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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23 views

How to calculate the cross-correlation of a halfwave?

I'm trying working on a vehicle modle and testing it with steering maneuvres standardised by the ISO. When it comes to data analysis the ISO standard says the following: 6.4 Time lag and Sine Time ...
0
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1answer
81 views

Convolution of two Uniform random variables

We have $X \sim \mathrm{Unif}[0,2]$ and $Y \sim \mathrm{Unif}[3,4]$. The random variables $X,Y$ are independent. We define a random variable $Z = X + Y$ and want to find the PDF of $Z$ using ...
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0answers
27 views

Convolution and differential equations

Consider the following system of differential equations: \begin{align} x_1'=f_1(x_1,x_2)\\ x_2'=f_2(x_1,x_2) \end{align} Assume that a solution $x(t)$ exists for $t\in (-T,T)$. Let $g:\mathbb{R}^2\to\...
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0answers
31 views

Is there a mistake in my calculations?

I am trying solve the differential equation $y''+3y'+2y=u(t-1)-u(t-2), y(0)=y'(0)=0$, by calculating the convolution of $f(t)=1$ and $g(t)=e^{-t}-e^{-2t}$. The problem is that I get two different ...
3
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0answers
53 views

Approximating two-dimensional convolution

I am trying to use discrete 2d-convolution to estimate continuous double convolution. The convolution integral is $$g(x,y)=(f\ast h)(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(u,v) h(...
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2answers
86 views

Inverse Laplace Transfrom of $s^{-1}e^{-a\sqrt{s} + b/s}$

I am trying to find the inverse Laplace transform for following function and it seems almost impossible for me to find the answer. Can anyone help me please with final answer and also the way to get ...
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0answers
20 views

2D convolution with one-dimensional function

I am somewhat stumped on what may be a very basic question. I have a 2D input function $F(x,y)$ and an impulse response $H(x)$ that is independent of $y$. The output function is a convolution $G(x,y)...
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0answers
20 views

Estimating certainty of angle for rotations using curl filters.

Background Inspired by this answer which manages to make a connection between (the angle of) a proper rotation (displacement field of a rotation) and the curl of a vector field. This question aims to ...
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0answers
24 views

Fourier transform to determine stability of fixpoint of equation with temporal convolution

Given the differential equation \begin{align} \frac{d v}{d t} = - v(t) + \kappa * v \end{align} where $\kappa$ is some linear temporal filter (like a sum of two exponentials, for instance) and $...
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1answer
385 views

Help for convolution of two Multivariate Gaussian PDFs

I am looking for a proof for convolution of two multivariate Gaussians (where each Gaussian has multi-dimensional mean and co-variance). I found a proof in here: http://www.tina-vision.net/docs/memos/...
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1answer
333 views

Convolution - Difference of two random variables with different distributions

This is a homework problem, but it isn't me looking for an easy way out. I've been thinking about this problem for a while now and even went to my professor's office hours and still don't quite ...
2
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1answer
29 views

Convolution of multiple correlated probability density functions

Following up this question, assume $X_1$ to $X_n$ are $n$ correlated random variables with know marginal cumulative/probability distribution functions of $f_1(X_1)$ to $f_n(X_n)$, and a joint ...
1
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1answer
149 views

Prove that $\int\delta(x-b)\delta(x-a)\ \mathsf dx =\delta(a-b)$.

Show that the convolution of two $\delta$ functions at different points is again a Dirac $δ$ function. Convolution of a Dirac $\delta$ function with a function $f$ is defined as : $$\int\delta(x-y)f(...
1
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1answer
29 views

the solution to heat equation in convolution form

Let $g\in C(\mathbb{R})\cap L^{\infty}(\mathbb{R})$. Let $u$ be defined as the function $$u(t,x)=\int_{\mathbb{R}}p_t(x-y)g(y)\,dy$$, where $$p_t(x)=\frac{1}{\sqrt{4\pi t}}e^{-\frac{|x|^2}{4t}},\quad ...
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0answers
76 views

Convolution with piecewise functions

Let $$x(t)=\begin{cases} \sqrt{\frac{A}{T}},\;\;t \in [0,T/2]\\ -\sqrt{\frac{A}{T}},\;\;t \in [T/2,T] \end{cases}$$ Now lets define two other functions: \begin{align*}h(t) &= x(T-t) \\ g(t) &= ...
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0answers
92 views

Convolutions and dot products

There is a convolution formula $$F(fg)= F(f) * F(g)$$ where $F$ represents a Fourier transform. Suppose I have something that looks like $$\int d^3 \vec{k}'\,\,\, \left[\vec{k}' F[f(x)](\vec{k}')\...
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1answer
131 views

Derivative of unit step function

The ramp function is given by r(t)=tu(t) If we differentiate ramp ,we get unit step function. That is, u(t)=1 So the derivative of unit step function is definitely 0 since u(t) is constant over the ...
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1answer
78 views

Show that the convolution of functions is differentiable.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$ be $2\pi$-periodic functions such that $f$ is bounded, $g$ is differentiable and $g'$ is continuous. How can I show that ...
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0answers
14 views

Autoconvolution Notation

Is there any special notation for a list (or function) convolved with itself? For example, the convolution of the list [1,1,1,1] with itself produces a new list: [1, 1, 1, 1] * [1, 1, 1, 1] = [1,...
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0answers
31 views

Function equation with sth like self-convolution

I would like to get the solution of $\int_{x}^{y}f(t,y)f(x,t)dt=af(x,y)$ for all $\{y>x>0\}$, satisfying $\int _{0}^{x}f(t,x)dt=1$. Or a more simple question may be if we assume $f(x,y)=g(x/y)/...
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3answers
30 views

Find Laplace inverse

Let $${{{{x^{\ast}(s) = \left( \frac{1}{(s+\mu_1 + \mu_2) (s + \hat{\lambda}_2) (s + \lambda_1 +\lambda_2 )}\right)}}}}$$ be the laplace transform in question, where $\mu_1,\mu_2, \lambda_1,\lambda_2, ...
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1answer
15 views

Explaining Disjointness of Events

Below is a copy and paste of Dartmouth's explanation of the sum of independent random variables: Suppose $X$ and $Y$ are two independent discrete random variables with distribution functions $m_1(x)...
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1answer
142 views

Solving forced undamped vibration using Laplace transforms

I'm heaving trouble solving the following undamped forced vibration problem using Laplace transforms: $$\ddot{q}(t) + \omega_n^2 q(t) = \cos(\omega t).$$ I will show what I have done so far, and I'd ...
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0answers
26 views

Any way to simplify this integral?

$$\int_{-\infty}^{\infty}e^{-y\left(x\right)}\left(\log\int_{-\infty}^{\infty}e^{-y\left(z\right)}N\left(x-z,v\right)dz\right)dx $$ where $y(x)\equiv\sum_{i=1}^{n}c_{i}x^{i}$, $n$ is even and ...
0
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1answer
31 views

Support of convolution in pde

Let $f$ be an square-integrable function on a bounded domain U. Moreover $\eta(x) = C e^{-\frac {1}{1-x^2}} $ on $[-1,1] $ and 0 elsewhere. C is taken in such a way that the integral over eta is ...
0
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1answer
143 views

sum of two independent variables Z=X+Y, find pdf for Z

There are two independent random variables. Find density function of Z=X+Y $\mathbf{\\ f_X(x)=\begin{cases} 1- \frac{x}{2} & 0\leq x\leq 2\\ 0 & \text{otherwise} \end{cases}} \ \hspace{20pt}...
2
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0answers
46 views

How can I find the convolution of these two functions?

Restrucutred this question, as it felt more natural to be asked in a different matter than I first wrote. Given the functions $$f(t) = \frac{e^{-2|t|}}{4},\quad g(t) = \frac{e^{-3|t|}}{6}$$ Find ...
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2answers
49 views

Extension of the convolution theorem

From the convolution theorem, we know that the multiplication in the frequency domain is equivalent to convolution in the time domain, and vice-versa. I am wondering if there is some kind of ...
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0answers
84 views

sum of n i.i.d. random variables with a 2 parameter exponential distribution

I know that the sum of n random variables with an exponential(λ) distribution is distributed gamma(n,λ). But what is the gamma distribution formula for a sum of n random variables with an exponential(...
2
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0answers
40 views

How to evaluate Integro-Differential Equation using Laplace convolution?

Can someone please explain how I begin to evaluate the following integro-differential equation? I know that it involves a convolution, but the $y(τ)$ within the integral is throwing me off. $$\int_0^...
0
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0answers
100 views

Convolution between a discrete stochastic signal and a continuous function

I'm trying to find the convolution between a discrete, stochastic signal (for which I have data at each t) and an exponential decay function (which is continuous in t). Now I know that one can ...
3
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4answers
417 views

Finding convolution of exponential and uniform distribution- how to set integral limits?

I am studying the convolution method for creating the density function of two independent random variables and I am struggling with understanding how the bounds for integrals are created. There is ...
0
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1answer
26 views

How to calculate convolution with logarithm numerically?

I'm trying to compute an optimisation problem, which has a cost function involving $$I=\int_0^1\log|x-y|\rho(y)dy$$ where $x\in[0,1]$ and $\rho$ is a probability density. Eventually, I will want to ...
5
votes
2answers
460 views

Number Theoretic Transform (NTT) example not working out

I'm reading up on the NTT, which is a generalisation of the DFT. I'm working in $\mathbb{F}_5$ with primitive root $w=2 \mod 5$. Suppose I want to compute the NTT of $x=(1,4)$. So far I have obtained: ...
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0answers
22 views

How to sketch commutative property of convolution

I've been asked to sketch why $$y(n)=\sum_{k=0}^{\infty}x(k)h(n-k) = \sum_{k=0}^{\infty}h(k)x(n-k)$$ are equivalent expressions for convolution. I could explain this pretty easily with a proof, ...
-1
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1answer
326 views

$X$ Poisson distribution, $Y$ geometric distribution - how to find $P(Y>X)$?

Suppose that $X$ has Poisson distribution with parameter $\lambda$ and that $Y$ has geometric distribution with parameter $p$ and is independent of $X$. What is the $P(Y>X)$ ? (the final formula ...
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0answers
46 views

How to approximate damped harmonic under random forcing

Given motion in response to random forcing, can a "simple" function be used to estimate future motion. Starting with the damped harmonic oscillator, we can write $${d^2 \over {dt^2}} h(t) + 2 \zeta \...
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2answers
2k views

probability mass function of sum of two independent geometric random variables

How could it be proved that the probability mass function of X + Y, where X and Y are independent random variables each geometrically distributed with parameter p; i.e. $p_X(n)=p_Y(n)=\left\{\begin{...
2
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1answer
231 views

Calculation of the convolution of Cauchy density function $\int_{-\infty}^{\infty}\frac{ab}{\pi^2}\frac{1}{y^2+a^2}\frac{1}{(x-y)^2+b^2}dy$

I tried to calculate the following integral, which is the convolution of Cauchy density function: $$\int_{-\infty}^{\infty}\frac{ab}{\pi^2}\frac{1}{y^2+a^2}\frac{1}{(x-y)^2+b^2}dy$$ I tried to use ...
3
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1answer
39 views

Inverse Laplace Transform of $F(s) = \frac{3s+8}{(s^2+2s+20)^2}$

Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor $$F(s) = \frac{3s+8}{(s^2+2s+20)^2}$$ tried to apply partial fractions to it and i just ...
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1answer
147 views

Gaussian smoothing kernel with different sigma values

I am not a mathematician by training, so excuse my lack of vocabulary or the imprecision in my question. I have a 1D distribution that I need to convolute, using a Gaussian kernel. However, all the ...
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0answers
75 views

Find the CDF of the sum of the inverse square of n random normal numbers

Question If I have n independent random normal numbers denoted $X_i$ each with mean $\mu_i$ and variance $\sigma_i$ (for $i = 1 ... n$). For each $X_i$ I have a weighting factor $w_i$. What is the ...
0
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0answers
28 views

Convolution basic

I am stuck with this question. Need some help. Solve for $y(t)$: $$ \frac{d^2y}{dt^2} + 3\frac{dy}{dt} + 2y = 2*x(t)$$ Given, $x(t) = \cos(tu(t))$, where $u(t)$ is unit step function, and with ...
2
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1answer
67 views

Does infinite repeated convolution with the same normal distribution converge?

According to Wikipedia, This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. ...
1
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1answer
105 views

Estimating the integral of convolution by a product of integrals

I'm reading the appendix of Evans' PDE book. One part of it says if $V$ is a compact subset of $\mathbb R^n$ and $\phi_\epsilon$ from $\mathbb R^n$ to $[0,\infty)$ has support inside $B(0,\epsilon)$. $...
2
votes
1answer
56 views

What is Convolution?

The definition of convolution is known as the integral of the product of two functions $$(f*g)(t)\int_{-\infty}^{\infty} f(t -\tau)g(\tau)\,\mathrm d\tau$$ But what does the product of the functions ...
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1answer
100 views

Simple Discrete Convolution Question

With the discrete step function $$ u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \\ \end{cases} $$ And the output $y[n]$ defined as a discrete convolution of the input $x[n]$ ...
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2answers
33 views

Evaluate convolution integral

Can someone tell me if I am calculating this integral correctly.
0
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1answer
169 views

Convolution integral involving two Heaviside functions

I am having trouble solving the following integral involving two Heaviside functions, obtained from a Laplace transform convolution: $\Large \int_0^t \frac{\tau}{\sqrt{\tau^2 - \alpha^2}} H(\tau - \...
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0answers
29 views

A small doubt regarding a previously asked limit of convolution

Previously, I asked this question to the forum. Pointwise limit of convolution Now, a question in this regard is coming to my mind. Suppose, we don't have the integral; i.e. we have the ...