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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548 views

How to compute Inverse Laplace transform using Convolution

How do you evaluate the inverse transform below using convolution ? $$ \mathcal{ L^{-1} } \left[ {\frac{s}{(s^2 + a^2)^2}} \right] $$ I tried $$\begin{align} \mathcal{ L^{-1} } \left[ ...
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1answer
124 views

What does it mean to convolve a matrix with a kernel?

I have a Matrix, M, of dimensions width x height. The problem is to apply the [-1, 0, 1] filter along the x and y axis (i.e. convolve the image with [-1, 0, 1] kernel along horizontal and vertical ...
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1answer
291 views

What is the Fourier transformation of a uniform B-Spline?

I'm looking for the Fourier transformation of the (constant) uniform B-Spline $$N_0(x) = \begin{cases}1 & 0 \leqslant x < 1 \\ 0 & otherwise \end{cases}$$ If $N_0(x)$ would also attain ...
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2answers
151 views

Finding spectrum using the convolution property

Using the convolution property, find the spectrum for $$w(t)= \sin(2\pi f_1 t) \cos(2\pi f_2 t).$$ I'm confused on how to solve this question. Can you give me any aproach?
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1answer
545 views

FFT with a real matrix - why storing just half the coefficients?

I know that when I perform a real to complex FFT half the frequency domain data is redundant due to symmetry. This is only the case in one axis of a 2D FFT though. I can think of a 2D FFT as two 1D ...
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1answer
611 views

Fourier transform of convolution of sinusoidal signals, or product of distributions (generalized functions)

I will unashamedly say that this was at least spurred by homework. However I have gone far beyond the syllabus of the course and still can't find an authoritative answer. And it seems an interesting ...
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2answers
2k views

Convolution with multiple step functions

This is a question from Bertsekas' Data Networks. It is question 2.2 on page 141. It is asking for the convolution of the following 2 functions. Function 1: $ s(t) = 1 $, when $0 \leq t \leq T$. It ...
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1answer
1k views

How to sketch the following discrete time signal?

i need to sketch y[n] where * denotes the convolution operator and delta is the unit impulse. I know how to sketch x[n-1] and delta[n-2] but i have problems with the convolution. In my script i only ...
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1answer
24 views

Apparent paradox commuting this convolution: where is the mistake?

Starting with some vector $x$, I am performing two operations: First, I convolve $x$ with another vector $g$ to compute $x*g$, where $~*~$ denotes convolution. Second, I pointwise multiply the result ...
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1answer
17 views

Holder continuity of $\frac{x}{|x|^3} \ast f$ with $f \in C^1_0$ in $\mathbb{R}^3$

Ok, so I need to show that for $f \in C^1_0(\mathbb{R}^3)$ the convolution with $k(x) := \frac{x}{|x|^3}$ is Holder continuous. The exponent doesn't matter much as long as I can bound it using ...
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1answer
16 views

if $f$ is in weak $L^p$ and $\phi$ is $C_0^{1}$ then $f \ast \phi$ is in weak $L^p$

Okay, so I'd like to know if what I wrote in the title is true. Suppose that $f \in L^{p,\infty}(\mathbb{R}^n)$ (weak $L^p$ space) and $\phi \in C_0^1(\mathbb{R}^n)$ [or even $C_0^{\infty}$ if it ...
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1answer
20 views

Using convolution to impose differentiablilty.

If I had a function $g$ that was not differentiable at a known point, is it possible to convolute it with say a $C^{\infty}$ function $f$, resulting in a differentiable function? Thanks in advance!
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1answer
43 views

convolution of function with itself 4 times

I have to compute the convolution of $ f(t) = \frac{1}{\pi}\frac{1}{t^2 + 1} $ with itself 4 times, i.e. $$ f \star f \star f \star f $$ I slightly doubt that doing it in steps, i.e. taking $f \star ...
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3answers
60 views

Convolution of maximum and minimum of uniform random variables

Let $X_1,\ldots, X_n$ be $n$ independent random variables uniformly distributed on $[0,1]$. Let be $Y=\min(X_i)$ and $Z=\max(X_i) $. Calculate the cdf of $(Y,Z)$ and verify $(Y,Z)$ has independent ...
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1answer
38 views

Probability Density of Convolution of Two Random Processes or Variables

Suppose that we have two stationary random processes $x(t)$ and $y(t)$ with probability density functions $f_{x}(x)$ and $f_{y}(y)$ respectively. Now suppose we form: $z(t) = x(t) \ast y(t)$ What is ...
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1answer
24 views

Question about mollifiers.

So here is my problem, Let $\rho \in C^\infty (\mathbb{R}^n,R)$ with $\rho\geq 0$, $\rho(x)= 0 \; \forall \|x\|\geq 1$ and $\int_{\mathbb{R}^n}\rho(x)dx=1$. Further, consider the linear map ...
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1answer
34 views

Laplace transform of convolution with no function of t

Instructions: Evaluate the given Laplace transform. Do not evaluate the integral before transforming. Problem Given: $\mathscr{L}\{\int_0^t e^{-\tau} cos\tau d\tau \}$ My Problem: To treat this as ...
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1answer
43 views

family of functions/sequences taken over reals instead of naturals

How does convergence for a sequence and a family of functions change when considering $n$ taken from $\mathbb{R}$ instead of the $\mathbb{N}$? for example, consider mollifiers which are defined as ...
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1answer
66 views

What is the solution of this differential equation? / How to solve it?

I have the following problem : $$m\ddot{x} + c\dot{x} + kx = f_f\delta(t-t_0) + f_c \sin(\omega t) + f_h \theta (2t_0-t)$$ where $x(t)$ is a function of time, $t>0$ and $t_0>0$ and where ...
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1answer
75 views

Sums of normal CDF's

This is my following problem: $$ CDF_A:F_A(x)=\Phi(x)^2 $$ $$ CDF_B:F_B(x)=1-\Phi(-x)^3 $$ $$ Defining: X=A+(-B) $$ I have those two CDF's and I want to calculate the probability that X is smaller ...
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2answers
34 views

Is convolution associative with regards to the complex unity?

Setup: I need to do a convolution with the function $\cfrac{i}{x}$, and I would like to get rid of the $i$. My functions to be convolved are all real valued. According to the ever-failable wikipedia, ...
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1answer
31 views

Sums of random variables expressed as convolution

If $X_1, \dots, X_n$ are a sequence of IID random variables, and $S_1, \dots, S_n$ is the sum of the first n random variables, i.e. $S_1 = X_1$ and $S_n = \sum_{i = 1}^{n} X_i$. Apparently, we can go ...
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1answer
78 views

Hölder's inequality. Understanding proof?!

I know how most standard textbooks show that $||f*g||_r \le ||f||_p||g||_q$ with $\frac{1}{r}+1=\frac{1}{p}+ \frac{1}{q}$, but I found a book where the hint $|f(x-y)g(y)|\le ...
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2answers
211 views

Evaluating the convolution integral of two sine functions

This is a homework problem, so I'm not looking for a worked out solution, merely to be pointed in the right direction. Convolve x(t) with h(t) where: $$ x(t) = sin(t) \\ h(t) = e^{-.1t}sin(2t)u(t) ...
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1answer
57 views

A question about stochastic ordering and convolution

Two probability density functions $f$ and $g$ are known to have distribution functions $F$ and $G$ respectively with $F(y)>G(y)$ for all $y$, say on $\mathbb{R}$. It is known that if we convolve ...
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1answer
66 views

convergence of convolutions and approximation of unity

Let $\phi : \mathbb{R}\rightarrow \mathbb{R}$ be an integrable function with $\int \phi(x)dx = 1$. Let us define $\phi_\delta = \delta^{−1}\phi(\delta^{-1}x)$. Show that for every continuous ...
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1answer
108 views

A self-convolution formula that counts bracket expressions

Problem: Consider an alphabet of size $m+2$, consisting of the two bracket symbols $\ [ \ ] \ $ plus $m$ non-bracket symbols ($m \ge 0$). Define $f_m(n)$ to be the number of length-$n$ strings on this ...
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1answer
35 views

Convolution convergent in $L^p$

Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow ...
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1answer
89 views

Convolution of distributions is not associative

I need some help with this exercise: It proposes to show that convolution of distributions is not associative: If $T=T_1$ (distribution given by f=1), $S=\delta'$, and $R=T_H$ (we denote as $H$ the ...
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1answer
31 views

Convolution composed with an invertible matrix

Let $T$ be an invertible $n \times n$ matrix and let $(h \circ T)(x)$ mean $h(Tx)$. Take functions $f,g$. Does it hold that $(f*g) \circ T = |det(T)| (f \circ T) * (g\circ T)?$ I have had some ...
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1answer
59 views

About the continuity of a convolution product

I need some help with this exercise: If $f\in L_p(\mathbb{R}^n)$ and $g\in L_q(\mathbb{R}^n)$, where $\frac{1}{p}+\frac{1}{q}=1$, Is their convolution $f\ast ...
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1answer
117 views

How can I write an ordinary function in terms of integral of delta Dirac function?

We have the result $(x * \delta)(t) = x(t)$. Here let $x(t)$ is a real valued function with respect to time and $\delta(t)$ is the unit impulse function. $$ x(t) * \delta (t) = \int_{-\infty}^\infty ...
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1answer
381 views

Integral of repeated convolution of the unit step function

Background Let $\theta$ be the unit step function: $$\theta(x) = \begin{array}{ll} \left\{ \begin{array}{ll} 0 & x \lt 0 \\ 1 & x\ge 0. \end{array}\right. \end{array} $$ Further, the ...
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1answer
144 views

convolve probit function with gaussian [duplicate]

I want to prove the following, however, not sure where to start. $\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$ Where $\Phi(\cdot)$ is the probit function, ...
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1answer
67 views

Norm of convolution of $n$ Gaussians

If $$f(x)=e^{-(\pi x)^2}$$ and $$\psi_n(x)=(f* f*\dots*f)(x)$$ ($n$ times convolution). Show that $$\lVert \psi_n(x)\rVert = 1$$ (norm in $L^1(\mathbb{R})$). I've tried using the Fourier ...
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1answer
350 views

Finding distribution of $X^2+Y^2$ where $X,Y\sim N(0,1)$

Assume I have two random independent standard normal variables $X,Y\sim N(0,1)$, How can I find the distribution of $Z=X^2+Y^2$? I thought integrating the convolution, i.e ...
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1answer
54 views

Analogous to convolution

A convolution of two functions $f$ and $g$ is defined as $$[f*g] = \int_{-\infty}^{\infty} f(\tau) g(t-\tau) d\tau.$$ I am interested on an analogous transformation of the form $$[f\star g] = ...
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1answer
83 views

Estimate of $|(f*g)(x)-f(x)|$ where $g$ is approximation to the identity

Let $f: \mathbb R \rightarrow \mathbb R$ continuous with compact support $[0,1]$. Assume $|f(x)| \leq M$ for all $x \in [0,1]$. Let $\epsilon > 0$. Then let $0< \delta <1$ s.t. $$ \forall ...
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1answer
53 views

Recovering function from convolution with a square function

Let $m : \mathbb{R} \to \mathbb{R}$ be a continuous function of compact support. Given $M$ as: $$ M(x) = \int_{\mathbb{R}} m(t) (t+x)^2\ \mathrm{d}t,$$ is it possible recover $m$? I thought it ...
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1answer
56 views

expanding convoluted integrand

I have a function on the form $$g(y) = \int_{-\infty}^{\infty}{e^{-v^2}f(y-v)dv}$$ I know that $g(y)$ is linear around $0$, $g(y\approx 0)\approx yG$, and I am interested in finding this gradient $G$. ...
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1answer
145 views

Sifting Property of Convolution

This is going to be a dumb question, but I can't figure it out, so here goes $ f(t)\quad \bigotimes \quad \delta \quad (t\quad -\quad { t }_{ o }) $ = $\int { f(\tau )\delta (t\quad -\quad { t }_{ ...
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1answer
291 views

How do I take the complex convolution of this impulse response and input?

I derived a an impulse response of $h[n] = (3/4) (-j3/4)^{n} u[n]$, where $u[n]$ is the unit step function. I have an input $x[n] = u[n-5]$. I can find a vector representation of the convolution of ...
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1answer
225 views

Graphical understanding of the convolution of discrete distributions.

I am studying convolution and trying to get a visual sense of the process. From the wikipedia page I understand what 2 continuous gaussian or 2 uniform distributions will produced when convolved, but ...
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2answers
252 views

Convolution of continuous function with $\mathcal{C}^{1} $ function

I'm having difficulty with the following problem: Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$ such that $\mbox{Supp}\left(f\right)$ is compact and let $g:\mathbb{R}\to\mathbb{R}$ ...
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1answer
173 views

PDE - (homogeneous) Heat equation - Solution?

today I have a question in PDE. It concerns the heat equation: Formulate the (homogeneous) heat equation for functions $f:(0,\infty)\times\mathbb{R^n} \longrightarrow\mathbb{C}$. Derive an equation ...
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1answer
295 views

Intuition behind the convolution of two functions

Suppose $f(x)$ and $g(x)$ are two functions. What is intuition or idea behind the convolution of $f$ and $g$? After taking the convolution we will get a new function. What is the geometric relation ...
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1answer
343 views

Double tophat convolved with a gaussian

I need some help calculating the analytically expression of this convolution. The functions in question are: 1) a gaussian ($g(x)$) 2) a "double tophat function" (in lack of a better name). i.e. ...
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2answers
134 views

Convolution and analyticity

Assume $f$ and $g$ are continuous and related to each other as $$ f(x) = \int _{0}^{x-1} \Big ( (x- y)^2 - 1\Big )^{3/2}g(y) \, dy, \qquad x>1. $$ If we happen to know that $f$ is real analytic ...
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1answer
104 views

Show the convolution of a $C_c^\infty (\Bbb R^n)$ function with a $L^p(\Bbb R^n)$ function is in $C^\infty(\Bbb R^n)$, $1\le p\le\infty$

Let $f \in L^p\left(\Bbb R^n\right)$ and $g \in C_c^\infty \left(\Bbb R^n\right)$. Show $f \ast g \in C^\infty\left(\Bbb R^n\right)$ for $1 \le p \le \infty$. Let $x=(x_1,x_2,\ldots,x_n)$ and ...
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2answers
890 views

Sum of random variables uniformly distributed (0,1) and (0,2)

I'm trying to get $P(0.9<Y<=1.8)$ for the sum of 2 random and uniform values x1,x2 (so that y=x1+x2) where $x1$~$u(0,1)$ and $x2$~$(0,2)$ and I'm trying to do the convolution for it. Seems like ...