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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2
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1answer
1k views

The distribution of the sum of two independent exponential distributions

I am trying to calculate the distribution of the sum of two independent log-uniform distributions but something doesn't add up. Suppose $a \sim \mathrm{uni}(0,1)$ and $b \sim \mathrm{uni}(0,1)$. ...
1
vote
1answer
503 views

a matrix inverse laplace transform problem

Let $\mathcal {L}^{-1}[\cdot]$ be an inverse Laplace transform. Let $A$ be a square matrix, and $I$ an identity matrix. Based on the fact that $\mathcal {L} ^{-1} [{(sI-A)}^{-1}] = e ^{tA}$, how can ...
0
votes
1answer
709 views

Integration Limits for calculating Convolution

to calculate the convolution.. I know how to divide it into intervals. but i don't know how to set the integration limits. i 've been working on it for 10 hours and become wrong results. can anyone ...
1
vote
2answers
738 views

Is convolution in spherical harmonics equivalent to multiplication in the spatial domain?

Spherical harmonic convolution is defined as: $$ ( k \star f )^l_m = \sqrt{ \frac{ 4 \pi }{2l+1} } h^l_m f^l_m $$ I have a function with RGB values for every $(\theta,\phi)$ in the spatial domain. ...
2
votes
2answers
163 views

Newtonian potential of a rotationally-invariant function

Lately I read up in the wikipedia article about the Newtonian potential, that for any compactly supported continuous function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ that is rotationally invariant ...
2
votes
1answer
119 views

The digit base and the NTT convolution

Suppose I'm using a number theoretic transform (NTT) in an integer field $GF(p)$. I assume that $2n$-th root of unity exists for such a $p$, and I want to compute a convolution of two $n$-length ...
2
votes
1answer
109 views

Property of Convolutions

Given $f(x), g(x): \mathbb{R} \to \mathbb{R}$ we can form the convolution $f * g$. Define $h(x) = f(cx)$ for some $c>0$. Can we express the convolution $h*g$ in terms of $f * g$ ? Thanks for all ...
3
votes
1answer
97 views

Computation of a Particular Convolution

Let $\xi_{1}, \xi_{2}, \xi_{3}$ be i.i.d. $N(0,1)$. I'm attempting to compute the density of $\max \{\xi_{1}, \xi_{2}\} + \xi_{3}$. I know the density of $\max \{\xi_{1}, \xi_{2}\} $ is $2\Phi(y) ...
3
votes
3answers
1k views

Laplacian 2D kernel - is it separable?

I'm wondering if the 2D laplacian kernel 0 1 0 1 -4 1 0 1 0 is also a separable kernel. How can I find that out?
0
votes
2answers
179 views

How to solve this 2-D deconvolution $g*f=δ$?

$g*f=δ$, where $*$ refers to convolution, $δ$ is impulse, $f$ and $g$ is 2-D matrix, $f$ is given and sum of all the elements in $f$ equals $0$, $g$ is unknown. i want to find $g$. i would appreciate ...
6
votes
2answers
678 views

Convolution of measures

We say that a family of measures $\mu_{t}\to \mu$ weakly if for any $g\in C_{0}$, $\int g d\mu_{t} \to \int g d\mu$. Show that if $\mu_{t}\to \mu$ weakly, then $\nu*\mu_{t}\to \nu*\mu$ weakly, where ...
1
vote
3answers
348 views

convolution of signals

I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals: $$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)} $$ where $u(t)$ is ...
1
vote
0answers
151 views

Cross Correlation

The cross-correlation function is defined as follows if $\bar{f}$ is the complex conjugate of $f$ and we assume that $f$ is real, such that $\bar{f} = f$. $$ \begin{align} f \star g &= ...
2
votes
2answers
630 views

Computing a convolution using FFT

I have two sequences of the same length, $(x_i), i=1, 2, \ldots, N$ and $(y_i), i=1, 2, \ldots, N$ and a function $K(t) = -t \times \exp(-t^2 / 2)/ \sqrt{2 \pi}$. I need to compute the following ...
4
votes
1answer
249 views

Proof that a convolution with $g(x)=\frac{1}{\pi} \frac{r}{r^2+x^2}$ is smooth

It is known that if $g_n: \mathbb{R} \rightarrow \mathbb{R}$, $n=1,2,...$, is in $C_c^{\infty}(\mathbb{R})$, $ \int_\mathbb{R} g_n(x)dx=1$, $supp(g_n) \subset (-r_n,r_n)$,where $0<r_n \rightarrow ...
0
votes
1answer
84 views

Coefficient of $z^{n-1}$ term in a generating function

This is a problem from Concrete Mathematics (Eq. 9.57 and 9.58 in 1995 edition). $G(z) = \sum_{k}g_k z^k = e^{\sum_{k} \frac{z^k}{k^2}}$ is a generating function of sequence $g_k$. By taking the ...
3
votes
1answer
143 views

Estimating number drawn from one distribution based on sum of that number and number drawn from another distribution

I have been working on this for several days and have been unable to come up with an answer. The problem is very simple to state, but it seems difficult to solve. A computer draws a number $x$ at ...
5
votes
1answer
337 views

On the closedness of $L^2$ under convolution

It is a direct consequence of Fubini's theorem that if $f,g \in L^1(\mathbb{R})$, then the convolution $f *g$ is well defined almost everywhere and $f*g \in L^1(\mathbb{R})$. Thus, $L^1(\mathbb{R})$ ...
1
vote
0answers
128 views

n-th self discrete convolution

Lets define discrete $ f_N(i) = 1,\space i = 1...N $ I need to find $ G_N^m = \underbrace {f_N * f_N * ... * f_N}_{m} $ For example $G_6^3$ have value (1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1) , ...
0
votes
1answer
141 views

Verify this distribution convolution: $E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$

In our class notes we are asked to verify the following equality: $$E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$$ where ...
1
vote
1answer
369 views

Define uniform B-spline basis functions via continuous convolution

I'm looking into different methods for defining uniform B-spline basis functions. One of those methods is using convolution. In the course notes of Dennis Zorin ("Bézier Curves and B-splines, ...
2
votes
1answer
426 views

Commutativity of Convolution in higher dimensions

I have a basic question about how to show that convolution in dimension $n$ is commutative - or maybe it is rather a question about change of variables .. So on $\mathbb{R}$ I know how to show ...
9
votes
1answer
455 views

The error term in Taylor series and convolution.

I've been wondering a lot why is the remainder of the Taylor expansion of a function, $R_n(x)$, expressed (in one of the many forms) as something very similar to aconvolution. Precisely: $$R_n(x) = ...
1
vote
1answer
417 views

Chirp Transform and Convolution

I was reading about the discrete fourier transform from the CLR algorithms book and I came upon an exercise whose hint confuses me. The exercise reads as follows: The chirp transform of a vector ...
3
votes
2answers
120 views

$\frac{1}{a^2+x^2}\ast \frac{1}{a^2+x^2}=\frac{2\pi /a}{4a^2+x^2} (a>0)$

I.e. $\displaystyle\int_{\mathbb{R}} \frac{1}{a^2+(x-y)^2}\cdot\frac{1}{a^2+y^2} dy=\frac{2\pi /a}{4a^2+x^2}$. If anyone feels like giving me a brief hint about how to get started on this I'd be ...
7
votes
1answer
869 views

Is this class of periodic functions closed under the (circular) convolution operation ? Help in proving.

I am studying the properties of a particular class of functions, and I'd appreciate some help in proving a property of that class. I started with a class of functions and made some modifications to ...
0
votes
3answers
252 views

Derivation of pmf from convolution

Suppose that a discrete random variable (with finite support) $Y$ is given by $Y = X_1 - X_2$, where $X_1$ and $X_2$ are both discrete random variables with finite support and with the same ...
1
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0answers
200 views

Convolution of two functions

for example I have somethink like that, $$ \begin{align*} f(x) &= \begin{cases} \frac{1}{3}x - \frac{2}{3} &\text{where }2 < x \leq 4, \\ \frac{-2}{3}x + \frac{10}{3} ...
2
votes
1answer
2k views

convolution of gaussian and sinc function

I have some data that I know is the convolution of a sinc function (fourier transform artifact) and a gaussian (from the underlying model). I would like to fit this data to a functional form of the ...
4
votes
3answers
317 views

Alternating sign Vandermonde convolution

The well-known Vandermonde convolution gives us the closed form $\sum_{k=0}^n {r\choose k}{s\choose n-k} = {r+s \choose n}$. For the case $r=s$, it is also known that $\sum_{k=0}^n (-1)^k {r \choose ...
2
votes
1answer
132 views

Integrability of the function $f_1(y)f_2(x-y)$ for almost all $x$. (convolution)

What I'm trying to prove is that for $f_1,f_2 \in \mathcal{L}_1$ the function $y \mapsto f_1(y)f_2(x-y)$ is integrable for almost all $x$, or: $$ F(x) = \int f_1(y)f_2(x-y)\,dy < \infty \text{ ...
3
votes
0answers
391 views

A special case of Young's inequality for convolutions

The problem: Suppose $f,g\in L^1(\mathbb{R})$. Let $x\in \mathbb{R}$ and $\phi_x(y) = f(y)g(x-y)$. Show that for almost all $x$, $\phi_x$ is integrable. For such $x$ let $\psi(x) = ...
8
votes
0answers
204 views

Is there a closed form solution of $f(x)^2+(g*f)(x)+h(x)=0$ for $f(x)$?

When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means ...
3
votes
1answer
262 views

Applications of Young's convolution inequality

Recall that the convolution of two functions is given by $$f*g(y)=\int f(x)g(y-x)dx.$$ The well known inequality known as Young's inequality, say that $$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$ provided ...
4
votes
1answer
198 views

Conditions for the Convolution $f \ast g$ to be Continuous at a Point

Let $f$ and $g$ be functions on $\mathbb{R}^n$. Let $x_0$ be a given point in the unit ball $B(0,1)$. I am looking for sufficient conditions for the convolution $$ (f \ast g)(x) = \int_{B(0,1)} ...
1
vote
0answers
84 views

Solution for this Convolution

We have $f(z)=z+ \sum_{n=2}^{\infty} a_{n}z^{n}$ where $a_{n}$ is a constant and $g(z)=z$, $(f*g)(z)$ is equal to what? i still wondering to confirm that $(f*g)(z)=z$.
2
votes
1answer
592 views

Young's inequality for discrete convolution

Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have $$\|f\star g\|_r\le\|f\|_p\|g\|_q$$ for $p$, $q$, $r$ satisfying ...
0
votes
3answers
410 views

LTI systems in convolution representation

LTI systems in state space representation are systems of the form: \begin{eqnarray} \dot{\mathbf{x}}(t)=\mathbf{Ax}(t)+\mathbf{Bu}(t) \end{eqnarray} \begin{eqnarray} ...
2
votes
1answer
189 views

Maximal ideals in a circular discrete convolution algebra

Let $G$ = $\mathbb{Z_n}$ and let $A$ = $\ell^1(G)$ with convolution, over $\mathbb{C}$. Let $A_{\mathbb{R}}$ denote the subring of real valued functions in $A$, so $A_{\mathbb{R}}$ is an algebra over ...
-3
votes
1answer
898 views

Is this formula for product of integrals correct?

$$\int_0^x f(x)dx \int_0^xg(x)dx=\int_0^xf(-x)*g(-x)dx+\frac12 \int_0^xf(2x)*g(2x)dx $$ Where * means convolution.
2
votes
2answers
286 views

Example of Convex Function

Knowning that $f(z)=z+a_2z^2+a_3z^3+...$ is a convex function, is it the derivative of f(z) is also a convex function?
2
votes
2answers
159 views

Z transform of a complex convolution

I found this paper on Hilbert Transform, which is a very nice read. I've studied signal processing, but from a more practical than mathematical perspective. Can someone explain to me how we arrive at ...
1
vote
2answers
2k views

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 ...
2
votes
0answers
225 views

FFT signal post processing

This is more a "post a suggestion" topic rather than a question. And thank you if you are willing to read this whole. I've been studing the code in the Nvidia Cuda SDK regarding how to operate a ...
0
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1answer
1k views

Convolution & DFTs: How much zero padding is necessary to avoid circular convolution?

When performing discrete [spatial] convolutions in the frequency domain, how much zero-padding is necessary to avoid the effects of circular convolution? I have a book that almost certainly answers ...
3
votes
1answer
98 views

Convolution on noncommutative group algebras

If $G$ is a non-Abelian locally compact group, and $f$ is in $L^1{(G)}$ and $u$ is in $L^{\infty}(G)$, and $f\ast u=0$ can it be concluded that $u\ast f=0$?
2
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1answer
1k views

LTI: How to calculate the step response of this impulse response?

i need to evaluate the convolution sum of x[n] * h[n]. x[n] is the step function u[n]. I know how the output should look like but i don't know how i can calculate it. I think the lower border is 0, ...
1
vote
1answer
1k views

How to sketch the following discrete time signal?

i need to sketch y[n] where * denotes the convolution operator and delta is the unit impulse. I know how to sketch x[n-1] and delta[n-2] but i have problems with the convolution. In my script i only ...
2
votes
3answers
2k views

Convolution of triangle function with itself

I'm trying to find the convolution $(A \ast A)(x)$, where $$A(x) = \begin{cases} 1 + x, & -1\leq x \leq 0,\\ 1 - x, & 0\leq x \leq 1,\\ 0, & \text{otherwise}, \end{cases}$$ so far ...
4
votes
1answer
343 views

“anti-gaussian” 2D convolution kernel

What is the 2D kernel, k, that when convoluted with a 2D signal, f, that is convoluted again with a gaussian 2D kernel, g, produces a result that is closest to the original signal, f'. Something like ...