# What is Convolution: Definition and 361 Discussions

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (

f

g

{\displaystyle f*g}
) that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reversed and shifted. The integral is evaluated for all values of shift, producing the convolution function.
Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation (

f

g

{\displaystyle f\star g}
) only in that either f(x) or g(x) is reflected about the y-axis; thus it is a cross-correlation of f(x) and g(−x), or f(−x) and g(x). For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator.
Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, engineering, physics, computer vision and differential equations.The convolution can be defined for functions on Euclidean space and other groups. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers.
Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.Computing the inverse of the convolution operation is known as deconvolution.

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1. ### Convolution vs Cross-correlation

Hello, Convolution is essentially superposition. Conceptually, a copy of the same mask/filter is essentially placed at every point in the signal (1D, 2D, ect.). Once all these convolution masks are in place, we just compute the sum and get the convolved signal. The integral formula for...
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### Fourier transform: duality property and convolution

Hello, First of all, I checked several other threads mentioning duality, but could not find a satisfying answer, and I don't want to revive years old posts on the subject; if this is bad practice, please notify me (my apologies if that is the case). For the past few days, I have had a lot of...
3. ### Solving this Differential Equation using Convolution

$s=c_1*\exp(-c_2*|(t)|)*r(t)$ But how can I solve $c_1+c_2$ ?
4. ### I Partial Derivative of Convolution

Hello, I am trying to calculate the partial derivative of a convolution. This is the expression: ##\frac{\partial}{\partial r}(x(t) * y(t, r))## Only y in the convolution depends on r. I know this identity below for taking the derivative of a convolution with both of the functions only...
5. ### Convolution of 2 signals

##x[n] = (\frac{1}{2})^{-2} u[n-4]## ##h[n] = 4^{n} u[2-n]## So I plotted x[k] and h[n-k] in picture but x[n] is 0 for n < 4, therefore ##y[n]## only has value for n >= 4. Therefore my sum is like that: ##y[n]=\sum_{k=4}^{\infty} 4^{n-k} (-\frac{1}{2})^k## ##y[n]=-4^{n}...
6. ### Python How to properly normalize convolution of Gaussian and Lorentzian

I'd like to plot the normalized convolution of a Gaussian with a Lorentzian (see the definitions in terms of full width half maximum (fwhm) in the attached image). Here is my attempt, but the print statements with np.trapz() do not return 1 in both cases, but rather ##\approx##0.2. I'd also like...
7. ### I Why doesn't sinc(x) converge to Gaussian upon repeated convolution?

Hello, I've read that repeated convolution tends, under certain conditions, to Gaussian distribution. I found this description helpful, and Wikipedia's version of this says: The central limit theorem states that if x is in L1 and L2 with mean zero and variance ##σ^2##, then...
8. ### I Domain of convolution vs. domain of Fourier transforms

Convolving two signals, g and h, of lengths X and Y respectively, results in a signal with length X+Y-1. But through convolution theorem, g*h = F^{-1}{ F{g} F{h} }, where F and F^{-1} is the Fourier transform and its inverse, respectively. The Fourier transform is unitary, so the output signal...

12. ### Engineering Finding the system output by convolution

Since there are initial conditions stated, I would have to craft the s equation in mind, in order to find the impulse by laplace inverse; which is this: ##(s^2Y(s)-sy(0)-y'(0))+8(sY(s)-y(0))+16Y(s)=x(s)## ##(s^2Y(s)+\frac{1}{2}s-1)+8(sY(s)-1)+16Y(s)=x(s)##...
13. ### Parametric distance of a line in a grid (Line Integral Convolution)

Hi, the above image is from the Line Integral Convolution paper by Cabral and Leedom. However, I am having a hard time implementing it, and I am quite certain I am misreading it. It is supposed to give me the distances of the lines like in the example below, but I am not sure how it can. First...
14. ### MHB Laplace Convolution: f(t)=-5t^2+9

f(t)=-5t^2+9\int_{0}^{t} \,f(t-u)sin(9u)du
15. ### I The signal is binned into time bins with a width ##δt##

Hi all. I would like to know about "binning window". This paper I'm reading says like this. Why do "convolving the data with the ##b(t)## before the sampling" and "binning into time bins with a width ##δt##" have the same meaning? I know I'm addicted to post to PF 😅 But this forum is so...
16. ### Semigroup property for convolution

Summary: Show that for this family of functions the following semigroup property with respect to convolution holds. Hi. My task is to prove that for the family of functions defined as: $$f_{a}(x) = \frac{1}{a \pi} \cdot \frac{1}{1 + \frac{x^{2}}{a^{2}} }$$ The following semigroup property...
17. ### Calculating Convolution Sum for Digital Signal Processing Class

Please see below my attempt to perform the convolution operation on two discrete-time signals as part of my Digital Signal Processing class. I suspect my folding operation, i.e. flipping one signal about k=0, might be the cause. Ostensibly the answer of the convolution sum evaluated at n=-2...
18. ### Convolution Help on tri(x,y) ** (step(x) * 1(y))

I have some confusion about this question. I am asked to do the 1D convolution of a function that is clearly 2-dimensional tri(x,y) ** (step(x) * 1(y)) where ** is the convolution. Furthermore my professor is not available for questions (have tried). I'm wondering if I simply ignore the bits...

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20. ### Output as the convolution of the Impulse response and input

As the title says, I am studying this topic for my control systems fundamentals course. I think I intuitively understand the meaning of the convolution integral that relates input, output and the impulse response, but I am failing to prove it graphically. For example, the intuitive explanation...
21. ### A Doubt about a time convolution master equation

I study of interaction between a system with a reservoir considering a weak coupling between them. I consider a bosonic bath, the initial state are separable and the operator of interaction between the system and bath is linear in the displacements of the oscillators. . In the book "Quantum...
22. ### C/C++ Maximising a convolution in C++ via a GSL routine

Consider an integral of the form $$\int_{-1}^1 dx f(x)g(x).$$ I'd like to use https://www.gnu.org/software/gsl/doc/html/min.html to find the maximum of the convolution ##f(x)g(x)## in the domain ##x \in [-1,1]##. The method initiates via a double function with parameters x and a void params...
23. ### Fourier transforms, convolution, and Fraunhofer diffraction

I've been exposed to this notion in multiple classes (namely math and physics) but can't find any details about how one would actually calculate something using this principle: Diffraction in optics is closely related to Fourier transforms and finding the Fraunhofer diffraction of an aperture...

31. ### Convolution - Fourier Transform

Homework Statement An LTI system has an impulse response h(t) = e-|t| and input of x(t) = ejΩt Homework Equations Find y(t) the system output using convolution Find the dominant frequency and maximum value of y(t) Ω = 2rad/s The Attempt at a Solution I have tried using the Fourier transform...
32. ### General solution for the heat equation of a 1-D circle

Homework Statement Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions ## g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta} ## Using the method of Green's functions, and ## S(\theta,t)= \frac{1}{\sqrt{4\pi...
33. ### A Convolution operator spectrum

Hi there, I am also familiar with Hilbert spaces and Functional Analysis and I find your question very interesting. I agree that the Fourier transform is a powerful tool for analyzing LTI systems and diagonalizing the convolution operator. As for your question about whether the same can be...
34. ### I Independence of variables in Convolution

Given a convolution: \begin{equation} \begin{split} g(x) * h(x) &\doteq \int_{-\infty}^{\infty} g(z) h(x-z) dz \end{split} \end{equation} Do ##z## and ##x## have to be independent? For example, can one set ##x=z+y## such that: \begin{equation} \begin{split} \int_{-\infty}^{\infty} g(z)...
35. ### A Understand Convolution, Singularity, Kernel, etc: Math Reading Guide

I'm reading a book on vortex methods and I came across the above mentioned terms, however, I don't understand what they mean in mathematical terms. The book seems to be quite valuable with its content and therefore I would like to understand what the author is trying to say using the above...
36. ### Mathematica Mathematica: Convolution Integral

Hi all! I'm new to Mathematica. I have written a code for performing a convolution integral (as follows) but it seems to be giving out error messages: My code is: a[x_?NumericQ] := PDF[NormalDistribution[40, 2], x] b[k_?NumericQ, x_?NumericQ] := 0.0026*Sin[1.27*k/x]^2 c[k_?NumericQ...
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48. ### Generalisation of Parseval's Theorem via Convolution Theorem

Homework Statement [/B] Suppose we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{C} whose Fourier coefficients are known. Parseval's theorem tells us that: \sum_{n = -\infty}^{\infty}|\widehat{f(n)}|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^{2}dx, where...
49. ### A Cauchy convolution with other distribution

I have a set of data which are probably convolutions of a Cauchy distribution with some other distribution. I am looking for some model for this other distribution so that a tractable analytic formula results. I know that the convolution Cauchy with Cauchy is again Cauchy, but I want the other...
50. ### A Convolution Questions: Expectation Value & PDF Method

Hi Two questions: 1) I saw this definition of expectation value: E[g(X)] = integral wrt x from -inf to inf of g(x)*f(x)*dx for some function g(x) of a random variable X and its density function f(x). Can this be used to derive why convolution gives the density of a random variable sum? 2) In...