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.

learn more… | top users | synonyms

1
vote
0answers
18 views

Applying the convolution theorem in the presence of a twiddle factor

The convolution theorem says that a 2-d cyclic convolution like $C = U \ast V$ can be evaluated more quickly than doing the raw sum $C_{i,j} = \sum_{a,b}^n U_{a,b} V_{i-a,j-b}$ for each point (assume ...
1
vote
0answers
49 views

Convolution of measures, why is the notation like this?

In both my book, and on Wikipedia they define convulution of two measures like this: $(\mu_1*\mu_2)(B)=\int_{\mathbb{R}^d}\mathcal{X}_B(x+y)d\mu_1(x)d\mu_2(y)$ It doesn't seem like a typo, but ...
0
votes
0answers
14 views

Is there an alternate name for the symplectic convolution?

Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution $$[F\star G](x,p) = \int \!dy\,dk\, ...
0
votes
0answers
18 views

Is the convolution of two waves again a wave?

Call a function $f:\mathbb{R}^{1+n}\to \mathbb{R}$ a wave, if it satisfies the wave equation $$\frac{1}{c^2} \frac{\partial^2 f}{dt^2} = \sum_{i=1}^n \frac{\partial^2 f}{dx_i^2} = div(grad\,f)$$ ...
2
votes
2answers
42 views

Why does convolution of delta function commute (test distribution perspective)?

If I understand correctly, for test functions $f$ we define $$ \langle\delta, f\rangle = f(0)$$ and convolution is defined as $$ \langle g * T, f\rangle = \langle T, g^- * f\rangle,$$ where $f$ ...
1
vote
0answers
27 views

Limit of sequence with discrete convolution $a_n = 1 - b * a$

I have a sequence \begin{equation} $a_n = 1 - \sum\limits_{i=1}^{n-1}b_na_{n-i}$ where $b_n\leq{c}^n$, and $c<1$ Based on multiple simulations with varying parameters, I think that the sequence ...
0
votes
0answers
15 views

Convolution of cosine and shifted unit step

I'm trying to understand the basics of a convolution and have troubles with the following task: $$ x_1 = \cos(2 \pi t ) \cdot u(t) $$ $$ x_2 = u(t-0,5)$$ The task is to compute the convolution $$x_1 ...
1
vote
1answer
19 views

What is the purpose of the requirement on mollification radius?

On page 66 of "Sobolev Spaces (Adams ed2)" in the proof of Lemma 3.16 (Mollification in $W^{m,p}(\Omega)$), it is mentioned that $\varepsilon < {\rm dist}(\Omega', \partial\Omega)$. However, I ...
1
vote
0answers
22 views

analyze convolution in spatial domain against multiplication in frequency domain

Lets say I have a image of $NxN$ and a separable filter that I want to apply on it. there are 2 ways to do that: 1. By convolution in spatial domain. 2. By multiplication in frequency domain. I need ...
0
votes
0answers
26 views

Convolution of delta-ish functions

I would like to compute the convolution of a function with itself, where the function is $f(x) = \frac{\delta(x)}{x}$. When there is a shift in the delta function it is easy to compute, but this one ...
0
votes
0answers
32 views

The Deconvolution Integral

The standard 1D continuous convolution integral is defined as: $$y(t) = h(t)*x(t) = \int^{+\infty}_{-\infty}h(\tau)\cdot x(t-\tau)\ d\tau$$ Using fourier transform, $$Y(j\omega) = X(j\omega)\cdot ...
0
votes
1answer
21 views

Conditional Expectation of Poisson Distribution

So I am not sure how to go about this, Say that $X_j\sim$Pois$(\theta)$, and are iid. Find the following: $$ E[X_1+2X_2+3X_3|\sum_{j=1}^nX_j] $$ I am aware that I am suppose to somehow make use of ...
0
votes
0answers
20 views

Multivariate probit gaussian convolution

For univariate normal distribution, we know the following formula exists $\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$ Is there a similar formula for ...
0
votes
0answers
22 views

existence of a function $f\in C_c^{\infty}(A_2)$ s.t. $f_{A_1}$ is constant 1

Let $A_1,A_2\subseteq \mathbb{R}^d$ two domains such that $A_1\subset \subset A_2$. Why exists a function $f\in C_c^{\infty}(A_2)$ such that $f_{|A_1}$ is constant 1? My idea is to define $f=\rho ...
1
vote
1answer
22 views

discrete convolution $f*g$ belongs to $\ell_\infty$, i.e. the sup norm is finite

Definitions. Fix any $\phi\in(0,1)$ and $\theta\in(0,1)$, and let us define functions \begin{equation}f(n)=\left\{\begin{array}{ll}n^{-\phi},&\text{ if }n\geq 1\\0,&\text{ ...
0
votes
1answer
38 views

Convolution of two functions.

$f(x)=2x/3$, $0<x<3$, $f(x)=0$ otherwise $g(x)=1$, $-1<x<3$, $g(x)=0$ otherwise I am trying to work out the convolution $h=f*g= \int(f(y)g(x-y))dy$ I am able to show that: $x - 3 ...
0
votes
1answer
23 views

Convolution of function with itself

I'm trying to tackle the following question: Let $\displaystyle g_a(x)=\begin{cases}1-\frac{|x|}{a},&x<0\\0,|x|\ge0\end{cases}$. Find $g_a\ast g_a$. So, I tried to compute it by ...
1
vote
1answer
29 views

Fourier transform this convolution

So we have that $$ g(t) = \frac{1}{T}\int_{t-T}^{t}f(\tau) d\tau $$ for $T>0$ and I'm asked to show that $\left| \hat{g}(w) \right|≤\left| \hat{f}(w) \right|$. The hint I get from the question is ...
1
vote
0answers
40 views

What is the output $y(t)$ when you have input $x(t) = \cos(2 \pi t) $ and frequency response response $h(t) = u(t) - u(t - 1/2)$?

The output $y(t)$ is the convolution of input $x(t)$ with impulse response $h(t)$: $$ y(t) = h(t) * x(t) $$ This is a linear, time invariant system. What is the output $y(t)$ in real form when you ...
0
votes
1answer
39 views

Calculate $\frac{x}{(1+x^2)^2}\ast \frac{1}{1+x^2}$ using Fourier transformations

Calculate $\left(\frac{x}{(1+x^2)^2}\ast \frac{1}{1+x^2}\right)(y)$ using Fourier transformations. I have found a solution, but my method was very long. How could I shorten the solution? ...
2
votes
2answers
70 views

Calculating the convolution of a piecewise function

Let $$f(x) = \begin{cases} \frac{1}{2}, & \text{if $\rvert x\lvert \le 1$ } \\ 0, & \text{otherwise} \end{cases}$$ I want to calculate the convolution of $f$ with itself. I am ...
0
votes
1answer
20 views

A question on convolutions

Let $f$ be an $L^2$ function on the line. If $f*g$ is an $L^2$ function for every $g$ in $L^2$ does it follows that $f$ is in $L^1$?
3
votes
2answers
76 views

$E(X\mid X+Y)$, where $X$ and $Y$ are independent $U(0,1)$.

Given that $X,Y\sim U(0,1)$ calculate: (1) $E(X\mid X+Y)$ I am stuck at this point: $$E(X\mid X+Y)=\int_0^1 xf_{X\mid X+Y}(x\mid x+Y) \, dx=\int_0^1 xf_{X, Y}(x, Y) \, dx$$
1
vote
1answer
51 views

Construct a nonnegative nonzero Schwartz function whose Fourier transform is nonnegative and compactly supported.

I tried the exercise with the hint that $\phi(x)=|\varphi\star\hat{\varphi}|^2$ could be the solution with $\varphi$ compactly supported and odd. Thus, \begin{align*} ...
2
votes
0answers
19 views

Conditions under which a convolution transformation is injective in the 1-d Torus

Let $X=[0,1)$ the 1-d torus. Given a bounded positive function $w\colon X\to\mathbb{R}$ with unit integral (I mean $w\geq 0$, $w\in L^\infty(X)$ and $\int_X w\; dx=1$), define \begin{align*} T_{w} ...
6
votes
1answer
151 views

Inf-convolution, two basic questions?

Let $E$ be a normed vector space. Given two functions $\varphi$, $\psi : E \to (-\infty, +\infty]$, one defines the inf-convolution of $\varphi$ and $\psi$ as follows: for every $x \in E$, ...
3
votes
0answers
35 views

Convolution of two signals

I have a problem with the convolution of two signals: $$x_{1}(t) = e^{2t}*u(-t)$$ $$x_{2}(t) = u(t-3)$$ $$x_1 \mathbin{\mathrm{(conv)}} x_2 = \int_{-\infty}^{+\infty} x_2(\tau) * x_1(t-\tau) \, ...
0
votes
1answer
34 views

Finding a limit involving Fourier series and Dirichlet's kernel

Find the limit $$\lim_{n\to\infty} \int_0^{2\pi} (x+\frac{\pi}{2})^2 \frac{\sin((n+\frac{1}{2})x + x\cos nx}{\sin\frac{x}{2}}\ dx$$ So we may define $f = (x+\frac{\pi}{2})^2$ and then look at the ...
0
votes
1answer
34 views

Convolution domains probability theory

Problem 1.4 here: ...
0
votes
2answers
33 views

Sum of X-Uniform(0,1) + Y-Uniform(0,2)

I'm trying to find the CDF the sum of $X$ and $Y$ (which are independent). $X$ is uniform distributed over $[0,1]$ and $Y$ over $[0,2]$. I've seen some similar questions which explain the situations ...
1
vote
0answers
27 views

Why are convolutions written with a minus sign [duplicate]

The convolution of two function $f$ and $g$ is defined[1] as $$(f*g)(x) = \int f(y) g(x-y) dy.$$ Terry Tao explains very nicely on MathOverflow that the convolution with a bump function can be ...
1
vote
0answers
22 views

Can someone explain about sufficient conditions of Convolution integral?

My text book, "continuous and discrete signals and systems 2/e by Soliman and Srinath, specifies sufficient conditions of convolution integral. $$y(t) = x(t) * h(t) = \displaystyle ...
1
vote
0answers
17 views

The validity of mollified method to prove the density of $C_0^\infty(\mathbb{R}^n)$

Let $X$ be a function space completed with converge topology (if possible $X$ is normed) such that $C_0^\infty(\mathbb{R}^n)\hookrightarrow X\hookrightarrow L^1(\mathbb{R}^n)$ is continuous dense ...
5
votes
2answers
236 views

Convolution with a polynomial is a polynomial. Why?

Let $P:\mathbb{R}\to\mathbb{R}$ such that $\deg P=N$. Let $f$, an integrable-$2\pi$-periodic function. Show that $f\star P$ is also a polynomial. So we can prove it for an arbitrary $x^n$ (Since ...
0
votes
0answers
35 views

Strong period of self-convolution of a strongly periodic cyclic function

Let $f : \mathbb{Z}_N \to F$ be a function with a $F$ a field. We say that $f$ has maximum period if the smallest positive integer $r$ (with $r \mid N$) such that $f(j) = f(j + r)$ for all $j \in ...
0
votes
0answers
22 views

Impulse Response from diffrence equation (and find the output when the input is given)

I'm a student in electronics, and in a few weeks, i have an exam on digital signal processing. The book isn't clear, and the lessons weren't clear either. I have this simple difference equation. I ...
2
votes
1answer
24 views

Convolution Problem

while working on a signal processing problem i've reached to the following: So my aproach was: Am I doing something wrong? Is it valid Y(f)=[X(f) x H(f)]*W(f)=X(f) x [H(f)*W(f)] If you could ...
0
votes
1answer
36 views

Convolution with additional cosine

I want to perform a convolution, but as a complication there is a cosine of the angle between any pair of vectors in the expression: \begin{equation} f(\theta^{\prime}) = \int d\theta ...
1
vote
1answer
48 views

Convolution of indicator functions with values in a finite field

This is something I haven't seen online yet, indicator functions with values in a finite field. Probably for a good reason, but I would like to know why, and if there are still things that can be ...
1
vote
1answer
34 views

Calculating the convolution of piecewise constant functions

Let $f(x) = \frac{1}{2}$ on $[-1,1]$. Find $f*f*f$ $(f*g)(x)=\int\limits_{-\infty}^\infty f(t)g(x-t)\,dt$. So $(f*f)(x)=\begin{cases} \frac{1}{4}x+\frac{1}{2} & -2\le x \le 0 \\ ...
1
vote
1answer
60 views

Fourier transform and convolution

Let $f \in L^1(\textbf{R})$ be such that $f'$ is continuous and $f' \in L^1(\textbf{R})$ . Find a function $g \in L^1(\textbf{R})$ such that $$ g(t) = \int_{-\infty}^{t}e^{u-t}g(u)\,du + f'(t) $$ ...
1
vote
0answers
39 views

Convolution and Fourier transform

Im stuck at a rather simple question. The problem is this Solve the integral $$ \int_{-\infty}^{\infty} \frac{\sin [5(t-u)]\sin 6t} {u (t-u)}du $$ And this is just the convolution of $$\frac{\sin ...
0
votes
1answer
46 views

How to get limit on integration for a convolution of two density functions

For two density functions: Suppose again that $Z = X + Y$. Find $f_Z(z)$ if $$f_X(x) = f_Y(x) = \begin{cases} x/2, & \text{if $0\lt x\lt 2$} \\ 0, & \text{otherwise} \end{cases}$$ I ...
0
votes
0answers
8 views

A question about convolution using a graphical approach

How do you convolve multiple dirac-delta functions a rect function? Is the below convolution correct? Thank you! Anthonya picture of graphical convolution
1
vote
3answers
42 views

Pointwise convergenve of mollified $f\in L^1_{loc}$

Let $\Omega\subseteq\mathbb{R}^n$ open, $f\in L^1_{loc}(\Omega)$, $\eta_\epsilon(x) = \dfrac{1}{\epsilon^n}\eta(\dfrac{x}{\epsilon})$ the usual scaled mollifier, i.e. $supp (\eta_\epsilon) ...
0
votes
0answers
20 views

How do you graph the convolution of two dirac delta functions and one rect function?

Would this result in one rect function between the two dirac delta functions? Or would it result in two rect functions centered at the location of the dirac-delta functions? Thank you very much! ...
1
vote
1answer
57 views

Inverse convolution of a distribution.

Notation. Let ${\mathcal{D}'}_+(\mathbb{R})$ be the set of distributions on $\mathbb{R}$ supported on $[0,+\infty[$. One easily derives the: Proposition. Let ...
1
vote
0answers
14 views

Complex filter factorizations with invariant points

Based on this question, using the same $z_0$: $$z_0 = e^{2\pi i / 8}$$ if we modify the sequence from previous question to look like this ($*$ denotes discrete convolution): $$\left(z_0^{[-2k,3k]} * ...
1
vote
0answers
18 views

Complex filter factorizations - continued

Continuing from this rather silly trivial question factoring real valued filters into shorter complex ones, hoping this won't be as trivial. If we modify it a bit: $$z_0 = e^{2\pi i / 8}$$ and ...
1
vote
0answers
23 views

Complex filter factorizations

There is a famous low pass filter $[1,2,1]$ in signal processing which can be factored in the sense of a convolution product over the real numbers : $[1,1] * [1,1]$. This is the only way to do it over ...