Convolution Definition and 25 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. 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...
  2. 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}...
  3. 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...
  4. 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...
  5. 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)...
  6. 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...
  7. 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...
  8. 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...
  9. 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)...
  10. 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...
  11. 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...
  12. kaniello

    I Convolution integral

    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...
  13. R

    Convolution of two Sinc functions

    Homework Statement Calculate the convolution of ##sinc(at)## and ##sinc(bt),## where ##a## and ##b## are positive real numbers and ##a>b.## Homework Equations Convolution integral The Attempt at a Solution The fact that ##a>b## tells us that the graph of ##sinc(at)## is ##a-b## times more...
  14. Summer95

    The Dirac Delta Function

    Homework Statement Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0## Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##. Then solve the differential equation with...
  15. R

    Convolution Calculation (piecewise function)

    Homework Statement Compute the convolution ##(f*h)(t)## where $$f(t) = \left\{\begin{matrix}1, \ \ for \ \ |t|<1 \\ 0, \ \ \ \ otherwise \end{matrix}\right.$$ and $$h(t) = \left\{\begin{matrix}2|t|-1, \ \ for \ \ |t|<1/2 \\ 0, \ \ \ \ otherwise \end{matrix}\right.$$ Homework Equations...
  16. R

    Convolution and Transfer Function

    Homework Statement The impulse response ##h(t)## of a linear time invariant system is real valued. Where * denotes the complex conjugate, the transfer function satisfies: $$H(- \nu) = H^* (- \nu)$$ Use this result to show that for such a system, given the input ##f(t)= \sin(2 \pi \nu t)##...
  17. R

    Convolution Integral Properties

    Homework Statement Either by using the properties of convolution or directly from the definition, show that: If $$F(t)=\int^t_{-\infty} f (\tau) d \tau$$ then $$(F * g) (t) = \int^t_{-\infty} (f * g) (\tau) d \tau$$ Homework Equations The convolution of ##f## with ##g## is given by...
  18. J

    Is there a mistake in my calculation or in my reasoning?

    Homework Statement y'' + 3y' + 2y = r(t), r(t) = u(t - 1) - u(t - 2), y(0) = y'(0) = 0. I need to solve this by convolution, which I know is commutative. The problem is that my calculation gives (f * g) =/= (g * f). Could someone please tell me where my mistake is? Homework Equations...
  19. E

    Fourier Transform and Convolution

    Considering two functions of ##t##, ##f\left(t\right) = e^{3t}## and ##g\left(t\right) = e^{7t}##, which are to be convolved analytically will result to ##f\left(t\right) \ast g\left(t\right) = \frac{1}{4}\left(e^{7t} - e^{3t}\right)##. According to a Convolution Theorem, the convolution of two...
  20. Italo Campoli

    Inverse Laplace Transform of a fractional F(s)

    Homework Statement [/B] Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor 2. The attempt at a solution tryed at first with partial fractions but that didnt got me anywhere, i know i could use tables at the 2nd fraction i got...
  21. wirefree

    Integration by sketching

    I greatly appreciate this chance to submit a query. I have the following integral: $$\int_{1}^t 2sin(t-\tau)e^{-2(t-1)} d\tau$$ and it has been suggested to me that if I sketch the two constituent functions and multiply them, I can read the answer off the paper. So here are my sketches: go...
  22. E

    Step Validity with the Fourier Transform of Convolution

    A convolution can be expressed in terms of Fourier Transform as thus, ##\mathcal{F}\left\{f \ast g\right\} = \mathcal{F}\left\{f\right\} \cdot \mathcal{F}\left\{g\right\}##. Considering this equation: ##g\left(x, y\right) = h\left(x, y\right) \ast f\left(x, y\right)## Are these steps valid...
  23. T

    Use convolution integral to find step response of a system

    Homework Statement An electrical network has the unit-impulse response :h(t)=3t⋅e-4t .If a unit voltage step is applied to the network, use the convolution integral to work out the value of the output after 0.25 seconds. Homework Equations Convolution integral: y(t)=f(t)*h(t) Unit step...
  24. Th3HoopMan

    Correlation between Iterative Methods and Convolution Codes

    Hey guys so I have this Calc 3 project and the end is throwing me for a loop. I've done the encoding part, and i've coded the standard iterative methods, but I don't see how the two correlate so I can use the iterative methods to decode a "y stream" with the inputs specified...
  25. E

    Convolution Dirac impulse and periodic signal

    Hi ☺️ i have to do a convolution with a periodic signal and a dirac impulse: x(t)=sen(πt)(u(t)−u(t−2)) h(t)=u(t−1)−u(t−3) The first is a periodic graph that intersect axis x in points 0 , 1 and 2 (ecc) The se ing is a rectangle ( Dirac impulse ) that intersect AxiS x in points 1 and 3. For...
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