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
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1answer
22 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
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0answers
23 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
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0answers
23 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
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1answer
24 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{ otherwise}\end{...
0
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1answer
39 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
28 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
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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
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0answers
41 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? Global ...
2
votes
2answers
95 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
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1answer
21 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
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2answers
77 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
54 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*} -\varphi\star\hat{\varphi}(x)...
2
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0answers
24 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
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1answer
185 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 $x \in E$, let$$(\...
3
votes
0answers
37 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) \, d\...
0
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1answer
39 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
36 views

Convolution domains probability theory

Problem 1.4 here: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/unit-ii/quiz-2/MIT6_041SCF13_quiz02....
0
votes
2answers
34 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
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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
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0answers
23 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 \int_{-\infty}^{\...
1
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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
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2answers
238 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 a ...
0
votes
0answers
37 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
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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
37 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 G(|\theta^{\...
1
vote
1answer
50 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 said....
1
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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 \\ \frac{-1}{4}x+\...
1
vote
1answer
63 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
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0answers
42 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 5t}...
0
votes
1answer
50 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
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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
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3answers
44 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) \subseteq ...
0
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0answers
27 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
65 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 $T,S\in{\mathcal{D}'}_+(\mathbb{R})$,...
1
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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
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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
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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 ...
1
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1answer
64 views

How to show convolution is smooth i.e. in $C^\infty$

$\newcommand{\Rn}{\mathbb{R}^N}$ $\newcommand{\dxi}[2]{\frac{\partial #1}{\partial x_{#2 } } } $ $\newcommand{\conv}[2]{#1 \star #2}$ $\newcommand{\ball}{B(0, {1 \over n})}$ I would like to show the ...
0
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0answers
25 views

Decimal multiplication ( primary school algorithms ) and relations to the Fourier Transform.

I suppose most of us are familiar with multiplication algorithms for decimals numbers we learn in primary school. 13*13 = 3*13 + 10*13 = 169 what actually is performed in this case (but of course they ...
1
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0answers
43 views

The convolution theorem - basic problem with the formula

I have a formular for the convolution theorem, and read several chapters in several scripts about it. This is the formula: $(f*g)(x)=\int_{\mathbb{R}^d}f(x-y)g(y)dy$ However much I read, I cannot ...
2
votes
1answer
109 views

Convolution algebra $L^1(G)$ for non sigma-finite $G$

Let's assume that $G$ is locally compact and Hausdorff topological group, hence it carries a Haar measure, $\mu$. We can than consider space of integrable functions $L^1(G)$ (class of functions to be ...
1
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1answer
34 views

Confusing Statement from a Derivation of the Convolution of Gaussians

I was reading a proof of the claim that the convolution of two Gaussians with arbitrary mean and variance results in another Gaussian. It can be found here. At the end of the proof, which I studied ...
1
vote
2answers
70 views

Convolution Theorem and Marginal Density Intuition.

In terms of marginal density, how does one know that summing over the $x$ (or rather along the linear line) values for the joint density of $(x,z-x)$ give us the density function of $z$? More ...
3
votes
2answers
105 views

Convolution of half-circle with inverse

I am trying to compute the function: $$f(\lambda)\equiv\int_{-1}^{1}\frac{\sqrt{1-x^2}}{\lambda-x}dx.$$ It arises as the convolution of the semi-circle density with the inverse function. When $\...
0
votes
1answer
42 views

Where is the convolution of two functions well defined?

I want to mollify a function $f\in L^1_{loc}(\Omega)$, $\Omega$ open in $\mathbb R^n$. So, we take a sequence of standard mollifiers $\{\phi_\epsilon\}$, for example, and do the convolution $f*\phi_\...
1
vote
2answers
64 views

Is convolution of sine-squared function, sinusoidal function?

Ladies, Gentlemen By sinusoidal function, I mean function of the form Asin(x) or Acos(x) for A real number. I make note that am beginner in convolution process. Regards
1
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0answers
17 views

Maximization of a convolution

Given a piecewise smooth, bounded, integrable, etc., causal function $h(t)$, such that $h(t)=0$ if $t<0$, my question is which bounded, causal, piecewise smooth, etc., function $c(t)$ will maximize ...
1
vote
1answer
59 views

Convolution of characteristic function

I am trying to figure out following problem. Let A ⊂ R. Then we can define the characteristic function: Let a be bigger than 0. I am trying to find a following convolution: \begin{align} \chi_{[-a,a]...