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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228 views

Convolution of functions with compact support

I have a question regarding convolution with compact support: Suppose $f \in L^1(\mathbb{R})$ and $g \in L^p(\mathbb{R})$, and both of them have compact support. Show that $f*g$ (convolution ...
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2answers
47 views

What is the name of this function similar to convolution?

The functions seems to be very near convolution function, but the only difference is that you integrate by $du$ in convolution, in contrast to $ds$ in this example: $g(t,u) ...
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1answer
58 views

About the continuity of a convolution product

I need some help with this exercise: If $f\in L_p(\mathbb{R}^n)$ and $g\in L_q(\mathbb{R}^n)$, where $\frac{1}{p}+\frac{1}{q}=1$, Is their convolution $f\ast ...
2
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1answer
65 views

Does negative distributive property of convolution over cross correlation holds?

Let $\star$ denote convolution binary operation and $\otimes$ denote cross correlation binary operation between two functions. Let $f,g,h$ be functions. Does this negative distribution property ...
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44 views

mathematical statistics - convolution and sums

Given integrable functions f and g on $\mathbb{R}$ define $f*g$, the convolution of $f$ and $g$ by $f*g(x) = \int_{-\infty}^\infty f(y)g(x-y) dy$, So i have to show that $M_Z(t) = M_X(T)M_Y(t)$ ...
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18 views

How to express this expression into a convolution, if possible

In order to speed-up some important numerical evaluation, I would like to express, if possible, $ \int d\mathbf{r}_1 \int d\mathbf{r}_2 \int d\mathbf{r}_3 n(\mathbf{r}_1) n(\mathbf{r}_2) ...
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0answers
77 views

n-fold convolutions of Poisson distribution

I understand the convolution between 2 random variables, i.e, it is the area under the product of $signal 1$ at time $t$ and $signal 2$ at time $t-\tau$ When I have a n fold convolution of a function ...
2
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1answer
110 views

Approximation in Sobolev Spaces

Consider the following proof in Lawrence Evans book 'Partial Differential Equations': How does it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in ...
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1answer
190 views

Evaluating the convolution using the convolution integral

I am having trouble evaluating the convolution of two signals using the convolution integral.I want to find the convolution of two signals x and h where, $$ x(t) = \begin{cases} e^{-at} ...
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1answer
2k views

Creating a System Impulse Response in Matlab

Preface: I'm extremely new to Matlab. Ok, so I have a sound file that I loaded in Matlab. Two variables are loaded: ...
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2answers
327 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
115 views

If $(f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d})$, prove that $\ast$ is commutative.

If $(f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d})$, prove that $\ast$ is commutative. I believe this is convolution. My attempt at the proof: $(f \ast g)(n) = ...
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3answers
92 views

How does the author get from one step to another?

I have to apply convolution theorem to find the inverse Laplace transform of a given function. I know that convolution is applied when the given function is multiplication of two functions. The ...
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0answers
102 views

Given a meromorphic function (via its Laurent series), how to obtain the (Taylor series of the) two holomorphic functions it is the quotient of?

Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...
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1answer
117 views

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

We have the result $(x * \delta)(t) = x(t)$. Here let $x(t)$ is a real valued function with respect to time and $\delta(t)$ is the unit impulse function. $$ x(t) * \delta (t) = \int_{-\infty}^\infty ...
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0answers
49 views

The role of conjugate reflected signal in an inner product

Let the conjugate reflected signal of $f(x)$ is $\tilde{f}(x) = \overline{f(-x)}$. Now the convolution between $f(x)$ and $g(x)$ is defined as the inner product $f*g(y) = \int f(x)g(x - y)dx$ and is ...
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1answer
146 views

Support of Convolution and Smoothing

I just want to know how it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in C^{\infty}(V)$ by using the translations, but I'm having difficulty seeing how it ...
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1answer
379 views

Integral of repeated convolution of the unit step function

Background Let $\theta$ be the unit step function: $$\theta(x) = \begin{array}{ll} \left\{ \begin{array}{ll} 0 & x \lt 0 \\ 1 & x\ge 0. \end{array}\right. \end{array} $$ Further, the ...
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0answers
29 views

Finding correlation of data with potentially hidden time lags

Let's say I have few independent variables plus multiple observables that I monitor over time for a system. I'd like to find out if there is any correlation between the observables and any of the ...
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2answers
318 views

The condition for Y to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$

I would like to know the condition for a random variable Y in order to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$, where $X_1$ and $X_2$ are iid. Any help would be ...
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2answers
88 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...
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1answer
144 views

convolve probit function with gaussian [duplicate]

I want to prove the following, however, not sure where to start. $\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$ Where $\Phi(\cdot)$ is the probit function, ...
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0answers
59 views

LTI system and convolution

I'm reading a rather informal text on (continuous) Linear Time Invariant (LTI) systems. It is just said to be a "black box" that transforms an input signal $x(t)$ into an output signal ...
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1answer
166 views

Convolution of Discrete Uniform ,$DU$, Distribution.

If $X\sim DU(k,a,h),\quad -\infty<a<\infty,h>0=1,2,\ldots$ then the probability function is $$P(X=a+jh)=\frac{1}{k},\quad j=0,1,\ldots,k-1$$ Let $Z\sim DU(r,0,s)$ and $Y\sim DU(s,0,1)$ , ...
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1answer
30 views

How to choose a phase for the deconvolution of an autocorrelation?

Say I have a function, $C=C\left(x\right)$, whose fourier transform is denoted by $c=c\left(k\right)$, i.e. $C\left(x\right)=\sum_{k=-\infty}^{\infty}c\left(k\right)\chi\left(x\right)$, where ...
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1answer
119 views

Convolution of two dimensional gaussian functions

I want to calculate the sum of two probability density functions. I know that it is: $P_{U+V} (x)= (P_{U} * P_{V})(x)$ If $P_{U}$ and $P_{V}$ are gaussian functions in one dimension, i.e. $P_{U}(x) ...
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1answer
67 views

Norm of convolution of $n$ Gaussians

If $$f(x)=e^{-(\pi x)^2}$$ and $$\psi_n(x)=(f* f*\dots*f)(x)$$ ($n$ times convolution). Show that $$\lVert \psi_n(x)\rVert = 1$$ (norm in $L^1(\mathbb{R})$). I've tried using the Fourier ...
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0answers
237 views

What is the physical significance of circular convolution?

I know the concept of linear convolution in both discrete and continuous domain for impulse response etc in linear time invariant systems. But why do we go for circular convolution?
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1answer
115 views

Convolution of an $L^1$ function and a function that tends to $0$ results in a function that tends to $0$

I'm trying to solve the following problem in review for a test, but have only partly succeeded: Let $K \in L^1(\mathbb{R})$ and $f$ be a bounded, measurable function on $\mathbb{R}$, with ...
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2answers
77 views

Proof of convolution

I would like to know how I could prove the following convolution: $$ D (f*g) =D f* g =f* Dg $$
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1answer
802 views

Is there an elementary proof of the convolution theorem?

Is there a way, without using much extra theory (other than the basic ideas used in textbooks deriving the Fourier transform for the first time, and ideally just using general theorems about ...
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1answer
47 views

Interpret convolution diagram

How do I interpret this "do convolutions" diagram? 1) How are the results computed? 2) When looking at this part: "x[n-k]" Do you interpret convolutions as delays or time reversals? $ y[n]= ...
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1answer
516 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
115 views

Why is this convolution true?

I am a little puzzled by how the following summation has been written as a convolution, with one of the inputs reversed in time. Consider the following sum on the LHS, and the convolution on the RHS. ...
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0answers
39 views

Question on change of variables during convolution/correlation

I am trying to understand how the following two statements are equivalent: $$ \sum_{l=-\infty}^{\infty} h^*[l] \ R_{xx}[m+l] = \sum_{i=-\infty}^{\infty} h^*[i-m] \ R_{xx}[i] $$ I get that we made ...
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0answers
72 views

Need a fast algorithm of adaptive convolution

Good morrow, gentlemen! I have to apply some kind of adaptive filter to my function $f(x).$ I present each point of my signal as a Gaussian, whose bandwidth depends on its location (not the point of ...
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0answers
55 views

Adaptive convolution

I have some 1D function $P_0(x)$ and a filter function $g_h(x)$. Also, i have a known function $h(x)$, that is the desired filter bandwidth in any point. So, I have to convolve my function with a ...
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0answers
46 views

Weakest Conditions for Convolution to be Differentiable

I was going through various posts about differrentiability of convolutions. What I would like to ask is: Suppose $f \in C^{1}(\mathbb{R})$. Then what conditions on the function $g$ would ensure that ...
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1answer
87 views

Fast evaluation of a variant of the convolution

Suppose $\{f_n\}$ and $\{g_n\}$ are finite sequences of complex numbers with $0\leq n \leq N-1$. The convolution $\{h_n\}$ of these two sequences is $$ h_n = \sum_{m = 0}^{N-1} f_m\; g_{n - m}\, . $$ ...
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0answers
42 views

Is there an expansion for element-wise scaled convolution?

If $x = a\cdot b$ is used to indicate $x_i = a_i\cdot b_i$, $y = a / b$ denotes $y_i = a_i / b_i$, and $a*b$ denotes convolution, then is there a simplification for this expression: $$ ...
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1answer
89 views

Analytic solution of the convolution of two discoutinous c.d.f s

I have a c.d.f of variable X with a mass point at the end point, $$F(x) = \begin{cases} 0 & x<a,\\ 1-\frac{m}{x+m-a} & a\le x < r-a,\\ 1 & x\ge r-a. \end{cases} $$ where m>0. Is it ...
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1answer
118 views

Is there a way to do this with fast convolution?

If you could please offer any advice, this puzzle is driving me mad: I've come across a problem that is trivial to compute in $\mathcal{O}(m^2)$ operations, but which very closely resembles a ...
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1answer
347 views

Finding distribution of $X^2+Y^2$ where $X,Y\sim N(0,1)$

Assume I have two random independent standard normal variables $X,Y\sim N(0,1)$, How can I find the distribution of $Z=X^2+Y^2$? I thought integrating the convolution, i.e ...
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0answers
195 views

Proof of a method to find the points of maximum slope

According to method described in a paper [1] if we want to find points of maximum slope in a signal $f(t)$, then one has to do following Convolve $f(t)$ with $g(t)$ where $g(t)=-cos(\omega ...
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1answer
54 views

Analogous to convolution

A convolution of two functions $f$ and $g$ is defined as $$[f*g] = \int_{-\infty}^{\infty} f(\tau) g(t-\tau) d\tau.$$ I am interested on an analogous transformation of the form $$[f\star g] = ...
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2answers
65 views

Which Method of Convolution (If Any) Is Most Appropriate Here?

I need to convolve (or otherwise get the impulse response h(t) of) the input signal $x(t) = 2u(t)$ and $y(t) = cos(4t) + 2e^{(t-1)}$. I have tried the Fourier Transform and the Laplace Transform, but ...
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2answers
33 views

What Does The Following System Do?

I have a system $y(t) = 0.5 \int^\infty _{-\infty} x(T)[d(t-T) - d(t+T) dT] $ Where d(x) is the Dirac Delta function (couldn't find the LaTEX representation - a little rusty there, so an edit to ...
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1answer
43 views

Question About Optimal Approach to Find Impulse Response?

I have the following O.D.E. relating a system's input and output, where x(t) is the input and y(t) is the output: $2x(t) = {d^2\over dt^2} [y(t)] + 6{d\over dt} [y(t)] + 8y(t)$ It is also known that ...
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1answer
106 views

Deconvolution of a convolution product with $Ax\ /\ (x^2+l^2)^{3/2}$

This is not a homework, and I have no idea whether it could be one. It is only a request for help, as I do not have any experience using Fourier transform. The origin of the problem is from physics. ...
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1answer
80 views

Understanding a diagram on Convolution

Could someone please explain what is happening at the "f*g" row and below? The image is located here as linked from the Wikipedia page. I want to teach myself about Fourier Transforms / Series, and ...