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
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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
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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
51 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
35 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
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1answer
52 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 ...
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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
31 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
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1answer
69 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
19 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
65 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
27 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
110 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
35 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
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2answers
73 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
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2answers
108 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
46 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
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2answers
65 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
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1answer
79 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]...
1
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0answers
19 views

Convolution of measures with a linear transformation

The task: Let $T:\mathbb R^d \rightarrow \mathbb R^d$ be a linear map. For $\mu, \nu \in M^1(\mathbb R^d)$ (so $\mu,\nu$ are probability measures), I have to show: $T(\mu\ast\nu)= T(\mu)\ast T(\nu)$ ...
0
votes
1answer
32 views

Convolution of shifted dirac delta's

what is the result of $\delta(t-t_0)*\delta(t+t_0):$ $\int_{-\infty}^\infty \delta(t+t_0-x)\delta(x-t_0)dx$ Im not sure how to proceed with that question. Normally when we use sifting property, ...
2
votes
1answer
84 views

Calculating convolution integral analytically

How can i compute convolution integral analytically, without using graphs. I hate using graphs, shiftings which are error prone. If this is possible can you explain what way i must follow? For ...
1
vote
1answer
16 views

How to calculate step response for $y''(t) - y(t) = x'(t) - x(t)$ in time domain?

How to calculate step response for $y''(t) - y(t) = x'(t) - x(t)$ in time domain? So, without Laplace or Fourier Transforms. This is what I tried: The Homogeneous solution of the differential ...
1
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2answers
65 views

Why is $\int e^{-t}u(t) dt = (1-e^{-t})u(t) + Constant$?

How do you solve $\int e^{-t}u(t) dt $? In which u(t) is the unit step function. $\int e^{-t}u(t) dt = (1-e^{-t})u(t) + Constant$ But what are the intermediate steps? Unit step u(t) = \begin{...
1
vote
1answer
29 views

can a square z-transform function be expressed as the convolution of itself in time-domain

Given $H(z) = (1+z^{-1}+z^{-2})^2$ I want to make use of the property of z-transform that $$x_1(n)\ast x_2(n) \rightarrow X_1(z)X_2(z)$$ so if $x_1(n) = x_2(n)$ then the $X_1(z)X_2(z) = X_1(z)^2 = ...
0
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0answers
14 views

A convolution identity (perhaps related to discrete fourier transform)

Let $M$ be a function of period $p$ defined by $M(0)=0$ and $M(n)=1/2-n/p$ for $n=1, \dots, p-1$. It is known that for $a<b$ we have $$(f*M)(b)-(f*M)(a)=\frac{f(a)+f(b)}{2}+\sum_{a<n&...
0
votes
1answer
64 views

Approximating convolution of two functions with Oh notation

Define the function $\|\cdot\|$ by $\|x\|=\min_{n\in \mathbb{Z}}|x-n|$. This is of course periodic with period $1$. Now let $f(x)=\|x\|^{-\alpha}$ and $g(x)=\|x\|^{-\beta}$, and assume $\alpha+\beta&...
0
votes
2answers
40 views

I'm unsure about $(\delta(t-a)*f)(t)=f(t-a)$, where $\delta$ is Dirac delta function.

I'm reading some notes from MIT OCW on function convolution defined by: $$(f*g)(t) = \int_{0^{-}}^{t^{+}} f(s)g(t-s) \ ds$$ It says that $(\delta(t-a)*f)(t)=f(t-a)$ and the proof is that: $(\delta(...
0
votes
3answers
71 views

Why is the convolution output in terms of 't' not $\tau$?

The convolution integral is defined as: $$y(t) = (h * x)(t) = \int^{+\infty}_{-\infty} h(\tau). x(t-\tau)\ d\tau$$ where $h(t)$ and $x(t)$ are functions in terms of time. Why is $y$ in terms of '$t$...
2
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1answer
87 views

Is the convolution of two continuous functions continuous?

The title is the question: Is it true, that the convolution of two continuous functions is continuous again?
3
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0answers
31 views

Is there a name for this “simplified” Volterra series?

Consider a nonlinear, time-invariant system of the following form: $g(t) = \left[h_1(t) \ast f(t)^1\right] + \left[h_2(t) \ast f(t)^2\right] + \left[h_3(t) \ast f(t)^3\right] + ...$ where $\ast$ ...
2
votes
1answer
39 views

Dominating function for using DCT with convolution

Given a function $f \in L^1$ and a (compactly supported) bounded kernel $k$, this answer suggests to use the Dominated Convergence Theorem to get $$ D(f \star k)(x) = f \star (Dk)(x). $$ My question ...
1
vote
1answer
15 views

Convolutions with differing arguments

I kind of want to clear this up once and for all: $g (t) * f(t) = \int g(u)f(t-u)du$ $g(at) * f(bt) = \int g(au)f(bt-u)du$ ???
0
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0answers
48 views

Convolution of exponential and rect functions

I have a convolution question in my signals and systems problem set that is puzzling me: $ f(t) = e^{-t/2T} u(t) $ and $ g(t) = rect(t/2T) $ find the convolution $f \ast g$ and I am assuming $T>0$...
1
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1answer
20 views

Convolution of scaled variable

A rather simple question... Is the following true? $$f * f(\frac{x}{a}) = \int_\mathbb{R} f(u)f(\frac{x}{a}-u)du$$ Or is it $$f * f(\frac{x}{a}) = \int_\mathbb{R} f(\frac{u}{a})f(\frac{x}{a}-u)du$$...
0
votes
1answer
43 views

Conditional expectation of an exponential RV, where conditioning is on sum of exponential RVs

I would like to find the conditional expectation of a random variable $q$ which is an exponential random variable with $\mbox{pdf}(q) = \lambda e^{-\lambda q}$ conditional on $q + v > k$, where $k$ ...
5
votes
1answer
48 views

$n$ fold convolution tends to zero a.e. if $\|f\|_{L^1}<\infty$.

Let $f\geq 0$ satisfy $\int_\mathbb{R} f < 1$. Let $f_n$ be the $n$ time convolution of $f$ by itself. Then I want to show $f_n \rightarrow 0$ a.e. as $n\rightarrow \infty$. We can clearly ...
2
votes
1answer
66 views

Differential operator applied to convolution

Suppose that $g\in \mathcal{S}(\mathbb{R^n})$ (Schwartz space) and $f\in L^p(\mathbb{R^n}).$ The idea is to prove that the differential operator $D^\alpha$ does not follow the Leibniz rule when ...
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0answers
34 views

autocorrelation after convolution

Suppose there are two time series. The first one is $x(t)$ where $t$ is the time. The second one is $y(t)=\int_{-\infty}^{+\infty} x(s)Q(t-s)\mathrm{d} s$ where $Q(s)$ is a weighting function and $\...
1
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1answer
32 views

Convolution of functions not in $L^1$

Is it possible to have two functions strictly outside $L^{1}(\mathbb R)$, $f,g: \int_{-\infty}^{\infty}|f(x)|dx=\int_{-\infty}^{\infty}|g(x)|dx=\infty$, and such that their convolution $f\ast g (x)=\...
1
vote
0answers
23 views

Find counterexample for $C^{*} $-Algebra axiom for $L^{1}(\mathbb{R})$ with convolution.

I want to find a counterexample for the $C^{*} $-Algebra axiom for $L^{1}(\mathbb{R})$ equipped with convolution as multiplication, that is I want to find a function $f \in L^{1}(\mathbb{R})$ such ...
0
votes
0answers
22 views

Apodization and convolution theorem

Suppose that we exponentially suppress high frequencies by multiplying the Fourier amplitude $\tilde{f(k)}$ by $e^{-\epsilon |k|}$$. Show that the original signal f(x) is smoothed by convolution with ...
2
votes
1answer
49 views

Show that $\lim_{n\to\infty}\|f_*^{(n)}\|_1^{\frac{1}{n}}=\|\hat{f}\|_\infty$

Let $1<p\leq 2$ and $f\in L^p(\mathbb{T})$, i.e. $f$ is $p-$th power integrable and is $1-$periodic. Define $$f_*^{(n)}=f*f*\dots*f\quad n\text{ times}$$ Show that $$\lim_{n\to\infty}\|f_*^{(n)}\|...
2
votes
2answers
80 views

Taylor series of a convolution

The derivation below is from Probability Theory: The Logic Of Science By E. T. Jaynes, chapter 7 "The Central Gaussian, Or Normal, Distribution", p.706 The Landon derivation. Text available online: ...