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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0
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1answer
29 views

Sign with Fourier transformation, convolution, periodicity

Let $x(t)$ be the sign with Fourier transformation $$X(\omega)=\delta(\omega)+ \delta(\omega-\pi)+\delta(\omega-5)$$ and let $h(t)=u(t)-u(t-2)$. Is $x(t)$ periodic? Is the convolution of $x(t)$ ...
2
votes
4answers
69 views

Calculating value of integral of convolution using Fourier transform

Calcuate the integral $$I=\int_{-\infty}^\infty\frac{\sin a\omega\sin b\omega}{\omega\cdot \omega}d\omega.$$ First I noticed that $$\mathcal{F}(\mathbb{1}_{[-h,h]})(\omega)=\frac{\sin h ...
2
votes
0answers
21 views

Prove the periodic total variation of f = $\sum_{n=0}^{N-1} |(f*h)[n]|$

Let $\Bbb{C}^N$ be the N-dimensional Euclidean space with its inner product defined as $$ \langle f,g\rangle=\sum_{n=0}^{N-1} f[n]g^*[n],\ \forall f,g \in \Bbb{C}^N$$ where $g^*[n]$ is the complex ...
0
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0answers
13 views

How can i explain the symmetry of the function in a more linguistic manner

To understand the convolution of these functions, please read the following wikipedia page I have the following expression $a = <f'(x),f''(x),\cdots>$ and $b = <g'(x),g''(x),\cdots >$ ...
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0answers
32 views

Convolutional Codes

i've been given a coding assignment that looks like this http://i.imgur.com/7wIEoHJ.png http://i.imgur.com/FINnNZZ.png I understand the concept of Jacobi and Gauss Seidel iteration, I know where ...
1
vote
0answers
38 views

uniform convergence and improper integral(convolution)

I want to show that $$\frac{d}{dx}(f*g)=(\frac{d}{dx}f)*g$$ where $f(x)=\frac{1}{\sqrt{x}}e^{-\frac{1}{x}}$, $g(y)$ is continuous and bounded. the convolutions are improper integrals. I'm now here ...
4
votes
2answers
134 views

Solve integral (convolution) equation

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$ and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$ How to find a closed expression for ...
1
vote
1answer
116 views

Convolution of Uniform Distribution and Square of Uniform Distribution

I am trying to find the CDF of $Z=X+Y$ whereby $X$ and $Y$ are random variables. Given that the CDF of Z is: $$F_Z(z)=\int F_X\left(z-y\right)f_Y(y)dy$$ Given that $X$ is uniform distribution over ...
1
vote
1answer
44 views

Random Gaussian variable raised to arbitrary power

Given $x$ that follows a distribution $P(x)=e^{\frac{-x^2}{2\sigma^2}}$ i.e. a random Gaussian variable, can I say anything about the distribution of $x^n$ for fixed $n$? Specifically, is there ever ...
3
votes
1answer
84 views

Discrete Fourier Transform question

Let $R_{M\times N}$ be a space of size $M\times N$. Define the 2D Discrete Fourier Transform of $f\in R_{M\times N}$ to be \begin{equation} ...
3
votes
1answer
47 views

Finding Limits and Its Convolution of Weighted Summation of Random Variables

I am trying to find the CDF of $Z=aX+bY$ whereby $X$ and $Y$ are random variables and $a$ and $b$ are positive integers. Given that the CDF of $Z$ is: $$F_Z(z)=\int ...
1
vote
1answer
101 views

PDF & CDF of a Sum of Weighted Independent Random Variables $Z=aX+bY$

From this question here, I learned that the Cumulative Distribution Function (CDF) of $Z=X+Y$ is: \begin{eqnarray*} F_Z \left( z \right) & = & \int F_X \left( z - y \right) dF_Y \left( y ...
1
vote
0answers
49 views

Convolution of Two Shifted Functions

I'm having some issues understanding the convolution of two rectangular functions. I have two rectangular pulses defined below and I need to find the convolution of them. $$ f(x)= \prod ({x-1\over ...
2
votes
1answer
148 views

About integrating product of two sinc function using Fourier transform

So the problem is which I think is pretty straight-foward by using Fourier transform and convolution property of two sinc functions and evaluating the convolution at 5. However, I got sinc(t) for ...
0
votes
0answers
9 views

Some variation of convolution formula

I am weak in convolution. We know $y(t) = x(t)*h(t)$. Now how about $y$ in the following case 1. $x(t)*h(-t)$ 2. $x(t-T)*h(-t)$ ] Then, let $y(t) = s(t)*s(-t)$, how about $y$ in the ...
1
vote
1answer
58 views

Solve $\int_{0}^{2\pi} f(t) \sin ^2 (t-\theta) dt = g(\theta)$ for unknown function $f$

Let $g(\theta)$ be a known real-valued function with domain $[0, 2\pi]$. Given that: $$\int_{0}^{2\pi} f(t) \sin ^2 (t-\theta) dt = g(\theta)$$ How would I solve for the unknown real-valued function ...
2
votes
1answer
39 views

Comparing Coefficients in Summations

Suppose I have the following equality: $$\sum_{k=0}^{n-a}\sum_{j=0}^{k}\binom{n}{k}\binom{k}{j}\frac{f(a,k)\cdot g(b,n-k)}{n!}=\sum_{k=0}^{n-a}\binom{n}{k}\binom{n-k}{a}\frac{z^k \cdot ...
1
vote
0answers
69 views

distribution of sum of double exponential random variables

I want to find out whether there is a concise expression (i.e. not a convolution) for the distribution of a random variable A which is the sum of $n$ i.i.d. rv's $B_i$, which are themselves double ...
1
vote
0answers
69 views

Holder continuity of the convolution of a Holder continuous function

Let $f(\theta, t)$ be a Holder continuous function for every $t$ on the interval $\theta \in (\alpha,\beta)$. It is known that the application of a singular operator to this function results in ...
0
votes
1answer
59 views

Properties of convolutions w.r.t. continuity and partial differentiability

Is there some good summary of properties of convolutions available out there? I'm interested in continuity and partial differentiability topics, like, when exactly do we have $f*g$ is continuous at ...
0
votes
0answers
3 views

Estimating certain singular discrete sums

I want to estimate sums of the following form: $S^d(\alpha,\beta,l):= \sum_{k \in \mathbb{Z}^d, k \notin \{0,l\}} \frac{1}{|k|^\alpha} \cdot \frac{1}{|k-l|^\beta}$, where $l \in \mathbb{Z}^d$ and ...
1
vote
1answer
52 views

convolution and associativity

Ok Let talk about this,... I am now so confused. 1-$$\mathcal{F}\Big\{c(x-x_0)b(x-x_0)\Big\}=\mathcal{F}\Big\{c(x-x_0)\Big\}\circ\mathcal{F}\Big\{b(x-x_0)\Big\}\\=\Bigg[e^{-2ix_0y}C(y) ...
0
votes
0answers
42 views

What is the distribution of sum of a Gaussian and and 2 r.v. Rayleigh distributed?

Let $Z=X+Y+W$; where $X∼N(0,σ_1^2)$ i.e. a Gaussian random variable and Y and W follow the Rayleigh distribution: $f_w(w)=\frac{w}{σ_2^2} . exp(−\frac{w^2}{2σ_2^2})$, $y\ge0$ What will be the ...
2
votes
0answers
70 views

Number Theoretic Transform (NTT) to speed up multiplications

I recently heard that the Number Theoretic Transform (NTT), which is a specialization of Fast Fourier Transformation (FFT) over the ring modulo an integer, can be used to speed up certain ...
0
votes
2answers
62 views

Convolution sum. Compute $y[n]=x[n]\ast h[n]$

Compute $y[n]=x[n]\ast h[n]$ $x[n]=(-\frac{1}{2})^2u[n-4]$ $h[n]=4^nu[2-n]$ In this question, when I try to calculate the convolution sum. I face with: ...
0
votes
0answers
56 views

Is this 3D kernel 2D+1D separable? 3D vs 2D+1D convolution.

I have a 3D image (dimensions: X x Y x Z), a 2D asymmetric kernel (dimensions: X x Y) and a 1D asymmetric kernel (dimensions: Z). I multiply the two kernels together and I get a 3D kernel. Is this new ...
1
vote
1answer
79 views

Solve 2nd order ordinary differential equation with unit-step driving function by Laplace transforms and convolution theorem. (5.6-42)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
1
vote
0answers
23 views

How to sum random variables

Let $Z_t = \psi_t |\lambda Z_{(t-1)} + (1-\lambda)\epsilon_t |$ be a random variable where $\epsilon~N(0,1)$ is a Gaussian distributed number, $Z_0 = z_0$ and $\psi \in [-1,1]$ a random variable, ...
1
vote
2answers
129 views

Convolution of a function with itself

Function $\phi (x)$ is defined as: $$\phi(x) = \begin{cases} 1 & \text{ if } 0 \leq x \leq 1\\0 & \text{otherwise} \end{cases} $$ How do I find the convolution of $\phi(x)$ with itself? I ...
1
vote
0answers
28 views

Convolution of which distribution will give a uniform distribution?

Suppose there are two IID random variables x1 and x2. What should be the distribution of these random variables so that the distribution of x1-x2 is a uniform distribution?
1
vote
1answer
47 views

Solve 2nd order ordinary differential equation by Laplace transforms and convolution of their inverse functions. (5.6-40)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
0
votes
1answer
7 views

How to apply convolution non-uniformly?

Suppose I wish to apply Gaussian blur everywhere, except some predefined region How calculate this with formula like below $\int\int I(x,y) g(x-u, y-v) dx dy$ What is $g()$ will be here?
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0answers
92 views

Sobolev norm of a convolution

Let $\eta$ be a rapidly decaying function such that it is radial and $(\mathscr{F}\eta)(\xi)=1$ for $\vert\xi\vert\leq 1$. (Here $\mathscr{F}$ is the Fourier transform). Let's put ...
1
vote
0answers
32 views

Proof of Convolution Properties

Let u,v, w $\in l_1(Z)$. I need to prove that the following are true: their convolution $u*v$ is also in $l_1(Z)$ if u and v are two probability vectors, then their convolution is also a ...
0
votes
0answers
68 views

Hessian Of Convolution's Quadratic Form

For the discrete inputs $\mathbf{x} \in \mathbb{C}^{M}$ and $\mathbf{y} \in \mathbb{C}^{N}$, I want to find the Hessian of $\Vert x \ast y \Vert_2^2$, where $\ast$ is the discrete convolution ...
1
vote
1answer
322 views

Calculate the convolution of the product of two simple functions. (5.6-12)

Synopsis: I cannot duplicate the answer given in a very reputable online symbolic integral calculator as shown in this link ($x$ is $\tau$) although my answer does appear very similar. This tells me ...
0
votes
1answer
77 views

Calculate the inverse Laplace transform by convolution. (5.6-26)

Synopsis: I cannot duplicate the answer in my text although I do get somewhat close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? The ...
1
vote
1answer
27 views

Calculate the inverse Laplace transform by convolution. (5.6-25)

Synopsis: I cannot duplicate the answer in my text although I do get somewhat close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? The ...
2
votes
1answer
46 views

A general theory of convolution product

in my childhood, I learned about convolution products for function over $\mathbb R$ (1). For quite a while now, I have played with polynomial rings, where also, the product is sometime called a ...
1
vote
1answer
90 views

Calculate the convolution of the product of a unit step function and t. (5.6-14)

Request Please check my work. I am not certain how to calculate the convolution of the unit step function. Given: Find the convolution of $f(t)=t$ and $g(t)=u(t-1)$. $$h(t)=(f*g)(t)=\int_0^t ...
0
votes
1answer
28 views

Calculate the convolution of the product of two sine functions. (5.6-13)

Request: I cannot duplicate the answer in the book although I do get very close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? Given: ...
0
votes
1answer
46 views

Calculate the convolution of the product of two identical sine functions. (5.6-7)

Request I am very new to this so please bear with me. I cannot duplicate the answer in the book although I do get very close. This tells me that my method is correct but I am making another kind of ...
1
vote
3answers
38 views

Convolution Product $\sin t *\sin t$ by complex replacement.

I want to compute $$\sin t * \sin t=\int_0^t \sin u \sin (t-u) \, d u.$$ I have already tried complex replacement: $$\int_0^t \sin u \sin (t-u) \, \mathrm d u = Im{\int_0^t e^{iu} e^{i(t-u)}\,d u ...
0
votes
1answer
22 views

Energy preservation in convolution

Is the energy preserved in convolution? I convolve two functions: $$g(t) = f_1(t) \cdot f_2(t)$$ provided that the integrals of $f_1$ and $f_2$ remain unchanged, is the integral of $g(t)$ always ...
1
vote
0answers
112 views

Solving a non-linear ODE in which one term is a convolution

Does anyone have any clue of how to solve the following ODE: \begin{align} \frac{F'(t)}{1-F(t)} &= p + (q-w)F(t) + w [F'(u) * e^{-\nu u}](t) \\ \frac{F'(t)}{1-F(t)} &= p + (q-w)F(t) + w ...
0
votes
0answers
14 views

Convolution, clarification about exercise

I'm studying convolution, and I have found this exercise, but I haven't understood a step. Let's consider this function: $$f(t)=\begin {cases}1, t\in [-1,1] \\ 0, t\not \in [-1,1]\end{cases} ...
2
votes
1answer
53 views

Calculate the convolution of two constants. (5.6-1)

Request I am very new to this so please bear with me. I cannot duplicate the answer in the book. I believe I may be making a methodical error. Please correct it for me. Given: Find the convolution ...
1
vote
1answer
91 views

Multiple self-convolution of rectangular function - integral evaluation

I am trying to find an $n$-multiple convolution of a rectangular function with itself. I have a function $f(x) = 1$ for $0<x<1$, 0 otherwise. I define $$ g_2 (y) = \int_{-\infty}^{\infty} ...
2
votes
0answers
39 views

Proving $\mu\ast K_n\to\mu$

Let $\{K_n\}$ be approximating unit and $\mu\in M(\mathbb{T})$. Show that $\mu\ast K_n\to \mu$ weakly means $$\int f(t)d(\mu\ast K_n)(t)\to\int f(t)d\mu(t)$$ Suppose $\mu\in L^1(\mathbb{T})$, I ...
1
vote
1answer
30 views

What is $f(t) * g(-t)$ (convolution)?

I know that the definition of convolution is the following: $$ f(t) * g(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) \mathrm d \tau $$ Then, which is the correct one between the two: $$ f(t) * ...