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Questions tagged [convolution]

Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.

Convolution is a commutative, associative, distributive operation between two functions that produces a third function. It is defined in the continuous domain as

$$(x \ast y)(t)=\int_{-\infty}^\infty{x(\tau)\space{}y(t-\tau)}\space{}d\tau$$

And in the discrete domain as

$$(x \ast y)[n]=\sum_{k=-\infty}^\infty{x[k]\space{}y[n-k]}$$

Its identity is the Dirac delta function $\delta(t)$ in continuous domain, and the Kronecker delta function $\delta[n]$ in discrete domain.

Reference: Convolution.

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How can I write an ordinary function in terms of integral of delta Dirac function?

We have the result $(x * \delta)(t) = x(t)$, where $*$ denotes convolution. Here let $x(t)$ is a real valued function with respect to time and $\delta(t)$ is the unit impulse function. $$ (x * \delta) (t) = \int_{-\infty}^\infty x(\tau)…
dexterdev
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Is possible get an estimate for the product between a polynomial and a convolution?

Let $f(x)$ a Schwartz function and $g(x)$ in $L^2(\mathbb{R})$. Is possible get \begin{align} \int_{\mathbb{R}}|x||(f*g)(x)|dx<\infty? \end{align} My attempt: By Holder's inequality, \begin{align} |(f*g)(x)|\leq \int_{\mathbb{R}}|f(x-y)g(y)|\,dy\leq…
eraldcoil
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Definition of Convolution

I am currently studying calculus, but I am stuck with the definition of convolution in terms of constructing the mean of a function. Suppose we have 2 functions, f and g. We want to create the mean of f for each x, interpreting g(t) as the “weight”…
IMM
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Convolution with multiple step functions

This is a question from Bertsekas' Data Networks. It is question 2.2 on page 141. It is asking for the convolution of the following 2 functions. Function 1: $ s(t) = 1 $, when $0 \leq t \leq T$. It is $0$ elsewhere. Function 2: $h(t) = \alpha e^{-a…
jrand
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Cancel one term of a function by convolution

I am trying to cancel one term within a function by convolution. $y[n] = x[n] + 2x[n-N]$ $y[n] * h[n] = x[n]$ here, $x$ is periodic and $N$ is a delay how can I find $h[n]$ such that convolution with $h[n]$ cancels the second term of $y[n]$ but…
Andre_van_stone
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Is there a general representation of $u[n-a]-u[n-b]$ in terms of unit impulses $\delta$?

Consider in the discrete-time domain, the unit-step function : $$ u[n]=\begin{cases} 1&\text{if $n\geq 0$}\\ 0&\text{if $n<0$} \end{cases} $$ We know that the first-order difference equation describes a relation between the unit-impulse function and…
Joumana L. Kincaid
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Can the delta function be even or odd?

Say I have the function, $$ \delta\left(\tau-\frac{T}{2}\right)\mathrm d\tau $$ where T = $$\frac{2π}{ω}$$ Is this even or odd? Or neither? Reason I ask is to find out whether I can cancel some cosine and sine terms for a convolution computation.
Macuser
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Understanding convolution

Take: $$ (u*v)(k) = \sum_{i=-\infty}^\infty u(i)v(k-i). $$ The $k$ is there, it's because you want to define $$ \ldots\ldots, (u*v)(-3), (u*v)(-2), (u*v)(-1), (u*v)(0), (u*v)(1), (u*v)(2), (u*v)(3), \ldots\ldots >$$ etc. The number in the…
user1095340
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associative law for convolution of distribution with test functions

Let ${\mathcal D} = C^\infty_c$, the space of test functions (smooth functions $\Bbb R \to \Bbb C$ with compact support). Let $\mathcal D^*$ be the space of distributions (continuous linear functionals ${\mathcal D} \to {\Bbb C}$). I am trying to…
Greg Grunberg
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Convolution and Smoothness Conditions

Suppose $f(x),g(x)\in L_1(\mathbb{R})$, with both $|f(x)| \leq 1$, $|g(x)| \leq 1$ and $|f(x)| \rightarrow 0$, $|g(x)| \rightarrow 0$ for $|x| \rightarrow \infty$. Given that we have two other functions $f_1,g_1$ such that for large enough $|x|$,…
user61038
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Decaying rate of a convolution between an integrable function and a Schwartz function

Suppose $f\in L^1(R^n)$ and $g\in S(R^n)$, where $S(R^n)$ is Schwartz space. Then, Can I have estimation like following? $$ |[f*g](x)|\leq\frac{1}{(1+|x|)^{s}}, $$ for some $s>n$. If it is correct, how to prove it? If it is not correct, what…
Lin Xuelei
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Convolution of two triangles

I want to convolve two triangles. The equation satisfied by one triangle is $$f(y) = \begin{cases} y + 1 & −1 < y < 0\\ \\ 1 − y & 0 \leq y < 1 \end{cases}.$$ So, the overall duration of a triangle is $\;-1\;$ to $\;1$. What I have found is that…
Xara
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Why$ f(x-x_0) \star g(x) = (f \star g)(x-x_0)$?

Convolution between 2 functions $f$ and $g$ is defined as $$(f\star g)(x) = \int_{-\pi}^\pi f(x') g(x-x') dx'.$$ Shift invariance of convolution is said to be the property that $$f(x-x_0) \star g(x) = (f \star g)(x-x_0).$$ Firstly, what does…
user10024395
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Commutativity of convolutions implies $\int\limits_{-\infty}^\infty f(t)dt = \int\limits_{-\infty}^\infty f(t-a)dt$?

As I understand it, to convolve $f$ and $g$ means to find $\displaystyle \int_{\mathbb R} f(a)g(t-a)da$, which is also apparently commutative, and therefore $\displaystyle \int_{\mathbb R}f(a)g(t-a)da = \displaystyle \int_{\mathbb R}f(t-a)g(a)da$…
user7634
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Finding the convolution of $2$ functions:

I'm trying to find the convolution $f*g$ where $f=g= \mathbb{1}_{\{ -1≤x≤1 \} }$ **Here's my attempt: $f*g = \int_{\mathbb{R} } \mathbb{1}_{\{ -1≤y≤1 \} }\mathbb{1}_{\{ -1≤x-y≤1 \} }dy$ We therefore have that $x \in [-2, 2]$, therefore $f*g =…
Sylvester Stallone
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