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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0
votes
1answer
20 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
17 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 ...
-2
votes
0answers
13 views

Convolution - Laplace transform

A friend of mine asked me for help. He needs these three exercizes about convolution: $$ \int\,\dfrac{ds}{s(s-1)} $$ $$ \int\,e^{3t} * \sin\,(5t)\,dt $$ $$ \int\,8t^2 * e^{8t}\,dt $$ When I ...
0
votes
0answers
12 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
13 views

How to compute this convolution in matlab?

The equation is as above, where $f_x$ and $f_y$ refer to axises in frequency domain, $x$ and $y$ refer to axises in space domain and $F$ refers to Fourier transform. My main problems lie on 1) how ...
-1
votes
0answers
12 views

linear convolution and circular convolution

Can anyone explain for me when circular convolution is equal to linear convolution. I get this example but i need more expanation esp on the underlined
0
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0answers
5 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
0answers
11 views

Convolution of Two Characteristic Functions on \mathbb{R}^1 [on hold]

Problem 2 from Jones, Lebesgue Integration on Euclidean Space, Chapter 11. On $\mathbb{R}^1$, let f= $\mathbb{1}_{(-a,a)}$ and g= $\mathbb{1}_{(-b,b)}$. Verify that: $(f*g)(x) =$ $2b$ if $\vert X ...
0
votes
1answer
13 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 < ...
0
votes
0answers
17 views

Periodic convolution of functions

Define the periodic convolution of functions in L2([0; 1]). What theorem of convolution do I use to define this and how do I solve this?
3
votes
1answer
44 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
16 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
votes
0answers
11 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
56 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: ...
1
vote
0answers
31 views

How much does convolution with a compact C^m kernel increase the order of continuity.

Let $f \in C^n$ and $g \in C^m$, with $g$ compactly supported and integrable. How much does the convolution $f\star g$ of $f$ with $g$ increase the order of continuity? Statement: I think that, under ...
0
votes
1answer
7 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 ...
0
votes
0answers
17 views

How to get the PDF for $p(a)$ $p(b)$ by using convolution?

Suppose there are three random variable $a$, $b$, $c$, and the PDF for each are $p(a)\ p(b)$ and $p(c) $ Also, $c$ = $a$ + $b$, $a$ and $b$ are two independent variable. and$$p(c)= \begin{cases} ...
1
vote
0answers
25 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
31 views

A question related to convolution

Let $f\ge0$ such that $\int_{{R^n}} {f(x)dx} = A < 1$, and ${f_k} = f*...*f$ a convolution of $k$ times. My problem is how to prove $f_k$ is integrable and $f_k\rightarrow0$ in $L^1(R^n)$? ...
1
vote
1answer
44 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
vote
0answers
24 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
37 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
20 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
78 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
20 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
28 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
28 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
36 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 ...
0
votes
0answers
11 views

convolution under differentiation

I have a continuous and differentiable time function $f(t)$ for $t\in [0, T] $ and a causal time function $g(t) $ for $ t \geq 0 $. Does the following make sense? $ \dot{f}(t) \ast g(t) = \frac d{dt} ...
1
vote
0answers
17 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
25 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 $x(t)$ ...
0
votes
0answers
13 views

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 ...
2
votes
4answers
60 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
20 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
12 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
votes
0answers
31 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
26 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
106 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
111 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
23 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
64 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
40 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 ...
0
votes
1answer
23 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
19 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
58 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
8 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
54 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
38 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 ...
1
vote
0answers
42 views

distribution of sum of double exponential random variables

I want to find out whether there is a concise expression (i.e. not a convolution) for the distribution of a random variable A which is the sum of $n$ i.i.d. rv's $B_i$, which are themselves double ...
1
vote
0answers
47 views

Holder continuity of the convolution of a Holder continuous function

Let $f(\theta, t)$ be a Holder continuous function for every $t$ on the interval $\theta \in (\alpha,\beta)$. It is known that the application of a singular operator to this function results in ...