All Questions
Tagged with convolution calculus
157
questions
1
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0
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33
views
Cross-correlation of a function with itself
I came up with the following question while writing on my thesis. We assume $f:\mathbb{R}\rightarrow\mathbb{R}$ to be a real valued $\mathcal{L}^1$-function. Then the cross-correlation of $f$ with ...
1
vote
0
answers
22
views
Prove that the convolution of the signals and its time reversal is an odd signal.
Suppose signal $g(t)$ is obtained by time reversal of signal $f(t)$ for all times $t$. Prove that the convolution of the signals $f$ and $g$ is an odd signal.
My attempt at proof
Given: $g(t)=f(-t)\...
- 1,210
0
votes
0
answers
54
views
Convolution of two characteristic functions
Given two intervals on XY-plane: $I_1$
lying on OX-axis, $I_2$
lying on OY-axis. Compute convolution of two characteristic functions of these intervals.
$$\chi_{I_1} \ast \chi_{I_2}.$$
MY ATTEMPT:
$(\...
- 39
0
votes
1
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32
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Numerical method to solve an 'almost' convolution integral
I'm trying to solve the following Volterra integral of the first kind for $w(x)$.
\begin{equation}
P(x)=\int_{-\infty}^{\infty}[d_1(x-x_0)+d_2(x)]w(x_0)dx_0
\end{equation}
My attempt so far: I'm aware ...
- 115
1
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0
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34
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What do I need to know for deriving the limits of sequences of discrete convolutions?
This is for a recreational math project which I hope to turn into a video.
If you start with a sequence $f_n$, finding the difference of that sequence $f_n-f_{n-1}$ is analogous to taking the ...
- 157
0
votes
0
answers
33
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Convolution of a Bessel function with a quadratic decay
I don't know if it is indeed doable but I am trying to compute analytically the convolution :
$$ f(\vec{r}) = \int \mathrm{d}\vec{r}_1 \cos(4\theta_1) \frac{K_0(\lambda \lVert \vec{r}-\vec{r}_1\rVert)}...
0
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1
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91
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Convolution of two uniform (solution with indicator function)
QUESTION
Find p.d.f of $Y=X_1+X_2$, where $X_1$and $X_2$ are two independent random variables $X_i$~Uniform(0,1), $i=1,2$
I read few question about convolution of two uniform in here. But I have ...
- 139
0
votes
1
answer
105
views
convolution of two exponential functions
I'm struggling with the convolution $f \ast g(x)$ where $$f(x)=e^{-x^2}$$ $$g(x)=e^{-5x^2}$$
I integrated $$\int_{-\infty}^{\infty} e^{-t^2}e^{-5(x-t)^2}\ dt = \int_{-\infty}^{\infty} e^{-5x^2+10tx-6t^...
- 23
0
votes
1
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149
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Evaluate the integral $\int e^{-it} \operatorname{sinc} t \,dt$
I need to find the integral
$$
\int_{-\infty}^x dt \, e^{-it}\text{sinc}(t)
$$
where
$$
\text{sinc}(t) = \frac{\text{sin}(t)}{t}.
$$
I need it to perform a convolution of the function $e^{-ix}\...
3
votes
1
answer
243
views
Solving $ y'' + a y = f(x) $ with zero initial conditions
Given the following initial value problem (IVP) $$ y'' + a y = f(x), \qquad y(0)= y'(0) = 0$$ show that $$y = \frac1a \int_{0}^{x}f(t)\sin(ax-at)\ {\rm d}t$$
My attempt
To solve the given initial ...
- 484
0
votes
1
answer
163
views
Tempered distribution by convolution with singular kernel
I am trying to learn more about convolutions with singularities and was hoping that someone could point me to a reference and perhaps check the following (i have it in my notes that this is from a MSE ...
- 505
1
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0
answers
106
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Why is convolution in the time domain equal to multiplication in the frequency domain? (intuitively, not mathematically)
I know the logic of how Fourier transforms work, and I know the logic of how convolution works, but I can't figure out why convolution in the time domain is equal to multiplication in the frequency ...
0
votes
1
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45
views
Reconstruct function as infinite sum of basis functions
I have some function $I(x)$ which I want to reconstruct as an infinite sum of 'simpler' basis functions $d(x)$. Each of these basis functions will have an (unknown) weighting $W(x)$ which I seek to ...
- 115
2
votes
1
answer
99
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Proper interpretation of convolution with product of functions
Suppose we have $\phi(x)\in C^\infty_c$ and:
$$\delta_a\{\phi\}=\int \delta(x-a)\phi(x)dx=\phi(a)$$
$$\delta_a*\phi=\phi(x-a)$$
The convolution leads to a shifting the input.
Suppose we multiply the ...
- 161
3
votes
1
answer
272
views
Continuity of Hilbert transform
Suppose $f : \mathbb{R} \to \mathbb{R}$, be a non-negative, bounded and continuous function, and its support is a compact interval in $\mathbb{R}$. Moreover, we have that $\int f(x) \, dx =1$. The ...
- 115