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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0answers
11 views

Periodic convolution of functions

Define the periodic convolution of functions in L2([0; 1]). What theorem of convolution do I use to define this and how do I solve this?
3
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1answer
43 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = ...
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0answers
15 views

Finding A Convolution

Let $f(x)=e^{- \mid x \mid}, g(x)= e^{-x^2}$ What is $(f*g)(\xi)$? I have been trying to find it, but I am stuck on finding the integral of $e^{y^2+y+\xi}$ Thank you!
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0answers
11 views

How to represent a periodic function as the sum of sinc functions in fourier transform

Suppose function $f(t)$ is 1-periodic. This means that in fourier transform, $F(\omega)$ is sum of impulse signals (dirac delta function and its shifts) at the multiples of $1$. Now we can form $g(t)$ ...
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0answers
53 views

How do I apply this PDE as an image filter?

I'm trying to preprocess a height map image with a helmholtz-type equation as described in this paper. The equation is: $$ddx(h') + ddy(h') + y(h'-h) = 0$$ I solved for h and got: ...
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0answers
31 views

How much does convolution with a compact C^m kernel increase the order of continuity.

Let $f \in C^n$ and $g \in C^m$, with $g$ compactly supported and integrable. How much does the convolution $f\star g$ of $f$ with $g$ increase the order of continuity? Statement: I think that, under ...
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1answer
7 views

Filter output of a signal

So I have a filter $$H(z) = 0.5 + 0.5z^3 = (1/2, 0, 0, 1/2)$$ and need to find the output of it on a cyclical signal $$x = (..., 3, -1, 2, 1, 5, 2, 3,-1, 2, 1, 5, 2, 3,...) $$ Would the output be ...
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0answers
17 views

How to get the PDF for $p(a)$ $p(b)$ by using convolution?

Suppose there are three random variable $a$, $b$, $c$, and the PDF for each are $p(a)\ p(b)$ and $p(c) $ Also, $c$ = $a$ + $b$, $a$ and $b$ are two independent variable. and$$p(c)= \begin{cases} ...
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0answers
25 views

Prove that the variance of a discrete random variable increases with a parameter

I have an infinite number of known probability density functions $f_1(x),f_2(x),f_3(x),...$. The PDFs $f_k(x)=\sum_{j=1}^k v(A+j-1)e^{-v(A+j-1)x}\binom{k}{j-1}q^{j-1}(1-q)^{k-j-1}$. Let ...
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1answer
30 views

A question related to convolution

Let $f\ge0$ such that $\int_{{R^n}} {f(x)dx} = A < 1$, and ${f_k} = f*...*f$ a convolution of $k$ times. My problem is how to prove $f_k$ is integrable and $f_k\rightarrow0$ in $L^1(R^n)$? ...
1
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1answer
40 views

Integral of $\int_{-\infty}^{+\infty}\left |{\frac{\sin{x}}{x(1+x^2)}}\right|^2\,dx $

So the first part of the questions asks us to find the Fourier Transform of $$ f(x) = \left\{ \begin{array}{ll} e^{y} & \quad {-\infty}<x < 0 \\ e^{-y} & ...
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0answers
24 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity. My Questions: ...
-1
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1answer
37 views

How to calculate convolution integral?

I know the formula for a convolution integral but how would you actually carry out one when you have two piece-wise defined functions? If you had $$ f(x) = \left\{ \begin{array}{ll} ...
1
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1answer
20 views

calculating probabilities about the sum of dependent discrete uniforms

Say I have the following information: $$ X_i \sim \text{Discrete Uniform}(1,13) $$ and I want to find $\mathbb P(X_1+X_2+X_3 \ge 25)$ for the cases where the $X_i$'s are dependent. What approximations ...
2
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2answers
75 views

Can we expect $g(f\ast h)= gf \ast gh$ for some $g\in C_{c}^{\infty}(\mathbb R)$?

Let $f,g:\mathbb R \to \mathbb C$ be nice functions so that their convolution make sense. My question: Is it possible to choose $0\neq g\in C_{c}^{\infty}(\mathbb R)$ (= the space of smooth ...
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0answers
20 views

existence of a function such that certain convolution hold

The question says: Let $f\in L^1(\mathbb{R},m)$ be defined by $f(x)=e^{-\vert x\vert}$. Find an integrable function $g$ solving $f\ast g=f$ or show that no such function exists. I missed the class ...
1
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1answer
28 views

Convolution with Heaviside function (integration)

To clarify notation, I use $u_n = 1$ when $x>n$, and $0$ otherwise. I am having troubles with the following convolution/integration: $u_2(t) \ast sin(\sqrt{2}t) = \int^t_0u_2(\tau) \cdot ...
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1answer
28 views

Inverse Laplace transform and convolution

Suppose we have two functions of $t$, $f(t)$ and $g(t)$. Letting $\mathcal{L}\{f(t)\} = F(s)$ and $\mathcal{L}\{g(t)\} = G(s)$, I know that: $\mathcal{L}\{f(t) \star g(t)\} = F(s) \cdot G(s)$, but ...
1
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1answer
36 views

Proving convolution identity

I am trying to prove the following identity: $$\int_0^x(f*g)(y)dy = (\int_0^xf(y)dy)*g(x) = f(x)*(\int_0^xg(y)dy)$$, where $(p*q)(t) = \int_0^tp(x)q(t-x)dx$. I thought that since I already know that ...
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0answers
11 views

convolution under differentiation

I have a continuous and differentiable time function $f(t)$ for $t\in [0, T] $ and a causal time function $g(t) $ for $ t \geq 0 $. Does the following make sense? $ \dot{f}(t) \ast g(t) = \frac d{dt} ...
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0answers
17 views

FFT of k*k matrix from FFT of a j*j matrix

FFT of matrix a j by j matrix, A $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ = $\begin{bmatrix}10 & -2\\-4 & ...
0
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1answer
25 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 $x(t)$ ...
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0answers
13 views

Correlation between Iterative Methods and Convolution Codes

Hey guys so I have this Calc 3 project and the end is throwing me for a loop. I've done the encoding part, and i've coded the standard iterative methods, but I don't see how the two correlate so I can ...
2
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4answers
60 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
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0answers
20 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 ...
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0answers
12 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
30 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
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0answers
24 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
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2answers
101 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
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1answer
110 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
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1answer
22 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
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1answer
62 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
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1answer
39 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 ...
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1answer
21 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 ...
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0answers
19 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
49 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 ...
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0answers
8 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 ...
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1answer
53 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
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1answer
38 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 ...
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0answers
40 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 ...
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0answers
47 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 ...
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1answer
53 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 ...
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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
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1answer
46 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
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0answers
28 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
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0answers
31 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
51 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
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0answers
25 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
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1answer
45 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
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0answers
18 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, ...