed Method:

step 1: Estimate the cumulant of the reference signal $\mathbf{x}(\mathbf{n})$, $\hat{c}_{3x}(\mathbf{m}_1, \mathbf{m}2)$, and the cyclic cumulant of the test signal $y(\mathbf{n})$, $\hat{c}{3y}(\mathbf{m}_1, \mathbf{m}_2)$. The asymptotically consistent estimator of the cyclic cumulant proposed in [3] can be employed. This estimator in the present set up takes the form

$${\widehat{c}}_{3y}({\mathbf{m}}_1,{\mathbf{m}}_2)=\frac{1}{T_y}\sum _{t\in W_y}y(t)y(t+{\mathbf{m}}_1)y(t+{\mathbf{m}}_2),(5)\backslash hat\{c\}\_\{3y\}(\backslash mathbf\{m\}\_1,\; \backslash mathbf\{m\}\_2)\; =\; \backslash frac\{1\}\{T\_y\}\; \backslash sum\_\{t\; \backslash in\; W\_y\}\; y(t)y(t+\backslash mathbf\{m\}\_1)y(t+\backslash mathbf\{m\}\_2),\; \backslash quad\; (5)$$

where $W_y$ is the set of all available data samples of signal $y(\mathbf{n})$, and $T_y$ is the cardinal number of $W_y$. Note that in practice the conventional estimator $\hat{c}_{3x}(\mathbf{m}_1, \mathbf{m}_2)$ for the (non-cyclic) cumulant of $\mathbf{x}(\mathbf{n})$ is implemented in the same way.

step 2: Define a partition ${q_n}$, $n = 1, 2, \dots, N$ of the scale (resolution) interval $(0, 1]$ with $q_n$ chosen as fractional numbers $L_n/M_n$, where $L_n \le M_n$ and $(L_n, M_n)$ are coprime integers.

step 3: For $n=1, 2, \dots, N$,

1. Compute the triple correlation $h_3^{(n)}(\mathbf{m}_1, \mathbf{m}_2)$ of the resampling filter $h(\mathbf{n})^{(n)}$, which should have gain $L_n$ and cutoff frequency $\pi/M_n$.

2. Estimate the cyclic cumulant $\hat{c}_{3y}^{(n)}(\mathbf{m}_1, \mathbf{m}_2)$ of $y^{(n)}(\mathbf{n})$, which is a resampled version of the reference signal $\mathbf{x}(\mathbf{n})$ at a sampling rate $M_n/L_n$ times lower than that of $\mathbf{x}(\mathbf{n})$.

This estimator can be implemented as in equation (3), using $L_n, M_n, h_3^{(n)}$ in place of $L, M, h_3$.

3. Compute the similarity index

$$f\left(\frac{L_n}{M_n}\right)\triangleq \sum _{{\mathbf{m}}_1,{\mathbf{m}}_2}\mathrm{\mid }{\widehat{c}}_{3,y}^{(n)}({\mathbf{m}}_1,{\mathbf{m}}_2)-{\widehat{c}}_{3,y}^{(n)}({\mathbf{m}}_1,{\mathbf{m}}_2){\mathrm{\mid }}^2\mathrm{.}(6)f\backslash left(\backslash frac\{L\_n\}\{M\_n\}\backslash right)\; \backslash triangleq\; \backslash sum\_\{\backslash mathbf\{m\}\_1,\; \backslash mathbf\{m\}\_2\}\; |\backslash hat\{c\}\_\{3,y\}^\{(n)\}(\backslash mathbf\{m\}\_1,\; \backslash mathbf\{m\}\_2)\; -\; \backslash hat\{c\}\_\{3,y\}^\{(n)\}(\backslash mathbf\{m\}\_1,\; \backslash mathbf\{m\}\_2)|^2.\; \backslash quad\; (6)$$

step 4: Obtain an estimate of the conversion ratio $L/M$ as the point $q_n = L_n/M_n$ of the global minimum of index $f(q_n)$ over all $n = 1, 2, \dots, N$. The estimate of the conversion ratio $L/M$ can be drawn arbitrarily close to the true ratio $L/M$ by repeatedly refining the partition $q_n$ of $(0, 1]$. Local refinement can be used, in a process of zooming into the neighborhood of the initial the global minimum.

Comment:

The similarity index $f(q_n)$ is in general a non-convex function of $q_n$. However, it has been observed that $f(q_n)$ exhibits a deep global minimum at $L/M$. Also in the neighborhood of the global minimum it assumes a convex form. Therefore, in the neighborhood of the global minimum the refinement process can be driven by fast minimization algorithms such as the Fibonacci and Golden Section methods, ([7]).

4. APPLICATIONS

Use of the proposed method for the estimation of the unknown sampling rate conversion ratio $L/M$ is investigated in the sequel, in an 1-D and a 2-D problem.

4.1. Velocity estimation via Doppler effect

A signal $x(t)$ reflected by an object moving with velocity $\mathbf{v}$ is observed at the source (detector) site as $y(t) = x(\alpha t)$, where $\alpha = \frac{c+v}{c}$ and $\mathbf{c}$ is the velocity of transmitted signal. This corresponds to a rescaling of the transmitted signal $x(t)$ by a factor $\alpha$. A similar relation with $\alpha = \frac{c-v}{c}$ holds in the case that the moving object itself is emitting the signal $x(n)$, rather than reflecting it.

Conventional methods assume that $x(t)$ is a narrowband signal, usually a sinusoid, and measure the frequency shift between the transmitted and the received signals as a means to compute $\alpha$ and then $\mathbf{v}$. This narrowband assumption is not always met, either because of physical constraints of the emitters, or when transmission of sinusoidal signals is to be avoided for security reasons.