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.

learn more… | top users | synonyms

1
vote
2answers
115 views

Why is this convolution true?

I am a little puzzled by how the following summation has been written as a convolution, with one of the inputs reversed in time. Consider the following sum on the LHS, and the convolution on the RHS. ...
0
votes
0answers
39 views

Question on change of variables during convolution/correlation

I am trying to understand how the following two statements are equivalent: $$ \sum_{l=-\infty}^{\infty} h^*[l] \ R_{xx}[m+l] = \sum_{i=-\infty}^{\infty} h^*[i-m] \ R_{xx}[i] $$ I get that we made ...
3
votes
0answers
74 views

Need a fast algorithm of adaptive convolution

Good morrow, gentlemen! I have to apply some kind of adaptive filter to my function $f(x).$ I present each point of my signal as a Gaussian, whose bandwidth depends on its location (not the point of ...
0
votes
0answers
59 views

Adaptive convolution

I have some 1D function $P_0(x)$ and a filter function $g_h(x)$. Also, i have a known function $h(x)$, that is the desired filter bandwidth in any point. So, I have to convolve my function with a ...
2
votes
0answers
47 views

Weakest Conditions for Convolution to be Differentiable

I was going through various posts about differrentiability of convolutions. What I would like to ask is: Suppose $f \in C^{1}(\mathbb{R})$. Then what conditions on the function $g$ would ensure that ...
0
votes
1answer
87 views

Fast evaluation of a variant of the convolution

Suppose $\{f_n\}$ and $\{g_n\}$ are finite sequences of complex numbers with $0\leq n \leq N-1$. The convolution $\{h_n\}$ of these two sequences is $$ h_n = \sum_{m = 0}^{N-1} f_m\; g_{n - m}\, . $$ ...
0
votes
0answers
42 views

Is there an expansion for element-wise scaled convolution?

If $x = a\cdot b$ is used to indicate $x_i = a_i\cdot b_i$, $y = a / b$ denotes $y_i = a_i / b_i$, and $a*b$ denotes convolution, then is there a simplification for this expression: $$ ...
0
votes
1answer
98 views

Analytic solution of the convolution of two discoutinous c.d.f s

I have a c.d.f of variable X with a mass point at the end point, $$F(x) = \begin{cases} 0 & x<a,\\ 1-\frac{m}{x+m-a} & a\le x < r-a,\\ 1 & x\ge r-a. \end{cases} $$ where m>0. Is it ...
4
votes
1answer
118 views

Is there a way to do this with fast convolution?

If you could please offer any advice, this puzzle is driving me mad: I've come across a problem that is trivial to compute in $\mathcal{O}(m^2)$ operations, but which very closely resembles a ...
1
vote
1answer
362 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 ...
2
votes
0answers
204 views

Proof of a method to find the points of maximum slope

According to method described in a paper [1] if we want to find points of maximum slope in a signal $f(t)$, then one has to do following Convolve $f(t)$ with $g(t)$ where $g(t)=-cos(\omega ...
1
vote
1answer
56 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] = ...
0
votes
2answers
66 views

Which Method of Convolution (If Any) Is Most Appropriate Here?

I need to convolve (or otherwise get the impulse response h(t) of) the input signal $x(t) = 2u(t)$ and $y(t) = cos(4t) + 2e^{(t-1)}$. I have tried the Fourier Transform and the Laplace Transform, but ...
0
votes
2answers
33 views

What Does The Following System Do?

I have a system $y(t) = 0.5 \int^\infty _{-\infty} x(T)[d(t-T) - d(t+T) dT] $ Where d(x) is the Dirac Delta function (couldn't find the LaTEX representation - a little rusty there, so an edit to ...
0
votes
1answer
43 views

Question About Optimal Approach to Find Impulse Response?

I have the following O.D.E. relating a system's input and output, where x(t) is the input and y(t) is the output: $2x(t) = {d^2\over dt^2} [y(t)] + 6{d\over dt} [y(t)] + 8y(t)$ It is also known that ...
0
votes
1answer
108 views

Deconvolution of a convolution product with $Ax\ /\ (x^2+l^2)^{3/2}$

This is not a homework, and I have no idea whether it could be one. It is only a request for help, as I do not have any experience using Fourier transform. The origin of the problem is from physics. ...
0
votes
1answer
83 views

Understanding a diagram on Convolution

Could someone please explain what is happening at the "f*g" row and below? The image is located here as linked from the Wikipedia page. I want to teach myself about Fourier Transforms / Series, and ...
0
votes
1answer
172 views

Gradient of Predictive Sparse Decomposition Cost function

I am trying to minimize the following Cost function with respect to $X_m$. $$ Energy = f(X) = \frac{1}{2}||I-\sum_{m=1}^{M}{C_m * X_m}||_2^2+\sum_{m=1}^{M}{||X_m-\phi(W_m * I)||_2^2}+\lambda|X|_1 $$ ...
1
vote
3answers
99 views

Convolution doubt

Can someone explain why the general formula of the convolution is this one: $$(f*g)(t)=\int_{-\infty}^{\infty}f(t-\tau)g(\tau)d\tau$$ But when both $f(\tau)$ and $g(\tau)$ are equal to zero for ...
7
votes
2answers
377 views

Convolution intuition: clarifying Terence Tao's “blurring”/“fuzz” interpretation

On this math.MO post, "What is convolution intuitively?", Terence Tao's answer (in the case where one function is a bump function) involves "blurring" and "fuzz." Could someone clarify his ...
1
vote
1answer
58 views

Solving $ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) $?

I need to find $F_{n}$ in : $$ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) , F_0 = 0 , n>=2 $$ This equation screams convolution , I think , but I find it as a quite long solution sometimes. ...
1
vote
1answer
84 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 ...
2
votes
1answer
131 views

Convoluted Lorentzian and Fourier Transformation

To describe a measurement, I have to calculate the convolution of three functions $f,g,h$: $f(x)=\frac{1}{(W/2)^2+x^2} \, , W>0$ $g(x)=e^{\beta x} \frac{(\beta x-2)e^{\beta x}+\beta ...
2
votes
0answers
214 views

Convolution of compactly supported functions

Let $f,g : \mathbb R \rightarrow \mathbb R$ continuous and compactly supported. I want to show that $f*g$ is continuous and compactly supported. I am 100% sure how to do it. I began as follows: ...
3
votes
1answer
112 views

Which mathematical tool or method should I use to compare two matrices most efficiently?

I have two matrices(the first one is mxm, while the second one is nxn, m>n). They store data pertaining to human speech. The second matrix contains a data segment that acts like an acoustic ...
1
vote
1answer
54 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 ...
2
votes
1answer
76 views

A continuous random walk of length 1

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity? For one ...
0
votes
1answer
254 views

Fourier Transform for a Convolution

Alright, so I am using the Convolution property of Fourier Transforms to find a function $f(x)$. So the obvious equation: $h(x) = f(x) \ast g(x)$. Definitions: $$g(x)=Rect\left[\frac x w ...
0
votes
1answer
55 views

convolution operation

The convolution operation is stated below and is equivalent to Now, lets say we have 2 functions namely x(t) and u1(t). If we convolve the x(t) with u1(t) where u1(t) is the unit doublet ...
3
votes
1answer
56 views

$f\in L^2(\mathbb{R}^3)$ implies $v(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}dy\in W^{2,2}$?

Let $f\in L^2(\mathbb{R}^3)$ be a function with compact support and define $v:\mathbb{R}^3\to\mathbb{R}$ by $$v(x)=\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}dy$$ Is true that $v\in W^{2,2}(\mathbb{R}^3)$ ...
1
vote
1answer
58 views

Signal Processing Convolution Summation Calculation

I am learning convolution of signals, and need to do a lot of summations and math. Because y[n]=Sum(x[k]h[n-k]) from negative infinity to infinity. I am always stuck at math procedures. Also, I am ...
2
votes
1answer
75 views

Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?

I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution $$v=\Gamma\star f$$ They claim that $v\in ...
2
votes
0answers
75 views

Solving Dirichlet problem by means of potential theory

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and consider the Dirichlet problem with $f\in H^{-1}(\Omega)$ $$\tag{1}-\Delta u=f$$ Is there a way to solve this problem by using ...
1
vote
0answers
153 views

Convolution property in terms of fft (matlab)

I am working on some signal processing and I have the following data: ...
1
vote
0answers
79 views

How to deconvolve from the result of a sort of double convolution integral?

Say that I have a probability density function defined on the unit circle, $f_{\Theta}(\theta)$, with $\theta \in \left[0,2\pi\right)$. I have a joint pdf, assuming independence, of ...
1
vote
0answers
20 views

How can convolution be interpreted as a recognizer?

In many textbooks it is said, that convolution can be interpreted as a pattern recognizer and that if kernel is located in region similar to it, then it gives greater response, than when it locates in ...
2
votes
3answers
172 views

Does the Convolution of Two Series Require Absolute Convergence?

Let $A=(a_n)_{n=0}^\infty$ and $B=(b_n)_{n=0}^\infty$ be sequences of real numbers. Let $C = (c_n)_{n=0}^\infty$ be the sequence such that $$c_n = {a_0}{b_n} + {a_1}{b_{n-1}} +\cdots+ {a_n}{b_0}.$$ ...
2
votes
1answer
201 views

Using FFT in matlab

I am not completely sure if this is where a MatLab question belongs, so if not, please direct me where I should ask. But onto my question. I am working on trying to deconvolution a signal with ...
0
votes
1answer
31 views

Convolution and integrating over G(t)

I'm struggling with the following expression in a statistics script: $$H(x) = \int_{-\infty}^\infty F(x-t) dG(t)$$ What does the dG(t) mean exactly? I've never seen that notation before. Background: ...
2
votes
0answers
252 views

How to express multiplication of two spherical harmonics expansions in terms of their coefficients?

Consider a spherical harmonics expansion/series like this: $$f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)$$ Presumably if we take two functions on ...
2
votes
1answer
57 views

Asymptotics at the origin of the convolution with an approximation to the identity.

In short, I am trying to find sufficient conditions for an approximation to the identity function $K_h$ so that, for $h$ small enough and fixed, the asymptotics at the origin of an $L^1 \cap L^2$ ...
1
vote
1answer
314 views

Convolution of distributions.

We are given with distributions $f,g \in D'(\Bbb R)$. If $suppf\subset (-\infty,a)$ and $supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined distribution. where $a$ and $b$ are real ...
3
votes
2answers
148 views

How to prove that operator is not compact in $L_2 (\mathbb{R})$

I have the operator $(Af)(x) = \int _{\mathbb{R}} e^{{-(x-t)^2}/2} f(t) dt$. It seems to me that it isn't compact and I'm looking for some general <=> criterion for integral operators to be ...
2
votes
1answer
119 views

The differentiability of convolutions

Yes, again, this type of question. Similar ones this and this. I come with another variant. Let $f\in\mathcal{S}$, i.e. Schwartz function, and $g\in L^{p}(\mathbb{R}^d),p\in[1,\infty]$. The following ...
3
votes
2answers
100 views

Problem of convolution.

If we are given with a polynomial $\mathcal P$ and a compactly supported distribution $g$. Can we prove that their convolution will be a polynomial again?
2
votes
2answers
133 views

What if the cauchy product of two series in $\mathbf{Z}$ is null

I have a problem I do not find a solution. Given two series $\left(a_n\right)_{n \in \mathbf{Z}}$ and $\left(b_n\right)_{n \in \mathbf{Z}}$ which have a cauchy product $\left(c_n\right)_{n \in ...
1
vote
0answers
122 views

Fourier transform of convolution in a finite range

Can anyone help me evaluate the Fourier transform of of the following function, $t \in \mathbb{R}$, $\lambda \in \mathbb{C}$, $g:\mathbb{R} \rightarrow \mathbb{R}$, $f(t) = \int_{t_0}^t ...
3
votes
2answers
1k views

convolution of exponential distribution and uniform distribution

Given $X$ an exponentially distributed random variable with parameter $\lambda$ and $Y$ a uniformly distributed random variable between $-C$ and $C$. $X$ and $Y$ are independent. I'm supposed to ...
1
vote
1answer
58 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 $G$. ...
2
votes
1answer
123 views

On using fourier transforms to solve the root of a convolution

In continuation of Lower bounds of laplace transform of characteristic functions. My question is: Can anyone point out where i'm going wrong in the derivation below. It's been a while ...