Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schartz's sense) or measures.

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0answers
166 views

Discrete convolution: where do I go from there?

I took this from the book Signals and Systems by Haykin. I have the following discrete system: $y[n] = u[n]*u[n-3]$, where $*$ is the discrete convolution and $u[n]$ is the unit step function, and ...
2
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2answers
113 views

convolution computation involving $e^{-x^2}$

In working a problem involving convolution, I have arrived at the following integral, but do not know how to compute it: $$2\int_0^{\infty}e^{-a(x-y)^2-by^2}dy$$ I thought that this integrand did not ...
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1answer
101 views

Function as a convolution product of other two

I need help with this: I have to prove that a function $f\in L_{2}(T)$ can be expressed as $f=g*h$ (convolution product) for some functions $g,h\in L_{2}(T)$ if and only if $(\hat{f}(n))_{n}\in ...
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2answers
242 views

Differentiability of convolution

First let me say that I have used the search bar and looked through all the "differentiability of convolution" questions that I saw, but none of them seem to cover this case. (If one of them did and I ...
3
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0answers
349 views

Infinite self-convolution for a function

I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. So consider a generic function $f : ...
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1answer
50 views

Need help with convolution problem.

I'm just learning about the convolution integral and am stuck on this example problem: Given $x_1(t) = \left \{\begin{array}{lr} 1 : 0 < x < 1 \\ 0: \text{elsewhere} \end{array} ...
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0answers
64 views

Fubini theorum for integrating 1 dimension of a 3d convolution

I have 3D volume that is convoluted with a 3D blur function. Both are positive and integrate to a finite value. I can see experimentally (meaning playing with matlab) that this is true: $\int_{-a}^{a ...
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1answer
205 views

A Mathematical way to represent a image kernel?

How to represent the calculation in this image mathematically? For example: With the discrete convolution and Fourier Transform. It tries to do a calculation on the original image (image A/input) ...
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1answer
319 views

What's the difference between convolution and crosscorrelation?

What's the difference between convolution and crosscorrelation? So why do you use '-' for convolution and '+' for crosscorrelation? Why do we need the "time reversal on one of the inputs" when doing ...
2
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2answers
428 views

Convolution: Laplace vs Fourier

Are there real world examples when it is better to use laplace instead of fourier to compute a convolution? And vice versa. Fourier can use negative numbers (as in 'integrates from minus infinity to ...
1
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1answer
328 views

Multiplying polynomial coefficients

Take: $u(x)$ and $v(x)$ to be integer polynomials, and then interpret them as sequences in the obvious way: i.e. you put the $i$th term to be the coefficient of $x^i$. Then you'll find that $u\ast ...
2
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2answers
162 views

Understanding convolution

Take: $$ (u*v)(k) = \sum_{i=-\infty}^\infty u(i)v(k-i). $$ The $k$ is there, it's because you want to define $$ \ldots\ldots, (u*v)(-3), (u*v)(-2), (u*v)(-1), (u*v)(0), (u*v)(1), (u*v)(2), ...
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1answer
129 views

Sifting Property of Convolution

This is going to be a dumb question, but I can't figure it out, so here goes $ f(t)\quad \bigotimes \quad \delta \quad (t\quad -\quad { t }_{ o }) $ = $\int { f(\tau )\delta (t\quad -\quad { t }_{ ...
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1answer
110 views

difference between convolution of two densities and mixture density?

I am wondering about the difference of the convolution of two probability density functions and the mixture of those two. This is not the same right? But what is the difference and how can it be ...
3
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1answer
124 views

Convergence of convolution of $L^p$ function with a sequence of distributions

let $h_n\in C_c^\infty (\mathbb{R}^d)$ s.t. $\int h_n dm = 1$ and $\operatorname{supp}(h_n)\to {0}$. I've proven that $h_n\to\delta_0$ in $\mathcal{D}'(\mathbb{R}^d)$, now I'm trying to show that for ...
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2answers
80 views

Convolution with sign function

I am having some trouble calculating the convolution $ (f*g)(t) $ between these two functions: $$ f(t)=e^{-t}1(t) $$ where $1(t)$ is the unit step function, and $$ g(t)=\mathrm{sgn}(t) $$ Using ...
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1answer
107 views

Convolution of dependent discrete random variables

We have a set $X_1, X_2, \ldots, X_n$ of correlated discrete random variable with a given correlation matrix. How can one compute the sum $X_1 + X_2 + \cdots+ X_n$ knowing the probability mass ...
2
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1answer
111 views

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
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1answer
125 views

$\mathcal{L}^p$ spaces and convolution

Suppose that $f \in \mathcal{L}^p$ and $g \in \mathcal{L}^q$, and $p,q$ are conjugate exponents. Then prove that (a) $h(x) = \int_{-\infty}^{\infty} f(t) g(x+t) \, dt$ defines a bounded continuous ...
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0answers
41 views

Cancellation of summations

I am working on some stuff related to the convolution property of the discrete Fourier transform. If we consider: $$\sum_{p = 0}^{N-1}\hat{s}_{p}e^{ik_{p}x_{m}} = \sum_{p = ...
3
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0answers
151 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
2
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2answers
456 views

Convolution of integer sequences

I read this somewhere: $U$ and $V$ are defined on the set $\mathbb{Z}$ of integers. The convolution of $u$ and $v$, noted as $(u*v) (k)\,\theta$ or $ (u \otimes v) (k)\theta$, is a new sequence whose ...
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1answer
255 views

How do I take the complex convolution of this impulse response and input?

I derived a an impulse response of $h[n] = (3/4) (-j3/4)^{n} u[n]$, where $u[n]$ is the unit step function. I have an input $x[n] = u[n-5]$. I can find a vector representation of the convolution of ...
3
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0answers
176 views

Convolution Exercise Homework

Put $\varphi(t)= 1- \cos \;t\;\;\;$ if $\;\;\;0 \leq t \leq 2 \pi$, $\varphi(t) = 0$ for all other real $t$. For $-\infty < x < \infty $, define $$ f(x)= 1,\;\;\;\;\;\;\;\;\;\;g(x) = ...
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1answer
883 views

How to determine the step response using convolution of the signal's impulse response?

The step response can be determined by recalling that the response of an LTI to any input signal is found by computing the convolution of that signal with the impulse response of the system. ...
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1answer
147 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
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1answer
209 views

Graphical understanding of the convolution of discrete distributions.

I am studying convolution and trying to get a visual sense of the process. From the wikipedia page I understand what 2 continuous gaussian or 2 uniform distributions will produced when convolved, but ...
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2answers
236 views

Convolution of continuous function with $\mathcal{C}^{1} $ function

I'm having difficulty with the following problem: Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$ such that $\mbox{Supp}\left(f\right)$ is compact and let $g:\mathbb{R}\to\mathbb{R}$ ...
2
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0answers
80 views

Convolution and Smoothness Conditions

Suppose $f(x),g(x)\in L_1(\mathbb{R})$, with both $|f(x)| \leq 1$, $|g(x)| \leq 1$ and $|f(x)| \rightarrow 0$, $|g(x)| \rightarrow 0$ for $|x| \rightarrow \infty$. Given that we have two other ...
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1answer
139 views

What is the distribution of empirical covariance between two independent normal distributions?

Suppose that we have two independent normal distributions $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ what is the distribution of empirical covariance (or empirical correlation if this make my ...
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2answers
286 views

How can you do convolution graphically?

I don't exactly remember whether I should get the common area under the curves of the functions being convolved or I should multiply them and get the area under the resulting curve.
1
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1answer
282 views

convolution square root of uniform distribution

I need to find a probability distribution function $f(x)$ such that the convolution $f * f$ is the uniform distribution (between $x=0$ and $x=1$). I would like to generate pairs of numbers with ...
0
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1answer
178 views

what's the distribution of the inverse of a random variable that follows a negative binomial distribution?

I was studying the method of moments estimation of parameters, and I encountered the following problem. I have a geometric distribution as following: $P(X=k) = p(1-p)^{k-1}$, and a sample size of n, ...
3
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4answers
810 views

Repeated convolution of probability distributions

Question Let $$S_k=\sum_{i=1}^k X_i$$ be the sum of $k$ independent random variables. I am interested in closed-form expressions of the pdf of $S_k$. In general, the pdf is given by the $k$-fold ...
7
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0answers
74 views

Properties of a continued fraction convolution operation

Usually the partial numerators of a continued fraction are all 1s. Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued ...
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1answer
987 views

Finding limits of integration in convolution

I am struggling to fully get how to choose proper limits of integration when calculating convolutions. Right now I am stuck on a problem where I have to show that when taking the Fourier transform of ...
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1answer
61 views

Convolution of $L_1$ functions is $L_1$

Let $f,g\in L_1(\mathbb R, m)$ where m is the Lebesgue measure. Prove that: a) $f(x-t)g(t) \in L_1(\mathbb R, m)$ as a fuction of $t$ almost all $x$ b) $h\in L_1(\mathbb R, m)$ where $h=(f \ast ...
2
votes
2answers
254 views

the fourier transform of a “double convolution”

Suppose I have a function $$ m(x) = f(x)\int_{-\infty}^{\infty} h(w)g(w-x)dw = f(x)h*g(x) $$ I want to find the Fourier transform of m(x) in terms of the Fourier transforms of $f,h,g$ but for the ...
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1answer
49 views

A problem about Convolution

If $\phi\in C_0^{\infty}(\mathbb{R}^N)$ and $\psi\in L_{loc}^1(\mathbb{R}^N)$ is defined by $\psi(x)=|x|^{2-N}$, $N\geq 3$ , does $\phi\star\psi$ is in the Schwartz space? Note: $\star$ stands for ...
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1answer
166 views

PDE - (homogeneous) Heat equation - Solution?

today I have a question in PDE. It concerns the heat equation: Formulate the (homogeneous) heat equation for functions $f:(0,\infty)\times\mathbb{R^n} \longrightarrow\mathbb{C}$. Derive an equation ...
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1answer
276 views

Intuition behind the convolution of two functions

Suppose $f(x)$ and $g(x)$ are two functions. What is intuition or idea behind the convolution of $f$ and $g$? After taking the convolution we will get a new function. What is the geometric relation ...
2
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1answer
73 views

Fourier transform of product

I would like to know the fourier transform of the product of the Cauchy probability distribution $f(x)=\frac{1}{\pi (1+x^2)}, -\infty<x<\infty$ with itself. I know that the fourier transform of ...
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1answer
83 views

Filter signal through convolution

I am a little bit unsure if I've set up the following problem correctly: Consider the signal $$f(t) = e^{-t}(\sin(5t) + \sin(3t) + \sin(t) + \sin(40t)) \quad 0 \leq t \leq \pi$$ Filter this signal ...
2
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0answers
60 views

Show compactness of an evolution operator

Consider the heat equation $$ u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$ with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$ and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$. 1.) ...
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1answer
85 views

Evolution operator

We call a function that assigns a starting value of a time-dependent differentialfunction to a solution of a later timevalue as the evolution operator $E(t)$. Look at the thermal equation $$ ...
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2answers
343 views

Find derivative of convolution with gaussian

Let $A(\sigma)$, $\sigma > 0$ be an operator that acts on bounded continuous functions $f$ on $\mathbb{R}$ by the rule $$ (A(t)f)(x) = \int\limits_{\mathbb{R}} f(y)\frac{1}{\sqrt{2 \pi ...
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1answer
300 views

Double tophat convolved with a gaussian

I need some help calculating the analytically expression of this convolution. The functions in question are: 1) a gaussian ($g(x)$) 2) a "double tophat function" (in lack of a better name). i.e. ...
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2answers
492 views

How to compute Inverse Laplace transform using Convolution

How do you evaluate the inverse transform below using convolution ? $$ \mathcal{ L^{-1} } \left[ {\frac{s}{(s^2 + a^2)^2}} \right] $$ I tried $$\begin{align} \mathcal{ L^{-1} } \left[ ...
3
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0answers
108 views

When does $|f*g|_{p}=|f|_{1}|g|_{p}$?

From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4 Suppose $1\le p\le \infty$, $f\in L^{1}(\mathbb{R}^{1})$, $g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ...
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1answer
122 views

Obtaining Impulse Response from Graph

I want to know how to solve those types of problems.. is it by inspection ? Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the ...