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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2
votes
0answers
26 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 ...
0
votes
0answers
25 views

Convert Infinite integral to sum

I want to convert an infinite integral to sum. I could not find much info on this online as my integral is from $\infty$ to $-\infty$. For example how would you convert the following? ...
-1
votes
2answers
15 views

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

I read that the convolution of two iid RVs is (f * g) (z) = f(z-y) g(y) dy. What is the general formula for more than two RVs. More specifically, I want the relative formula for three RVs.
0
votes
1answer
7 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
32 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
27 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
27 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
14 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
votes
0answers
7 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
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
45 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
14 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
57 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
32 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 ...
-1
votes
0answers
19 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
30 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
38 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
12 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
112 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
24 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
66 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
21 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 ...