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 (



{\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 (



{\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. F

    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...
  2. F

    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. J

    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. J

    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. Yalanhar

    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. S

    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. S

    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. S

    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...
  9. A

    A Convolution of two geometric distributions

    I'm trying to derive the convolution from two geometric distributions, each of the form: $$\displaystyle \left( 1-p \right) ^{k-1}p$$ as follows $$\displaystyle \sum _{k=1}^{z} \left( 1-p \right) ^{k-1}{p}^{2} \left( 1-p \right) ^{z-k-1}.$$ with as a result: $$\displaystyle \left( 1-p \right)...
  10. M

    Engineering Image Processing: Convolution vs Filtering

    Hi, So my question is perhaps better asked as: - What is the point of convolution (in 2D image processing)? - Why would we use that operation in image processing? - What is so special about that flipped version of the kernel? Context: In an image processing class, I was learning about the...
  11. agnimusayoti

    Convolution Problem (ML Boas, Ch 8, Sec 10, Prob 18)

    I can use the convolution integrals and get the idea of this concept for t<a. But, I can't get the answer for t>a. MY idea is substitute ##f(t) = 0## to the ODE, then I have second order linear differential equations with right hand is zero. So, the solution is $$y=Ae^{i\omega t} + Be^{-i\omega...
  12. jisbon

    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. Avatrin

    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. A

    MHB Laplace Convolution: f(t)=-5t^2+9

    f(t)=-5t^2+9\int_{0}^{t} \,f(t-u)sin(9u)du
  15. arcTomato

    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. A

    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. wirefree

    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. snatchingthepi

    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...
  19. R

    Laplace inverse convolution

  20. cnh1995

    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

    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/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. M

    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...
  24. J

    MHB "Approximation to the Identity" and "Convolution" Proof

    Problem: Let $\phi(x), x \in \Bbb{R}$ be a bounded measurable function such that $\phi(x) = 0$ for $|x| \geq 1$ and $\int \phi = 1$. For $\epsilon > 0$, let $\phi_{\epsilon}(x) = \frac{1}{\epsilon}\phi \frac{x}{\epsilon}$. ($\phi_{\epsilon}$ is called an approximation to the identity.) If $f \in...
  25. K

    Convolution Problem -- Triangular and Rectangular pulses

    Homework Statement Homework Equations y(t)=x(t)*h(t)=∫x(λ)⋅h(t-λ)⋅dλ The Attempt at a Solution [/B] Is what I have the correct interpretation or or am I wrong? Thanks
  26. H

    I What type of convolution integral is this?

    Convolution has the form (f\star g)(t) = \int_{-\infty}^{\infty}f(\tau)g(t-\tau)d\tau However, I for my own purposes I have invented a similar but different type of "convolution" which has the form (f\star g)(t) = \int_0^{\infty}f(\tau)g(t/\tau)d\tau So instead of shifting the function g(t)...
  27. A

    Trying to intuit the unit impulse response

    Homework Statement Hi there, I've been trying to gain some intuition on how the convolution sum works, but as I dig deeper I am realizing that there is an issue with my intuition of signals and systems, in particular the unit impulse response. My issue is trying to understand how a unit...
  28. J

    Convolution theorem property about signals

    Homework Statement Homework Equations Laplace and then inverse laplace. The Attempt at a Solution Laplace of U(t-to) = 1/s e^(-tos) x(t)-->X(s) Laplace inverse 1/s means integration. e^(-tos) means delay on x(t) by to. I think answer should be C Book answer is D. How am I wrong?
  29. jsg94

    Determine impulse response given input and output signals

    Homework Statement Hello everyone, In the following problem I have to find the unknown impulse response g1(t) given the input and output signals, as shown below: (the answer is already there, at the moment I am trying to understand how to get there). Homework Equations [/B] I have...
  30. M

    MHB Calculate density using convolution formula

    Hey! :o Let $X_1, X_2, X_3$ be i.i.d. with $X_1 \sim U[0, 1]$. I want to determine the density of $S=X_1+X_2+X_3$ using the convolution formula. I have done the following: Since $X_1, X_2, X_3$ are i.i.d. we have that they are independent identically distributed random variables. Since $X_1...
  31. Jonski

    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. G

    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. M

    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. redtree

    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. vktsn0303

    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. Q

    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...
  37. M

    Convolution Fourier Transform

    Homework Statement $$u_{xx}=u_t+u_x$$ subject to ##u(x,0)=f(x)## and ##u## and ##u_x## tend to 0 as ##x\to\pm\infty##. Homework Equations Fourier Transform The Attempt at a Solution Taking the Fourier transform of the PDE yields $$ (\omega^2-i\omega) F\{u\}=...
  38. G

    Laplace transform of derivative of convolution

    Prelude Consider the convolution h(t) of two function f(t) and g(t): $$h(t) = f(t) \ast g(t)=\int_0^t f(t-\tau) g(\tau) d \tau$$ then we know that by the properties of convolution $$\frac{d h(t)}{d t} = \frac{d f(t)}{d t} \ast g(t) = f(t) \ast \frac{d g(t)}{d t}$$ Intermezzo We also know that...
  39. A

    Convolution integral problem

    Homework Statement Hi all, I hope you all can help me so I'm studying for my signals course and I encounter this example in the book, and the answer is there but the solution isn't... The convolution integral exists for 3 intervals and I could evaluate the first two just fine... however I can't...
  40. O

    Convolution Properties

    How were they able to simplify the following? I understand the distributive property and how the convolution component of the delta dirac function worked but I do not understand how the second term convoluted becomes what it is. Thank you for your time
  41. kaniello

    I Calculating the Convolution Integral for General Math Community

    Dear "General Math" Community, my goal is to calculate the following integral $$\mathcal{I} = \int_{-\infty }^{+\infty }\frac{f\left ( \mathbf{\vec{x}} \right )}{\left | \mathbf{\vec{c}}- \mathbf{\vec{x}} \right |}d^{3}x $$ in the particular case in which f\left ( \mathbf{\vec{x}} \right...
  42. Uriel

    A Problem with a convolution algorithm

    Hi. I've been reading "Statistical Mechanics Algorithms and Computations". And I came to a problem while processing Algorithm 1.26 (I attach a link at the end). I don't get why the weights are the way they are, specially I can't understand the sequence {1/2l,1/l,...,1/l,1/2l}. Does anyone...
  43. E

    Convolution Neural Network

    Hi I am using a convolution neural network (with inversion) to simulate images with the same "texture" as the input image, using a random image to start with. The activations of the CNN are first learned with an example or source image. A cost function then minimizes the difference between the...
  44. Pouyan

    Finding inverse of a Laplace transform by convolution

    Homework Statement find the inverse Laplace transform of the given function by using the convolution theorem Homework Equations F(s) = s/((s+1)(s2)+4) The theorem : Lap{(f*g)(t)} = F(s)*G(s) The Attempt at a Solution I know how to find it the answer is : we have 1/(s+1) * s/(s+4) and the...
  45. B

    Convolution of two functions

    Homework Statement I have the two functions below and have to find the convolution \beta * L Homework Equations Assume a<1 \beta(x)=\begin{cases} \frac{\pi}{4a}\cos\left(\frac{\pi x}{2a}\right) & \left|x\right|<a\\ 0 & \left|x\right|\geq a \end{cases} L(x)=\begin{cases} 1 &...
  46. R

    I Derive Convolution Expression for Z_PDF(z)

    Hi Can I derive the expression for Z_PDF(z) where: Z = t(X,Y) = X + Y By starting with: Z_PDF(z)*|dz| = X_PDF(x)*|dx| * Y_PDF(y)*|dy| Z_PDF(z) = X_PDF(x) * Y_PDF(y) * |dx|*|dy|/|dz| and then substitute the deltas with derivatives and x and y with expressions of z?
  47. FeDeX_LaTeX

    I Discrete Convolution of Continuous Fourier Coefficients

    Suppose that we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{R}, whose continuous Fourier coefficients \hat{f} are known. The convolution theorem tells us that: $$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$ where \ast denotes the...
  48. FeDeX_LaTeX

    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. D

    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. R

    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...