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Laplace transform | Do signals with a Fourier transform with discontinuities or zero amplitude (in some frequencies) have Laplace transform? | https://dsp.stackexchange.com/questions/20075/do-signals-with-a-fourier-transform-with-discontinuities-or-zero-amplitude-in-s | <p>I am reading a book on Laplace transform, and in the section on the convergence of Laplace transform for various signals the following theorem is stated, without any proof :</p>
<p><em>If a signal's Fourier transform be zero in some frequencies, or have discontinuities, it will not have Laplace transform (like sinc... | <p>First of all, it is important to distinguish between the unilateral and the bilateral Laplace transform. For causal signals we can use the unilateral Laplace transform. If for such a signal the Fourier transform exists, then also its Laplace transform exists. This is simply the case because here the Fourier transfor... | 134 |
Laplace transform | How to compute the Laplace transform of a discrete signal? | https://dsp.stackexchange.com/questions/45030/how-to-compute-the-laplace-transform-of-a-discrete-signal | <p>Assume I have a discrete random signal, $f(t)$ for which I want to calculate the laplace transform. </p>
<p>How can I do it in matlab without using <code>sym</code> variables, for example consider I have this discrete signal <code>f(t)</code>:</p>
<pre><code>>> t=linspace(0,1000, 10000);
>> f=t.*cos(t)... | <p>This requires use of the MATLAB symbolic toolbox</p>
<pre><code>>> syms x
>> f = x * cos(x);
>> t = linspace(0, 1000, 1000); % Or whatever values you want to evaluate the Laplace Transform over
>> L = double(laplace(f, t)); % Simulataneously compute the Transform and convert it from 'syms' ... | 135 |
Laplace transform | Can I use Fourier transforms instead of Laplace transforms (analyzing RC circuit)? | https://dsp.stackexchange.com/questions/26775/can-i-use-fourier-transforms-instead-of-laplace-transforms-analyzing-rc-circuit | <p>I don't study electrical engineering or something related but I was assigned a problem on transfer functions, impulse responses, and in general, everything related to <a href="https://dsp.stackexchange.com/questions/536/what-is-meant-by-a-systems-impulse-response-and-frequency-response">this post</a>. (Specifically,... | <p>Both transforms have a large overlap in their applications. So you can use both to analyze an RC circuit. However, with the unilateral Laplace transform it's much more straightforward to take initial conditions into account, such as an initially charged capacitor. This has to do with the unilateral Laplace transform... | 136 |
Laplace transform | How is the simplified version of the Bromwich inverse Laplace transform integral derived? | https://dsp.stackexchange.com/questions/41525/how-is-the-simplified-version-of-the-bromwich-inverse-laplace-transform-integral | <p>I do not understand how the last equality is derived from the previous.
Apparently the first term in the integral (involving $\mathrm{cos}$) is equivalent to the second (involving $\mathrm{sin}$)!! How so??</p>
<p>I DO understand how the integral range is halved (since $F(s)^*=F(s^*)$; where $F(s)$ is the Laplace t... | <p>I agree that the derivation is unclear, yet the final result is correct (for $t>0$, see below). There are two conditions that are necessary for the final result to be true:</p>
<ol>
<li>$f(t)$ is real-valued</li>
<li>$f(t)$ is causal</li>
</ol>
<p>The step from line 2 to line 3 in the derivation assumes that $f... | 137 |
Laplace transform | Why are we still using Continuous Time Fourier Transform when we have Laplace Transform? | https://dsp.stackexchange.com/questions/36709/why-are-we-still-using-continuous-time-fourier-transform-when-we-have-laplace-tr | <p>I've read that <strong>Laplace Transform</strong> is more versatile and can cover a broad range of signals compared to <strong>Continuous Time Fourier Transform</strong>. Then why are we still using <strong>Continuous Time Fourier Transform</strong> ?</p>
| 138 | |
Laplace transform | Laplace Transform of Cosine, Poles and Mapping to Frequency Domain | https://dsp.stackexchange.com/questions/37265/laplace-transform-of-cosine-poles-and-mapping-to-frequency-domain | <p>I am trying to understand the connection between Laplace transform ($s$-plane), and frequency domain calculation.</p>
<p>Let's take the Fourier transform of $\cos(\omega_0t)$, which equals to $\pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$. So clearly the frequency domain has only two non-zero values a... | <p>You're comparing the transforms of two different functions. You consider the Fourier transform of the function $x_1(t)=\cos(\omega_0 t)$, but you took the Laplace transform of the function $x_2(t)=\cos(\omega_0t)u(t)$, where $u(t)$ is the unit step function:</p>
<p>$$X_1(j\omega)=\int_{-\infty}^{\infty}x_1(t)e^{-j\... | 139 |
Laplace transform | Why Fourier transform is not sufficient and we have to use Laplace transform? | https://dsp.stackexchange.com/questions/32357/why-fourier-transform-is-not-sufficient-and-we-have-to-use-laplace-transform | <ul>
<li><p>Is there an easy way to explain the motivation behind the use of Laplace transform instead of Fourier transform?</p></li>
<li><p>Isn't that any periodic function can be represented by sines and cosines? - Why to introduce exponential idea? </p></li>
<li>Why not using differential equations with Fourier tran... | <p>The Laplace Transform is more representative of real systems that have a starting point, which is why the integral starts at 0, and also why the unit step function is generally talked about alongside the Laplace Transform. With the Laplace Transform, we can examine the transient and steady-state behavior of a system... | 140 |
Laplace transform | Why do singularities on the imaginary axis affect the Fourier transform differently than the Laplace transform? | https://dsp.stackexchange.com/questions/91712/why-do-singularities-on-the-imaginary-axis-affect-the-fourier-transform-differen | <p>(Please note that I'm aware there are already several questions asking about the difference between the two transforms. However, none of them that I could find touch on this specific issue of the affect of singularities.)</p>
<p>I was reading <a href="https://dsp.stackexchange.com/a/15356/56502">this answer</a> whi... | <p>The Laplace transform of a function <span class="math-container">$f(t)$</span> is defined as:</p>
<p><span class="math-container">$$F(s) = \int_{0^-}^{\infty} e^{-st} f(t) \, dt,$$</span></p>
<p>where <span class="math-container">$s$</span> is a complex variable <span class="math-container">$s = \sigma + j\omega$</s... | 141 |
Laplace transform | Is the Laplace transform a special case of Fourier transform? (Not the other way around) | https://dsp.stackexchange.com/questions/64624/is-the-laplace-transform-a-special-case-of-fourier-transform-not-the-other-way | <p>Always had a thought about why Laplace transform reveals the transient properties of the system?
My doubt is based on the following fact,
Fourier transform is given as </p>
<p><span class="math-container">\begin{equation}
\mathscr{F}\left\lbrace f(t)\right\rbrace = \int_{-\infty}^\infty f(t) e^{ -j \omega t} dt
\e... | <p>The Fourier Transform is the Laplace Transform with the complex variable s restricted to be the imaginary axis on the s plane. For this reason the Fourier Transform only exists when the imaginary axis is within the region of convergence. The variable s is called a "complex frequency" as it is the frequency variable ... | 142 |
Laplace transform | Laplace Transform of $f(t+a), a>0$ where $f(t)$ is not periodic | https://dsp.stackexchange.com/questions/41211/laplace-transform-of-fta-a0-where-ft-is-not-periodic | <p>For $a > 0$, is there any known representation of the Laplace transform of $f(t+a)$ in terms of the Laplace Transform of $f(t) $</p>
<p>Note: In my application, $f(t)$ is not a periodic function and the functional form of $f(t)$ is not actually known a-priori, because I have to couple it to another set of equati... | <p>Let $s = \sigma + j\omega$, the inverse Laplace transform of $f(t+a)$ is given by
$$f(t+a) = \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} F(s)e^{s(t+a)} \mathrm{d}s = \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} F(s)e^{sa}e^{st} \mathrm{d}s.$$</p>
<p>Hence the <strong>bilateral</strong> Laplac... | 143 |
Laplace transform | Linear linearly time varying systems Laplace transform | https://dsp.stackexchange.com/questions/87456/linear-linearly-time-varying-systems-laplace-transform | <p>Suppose that for a system <span class="math-container">$S$</span> if we have <span class="math-container">$t_{2} = t_{1}+t_{0}\rightarrow h(t,t_{2}) =h(t,t_{1})+h(t,t_{0}) $</span> .Then if we take the double Laplace transform to<span class="math-container">$t,t_{2}$</span> we will get:</p>
<p><span class="math-cont... | 144 | |
Laplace transform | How can we prove the correctness of the integration property of the Laplace transform? | https://dsp.stackexchange.com/questions/72903/how-can-we-prove-the-correctness-of-the-integration-property-of-the-laplace-tran | <p>I was going through an Electrical Engineering textbook for understanding the Laplace transform and came across the following proof for one of the properties of the Unilateral Laplace transform.</p>
<p>Integration property of the unilateral Laplace transform:
<a href="https://i.sstatic.net/sjRS6.png" rel="nofollow no... | <p>You are right that the argument in the proof is not correct, or at least misleading. The fact that <span class="math-container">$e^{-st}$</span> becomes zero as <span class="math-container">$t\to\infty$</span> is true for any <span class="math-container">$s$</span> with <span class="math-container">$\textrm{Re}\{s\}... | 145 |
Laplace transform | Laplace transform of a finite duration signal | https://dsp.stackexchange.com/questions/59694/laplace-transform-of-a-finite-duration-signal | <p>Consider the following signal:
<span class="math-container">$$ x(t) = e^{-2t}[u(t) - u(t-5)] $$</span></p>
<p>This signal exists only from 0 to 5 time units. Elsewhere, it is zero.</p>
<p>Now, let's find the laplace transform of this signal using Linearity and Time shift properties.</p>
<p><span class="math-conta... | <p>The property claimed by Schaum and Oppenheim is also true for the given example. Note that the Laplace transform</p>
<p><span class="math-container">$$X(s)=\frac{1-e^{-5(s+2)}}{s+2}\tag{1}$$</span></p>
<p>has <em>no</em> pole at <span class="math-container">$s=-2$</span>:</p>
<p><span class="math-container">$$\li... | 146 |
Laplace transform | Laplace transform of a time domain sampled data MATLAB | https://dsp.stackexchange.com/questions/45287/laplace-transform-of-a-time-domain-sampled-data-matlab | <p>I have two sets of one second voltage data sampled with 4000Hz and I can plot all the voltage points vs time points in MATLAB. So it means I have a data matrix with with length of 4000 one column for the time in seconds the other for the voltage. </p>
<p>Now I have simultaneously sampled two data matrix in time dom... | <p>Since you are sampling a real signal, you can just use the a Fourier transform, the fft function in MATLAB. You can divide the FFT of the output by the input, and then fit a curve to the result to approximate the transfer function</p>
| 147 |
Laplace transform | The Laplace transform - Steven W. Smith Book - impulse response cancellation method | https://dsp.stackexchange.com/questions/80628/the-laplace-transform-steven-w-smith-book-impulse-response-cancellation-met | <p>While studying the Laplace transform using <a href="http://www.dspguide.com/pdfbook.htm" rel="nofollow noreferrer">Steven W. Smith Book</a> I found something uncomprehending. In the 32th chapter - The Laplace Transform, page 590, last paragraph describes the cancelling phenomena when an impulse response is cancelle... | <p>Peter's comment is correct, it's about the integral of the product <span class="math-container">$p(t)h(t)$</span>:</p>
<p><span class="math-container">$$I=\int_{-\infty}^{\infty}p(t)h(t)dt\tag{1}$$</span></p>
<p>The impulse response <span class="math-container">$h(t)$</span> has the following form:</p>
<p><span clas... | 148 |
Laplace transform | Is it possible to take Fractional Fourier transform of Laplace transform? | https://dsp.stackexchange.com/questions/95963/is-it-possible-to-take-fractional-fourier-transform-of-laplace-transform | <p>Let <span class="math-container">$L_t\{f(x, t)\}$</span> denotes the Laplace transform (two-sided) of <span class="math-container">$f(x,t)$</span> with respect to <span class="math-container">$t$</span>. That is,</p>
<p><span class="math-container">$L_t\{f(x, t)\}(s)=\int_{-∞}^{+∞}f(x, t) e^{-st} dt$</span></p>
<p>a... | 149 | |
Laplace transform | Connection from Fourier to Laplace Transform | https://dsp.stackexchange.com/questions/78924/connection-from-fourier-to-laplace-transform | <p>I have a basic understanding of Laplace and Fourier but having trouble making a connection. Every time I attempt to look at reasons these are connected I'm told about the s-plane and regions of convergence which confuse me quite a lot trying to wrap my head around. What I would like to know is whether my understandi... | <p>Yes, the OP’s intuition for Fourier seems correct and more specifically I call attention to the basic "correlation" structure and add more meaning to what <span class="math-container">$e^{j\omega t}$</span> is as the true "single frequency component" in signal processing (as opposed to using a si... | 150 |
Laplace transform | Relation between Laplace and Fourier transforms | https://dsp.stackexchange.com/questions/28100/relation-between-laplace-and-fourier-transforms | <p>I know that <span class="math-container">$$X_L(s) \Big|_{s=j\omega}=X_F(\omega)$$</span> if <span class="math-container">$x(t)$</span> is one sided and absolutely integrable and hence the imaginary axis of the Laplace transform is the Fourier transform.</p>
<p>But Fourier transform also has imaginary and real parts... | <p>The Laplace transform evaluated at $s=j\omega$ is equal to the Fourier transform if its region of convergence (ROC) contains the imaginary axis. This is also true for the bilateral (two-sided) Laplace transform, so the function need not be one-sided.</p>
<p>As for real and imaginary parts, since $s$ is a <em>comple... | 151 |
Laplace transform | Laplace transform of product of signal and impulse train | https://dsp.stackexchange.com/questions/40433/laplace-transform-of-product-of-signal-and-impulse-train | <p>I'm reading 'Discrete Time Control Systems' book by Ogata and came across a few statements (specifically, (3-1) and (3-2)) which I have not been able to understand.</p>
<p>It is said that any continuous signal can be sampled and the output represented as
$$y(t) = \sum_{n=- \infty}^{+\infty}x(nT)\delta(t-nT) $$ </p... | <p>since no one else seems to have said it, if the ideally-sampled $x(t)$ is defined as</p>
<p>$$x_\text{s}(t) \triangleq \sum_{n=-\infty}^{+\infty}x(nT)\delta(t-nT) $$</p>
<p>and we define discrete-time samples as $x[n] \triangleq x(nT)$, the Laplace transform of</p>
<p>$$\begin{align}
X_\text{s}(s) &= \sum_{n=... | 152 |
Laplace transform | Why is the ROC of Laplace transform independent of imaginary part of s? | https://dsp.stackexchange.com/questions/56793/why-is-the-roc-of-laplace-transform-independent-of-imaginary-part-of-s | <p>An integral is defined as converging if it yields a finite value upon application of limits of integration. It is divergent otherwise.</p>
<p>Now sticking to the mathematical notation of Laplace transform, we have for a causal function <span class="math-container">$x(t) = u(t)$</span>:
<span class="math-container">... | <p>I think your misunderstanding is that you manipulate an expression, which is only valid for <span class="math-container">$\text{Re}\{s\}>0$</span>, in order to show for which values of <span class="math-container">$s$</span> it might be valid.</p>
<p>Note that the integral</p>
<p><span class="math-container">$$... | 153 |
Laplace transform | Impulse response of a causal LTI system without using Laplace transform | https://dsp.stackexchange.com/questions/93642/impulse-response-of-a-causal-lti-system-without-using-laplace-transform | <p>I have this differential equation that models a causal LTI system:
<span class="math-container">$$
\ddot{v}(t) - \dot{v}(t) - 2v(t) = \ddot{u}(t) + 2\dot{u}(t) + u(t)
$$</span></p>
<p>I was asked to find the impulse response both by using Laplace transform and by solving the ODE.</p>
<p>The first method is quite sim... | <p>The problem with your first approach is that you assume</p>
<p><span class="math-container">$$f(t)\delta'(t)\stackrel{?}{=}f(0)\delta'(t)\tag{1}$$</span></p>
<p>which is wrong.</p>
<p>The correct equation is</p>
<p><span class="math-container">$$f(t)\delta'(t)=f(0)\delta'(t)-f'(0)\delta(t)\tag{2}$$</span></p>
<p>Equ... | 154 |
Laplace transform | What is relationship between the Laplace transform of the ideally-sampled signal and that of the original continuous signal? | https://dsp.stackexchange.com/questions/95098/what-is-relationship-between-the-laplace-transform-of-the-ideally-sampled-signal | <p>Suppose a continuous signal <span class="math-container">$x(t)$</span>, the Laplace transform of <span class="math-container">$x(t)$</span> is <span class="math-container">$X(s)$</span>. Suppose the ideally-sampled signal of <span class="math-container">$x(t)$</span> is <span class="math-container">$\hat{x}(t)=\sum\... | <p>Yes it follows the same because Laplace is nothing but making your non convergence or unstable signal to stable by ROC(Sigma) and taking Fourier transform of it. So it follows the same.</p>
| 155 |
Laplace transform | Can a Fourier Transform exist even if the j$\omega$ axis is not in the Region of Convergence in it's Laplace Transform | https://dsp.stackexchange.com/questions/53875/can-a-fourier-transform-exist-even-if-the-j-omega-axis-is-not-in-the-region-of | <p>A couple of confusions have been occurred. The Signal I'm considering is <strong>f(t) = sin(t)*u(t)</strong></p>
<ol>
<li><p>Fourier Transform of it can be derived.
<em><span class="math-container">$-i \pi (\delta (\omega -1)-\delta (\omega +1))$</span></em></p></li>
<li><p>According to my mathematica code, the RO... | <p>You're right that the Laplace transform is <em>not</em> more general than the Fourier transform. They are just different. There are several (theoretically) important functions for which the Laplace transform doesn't exist, but the Fourier transform does. A few examples are</p>
<ol>
<li><span class="math-container">... | 156 |
Laplace transform | Determining Stability of a continuous time system using Laplace Transform | https://dsp.stackexchange.com/questions/60662/determining-stability-of-a-continuous-time-system-using-laplace-transform | <p>I'm following Oppenheim's book. In exapmles, the laplace transforms of of the following signals </p>
<p><span class="math-container">$e^{-t}u(t)$</span> and <span class="math-container">$e^{-t-1}u(t+1)$</span> </p>
<p>is given as <span class="math-container">$\frac{s}{(s+1)}$</span> and <span class="math-container... | <p>ROC is a strip of plane which does not contain any poles. The region is determined by the location of poles. </p>
<p>The first signal, <span class="math-container">$x(t) = e^{-t} u(t)$</span> , has the Laplace transform <span class="math-container">$X(s) = \frac{1}{s+1}$</span> and its ROC is <span class="math-cont... | 157 |
Laplace transform | LTI system with Laplace transform | https://dsp.stackexchange.com/questions/42004/lti-system-with-laplace-transform | <p>Given the input $$x(t)=u(t)$$ and the corresponding output signal measured as $$y(t)= 2 e^{-3t} u(t)$$ determine the impulse response $h(t)$.</p>
<p>This what have done so far:
$$ h(t)= \mathscr{L}^{-1} \left\{ \frac{Y(s)}{X(s)} \right\} = \frac{2/(s+3)}{1/s}
= \frac{2s}{s+3} $$. </p>
<p>I need to find the Laplace... | <p>Your approach is correct. Rewrite $H(s)$ as</p>
<p>$$H(s)=\frac{2s}{s+3}=2\frac{s+3-3}{s+3}=2-\frac{6}{s+3}\tag{1}$$</p>
<p>and use basic Laplace transform identities to obtain $h(t)$ from $(1)$.</p>
<p>Note that you don't need to use the Laplace transform. A time domain approach as suggested in <a href="https://... | 158 |
Laplace transform | Can use of Fourier transform be minimized completely with the help of Laplace and Z transform? | https://dsp.stackexchange.com/questions/31415/can-use-of-fourier-transform-be-minimized-completely-with-the-help-of-laplace-an | <p>Fourier transform has different types like continuous Fourier transform (CFT), Discrete time Fourier transform (DTFT) and Discrete Fourier transform ( DFT).</p>
<p>CFT can be used in case of continuous aperiodic signals while DFT for discrete aperiodic signals . </p>
<p>On the other hand, Laplace transform can be... | <p>The answer to your last question is definitely 'no'. The point hotpaw2 makes in <a href="https://dsp.stackexchange.com/a/31416/4298">his answer</a> is very relevant: the FFT is an efficient implementation of the DFT, and there are no equivalently efficient implementations for the numerical computation of the $\mathc... | 159 |
Laplace transform | Inverse Laplace Transform | https://dsp.stackexchange.com/questions/60664/inverse-laplace-transform | <p>A system given by <span class="math-container">$\frac{s-1}{(s+1)(s-2)}$</span> has to be inverse transformed so that it is anticausal and nonstable. The answer given is <span class="math-container">$h(t)=-\frac{1}{3}(2e^{-t}+e^{2t})u(-t)$</span></p>
<p>Why the minus sign at the beginning?</p>
| <p>First you have to remember a Laplace transform property:</p>
<p><span class="math-container">$$ e^{a t} u(t) \longleftrightarrow \frac{1}{s-a} ~~~,~~~ \mathcal{Re}\{s\} > \mathcal{Re}\{a\} \tag{1} $$</span></p>
<p><span class="math-container">$$ -e^{a t} u(-t) \longleftrightarrow \frac{1}{s-a} ~~~,~~~ \mathca... | 160 |
Laplace transform | Meaning and unit of frequency in Laplace (Fourier) transform | https://dsp.stackexchange.com/questions/43733/meaning-and-unit-of-frequency-in-laplace-fourier-transform | <p>Imagine transfer function obtained by Laplace transform, for example:</p>
<p>$G(s) = \dfrac{1}{s+1}$</p>
<p>Now, I would like to do some frequency analysis, so I replace the $s$ with $\omega i$ (let's consider this operation valid for this example).</p>
<p>What is the unit of the $\omega$? So far what I have seen... | <p>If you are dealing with the Laplace transform $G(s)$ of a <strong>time</strong> domain signal $g(t)$ and its evaluation on the imaginary axis to get the Fourier transform $G(j\omega)$ (assuming it exists) then the unit of your frequency $\omega$ is <strong>radians per second</strong> assuming the unit of the time wa... | 161 |
Laplace transform | Why can you use the one-sided laplace transform to solve differential equation describing a causal LTI-system? | https://dsp.stackexchange.com/questions/71810/why-can-you-use-the-one-sided-laplace-transform-to-solve-differential-equation-d | <p>In an example, an equation describing a causal LTI-system is</p>
<p><span class="math-container">$$
(D^2 + 5D + 6) y(t) = (D+1) x(t)
$$</span></p>
<p>where <span class="math-container">$y(t) = y_{zs}(t) + y_{zi}(t)$</span> and the initial conditions are <span class="math-container">$y(0^-) = 2, \dot{y}(0^-) = 1$</sp... | <blockquote>
<p>If it is not equal to 0 for t<0 how can we know we can take the one-sided laplace transform of the LHS?</p>
</blockquote>
<p>Unfortunately, the way the problem is framed it is very difficult to do in a rigorous manner. You <em>can</em>, by remembering that <span class="math-container">$0^-$</span> i... | 162 |
Laplace transform | How can I plot a 3D graph of a given Laplace Transform of a function? | https://dsp.stackexchange.com/questions/40628/how-can-i-plot-a-3d-graph-of-a-given-laplace-transform-of-a-function | <p>Let's say I have a function called $f(t)$ in time domain as: </p>
<p>$$f(t) = \exp(-3t)\cos(5t)$$</p>
<p>And the Laplace transform of this function call it $F(s)$ becomes:</p>
<p>$$F(s)=\frac{(s + 3)}{(s + 3)^2 + 25}$$</p>
<p>I want to plot the 3D plot of $\lvert F(s)\rvert$ as a surface above the $s$-plane.</p>... | <pre><code>[X,Y] = meshgrid(-10:.1:10);
s=X+j*Y;
Z= abs((s+3)./((s+3).^2+25));
mesh(X,Y,Z)
</code></pre>
<p><a href="https://i.sstatic.net/q999y.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/q999y.png" alt="3d plot"></a></p>
| 163 |
Laplace transform | Laplace transform of $f\left(\frac{t - b}{a}\right)$ | https://dsp.stackexchange.com/questions/38991/laplace-transform-of-f-left-fract-ba-right | <p>Consider the function $f\left(\frac{t - b}{a}\right)$. We want want to calculate its Laplace transform. There are two approaches:</p>
<ul>
<li><p>Firstly, </p>
<ol>
<li>let $g(t) = f\left(\frac ta\right)$. </li>
<li>Then $\mathcal{L}\left\{f\left(\frac{t-b}{a}\right)\right\} = \mathcal{L}\left\{g(t - b)\right\} = ... | <p>Your two alternate derivations of the Laplace transform (specifically the bilateral I'm referring to) of the signal $f((t - b)/a)$ seems right, both resulting in, the same , as a check.</p>
<p>$$
\mathcal{L}\{f((t - b)/a)\} = |a|e^{-bs}F(as)
$$</p>
<p>However, when you assume the signal $f(t) = e^t u(t)$ whose Lap... | 164 |
Laplace transform | When to use Fourier, Laplace and Z transforms? | https://dsp.stackexchange.com/questions/64539/when-to-use-fourier-laplace-and-z-transforms | <p>If we have an LTI system, with an input signal <span class="math-container">$x(t)$</span>, impulse response <span class="math-container">$h(t)$</span> and output <span class="math-container">$y(t)$</span>, I was under the assumption that if the input and impulse response were continuous in time, then you would use t... | <p>It's natural consequence of applying a transform to a convolution relation. The output <span class="math-container">$y(t)$</span> of an (continuous-time) LTI system is described by a convolution integral :</p>
<p><span class="math-container">$$y(t) = h(t)\ast x(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau $$... | 165 |
Laplace transform | Why is a negative exponent present in Fourier and Laplace transform? | https://dsp.stackexchange.com/questions/19004/why-is-a-negative-exponent-present-in-fourier-and-laplace-transform | <p>could anyone explain why there is a need of negative exponent in fourier and laplace transform.I looked through the web but I couldn't get anything.Does anything happen if a positive exponent is placed in these transforms.</p>
<p>While looking through <a href="http://1drv.ms/1tbV45S" rel="noreferrer">http://1drv.ms... | <p>Matt is correct that the sign is convention. I think that there is a reason for it beyond that though.</p>
<p>If we look at complex frequencies in the complex plane, they look like a constant vectors that rotate in one direction or another. Positive frequencies rotate counter-clockwise, negative frequencies rotat... | 166 |
Laplace transform | Why does subbing $s = j\omega$ into the Laplace transform of a cosine wave yield a purely imaginary result? | https://dsp.stackexchange.com/questions/58364/why-does-subbing-s-j-omega-into-the-laplace-transform-of-a-cosine-wave-yield | <p>The Laplace transform of a cosine starting at <span class="math-container">$t=0$</span> is given by</p>
<p><span class="math-container">$$F(s) = \frac{s}{s^2 + \omega_0^2}$$</span></p>
<p>If I sub in <span class="math-container">$s = j\omega$</span>, I get the Fourier transform of a cosine starting at <span class=... | <blockquote>
<p>If I sub in <span class="math-container">$s=j\omega$</span>, I get the Fourier transform of a cosine wave starting at <span class="math-container">$t=0$</span>.</p>
</blockquote>
<p>No, you don't. You can't just set <span class="math-container">$s=j\omega$</span> in an expression for the Laplace tran... | 167 |
Laplace transform | Name of property of Laplace transform | https://dsp.stackexchange.com/questions/84473/name-of-property-of-laplace-transform | <p><span class="math-container">\begin{align}
L[e^{-at}u(t)] &= \frac{1}{s+a}\\
L[\cos(\omega_{o}t)u(t)] &= \frac{s}{s^{2}+\omega^{2}_{o}}\\
L[e^{-at}\cos(\omega_{o}t)u(t)] &= \frac{s+a}{(s+a)^{2}+\omega_{o}^2}
\end{align}</span>
Everywhere <span class="math-container">$e^{-at}$</span> is multiplied with a ... | <p>That's the <a href="https://www.tutorialspoint.com/time-scaling-and-frequency-shifting-properties-of-laplace-transform" rel="nofollow noreferrer">Frequency Shifting Property</a></p>
| 168 |
Laplace transform | How to calculate the steady state response $y_{ss}(t)$ of a LTI system given the Laplace transform $Y(s)$? | https://dsp.stackexchange.com/questions/36020/how-to-calculate-the-steady-state-response-y-sst-of-a-lti-system-given-the | <p>I am given the Laplace transform of the output of a LTI system: $$Y(s) = \frac{1}{s((s+2)^2+1)}$$ Asked is what the steady state response $y_{ss}(t)$ would be. I think that $y_{ss}(t) = \lim_{t\to\infty} y(t)$, since after waiting infinit long, the system should be in steady state. (Right?)</p>
<p>I thought to use ... | <p>For these calculations, it is better to give the <em>Wolfram Alpha</em> answers:</p>
<p><a href="http://www.wolframalpha.com/input/?source=frontpage-immediate-access&i=inverse%20Laplace%20transform%201%2F(s*(s%5E2%20%2B%204*s%20%2B%205))" rel="nofollow noreferrer">inverse Laplace transform 1/(s*(s^2 + 4*s + 5))... | 169 |
Laplace transform | Finding and displaying Laplace or Z transform ROC(region of convergence) using MATLAB | https://dsp.stackexchange.com/questions/83285/finding-and-displaying-laplace-or-z-transform-rocregion-of-convergence-using-m | <p>Is there any way, we can use MATLAB for finding and displaying Laplace or Z transform Region of convergence?</p>
| <p>Matlab can only compute expressions for the uni-lateral (one-sided) versions of the Laplace transform and Z-transform. It doesn't explicitly determine the ROCs, but since both transforms are uni-lateral, there's only one possible choice for the ROCs: let <span class="math-container">$p_k$</span> be the poles of the ... | 170 |
Laplace transform | Wavelet transform in control systems | https://dsp.stackexchange.com/questions/18053/wavelet-transform-in-control-systems | <p>In control systems, the Laplace transform is often used to analyze the stability and the performance of <a href="http://en.wikipedia.org/wiki/LTI_system_theory" rel="nofollow">LTI system</a>. For instance, the LTI system is stable if and only if the <a href="http://en.wikipedia.org/wiki/Transfer_function" rel="nofol... | <p>In the paper <a href="http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6811177" rel="nofollow">Haar-Based Stability Analysis of LPV Systems</a>, the Haar wavelet transform theory have been used to design linear matrix inequalities (LMIs) to analyze the stability of <a href="https://en.wikipedia.org/wiki/Linear_p... | 171 |
Laplace transform | Transfer function and Laplace domain | https://dsp.stackexchange.com/questions/84485/transfer-function-and-laplace-domain | <p>If we give a input <span class="math-container">$x(t)=u(t)$</span> to a system <span class="math-container">$\mathcal{S}$</span> we get an output <span class="math-container">$y(t) = e^{-t} u(t)$</span>.<br />
After we Laplace-transform both the input and the output we get the transfer function
<span class="math-con... | <p>First of all it's important to understand that this is all about <em>linear and time-invariant (LTI)</em> systems. Otherwise, you can't generally use a transfer function to characterize a system. So if you have some input <span class="math-container">$x(t)$</span> and some corresponding system output <span class="ma... | 172 |
Laplace transform | Step response of a given input and output (Laplace or Fourier) | https://dsp.stackexchange.com/questions/81291/step-response-of-a-given-input-and-output-laplace-or-fourier | <p>I am trying to calculate the step response of the following given:
Should I use Laplace transform or Fourier transform?
<a href="https://i.sstatic.net/36JZu.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/36JZu.jpg" alt="enter image description here" /></a></p>
| 173 | |
Laplace transform | Laplace transform of this simple parallel RLC circuit? (For audio speaker simulation ...) | https://dsp.stackexchange.com/questions/91516/laplace-transform-of-this-simple-parallel-rlc-circuit-for-audio-speaker-simula | <h2>SPEAKER AS RLC CIRCUIT</h2>
<p><a href="https://circuitdigest.com/electronic-circuits/simulate-speaker-with-equivalent-rlc-circuit" rel="nofollow noreferrer">I read this article here</a> which demonstrates a simulation of a speaker as a simple RLC circuit where the RLC components are in parallel:</p>
<p><a href="ht... | <p>Do what r b-j suggests: get a Laplace domain equivalent first, and then transform from <span class="math-container">$s$</span> to <span class="math-container">$z$</span>.</p>
<p>The <a href="https://circuit-analysis.github.io/chapter-11.html" rel="nofollow noreferrer">equivalent circuit</a> will be something like:</... | 174 |
Laplace transform | How to transform a Fractional Order Laplace Transfer Function into a digital filter? | https://dsp.stackexchange.com/questions/45918/how-to-transform-a-fractional-order-laplace-transfer-function-into-a-digital-fil | <p>I'm working with loudspeaker impedance analysis. Electrical behavior of loudspeakers can be modeled with RLC networks. But real loudspeakers have components, that exhibit some non-linear and frequency dependent behaviors, that make them difficult to model with simple LTI systems.</p>
<p>One of the problems with lou... | 175 | |
Laplace transform | Question about z transform | https://dsp.stackexchange.com/questions/27385/question-about-z-transform | <p>After studying z transform from different books and literature on internet I want to ask few which makes me confuse. </p>
<p>a) From the Discrete Time Fourier Transform we have drive equation for z transform. $$ X(z)= \sum _ {n=-\infty}^{+\infty} x[n]z^{-n}$$ where $z$ is represented in polar form $z=re^{j\omega}$
... | <p>I think it is common (in signal processing books) to write the z transform in polar form, to make clear its relationship with the fourier transform, that is
z-transform equal to fourier transform on the unit circle, that is when r=1, then:</p>
<p>Ztransf-> $z=r*e^{jw}=e^{jw}$<- fourier transform or just</p>
<... | 176 |
Laplace transform | Laplace of step and integration are same? | https://dsp.stackexchange.com/questions/42723/laplace-of-step-and-integration-are-same | <p>Why do we have Laplace transform of a step function and integrator is same.</p>
<p>\begin{align}
\mathcal L\left[u(t)\right] &= \frac 1s\\
\mathcal L \left[ \int dt\right] &= \frac 1s
\end{align}</p>
<p>Please clear my doubt on this.</p>
| <p>This is because the impulse response of an integrator is $h(t)=u(t)$. The output which is the convolution with the impulse respoponse is
$$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$$
and with $h(t)=u(t)$ it becomes
$$\begin{align}
y(t)&=\int_{-\infty}^{\infty}x(\tau)u(t-\tau)d\tau\\
&=\int_{-\infty}^... | 177 |
Laplace transform | ROC of the function in the problem 9.14 of Oppenheim's Signals and Systems textbook | https://dsp.stackexchange.com/questions/63377/roc-of-the-function-in-the-problem-9-14-of-oppenheims-signals-and-systems-textb | <blockquote>
<p><a href="https://i.sstatic.net/7paLk.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/7paLk.jpg" alt="Problem 9.14"></a></p>
</blockquote>
<hr>
<p>I have solved the problem 9.14 in Oppenheim's Signals and Systems textbook, but my solution and the one in Slader is different. Problem is ... | <p>You're right, the ROC of the Laplace transform of a two-sided signal is a strip in the complex plane. In your case, the imaginary axis is inside the ROC, and the ROC is limited by the poles in the right and left half-planes. If the ROC were the right half-plane, the signal would be right-sided, which is clearly not ... | 178 |
Laplace transform | From where this Laplace transform for tracking error came? | https://dsp.stackexchange.com/questions/44939/from-where-this-laplace-transform-for-tracking-error-came | <blockquote>
<p>System for estimating the tracking error in an A-D converter.
<a href="https://i.sstatic.net/9Evhk.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9Evhk.png" alt="enter image description here"></a></p>
<p>Let $e_i(t)$ be ramp function with slope $e_i'$ and assume that the samplin... | <p>You correctly noted
$$
\frac{E_o(s)}{E_i(s)} = \frac{1/sC}{R+1/sC}.
$$</p>
<p>But it also says $e_i(t)$ is a ramp with slope $e_i'$, so its Laplace transform is given by $E_i(s) = \frac{e_i'}{s^2}$.</p>
<p>Now just do some algebra:</p>
<p>$$
\frac{E_i(s)-E_o(s)}{E_i(s)} = 1-\frac{E_o(s)}{E_i(s)} = 1-\frac{1/sC}{R... | 179 |
Laplace transform | Why Z-transform is considered as separate transform? | https://dsp.stackexchange.com/questions/24099/why-z-transform-is-considered-as-separate-transform | <p>The mathematical formula of the Laplace and Z transforms are same with just one difference. I.e. in the first we use $t$ for continuous-time signal and in the latter uses $n$ for discrete-time signal. I don't think that there are any other differences. </p>
<p>While discussing Fourier transform for continuous-time... | <p>There is indeed a transform called <em>discrete Laplace transform</em> and it is of course closely related to the $\mathcal{Z}$-transform. The (unilateral) discrete Laplace transform of a sequence $f_n$ is defined by (cf. <a href="https://books.google.nl/books?id=wCIGCAAAQBAJ&lpg=PA78&ots=FDqF81ObPp&dq=%... | 180 |
Laplace transform | Can we tell if a system is linear and time-invariant from its frequency response? | https://dsp.stackexchange.com/questions/78176/can-we-tell-if-a-system-is-linear-and-time-invariant-from-its-frequency-response | <p>Given a system with a known frequency response in the S-domain. Is there a way to find whether the system is linear and time invariant?</p>
<p>My current understanding is that we need to take the inverse Laplace transform of the system and prove linearity in the time domain.</p>
<p>Edit:<br />
As per the comments, g... | <blockquote>
<p>Given a system with a known frequency response in the S-domain. Is there a way to find whether the system is linear and time invariant?</p>
</blockquote>
<p>If by "known frequency response in the s-domain" you mean a Laplace transfer function* as a ratio of polynomials in s -- yes. Laplace tr... | 181 |
Laplace transform | Bilateral $\mathcal Z$-transform of exponential | https://dsp.stackexchange.com/questions/25489/bilateral-mathcal-z-transform-of-exponential | <p>We all know that $a^nu(n)$ has unilateral $\mathcal Z$-transform. But what is the $\mathcal Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'.</p>
<p>But bilateral Laplace transform of $e^t$ doesn't exist. Both are exponentials in discrete and continuous domain respectively. Conside... | <p>In complete analogy with the bilateral Laplace transform of $x(t)=e^{-at}$ (which doesn't exist), the bilateral $\mathcal{Z}$-transform of $a^n$ doesn't exist either. The series</p>
<p>$$\sum_{n=-\infty}^{\infty}a^nz^{-n}$$</p>
<p>converges nowhere, simply because $a^n$ grows without bounds for $n\rightarrow -\inf... | 182 |
Laplace transform | Laplace domain transfer function from system sampled at discrete times | https://dsp.stackexchange.com/questions/88214/laplace-domain-transfer-function-from-system-sampled-at-discrete-times | <p>I'm trying to understand an analysis of a sampled continuous time system in the Laplace domain. The source analysis is <a href="http://bwrcs.eecs.berkeley.edu/Classes/icdesign/ee240_sp10/lectures/Lecture22_Offset_Cancel_2up.pdf" rel="nofollow noreferrer">here</a> (PDF page 6, slide marked 11); I'll explain further b... | <p>There are two ways of analyzing the given system. First, we could simply ignore the sampling process and treat the system as a continuous-time system. This is possible if the input signals are sufficiently band-limited, and if the sampling rate satisfies the Nyquist criterion. In this case we simply have the differe... | 183 |
Laplace transform | Discrete version of this transform? | https://dsp.stackexchange.com/questions/86922/discrete-version-of-this-transform | <p>I have the following transform for <span class="math-container">$t>0, a_i>0$</span>
<span class="math-container">$$f(t)=\sum_{i=0}^d a_i \exp(-t a_i)$$</span></p>
<p>And I need to invert it for a set of target values <span class="math-container">$b$</span>:</p>
<p>Find <span class="math-container">$(t_0,t_1,\l... | 184 | |
Laplace transform | Why do these 2 methods give different solutions? | https://dsp.stackexchange.com/questions/40087/why-do-these-2-methods-give-different-solutions | <p><a href="https://i.sstatic.net/bjZR8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/bjZR8.png" alt="enter image description here"></a></p>
<p>I need to solve what is underlined in red for $x_i$, nut currently I'm interested in the right side of the equation only.</p>
<p>On the left I sarted by doin... | <p>The problem is that you took the derivative of the function</p>
<p>$$\hat{x}_u(t)=2e^{-3t}-e^{-4t}\tag{1}$$</p>
<p>whereas using the Laplace transform you implicitly assumed that $x_u(t)$ equals zero for $t<0$:</p>
<p>$$x_u(t)=\hat{x}_u(t)u(t)=(2e^{-3t}-e^{-4t})u(t)\tag{2}$$</p>
<p>where $u(t)$ is the unit st... | 185 |
Laplace transform | S domain vs frequency domain? | https://dsp.stackexchange.com/questions/84250/s-domain-vs-frequency-domain | <p>Laplace domain is also known as <em>"s domain"</em>.</p>
<p>Is there any difference between <em>"s domain"</em> and <em>"frequency domain"</em>? Can we use both terms interchangeably?</p>
<p>If we want to convert a time domain signal to frequency domain, can we use Laplace transform?</p... | <p>The s domain is synonymous with the "complex frequency domain", where time domain functions are transformed into a complex surface (over the s-plane where it converges, the "Region of Convergence") showing the decomposition of the time domain function into decaying and growing exponentials of the... | 186 |
Laplace transform | Is there an analogy of the Fourier-decomposition in the Laplace space to decompose a signal to a few components? | https://dsp.stackexchange.com/questions/56644/is-there-an-analogy-of-the-fourier-decomposition-in-the-laplace-space-to-decompo | <p>I have a signal from which I know, that it is the sum of a few, exponentially decaying components. I want to find these components.</p>
<p>If it would be a sum of some sinusiod waves, it would be easy to Fourier-transform it, and then find the <a href="https://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow nor... | 187 | |
Laplace transform | A question about the meaning of pole in time domain | https://dsp.stackexchange.com/questions/40629/a-question-about-the-meaning-of-pole-in-time-domain | <p>Lets say I have a transfer function $H(s)$ of a system defined in $s$-domain as:
$$H(s) = \frac{1}{s - (-1-j)}$$</p>
<p>So I conclude that the pole on the $s$-plane is where $s = 1+j$. So far so good.</p>
<ul>
<li><p>Now does that mean if the Laplace transform of the input to the system is $s = 1+j$ the system goe... | <p>Let $H(s)$ be a transfer function of the form
$$H(s) = \frac{1}{s-p}$$
where $p$, which is a pole of $H(s)$, can be written as a complex number $a+jb$. Taking the inverse Laplace transform of $H(s)$ gives the corresponding impulse response $h(t)$ (that is, the output of your system when given $\delta(t)$ as input). ... | 188 |
Laplace transform | LTI system response to periodic input | https://dsp.stackexchange.com/questions/30712/lti-system-response-to-periodic-input | <p>I'm trying to find the zero-state response (ZSR) of an LTI system to a one sided periodic input, like a square wave that is equals to zero for $t < 0$.</p>
<p>I know that I can use the Fourier series of said input function to find the steady-state (SS) response, however I'm having trouble understanding how to us... | <p>You can use the Laplace transform, but can also simply use convolution in the time domain. In any case, you will need the system's impulse response $h(t)$. Let the input signal $x(t)$ satisfy $x(t)=0$ for $t<0$, and $x(t+T)=x(t)$ for $t>0$ and $T>0$, as required. Furthermore, let $f(t)$ be the first period ... | 189 |
Laplace transform | DFT/FFT Transfer function | https://dsp.stackexchange.com/questions/27963/dft-fft-transfer-function | <p>I want play and record a sine sweep.
When i have both signals the recorded one and the send one i can create a Transferfunction.
That is what i know so far.</p>
<p>$$
H_0 = \frac{OUT}{IN} = \frac{Y}{X}
$$</p>
<p>Where i'm stuck is that when i read about the Transfer function it is all about the $Laplace \, transf... | <p>Your question is fairly broad, let me answer it step by step. </p>
<p>First of all $H(s)$ is indeed called the transfer function, and is the laplace transform of the impulse response. It's useful for finding the poles and zeroes, which is what the fourier transform can't do alone. $H(\omega)$ is the frequency respo... | 190 |
Laplace transform | Why is z (and not ω) the variable of interest for discrete time systems? | https://dsp.stackexchange.com/questions/75831/why-is-z-and-not-%cf%89-the-variable-of-interest-for-discrete-time-systems | <p>A continuous time domain system is well described by the Laplace transform. It allows to express any continuous signal x(t) as the integral sum of weighted complex and exponentially growing/decaying sine waves <span class="math-container">$e^{st} = e^{\sigma t} \cdot e^{j\omega t}$</span>:</p>
<p><a href="https://i.... | <p>The z-transform is the discrete version of the Laplace transform and in both cases z and s are the set of all complex numbers, and as such we map with the transform the time domain function into the domain of complex frequencies; signals that change in rotation only which is the Fourier Transform and in addition to ... | 191 |
Laplace transform | What is the difference between delay and difference properties of z-transform? | https://dsp.stackexchange.com/questions/82466/what-is-the-difference-between-delay-and-difference-properties-of-z-transform | <p>I'm working on a discrete updating algorithm as follows:</p>
<p><span class="math-container">$x[n+1]=Kx[n]$</span><br />
Here <span class="math-container">$K$</span> is a constant.</p>
<p>The continuous counterpart of this algorithm translates to:</p>
<p><span class="math-container">$\dot{x(t)}=Kx(t)$</span></p>
<p>... | <p>If by Laplace transform
<span class="math-container">$$\dot{x(t)}=Kx(t)$$</span>
becomes
<span class="math-container">$$sX(s)=KX(s)$$</span>
The by analogy, using the z-transform
<span class="math-container">$$x[n+1]=Kx[n]$$</span>
becomes
<span class="math-container">$$zX(z)=KX(z)$$</span>
This is simply using the ... | 192 |
Laplace transform | Not getting the same step response from Laplace transform and it's respective difference equation | https://dsp.stackexchange.com/questions/88753/not-getting-the-same-step-response-from-laplace-transform-and-its-respective-di | <p>I am trying to simulate a plant on a microcontroller. The transfer function of the plant is</p>
<p><span class="math-container">$$ G_{p} \left( s \right) = \frac{2}{\left( s + 3 \right) \left( s - 1 \right)} \tag{1} \label{1}$$</span></p>
<p>The step response for this function from Octave is</p>
<p><a href="https://... | <p>The difference equation as written in your question is wrong, but I see that you implemented the correct version, also using delayed versions of the output to compute the current output.</p>
<p>The problem is that you used truncated values for the coefficients. You need to represent the coefficients with high accura... | 193 |
Laplace transform | Why is the digital frequency response taken on the unit circle, while the analog is taken along the imaginary axis? | https://dsp.stackexchange.com/questions/95816/why-is-the-digital-frequency-response-taken-on-the-unit-circle-while-the-analog | <p>For digital signals, the fourier transform is taken along the unit circle of the Z-transform.<br />
The equivalent to the Z-transform in continuous signals is the Laplace transform, but in that case the fourier transform is taken along the imaginary axis.</p>
<p>Why the difference? Why don't we take the z-transform ... | <p>One way to see this is to consider the Laplace transform of a sampled signal:</p>
<p><span class="math-container">$$x_d(t)=\sum_{n=0}^{\infty}x(nT)\delta(t-nT)\tag{1}$$</span></p>
<p>where I've assumed that <span class="math-container">$x(t)$</span> starts at <span class="math-container">$t=0$</span>. <span class="m... | 194 |
Laplace transform | Creating a digital filter, from Laplace to $\mathcal Z$-transform (zero order hold) to code? | https://dsp.stackexchange.com/questions/18329/creating-a-digital-filter-from-laplace-to-mathcal-z-transform-zero-order-ho | <p>I'm trying to create a digital filter in code(C) but any language is fine. Now I've got an analogue filter that I have represented by an equation in the Laplace domain and I want to try and implement it digitally. </p>
<p>So my filter has this form in the Laplace domain:
$$\frac{as+b}{cs^2+ds}$$</p>
<p>I then use ... | <p>The example I looked at used a tustin or bilinear conversion not a zero order hold(the default for matlabs "c2d" command). So this is more an answer to what i wanted to do rather than the question that i asked above.</p>
<p>I solved the following (converting the s domain function into code) by taking the s domain f... | 195 |
Laplace transform | Validity of applying Heaviside function for signal processing applications | https://dsp.stackexchange.com/questions/66998/validity-of-applying-heaviside-function-for-signal-processing-applications | <p>I wasn't sure if this question was more suitable for math.stackexchange, but I suspect it's more-so a signal processing question (albeit, a theoretical one) than a mathematical one.</p>
<p>I am currently studying the textbook <em>An Introduction to Laplace Transforms and Fourier Series</em>, second edition, by Phil ... | 196 | |
Laplace transform | Fourier transform of unit step | https://dsp.stackexchange.com/questions/67974/fourier-transform-of-unit-step | <p>I was reading pdf by caltech and in one of its section, Fourier transform of Unit step signal is calculated but I am confused, how this can be possible if region of convergence for Laplace transform (<span class="math-container">$1/s$</span>) of unit step signal does not contain imaginary axis?</p>
<p>And if above ... | <p>The Fourier transform can be generalized for functions that are not absolutely integrable. We can define a Fourier transform for functions with a constant envelope (e.g., sine, cosine, complex exponential), and even for functions with polynomial growth (but not with exponential growth). In these cases we must be pre... | 197 |
Laplace transform | Visualising a Z-transformed Transfer Function? | https://dsp.stackexchange.com/questions/22494/visualising-a-z-transformed-transfer-function | <p>For designing any analog filter and various other outputs of filter we use <strong>laplace transform</strong>,I can visualise a laplace transform like for ex.<br>
<code>s[X(s)]</code> can be implemented as differentiator fetched with signal <code>x(t)</code>while implementing differentiators we generally use
capa... | <p>in continuous-time functions, like $x(t)$, the operation of the derivative makes sense, it is well defined. the three components to a continuous-time LTI filter are adders (two or more signals being added), scalers (a signal is simply multiplied by a constant), and integrators (the $s^{-1}$ operators). the first t... | 198 |
Laplace transform | What is the significance of Z-transform? | https://dsp.stackexchange.com/questions/22556/what-is-the-significance-of-z-transform | <p>As we have in Laplace transform that the roots decide the stability of the system i.e. if the roots are complex and lie in the left side of the plane you get a sinusoidal response with decreasing amplitude </p>
<p>similarly is there any significance of the roots , zeros and ROC of the z-transform and the stability... | <p>First of all, I think you're reading the wrong books. Almost any basic text on DSP has a chapter on the $\mathcal{Z}$-transform and its significance to describe linear time-invariant (LTI) discrete-time systems. If you're looking for good (and free) books, take a look at <a href="https://dsp.stackexchange.com/questi... | 199 |
wavelet transform | Continuous Wavelet Transform vs Discrete Wavelet Transform | https://dsp.stackexchange.com/questions/76624/continuous-wavelet-transform-vs-discrete-wavelet-transform | <p>The discrete wavelet transform is applied in many areas, such as signal compression, since it is easy to compute. I notice that, However, the continuous wavelet transform (CWT) is also applied to different subjects. In my opinion, the CWT is redundant and hence difficult to compute. So what are the advantages of the... | <p>On the one hand with the DWT, only a restricted choice of wavelets is available: those that implement 2-band perfect reconstruction (Daubechies, Symmlets, Coiflets, Spline). They are non-redundant, and often orthogonal or close to orthogonal, which simplifies some computations, inversion or statistical analysis, for... | 200 |
wavelet transform | Synchrosqueezing Wavelet Transform explanation? | https://dsp.stackexchange.com/questions/71398/synchrosqueezing-wavelet-transform-explanation | <p>How does Synchrosqueezing Wavelet Transform work, intuitively? What does the "synchrosqueezed" part do, and how is it different from simply the (continuous) Wavelet Transform?</p>
| <p>Synchrosqueezing is a powerful <em>reassignment</em> method. To grasp its mechanisms, we dissect the (continuous) Wavelet Transform, and how its pitfalls can be remedied. Physical and statistical interpretations are provided.</p>
<p>If unfamiliar with CWT, I recommend <a href="https://ccrma.stanford.edu/%7Eunjung/my... | 201 |
wavelet transform | Implementing Continuous Wavelet Transform | https://dsp.stackexchange.com/questions/37528/implementing-continuous-wavelet-transform | <p>I need to implement the discretized continuous wavelet transform from scratch. Could someone please point me to useful papers and references available online for this?</p>
| <p>In 1D, some of the standard references are:</p>
<ul>
<li><a href="http://www.sciencedirect.com/science/article/pii/S0165168402001408" rel="nofollow noreferrer">Continuous wavelet transform with arbitrary scales and $O({N})$ complexity</a>, A. Muñoz and R. Ertl\'e and M. Unser, Signal Processing, 2002</li>
<li><a hr... | 202 |
wavelet transform | Daubechies wavelet transform | https://dsp.stackexchange.com/questions/28629/daubechies-wavelet-transform | <p>i have N samples obtained by sampling a signal with lot of frequency contents. How will i apply daubechies wavelet transform to obtain the frequency and its location? i need to write a program which will process the signal and gives the frequency and location as the result.</p>
| <p>Looks like you need a general explanation of the discrete wavelet transform (DWT). DWT breaks a signal down into subbands distributed evenly in a logarithmic frequency scale, each subband sampled at a rate proportional to the frequencies in that band. The traditional Fourier transformation has no time domain resolut... | 203 |
wavelet transform | Opposite of wavelet transform? | https://dsp.stackexchange.com/questions/24766/opposite-of-wavelet-transform | <p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/c4/STFT_and_WT.jpg/500px-STFT_and_WT.jpg" alt="STFT and Wavelet"></p>
<p>Wavelet transform gives good time resolution for high-frequency events and good frequency resolution for low-frequency events. </p>
<p>=> I want to have complete opposite of wav... | <p>wavelet transform also serves to you what you want to do, all you have to do is to first apply a low-pass your signal so as to keep the frequency range you want to scan only, another approach is to transform Gabor where you have to define the size and shape of the analysis window, but I would recommend better using... | 204 |
wavelet transform | Disadvantages of wavelet transform | https://dsp.stackexchange.com/questions/15148/disadvantages-of-wavelet-transform | <p>I have a question related to wavelet transform: we know that while the Fourier transform is good for a spectral analysis or which frequency components occurred in signal, it will not give information about at which time it happens. That's why the wavelet transform is suitable for the time-frequency analysis. It is ... | <p>If you consider the whole set of potential wavelet transforms, then you have a lot of flexibility. </p>
<p>For instance, should you use 1D continuous complex wavelet transforms, by analyzing the modulus and the phase of the scalogram, and provided you use well-chosen wavelets (potentially different for the analysis... | 205 |
wavelet transform | Continuous wavelet transform | https://dsp.stackexchange.com/questions/58615/continuous-wavelet-transform | <p>Continuous wavelet transformation has been quite widely used for various applications. Most of the papers that I found were using CWT for non-stationary signals. Can we use CWT for stationary signal analysis? if not what are the drawbacks in using Continuous wavelet transform?</p>
| <p>Stationarity is a multi-fold concept in signal processing. It can denote a wide range of behavior, encompassing deterministic or stochastic aspects. Beyond that, the main question is: do you know if your signal is stationary, and how?</p>
<p>If you actually know how, it is probably wiser to use the generation proce... | 206 |
wavelet transform | Implementing Wavelet Transform using Equations | https://dsp.stackexchange.com/questions/8781/implementing-wavelet-transform-using-equations | <p>I want to implement Wavelet Transform from the scratch, that mean breaking the wavelet transform into its equations to implement in any Programming language. Matlab Comes with built-in functions to implement Wavelet Transform but It is really hard to understand which processes are exactly involved in the implementat... | <p>I found the <a href="http://grail.cs.washington.edu/pub/stoll/wavelet1.pdf" rel="nofollow">Wavelets for Computer Graphics: A Primer</a> a good introduction to the Haar wavelet and its role in image processing. </p>
| 207 |
wavelet transform | Wavelet Transform | https://dsp.stackexchange.com/questions/2149/wavelet-transform | <p>I want to perform 2D haar discrete wavelet transform and inverse DWT on an image.<strong>Will you please explain 2D haar discrete wavelet transform and inverse DWT in a simple language and an algorithm using which I can write the code for 2D haar dwt</strong>?The information given in google was too technical.I under... | <blockquote>
<p>Will you please explain 2D haar discrete wavelet transform and inverse
DWT in a simple language</p>
</blockquote>
<p>It is useful to think of the wavelet transform in terms of the <a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform">Discrete Fourier Transform</a> (for a number of reaso... | 208 |
wavelet transform | stationary vs. undecimated wavelet transform | https://dsp.stackexchange.com/questions/27836/stationary-vs-undecimated-wavelet-transform | <p>I have a little bit confused on the difference between stationary wavelet transform and un-decimated wavelet transform.</p>
<p>So, can anyone tell me, if there is a difference between them?</p>
| <p>The translation invariant version of the DWT is known by a variety of names, including stationary wavelet transform (SWT), redundant wavelet transform, algorithm à trous, quasi-continuous wavelet transform, translation-invariant wavelet transform, shift invariant wavelet transform, cycle spinning, maximal overlap wa... | 209 |
wavelet transform | Wavelet Transform and STFT | https://dsp.stackexchange.com/questions/54551/wavelet-transform-and-stft | <p>How wavelet transform is different from STFT. </p>
<p>I'm not able to understand what is resolution in frequency domain means?</p>
| <p><a href="https://i.sstatic.net/SyTNn.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/SyTNn.png" alt="STFT vs CWT"></a></p>
<p>In the STFT, you apply windowing and Fourier transform on the signal using sliding patches and then combine the resulting transforms, which will help you eventually end up wit... | 210 |
wavelet transform | Transfer functions from wavelet transform | https://dsp.stackexchange.com/questions/30895/transfer-functions-from-wavelet-transform | <p>So I have this problem where I need to measure the phase of a signal and correct for a delay associated with the travel time of the signal while simultaneously determining the transfer function of my system (with the delay corrected).</p>
<p>So I thought I probably need a wavelet transform so that I can determine w... | 211 | |
wavelet transform | Relationship between windowed fourier transform and wavelet transform | https://dsp.stackexchange.com/questions/13779/relationship-between-windowed-fourier-transform-and-wavelet-transform | <p>I was reading on windowed fourier transform and wavelet transform, and i was thinking that the windowed fourier transform is a subset of wavelet transform. Is that true?</p>
| <p>Define the Fourier transform as <span class="math-container">$$ x(t) = \mathscr{F}^{-1}\big\{ X(\omega) \big\} \triangleq\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(\omega) e^{j\omega t} \ d\omega $$</span></p>
<p>and <span class="math-container">$$ X(\omega) = \mathscr{F}\big\{ x(t) \big\} = \int_{-\infty}^{\inft... | 212 |
wavelet transform | Does the Fast Wavelet Transform produce the same coefficient as the Discrete Wavelet Transform? | https://dsp.stackexchange.com/questions/71394/does-the-fast-wavelet-transform-produce-the-same-coefficient-as-the-discrete-wav | <p>Does the Fast Wavelet Transform(FWT) produce the same coefficients as the Discrete Wavelet Transform(DWT) if configured for the same depths? Or is the the FWT just an approximation of the DWT?</p>
| <p>If the discrete wavelet transform can be implemented with a FIR filter bank, with appropriate extensions, yes, up to numerical precision, coefficients will be the same.</p>
<p>If the discrete wavelet transform possesses a non finite support, then a FIR filter bank implementation would require filter truncation, and... | 213 |
wavelet transform | Should i use window with hop_size in Wavelet Transform or Discrete Wavelet Transform? | https://dsp.stackexchange.com/questions/78846/should-i-use-window-with-hop-size-in-wavelet-transform-or-discrete-wavelet-trans | <p>I have a signal (audio - voice) with 1 second of duration with sample rate of 50000 Hz. It is big signal and I wish extract some features and apply pattern recognition or classification.</p>
<p>My question is if the Wavelet transform or Discrete Wavelet transform is a time frequency representation (or time scale). S... | <p>Basically, an analysis linear filter-bank is composed of several branches of convolutive filters, each branch with its own hop. The theory consists in finding under which the filter-bank is invertible, how to design the filters and choose the hops.</p>
<p>Each level of a dyadic discrete wavelet transform is a filt... | 214 |
wavelet transform | Clarification regarding discrete wavelet transform | https://dsp.stackexchange.com/questions/61728/clarification-regarding-discrete-wavelet-transform | <p>One of the books on "Conceptual Wavelets" by Fugal explains some major differences between the undecimated discrete wavelet transform (UDWT) vs. discrete wavelet transform (DWT). In UDWT the scale of wavelet is increased continuously just like the continuous wavelet transform, but the scale increases in dyads (power... | <p>Assuming that you have a sufficient number of samples, throwing away odd / even samples does not matter.</p>
<p>The DWT can be thought of as measuring time/frequency content with varying levels of time/frequency resolution. For non frequency varying signals (like chirps), the odd and even samples both contain the ... | 215 |
wavelet transform | Shifting of Shift-Invariant Wavelet Transforms | https://dsp.stackexchange.com/questions/14086/shifting-of-shift-invariant-wavelet-transforms | <p><strong>Main Question: Why would iterative wavelet/inverse-wavelet transforms cause a shift along the x-axis for undecimated (shift-invariant) wavelet transforms?</strong></p>
<p>I am attempting to remove backgrounds from signals using an iterative wavelet transform method similar to this approach which I found in ... | 216 | |
wavelet transform | Nonlinear wavelets transform? | https://dsp.stackexchange.com/questions/12926/nonlinear-wavelets-transform | <p>Is wavelet a Nonlinear transform, or Not?<br>
specifically, continuous wavelet transform with morlet function.<br>
I am studying behavior of a dynamic system, and it has nonlinear behaviour. can I employ wavelet transform? </p>
| <p>A transform being linear has very little to do with its ability to analyze linear or nonlinear systems.</p>
<p>The wavelet transform $W[s(t)]$ of a signal $s(t)$ is linear because $$W[a s_1(t) + b s_2(t)]=a W[s_1(t)]+b W[s_2(t)]$$ for real or complex $a$ and $b$.</p>
<p>The signal you're analyzing is just a signal... | 217 |
wavelet transform | Where is the mother wavelet defined in the Fast Wavelet Transform? | https://dsp.stackexchange.com/questions/71263/where-is-the-mother-wavelet-defined-in-the-fast-wavelet-transform | <p>Referring to the <a href="https://en.wikipedia.org/wiki/Fast_wavelet_transform#cite_note-1" rel="nofollow noreferrer">Fast Wavelet Transform</a>, this transform is implemented as a QMF filter bank. This algorithm consists of high/low pass filtering and subsampling. However, a wavelet transform is typically defined b... | <p>TL;DR: the wavelet appears at the end of the synthesis filter bank, iterated infinitely.</p>
<p>Theoretically founded, practical and fast DSP tools are often derived from continuous theory: think about how the DFT is derived from the continuous Fourier transform, by discretizing both in time (like the Discrete-time ... | 218 |
wavelet transform | Advantage of STFT over wavelet transform | https://dsp.stackexchange.com/questions/79586/advantage-of-stft-over-wavelet-transform | <p>I have learned about STFT and wavelet transform recently, and wavelet transform seems better than STFT in my opinion.
So, I wonder if there is any advantage of using STFT than WT, and if so, what are practical applications of STFT?</p>
| <p>Wavelet transforms and short-term/short-time Fourier transforms are broad names for classes of transformations that are not totally distinct and may overlap (pun intended).</p>
<p>Both can be efficient for non-stationary features of data, and they both have merits or drawbacks, depending on their parameters and sign... | 219 |
wavelet transform | Discrete Wavelet Transform (DWT) and wavelet family | https://dsp.stackexchange.com/questions/76594/discrete-wavelet-transform-dwt-and-wavelet-family | <p>I have just started reading about wavelets for a data compression problem that I want to perform. I am reading about Discrete Wavelet Transform (DWT) but I can't understand where the wavelet family that has to be set is used.</p>
<p>This is the DWT schema</p>
<p><a href="https://i.sstatic.net/rr56r.png" rel="nofollo... | <p>There are actually four filters involved:</p>
<ul>
<li>2 for the decomposition of signals [the h[n] and g[n] in the diagram above]</li>
<li>2 for the reconstruction of signals</li>
</ul>
<p>The diagram you are showing is only for signal decomposition. There is a corresponding diagram for signal reconstruction which ... | 220 |
wavelet transform | Wavelet transform of a spatial convolution | https://dsp.stackexchange.com/questions/52839/wavelet-transform-of-a-spatial-convolution | <p>Does anyone know if there exist a kind of convolution theorem for the discrete wavelet transform (decimated or undecimated)? </p>
<p>In other words can I find a simple form of
<span class="math-container">$W\left[ \int f(t) g(x-t) \, dt\right] $</span> where <span class="math-container">$W$</span> is the discrete w... | <p>I cannot say I have a clear understanding of this at this time. However, a few pointers. I'd love to see somebody provide a detailed account. Others bits at: <a href="https://dsp.stackexchange.com/a/31590/15892">Multiplication in the wavelet domain, what does it look like in real space?</a></p>
<ul>
<li><p>The non... | 221 |
wavelet transform | Discrete wavelet transform disadvantages | https://dsp.stackexchange.com/questions/61213/discrete-wavelet-transform-disadvantages | <p>I read in <a href="https://books.google.com.eg/books?id=49FBDwAAQBAJ&pg=PA79&lpg=PA79&dq=DWT+shift+variance+property+due+to+the+downsampling+process+lack+of+directional+selectivity.&source=bl&ots=wnhZpeTQcY&sig=ACfU3U3heH_2sefjO995Jqn52pJ7udyrug&hl=en&sa=X&ved=2ahUKEwjtqtG9xZnlAhU... | <p>A <span class="math-container">$2$</span>-channel stage of a wavelet transform, combines two filters in parallel, followed by a down-sampling by two. The later is the cause for shift-invariance, as the filters are time-invariant. Signals <span class="math-container">$$x_0[n] = \{\ldots,0,1,0,1,0,1,\ldots\}$$</span> ... | 222 |
wavelet transform | Wavelet transform 3D plot for CoP | https://dsp.stackexchange.com/questions/31936/wavelet-transform-3d-plot-for-cop | <p>I'm trying to perform wavelet transform and make a 3D plot like :</p>
<p><a href="https://i.sstatic.net/GHooq.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GHooq.gif" alt="enter image description here"></a></p>
<p>With the wavelet transform function :</p>
<p>$$
\textrm{CWT}_x^\psi (\tau, s)=\frac... | <p>Basically, I see a plot of a 2D function of discretized scale and translation parameters. Instead of a smooth 2D surface, it looks like 1D plots of coefficients at all scales $s_n$, put behind each other along each location $\tau_n$ on the translation axis. And each 1D plot is colored in a level-set fashion: the "... | 223 |
wavelet transform | 3 Band Wavelet Transform In MATLAB | https://dsp.stackexchange.com/questions/9777/3-band-wavelet-transform-in-matlab | <p>I am currently working on an audio watermarking project in MATLAB. I currently have a code I am using to construct a nxn 3 Band Wavelet Transform matrix. However, when I try to construct a matrix that is larger in size, I get the error "Maximum variable size allowed by the program is exceeded" or "Out of Memory."</p... | 224 | |
wavelet transform | Discrete wavelet transform | https://dsp.stackexchange.com/questions/29138/discrete-wavelet-transform | <p>I am unable to understand the <strong>discrete wavelet transform</strong> on images. I followed Robi Polikar's tutorial and got a brief idea about the theory. But I'm unable to understand w.r.t images.</p>
<p>Using Matlab's <code>ndwt2('chess.jpg', 2, 'haar')</code> function on the chess board , I obtained the othe... | <p>One can implement the standard discrete wavelet transform (DWT) on an image (<code>dwt2</code> in Matlab) with a series of filtering and decimation operations, on the rows and the columns. And the wavelet by itself results from the iteration at different levels. </p>
<p>Start with the Haar wavelet. In 1D, it can b... | 225 |
wavelet transform | Difference between a wavelet transform and a wavelet decomposition | https://dsp.stackexchange.com/questions/10675/difference-between-a-wavelet-transform-and-a-wavelet-decomposition | <p>I'm confused about the difference between a wavelet transform and a wavelet decomposition is. For example</p>
<pre><code>load woman
[cA1,cH1,cV1,cD1] = dwt2(X,'db1');
[c,s] = wavedec2(X,2,'db1');
</code></pre>
<p>What's the difference between these two matlab commands, and when would you want to do one over the o... | <p>I don't think there is any difference. The documentation for <a href="http://www.mathworks.com/help/wavelet/ref/dwt2.html" rel="noreferrer">dwt2</a> says</p>
<blockquote>
<p>Single-level discrete 2-D wavelet transform</p>
<p>The dwt2 command performs a single-level two-dimensional wavelet decomposition...</... | 226 |
wavelet transform | Difference between "Discrete Wavelet Transform" and "Discrete Wavelet Decomposition" | https://dsp.stackexchange.com/questions/59382/difference-between-discrete-wavelet-transform-and-discrete-wavelet-decomposit | <p>I have a rough overview on Discrete Wavelet Transform (DWT). However, I am confused about Discrete Wavelet <em>Decomposition</em> and did not find a good reference yet which explains this well. What is it actually about? Is it somehow part of DWT or an inverse operation to it?</p>
| <p>The <strong>discrete wavelet transform</strong> should denote "the operations" that, applied to some data, yield a <strong>discrete wavelet decomposition</strong>. The first one can be seen as a matrix operator, while the second relates to the actual wavelet coefficients, or the structure of thereof, that you would ... | 227 |
wavelet transform | Intuition behind the Continuous Wavelet Transform? | https://dsp.stackexchange.com/questions/15662/intuition-behind-the-continuous-wavelet-transform | <p>I was thinking sometime back about how to explain the Continuous Wavelet Transform ELI5. So this is what I came across.</p>
<p>The correlation of two exact signals is 1. So if I have an input signal $f(x)$ made up of an array of frequencies, how can I find out what frequencies exist at what points? Well, slide a si... | <p>I think that the best way to explain the CWT is to start by explaining the Fourier Transform, then move on to explaining the Short-Time Fourier Transform, and then finally explain the CWT as a variation of the STFT.</p>
<p>The Fourier Transform exploits the fact that any decently behaved function can be represented... | 228 |
wavelet transform | Reading the Wavelet transform plot | https://dsp.stackexchange.com/questions/7911/reading-the-wavelet-transform-plot | <p>I am having trouble understanding on how to read the plot plotted by a wavelet transform,</p>
<p>here is my simple Matlab code,</p>
<pre><code>load noissin;
% c is a 48-by-1000 matrix, each row
% of which corresponds to a single scale.
c = cwt(noissin,1:48,'db4','plot');
</code></pre>
<p><img src="https://i.ssta... | <p>This is the example that i think is the best to understand Wavelet plot.</p>
<p>Have a look at the image below,
The Waveform (A) is our original Signal, Waveform (B) shows a Daubechies 20 (Db20) wavelet about 1/8 second long that starts at the beginning (t = 0) and effectively ends well before 1/4 second. The zero ... | 229 |
wavelet transform | Implementing wavelet transform for finding transients in the power supply | https://dsp.stackexchange.com/questions/28018/implementing-wavelet-transform-for-finding-transients-in-the-power-supply | <p>I am new to the concept of wavelet transforms. Can somebody please help me in understanding this ? and also how to implement it in c. Is Short term Fourier transform more efficient than Wavelet Transform for finding Transients ?</p>
| <p>I would say that a matching Mother wavelet could be the best for detecting a transient; but both the selection and implementation would be much slower. The old adage: do you want quantity or quality :) or Bandwidth vs. noise. Life doesn't come in our neat intellectual packets.
BTW: the simplest technique is an isol... | 230 |
wavelet transform | Bandpass filter using wavelet transform | https://dsp.stackexchange.com/questions/36914/bandpass-filter-using-wavelet-transform | <p>I'm working on a speech recognition project. The first step of this project is to find phoneme in the speech signal. To do that, I found <a href="https://www-users.cs.york.ac.uk/~suresh/papers/PSOS.pdf" rel="nofollow noreferrer">this paper</a> that discusses about it.</p>
<p>In the paper, wavelets are used to visua... | <p>If you refer to the documentation for <a href="https://fr.mathworks.com/help/wavelet/ref/wavedec.html" rel="nofollow noreferrer"><code>wavedec</code></a>, a signal $x$ is decomposed on one level into two sets of coefficients: $cA_1$ and $cD_1$. They correspond to a low-pass and a high-pass filter applied to $x$, fol... | 231 |
wavelet transform | Wavelet transform in control systems | https://dsp.stackexchange.com/questions/18053/wavelet-transform-in-control-systems | <p>In control systems, the Laplace transform is often used to analyze the stability and the performance of <a href="http://en.wikipedia.org/wiki/LTI_system_theory" rel="nofollow">LTI system</a>. For instance, the LTI system is stable if and only if the <a href="http://en.wikipedia.org/wiki/Transfer_function" rel="nofol... | <p>In the paper <a href="http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6811177" rel="nofollow">Haar-Based Stability Analysis of LPV Systems</a>, the Haar wavelet transform theory have been used to design linear matrix inequalities (LMIs) to analyze the stability of <a href="https://en.wikipedia.org/wiki/Linear_p... | 232 |
wavelet transform | Why are wavelet transforms implemented in Python/Matlab often called Continuous wavelet transform when they take discrete-time input? | https://dsp.stackexchange.com/questions/86713/why-are-wavelet-transforms-implemented-in-python-matlab-often-called-continuous | <p>The implementations of Synchrosqueezing wavelet transform in Python (<a href="https://github.com/OverLordGoldDragon/ssqueezepy" rel="nofollow noreferrer">ssqueezepy</a>) and <a href="https://www.mathworks.com/help/wavelet/gs/wavelet-synchrosqueezing.html" rel="nofollow noreferrer">MATLAB</a> both write in their do... | <p>Good question.</p>
<h2>From nomenclature standpoint</h2>
<p>Sampling a continuous-time result (called <em>discretization</em>) most often inherits the original name. For example, we still say "IIR filters", though they're surely finite on a computer.</p>
<p>The following are my observations that are <em>so... | 233 |
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