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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1answer
12 views

Convolution CDF formula?

In reference to this post, the pdf of dependent random variables $A+B$ is given by: $$f_{A+B}(z) = \int_{-\infty}^{\infty} f_{A,B}(a,z-a) \mathrm da = \int_{-\infty}^{\infty} f_{A,B}(z-b,b) \mathrm ...
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1answer
22 views

How to obtain the convolution directly (not graphical) of the two functions $e^{-t}u(t)$ and $e^{-2t}u(t)$?

I'm having trouble solving this convolution integral graphically. I don't understand where I stop sliding my function $h(t-\lambda)$ since $x(t)$ doesn't have a boundary as lambda approaches infinity ...
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0answers
25 views

Equality condition for convolution's $L^p$ norm.

Suppose that $1< p< \infty$, $f\in L^1(R)$, and $g\in L^p(R)$ and that $\|f*g\|_p=\|f\|_1\|g\|_p$. Show that then either $f=0$ a.e or $g=0$ a.e I have solved for $g=0$ a.e. if $||f||_1>0$ ...
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0answers
42 views

convolve chirp with rect

I'm trying to evaluate $$g[x] = f[x] \ast f[x]$$ where * is the convolution operator and $$f[x] = RECT(\frac{x-2.5}{5}) \cdot exp (+i \pi x^2)$$ I assume the best approach to this equation is: ...
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0answers
20 views

estimate for derivative of convolution

Let $u\in L^\infty(\mathbb R\times (0,\infty))$ be a function such that $$u(x+z,t)-u(x,t)\leq c\left(1 + \frac 1 c\right)z\tag{$*$}$$ for some constant $c\in\mathbb R$ and almost all $x,z\in\mathbb ...
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1answer
17 views

what does support of convolution of functions says geometrically?

Let $f,g \in L^{1}(\mathbb R)$ we define $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy $ for all most all $x,$ and denote $\text{supp} (f)$ the support of $f.$ Fact: If $A$ is the closure of $\{x+y: ...
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1answer
16 views

convolution of non-zero functions

Let $f,g$ be two continuous functions with compact support. Show that if $f$ and $g$ are not identically $0$, then neither is $f\ast g$. This statement seems rather elementary, and I would prefer if ...
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0answers
26 views

Convolution of convex polygons and a Gaussian

I need to find the closest solutions for convolution of convex polygons/circles with a Gaussian function for computer graphics purposes. I was only able to find solutions for rectangles, like this ...
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2answers
32 views

Combining two convolution kernels

Is it possible to combine two convolution kernels (convolution in terms of image processing, so it's actually a correlation) into one, so that covnolving the image with the new kernel gives the same ...
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0answers
15 views

Convolution Theorem of a product of 3 functions of x

I am trying to evaluate the integral over a product of f(t), g(t) and h(t) using the convolution theorem. $$\int_0^\infty f(t) g(t)h(t) dt$$ So after taking the Laplace of each of f(t) g(t) & h(t) ...
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0answers
17 views

Lalplace Transform and Convolution theorem

How would one go about a problem of this nature $$\int_0^\infty f(t) g(t) dt$$ Using the convolution theorem. I have taken the Laplace transform of both f(t) and g(t) to get F(s) and G(s) however ...
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1answer
15 views

Deconvolution of two delta functions (solving $y(t) = A x(t-a) + B x(t-b)$)

I would like to calculate $x(t)$, when only $y(t)$ with $y(t) = A x(t-a) + B x(t-b)$ is known. Since this is a linear shift invariant operation (convolution), the inverse relation must be of the ...
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1answer
14 views

Using dirac delta functions to get $h(t)$ that satisfies $[u(t+1/2)-u(t-1/2)] \ast h(t) = [u(t+6)-u(t+2)]$ where $u(t)$ is unit-step function

Using dirac delta functions, how does one get $h(t)$ that satisfies $[u(t+1/2)-u(t-1/2)] \ast h(t) = [u(t+6)-u(t+2)]$ where $u(t)$ is unit(heaviside)-step function and $\ast$ is convolution?
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2answers
23 views

convolution of $\delta(t+4) \ast \delta(t-1)$?

How does one solve convolution of $\delta(t+4) \ast \delta(t-1)$ where $\delta$ is dirac delta function? In ordinary function convolution, tricks are obvious but does dirac delta function share the ...
4
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1answer
103 views

Proof of Faà di Bruno's formula using a convolution identity for Bell polynomials?

I have noticed there is an identity for Bell polynomials that can apply of Faà di Bruno's formula. This is a convolution identity that states: $$ (x \ast y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j ...
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1answer
22 views

Laplace transform convolution

$x(t) = cos(3πt)$ h(t) = $\exp(-2t)u(t)$ Find y(t) = x(t) * h(t) (ie convolution) Y(s) = X(s)H(s) and then take inverse laplace tranform of Y(s) $ L(x(t)) = \frac{s}{s^2+9π^2} $ $ L(h(t)) = ...
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1answer
11 views

Matlab: Impulse response of linear time invariable (LTI) sine-signal

I'm preparing for a lab in a Signals and Systems course in my university, 5th semester. I've found old exercise material from the class and since I know some Matlab and have dealt with LTI systems ...
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2answers
25 views

Convolution with delta function

I am merely looking for the result of the convolution of a function and a delta function. I know there is some sort of identity but I can't seem to find it. $\int_{-\infty}^{\infty} ...
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0answers
10 views

Calculating the probability mass function of the sum of two independent, non-similar, geometric random variables using convolution

Given two independent geometric random variables (where $p_1,p_2 \in [0,1]$): $$\mathbf{X_1} \sim \text{geometric}(p_1)$$ $$\mathbf{X_2} \sim \text{geometric}(p_2)$$ I want to find the probability ...
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0answers
20 views

Convolution of functions

If $f$ and $u$ are functions of $x$ defined in $(-\infty, \infty)$, then $f\ast u=\int f(x-t) u(t) dt $. But what is the result of $f \ast u_{xx}$?
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3answers
96 views

convolution of characteristic functions

Suppose $A$ and $B$ are measurable subsets of $\mathbb{R}$ of finite positive measure. Show that the convolution $\chi_A*\chi_B$ is continuous and not identically $0$. Use this to prove that $A+B$ ...
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1answer
27 views

Partial derivative of convolution

I have a convolution: $$g(x,\alpha) = \int_D \phi(t)f(x-t,\alpha)dt,$$ where $D$ is compact. I need to calculate $\frac{\partial}{\partial \alpha}g(x,\alpha)$. Under what conditions: ...
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0answers
21 views

If $\int^{s'}_{0} f(\gamma(s))ds\le g(s')$ is it true then that $\int^{s'}_{0} f_{n}(\gamma(s))ds\le g(s')$ for $f_{n}$ a convolution?

Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a $L^{2}$ function such that for any curve $\gamma:\mathbb{R}\rightarrow \mathbb{R}^{n}$ the following estimate is true $$\int^{s'}_{0} ...
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0answers
10 views

Convolution integral and inversion from Laplace space.

I am stuck with the following problem: I have in Laplace space: $\bar{f}=\frac{1}{s+b\sqrt{s}}s^{a-1/2}$ $...1$ I want to invert it. I used the Duhamell's Principle to get the ...
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2answers
24 views

Reference for “Approximation of identity” of a convolution

I need a textbook reference for the "Approximation of identity" of a convolution: http://en.wikipedia.org/wiki/Mollifier#Properties I would appreciate any pointers. The wikipedia article refers to ...
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0answers
36 views

Maximum of a convolution

I have a function $f:{\mathbb R}\rightarrow {\mathbb R}_+$ which has a unique maximum at $x=0$. $f$ can be symmetric or asymmetric. I am interested on the mollified-f function ...
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0answers
24 views

Convolution of a Dirac impulse with a periodic signal

I have to do a convolution of a periodic signal with a Dirac impulse. $\quad \quad x(t)=\sin(π\, t)(u(t)−u(t−2))$ $\quad \quad h(t)=u(t−1)−u(t−3)$ The first is a periodic signal that intersects the ...
2
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0answers
20 views

Help proving this Convolution Integral:

Can someone give me the steps for proving the following integral: ...
2
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3answers
99 views

Entropy of convolution of measures

Let $G$ be a countable, discrete group, and let $\mu_1,\mu_2$ be probability measures on the group $G$. We define the entropy of $\mu_i$ as $H(\mu_i)=\sum\limits_{g \in G}-\mu_i(g)\log(\mu_i(g))$ ...
0
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1answer
29 views

How to use matlab to a simple convolution problem (a convolution of the solution of the diffusion equation with continuous sources)?

I assume this is a simple problem, but since I don't have a strong maths background, I have trouble to implement this in matlab. Could anyone help me with it? Many thanks! The concentration of a ...
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1answer
21 views

Random Distance on Torus

Let $U=(X_U, Y_U)$ and $V=(X_V, Y_V)$ be two independent random points in $[0,1] \times [0,1]$, where each possible position is equally likely. Now I am interested in the probability that these two ...
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0answers
9 views

Spherical harmonics and convolutions on $S^3$

Thinking of Hopf fibration of $S^3$ I got these two questions: Do $S^3$ spherical harmonics have a simpler expression in the Hopf coordinates? In $S^2$ we can convolve only with zonal functions. It ...
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1answer
46 views

Calculating a limit of integrals

I am having a problem with the following exercise: Show that for every bounded borelian function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$, $\underset{n}{lim} \frac{n}{\sqrt{2\pi}} ...
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0answers
25 views

Using Convolution to find density of sum of non-independent normal densities

$X_1 \sim N(\mu_1, \sigma_1^2)$ and $X_2 \sim N(\mu_2, \sigma_2^2)$. The $X_i's$ are not independent. Let $Y = X_1$ + $X_2$. Then, $ \begin{align*} f_Y(y) &= \int_{0}^{y} ...
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0answers
11 views

Giving a bound for |f(x) \star \phi_k(x) -f(x)|

Here is the problem: Let $\phi(x) \in S$, where $S$ is the Schwartz class, such that $\displaystyle\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} \phi=1$. Also, for some $N\in\mathbb{N}$, ...
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1answer
34 views

Differentiability of the convolution $\int_0^tf(t-s)g(s)\;ds$

Given two continuously differentiable functions $f,g:[0,\infty)\to\mathbb{R}$. I want to know what we can tell about the differentiability of $$(f\ast g)(t)=\int_0^tf(t-s)g(s)\;ds$$ Especially, why ...
0
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1answer
41 views

Writing a function in terms of the rect and delta functions.

Say I have a function that is equal to 1 at two unit area squares. One is centered at $(-3,0)$ and the other at $(3,0)$. I am trying to find a formula for this function using only the rect function ...
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0answers
25 views

2D convolution: how to eyeball it?

I have a question of doing simple convolution in 2d by just "eye-balling" it without doing the actual computation. In 1D discrete time, when we have a simple input ...
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1answer
46 views

Convolution of cosine with exponential

As part of an exercise, I'm trying to find the output of a cosine wave entering a low-pass filter by using a convolution integral. The impulse response of the filter is $h(t) = ...
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1answer
24 views

Convolution integral identity proof

How can one show that $$ f * g = t^{m+n+1} \int_0^1 u^m(1-u)^n du $$ where $f(t) = t^m$, $g(t) = t^n$ and $$f*g = \int_0^t f(\tau)g(t-\tau)d\tau = \int_0^t f(t-\tau)g(\tau)d\tau $$ I tried with the ...
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0answers
14 views

How injective is the Laplace transform?

Denote the Laplace transform by $\mathcal{L}$, and assume $\mathcal{L}[f]$ and $\mathcal{L}[g]$ exist for some functions $f$ and $g$. Then we know that $\mathcal{L}[f*g]=\mathcal{L}[f]\mathcal{L}[g]$. ...
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0answers
5 views

Can I calculate correlation with convolution?

You know the correlation is the degree of similarity between two difference signals, and the convolution is used for calculate the output of a system or signal, so can I use convolution for calculate ...
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0answers
15 views

Fourier Transform Identity?

$f(x) \in \mathbb{R}$ and $g(x) \in \mathbb{R}$ $$\int\int \mathop{dx \, dy} f(x)f(y)g(x-y) = \int dk \, \left| \tilde{f}(k)\right|^2\tilde{V(k)} $$ All integrals are over all space. Is this true? ...
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1answer
22 views

Find convolution of u[n]-u[n-2] and u[n]-u[n-2]

Question: Find convolution of $u[n]-u[n-2]$ and $u[n]-u[n-2]$ I have found that $u[n]\cdot u[n]=n$, $u[n]\cdot u[n-2]=n-2$, $u[n-2]\cdot u[n-2]=n-4$ Use linear property, my answer is: ...
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0answers
7 views

Approximation of $C^1$ by $C^1_b$

Can we approximate a function which is $C^1$ with functions that are $C^1$ with bounded first derivative? Thank you in advance.
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3answers
62 views

How to find the impulse response with input and output given?

The Question: A CT signal x(t), which is non-zero only over the time interval, t = [-2,3] is applied to an LTIC system with impulse response h(t). The output y(t) is observed to be non-zero only over ...
2
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0answers
90 views

Convolution of two Gaussians or two sinc functions using direct integration

I tried to solve the following to problems from Gaskil's book Linear Systems, Fourier Transforms, and Optics. But I'm struggling to get the right results. My experience with calculating convolutions ...
4
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0answers
50 views

Finding convolution identites

Suppose I have the following definition: $$\frac{x^2/2!}{e^x-1-x}=\sum_{k=0}^{\infty}A_k\frac{x^k}{k!}$$ I want to find a convolution identity for these coefficients $A_k$, but I've never studied ...
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1answer
19 views

Is it possible to find a solution to this integral equation?

I have an integral equation of the following form: $y(t)=\lambda x(t) + x(t)\int_{-\infty}^{\infty}K(t,s)x(s)ds$ I haven't been able to find any discussion online of integral equations with the ...
0
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1answer
32 views

Mollification of $L^{\infty}$ functions

We know when $1\leq p<\infty$ , the mollification function $f^{\epsilon}=\phi_{\epsilon}*f$ for $L^{p}(R^n)$ functions converge to $f$ in $L^{p}$ norm, when $p=\infty$ it might be wrong. But who ...