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.

learn more… | top users | synonyms

0
votes
1answer
30 views

Convolution of a signal with the butterworth filter.

Let $f(t)$ be a signal that is $0$ when $t<0$ or $t>1$. Show that, for the Butterworth filter, one has $$Ae^{-\alpha t}\int_{0}^{\min\{t,1\}}e^{\alpha\tau}f(\tau)d\tau$$ My attempt: \begin{...
0
votes
0answers
6 views

Gaussian Smoothing Kernel Sigma from x-axis

I already asked a question and got great swift answers: Gaussian smoothing kernel with different sigma values However I have come across another problem. As I understand it, Gaussian kernels used ...
0
votes
1answer
17 views

Proof involving convolutions

Suppose that $f(t)=0$ for all $|t|\geq a>0$ and that $g(t)=0$ for all $|t|\geq b>0$. Show that $f*g(t)=0$ for all $|t|\geq a+b$. Without loss of generality, assume $b\leq a$. Then $$f*g(t)=\...
1
vote
1answer
24 views

Convolution of Measures, several definitions

I'm familiar with this definition of convolution : $$f*g(x)=\int f(x-y)g(y) dy$$ But anyone help me see the link between this one and the two definitions hereafter? : Definition 1 : $$f*g(A)=\...
1
vote
2answers
28 views

Convolution of two piecewise functions

$$\phi(t)=\begin{cases}1&t\in[0,1)\\0&\text{otherwise}\end{cases}$$ and $$\psi(t)=\begin{cases}1&t\in[0,1/2)\\-1&t\in[1/2,1)\\0&\text{otherwise}\end{cases}$$ I know that $\mathcal{...
0
votes
1answer
26 views

Convolution of $h(t) = u(t+2) - u(t-2)$ and $ f(t) = tu(t) - tu(t-2)$.

Could someone please explain how I perform the convolution? My professor only taught me how to use the table, but I have been teaching myself from the book. I know that convolution is associative and ...
0
votes
0answers
40 views

Calculating a function from its auto-correlation

How do I calculate a function if I know its auto-correlation? To be more specific, I have a function of one variable, let's call it $g(x)$, and I know it's the cross-correlation of a function $f(x)$ ...
0
votes
0answers
20 views

Classes of binary operations between functions

Let $f,g : D\to \mathbb{R}$ be two functions defined from a domain $D\in \mathbb{R}$ to $\mathbb{R}$. I am looking for classes of binary operations $\circ$ between $f$ and $g$ that produce an $h:=f\...
1
vote
0answers
37 views

Proof verification for the Inverse Fourier transform of a Convolution

Below is a word by word copy from my textbook of a certain derivation for the Fourier transform of a Convolution: Let $g_1(\alpha)$ and $g_2(\alpha)$ be the ...
0
votes
0answers
30 views

Proving that the Laplace Transform is an isomorphism with convolution

My question is primarily more about the convolution integral/theorem than the proof in question, but I wanted to give some idea of why I'm asking. The Laplace transform of the convolution $$(f\star g)...
0
votes
0answers
18 views

finding image and kernel from convolution result

given a one dimension image/(signal) I and three kernels K,L,J. given the results of the convolution of I with each kernel, for example \begin{matrix}I*K =[ 1/3&2/3&1&1&1&1&1&...
0
votes
0answers
20 views

Can any integral of this form be written as a sum of convolutions?

Does the second equality always hold? $$ I(x) \equiv \int dy F(y,x-y) = \sum_{i=1}^{N}\int dy f_i(y)g_i(x-y) $$ Motivation: The first integral is not obviously a convolution that I could calculate ...
0
votes
2answers
59 views

Find the integral $\int\limits_{-\infty}^{+\infty}{\frac{dt}{(1+(x-t)^2)(1+t^2)}}$

My problem is to calculate auto-convolution of $f(x) = \frac{1}{1+x^2}$. I know that $$(f \star f)(x) = \int\limits_{-\infty}^{+\infty}{\frac{dt}{(1+(x-t)^2)(1+t^2)}} = \frac{2 \pi}{x^2+4},$$ but I ...
0
votes
0answers
44 views

Solve a function consist of two functions in a sigma summation

I'm still new in here. I have been struggling to solve an equation. The equation is like: f(T)= $\sum_{i=0}^t [g(T-i)*d(i)]$ It is an equation describing the relationship between f(T), g(T) and d(t)....
1
vote
1answer
31 views

Convolution form of the renewal density

In Asmussen's Applied Probability and Queues, Proposition 2.7 makes a claim about the form of the renewal density in terms of the density of the interarrival distribution $F$, namely: The renewal ...
2
votes
0answers
22 views

Estimating Self-Convolution of Surface Measure on Sphere

Let $n\geq 2$, and let $\sigma$ denote the standard surface measure on the $n$-sphere $S^{n-1}$, normalized to have total measure $1$. According to the exercise at the end of Terence Tao's blog post ...
1
vote
1answer
17 views

Interchanging integral and derivative operations in the context of Duhamel's formula

I'll give you the whole context: In solving the heat equation $u_t = ku_xx$ with bounds $u(x,0)=0, u(0,t)=0, u(l,t)=f(t)$, let $v(x,t)$ be the solution for the special case $f(t)=1$. Use the Laplace ...
0
votes
0answers
32 views

Help understanding the math of a basic convolution

We are learning about convolutions and I am totally totally lost. I think I understand the idea behind them, basically you're summing the output of the two signals at every point in time. Its the math ...
2
votes
1answer
43 views

Convolution of convex function and Gaussian is convex

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function, i. e., for all $x_1, x_2 \in \mathbb{R}$ and $t \in [0, 1]$, $$ \qquad f(tx_1+(1-t)x_2)\leq t f(x_1)+(1-t)f(x_2).$$ I want to prove that the ...
1
vote
0answers
40 views

How to calculate the volume of intersection of two cylinders which have radius $r$ and $R$ $(r\le R)?$

The cylinders have their axis meet at right angle. I try to use integral to derive the equation but I have a hard time visualizing the edge of intersection. Could someone help?
0
votes
0answers
34 views

Additive combinatorics: Switching $\mathbb{Z}/N\mathbb{Z}$ with $\mathbb{Z}$

For a positive integer $N$, denote $\mathbb{Z}_N = \mathbb{Z}/N\mathbb{Z}$. Now let $F$ be a field and let $f_1, \ldots, f_s : \mathbb{Z}_N \to F$. Can we find functions $\tilde{f_1}, \ldots, \tilde{...
0
votes
0answers
45 views

Transform with tensor product

I'm new to Laplace and Fourier transforms when convolution is involved, and I've never seen an example involving a tensor product. I'd like to see how the Fourier transforms of the following would ...
0
votes
1answer
27 views

Problem with convolution, insecure

$$f(t)= t^2\cdot u(t),\quad g(t)=t^4\cdot u(t)$$ I know that I need to use convolution theorem to solve this problem, but I really don't know what to do with step functions. Do I need to include ...
1
vote
1answer
34 views

Convolution using Integration

Using integration, how would I solve f(t) convolve g(t) given that $$f(t)=u(t)-u(t-5)$$ and $$g(t)=2[u(t)-u(t-1)]$$ I know it should be $$\int_0^6 f(\tau) \ast g(t-\tau)~ d\tau = \int_0^6(u(\...
0
votes
0answers
25 views

Square root of Bezier curve via deconvolution

I calculate the product of two Bezier curves via convolution as described in Sanchez-Reyes 2003. I would also like to calculate the square root of a Bezier curve (I have not seen this published ...
0
votes
0answers
27 views

Approximation of Sobolev function with convolution

As a homework exercise in Sobolev spaces course we have the following: Suppose $u \in W^{1,p}(\mathbb{R}^n_+)$ and $u_\varepsilon(x)=u(x+2\varepsilon e_n)$, where $e_n=(0,\dots, 0,1)$ and $\mathbb{R}^...
1
vote
1answer
71 views

Convolution $f * g$

Assume that $f$ is in $L^1 (\mathbb{R})$ and $g(x)= e^{2iπx}$. Compute $f * g$ I just need a hint and not the entire answer. How can I compute the convolution when I don't know what $f$ is?
1
vote
1answer
48 views

Graphical Convolution

For the first part of the above problem, I copied an example from my book and I got the answer to be $$t(t-1)+t(t-2)=t^2-3t$$ considering that the integral is the sum of the area of the rectangles ...
0
votes
1answer
27 views

Why is the inf-convolution of lower semicontinuous functions continuous?

I'm confusing now about the continuity of inf-convolution. I understand that the inf-convolution of lower semicontinuous functions is semiconcave and so it's locally Lipschitz continuous (in ...
0
votes
0answers
26 views

convolution properties of distributions

Let $f,g,h \in D'(R^n)$. How we define the convolution of these functions? I'm trying to show some properties of convolutions such as $\delta\ast f=f$ $(f\ast g)' = f'\ast g=f\ast g'$ $(f\ast g) \...
0
votes
0answers
19 views

Meaning of standard integral convolution

In this paper, in the proof of Lemma $13$, there is this sentence: Now, we find a $1$ Lipschitz $\bar{g} \in C^1(\mathbb{R}^n)$ with $\| f - \bar{g}\|_{|\infty} < \epsilon / 2K$, using the ...
1
vote
2answers
38 views

Fourier series and convolution

Let $f$ and $g$ be $2\pi$-periodic, piece-wise smooth functions having Fourier series $f(x)=\sum_n\alpha_ne^{inx}$ and $g(x)=\sum_n\beta_ne^{inx}$, and define the convolution of $f$ and $g$ to be $f\...
0
votes
0answers
26 views

Sum of two random variables - uniform distributions [duplicate]

I have two continuous uniform random variables I need to add. I read that to get the sum of two pdfs you convolve them. I'm getting a bit confused on the limits of integration though. If both RVs are ...
1
vote
1answer
24 views

Convolution for two random variables

In the textbook i'm currently reading it is said that for two independent random variables $X$ and $Y$ density function of variable $Z=X+Y$ can be found from the equation: $$ g(z) = \int_{-\infty}^\...
1
vote
0answers
37 views

Convolution of two $L^\infty$ function with compact support.

I have the following lemma without proof: Lemma. Let $f, g \in L^\infty(\mathbb R^n)$ with compact supports. Then $f \ast g \in C(\mathbb R^n)$. Is this even true? I get this: $$ \begin{align*} ...
3
votes
2answers
67 views

Convolution Integral to Evaluate Fourier Transform

According to Mathematica with Fourier transform convention $$\widehat{f}(\xi)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}f(x)e^{i\pi x}dx$$ The Fourier transform of the function $f(x):=|x|^{-1/2}e^{-|x|}$ ...
2
votes
0answers
39 views

Conditions under which an Convolution operator is normal.

I have a possibly complex valued convolution operator given by $ \int_{\mathbb{R}}K(x-y)f(y)dy$ I know that the operator is self-adjoint if $K(x)=\overline{K(-x)}$ holds. But are there softer ...
1
vote
0answers
18 views

Convolution of basis functions is a member of the same set of basis functions?

Suppose $\left\lbrace \Phi_i\right\rbrace_{i=0}^{\infty}$ is a complete basis of $\ell_1$. So if $M\in\ell_1$ we can write it as a linear combination of the basis functions $M=\sum_{i=0}^\infty m_i\...
0
votes
0answers
19 views

Convergence of convolution with an even summability kernel

Suppose that $f(\theta) : [0, 2\pi] \rightarrow \mathbb{R}$ is a monotone increasing real valued function and $\{k_n\}$ is an even summability kernel. I want to show that $f \star k_n$ converges ...
1
vote
0answers
18 views

Applying the convolution theorem in the presence of a twiddle factor

The convolution theorem says that a 2-d cyclic convolution like $C = U \ast V$ can be evaluated more quickly than doing the raw sum $C_{i,j} = \sum_{a,b}^n U_{a,b} V_{i-a,j-b}$ for each point (assume ...
1
vote
0answers
51 views

Convolution of measures, why is the notation like this?

In both my book, and on Wikipedia they define convulution of two measures like this: $(\mu_1*\mu_2)(B)=\int_{\mathbb{R}^d}\mathcal{X}_B(x+y)d\mu_1(x)d\mu_2(y)$ It doesn't seem like a typo, but ...
0
votes
0answers
15 views

Is there an alternate name for the symplectic convolution?

Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution $$[F\star G](x,p) = \int \!dy\,dk\, ...
0
votes
0answers
18 views

Is the convolution of two waves again a wave?

Call a function $f:\mathbb{R}^{1+n}\to \mathbb{R}$ a wave, if it satisfies the wave equation $$\frac{1}{c^2} \frac{\partial^2 f}{dt^2} = \sum_{i=1}^n \frac{\partial^2 f}{dx_i^2} = div(grad\,f)$$ ...
2
votes
2answers
45 views

Why does convolution of delta function commute (test distribution perspective)?

If I understand correctly, for test functions $f$ we define $$ \langle\delta, f\rangle = f(0)$$ and convolution is defined as $$ \langle g * T, f\rangle = \langle T, g^- * f\rangle,$$ where $f$ ...
1
vote
0answers
29 views

Limit of sequence with discrete convolution $a_n = 1 - b * a$

I have a sequence \begin{equation} $a_n = 1 - \sum\limits_{i=1}^{n-1}b_na_{n-i}$ where $b_n\leq{c}^n$, and $c<1$ Based on multiple simulations with varying parameters, I think that the sequence ...
0
votes
0answers
15 views

Convolution of cosine and shifted unit step

I'm trying to understand the basics of a convolution and have troubles with the following task: $$ x_1 = \cos(2 \pi t ) \cdot u(t) $$ $$ x_2 = u(t-0,5)$$ The task is to compute the convolution $$x_1 ...
1
vote
1answer
19 views

What is the purpose of the requirement on mollification radius?

On page 66 of "Sobolev Spaces (Adams ed2)" in the proof of Lemma 3.16 (Mollification in $W^{m,p}(\Omega)$), it is mentioned that $\varepsilon < {\rm dist}(\Omega', \partial\Omega)$. However, I ...
1
vote
0answers
24 views

analyze convolution in spatial domain against multiplication in frequency domain

Lets say I have a image of $NxN$ and a separable filter that I want to apply on it. there are 2 ways to do that: 1. By convolution in spatial domain. 2. By multiplication in frequency domain. I need ...
0
votes
0answers
27 views

Convolution of delta-ish functions

I would like to compute the convolution of a function with itself, where the function is $f(x) = \frac{\delta(x)}{x}$. When there is a shift in the delta function it is easy to compute, but this one ...
0
votes
0answers
36 views

The Deconvolution Integral

The standard 1D continuous convolution integral is defined as: $$y(t) = h(t)*x(t) = \int^{+\infty}_{-\infty}h(\tau)\cdot x(t-\tau)\ d\tau$$ Using fourier transform, $$Y(j\omega) = X(j\omega)\cdot H(...