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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Practical convolution of poisson and log-normal distribution

Hi guys, Im trying to make a Loss Distribution (in Excel), following the "Loss Distribution Approach". I do understand that the main idea is that we have a severity distribution and a frequency ...
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13 views

Inverse Fast Fourier Transform to find the voltage across a capacitor of a RC circut

Fourier transform of a RC circuit The following example of a RC circuit describes the use of the fourier transform in order to receive the output voltage across the capacitor. My questions ...
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28 views

Finding $\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$ using Plancherel’s theorem

Suppose $G(\mathbb R)\ni f(x),\mathcal{F}[f](\omega)=\frac{1}{1+|w|^3}$ find the value of $$\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$$ I thought using Plancherel’s theorem \begin{align} ...
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1answer
20 views

Convolution/Deconvolution $\stackrel{?}{=}$ Coding/Decoding

In a strict mathematical sens, can a convolution/deconvolution be equivalent to a coding/decoding process ? I just got the remark from a reviewer that it's strictly different, it's a little surprising ...
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5 views

Using differentiating Gaussian kernel to filter a curve

I saw an equation using a differentiating Gaussian kernel to filter a curve and not quite understand. Suppose we have a curve $\gamma_{t} = [x_{t}, y_{t}]$ with $t = 1, 2, ..., T$. From [1], we know ...
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1answer
47 views

Differential equation with fourier transform and convolution

We have differential equation $3s(t)-2s''(t)=r(t)\,$ and $s(t)$ is convolution $s=g*r\,$ where $g(t)=ae^{-b\left | t \right |}\,$ $\\a,b\in\mathbb R+$ Solve constans a and b. I tried to solve ...
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33 views

Sort of Convolution

I was wondering if the following convolution I am considering already has a name or well-studied. Thanks for your help. If the condition of the positive radius of convergence is not enough, then I ...
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1answer
291 views

Dirac delta convolution with function

I've come into a bit of a snag, and thought some more talented mathematicians could maybe help. I am trying to do the following integral: $$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$ where ...
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31 views

Evaluating convoluted integrals of complex exponentional and rational

I want to evaluate the following integral: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda\int_0^{t-\lambda} d\tau \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \int_0^{t-\lambda} d\tau ...
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20 views

L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
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Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
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1answer
24 views

Convolution Case

The * denotes convolution and u[n] as the heaviside function. $x[n]= u[n]α^n$ Determine a sequence $h[n]$ such that-: $x[n]∗h[n]=α^n(u[n+2]−u[n−2])$ I am trying this problem for quite awhile now. ...
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1answer
19 views

Uncertain with piecewise result for convolution integral

I have two equations $$x(t) = u(t) - 2u(t-2) + u(t-5)$$ $$h(t) = e^{2t}u(1-t)$$ where $u(t)$ is the unit step function. I'm attempting to find the convolution of the two: $$y(t) = h(t)*x(t)$$ I ...
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1answer
688 views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
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2answers
49 views

Integration of dirac function explanation

I have a problem that need your help. I have a gray image. We denotes $I(x)$ is gray level of a pixel in the image and $f(z)$ is a function of $z$(ie: histogram function...)-where $z$ is the set of ...
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1answer
50 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
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2answers
23 views

convolution problem given $h(x)=1/2$ for $0<x<2$ and $0$ otherwise

I have a convolution problem in the form $$g(x)= \int_{-\infty}^\infty h(y)h(x-y)\,dy$$ where they give me the function $h(x)=1/2$ for $0<x<2$ and $0$ otherwise. I have never done a ...
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2answers
35 views

Computing a messy convolution

Consider the functions $$ x(t) = u(t - \frac{1}{2}) - u(t - \frac{3}{2}) $$ and $$ h(t) = tu(t) $$ where $u(t) = 1$ if $t \geq 0$ and $u(t) = 0$ if $t < 0$. I'm trying to compute $$ (x*h)(t) ...
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1answer
38 views

Calc $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$

The answer to this integral:$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$$ is ...
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3answers
39 views

Divisor function convolution

I need some help to prove that $$ (d*d)(p^k) = \frac{(k+3)(k+2)(k+1)}{6} \qquad \forall p \in \mathcal{P},\quad \forall k \in \mathbb{N}, $$ where $d$ is the divisor function and $\mathcal{P}$ the set ...
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1answer
55 views

problem with convolution

I'm struggling with this kind of problem: I have an assumption that $f$ and $g$ are in $L^2(R)$, and I should prove that $f\star g \rightarrow 0$ when $|x| \rightarrow \infty$. I think (but I'm not ...
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42 views

Distribution of the sum of many lognormal random numbers from same distribution

In my application I have to sum up a lot (between 1000 and 2000) lognormally distributed random numbers and use their sum. All random numbers that I sum up follow the same distribution. The current ...
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19 views

Function smoothing using convolution

I have a function $\hat f$ which is an estimator of an unknown function $f$. The estimator $\hat f$ looks pretty irregular (see the red line). I would like to smooth it with some kernel function ...
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28 views

understanding discrete-time convolution

I'm trying to understand the discrete-time convolution for LTIs and its graphical representation. standard explanations (like: this one) start with the idea of decomposing an input signal $x[t]$ into ...
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2answers
411 views

How to calculate a 1D convolution summation?

I hope I said that right. I'm trying to follow along with a convolution example but maybe I am in over my head. I don't understand how in this example they get the values on the right. For example, I ...
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2answers
76 views

Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
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1answer
25 views

When convolution of two functions has compact support?

It is well-known that, if $f$ and $g$ are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous (Hörmander 1983, Chapter 1). Next, ...
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Convolution with itself equals itself times a function

Consider the case that $f \in L^1(\mathbb{R})$ and $g \in L^1_{loc}(\mathbb{R})$. Then look at the equation $$ f*f=g\cdot f. $$ I know that if $g$ is constant, then $f=0$. But what about other $g$'s? ...
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using convolution for power series solution method for DE's

Say I have a homogeneous linear differential equation of the form $y''+py'+qy=0$ and I want to solve it using the power series solution method. So I use the substitution ...
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Autocorrelation of Raised Cosine Function

Let us define the raised cosine function as follows: $f \left( x \right) = \dfrac{\left( 1 + \cos \left( x \right) \right)}{2}$, for $- \pi < x < \pi$. $f \left( x \right) = 0$, elsewhere. I ...
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35 views

Avoiding FFTs by reusing prior FFT results

Background From a mathematical point of view, the formulas similar to the following were produced: $F_1(f) = \mathcal{F}(T(t))$ $F_2(f) = \mathcal{F}(T(t)\times sin\Theta t)$ $F_3(f) = ...
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1answer
30 views

Convolution of Strictly Convex Function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^2$, strictly convex function, and $\theta_\epsilon$ the standard approximation to identity ...
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56 views

Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
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1answer
35 views

What is the generalization for a convolution in $\mathbb C$?

Since the integration range of "the" convolution is $\mathbb R$, what is a sensible generalization in complex numbers? Would one still integrate over $\mathbb R$, or some other path, or over the ...
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25 views

Proof commutativity of (differential) convolution operater

I tried to proof a claim and I'm not sure if I did it right. It would be great if someone could have a look at it! First I give a definiton: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define ...
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1answer
14 views

Show separability of discrete convolution.

Given two functions $I, H$ we define the discrete convolution as $$ I' (u,v) = I(u,v) \ast H(u,v) = \sum_{i = -\infty}^\infty \sum_{j = -\infty}^\infty I(u-i, v-j) H(i,j)$$ Now, I need to show that ...
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1answer
36 views

Fourier transform of a Laplace transform

Is there an easy way to find the Fourier transform of a Laplace transform of function? $$ F[L[f(t)]_{s}] $$ Where my $f(t)$ is $\sqrt{t}$. However, Before finding the Fourier transform I do the ...
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1answer
25 views

Laplace transform of a product of two functions

I have read questions and answers about this topic and i am still confused, using this formula we can calculate the Laplace transform of a product of two functions: $$ L[a_{(t)} b_{(t)}]={{1}\over{2 ...
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1answer
22 views

Bound on the derivative of a cut-off function

Let $\rho$ be a smooth function in $\mathbb R^n$ such that $0 \leq \rho \leq 1$ and $\rho$ is supported in the unit disk and let $\rho_\epsilon(x) = \epsilon^{-n}\rho(\epsilon^{-1}\|x\|)$. If $f$ is ...
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1answer
34 views

Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
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2answers
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Is the convolution operation some kind of group operation?

I'm just curious but will the convolution operation be any sort of group operation? A motivating example would be to see that the natural exponential family of distribution functions are closed under ...
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2answers
52 views

Density of sum of two uniform random variables

I have two uniform random varibles. $X$ is uniform over $[\frac{1}{2},1]$ and $Y$ is uniform over $[0,1]$. I want to find the density funciton for $Z=X+Y$. There are many solutions to this on this ...
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1answer
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Probability of a single variable from a Moment Generating Function

This is from the A/S/M study guide, and the answer is listed, I just don't understand how he's arriving at the answer... I'm sure it's something simple I am missing! I have two identically ...
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2answers
147 views

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
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1answer
27 views

Details about Generalized Convolution (Number Theory - Apostol)

In "Introduction to analytic Number Theory" by Apostol there is chapter about generalized convolution. Let F denote a real or complex-valued function defined on the positive real axis such that ...
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2answers
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Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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34 views

Re-expressing the Integral $\int h(t-\tau)x(t-\tau)x(\tau)^2\,\text{d}\tau$

Given the integral: $$ \int^{t}_{0} h(t-\tau)x(t-\tau)x(\tau)^2\,\text{d}\tau = \int^{t}_{0} h(t-\tau)F(\tau)\,\text{d}\tau + K $$ Can you find $F(\tau)$ so it is not a function of $t$ and find $K$ ...
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Variance reduction factor using gaussian filtering

I am currently trying to find the variance reduction ratio using gaussian filtering. For a simpler filter (as mean filtering for example), I am able to calculate it easily to find the well known ...
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1answer
106 views

Why does convolution not maintain asymptotic equality of coefficients?

Assume I have four (generating) functions $f$, $f'$, $g$ and $g'$. If that is interesting, we can assume that they all share the same radius of convergence $\rho > 0$. In addition, we know that ...
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Do asymptotically equivalent coefficients survive convolution at least in Θ?

This is a follow-up question to this one where I asked if asymptotic equivalence of coefficients carried over after convolution, resp. why this was not the case. Answerer Daniel Fischer proposed that ...