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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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 $$ g(t) = \int_{-\infty}^{t}e^{u-t}g(u)\,du + f'(t) $$ ...
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0answers
40 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 ...
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1answer
48 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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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
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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) ...
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0answers
23 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! ...
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1answer
59 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 ...
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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]} * ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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1answer
108 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 ...
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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 ...
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2answers
66 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 ...
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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 ...
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1answer
40 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 ...
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2answers
62 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
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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 ...
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1answer
55 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} ...
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0answers
18 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
27 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
81 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
64 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) = ...
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1answer
26 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
12 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 ...
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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 ...
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2answers
37 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: ...
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3answers
60 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 ...
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1answer
67 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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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
22 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
35 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
46 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 ...
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1answer
16 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} ...
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1answer
40 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
43 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 ...
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1answer
60 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
30 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
31 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 ...
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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
18 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 ...
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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 ...
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2answers
76 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
106 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 = ...
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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 ...
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1answer
15 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 ...