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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1
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1answer
28 views

Convolution, g(t)=sin(t)

I have two functions: $f(t)=(t+\pi)\theta(t+\pi)-2t\theta(t)+(t+\pi)\theta(t-\pi)$, (looks like $-|x|+1, -\pi < x < \pi$ ) and $g(t) = \sin{(t)}$ Could someone please point me in the right ...
-3
votes
1answer
84 views

Fourier Analysis question on convolution [closed]

Let $f(x) = \operatorname{sinc}(x)^2$. Find $(f*f)(x)$? This is what I tried $f(x)=\mathrm{sinc}(x)^2$ $$ \begin{align} f\ast f(x) &=\int f(u)\cdot f(x-u)\,\mathrm{d}u\\ ...
1
vote
1answer
85 views

Convolution of Two Random Variables

I have been working a few hours on this particular problem. Please excuse my lack of formatting. This is the question: Let $X$ and $Y$ be random variables with density function $f(x) = 2x$ on $[0, ...
1
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1answer
17 views

Convolution of derivatives

When transforming nonlinear PDE to its Fourier space, I encounter the following problem: Consider the equation $u_t=(u^3-u)_{xx}$. Then, when transforming to Fourier space we get \begin{equation*} ...
1
vote
1answer
32 views

What is this Toeplitz like matrix called and how do I represent it as a convolution?

I have a matrix that is used to represent the Green's function in a popular class of fast E & M solvers (CG-FFT). The matrix represents distances, that are later filled in using the appropriate ...
0
votes
0answers
26 views

Convolution of two equations f*g(t)

I am trying to calculate the convolution of: $$ f*g(t) $$ where $f(t) = \delta''(t) -\delta(t) $ and $ g(t)=e^{-t^2}+e^{-t} \theta (t) $ Maybe someone can give me a solution or point me in the ...
0
votes
0answers
65 views

How to convert infinite intergral to sum

How to convert Wiener filter formulas from integral to sum? They are for images therefore it must be possible to convert them to sums. Any help will be appreciated: I could not find much info on ...
2
votes
1answer
27 views

Is convolution a coercive bilinear form in $L^2$ -space?

This is one of the problems in functional analysis course I'm having. Suppose $f,g \in L^2(0,10)$. Then define a bilinear form $$ B:(f,g)\mapsto \int_0^{10} f(x)g(10-x) dx. $$ Now I have to find out ...
5
votes
2answers
748 views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
1
vote
2answers
52 views

Sums of independent random variables (more than two) [closed]

I read that the convolution of two iid random variables is $$(f * g) (z) = \int f(z-y) g(y) dy$$ What is the general formula for more than two RVs? For example, for three RVs.
0
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1answer
30 views

Graphs of functions defined by convolution

A sequence of functions on the real line is defined as $$f_0=\chi_{[-1,1]},\qquad f_{n+1}=f_n*f_0, n=0,1,2,\dots $$ Here * means convolution. I tried to draw the graphs of the functions and see what ...
0
votes
1answer
198 views

Sum of two uniform independent random variables

I would like to find the cdf of $Z=X_1+X_2$, with $X_1\sim U(0,1) $, $X_2\sim U(0,2)$ I always prefer to find the cdf instead of the pdf with convolution, and this time I am having troubles with the ...
0
votes
1answer
52 views

Convolution of an integrable function an $L^\infty$ function [duplicate]

Let $f$ be an integrable function on $\mathbb{R}$, and $g$ be an $L^\infty$ function on $\mathbb{R}$. Then, the convolution $f*g$ is said to be continuous and bounded on R. I managed to show that it ...
0
votes
1answer
152 views

The maximum value (peak) of multiple self-convolution of rectangular function

In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula ...
0
votes
0answers
25 views

Convolution in linear phase FIR filter

$G(z)$ is a linear phase FIR filter. is it possible to design another a realizable filter $H(z)$ such that it will undo the effect of the filter. That is $y(n)=g(n)*x(n)$ and the one going to be ...
0
votes
1answer
33 views

Discrete convolution equation

Let $x_1 = (x_1^k)_{k =-\infty}^{+\infty}$, $x_2 = (x_2^k)_{k=-\infty}^{+\infty}$, $x_3 = (x_3^k)_{k=-\infty}^{+\infty}$ be three sequences of real numbers such that $x_j^k = 0$ for $k < -m_j < ...
3
votes
1answer
50 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = ...
1
vote
0answers
17 views

Finding A Convolution

Let $f(x)=e^{- \mid x \mid}, g(x)= e^{-x^2}$ What is $(f*g)(\xi)$? I have been trying to find it, but I am stuck on finding the integral of $e^{y^2+y+\xi}$ Thank you!
0
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0answers
153 views

How to represent a periodic function as the sum of sinc functions in fourier transform

Suppose function $f(t)$ is 1-periodic. This means that in fourier transform, $F(\omega)$ is sum of impulse signals (dirac delta function and its shifts) at the multiples of $1$. Now we can form $g(t)$ ...
3
votes
0answers
81 views

How do I apply this PDE as an image filter?

I'm trying to preprocess a height map image with a helmholtz-type equation as described in this paper. The equation is: $$ddx(h') + ddy(h') + y(h'-h) = 0$$ I solved for h and got: ...
0
votes
1answer
14 views

Filter output of a signal

So I have a filter $$H(z) = 0.5 + 0.5z^3 = (1/2, 0, 0, 1/2)$$ and need to find the output of it on a cyclical signal $$x = (..., 3, -1, 2, 1, 5, 2, 3,-1, 2, 1, 5, 2, 3,...) $$ Would the output be ...
1
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0answers
31 views

Prove that the variance of a discrete random variable increases with a parameter

I have an infinite number of known probability density functions $f_1(x),f_2(x),f_3(x),...$. The PDFs $f_k(x)=\sum_{j=1}^k v(A+j-1)e^{-v(A+j-1)x}\binom{k}{j-1}q^{j-1}(1-q)^{k-j-1}$. Let ...
2
votes
1answer
41 views

Norm convergence of approximations to the identity

Let $\varphi \in L^1(\mathbb{R}^d)$ be such that $$\int_{\mathbb{R}^d} \varphi(x) \, dx = 1.$$ For each $\varepsilon>0$, let $\varphi_\varepsilon:= \varepsilon^{-d} \varphi\left( \dfrac x ...
1
vote
1answer
53 views

Integral of $\int_{-\infty}^{+\infty}\left |{\frac{\sin{x}}{x(1+x^2)}}\right|^2\,dx $

So the first part of the questions asks us to find the Fourier Transform of $$ f(x) = \left\{ \begin{array}{ll} e^{y} & \quad {-\infty}<x < 0 \\ e^{-y} & ...
1
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0answers
28 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity. My Questions: ...
-1
votes
1answer
71 views

How to calculate convolution integral?

I know the formula for a convolution integral but how would you actually carry out one when you have two piece-wise defined functions? If you had $$ f(x) = \left\{ \begin{array}{ll} ...
1
vote
1answer
38 views

calculating probabilities about the sum of dependent discrete uniforms

Say I have the following information: $$ X_i \sim \text{Discrete Uniform}(1,13) $$ and I want to find $\mathbb P(X_1+X_2+X_3 \ge 25)$ for the cases where the $X_i$'s are dependent. What approximations ...
2
votes
2answers
84 views

Can we expect $g(f\ast h)= gf \ast gh$ for some $g\in C_{c}^{\infty}(\mathbb R)$?

Let $f,g:\mathbb R \to \mathbb C$ be nice functions so that their convolution make sense. My question: Is it possible to choose $0\neq g\in C_{c}^{\infty}(\mathbb R)$ (= the space of smooth ...
0
votes
0answers
23 views

existence of a function such that certain convolution hold

The question says: Let $f\in L^1(\mathbb{R},m)$ be defined by $f(x)=e^{-\vert x\vert}$. Find an integrable function $g$ solving $f\ast g=f$ or show that no such function exists. I missed the class ...
1
vote
1answer
64 views

Convolution with Heaviside function (integration)

To clarify notation, I use $u_n = 1$ when $x>n$, and $0$ otherwise. I am having troubles with the following convolution/integration: $u_2(t) \ast sin(\sqrt{2}t) = \int^t_0u_2(\tau) \cdot ...
0
votes
1answer
38 views

Inverse Laplace transform and convolution

Suppose we have two functions of $t$, $f(t)$ and $g(t)$. Letting $\mathcal{L}\{f(t)\} = F(s)$ and $\mathcal{L}\{g(t)\} = G(s)$, I know that: $\mathcal{L}\{f(t) \star g(t)\} = F(s) \cdot G(s)$, but ...
1
vote
1answer
52 views

Proving convolution identity

I am trying to prove the following identity: $$\int_0^x(f*g)(y)dy = (\int_0^xf(y)dy)*g(x) = f(x)*(\int_0^xg(y)dy)$$, where $(p*q)(t) = \int_0^tp(x)q(t-x)dx$. I thought that since I already know that ...
1
vote
0answers
23 views

FFT of k*k matrix from FFT of a j*j matrix

FFT of matrix a j by j matrix, A $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ = $\begin{bmatrix}10 & -2\\-4 & ...
0
votes
1answer
31 views

Sign with Fourier transformation, convolution, periodicity

Let $x(t)$ be the sign with Fourier transformation $$X(\omega)=\delta(\omega)+ \delta(\omega-\pi)+\delta(\omega-5)$$ and let $h(t)=u(t)-u(t-2)$. Is $x(t)$ periodic? Is the convolution of $x(t)$ ...
2
votes
4answers
80 views

Calculating value of integral of convolution using Fourier transform

Calcuate the integral $$I=\int_{-\infty}^\infty\frac{\sin a\omega\sin b\omega}{\omega\cdot \omega}d\omega.$$ First I noticed that $$\mathcal{F}(\mathbb{1}_{[-h,h]})(\omega)=\frac{\sin h ...
2
votes
0answers
22 views

Prove the periodic total variation of f = $\sum_{n=0}^{N-1} |(f*h)[n]|$

Let $\Bbb{C}^N$ be the N-dimensional Euclidean space with its inner product defined as $$ \langle f,g\rangle=\sum_{n=0}^{N-1} f[n]g^*[n],\ \forall f,g \in \Bbb{C}^N$$ where $g^*[n]$ is the complex ...
0
votes
0answers
14 views

How can i explain the symmetry of the function in a more linguistic manner

To understand the convolution of these functions, please read the following wikipedia page I have the following expression $a = <f'(x),f''(x),\cdots>$ and $b = <g'(x),g''(x),\cdots >$ ...
0
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0answers
35 views

Convolutional Codes

i've been given a coding assignment that looks like this http://i.imgur.com/7wIEoHJ.png http://i.imgur.com/FINnNZZ.png I understand the concept of Jacobi and Gauss Seidel iteration, I know where ...
1
vote
0answers
47 views

uniform convergence and improper integral(convolution)

I want to show that $$\frac{d}{dx}(f*g)=(\frac{d}{dx}f)*g$$ where $f(x)=\frac{1}{\sqrt{x}}e^{-\frac{1}{x}}$, $g(y)$ is continuous and bounded. the convolutions are improper integrals. I'm now here ...
4
votes
2answers
149 views

Solve integral (convolution) equation

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$ and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$ How to find a closed expression for ...
1
vote
1answer
136 views

Convolution of Uniform Distribution and Square of Uniform Distribution

I am trying to find the CDF of $Z=X+Y$ whereby $X$ and $Y$ are random variables. Given that the CDF of Z is: $$F_Z(z)=\int F_X\left(z-y\right)f_Y(y)dy$$ Given that $X$ is uniform distribution over ...
1
vote
1answer
87 views

Random Gaussian variable raised to arbitrary power

Given $x$ that follows a distribution $P(x)=e^{\frac{-x^2}{2\sigma^2}}$ i.e. a random Gaussian variable, can I say anything about the distribution of $x^n$ for fixed $n$? Specifically, is there ever ...
3
votes
1answer
116 views

Discrete Fourier Transform question

Let $R_{M\times N}$ be a space of size $M\times N$. Define the 2D Discrete Fourier Transform of $f\in R_{M\times N}$ to be \begin{equation} ...
3
votes
1answer
62 views

Finding Limits and Its Convolution of Weighted Summation of Random Variables

I am trying to find the CDF of $Z=aX+bY$ whereby $X$ and $Y$ are random variables and $a$ and $b$ are positive integers. Given that the CDF of $Z$ is: $$F_Z(z)=\int ...
2
votes
1answer
212 views

PDF & CDF of a Sum of Weighted Independent Random Variables $Z=aX+bY$

From this question here, I learned that the Cumulative Distribution Function (CDF) of $Z=X+Y$ is: \begin{eqnarray*} F_Z \left( z \right) & = & \int F_X \left( z - y \right) dF_Y \left( y ...
1
vote
0answers
187 views

Convolution of Two Shifted Functions

I'm having some issues understanding the convolution of two rectangular functions. I have two rectangular pulses defined below and I need to find the convolution of them. $$ f(x)= \prod ({x-1\over ...
2
votes
1answer
1k views

About integrating product of two sinc function using Fourier transform

So the problem is which I think is pretty straight-foward by using Fourier transform and convolution property of two sinc functions and evaluating the convolution at 5. However, I got sinc(t) for ...
0
votes
0answers
12 views

Some variation of convolution formula

I am weak in convolution. We know $y(t) = x(t)*h(t)$. Now how about $y$ in the following case 1. $x(t)*h(-t)$ 2. $x(t-T)*h(-t)$ ] Then, let $y(t) = s(t)*s(-t)$, how about $y$ in the ...
1
vote
1answer
59 views

Solve $\int_{0}^{2\pi} f(t) \sin ^2 (t-\theta) dt = g(\theta)$ for unknown function $f$

Let $g(\theta)$ be a known real-valued function with domain $[0, 2\pi]$. Given that: $$\int_{0}^{2\pi} f(t) \sin ^2 (t-\theta) dt = g(\theta)$$ How would I solve for the unknown real-valued function ...
2
votes
1answer
47 views

Comparing Coefficients in Summations

Suppose I have the following equality: $$\sum_{k=0}^{n-a}\sum_{j=0}^{k}\binom{n}{k}\binom{k}{j}\frac{f(a,k)\cdot g(b,n-k)}{n!}=\sum_{k=0}^{n-a}\binom{n}{k}\binom{n-k}{a}\frac{z^k \cdot ...