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convolution | Convolution of shifted signal | https://dsp.stackexchange.com/questions/52560/convolution-of-shifted-signal | <p>If <span class="math-container">$y(t) = x(t)*h(t)$</span>, then what is the expression for <span class="math-container">$y(t+a)$</span>?</p>
<p>Is it <span class="math-container">$x(t+a)*h(t+a)$</span> or <span class="math-container">$x(t+a)*h(t)$</span>?</p>
| <p>From your confusion of <span class="math-container">$x(t+a) \star h(t+a)$</span> vs <span class="math-container">$h(t) \star x(t+a)$</span>
I guess that a little help on the argument manipulations on functions and convolutions could be appropriate here, moving in simple examples:</p>
<p>First let us express the us... | 434 |
convolution | What are three different techniques to convolve a signal | https://dsp.stackexchange.com/questions/63829/what-are-three-different-techniques-to-convolve-a-signal | <p>I have studied convolution in time domain and frequency domain. But doesn't know the 3rd method and technique </p>
| 435 | |
convolution | Why does linear convolution with itself converges to Gaussian, but not circular? | https://dsp.stackexchange.com/questions/83945/why-does-linear-convolution-with-itself-converges-to-gaussian-but-not-circular | <p>If you have a discrete signal, say</p>
<blockquote>
<p>[1,0,0,1]</p>
</blockquote>
<p>and you convolve it linearly with itself, you will quickly get a Gaussian distribution.</p>
<p>Why does it not hold for circular convolution?</p>
| <p>Convolving <span class="math-container">$x = [1,0,0,1]$</span> with itself repeatedly will not generate a "Gaussian distribution".</p>
<p><span class="math-container">\begin{align}
x \star x &= [1,0,0,2,0,0,1]\\
x \star x \star x &= [1,0,0,3,0,0,3,0,0,1]
\end{align}</span></p>
<p>and so on.</p>
<p>... | 436 |
convolution | Is Periodic Convolution Same as Circular Convolution? | https://dsp.stackexchange.com/questions/38450/is-periodic-convolution-same-as-circular-convolution | <p>I know exactly what is periodic convolution. But i don't know if circular convolution means to be the same thing!</p>
| <p>You did not give specific details on how you define periodic or circular. Assuming standard definitions, the answer is <strong>yes</strong>.</p>
<p>Standard infinite-support data is treated with standard convolution. When the support is finite, the "outer unknown samples" require assumptions. </p>
<p>Zero-valued ... | 437 |
convolution | A question about time-shifted convolution | https://dsp.stackexchange.com/questions/87358/a-question-about-time-shifted-convolution | <p>I have 2 time-shifted signals,<span class="math-container">$f(t-t_0)$</span> and <span class="math-container">$g(t-t_0)$</span>. Assuming that <span class="math-container">$y=f(t)*g(t)$</span>, which is the convolution of the signals.</p>
<p>By definition, I have <span class="math-container">$f(t-t_0)*g(t-t_0)=\int_... | <p>The mistake lies in the use of bad notation for convolution:</p>
<p><span class="math-container">$$y(t)=f(t)*g(t)\qquad \text{(bad!)}\tag{1}$$</span></p>
<p>The correct way to write convolution is</p>
<p><span class="math-container">$$y(t)=(f*g)(t)\tag{2}$$</span></p>
<p>As long as you don't play around with the var... | 438 |
convolution | Simplification of convolution | https://dsp.stackexchange.com/questions/87361/simplification-of-convolution | <p>Let's assume that there are two convolutions:</p>
<p><span class="math-container">$y_1 = (h_2[n]\cdot x[n])*(\bar{h_1}[n]\cdot\bar{x}[n])$</span></p>
<p><span class="math-container">$y_2 = (h_3[n]\cdot x[n])*(\bar{h_2}[n]\cdot\bar{x}[n])$</span></p>
<p>where "<span class="math-container">$\bar{x}$</span>" ... | <p>Here is the simplest approach I could think of:</p>
<p><span class="math-container">$$x_2[n]= h_2[n]x[n]$$</span></p>
<p><span class="math-container">$$\bar{x_2}[n] = \bar{h_2}[n]\bar{x}[n]$$</span></p>
<p>Thus we eliminate one product as we can get <span class="math-container">$\bar{x_2}[n]$</span> from <span class... | 439 |
convolution | Implemenation of Length N Moving average filter using block covolution algorithm(overlap and save algorithm) | https://dsp.stackexchange.com/questions/68515/implemenation-of-length-n-moving-average-filter-using-block-covolution-algorithm | <p>I want to implement a Length N moving average filter using block convolution algorithm(overlap save algorithm)</p>
<p>I am copying block convolution algorithm(overlap save algorithm) from Proakis, DSP using MATLAB 3rd Ed</p>
<pre><code>function [y] = ovrlpsav(x,h,N)
% Overlap-Save method of block convolution
% -----... | <p>in psudo pyhon code</p>
<pre><code>const tail = 5
'inside loop where you feed it values
list.push(entry)
list.rotate.left(1)
if len(list)>tail list.pop()
avg = sum(lst) / len(lst)
print avg
</code></pre>
| 440 |
convolution | Question about convolution of shifted function-simple | https://dsp.stackexchange.com/questions/43339/question-about-convolution-of-shifted-function-simple | <p>I memorized the convolution formula, all good. I started doing examples and in one of them I had to calculate the convolution of a shifted function with another one.</p>
<p>I know the property $$H(x)=f(x)\ast g(x)\\H(x-k)=f(x)\ast g(x-k)=f(x-k)*g(x)$$</p>
<p>So I had no trouble finding the result but I had trouble... | <p>Considering a continuous-time LTI system with impulse response <span class="math-container">$h(t)$</span>, its output to any valid input <span class="math-container">$x(t)$</span> is described by a <strong>convolution integral</strong>:</p>
<p><span class="math-container">$$y(t) = x(t) \star h(t) = \int_{-\infty}^{\... | 441 |
convolution | Fusing convolutions | https://dsp.stackexchange.com/questions/89456/fusing-convolutions | <p>I am attempting to combine two consecutive 2D convolutions into a single 2D convolution. Ideally, a convolution operation is a linear transformation, so it should be possible to merge two linear transformations into one.</p>
<p>In practice, I have performed all the necessary calculations to derive the fused kernel. ... | 442 | |
convolution | Convolution of delta function in discrete time with parameter | https://dsp.stackexchange.com/questions/46363/convolution-of-delta-function-in-discrete-time-with-parameter | <p>If I have a delta function of the following type</p>
<p>$$ n\delta(5n-20) * 3^{5n}u[n] $$ How can I calculate this convolution?</p>
<p>I have thought of the property $$ \delta[n-n_0] *x[n] = x[n-n_0] $$</p>
<p>Then if I say $$ k=5n $$ and $$ t[n]= 3^{5n}u[n] $$</p>
<p>It should be that the convolution is equal ... | <p>You can use the following argumentation to find the result. The discrete time unit-sample function $\delta[n]$ has the following property for integer $M$:
$$ \delta[Mn] = \delta[n] $$ and more generally you can conlcude that for integer $M$ and $d$ we have
$$ \delta[M(n-d)] = \delta[n-d] $$</p>
<p>Therefore you can... | 443 |
convolution | Sample time in convolution | https://dsp.stackexchange.com/questions/63927/sample-time-in-convolution | <p>I have a signal sampled at a period of 0.01 and a channel impulse response sampled at 0.1250. I convolved the signal with the channel impulse response.</p>
<ul>
<li>It is right to perform the convolution between two signals with diffrent sample times?</li>
<li>If it is, how can I calculate the sample time of the ... | <blockquote>
<p>It is right to perform the convolution between two signals with diffrent sample times?</p>
</blockquote>
<p>No, aside from a very few special use cases (if you have to ask, not yours), you need to resample either of them so that both are at the same sample rate.</p>
<p>Also notice that sampling a ch... | 444 |
convolution | Why is circular convolution used in DSP? Why not linear convolution? | https://dsp.stackexchange.com/questions/35155/why-is-circular-convolution-used-in-dsp-why-not-linear-convolution | <ol>
<li><p>Why are we using circular convolution in DSP? </p></li>
<li><p>What's the main solid reason for the use of it in digital
processing? </p></li>
<li><p>Why does the concept of circular convolution come more often than
linear convolution?</p></li>
</ol>
| <p>Given a discrete-time <strong>LTI</strong> system with impulse response <span class="math-container">$h[n]$</span>, one can compute its response to any input <span class="math-container">$x[n]$</span> by a <strong>convolution</strong> sum: <span class="math-container">$$y[n] = x[n] \star h[n] = \sum_{k=-\infty}^{\i... | 445 |
convolution | Convolution in Continious time | https://dsp.stackexchange.com/questions/13309/convolution-in-continious-time | <p><img src="https://i.sstatic.net/Md1Ow.png" alt="enter image description here"></p>
<p>Dear all,</p>
<p>Please find attached image,
I am not able to understand why is the underlined term is to be included.
Shouldn't it be simply the exponential term?</p>
| <p>The $u(t-2)$ term needs to be included because the upper limit of the definite integral is $t-2$, not $\infty$:</p>
<p>\begin{align*}
h(t) &= e^{-(t-2)}u(t-2)\\
y(t) &= x(t) * h(t)= \int_{-\infty}^{\infty} x(t')h(t-t') dt'\\
&= \int_{-\infty}^{\infty} x(t')e^{-(t-t'-2)}u(t-t'-2) dt'\\
u(t-t'-2)&= \... | 446 |
convolution | Is there a convolution mistake in my method? | https://dsp.stackexchange.com/questions/19934/is-there-a-convolution-mistake-in-my-method | <p>I have the input signal $x(t)$ <img src="https://i.sstatic.net/XxdCL.png" alt="enter image description here"></p>
<p>And impulse response $h(t)=20 e^{-1000t} u(t)$ in which u(t) is the unit step function.</p>
<p>When I try a convolution, I thought the solutions would be something like:</p>
<p>$
\begin{array}{ll}
... | <p>You just multiplied the two functions and integrated them but you didn't <em>convolve</em> them. You must compute</p>
<p>$$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau=2\int_0^2h(t-\tau)d\tau-\int_2^3h(t-\tau)d\tau$$</p>
<p>Alternatively, you can compute the step response</p>
<p>$$a(t)=\int_{-\infty}^th(\tau... | 447 |
convolution | Convolution - Massive spike in noise section | https://dsp.stackexchange.com/questions/31642/convolution-massive-spike-in-noise-section | <p>I am computing the correlation with MATLAB using the convolution fonction with a signal flipped before feeded into the function. Here are the results.</p>
<p><a href="https://i.sstatic.net/sTydl.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/sTydl.jpg" alt="enter image description here"></a></p>
<p... | <p>If your searched-for signal is partially similar to itself at some offset, that can also show up when correlated against an actual input signal containing it. </p>
<p>Test your searched-for signal by autocorrelation against itself to check for this problem. If you want a sharp convolution peak against test input,... | 448 |
convolution | Identifying origin in convolution table | https://dsp.stackexchange.com/questions/44401/identifying-origin-in-convolution-table | <p>I am taking the convolution of $x(n)=\left\{2,1,-1,-2,3 \right\}$ with $n=0$ at the third position with $h(n)=\left\{1,2,0,3 \right\}$ with $n=0$ at the second position. The answer is $y(n)=\left\{2,5,1,-10, -10, -3, 6, -9 \right\}$ with the $n=0$ at the fourth position. I studied convolution more than a decade ago ... | <p>Given a signal <span class="math-container">$x[n]$</span>, guaranteed to be zero outside of the domain <span class="math-container">$N_1 \leq n \leq N_2$</span>, and the signal <span class="math-container">$h[n]$</span> being zero outside of <span class="math-container">$M_1 \leq n \leq M_2$</span>, then their convo... | 449 |
convolution | Difficulties while understanding Convolution | https://dsp.stackexchange.com/questions/49578/difficulties-while-understanding-convolution | <p>I have just started convolution sum. I am able to do Mathematical convolution sum. But I am unable to make a figure of it in my mind. So I tried to go through the graphical convolution.
Below is the picture
<a href="https://i.sstatic.net/Zwgjn.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Zwgjn.jpg"... | <p>The convolution sum is
\begin{equation}
y[n] = \sum_k x[k]h[n-k]
\end{equation}
If you take the term corresponding to $k=0$, the contribution is $x[0]h[n]$. Similarly the contribution of $k=1$ is $x[1]h[n-1]$. Here, $h[n]$ is delayed by $n=1$. Similarly you can see how each term of the sum contributes to overall su... | 450 |
convolution | Convolution - change of variables | https://dsp.stackexchange.com/questions/49906/convolution-change-of-variables | <p>I am struggling trying to understand the 3rd line of the expression below. The second line can be simply visualized by letting the kernel size be 3 and $i=2$ for eg. </p>
<p>Then
$ y_2 = \sum_a x_{2+a-1}k_a = x_2k_1 + x_3k_2 + x_4k_3$</p>
<p>In the 3rd line, there is a change of variable letting $u=i+a-1$. What h... | 451 | |
convolution | Using Matlab's conv() function | https://dsp.stackexchange.com/questions/52891/using-matlabs-conv-function | <p><a href="http://matlabtricks.com/post-3/the-basics-of-convolution" rel="nofollow noreferrer">Here</a> is an example of convolution given:</p>
<p><a href="https://i.sstatic.net/xWkbq.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/xWkbq.png" alt="enter image description here"></a></p>
<p>I have two q... | <p>For your second question, <em>valid</em> means the range of output where there is full overlap between the convolving signals. </p>
<p>Note that for two causal signals, <span class="math-container">$x[n]$</span> of length <span class="math-container">$L_x$</span> and <span class="math-container">$h[n]$</span> of le... | 452 |
convolution | Graphical DT Convolution | https://dsp.stackexchange.com/questions/56392/graphical-dt-convolution | <p>Attached is the question 1 part B, solution and my attempt. My attempt seems to be not correct. can someone comment?<a href="https://i.sstatic.net/AUfW9.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/AUfW9.png" alt="Question"></a></p>
<p><a href="https://i.sstatic.net/SFjWF.png" rel="nofollow norefe... | <p>To elaborate the point of my comment:</p>
<p>What you did there is a kind of circular convolution, where you thought of the impulse response <span class="math-container">$h(t)$</span> as periodic, i.e. the system triggered by <span class="math-container">$\delta(0)$</span> would oscillate until infinity. The period... | 453 |
convolution | Is there a process for deriving special cases of convolution? | https://dsp.stackexchange.com/questions/63741/is-there-a-process-for-deriving-special-cases-of-convolution | <p>For my signals and systems homework, I was met with the following problem:</p>
<p>Consider a CT system with impulse response given by</p>
<p><span class="math-container">$$h(t) = e^{-|t|}$$</span></p>
<p>and a step input <span class="math-container">$x(t) = u(t)$</span>. Using the definition of convolution, determ... | <p>Well, indeed there are special cases for convolutions but yours is quite straightforward. You have to consider two cases: <span class="math-container">$t<0$</span> and <span class="math-container">$t>0$</span>. </p>
<p>If you sketch the convolution process for these two cases, by sliding your time shifted and... | 454 |
convolution | Confusions regarding convolution? | https://dsp.stackexchange.com/questions/63971/confusions-regarding-convolution | <p>As far as i have studied and understood, convolution is the process by which we can get/determine output of LTI systems
While reading one web link about convolution, i came across certain notation ,that i couldn't understand as shown highlighted in attached photo</p>
<p><strong>1)Are these notations referring to de... | <blockquote>
<p>1)Are these notations referring to delay/shift in input and impulse response?</p>
</blockquote>
<p>Yes</p>
<blockquote>
<p>2)Also it mentions that **Periodic or circular convolution is also called as fast convolution</p>
</blockquote>
<p>That's a bit of a misrepresentation. For a non-trivial leng... | 455 |
convolution | Properties of convolution same as LTI systems? | https://dsp.stackexchange.com/questions/66460/properties-of-convolution-same-as-lti-systems | <p>I have attached a snapshot from "signal processing first "
Where author says that properties of LTI systems are same as properties of convolution as shown in attached snapshot </p>
<p>Are these properties exactly same for both cases,convolution and LTI systems?and we don't need to apply any scaling etc?</p>
<p><a ... | <p>The author simply means that the response of an LTI system <span class="math-container">$h[n]$</span> can be found by convolving the input sequence <span class="math-container">$x[n]$</span> with the <span class="math-container">$h[n]$</span>. And in doing that all properties of convolution will hold true for LTI sy... | 456 |
convolution | How to index output of convolution product | https://dsp.stackexchange.com/questions/68530/how-to-index-output-of-convolution-product | <p>Unfortunately indexing the output of convolution product confuses me. Suppose <span class="math-container">$x[n]$</span> is of length <span class="math-container">$15$</span> (i.e. <span class="math-container">$0 <= n <= 14$</span>) and <span class="math-container">$y[n]$</span> has the length of <span class="... | <p>If <span class="math-container">$x[n]$</span> and <span class="math-container">$y[n]$</span> are both causal and starting at index <span class="math-container">$0$</span>, then the result of convolution will also be causal and it will start at index <span class="math-container">$0$</span>. Just plug in <span class="... | 457 |
convolution | Prove that the twisted Convolution of $f$ and $g$ lies in $L^2(C^n)$? | https://dsp.stackexchange.com/questions/76627/prove-that-the-twisted-convolution-of-f-and-g-lies-in-l2cn | <p>Edit,this is the <strong>exact phrasing</strong> of my question</p>
<blockquote>
<p>Let <span class="math-container">$\lambda\in\mathbb{R^n}$</span>.Prove that</p>
<p><span class="math-container">$f\star_{\lambda}g\in L^2(\mathbb{C^n})$</span> for all functions <span class="math-container">$f$</span> and <span class... | 458 | |
convolution | Convolution output signal transient correction | https://dsp.stackexchange.com/questions/89536/convolution-output-signal-transient-correction | <p>I'm just starting out in signal analysis and I've come across this effect. When I use convolution of a sinusoid with any other, either triangular pulse, rectangular pulse or decreasing exponential (regardless of whether they are causal) a transient segment appears in the output signal. Is this effect of the convolut... | <blockquote>
<p>Is this effect of the convolution?</p>
</blockquote>
<p>Yes.</p>
<p>If you convolve two finite signals of length <span class="math-container">$N$</span> and <span class="math-container">$K$</span>, the resulting signal will have a length of <span class="math-container">$M = N + K + 1$</span>. Let's take... | 459 |
convolution | What is convolution of two sine waves (tones)? | https://dsp.stackexchange.com/questions/18480/what-is-convolution-of-two-sine-waves-tones | <p>Convolution of two sine waves (or tones as called in audio) is theoretically not defined as the integral is infinite. Taking finite duration windowed sine waves and doing there convolution computationally always contains a fundamental frequency equal to that of the lower frequency sine wave. I am not getting an intu... | <p>Convolution in the time domain is equivalent to multiplication in the frequency domain. </p>
<p>If you window two sinusoids in the time domain to get finite length waveforms, and the two sinusoids are exactly integer periodic in the window width, then the DFT will be impulses. If the frequencies are different, th... | 460 |
convolution | Convolution in frequency domain, what is the output bandwidth | https://dsp.stackexchange.com/questions/43828/convolution-in-frequency-domain-what-is-the-output-bandwidth | <p>If two signals do convolution in frequency domain, would the output width/BW be related to the width/BW input signals?
Would it be addition or any other?</p>
| <p>For signals <span class="math-container">$x(t)$</span> and <span class="math-container">$y(t)$</span> having domains <span class="math-container">$ t_1 < t < t_2 $</span> and <span class="math-container">$ t_3 < t < t_4 $</span>, respectively, their convolution in the time-domain <span class="math-contai... | 461 |
convolution | A better explaination for 'there's no flipping of the impulse response' | https://dsp.stackexchange.com/questions/8530/a-better-explaination-for-theres-no-flipping-of-the-impulse-response | <p>I showed the question about flipping the impulse response to some math teachers and electrical engineers at an university. They read Dilip's answer and thought that it was hard to understand what Dilip meant and it was simply not true...</p>
<p>Could someone confirm that it really isn't true what Dilip Sarwate wrot... | <p>Flipping the impulse response is really just a matter of perspective. The LTI system doesn't care about perspective. In any case, here is a graphic showing a system that takes an input of color weighted impulses.</p>
<p><img src="https://i.sstatic.net/Z2f2T.jpg" alt="enter image description here"></p>
| 462 |
convolution | Is there any meaning to the extra-points introduced by a convolution product? | https://dsp.stackexchange.com/questions/94494/is-there-any-meaning-to-the-extra-points-introduced-by-a-convolution-product | <p>Mathematically speaking, when convolving a <span class="math-container">$N$</span>-points signal with a <span class="math-container">$M$</span>-points signal, the resulting signal contains <span class="math-container">$N+M-1$</span> points.</p>
<p>In practice, the <span class="math-container">$N$</span>-points signa... | <p>Sure, the samples at the end contain the convolution of the last samples of your signal with the filter. Imagine your filter being a bit boring, having the (odd-length) impulse response</p>
<p><span class="math-container">$$(\underbrace{0,0, \ldots, 0}_{(M-1)/2\text{ zeros}}, 1,\underbrace{0,0, \ldots, 0}_{(M-1)/2\t... | 463 |
convolution | This isn't a typical convolution, what is it then? | https://dsp.stackexchange.com/questions/94669/this-isnt-a-typical-convolution-what-is-it-then | <p>In the numerous PlayStation emulators there are out there, this small piece of code is used to resample the input to feed to the reverb, which only works at 22050 Hz:</p>
<p><a href="https://github.com/libretro-mirrors/mednafen-git/blob/master/src/psx/spu_reverb.inc#L69" rel="nofollow noreferrer">https://github.com/... | <p>I assume that it is a 2x downsampler (from 44100 to 22050) implemented using a halfband lowpass filter (~50% of the taps are zero).</p>
<p>The code appears to produce only a single output sample (must be called repeatedly with the *src pointer shifted appropriately each time). Thus it is not what I would think of as... | 464 |
convolution | What is the physical meaning of the convolution of two signals? | https://dsp.stackexchange.com/questions/4723/what-is-the-physical-meaning-of-the-convolution-of-two-signals | <p>If we convolve 2 signals we get a third signal. What does this third signal represent in relation to the input signals?</p>
| <p>There's not particularly any "physical" meaning to the convolution operation. The main use of convolution in engineering is in describing the output of a <a href="http://en.wikipedia.org/wiki/LTI_system_theory">linear, time-invariant (LTI)</a> system. The input-output behavior of an LTI system can be characterized v... | 465 |
convolution | Why the delta at the end of the approximation? | https://dsp.stackexchange.com/questions/16410/why-the-delta-at-the-end-of-the-approximation | <p>The equation to approximate an input signal with a unit impulse in Continuous Time, is shown below, before we take the limit $\hat{x}(t)=\frac{lim}{\Delta\rightarrow0}\sum^{\infty}_{-\infty}x(k\Delta)\delta_\Delta(t-k\Delta)\Delta$ <-- why is there a final $\Delta$ multiplying the $\delta_\Delta(t-k\Delta)$?</p>
| <p>Here the signal is in continuous time domain. We can approximate any signal with weighted integral of unit impulse. As the signal is in continuous time domain <strong>integration</strong> is used instead of <strong>summation</strong>.</p>
<p>The actual equation is ,
$$x(t)=\int_{-\infty}^{\infty}{x(t_0)\delta(t-t_0... | 466 |
convolution | Unit step function present in convolution result | https://dsp.stackexchange.com/questions/28671/unit-step-function-present-in-convolution-result | <p>We are currently learning about convolution in my signals and systems class, and one of our textbook problems is to compute the convolution of $x(t) = e^{-\alpha t} u(t)$ and $y(t) = e^{-\beta t} u(t)$. After working out the integral, I obtained the following result</p>
<p>$$
x(t) * h(t) = \frac{e^{-\alpha t} - e^{... | <p>A step by step derivation of this convolution would start with the following:</p>
<p>$$
\begin{align}
f(t)
&= \int_{-\infty}^\infty x(\tau) h(t-\tau) \,d\tau \\
&= \int_{-\infty}^\infty e^{-\alpha\tau} u(\tau) e^{-\beta(t-\tau)} u(t-\tau) \,d\tau
\end{align}
$$
And this is probably where you probably we... | 467 |
convolution | Does the output of this system have a closed form expression? | https://dsp.stackexchange.com/questions/28831/does-the-output-of-this-system-have-a-closed-form-expression | <p>\begin{align}
x(n)&=\left(-1\right)^n u(n)\\
h(n)&= u(n-1)+u(n-2)\\
y(n)&= x(n)*h(n)\\
y(n)&= \sum_{k=-\infty}^\infty \left(-1\right)^k u(k)\left[u(n-k-1)+u(n-k-2)\right]
\end{align}
Graphically it looks like the output is $u(n-1)$ but I can't do the math to get it into the closed form. </p>
<p>Edi... | <p>You figured out the solution by yourself, so this is just about how to show it formally. First, split the sum into two sums:</p>
<p>$$y[n]=\sum_{k=-\infty}^{\infty}(-1)^ku[k]u[n-1-k]+\sum_{k=-\infty}^{\infty}(-1)^ku[k]u[n-2-k]\tag{1}$$</p>
<p>The unit steps give you upper and lower summation indices, but note that... | 468 |
convolution | What does the convolution mean, what is the convolution philosophy | https://dsp.stackexchange.com/questions/62233/what-does-the-convolution-mean-what-is-the-convolution-philosophy | <p>I would like to know why the convolution is necessary.</p>
<p>that is, who said that multiplying numbers with others and then adding them would tell us something?</p>
<p>If you could give me analogies without mathematics, it would be a great help for non-mathematicians.</p>
| <p>I'm somewhat in the same situation as you @Jony. But I'm working currently on a project involving audio processing, and I had to read a lot about convolution and related topics in the last few days.</p>
<p>I'm not sure I gained an <em>intuitive</em> understanding of convolution--esp. convolution vs. correlation-- b... | 469 |
convolution | Why is "only the term for k = n is nonzero" relevant? | https://dsp.stackexchange.com/questions/96007/why-is-only-the-term-for-k-n-is-nonzero-relevant | <p>In the book Applied Digital Signal Processing by Manolakis and Ingle, page 45, section 2.5.1 "Properties of convolution", the author says:</p>
<p>"If we consider a system with impulse response <span class="math-container">$h[n] = \delta[n]$</span>, direct substitution into the convolution formula give... | <p>Fix the value of <span class="math-container">$n$</span>, say <span class="math-container">$n=2$</span>, and Write out the sum explicitly instead of using mathematical gobblydegook like summation signs <span class="math-container">$\displaystyle \sum$</span>, like so:</p>
<p><span class="math-container">$$\cdots + x... | 470 |
convolution | Convolution for audio, how to control amplitude? | https://dsp.stackexchange.com/questions/68748/convolution-for-audio-how-to-control-amplitude | <p>I am implementation a convolution reverb method for audios array in Python using <code>scipy.signal.convolve</code>. I choose the fft mode, which essentially a product of the rfft of the two signals then ifft back.</p>
<p>What I am not sure is how to perform the convolution without changing the amplitude of the orig... | 471 | |
convolution | Matched filter - do I need to know the signal I am trying to find? | https://dsp.stackexchange.com/questions/68876/matched-filter-do-i-need-to-know-the-signal-i-am-trying-to-find | <p>I need to identify a matched filter, and so have gone through the mathematics to do this as listed here <a href="https://en.wikipedia.org/wiki/Matched_filter" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Matched_filter</a> which derives the optimal filter <span class="math-container">$h$</span> to apply t... | <p>As stated by others, the signal <span class="math-container">$s[n]$</span> has to be known to calculate the corresponding matched filter.</p>
<p>An alternative is to use the power of deep learning to emulate a matched filter response as shown in <a href="https://arxiv.org/abs/1701.00008" rel="nofollow noreferrer">th... | 472 |
convolution | PCA with CNN Tensorflow | https://dsp.stackexchange.com/questions/69130/pca-with-cnn-tensorflow | <p>I need to improve my model of Convolutional Neural Network (CNN). The goal is to recognize facial expression.
I've been using some strategies like dropout for regularization and Adam optimazer, but i can get a good accuracy. Therefore I'm trying to implement PCA.</p>
<p>I use to filters for convolution, max pooling ... | 473 | |
convolution | Twisted convolution - simplifying multiple integral via substitution | https://dsp.stackexchange.com/questions/69561/twisted-convolution-simplifying-multiple-integral-via-substitution | <p>I need help with understanding a proof from a paper. Following is the description:</p>
<blockquote>
<p>Let</p>
<p><span class="math-container">$$r_{1}(t) = \int \int h_{1}\left(\tau, \nu\right) e^{j 2\pi \nu (t - \tau)} s\left(t - \tau\right) d\tau d\nu$$</span></p>
<p><span class="math-container">$$r(t) = \int \int... | <p>Since these are two different integral equations, during substitution you cannot use the same variables for both the integrals.
On substituting <span class="math-container">$r_1(t-\tau)$</span> to <span class="math-container">$r(t)$</span> we get,
<span class="math-container">$$r(t) = \int \int h_2(\tau,\nu)e^{j 2\p... | 474 |
convolution | Linear convolution in the DFT domain | https://dsp.stackexchange.com/questions/71142/linear-convolution-in-the-dft-domain | <p>Let's say I have 2 sequences a and b in the time domain. Both are length N. A and B are the DFT of a and b. If I do a circular convolution of A and B in freq domain (A o B), then the IDFT of the result will be a x b in the time domain. Correct?
Now what happens if I do linear convolution of A and B in the freq domai... | 475 | |
convolution | Finding the convolution $a^n u[n]$ with $b^n u[n]$ | https://dsp.stackexchange.com/questions/71154/finding-the-convolution-an-un-with-bn-un | <p>Hi I'm trying to solve the problem when studying for an upcoming test. The given solution is <span class="math-container">$$y[n]= \frac{1}{b-a}(b^{n+1}-a^{n+1}) \quad \text{for } n\ge0 \ .$$</span> However, I'm not sure how to reach this point. We have to also verify using the z-transform. Any help would be apprecia... | <p>Just do it the hard way by writing out the individual formulas for the general convolution sum
<span class="math-container">\begin{align}
y[n] &= \sum_{k=-\infty}^\infty a[k]b[n-k]\\
&= \sum_{k=0}^\infty a[k]b[n-k] &\scriptstyle{\text{because }a[k]=0~\text{whenever } k < 0,}\\
&= \sum_{k=0}^n a[k]... | 476 |
convolution | How to compute inner product of Wavelet transform convoluted with signal | https://dsp.stackexchange.com/questions/71735/how-to-compute-inner-product-of-wavelet-transform-convoluted-with-signal | <p>I have two datasets <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> in a sparse wavelet basis, and I have two filters <span class="math-container">$f_1$</span> and <span class="math-container">$f_2$</span>. I’d like to compute the inner product of the convolutions <span class=... | 477 | |
convolution | Convolution of two sets | https://dsp.stackexchange.com/questions/72626/convolution-of-two-sets | <p>In my systems and signals course I had been asked a question about finding the convolution of two sets.
I was given:</p>
<p><span class="math-container">\begin{align}
x[n] &= \{3,2,1\}\\
h[n] &= \{1,-2,3\}\\
\text{Find}\quad y[n] & = x[n]\star h[n]
\end{align}</span></p>
<p>I have no idea on how to f... | <p>If it is difficult for you to remember or calculate the convolution of two sequences then you may try doing it as polynomial multiplication. <br/>
Think of <code>x[n]</code> and <code>h[n]</code> as polynomial coefficients. So we have <br/></p>
<pre><code>Px = 3x^2 + 2*x + 1
Ph = 1x^2 - 2*x + 3
</code></pre>
<p>Reme... | 478 |
convolution | Covolution of DTFT | https://dsp.stackexchange.com/questions/6211/covolution-of-dtft | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://dsp.stackexchange.com/questions/6199/convolution-and-dtft">Convolution and DTFT</a> </p>
</blockquote>
<p>$x_1(n)=x_2(n)=1$<br>
where $0 ≤ n ≤ N-1$</p>
<p>1)The linear convolution of the signal gives a triangle how you write it in mathem... | 479 | |
convolution | Reconstruction from Convolution | https://dsp.stackexchange.com/questions/8389/reconstruction-from-convolution | <p>I am trying to implement Wavelet transform in an embedded environment. The problem I face goes like this:</p>
<ol>
<li><p>Two vectors, A and B. A is input samples. B is vector co-efficients.</p></li>
<li><p>I take convolution of A and B, call it C.</p></li>
<li><p>Now I go back to B and up-sample the co-efficients.... | 480 | |
convolution | Circular Convolution in Scilab | https://dsp.stackexchange.com/questions/10029/circular-convolution-in-scilab | <p>I am quite familiar with signal processing using Matlab. Now I am working in its open source alternative 'Scilab.' In Matlab, cconv() function is used to perform circular convolution between two sequences. I want to do circular convolution in Scilab. But I could not find an equivalent function for circular convoluti... | <p>I can't tell about SciLab but if it is not implemented, you can always get the circular convolution by</p>
<p>IDFT(DFT(x1) * DFT(x2))</p>
<p>where x1 and x2 are your signal vectors and the multiplication is element-wise. If the signals have a different length, zeropad the shorter one to the length of the longer on... | 481 |
convolution | Simplification of a convolution-like expression | https://dsp.stackexchange.com/questions/10684/simplification-of-a-convolution-like-expression | <p>I am trying to further reduce the following expression (if there's a nicer way to write it):</p>
<p>$\sum_{k=-\infty}^{+\infty}\sum_{p=-\infty}^{+\infty} \sum_{l=-\infty}^{+\infty} w(k) \varphi_\mathrm{y}(l-p)\varphi_\mathrm{x}(k-n-l+p)$</p>
<p>where $\varphi_\mathrm{y}$ and $\varphi_\mathrm{x}$ are the autocorrel... | <p>As you say, since $\varphi_x$ is an autocorrelation we have that
$$\sum_{k=-\infty}^{\infty} w(k) \varphi_\mathrm{x}(k-n-l+p) =
(w \star \varphi)\bigr |_{n+l-p} = g(n+l-p)$$
where $g = w \star \varphi$.
So, since $\varphi_y$ is also an autocorrelation, we have
$$\begin{align}
\sum_{l=-\infty}^{\infty} g(n+l-p)\varp... | 482 |
convolution | Savitzky-Golay Filter Coefficients and Wikipedia | https://dsp.stackexchange.com/questions/10698/savitzky-golay-filter-coefficients-and-wikipedia | <p>I am trying to implement a Savitzky-Golay filter by following <a href="https://en.wikipedia.org/wiki/Savitzky%E2%80%93Golay_smoothing_filter" rel="nofollow">this Wikipedia</a> page and in the first instance I have written the following Octave software code to create the convolution coefficients</p>
<pre><code>clear... | 483 | |
convolution | Convolving time reversed discrete input with time reversed discrete impulse response | https://dsp.stackexchange.com/questions/12690/convolving-time-reversed-discrete-input-with-time-reversed-discrete-impulse-resp | <p>Question 3.1 d from Chapter 3 of Oppenheim's Signals & Systems
I have </p>
<p>$$x[n]=(-1)^n{u[-n]-u[-n-8]}$$
$$h[n]=u[n]-u[n-8]$$</p>
<p>and the question asks for $y[n]$ which is the convolution of $x[n]$ and $h[n]$ namely </p>
<p>$$y[n]=x[n]\star h[n]$$</p>
<p>I'm stuck with the limits on the sum </p>
<p>$... | <p>Here is another example of how incredibly poor choice of notation misleads students everywhere. I do not possess a copy of any of the various tomes on signal processing that seem to be revered as the fifth Gospel on this site, and so
I do not know if the book cited by the OP actually has written $(1)$,
but I assert ... | 484 |
convolution | How do I combine a copy of a signal with the original to achieve a chorus effect in a .wav file using Python? | https://dsp.stackexchange.com/questions/14514/how-do-i-combine-a-copy-of-a-signal-with-the-original-to-achieve-a-chorus-effect | <p>Can you please explain or point the right way to think about how to implement two sound effects in a .wav file.</p>
<p>So I read a .wav file and converted it to int values (in <code>numpy</code> array) </p>
<pre><code>array([59383, 58290, 60629, ..., 52606, 51982, 56274], dtype=uint16)
</code></pre>
<p>First, I a... | 485 | |
convolution | Finding linear convolution of two time series | https://dsp.stackexchange.com/questions/16414/finding-linear-convolution-of-two-time-series | <p>PROBLEM</p>
<p>Two real, causal time series</p>
<p>$$f_k = a^k \quad \text{and} \quad g_k = b^k$$</p>
<p>where $a \neq b$, $|a| < 1$, and $|b| < 1$, are given for $k = 0,1,2,...$</p>
<p>Find the linear convolution $h_n$ of the time series $f_k$ and $g_k$.</p>
<p>ATTEMPTED SOLUTION</p>
<p>I think perhaps ... | <p>There is a small mistake in your solution. Here $f_k$ and $g_k$ are infinite duration signal, ie. $n \to \infty$. You made a mistake that you took <strong>n</strong> in $h_n$ and number of coefficients in $f_k$ or $g_k$ as same, but it is different.</p>
<p>$$h_n = \sum_{k=0}^{\infty} f_k g_{n-k} = \sum_{k=0}^{\inf... | 486 |
convolution | How to derive the answer to this convolution problem? | https://dsp.stackexchange.com/questions/18136/how-to-derive-the-answer-to-this-convolution-problem | <p>I came across this below question, (which was a homework assignment question for Signal Processing class, which my friend mailed me for help solving), mulled over it for an hour and had no idea how to proceed with solving it. </p>
<p>Let $C(x) = A(x)B(x)$ where:</p>
<p>$$A(x)=\sum_{n=0}^{N_1}a(n)x^n$$
$$B(x)=\s... | <p><strong>HINT:</strong> what is the highest power of $x$ after multiplying $A(x)$ and $B(x)$? This gives you directly the value of $N_3$. Then rewrite $B(x)$ as</p>
<p>$$B(x)=\sum_{n=0}^{2N_2}\hat{b}_nx^n$$</p>
<p>and you can use normal convolution of $a(n)$ and $\hat{b}(n)$ to derive $c(n)$. Now you just need to e... | 487 |
convolution | Convolution of h(n) & x(n) | https://dsp.stackexchange.com/questions/18408/convolution-of-hn-xn | <p>can anyone please help me with finding Convolution for following problem .....</p>
<p>1) $h(n) = \delta[\sin(1 + |n|)]\quad \text{and}\quad x(n) = \sin(n^2)$</p>
<p>2) $h(n) = \delta[\sin(2\pi n/N)] \quad \text{and}\quad x(n) = n^2$ where $N$ is odd.</p>
| <p><strong>HINT:</strong></p>
<p>I assume that $n$ is an integer and that we're talking about discrete-time signals. You need to simplify the expressions for $h(n)$ by noting that $\delta(n)=1$ for $n=0$ and zero otherwise. So, for which values of $n$ does $\sin(1+|n|)$ equal zero, and for which values of $n$ does $\s... | 488 |
convolution | Implementing overlap add method for convolution with a slice of the filter | https://dsp.stackexchange.com/questions/19591/implementing-overlap-add-method-for-convolution-with-a-slice-of-the-filter | <p>In <a href="http://en.wikipedia.org/wiki/Overlap%E2%80%93add_method" rel="nofollow">overlap add method for convolution</a> a filter $x$ of length $M$ will get convolved with a signal $y$ of length $N$ where $M << N$, i.e., $z = x * y $.</p>
<p>Here the signal $y$ is sliced into subparts each of length $M$.</p... | <p>What you are describe is typically called "Partitioned Convolution". The benefit is reduced latency at some cost to efficiency (unless the larger size FFT is considerably less efficient e.g. due to cache size). This reduction in efficiency may be outweighed by the increased ability to use multiple cores for the pa... | 489 |
convolution | convolution of a real and complex signal | https://dsp.stackexchange.com/questions/19890/convolution-of-a-real-and-complex-signal | <p>How do we define convolution of: $$x(t)= \mathrm{sinc}(W t)$$ and $$y(t)= -j\mathrm{sinc}(2W t)$$ </p>
<p><strong>Answer:</strong> In the frequency domain, both of them are rectangular functions and multiplication of them is equal to the rectangular signal with smaller bandwidth i.e $x(t)$, which means convoluted s... | <p>You are confusing several different notions. First of all, two signals $x(t)$ and $y(t)$ are orthogonal if</p>
<p>$$\int_{-\infty}^{\infty}x(t)y^*(t)dt=0\tag{1}$$</p>
<p>From (1) it is clear that if two real-valued functions $x(t)$ and $y(t)$ are <em>not</em> orthogonal, i.e. the integral in (1) is not equal to ze... | 490 |
convolution | Convolution of one signal with an evenly spaced signal | https://dsp.stackexchange.com/questions/23189/convolution-of-one-signal-with-an-evenly-spaced-signal | <p>I do not specialize in signal processing so I wonder if there is any references to the following procedure.</p>
<p>Let $[n]=\{0,1,\ldots,n\}$.</p>
<p>Consider the function $f:[n] \to \mathbb{R}$ and $g:[kr] \to \mathbb{R}$, such that $\operatorname{supp} g = \{0,r,2r,3r,\ldots,kr\}$. Assume for simplicity, $n/r$ i... | <p>This is known as <a href="http://www.ece.unm.edu/faculty/bsanthan/ece539/note3.pdf" rel="nofollow">polyphase decomposition</a>. It is often used as en efficient implementation of filtering combined with decimation or interpolation.</p>
| 491 |
convolution | Limits of convolution | https://dsp.stackexchange.com/questions/26379/limits-of-convolution | <p>Say we have a function of time ($S(t)$) of the length $T$, and then a customized impulse response (say $I(t)$) of the length $T+N$. The question is, when $S(t)$ is convolved with $I(t$), what are the possible frequency domain (or alternately time domain) representation of all of the possible resulting functions?</p>... | <p>If the function is zero-valued throughout then so will be the output. But if you exclude this trivial homogeneous case, then your question can be reformulated as:</p>
<p>Given any function $f$ of compact support and any list $l$ of frequency domain zeros, does a function $g$ always exist that equals $f$ within the ... | 492 |
convolution | Approaching the response of an RC circuit via y=ifft(fft(x)fft(h)). Why is this valid? | https://dsp.stackexchange.com/questions/26924/approaching-the-response-of-an-rc-circuit-via-y-ifftfftxffth-why-is-this | <p>Some textbooks like (Numerical recipes the art of scientific computing) derive the DFT as a Riemann sum of the CTFT. With this in mind it would be natural then to approximate the identity $$y(t)=x*h=\mathcal{F}^{-1}\big\{XH\big\}$$</p>
<p>with the mathlab code y=ifft(fft(x).*fft(h)) which roughly means that my resp... | 493 | |
convolution | Convolution & FT properties | https://dsp.stackexchange.com/questions/28958/convolution-ft-properties | <p>I have been trying hardly to check if these equalities are true or false. However, I have not been able to conclude anything. Could you help me, please?</p>
<p>$$y[2n]=h[n]\star x[2n] $$</p>
<p>$$\mathfrak{F}(x[-n])e^{-jp\pi Fm}=\mathfrak{F}(x[-n+m]) $$</p>
| <p>I'll try to get you started.</p>
<p>For the first part:</p>
<ol>
<li>Write out the convolution sum for $y[n]=h[n]\star x[n]$</li>
<li>Replace $n$ by $2n$, which gives you an expression for $y[2n]$</li>
<li>Write out the convolution sum for $h[n]\star x[2n]$</li>
<li>Check if both expressions are equal. If you've d... | 494 |
convolution | Do I have to flip my kernel when performing an FFT-based convolution? | https://dsp.stackexchange.com/questions/29065/do-i-have-to-flip-my-kernel-when-performing-an-fft-based-convolution | <p>I have a 2D image and I convolve it with a 2D kernel image using FFT. So far I was always using symmetric kernels (e.g., Gaussian with stddev_x = stddev_y). As a result, I never bothered thinking whether I have to flip my kernel image or not because it wouldn't have made any difference. However, now I want to convol... | <p>Flipping & dragging is an animative method used in graphical computation of convolution in time (or space) domain. It is the result of an argument manipulation in $h[n-k]$ (or $x[n-k]$) signal animated as a function of n, but drawn on an axis of k, in the convolution sum: $$y[n] = x[n]*h[n]=\sum {x[k]h[n-k] }$$ ... | 495 |
convolution | Concept check on doing multiply in time with circular convolution in frequency domain | https://dsp.stackexchange.com/questions/29100/concept-check-on-doing-multiply-in-time-with-circular-convolution-in-frequency-d | <p>So I think I have my thinking correct on what I'm trying to do, but I just want to make sure I'm not assuming something that isn't true. Below is what I'm currently doing in the time domain.</p>
<p>$r(t)$ is my received signal<br>
$T$ is my period<br>
$MultCurrent = r(n)\cdot r(n-T)$<br>
$MultPrevious = r(n-T)\cdot... | 496 | |
convolution | An effective algorithm for convolution of very many long signals | https://dsp.stackexchange.com/questions/32445/an-effective-algorithm-for-convolution-of-very-many-long-signals | <p>I need to calculate the following linear convolution </p>
<p>$$ y[n] = h_1[n] \circledast h_2[n] \circledast h_3[n] \circledast \cdots \circledast h_{k-1}[n] \circledast h_k[n]$$ </p>
<p>where $k$ exceeds $5000$ and the length of each $h_i[n]$ exceeds $100,000$. I tried the method which is based on the circular c... | <p>So, I'm becoming a bit notorious with the whole "by using commonly available tools, how complex is your problem <em>really</em>?" business, but oh well:</p>
<p>I went ahead and simulated the "last" of your convolutions, i.e. I went ahead and picked up a FFT FIR (which is convolution by the overlap-save method Jason... | 497 |
convolution | Convolution and DFT size | https://dsp.stackexchange.com/questions/32449/convolution-and-dft-size | <p>Assuming that filtered signal $y(n)$ is given as</p>
<p>$$
y(n)=x(n)\star h(n)=\sum_{m=0}^{2N-1}x(m)h(n-m), \quad n \in[0,1,\ldots, 2N]
$$</p>
<p>where $\star$ is convolution parameter. $x(n)$ is discrete signal of lenght $2N$ and $h(n)$ is filter of the same lenght as $x(n)$. If I want to take for example $N$-po... | <p>Note that if $x[n]$ and $y[n]$ are non-zero in the interval $n\in [0,2N-1]$ (and, for convenience, they are defined as being zero outside that interval), the (linear) convolution sum can be written as</p>
<p>$$y[n]=(x\star h)[n]=\sum_{m=0}^nx[m]h[n-m],\qquad 0\le n<4N-1\tag{1}$$</p>
<p>Note that the upper limit... | 498 |
convolution | Convolving complex exponential with box function (discrete) | https://dsp.stackexchange.com/questions/34195/convolving-complex-exponential-with-box-function-discrete | <p>If I have $$x[n] = u[n] - u[n-4]$$ where $u[n]$ is the unit step function, and $$h[n] = e^{-i\frac{\pi}{2}n}$$</p>
<p>does $x[n] * h[n] = 0$? </p>
<p>I tried doing the convolution sum and I got: $1 - i - 1 + i = 0$ and I also just tried plotting points and think I got 0 also, but I'm not sure if I did either of th... | <p>The discrete-time Fourier transform (DTFT) of a length $N$ "discrete box" sequence starting at $n=0$ is</p>
<p>$$\sum_{n=0}^{N-1}e^{-jn\omega}=e^{-j(N-1)\omega/2}\frac{\sin\left(\frac{N\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}\tag{1}$$</p>
<p>The numerator of $(1)$ has zeros at frequencies</p>
<p>$$\o... | 499 |
spectral analysis | Spectral Analysis vs. Spectral Line Analysis | https://dsp.stackexchange.com/questions/26021/spectral-analysis-vs-spectral-line-analysis | <p>We usually talk about "spectral analysis" but some resources (<a href="http://www.sciencedirect.com/science/article/pii/016516849390090W" rel="nofollow">this paper</a> or <a href="http://www.alma.inaf.it/images/Moments.pdf" rel="nofollow">this doc</a>) talk about "spectral line analysis".</p>
<p>Does this make sens... | <p>Spectrum analysis is more general: it involves looking at the entire spectrum of a given signal.</p>
<p>Spectral line analysis assumes that the spectrum contains several peaks (lines) of interest at specific frequencies. The aim then is to find the precise frequency, magnitude, and phase of those peaks (lines).</p... | 500 |
spectral analysis | Unexpected peaks in spectral analysis | https://dsp.stackexchange.com/questions/35593/unexpected-peaks-in-spectral-analysis | <p>I am doing the spectral analysis of some antenna recorded signals by computing PSD (logarithmic absolute value of the FFT) and the spectrogram of the signal. The problem is that I get some unexpected spikes in the signal that appear through all the signal, as it can be seen in the next figure:</p>
<p><a href="https... | <p>The large spike at ~9MHz could be some form of interference from electronics near the receiver or internal to the receiver (maybe from using a poorly conditioned power source). My best guess for the spike at DC is a DC bias from quantization. If the digitizer truncates instead rounds, the resulting digital signal ... | 501 |
spectral analysis | Spectral analysis of positive signals | https://dsp.stackexchange.com/questions/9220/spectral-analysis-of-positive-signals | <p>Suppose that I have a sensor that can acquire samples $X[k]$ of the Fourier transform of an unknown signal $Y[t]$. An example is MRI, where the acquired data is in $k-$space. Now suppose that the unknown signal $Y[t]$ is known to be real and non-negative. My question is: is there a principled way to incorporate this... | <p>To give a complete answer to this question you're going to need to provide more details about the kind of models you're considering in the first place. But yes, in many cases you can augment those models with <em>a priori</em> constraints on $Y[t]$, such as $0 \leq Y[t] \leq 1$. </p>
<p>For example, if the standard... | 502 |
spectral analysis | Chi-squared distribution variable spectral analysis | https://dsp.stackexchange.com/questions/56226/chi-squared-distribution-variable-spectral-analysis | <p>I an interested in spectral analysis of a random signal.</p>
<p>Assume the signal <span class="math-container">$n(t)\sim\mathcal{N}(0,\sigma^2)$</span> is followed by the white Gaussian noise with zero mean and <span class="math-container">$\sigma^2$</span> variance.</p>
<p>The test signal is <span class="math-con... | 503 | |
spectral analysis | Spectral analysis on polling data | https://dsp.stackexchange.com/questions/66314/spectral-analysis-on-polling-data | <p>First-time questioner...</p>
<p>I'm writing an application (hobby/non-commercial, asp.net c#) to process polling data (answers to a single yes/no question) automatically collected from a website. The answers accumulate (yes = +1, no = -1) in a running total that ticks upwards and downwards over time. The applicatio... | 504 | |
spectral analysis | Power spectral analysis in baseband vs bandpass | https://dsp.stackexchange.com/questions/85115/power-spectral-analysis-in-baseband-vs-bandpass | <p>I have two signals <span class="math-container">$x_1(t)$</span> and <span class="math-container">$x_2(t)$</span> both with a baseband bandwidth of <span class="math-container">$W$</span> Hz. <span class="math-container">$x_1(t)$</span> is transmitted over carrier frequency <span class="math-container">$f_1$</span> H... | <p>It should not matter whether you perform your PSD estimation in baseband or not. Basebanding a signal is carried out with a complex demodulation, which simply shifts the frequency of the measurement. For a bandlimited signal, appropriate low-pass filtering will practically prevent amplitude distortion. Just make sur... | 505 |
spectral analysis | How to use function spectrum in R to perform spectral analysis | https://dsp.stackexchange.com/questions/47271/how-to-use-function-spectrum-in-r-to-perform-spectral-analysis | <p>I intend to perform spectral analysis with <code>R stat:spectrum</code> and then plot power spectrum.</p>
<p>My code is listed here:</p>
<pre><code> t <- seq(0,1024,by=0.1)
x <- cos(2*pi*t) + 0.75*sin(2*pi*4*t) + 2*sin(2*pi*6*t)
x.spec <- spectrum(x,log="no",span=5,plot=FALSE)
del <- 0.1
spx &... | <p>Hi: Your sampling frequency is $f_{s} = 10 $ but the frequency of the third component is $f_{3} = 6$.. Therefore, the nyquist criterion for the maximum frequency is not being met because $2 * 6 = 12$ which is not less than $f_{s} = 10$. I'm new at this but hopefully this is the reason why. Someone will surely corre... | 506 |
spectral analysis | Most appropriate spectral analysis method for a recording of a saxophone note? | https://dsp.stackexchange.com/questions/35224/most-appropriate-spectral-analysis-method-for-a-recording-of-a-saxophone-note | <p>I'm currently doing an investigation where I am attempting to resonant frequencies of an Alto Saxophone at various different notes. I have taken audio recordings of the notes with a sample rate of $44100\textrm{ Hz}$, and exported them into CSV files. Now, I am attempting to analyze this data using the signal proces... | <p>The plot OP shared looks similar to log magnitude spectrum(black line) plotted with its envelope(red line). so from what OP says, OP need to plot log magnitude spectrum of recorded audio/signal. One thing to remember for this analysis is that same note should be played for entire duration of the signal to be analyse... | 507 |
spectral analysis | Minimum discernible frequency in power spectral analysis | https://dsp.stackexchange.com/questions/76662/minimum-discernible-frequency-in-power-spectral-analysis | <p>Say I have a signal of length <code>20s</code> that contains signal from various (unknown) biological sources, e.g. heartbeat (~0.2Hz), respiration (~1Hz), and possibly som very-low frequency oscillations (~0.05Hz). My sampling frequency is <code>fs=10Hz</code>. I conduct a periodogram PSD estimate (hanning window, ... | <p>If the goal is to map out a <em>range</em> of low frequencies, then CWT is preferred over STFT, as it zooms logarithmically and will provide far more detail (<a href="https://overlordgolddragon.github.io/test-signals/" rel="nofollow noreferrer">examples</a>). If the goal is a few specific frequencies, then a targete... | 508 |
spectral analysis | Will noise cancellation with playstation eye improve my spectral analysis? | https://dsp.stackexchange.com/questions/54487/will-noise-cancellation-with-playstation-eye-improve-my-spectral-analysis | <p>Im doing some beginners spectral analysis for a homemade reactive RGB LED project and was thinking of ways to improve the accuracy of the audio analysis. Im using a Playstation eye with a Raspberry Pi Zero W (limited performance) and ive read several articles to try to better my understanding. According to this <a h... | <p>As I read your prompt, it seems that the real question you’re asking is why you can’t record frequency peaks greater than 7kHz. The second question seems to be whether you can improve this performance through noise cancellation techniques.</p>
<p>If that’s the case, and if you’re right about the 2nd and 3rd channel... | 509 |
spectral analysis | Wigner-Ville Distribution (WVD) vs STFT for Spectral Analysis | https://dsp.stackexchange.com/questions/86211/wigner-ville-distribution-wvd-vs-stft-for-spectral-analysis | <p>What are the Pros and Cons of using Wigner-Ville Distribution for spectral analysis of a signal vs taking the STFT. When is it appropriate to use one over the other and is WVD used in real-time implementations?</p>
<p><strong>Update:</strong> Since no answers were received for the first few days of this post, I trie... | <p>I preface this answer with that I know little about WVD and never worked with it, but do know time-frequency, and synchrosqueezing, which shares similarities. Part of my answer will be for SSQ.</p>
<h3>Re: ChatGPT</h3>
<blockquote>
<p>The WVD is a time-frequency representation that provides a high degree of resoluti... | 510 |
spectral analysis | Which sample autocovariance sequence estimate should I use for spectral analysis? | https://dsp.stackexchange.com/questions/95335/which-sample-autocovariance-sequence-estimate-should-i-use-for-spectral-analysis | <p>Is there any reason to choose a biased autocovariance sequence estimate over an unbiased estimate for power spectral density analysis? Why would we want to introduce bias into our estimate of the power spectral density?</p>
| <p><strong>TL;DR</strong>
While bias does decrease resolution in spectral estimators, there are two primary reasons the biased autocovariance sequence should be used. The first is that the unbiased ACS is indefinite, which can lead to negative spectral estimates, where as the biased ACS is positive definite (see <a hre... | 511 |
spectral analysis | How do I perform Spectral Analysis (FFT, Windowing, Detrending) on Sonic Anemometer Data? | https://dsp.stackexchange.com/questions/23900/how-do-i-perform-spectral-analysis-fft-windowing-detrending-on-sonic-anemome | <p>I am having trouble with all of the forums that I have looked at thus far.
I have one day's worth of Sonic Anemometer data. I want to see the spectral analysis of this data. When the anemometer data was being logged it appears that the frequency ranges between 31 and 32 hz. I am looking to analyze this data in thirt... | <p>take fft of data</p>
<p>let's say data is <code>x = [ 3 4 5 6 6 9 7 ]</code></p>
<p>take fft</p>
<pre><code>y = fft(x);
shift so -fs/2 to fs/2
y = fftshift(x);
plot mag
plot(abs(y))
</code></pre>
<p>see link for help
<a href="https://youtu.be/z10ISekkOQs" rel="nofollow noreferrer">fft Help</a></p>
| 512 |
spectral analysis | Non-Linear, Non-Stationary spectral analysis methods! When and where? | https://dsp.stackexchange.com/questions/59797/non-linear-non-stationary-spectral-analysis-methods-when-and-where | <p>I have been reading about non-linear non-stationary signal analysis methods and it seems to do this type of analysis the go-to method is the Empirical Mode Decomposition (EMD), then Hilbert Transform (HT) to get instantaneous phase and frequency.</p>
<p>However I have been doing some analysis on a non-linear, non-s... | <p>Non-linearity and non-stationarity are <strong>non-properties</strong>. Without more details, they do not say much about the methods that may perform well, and moreover the choice depends a lot on what you really do: analysis, feature extraction, enhancement, filtering, component separation, restoration? What follo... | 513 |
spectral analysis | Spectral analysis of a cross correlation function | https://dsp.stackexchange.com/questions/68308/spectral-analysis-of-a-cross-correlation-function | <p>So it's widely known that the Cross Correlation of 2 signals helps us in figuring out the time delay in those signals by analyzing the peak of the correlation coefficient in the time domain.</p>
<p>For something I am working on, due to a short baseline in comparison to the wavelength of the signal in between two of... | <p>cross-correlation would determine the time-delay. but the scale of signals' length and the delay value should be considered precisely. I mean if you have two long signal of, for say, 100sec and you are looking for a time delay in an order of 1msec, you don't need to do cross-correlation over the whole 100sec. Instea... | 514 |
spectral analysis | Spectral analysis with high selectivity but low number of points | https://dsp.stackexchange.com/questions/73360/spectral-analysis-with-high-selectivity-but-low-number-of-points | <p>I'm trying to create spectral plots similar to one below (it's from the <a href="https://www.analog.com/media/en/technical-documentation/data-sheets/AD1896.pdf" rel="nofollow noreferrer">datasheet</a> of an AD1986 sample rate converter, page 8):</p>
<p><a href="https://i.sstatic.net/3Zoev.png" rel="nofollow noreferr... | <p>The best way to do ae this measurement: Use a sine wave generator that is phase locked to your data acquisition clock with a frequency that's an integer multiple of your sample rate divided by the FFT length. If you do this, the period of your sine wave becomes an integer number of samples, you don't get any spectra... | 515 |
spectral analysis | How precisely can power spectral analysis be replicated? | https://dsp.stackexchange.com/questions/96123/how-precisely-can-power-spectral-analysis-be-replicated | <p>I was tasked with comparing my own PSA algorithm's results with those of a different lab. The data which is being analyzed is NREM epochs of a sleep EEG. The match I have obtained is, to my eyes, very decent in terms of the overall trend.</p>
<p>PSA is calculated as follows: divide each <span class="math-container">... | <blockquote>
<p>the code which the other lab uses is proprietary and hence unknown to any of us</p>
</blockquote>
<p>Hard to say. Spectral analysis has a lot of parameters you can tweak: window shape, overlap, hop size, smoothing, framing, FFT size, padding, etc. Could also be a non-standard way of doing things or just... | 516 |
spectral analysis | Comparison of WVD vs STFT Spectral analysis in the presence of Noise | https://dsp.stackexchange.com/questions/86297/comparison-of-wvd-vs-stft-spectral-analysis-in-the-presence-of-noise | <p>This question is an extension to the question about WVD vs STFT originally posted <a href="https://dsp.stackexchange.com/questions/86211/wigner-ville-distribution-wvd-vs-stft-for-spectral-analysis/86287?noredirect=1#comment182690_86287">Here</a>. During the QA it was pointed out that the WVD only works for noiseless... | 517 | |
spectral analysis | Spectral Analysis of a Time Series with Missing Data Points | https://dsp.stackexchange.com/questions/22930/spectral-analysis-of-a-time-series-with-missing-data-points | <p>I use a PC to record time series of some physical property. The problem is that, for some reason, I did not record the time series as a whole, rather I record first segment, then second, third, etc. Each segment is 500,000 points, representing ~30 min. However, they are not strictly consecutive: between each segment... | <p>Given <span class="math-container">$ \left\{ x \left[ n \right] \right\}_{n \in M} $</span> where <span class="math-container">$ M $</span> is the set of indices given for the samples of <span class="math-container">$ x \left[ n \right] $</span>.</p>
<p>The trivial solution (Which it would be great to have a faster ... | 518 |
spectral analysis | Amplitude ratio is larger than 0.5 on a RC-filter if RC = 1.7684e-03 | https://dsp.stackexchange.com/questions/84469/amplitude-ratio-is-larger-than-0-5-on-a-rc-filter-if-rc-1-7684e-03 | <p>I have two signals</p>
<pre><code>>> t = linspace(0, 10, 10000); % 0 to 10 seconds
>> u1 = 30*sin(2*pi*5*t); % Sine signal 1
>> u2 = 10*sin(2*pi*90*t); % Sine signal 2
>> u = u1 + u2; % Combine
>> spa(u, t); % Spectral analysis
>> grid on
</code></pre>
<p>And the spectral analysis... | 519 | |
spectral analysis | Fourier Like Spectral Analysis with Uneven Intervals and Redesigned DFT Matrix | https://dsp.stackexchange.com/questions/54569/fourier-like-spectral-analysis-with-uneven-intervals-and-redesigned-dft-matrix | <p>I intended to use a discrete Fourier transform (DFT) on a time series sampled at uneven intervals. What I did was to calculate a DFT matrix where the elements are the values at the uneven locations like this</p>
<p><span class="math-container">$$
{\displaystyle {\begin{aligned}X_{k}&= \frac{1}{\sqrt{N}}\sum _{n=... | <p>I have derived the DFT for data whihc is sampled in a non uniform manner:</p>
<h1>The DFT Matrix for Non Uniform Time Samples Series</h1>
<h2>Problem Statement</h2>
<p>We have a signal <span class="math-container">$ x \left( t \right) $</span> defined on the interval <span class="math-container">$ \left[ {T}_{1}, {T... | 520 |
spectral analysis | STFT frequency domain processing destroys window shape, makes synthesis glitchy | https://dsp.stackexchange.com/questions/88402/stft-frequency-domain-processing-destroys-window-shape-makes-synthesis-glitchy | <p>I have a signal processing chain which goes:</p>
<ul>
<li>
<ol>
<li>Every 256 samples, take 512 samples of input buffer</li>
</ol>
</li>
<li>
<ol start="2">
<li>Apply a Hann window to the samples</li>
</ol>
</li>
<li>
<ol start="3">
<li>Copy the windowed samples into the real components of a complex array</li>
</ol>... | <p>Frequency domain processing is complicated primarily because of "time domain aliasing".</p>
<p>Most frequency domain manipulations can easily be expressed as a linear filtering operation, i.e.</p>
<p><span class="math-container">$$H[k] = \frac{Y[k]}{X[k]}$$</span></p>
<p>where <span class="math-container">... | 521 |
spectral analysis | Clarification concerning power spectral density | https://dsp.stackexchange.com/questions/43155/clarification-concerning-power-spectral-density | <p>Many books in signal processing, e.g. Papoulis [1], define power spectral density (PSD) as:</p>
<p>$$S(\omega)=\sum_{k=-\infty}^{\infty}R_{xx}(k)e^{-j\omega k}$$</p>
<p>Which is the fourier transform of the correlation function:
$$R_{xx}(\tau)=E[x(t)x(t-\tau)]$$</p>
<p>However, some authors in the context of tim... | <p>I haven't seen Jenkins' context but for <em>zero mean</em> WSS random processes the <em>autocorrelation function</em> and the <em>auto covariance function</em> will the the same, hence you can use either of the definitions when the mean is zero.</p>
<p>For nonzero mean processes, see the relation between autocorre... | 522 |
spectral analysis | Discrete Fourier transform of a finite length signal which saturates at non-zero value | https://dsp.stackexchange.com/questions/72120/discrete-fourier-transform-of-a-finite-length-signal-which-saturates-at-non-zero | <p>I am performing spectral analysis of a finite length signal that saturates to a non-zero value. The signal (<span class="math-container">$s(t)$</span>) can, practically, be write as
<span class="math-container">$s(t) = f(t) \big(1-H(t-t_0)\big) $</span>, where <span class="math-container">$t_0$</span> is the time at... | <p>You can & should window your signal. Choice of window depends on the specific requirements of your application. Wikipedia has a really good overview on properties and trade-offs: <a href="https://en.wikipedia.org/wiki/Window_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Window_function</a></p... | 523 |
spectral analysis | Need help understanding FFT for timeseries analysis? | https://dsp.stackexchange.com/questions/93411/need-help-understanding-fft-for-timeseries-analysis | <p>I am studying the seismo-ionospheric anomalies associated with the Japan earthquake using total electron content derived from a GPS signal. I am trying to detect acoustic-gravity waves in the signal caused by the earthquake if possible.</p>
<p>I am trying to perform spectral analysis for this task using matplotlib's... | <p>I think you are approaching this backwards. In general I would recommend</p>
<ol>
<li>Start with a mathematical model of your system</li>
<li>Analyze that model and identify certain properties or behaviors that might be identifiable in that data</li>
<li>Create some hypothesis around these properties</li>
<li>Pick t... | 524 |
spectral analysis | Cepstrum analysis - Cut off frequencies displayed | https://dsp.stackexchange.com/questions/38837/cepstrum-analysis-cut-off-frequencies-displayed | <p>My inquiry is regarding the so called <a href="https://en.wikipedia.org/wiki/Cepstrum" rel="nofollow noreferrer">cepstrum</a> analysis.</p>
<p>By conducting some experiments, i have real time data at my disposal. The acquisition rate is 1600Hz.</p>
<p>I wanted to try cepstrum analysis just out of curiosity to see ... | <p>Presence of frequencies above cutoff in Cepstrum could be because of - </p>
<ul>
<li>Is lowpass filter used a perfect lowpass filter(does it removes all the frequencies above cutoff or just attenuates them) ?</li>
<li>If lowpass filter is not perfect, what are the relative magnitudes of high frequencies(abov... | 525 |
spectral analysis | what is meant by biased and unbiased Autocovariance sequence estimate? | https://dsp.stackexchange.com/questions/48407/what-is-meant-by-biased-and-unbiased-autocovariance-sequence-estimate | <p>In nonparametric spectral estimation method what is meant by biased and unbiased Autocovariance sequence estimate and why most commonly biased one is used?
I have the answer to the second question from Peter Stoica (SPECTRAL ANALYSIS OF SIGNALS)but not getting what he wants to say. </p>
| <p>In spectral estimation, one estimates "spectra" and other values. And one is interested in knowing whether the estimate $\hat{x}$ converge to the "true" but unknown value $x$. It is unknown, for instance because one has only access to a finite quantity $N$ of samples to compute with, a limited number of realization... | 526 |
spectral analysis | Indian Space Research Organization - Extent of spectral spread 1/3rd Octave Analysis | https://dsp.stackexchange.com/questions/74666/indian-space-research-organization-extent-of-spectral-spread-1-3rd-octave-anal | <p>I am a scientist at Indian Space Research Organization (ISRO) and will be representing my team. We work on Telemetry and Data Handling units for our Launch Vehicles.</p>
<p>We are preparing for our upcoming SSLV Launch (Small Satellite Launch Vehicle) and need some information to confirm if we are at par with the in... | <p>That's not really a signal processing question, but here are are some pointers.</p>
<p>3rd octave filters are well defined in ANSI S1.11-2004 and any decent commercial implementation should be fully compliant to the standard. The implementation is rather simple, so compliance isn't much of a problem. It's easy enou... | 527 |
spectral analysis | DCT - Measures of energy compaction gain achieved using DCT over FFT | https://dsp.stackexchange.com/questions/87162/dct-measures-of-energy-compaction-gain-achieved-using-dct-over-fft | <p>The discrete cosine transform(DCT) is a popular choice for spectral analysis in audio, video, image compression algorithms. This is primarily due its efficient "spectral compaction" property in comparison with Fast Fourier Transform(FFT). Is there a way to measure approximate "spectral compaction-gain... | <p>In <em>Discrete-Time Signal Processing</em> by <em>Oppenheim</em>, <strong>chapter 8.5</strong>, there is a quantification of mean squared error by setting coefficients to 0 for both DCT and DFT:</p>
<p><a href="https://i.sstatic.net/UumIR.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UumIR.png" alt... | 528 |
spectral analysis | Whats a good grad level audio DSP textbook? | https://dsp.stackexchange.com/questions/95762/whats-a-good-grad-level-audio-dsp-textbook | <p>Whats a good rigorous grad level textbook on DSP focused on audio/spectral analysis? Coming from a half proof based half applied math/physics background; have done several research projects where I worked with spectra and did audio analysis but never really understood all the details I always see in posts on this si... | <p>There aren't very many graduate-level DSP books that are specifically about audio. But there are important DSP books that you should have for graduate-level work in DSP, whether it's audio or some other signal class.</p>
<p>This would include, what we used to call <em>"scripture"</em>, <a href="https://ra... | 529 |
spectral analysis | Weak sense stationarity sufficient for multitaper spectrum analysis? | https://dsp.stackexchange.com/questions/19559/weak-sense-stationarity-sufficient-for-multitaper-spectrum-analysis | <p>This is a general question on multitaper spectral analysis of signals and stationarity</p>
<p>I work with time series which are non stationary. In this regard I have two questions</p>
<ol>
<li><p>Is weak sense stationarity of second order sufficient to apply multitaper spectrum to compute valid estimates from cont... | <p>The answers to the questions are </p>
<ol>
<li>Yes WSS is sufficient. </li>
<li><ul>
<li><strong>Quasiperiodic</strong> </li>
</ul>
<p><img src="https://i.sstatic.net/SIYxV.jpg" alt="enter image description here"></p>
<ul>
<li><strong>Semiperiodic</strong></li>
</ul>
<p><img src="https://i.sstatic.net/9ROKX.png"... | 530 |
spectral analysis | What math should I study to really understand signal processing? | https://dsp.stackexchange.com/questions/91794/what-math-should-i-study-to-really-understand-signal-processing | <p>I am reading an elementary book on signal processing - "<a href="https://books.google.co.in/books/about/Signals_Systems.html?id=LwQqAQAAMAAJ&redir_esc=y" rel="nofollow noreferrer">Signals and Systems</a>".</p>
<p>It never struck until recently, the math involving signal processing seemingly has a lot m... | <p>The entire series of books titled <em>Signal Processing</em> and authored by Oppenheim, Willsky, plus some random grad student(s), is designed to give a student who has the typical mathematical knowledge of a 2nd-year EE student a tool kit to do practical work. It does not, and can not, give a full mathematical gro... | 531 |
spectral analysis | What is spectral whitening? | https://dsp.stackexchange.com/questions/10183/what-is-spectral-whitening | <p>What is meant by "spectral whitening" in DSP?</p>
<p>What effect does spectral whitening have when used in image processing? (visually or otherwise...)</p>
<p>Where might spectral whitening be useful in audio processing or analysis? What would a spectrally whitened audio signal sound like?</p>
| <blockquote>
<p>What is meant by "spectral whitening" in DSP?</p>
</blockquote>
<p>Spectral whitening is usually an attempt to make the spectrum of the signal "more uniform". One reason this might be a good thing to do is that it can have the effect of making the autocorrelation of the signal "... | 532 |
spectral analysis | 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... | 533 |
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