Questions tagged [convolution]

Convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions.

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Filtering relationship for sources propagating over a sensor array

A sensor array $y$ measures the superposition of $N$ sources $s$, at time $t$ and position $x$ we have : $$y(t,x)=\sum_{i=1}^{N} s_i(t,x)$$ The sources $s$ travel at constant speed over the array of $...
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Calculating cross-correlation using Walsh-Hadamard transform

I am trying to implement MLS method of measuring impulse responses. There is an article describing the method: http://www.commsp.ee.ic.ac.uk/~mrt102/projects/mls.... As I understand, to get an impulse ...
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Physically, what does the usage of two variables mean for convolution

My intuition of convolution is that it is just a way to depict multiplication of two signals where each signal is made up of various frequencies and phases. Since it isn't easy to find the value of $\...
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Image convolution using FFT

I just can't get my head around Fourier transform and convolution in 2D. I am trying to implement image convolution using fast Fourier transform (in julia). So the first thing I need to do is to pad ...
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+200

Combining signals in MFCC space

I'm looking at a data augmentation method for training up a neural network of speech data. Currently I have two version of augmentation. The first method works by taking an audio file and mixing in a ...
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bit reversed order of FFT matrix for channel estimation

Although the question seems similar to this one HERE, but what I want to ask about is little bit different. Assume we have a signal $X$ whose length is $N$x$1$, we convert that signal into time ...
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Channel estimation using bit reversed order of FFT matrix

Is it possible to use the bit-reverse order of FFT instead of FFT to estimate the channel? Based on my reading, I found that bit reversed order of FFT matrix can diagonalize the channel, but ...
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Scaling the input vs scaling the impulse response for an LTI system

Two different cases: We pass $x(t)$ to an LTI system with impulse response $h(2t)$ and get the output $y(t)$. We pass $x(2t)$ to an LTI system with impulse response $h(t)$ and get the output $z(t)$. ...
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Upsampling class activation maps for discriminative feature localisation

I am currently reading a paper by on learning deep features for discriminative localization where the authors propose to use class activation maps to learn discriminative localised features. The ...
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convolution of two exponential signals with imaginary numbers

I can solve problems without imaginary numbers, but when exponential contain imaginary numbers, I can't solve the problem. For example, $x(t)=e^{3jt}+e^{4jt}$, $y(t)=(e^{-3t}-e^{-4t})u(t)$ (where $j$ ...
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Unit impulse response of a cascade interconnection of three discrete-time systems

I am nearly at the end of finishing a problem in my textbook but I couldn't understand something in the answer; I did everything to the point I found the overall response of the system in terms of $...
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Edge artifacts in frequency-domain Moving Average filter

I am currently trying to implement in python an algorithm to identify and filter out one specific periodic EEG artifact -- the grad fMRI artifact -- as described in this paper, and I am incurring in ...
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Finding Interval of Integration

If we let : $$ x(t)=\begin{cases} 1&\text{if $0<t<1$}\\ 0&\text{if otherwise} \end{cases} $$ and $$ h(t)=x(t/a)=\begin{cases} 1&\text{if $0<t<a$}\\ 0&\text{if otherwise}\...
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Computing Discrete Convolution in terms of unit step function

Consider : $$ x[n]:=\begin{cases} 1&\text{if $3\leq n\leq 8$}\\ 0&\text{if otherwise} \end{cases} \quad\text{and}\quad h[n]:=\begin{cases} 1&\text{if $4\leq n \leq 15$}\\ 0&\text{if ...
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Confused about shifting convolutions

We know that for two signals $x[n]$ and $h[n]$ such that : $$ y_{1}[n]=(x*h)[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]=\sum_{k=-\infty}^{\infty}h[k]x[n-k] $$ We can deduce that :$$y_{2}[n]=x[n+2]*h[n]=y_{...
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What is the best way to spatialise a real-time stream of a speech? Developing a C++ engine

I'm working on a mobile application in C++. The app brings listeners to a virtual room to listen to real-time streams, such as interviews, talks, lectures. The idea is to make sound immersive, to give ...
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Determine g(t) for minimal symbol error probability, given h(t), K=1 and Minimum distance receiver

Determine $g(t)$ for minimal symbol error probability, given $h(t)$, $K=1$ and Minimum distance receiver. $h(t)= \begin{cases} 0, & \text{$t<0$}\\ 2, & \text{$t\in(0\;, 10^{-3}...
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Constructing a Gaussian kernel in the frequency domain

I'm currently learning about Fourier transform, but find the differences between spatial domain and frequency domain a bit confusing at times. Let's say I would like to perform convolution of an image ...
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Why there is static noise and voice distortion after convolving it with impulse response?

I have been working on the decorrelation of audio signals while following this paper : The Decorrelation of Audio Signals and Its Impact on Spatial Imagery So far I have generated impulse responses in ...
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Is the expansion of this expression correct?

\begin{align*} &(\delta[n]+\delta[n-1])*(\delta[n]+\delta[n-1])\\ \\ =\;&\delta[n]*\delta[n]+2(\delta[n]*\delta[n-1])+\delta[n-1]*\delta[n-1]\\ \\ =\;&(\delta[n]+2(\delta[n]*\delta[n-1])+\...
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UPDATE : How to continue Computing the Convolution

$$ x(t):=\begin{cases} 1&\text{if $0<t<T$}\\ \\ 0&\text{if otherwise} \end{cases} \qquad\text{and}\qquad h(t):=\begin{cases} t&\text{if $0<t<2T$}\\ \\ 0&\text{if otherwise} ...
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Where is the mistake in the convolution?

Let : $$ x[n]=\begin{cases} 1&\text{if $0\leq n\leq 4$}\\ \\ 0&\text{if otherwise} \end{cases} \qquad \text{and} \qquad h[n]=\begin{cases} \alpha^{n}&\text{if $0\leq n\leq 6$}\\ \\ 0&\...
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Discrete Convolution of two piecewise sequences having this specific form

Assume I have the following two sequences : $$ x[n]=\begin{cases} \alpha&\text{if $a\leq n\leq b$}\\ \\ \tag{1} 0&\text{if otherwise} \end{cases} \qquad \text{and} \qquad h[n]=\begin{cases} \...
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Finding the impulse response given response to another signal

I was trying to solve this question : I respresented $x(t) = u(t+1)-u(t-1)$ writing the convolution as $[u(t+1)-u(t-1)]*h(t) = y(t)$ I then used the property of differentiation to convert from the ...
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Initial rest condition applied on $x(t)$ vs $h(t)$

Define the LTI system $\mathcal{H} : x\mapsto y$ Define the convolution for continuous-time system : $$ (x*h)(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\;\text{d}\tau $$ The initial rest condition ...
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Circular wrapping of an asymmetric function (in DFT calculations)

Convoluting a signal (using discrete FT) for a given interval [a, b] with a Gaussian can be done by circular wrapping as shown in Numerical Recipes. I found a shortcut for circular wrapping so that ...
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Convolution of discrete signals using convolution sum

I want to perform the convolution of the following discrete signals: $$h[n]=u[n-2] $$ and $$x[n] = (0.5)^nu[n+2]$$. That's what I've done so far: $$\sum_{k=-\infty}^{\infty} (0.5)^ku[k+2]*u[n-2-k]$$ ...
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Deconvolution of sidelobes in a point spread function?

It seems that most deconvolution algorithms mainly handle the main lobe of a point spread function (PSF) and assume that sidelobes can be safely neglected. For a direct algorithm trying to perform a ...
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PID and convolution

In that formula : $$ u(t)=K_{\mathrm{P}} e(t)+K_{\mathrm{I}} \int_{0}^{t} e(\tau) \mathrm{d} \tau+K_{\mathrm{D}} \frac{\mathrm{d} e(t)}{\mathrm{d} t} $$ I know from the formulae collection we refer to ...
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Filter length (N) has no effect on the reconstructed signal & Toeplitz Matrix

Hello Signal Processing community, I implemented a filtering algorithm called Minimal Mean Squared Error in Frequency domain with N-coefficient and Signal Detection (FMSENSD) in MATLAB, I got some ...
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Delay a signal in time vs in frequency

I have a signal h that I want to delay for a time t. I know that I could use two possible approaches: $h_{\text{delay}}[k] = \...
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How do I perform 2D Fourier domain multiplication if the filter mask doesn't match the image size?

Let's say I have an image that is 512 x 512 pixels. I've been tasked with creating two ideal half-band low-pass filters that will filter the image. The first filter is 8 x 8, and the second one is 16 ...
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Feature Based Methods for Quality Inspection Problem

I am currently working on an image classification problem to classify thermal images (can be interpreted as false-color images) of certain products as with or without defect. From an initial search in ...
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Filter Bank and Auto-Encoder

I'm trying to find an intuition behind auto-encoder using an analogy with filter banks. I can comprehend the encoder and analysis filters in a filter bank as extracting features from the input signal ...
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How can convolution be a linear and invariant operation?

I'm having a slight breakdown right now with a seemingly simple question. Say I have a system that convolves an input function with itself to produce an output function: $g(x) = f(x) ∗ f(x)$ I've ...
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Couting exact number of mutiplications in Separable filters with Zero Padding

I'm convolving an Image $I$ of dimension $2\times 3$ with a kernel of size $2 \times 2$ and counting the exact number of multiplications.My image and kernel are given as follows $$ I=\left[\begin{...
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Why is FFT-based convolution efficient only for signals of large size?

According to the documentation of scipy.signal.fftconvolve This is generally much faster than convolve for large arrays (n > ~500), but can be slower when only ...
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What is the purpose of wrapping the negative times of a response function in discrete convolution?

I am trying to rationalize a figure given in the Numerical Recipes in C in the section of Fourier based convolution and deconvolution. The authors show the example of a continuous convolution with a ...
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Reconstructing signal from frames

Edit: Please treat code here as python-esque pseudo-code; it would syntactically fall closest to MATLAB or Python+numpy+scipy users. If I have a signal in frames, and I want to put it back together - ...
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Convolution of two sets

In my systems and signals course I had been asked a question about finding the convolution of two sets. I was given: \begin{align} x[n] &= \{3,2,1\}\\ h[n] &= \{1,-2,3\}\\ \text{Find}\quad ...
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What is the correct length for obtaining a true linear convolution from DFT?

In the linear convolution of two equal length sequences M and N, the length of the output is length(A)+length(B)-1, and if we apply the DFT property of converting convolution into multiplication, the ...
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Possibilities in Audio Convolution Math

New Python user here! I am a musician working on a program to streamline a process of using convolution math on a folder of .wav samples. The idea is that each audio sample will be multiplied with ...
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multiplication of a function with a Fourier-transformed equals to Fourier-transformed with a function

I already showed b item using the fact that it is $h\left(0\right)=\int \:f\left(t\right)g\left(0-t\right)dt$ I struggle a lot of hours trying to find the trick in item C. Can anyone help please ?
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Inverse system of sinc?

I'm doing some self-study for an important exam I'll have in late March and came across the following question: So, using the convolution properties, if I want to find an identity system so that the ...
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Concrete use of convolution algebraic properties [closed]

There are 3 main properties of convolution in every image processing lecture notes I read : Commutativity: $f\star h = h\star f$ Associativity: $f\star (h_1\star h_2) = (f\star h_1)\star h_2$ ...
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Mathematical expression of Multicarrier frequency equalizer

I have a signal $X$ with length of $N$, multiplied with any unitary matrix, i.e the transpose of DCT matrix as: $x = D' X$ where $D'$ is the transpose of the of DCT matrix. Then let's add $CP$ guard ...
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Normalization factor in the convolution theorem

Maybe it's a trivial question, but I couldn't find any explanation for this. According to the convolution theorem, in the continues case we add normalization factor, i.e. $$ \mathcal F\left\{h\star g\...
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Circular Convolution as Cyclic Shift Operator

Given the following signal vectors: $$ γ=[ψ_0,0,ψ_1,0,ψ_2,0,…,ψ_{N-1},0]^T\in \mathbb{R}^{2N}, ϕ=[1,\frac{1}{2},0,…,0,\frac{1}{2}]^T \in \mathbb{R}^{2N}$$ I want to show that the convolution of $γ$ ...
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Convolution output signal is very large, do I need to scale inputs or outputs?

I am writing some simple matlab code to refresh my convolution knowledge (it has been awhile) in regards to DSP. I have an input signal x and a filter - ...
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Impulse response of forward difference cascaded with one sample delay

Below is the excerpt from Discrete Time Signal Processing by Alan Oppenheim. I don't get how $(\delta[n+1] - \delta[n]) * \delta[n-1])$ becomes $\delta[n] - \delta[n-1]$. The convolution sum operator ...

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