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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0
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1answer
130 views

Obtaining Impulse Response from Graph

I want to know how to solve those types of problems.. is it by inspection ? Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the ...
0
votes
1answer
126 views

Probability density of vector sum

Consider two unit $\mathbb R^2$ vectors $v$ and $w$. Then $v+w$ lies within a (closed) circle with radius 2, that is, in the region $x^2+y^2\leq4$. Intuitively, the probability of $v+w$ lying close ...
2
votes
1answer
167 views

Fourier Transform of an Operator

I need to calculate the fourier transform of an Operator. meaning I need to calculate the transform of the Operator's corresponding convolution kernel. so the question is: 1.given a 2d fourier ...
1
vote
0answers
170 views

convolution of L1 function with a harmonic oscillation

I have to show that the convolution of a function $f \in L^1(\mathbf{R})$ with the harmonic oscillation $\phi_\omega (t) = \exp(2 \pi i t \omega)$ is equal to the Fourier Transform of $f$, ...
6
votes
1answer
224 views

Convolution between two distributions

I want to define the convolution $*$ between two distributions $S$ and $T$. For a test function $\varphi$, can I say: $$\langle S * T, \varphi \rangle \doteqdot \langle S, T*\varphi \rangle $$ where ...
0
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0answers
248 views

Convolution of two functions (pdfs)

I want to convolve two signals . The range of each of the signal is 0 to 1. ...
4
votes
2answers
215 views

Convoluting two surfaces

I'm wondering how the concept of convolution can be extended to 2D. As example, let us take a constant function $z=f(x,y)=1$ with support on $[0..1]^2 \in \mathbb{R}^2$ (see Fig. 1). If we now ...
2
votes
2answers
2k views

Convolution of two triangles

I want to convolve two triangles. The equation satisfied by one triangle is $$f(y) = \begin{cases} y + 1 & −1 < y < 0\\ \\ 1 − y & 0 \leq y < 1 \end{cases}.$$ So, the overall ...
0
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0answers
185 views

What is the purpose and usage of convolution?

I am curious of what the purpose and usage of convolution are. Why is convolution created? In layman's term (and in mathematical term), what defines convolution?
1
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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 ...
3
votes
3answers
2k views

Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
1
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0answers
62 views

Convolutions of Path Integrals of Gaussian Functions

I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
1
vote
1answer
203 views

Convolution Laplace transform

Find the inverse Laplace transform of the giveb function by using the convolution theorem. $$F(x) = \frac{s}{(s+1)(s^2+4)}$$ If I use partial fractions I get: $$\frac{s+4}{5(s^2+4)} - ...
1
vote
0answers
50 views

Maxium value of discrete convolution

I'm trying to calculate the maximum possible short-term energy $E[n]$ of a sampled signal $s$ in terms of $N$ and $\text{bitdepth}$. $$ E[n] =\sum_{m=-\infty}^{\infty} s^2[n]w[n-m] $$ where $$ w(n) ...
5
votes
1answer
107 views

Compactness of Convolutions of Compact Measures

This regards measures on $d$-dimensional Euclidean space $\mathbb R^d$ and their associated densities. A super-level set of a density $f : \mathbb R^d \to \mathbb R^+$ at level $t$ is the set $\{x \in ...
2
votes
3answers
419 views

Convolution of an integrable function of compact support with a bump function.

Let $f\in L^1(\mathbb{R})$ be of compact support and $\psi(x)=C \exp(-(1-x^2)^{-1})$ where $C$ is chosen so that $\int_{\mathbb{R}} \psi =1$. Show that the convolution $f*\psi(x)=\int_{\mathbb{R}} ...
3
votes
0answers
261 views

Convolution theorem in 3D

Suppose to have a 3-dimensional discrete grid. I would like to convolve it with a 3-dimensional tensor (a 3x3x3 "cube"), applying the convolution theorem. Hence, I should apply a Fourier transform to ...
1
vote
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 ...
2
votes
0answers
190 views

Poisson exponentiation distribution family and convolution

Assume $\xi_i \sim \mathbb{F}_{\lambda_i}(x)$ are random variables from Poisson distribution. Consider random variables $\eta_i \sim \tilde{F}_{\lambda_i,t}(x)$, where $\tilde{F}_{\lambda_i,t}(x) = ...
-1
votes
1answer
282 views

How does convolution/deconvolution with gaussian affect signal mean

I have a discrete signal (an image actually), which I am convolving/deconvolving with a zero-mean Gaussian kernel. I would like some proof that these operations do not alter the signal mean. Well, it ...
8
votes
1answer
485 views

How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
1
vote
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 ...
1
vote
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 ...
2
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0answers
69 views

bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
1
vote
1answer
148 views

How to compute the convolution of two functions which diverge at infinity?

How to compute the convolution of two functions which diverge at infinity? e.g. $e^{x^2}*e^{x^4}$ We can't directly write as $\int_{-\infty}^\infty e^{t^2}e^{(x-t)^4}~dt$ or $\int_{-\infty}^\infty ...
0
votes
2answers
116 views

Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$

Let $C_c^\infty$ denotes the set of real valued function with compact support. Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$. If ...
0
votes
0answers
1k views

how does one convolve two matrices

so in OpenCV I retrieve a Gabor kernel for image processing which is a 10:10 matrix. I have a gray matrix of the original image. How do I convolve the two and get the output of the convolution? I'm ...
0
votes
1answer
332 views

Deriving complex form of Fourier series

I have encountered a problem where I get the correct outcome, but I am uncertain as to whether or not my steps are logically justified. I would really appreciate some input regarding this! The ...
1
vote
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 ...
5
votes
3answers
275 views

if convolution of $f$ with itself remains same, then $f=0$ a.e?

I'm trying to answer the question above.. But I'm not certain in either way. I tried to prove it by giving counter examples.. But it always failed.. Then i also tried to draw contradictions But ...
1
vote
0answers
173 views

Convolution with a special approximation to the identity function

I'm working my way through Stein and Shakarchi's Real Analysis, and I'm having some trouble figuring this exercise out. Given the function $K_\delta$ that satisfies the normal approximation to the ...
1
vote
1answer
251 views

Convolution converges in infinity norm?

Assume $\phi$ to be a nonnegative continuous function on the real line with compact support. Also assume that integral of $\phi$ over $\mathbb{R}$ is normalized to $1$. Let $\phi_e(x) = ...
1
vote
1answer
180 views

Should mean be subtracted before convolution?

Suppose I have a signal in the form of N real numbers. I am searching for a continuous pattern within this signal. The pattern is represented by M real numbers. (M < N) To find the pattern's ...
1
vote
0answers
155 views

Consider the correlation of two functions, what is the derivative of the result with respect to one of those functions?

I have a problem that comes up from time to time in signal processing applications. Let $f(x)\geq0\, \forall x$ and $g(x)$ be real functions with finite range and support. Let $I(f(x),g(x)) = ...
1
vote
3answers
857 views

Approximate a convolution as a sum of separable convolutions

I want to compute the discrete convolution of two 3D arrays: $A(i, j, k) \ast B(i, j, k)$ Is there a general way to decompose the array $A$ into a sum of a small number of separable arrays? That is: ...
3
votes
1answer
261 views

How to show that if $f$ or $g$ is continuous, then the convolution $f \star g$ of those functions is continuous?

How to show that $f \star g$ is continuous if $f$ or $g$ is continuous? Do you use $\epsilon - \delta $ - approach in the proof? some hint. I define $$(f \star g)(x)=\frac{1}{2\pi} \int_{- \pi}^{\pi} ...
1
vote
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?
1
vote
0answers
96 views

Integrability and differentiability of convolution of the fundamental solution and an integrable function

Define a function $\Gamma(\cdot)$ as $$ \Gamma(x-y)=\frac{1}{2\pi}\log\|x-y\|,\quad x\neq y $$ where $x=(x_1,x_2),y=(y_1,y_2)\in R^2$, and $\|x-y\|^2=(x_1-y_1)^2+(x_2-y_2)^2$. Note that $\Gamma(x-y)$ ...
1
vote
1answer
196 views

Integrating a function within a convolution, variable substitution

Let $f(x)= \begin{cases} 1 & |x|\lt 1 \\ 0 & |x|\ge 1 \end{cases}$ I want to find the convolution $f(x) \ast f(x) = \int\limits^{\infty}_{-\infty}f(y)f(x-y)\,\mathrm{d}y$ I started out by: ...
1
vote
1answer
313 views

Computing convolution density function with Maple

I would like to know how to compute a probability function of a convolution of Negative Binomial distribution with Maple. Here is an easy example of what I want to do : ' ...
3
votes
1answer
200 views

Is this a correct way to convert an convolution equation into differential/difference equation?

For functions $f,g,h$ that are defined over $\mathbb{R}$, suppose we have a convolution equation: $$ f = g * h. $$ I would like to convert it into a differential equation. Is it correct that $$ ...
4
votes
1answer
246 views

How can I compute $\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$

If $f(x)=\text{arccot}(x)$ for non-negative $x$ and $0$ otherwise, how can I calculate $$\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$$ for $y\in\mathbb{R}$?
1
vote
3answers
2k views

Exact deconvolution of two matrices using numerical techniques

Suppose that I am given two $n \times m$ matrices $\bf{A}$ and $\bf{C}$, and let $\bf{B}$ be a matrix that is convolved with $\bf{A}$, such that: $\bf{A} * B = C$ In the above, $*$ is the ...
0
votes
1answer
212 views

Point spread function (PSF) expressed as a convolution and a sum

The Wikipedia article on the Point Spread Function (link) discusses how an imaging system can be conceptually described using linear system theory. A convolution of the PSF with the image in the ...
7
votes
2answers
2k views

convolution of a function with itself equals itself

In a homework question, I was asked to show: (1) in $L^1(R)$, if $f*f = f$, then $f$ must be a zero function. (2) In $L^2(R)$, find a function $f*f=f$. I don't know how to proceed. for (1), $f*f=f$ ...
0
votes
1answer
132 views

how to compute the convolution of two measures explicitly

Here is my example:u and v are the surface measures on the spheres {${x;|x|=a}$} and {${x;|x|=b}$} in $\mathbb{R}^{3}$.Then what's $u\ast v$ ? And what if in $\mathbb{R}^{n}$?
1
vote
1answer
275 views

analytically calculate value of convolution at certain point

i'm a computer science student and i'm trying to analytically find the value of the convolution between an ideal step-edge and either a gaussian function or a first order derivative of a gaussian ...
2
votes
2answers
593 views

Is convolution operator compact?

I know convolution is not a Hilbert–Schmidt integral operator, but it needs more to tell if convolution is compact or not.
0
votes
0answers
333 views

Circular to linear convolution with matrices

I know how to perform a circular convolution with vectors (http://engineering-matlab.blogspot.it/2010/12/matlab-program-for-implementing_5864.html) and I know that circular convolution can be obtained ...
1
vote
1answer
544 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 ...