Tagged Questions

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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16 views

Entropy of convolution of measures

Let $G$ be a countable, discrete group, and let $\mu_1,\mu_2$ be probability measures on the group $G$. We define the entropy of $\mu_i$ as $H(\mu_i)=\sum\limits_{g \in G}-\mu_i(g)\log(\mu_i(g))$ ...
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1answer
13 views

How to use matlab to a simple convolution problem (a convolution of the solution of the diffusion equation with continuous sources)?

I assume this is a simple problem, but since I don't have a strong maths background, I have trouble to implement this in matlab. Could anyone help me with it? Many thanks! The concentration of a ...
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1answer
19 views

Random Distance on Torus

Let $U=(X_U, Y_U)$ and $V=(X_V, Y_V)$ be two independent random points in $[0,1] \times [0,1]$, where each possible position is equally likely. Now I am interested in the probability that these two ...
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0answers
14 views

inverse sine transform, laplace transform and convolution theorem

when we apply Laplace and Sine Fourier transform to semi-infinite beam equation with boundary conditions $y(0,t)=V(t),~~~y_{xx}(0,t)=0$, we have, $\bar{Y_{s}}(\zeta,s)={\sqrt{2 \over ...
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0answers
9 views

Spherical harmonics and convolutions on $S^3$

Thinking of Hopf fibration of $S^3$ I got these two questions: Do $S^3$ spherical harmonics have a simpler expression in the Hopf coordinates? In $S^2$ we can convolve only with zonal functions. It ...
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1answer
43 views

Calculating a limit of integrals

I am having a problem with the following exercise: Show that for every bounded borelian function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$, $\underset{n}{lim} \frac{n}{\sqrt{2\pi}} ...
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0answers
19 views

Using Convolution to find density of sum of non-independent normal densities

$X_1 \sim N(\mu_1, \sigma_1^2)$ and $X_2 \sim N(\mu_2, \sigma_2^2)$. The $X_i's$ are not independent. Let $Y = X_1$ + $X_2$. Then, $ \begin{align*} f_Y(y) &= \int_{0}^{y} ...
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0answers
11 views

Giving a bound for |f(x) \star \phi_k(x) -f(x)|

Here is the problem: Let $\phi(x) \in S$, where $S$ is the Schwartz class, such that $\displaystyle\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} \phi=1$. Also, for some $N\in\mathbb{N}$, ...
2
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1answer
33 views

Differentiability of the convolution $\int_0^tf(t-s)g(s)\;ds$

Given two continuously differentiable functions $f,g:[0,\infty)\to\mathbb{R}$. I want to know what we can tell about the differentiability of $$(f\ast g)(t)=\int_0^tf(t-s)g(s)\;ds$$ Especially, why ...
0
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1answer
34 views

Writing a function in terms of the rect and delta functions.

Say I have a function that is equal to 1 at two unit area squares. One is centered at $(-3,0)$ and the other at $(3,0)$. I am trying to find a formula for this function using only the rect function ...
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0answers
34 views

Proof using convolution?

there. I am a novice in graduate school. This is the first time I learn PDE in graduate level. I found it so hard. I am going to have a test next week and I am so worried about it. Since I always ...
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0answers
22 views

2D convolution: how to eyeball it?

I have a question of doing simple convolution in 2d by just "eye-balling" it without doing the actual computation. In 1D discrete time, when we have a simple input ...
1
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1answer
36 views

Convolution of cosine with exponential

As part of an exercise, I'm trying to find the output of a cosine wave entering a low-pass filter by using a convolution integral. The impulse response of the filter is $h(t) = ...
1
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1answer
23 views

Convolution integral identity proof

How can one show that $$ f * g = t^{m+n+1} \int_0^1 u^m(1-u)^n du $$ where $f(t) = t^m$, $g(t) = t^n$ and $$f*g = \int_0^t f(\tau)g(t-\tau)d\tau = \int_0^t f(t-\tau)g(\tau)d\tau $$ I tried with the ...
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0answers
14 views

How injective is the Laplace transform?

Denote the Laplace transform by $\mathcal{L}$, and assume $\mathcal{L}[f]$ and $\mathcal{L}[g]$ exist for some functions $f$ and $g$. Then we know that $\mathcal{L}[f*g]=\mathcal{L}[f]\mathcal{L}[g]$. ...
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0answers
4 views

Can I calculate correlation with convolution?

You know the correlation is the degree of similarity between two difference signals, and the convolution is used for calculate the output of a system or signal, so can I use convolution for calculate ...
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0answers
15 views

Fourier Transform Identity?

$f(x) \in \mathbb{R}$ and $g(x) \in \mathbb{R}$ $$\int\int \mathop{dx \, dy} f(x)f(y)g(x-y) = \int dk \, \left| \tilde{f}(k)\right|^2\tilde{V(k)} $$ All integrals are over all space. Is this true? ...
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0answers
17 views

Find convolution of u[n]-u[n-2] and u[n]-u[n-2]

Question: Find convolution of $u[n]-u[n-2]$ and $u[n]-u[n-2]$ I have found that $u[n]\cdot u[n]=n$, $u[n]\cdot u[n-2]=n-2$, $u[n-2]\cdot u[n-2]=n-4$ Use linear property, my answer is: ...
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0answers
6 views

Approximation of $C^1$ by $C^1_b$

Can we approximate a function which is $C^1$ with functions that are $C^1$ with bounded first derivative? Thank you in advance.
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3answers
42 views

How to find the impulse response with input and output given?

The Question: A CT signal x(t), which is non-zero only over the time interval, t = [-2,3] is applied to an LTIC system with impulse response h(t). The output y(t) is observed to be non-zero only over ...
2
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0answers
62 views

Convolution of two Gaussians or two sinc functions using direct integration

I tried to solve the following to problems from Gaskil's book Linear Systems, Fourier Transforms, and Optics. But I'm struggling to get the right results. My experience with calculating convolutions ...
4
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0answers
50 views

Finding convolution identites

Suppose I have the following definition: $$\frac{x^2/2!}{e^x-1-x}=\sum_{k=0}^{\infty}A_k\frac{x^k}{k!}$$ I want to find a convolution identity for these coefficients $A_k$, but I've never studied ...
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1answer
18 views

Is it possible to find a solution to this integral equation?

I have an integral equation of the following form: $y(t)=\lambda x(t) + x(t)\int_{-\infty}^{\infty}K(t,s)x(s)ds$ I haven't been able to find any discussion online of integral equations with the ...
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1answer
30 views

Mollification of $L^{\infty}$ functions

We know when $1\leq p<\infty$ , the mollification function $f^{\epsilon}=\phi_{\epsilon}*f$ for $L^{p}(R^n)$ functions converge to $f$ in $L^{p}$ norm, when $p=\infty$ it might be wrong. But who ...
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1answer
36 views

Convolution theorem - proof

I'm trying to understand a proof of convolution theorem given here: http://www-structmed.cimr.cam.ac.uk/Course/Convolution/convolution.html In section named "Proof of second statement of convolution ...
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0answers
15 views

Practical convolution of poisson and log-normal distribution

Hi guys, Im trying to make a Loss Distribution (in Excel), following the "Loss Distribution Approach". I do understand that the main idea is that we have a severity distribution and a frequency ...
2
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0answers
24 views

Inverse Fast Fourier Transform to find the voltage across a capacitor of a RC circut

Fourier transform of a RC circuit The following example of a RC circuit describes the use of the fourier transform in order to receive the output voltage across the capacitor. My questions ...
2
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1answer
29 views

Finding $\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$ using Plancherel’s theorem

Suppose $G(\mathbb R)\ni f(x),\mathcal{F}[f](\omega)=\frac{1}{1+|w|^3}$ find the value of $$\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$$ I thought using Plancherel’s theorem \begin{align} ...
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1answer
28 views

Convolution/Deconvolution $\stackrel{?}{=}$ Coding/Decoding

In a strict mathematical sens, can a convolution/deconvolution be equivalent to a coding/decoding process ? I just got the remark from a reviewer that it's strictly different, it's a little surprising ...
1
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1answer
51 views

Differential equation with fourier transform and convolution

We have differential equation $3s(t)-2s''(t)=r(t)\,$ and $s(t)$ is convolution $s=g*r\,$ where $g(t)=ae^{-b\left | t \right |}\,$ $\\a,b\in\mathbb R+$ Solve constans a and b. I tried to solve ...
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35 views

Sort of Convolution

I was wondering if the following convolution I am considering already has a name or well-studied. Thanks for your help. If the condition of the positive radius of convergence is not enough, then I ...
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0answers
32 views

Evaluating convoluted integrals of complex exponentional and rational

I want to evaluate the following integral: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda\int_0^{t-\lambda} d\tau \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \int_0^{t-\lambda} d\tau ...
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0answers
25 views

L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
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1answer
25 views

Convolution Case

The * denotes convolution and u[n] as the heaviside function. $x[n]= u[n]α^n$ Determine a sequence $h[n]$ such that-: $x[n]∗h[n]=α^n(u[n+2]−u[n−2])$ I am trying this problem for quite awhile now. ...
0
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1answer
24 views

Uncertain with piecewise result for convolution integral

I have two equations $$x(t) = u(t) - 2u(t-2) + u(t-5)$$ $$h(t) = e^{2t}u(1-t)$$ where $u(t)$ is the unit step function. I'm attempting to find the convolution of the two: $$y(t) = h(t)*x(t)$$ I ...
2
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2answers
52 views

Integration of dirac function explanation

I have a problem that need your help. I have a gray image. We denotes $I(x)$ is gray level of a pixel in the image and $f(z)$ is a function of $z$(ie: histogram function...)-where $z$ is the set of ...
1
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1answer
55 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
1
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2answers
28 views

convolution problem given $h(x)=1/2$ for $0<x<2$ and $0$ otherwise

I have a convolution problem in the form $$g(x)= \int_{-\infty}^\infty h(y)h(x-y)\,dy$$ where they give me the function $h(x)=1/2$ for $0<x<2$ and $0$ otherwise. I have never done a ...
1
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2answers
36 views

Computing a messy convolution

Consider the functions $$ x(t) = u(t - \frac{1}{2}) - u(t - \frac{3}{2}) $$ and $$ h(t) = tu(t) $$ where $u(t) = 1$ if $t \geq 0$ and $u(t) = 0$ if $t < 0$. I'm trying to compute $$ (x*h)(t) ...
2
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1answer
40 views

Calc $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$

The answer to this integral:$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$$ is ...
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3answers
39 views

Divisor function convolution

I need some help to prove that $$ (d*d)(p^k) = \frac{(k+3)(k+2)(k+1)}{6} \qquad \forall p \in \mathcal{P},\quad \forall k \in \mathbb{N}, $$ where $d$ is the divisor function and $\mathcal{P}$ the set ...
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1answer
55 views

problem with convolution

I'm struggling with this kind of problem: I have an assumption that $f$ and $g$ are in $L^2(R)$, and I should prove that $f\star g \rightarrow 0$ when $|x| \rightarrow \infty$. I think (but I'm not ...
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0answers
26 views

Function smoothing using convolution

I have a function $\hat f$ which is an estimator of an unknown function $f$. The estimator $\hat f$ looks pretty irregular (see the red line). I would like to smooth it with some kernel function ...
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0answers
54 views

Distribution of the sum of many lognormal random numbers from same distribution

In my application I have to sum up a lot (between 1000 and 2000) lognormally distributed random numbers and use their sum. All random numbers that I sum up follow the same distribution. The current ...
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0answers
33 views

understanding discrete-time convolution

I'm trying to understand the discrete-time convolution for LTIs and its graphical representation. standard explanations (like: this one) start with the idea of decomposing an input signal $x[t]$ into ...
1
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1answer
34 views

When convolution of two functions has compact support?

It is well-known that, if $f$ and $g$ are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous (Hörmander 1983, Chapter 1). Next, ...
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2answers
87 views

Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
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0answers
121 views

Convolution with itself equals itself times a function

Consider the case that $f \in L^1(\mathbb{R})$ and $g \in L^1_{loc}(\mathbb{R})$. Then look at the equation $$ f*f=g\cdot f. $$ I know that if $g$ is constant, then $f=0$. But what about other $g$'s? ...
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0answers
21 views

using convolution for power series solution method for DE's

Say I have a homogeneous linear differential equation of the form $y''+py'+qy=0$ and I want to solve it using the power series solution method. So I use the substitution ...
0
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0answers
36 views

Autocorrelation of Raised Cosine Function

Let us define the raised cosine function as follows: $f \left( x \right) = \dfrac{\left( 1 + \cos \left( x \right) \right)}{2}$, for $- \pi < x < \pi$. $f \left( x \right) = 0$, elsewhere. I ...