# Tagged Questions

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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### 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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### LTI: How to calculate the step response of this impulse response?

i need to evaluate the convolution sum of x[n] * h[n]. x[n] is the step function u[n]. I know how the output should look like but i don't know how i can calculate it. I think the lower border is 0, ...
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### Does Gaussian convolution respects order?

Assume that we have two continuous integrable functions $f,g \in L^1(\mathbb{R})$ such that, for some $x_0 \in \mathbb{R}$, we have, $$f(x_0) \leq g(x_0) \; \; \; \; (1).$$ Now let us define the ...
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### Convolution of normal distribution not equal to product with constant?

Convolution of a normal distribution says: If, $X \sim \mathcal{N}(\mu, \sigma^2)$, then $X+X\sim\mathcal{N}(\mu+\mu, \sigma^2+\sigma^2)=\mathcal{N}(2\mu,2\sigma^2)$ However, Multiplication of a ...
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### Uniqueness result for convolution

I have seen the convolution operator in different settings, and I was wondering about the following: Suppose $h=f\ast g$ for an unordered pair of functions $(f,g)$. Does there exist a pair of ...
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### Calculating the convolution of a piecewise function

Let $$f(x) = \begin{cases} \frac{1}{2}, & \text{if \rvert x\lvert \le 1 } \\ 0, & \text{otherwise} \end{cases}$$ I want to calculate the convolution of $f$ with itself. I am ...
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### Convolution Problem

while working on a signal processing problem i've reached to the following: So my aproach was: Am I doing something wrong? Is it valid Y(f)=[X(f) x H(f)]*W(f)=X(f) x [H(f)*W(f)] If you could ...
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### Dominating function for using DCT with convolution

Given a function $f \in L^1$ and a (compactly supported) bounded kernel $k$, this answer suggests to use the Dominated Convergence Theorem to get $$D(f \star k)(x) = f \star (Dk)(x).$$ My question ...
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### 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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### Integration of dirac function explanation

I have a problem that need your help. I have a gray image. We denotes $I(x)$ is gray level of a pixel in the image and $f(z)$ is a function of $z$(ie: histogram function...)-where $z$ is the set of ...
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### Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$A_f(g) = \int f(\tau)g(t-\tau)d\tau$$ and apply it to $g(t)=e^{ikt}$. ...
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### Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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### How can this integral be rewritten with convolutions?

I've got $f:\mathbb{R}\rightarrow\mathbb{R}$ bounded and I'm trying to write `$\mathtt{f}$,' a discrete version of $f$, where each element in the domain takes on the average of the corresponding ...
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### What is the distribution of empirical covariance between two independent normal distributions?

Suppose that we have two independent normal distributions $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ what is the distribution of empirical covariance (or empirical correlation if this make my ...
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### Fourier transform of product

I would like to know the fourier transform of the product of the Cauchy probability distribution $f(x)=\frac{1}{\pi (1+x^2)}, -\infty<x<\infty$ with itself. I know that the fourier transform of ...
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### 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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### Commutativity of Convolution in higher dimensions

I have a basic question about how to show that convolution in dimension $n$ is commutative - or maybe it is rather a question about change of variables .. So on $\mathbb{R}$ I know how to show ...
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What I'm trying to prove is that for $f_1,f_2 \in \mathcal{L}_1$ the function $y \mapsto f_1(y)f_2(x-y)$ is integrable for almost all $x$, or: $$F(x) = \int f_1(y)f_2(x-y)\,dy < \infty \text{ ... 1answer 208 views ### Maximal ideals in a circular discrete convolution algebra Let G = \mathbb{Z_n} and let A = \ell^1(G) with convolution, over \mathbb{C}. Let A_{\mathbb{R}} denote the subring of real valued functions in A, so A_{\mathbb{R}} is an algebra over ... 1answer 30 views ### Approximation estimates in Sobolev spaces Let's consider a bounded domain \Omega \subset \mathbb{R}^d, d =2,3, and let \varphi be in H^1(\Omega) \cap W^{1,\infty}(\Omega). Is it there a smooth (at least W^{2,4}(\Omega)\cap W^{1,\... 1answer 44 views ### Convolution - Hölder inequality I wonder if you guys can help me out with a question(not homework). I have \phi(x)=\int_\mathbb{R} |f(t)g(x-t)|dt where f \in L^1(\mathbb{R})  and g \in L^p(\mathbb{R}) and p and p' are ... 1answer 110 views ### Convolution algebra L^1(G) for non sigma-finite G Let's assume that G is locally compact and Hausdorff topological group, hence it carries a Haar measure, \mu. We can than consider space of integrable functions L^1(G) (class of functions to be ... 1answer 84 views ### Is the convolution of two continuous functions continuous? The title is the question: Is it true, that the convolution of two continuous functions is continuous again? 1answer 66 views ### Differential operator applied to convolution Suppose that g\in \mathcal{S}(\mathbb{R^n}) (Schwartz space) and f\in L^p(\mathbb{R^n}). The idea is to prove that the differential operator D^\alpha does not follow the Leibniz rule when ... 1answer 328 views ### Find the PDF of X1 +X2 +X3. The problem said: If X1,X2,X3 are independent random variables that are uniformly distributed on (0,1), find the PDF of X1 +X2 +X3. The theory I have said: Following the theory and the ... 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. ... 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$$But what does the product of the functions ... 3answers 96 views ### Adding two random variables with convolution I am trying to understand the purpose of convolution of two probability functions. Also when it is appropriate to use the convolve function on two independent probability distributions. ... 2answers 121 views ### When is convolution associative? Convolution is associative on e.g. integrable function on \mathbb{R}, but not on distributions. What about the convolution of measures on an unimodular group G? 1answer 41 views ### Convolution of various functions There is asked in an example to do convolution  h_1(t)*h_2(t) + h_3(t)*h_4(t)  where h_1(t) = e^{-2t}u(t) h_2(t) = 2e^{-t}u(t)  h_3(t) = e^{-3t}u(t)  h_4(t) = 4\delta(t)  and then the ... 1answer 65 views ### A general theory of convolution product in my childhood, I learned about convolution products for function over \mathbb R (1). For quite a while now, I have played with polynomial rings, where also, the product is sometime called a ... 1answer 257 views ### Calculate the convolution of two constants. (5.6-1) Request I am very new to this so please bear with me. I cannot duplicate the answer in the book. I believe I may be making a methodical error. Please correct it for me. Given: Find the convolution ... 1answer 191 views ### Multiple self-convolution of rectangular function - integral evaluation I am trying to find an n-multiple convolution of a rectangular function with itself. I have a function f(x) = 1 for 0<x<1, 0 otherwise. I define$$ g_2 (y) = \int_{-\infty}^{\infty} f(y+...
Let $f,g$ be two continuous functions with compact support. Show that if $f$ and $g$ are not identically $0$, then neither is $f\ast g$. This statement seems rather elementary, and I would prefer if ...