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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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: $T(\mu\ast\nu)= T(\mu)\ast T(\nu)$ ...
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1answer
31 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, ...
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1answer
82 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 ...
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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 ...
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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{...
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1answer
28 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 = ...
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0answers
13 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&...
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1answer
63 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&...
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2answers
38 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(...
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3answers
64 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 '$t$...
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1answer
79 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?
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0answers
28 views

Is it possible to do this integral using the residue theorem? $ H(u,a)= \frac{a}{\pi} \int_{-\infty}^{\infty} \frac{e^{-x^2}}{(u-x)^2+a^2} dx $

$ H(u,a)= \frac{a}{\pi} \int_{-\infty}^{\infty} \frac{e^{-x^2}}{(u-x)^2+a^2} dx $ Someone asked a question that involves this integral on another math forum. I put it into wolfram alpha to see what ...
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0answers
30 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
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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 ...
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1answer
12 views

Convolutions with differing arguments

I kind of want to clear this up once and for all: $g (t) * f(t) = \int g(u)f(t-u)du$ $g(at) * f(bt) = \int g(au)f(bt-u)du$ ???
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0answers
47 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 $T>0$...
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1answer
17 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 $$f * f(\frac{x}{a}) = \int_\mathbb{R} f(\frac{u}{a})f(\frac{x}{a}-u)du$$...
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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 $k$ ...
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1answer
47 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
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1answer
65 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
32 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 $\...
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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)=\...
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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 ...
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0answers
20 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
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1answer
36 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)}\|...
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2answers
79 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: ...
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1answer
126 views

Convolution of two DIFFERENT rectangular pulses

I'm looking for the convolution of $\mathcal{X}_{[0,1/2]}$ and $\mathcal{X}_{[0,1]}$ and I'm having trouble. $\begin{align} \int \mathcal{X}_{[0,1]}(s)\mathcal{X}_{[0,1/2]}(t-s)ds = \int_0^{1/2}\...
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0answers
10 views

Convolutions of L^p functions

Denote by (fg) the convolution of f and g. If g is square integrable and (fg) is square integrable for every square integrable f can we conclude that g in integrable? This is a converse to the well-...
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1answer
16 views

energy of a convolution

I have to find the energy of $y(t)$ $$h(t)=ho\;sinc^3(t/T)\\ x(t)=V_0+V_1\;sin(3\pi\; t/T)\\ y(t)=x*h\;(t) $$ Where "$*$" is the convolution and $sinc(t)=\frac {sin(\pi t)}{\pi t}$ I think that the ...
2
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1answer
314 views

Find the PDF of X1 +X2 +X3.

The problem said: If X1,X2,X3 are independent random variables that are uniformly distributed on (0,1), find the PDF of X1 +X2 +X3. The theory I have said: Following the theory and the ...
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0answers
22 views

How to calculate the cross-correlation of a halfwave?

I'm trying working on a vehicle modle and testing it with steering maneuvres standardised by the ISO. When it comes to data analysis the ISO standard says the following: 6.4 Time lag and Sine Time ...
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1answer
76 views

Convolution of two Uniform random variables

We have $X \sim \mathrm{Unif}[0,2]$ and $Y \sim \mathrm{Unif}[3,4]$. The random variables $X,Y$ are independent. We define a random variable $Z = X + Y$ and want to find the PDF of $Z$ using ...
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0answers
27 views

Convolution and differential equations

Consider the following system of differential equations: \begin{align} x_1'=f_1(x_1,x_2)\\ x_2'=f_2(x_1,x_2) \end{align} Assume that a solution $x(t)$ exists for $t\in (-T,T)$. Let $g:\mathbb{R}^2\to\...
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0answers
31 views

Is there a mistake in my calculations?

I am trying solve the differential equation $y''+3y'+2y=u(t-1)-u(t-2), y(0)=y'(0)=0$, by calculating the convolution of $f(t)=1$ and $g(t)=e^{-t}-e^{-2t}$. The problem is that I get two different ...
3
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0answers
50 views

Approximating two-dimensional convolution

I am trying to use discrete 2d-convolution to estimate continuous double convolution. The convolution integral is $$g(x,y)=(f\ast h)(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(u,v) h(...
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2answers
86 views

Inverse Laplace Transfrom of $s^{-1}e^{-a\sqrt{s} + b/s}$

I am trying to find the inverse Laplace transform for following function and it seems almost impossible for me to find the answer. Can anyone help me please with final answer and also the way to get ...
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0answers
20 views

2D convolution with one-dimensional function

I am somewhat stumped on what may be a very basic question. I have a 2D input function $F(x,y)$ and an impulse response $H(x)$ that is independent of $y$. The output function is a convolution $G(x,y)...
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0answers
20 views

Estimating certainty of angle for rotations using curl filters.

Background Inspired by this answer which manages to make a connection between (the angle of) a proper rotation (displacement field of a rotation) and the curl of a vector field. This question aims to ...
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0answers
23 views

Fourier transform to determine stability of fixpoint of equation with temporal convolution

Given the differential equation \begin{align} \frac{d v}{d t} = - v(t) + \kappa * v \end{align} where $\kappa$ is some linear temporal filter (like a sum of two exponentials, for instance) and $...
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1answer
342 views

Help for convolution of two Multivariate Gaussian PDFs

I am looking for a proof for convolution of two multivariate Gaussians (where each Gaussian has multi-dimensional mean and co-variance). I found a proof in here: http://www.tina-vision.net/docs/memos/...
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1answer
306 views

Convolution - Difference of two random variables with different distributions

This is a homework problem, but it isn't me looking for an easy way out. I've been thinking about this problem for a while now and even went to my professor's office hours and still don't quite ...
2
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1answer
27 views

Convolution of multiple correlated probability density functions

Following up this question, assume $X_1$ to $X_n$ are $n$ correlated random variables with know marginal cumulative/probability distribution functions of $f_1(X_1)$ to $f_n(X_n)$, and a joint ...
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1answer
148 views

Prove that $\int\delta(x-b)\delta(x-a)\ \mathsf dx =\delta(a-b)$.

Show that the convolution of two $\delta$ functions at different points is again a Dirac $δ$ function. Convolution of a Dirac $\delta$ function with a function $f$ is defined as : $$\int\delta(x-y)f(...
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1answer
29 views

the solution to heat equation in convolution form

Let $g\in C(\mathbb{R})\cap L^{\infty}(\mathbb{R})$. Let $u$ be defined as the function $$u(t,x)=\int_{\mathbb{R}}p_t(x-y)g(y)\,dy$$, where $$p_t(x)=\frac{1}{\sqrt{4\pi t}}e^{-\frac{|x|^2}{4t}},\quad ...
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0answers
76 views

Convolution with piecewise functions

Let $$x(t)=\begin{cases} \sqrt{\frac{A}{T}},\;\;t \in [0,T/2]\\ -\sqrt{\frac{A}{T}},\;\;t \in [T/2,T] \end{cases}$$ Now lets define two other functions: \begin{align*}h(t) &= x(T-t) \\ g(t) &= ...
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0answers
82 views

Convolutions and dot products

There is a convolution formula $$F(fg)= F(f) * F(g)$$ where $F$ represents a Fourier transform. Suppose I have something that looks like $$\int d^3 \vec{k}'\,\,\, \left[\vec{k}' F[f(x)](\vec{k}')\...
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1answer
122 views

Derivative of unit step function

The ramp function is given by r(t)=tu(t) If we differentiate ramp ,we get unit step function. That is, u(t)=1 So the derivative of unit step function is definitely 0 since u(t) is constant over the ...
0
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1answer
73 views

Show that the convolution of functions is differentiable.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$ be $2\pi$-periodic functions such that $f$ is bounded, $g$ is differentiable and $g'$ is continuous. How can I show that ...
0
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0answers
13 views

Autoconvolution Notation

Is there any special notation for a list (or function) convolved with itself? For example, the convolution of the list [1,1,1,1] with itself produces a new list: [1, 1, 1, 1] * [1, 1, 1, 1] = [1,...
1
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0answers
31 views

Function equation with sth like self-convolution

I would like to get the solution of $\int_{x}^{y}f(t,y)f(x,t)dt=af(x,y)$ for all $\{y>x>0\}$, satisfying $\int _{0}^{x}f(t,x)dt=1$. Or a more simple question may be if we assume $f(x,y)=g(x/y)/...