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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0answers
31 views

Convolution of complex-valued probability distributions

This may be an elementary question, but I am wondering: suppose that I have two complex-valued random variables $X$ and $Y$ with corresponding density functions $f_X(x)$ and $f_Y(y)$. Obviously ...
0
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0answers
111 views

Trying to figure out Fourier transform of {(0.5^n)(u(n))

I'm working in a signals class for continuous signals, and we have this problem shown above. I have tried using this function f_1 X f_2 = F_1 * F_2, where I'm ...
0
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1answer
84 views

Using Mollifiers

If we take $f$ to be a smooth function, then how does it follow that we can write $f^{\epsilon}(x)-f(x) = \int_{B(0,1)}\eta(y)(f(x-\epsilon y)-f(x))dy$ where $f^{\epsilon} := \eta_{\epsilon}\ast f$ ...
2
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1answer
133 views

Convolution and uniform continuity

If $f\in L^{\infty}(\mathbb{R}^n)$ and $f$ is continuous at $x$, then $$\lim_{k\to\infty}(f*\phi_k)(x)=cf(x)$$ If $f\in L^{\infty}(\mathbb{R}^n)$ and is uniformly continuous, then $f*\phi_k\to cf$ ...
3
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1answer
72 views

Interesting inequality $\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$ over $L^p$

Consider the function $$F(x)=\int_0^\infty \frac{f(y)}{x+y} \, dy, \quad0<x<\infty$$ Prove that if $1<p<\infty$, $$\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$$ and show that the constant ...
0
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1answer
61 views

Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$)

Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$) Here's a theorem from my real analysis book: Assume $1\le p\le \infty$ and ...
2
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1answer
127 views

$L^p$ Spaces, Young's Theorem, Convolutions, and Minkowski's Inequality

I need to show \begin{align} \|f*g\|_p \le \|f\|_p\|g\|_1 \end{align} By using the generalized Minkowski inequality instead of just Young's Theorem. I have spent a lot of time, but I keep hitting a ...
1
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1answer
33 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 ...
2
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0answers
300 views

How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
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1answer
46 views

Maximum of one exponential and one uniformly distributed random variable

If X and Y are independent random variables with X exponentially distributed with mean 1 and Y uniformly distributed in [0,1] , how do I find the distribution of Max(X,Y)
2
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1answer
60 views

Young's inequality

Let $U \in L^1(\mathbb{R}^d)$ and $\rho \in L^1(\mathbb{R}^d)$ such that $\rho \ge 0$ and the support of $\rho$ is included in $B(0,1)$ (the euclidean unit ball of $\mathbb{R}^d$). Is there a way to ...
1
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1answer
55 views

What is the distribution of $X+Y$ where $X \sim U(0,\frac{L}{2})$ and $Y \sim U(\frac{L}{2},L)$?

I started along these lines: Let $Z = X + Y$ where $\frac{L}{2}< z < \frac{3L}{2}$, then, $$f_{X+Y}(z)=f_{Z}(z) = \int f_{X}(x)f_{Y}(z-x)dx$$ However, I am not sure how to fill in the bounds ...
0
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0answers
29 views

Using matrix of ones for 2d convolution to accomplish summation

So I was recently working on an image processing project where we used a [7,1] ones matrix to accomplish coherent integration, ultimately a summation. It didn't and still doesn't make sense to me how ...
0
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1answer
237 views

Convolution of functions with compact support

I have a question regarding convolution with compact support: Suppose $f \in L^1(\mathbb{R})$ and $g \in L^p(\mathbb{R})$, and both of them have compact support. Show that $f*g$ (convolution ...
1
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2answers
48 views

What is the name of this function similar to convolution?

The functions seems to be very near convolution function, but the only difference is that you integrate by $du$ in convolution, in contrast to $ds$ in this example: $g(t,u) ...
1
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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 ...
2
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1answer
66 views

Does negative distributive property of convolution over cross correlation holds?

Let $\star$ denote convolution binary operation and $\otimes$ denote cross correlation binary operation between two functions. Let $f,g,h$ be functions. Does this negative distribution property ...
0
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0answers
45 views

mathematical statistics - convolution and sums

Given integrable functions f and g on $\mathbb{R}$ define $f*g$, the convolution of $f$ and $g$ by $f*g(x) = \int_{-\infty}^\infty f(y)g(x-y) dy$, So i have to show that $M_Z(t) = M_X(T)M_Y(t)$ ...
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0answers
18 views

How to express this expression into a convolution, if possible

In order to speed-up some important numerical evaluation, I would like to express, if possible, $ \int d\mathbf{r}_1 \int d\mathbf{r}_2 \int d\mathbf{r}_3 n(\mathbf{r}_1) n(\mathbf{r}_2) ...
0
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0answers
79 views

n-fold convolutions of Poisson distribution

I understand the convolution between 2 random variables, i.e, it is the area under the product of $signal 1$ at time $t$ and $signal 2$ at time $t-\tau$ When I have a n fold convolution of a function ...
2
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1answer
111 views

Approximation in Sobolev Spaces

Consider the following proof in Lawrence Evans book 'Partial Differential Equations': How does it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in ...
3
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1answer
200 views

Evaluating the convolution using the convolution integral

I am having trouble evaluating the convolution of two signals using the convolution integral.I want to find the convolution of two signals x and h where, $$ x(t) = \begin{cases} e^{-at} ...
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1answer
2k views

Creating a System Impulse Response in Matlab

Preface: I'm extremely new to Matlab. Ok, so I have a sound file that I loaded in Matlab. Two variables are loaded: ...
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2answers
358 views

How to calculate a 1D convolution summation?

I hope I said that right. I'm trying to follow along with a convolution example but maybe I am in over my head. I don't understand how in this example they get the values on the right. For example, I ...
1
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2answers
115 views

If $(f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d})$, prove that $\ast$ is commutative.

If $(f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d})$, prove that $\ast$ is commutative. I believe this is convolution. My attempt at the proof: $(f \ast g)(n) = ...
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votes
3answers
93 views

How does the author get from one step to another?

I have to apply convolution theorem to find the inverse Laplace transform of a given function. I know that convolution is applied when the given function is multiplication of two functions. The ...
3
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0answers
102 views

Given a meromorphic function (via its Laurent series), how to obtain the (Taylor series of the) two holomorphic functions it is the quotient of?

Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...
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1answer
120 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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0answers
52 views

The role of conjugate reflected signal in an inner product

Let the conjugate reflected signal of $f(x)$ is $\tilde{f}(x) = \overline{f(-x)}$. Now the convolution between $f(x)$ and $g(x)$ is defined as the inner product $f*g(y) = \int f(x)g(x - y)dx$ and is ...
1
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1answer
150 views

Support of Convolution and Smoothing

I just want to know how it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in C^{\infty}(V)$ by using the translations, but I'm having difficulty seeing how it ...
1
vote
1answer
385 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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0answers
30 views

Finding correlation of data with potentially hidden time lags

Let's say I have few independent variables plus multiple observables that I monitor over time for a system. I'd like to find out if there is any correlation between the observables and any of the ...
5
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2answers
318 views

The condition for Y to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$

I would like to know the condition for a random variable Y in order to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$, where $X_1$ and $X_2$ are iid. Any help would be ...
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2answers
89 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...
1
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1answer
150 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, ...
1
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0answers
59 views

LTI system and convolution

I'm reading a rather informal text on (continuous) Linear Time Invariant (LTI) systems. It is just said to be a "black box" that transforms an input signal $x(t)$ into an output signal ...
-2
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1answer
168 views

Convolution of Discrete Uniform ,$DU$, Distribution.

If $X\sim DU(k,a,h),\quad -\infty<a<\infty,h>0=1,2,\ldots$ then the probability function is $$P(X=a+jh)=\frac{1}{k},\quad j=0,1,\ldots,k-1$$ Let $Z\sim DU(r,0,s)$ and $Y\sim DU(s,0,1)$ , ...
0
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1answer
30 views

How to choose a phase for the deconvolution of an autocorrelation?

Say I have a function, $C=C\left(x\right)$, whose fourier transform is denoted by $c=c\left(k\right)$, i.e. $C\left(x\right)=\sum_{k=-\infty}^{\infty}c\left(k\right)\chi\left(x\right)$, where ...
0
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1answer
121 views

Convolution of two dimensional gaussian functions

I want to calculate the sum of two probability density functions. I know that it is: $P_{U+V} (x)= (P_{U} * P_{V})(x)$ If $P_{U}$ and $P_{V}$ are gaussian functions in one dimension, i.e. $P_{U}(x) ...
1
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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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0answers
245 views

What is the physical significance of circular convolution?

I know the concept of linear convolution in both discrete and continuous domain for impulse response etc in linear time invariant systems. But why do we go for circular convolution?
4
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1answer
116 views

Convolution of an $L^1$ function and a function that tends to $0$ results in a function that tends to $0$

I'm trying to solve the following problem in review for a test, but have only partly succeeded: Let $K \in L^1(\mathbb{R})$ and $f$ be a bounded, measurable function on $\mathbb{R}$, with ...
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2answers
81 views

Proof of convolution

I would like to know how I could prove the following convolution: $$ D (f*g) =D f* g =f* Dg $$
2
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1answer
827 views

Is there an elementary proof of the convolution theorem?

Is there a way, without using much extra theory (other than the basic ideas used in textbooks deriving the Fourier transform for the first time, and ideally just using general theorems about ...
0
votes
1answer
49 views

Interpret convolution diagram

How do I interpret this "do convolutions" diagram? 1) How are the results computed? 2) When looking at this part: "x[n-k]" Do you interpret convolutions as delays or time reversals? $ y[n]= ...
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1answer
598 views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
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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
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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
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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
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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 ...