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
40 views

How to find solution of the integral equation?

$$y(t) + t \int_0^t y(v)dv = 1 + \int_0^t vy(v)dv$$ I found the answer to be $y(t) = \cos{t}$. I have no idea how they go this answer. I would appreciate any suggestions how to solve this.
1
vote
1answer
41 views

Does convolution preserve strict log-concavity?

Suppose $f, g$ are strictly log-concave functions. Then the convolution $f * g$ will also be log-concave. However, will it also be strictly log-concave? Thanks!
0
votes
1answer
82 views

Convolution of discrete uniform distributions

For two independent, discrete, uniformly distributed random variables $X$ and $Y$, I wish to obtain the distribution of the sum $Z=X+Y$. I have the densities: $$f_X(x)=\left\{\begin{matrix} ...
0
votes
0answers
43 views

Product of two random variables - Resulting Distribution and Correlation?

Let $X \sim \mathcal{N}(0,1)$ and let $Z$ be a random variable independent of $X$ such that \begin{align*} P(Z=z) = \begin{cases}\frac{1}{2} & z=-1\\ \frac{1}{2} & z = 1\\ 0 & ...
1
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0answers
50 views

What is a convolution kernel?

What is a convolution kernel? (in measure theory, probability theory) In which book can I read about kernels on measurable spaces and convolution kernels? Thank you!
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0answers
23 views

Convolution integral

I got unfortunally stuck by performing a (quite simple?) convolution integral. Given are those functions: $$f_1(t) = k_1\cdot e^{b_1\cdot t}$$ and $$f_2(t) = k_2\cdot t$$ where $k_1, k_2$ and $b_1$ ...
2
votes
1answer
43 views

Calculating PDF of $Z$ from $X,Y$ when $Z=X+Y$, given the PDFs of $X$ and $Y$

A Student is taking an exam which has two parts, X and Y, with each part given a score from 200 to 800. The students probability distribution for each part is given by $$ f_X(x)= \begin{cases} ...
1
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0answers
28 views

Inverse Laplace transformation correct?

I'm actually on the way to solve a little bit complicated differential-equation. Therefore I used the Laplace transformation. I've already solved it but I am actually not sure, whether my solution ...
0
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1answer
49 views

Exponential decay convolved with a gaussian

I need to convolve an exponential decay (defined as the exponential $Ae^{-\lambda t}$ from $0$ to $+\infty$) with a Gaussian of known standard deviation $\sigma$, in other words I need to compute the ...
0
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0answers
18 views

Is it true that for every signed probability distribution `f`, there are positive distributions `g` and `h` st. `fg=h`?

While reading the article Half of a Coin: Negative Probabilities, I came across the following theorem: For every generalized g.f. f (of a signed probability distribution) there exist two ...
0
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1answer
275 views

Convolution Theorem involving a constant.

Should one have f(x) and g(x), and wants $f(x) \ast g(x) $ from what i understand this can be quite difficult, however should $f(x)=\alpha$, a constant, what is $f(x) \ast g(x) $?
0
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0answers
32 views

Laplace transform involving two functions of t

I need to solve the following $$ \int_0^{\infty} f(t)g(t)e^{rt} dt$$ Where $$g(t)=t^n$$ Letting r=-s we have the definition of $$ \mathcal{L} [ f(t)g(t) ]$$ and am unsure how to continue.
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0answers
42 views

Discretization of a convolution integral for constrained optimization problem

I'm working on a constrained optimization problem in which an unknown forcing function, $u(\eta)$, is in the integrand of a convolution integral. To find an optimal shape for $u(\eta)$, the integral ...
0
votes
2answers
84 views

Finding convolution of exponential distribution

So this is a probability question, and I am asked to find $P(0.6 < Y <= 2.2)$ where $Y = X_1 + X_2$ X1~U(0,1) and X2~exp(2). Our professor worked it out, but I do not understand his ...
0
votes
1answer
37 views

Solving convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside step function

How does one solve convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside (unit) step function? I tried using Fourier transform of both functions to ...
1
vote
1answer
42 views

Solve initial value problem with unspecified right-hand side $g(t)$

Consider the initial value problem $$y''-6y'+9y=g(t),\quad y(0)=1,\ y'(0)=3.$$ 1) Use the Convolution Theorem to find the solution to the IVP for any piecewise continuous function $g(t)$ that is of ...
0
votes
1answer
77 views

Young's inequality for convolutions

Let's assume that the convolution $f * g$ is continuous with $\lim_{|x| \to \infty}(f*g)(x) = 0$ and that $f, g \in L^2$. Then the following inequality holds $$ \| f * g \|_{\infty} \leq \| f \|_2 ...
1
vote
0answers
22 views

Extend bivariate to multivariate convolution formula?

In reference to this post, the pdf for dependent random variables $X_1+X_2$ is given by: $$f_{X_1+X_2}(z) = \int_{-\infty}^{\infty} f_{X_1,X_2}(x,z-x) \mathrm dx$$ How does this formula extend to ...
2
votes
0answers
21 views

Convolution of two bernoulli distributions

Find the probability mass function of the sum of X ∼ Bernoulli(p) and an independent Y ∼ Bernoulli(q) variable. I started by letting Z=X+Y So $$P_z(Z)= \sum_{i=0}^{1}f_x(x) f_y(z-x) $$ $$ ...
0
votes
0answers
15 views

Approximation technique of common probability distributions that can be convoluted and integrated fast

I am looking for a approximation technique of functions with two conditions: It is possible to perform a fast approximate convolution with the approximate functions. It is possible to numerically ...
1
vote
1answer
39 views

Convolution with heaviside function, argument of the heaviside carry on to the dirac function?

So I have this equation to demonstrate: $$ x(t)*u(t)= \int_{-\infty}^t x(\tau)d\tau $$ , where $u(t)=\int_{-\infty}^t \delta(\tau)d\tau$ I opened the convolution as $ \int_{-\infty}^\infty ...
2
votes
1answer
177 views

Convolution of sine and unit step function

I started studying signal convolution recently and the first sample problem I got is to find convolution of sine and unit step function (Heaviside function). Here is what I have right now. ...
0
votes
1answer
51 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 ...
1
vote
1answer
60 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
40 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 $||f||_1>0$ ...
1
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0answers
46 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: ...
0
votes
0answers
28 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 ...
1
vote
1answer
27 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: ...
2
votes
1answer
54 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 ...
0
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0answers
46 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 ...
1
vote
3answers
256 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 ...
0
votes
1answer
25 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 ...
1
vote
1answer
32 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?
1
vote
2answers
38 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 ...
5
votes
1answer
181 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 ...
1
vote
1answer
30 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)) = ...
0
votes
1answer
215 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 ...
1
vote
2answers
57 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} ...
0
votes
0answers
38 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 ...
3
votes
3answers
309 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 $A+B$ ...
2
votes
1answer
41 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: ...
0
votes
0answers
25 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 ...
1
vote
2answers
31 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 ...
1
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0answers
43 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 ...
0
votes
0answers
40 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
33 views

Help proving this Convolution Integral:

Can someone give me the steps for proving the following integral: ...
3
votes
3answers
129 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
votes
1answer
74 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 ...
0
votes
1answer
26 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 ...
0
votes
0answers
27 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 ...