All Questions
Tagged with convolution dirac-delta
70
questions
1
vote
1
answer
76
views
Proof review Tempered Distributions: $(T*\varphi)^\wedge = (2\pi)^{n/2} \hat{\varphi}\hat{T}$ and computing $(x^\alpha)^\wedge$
I am learning about tempered distributions on schwartz space. But there are 2 questions in this text where Im unsatisfied with my solution... I ask kindly for your review of my solutions below, and if ...
- 3,266
0
votes
0
answers
34
views
Trouble visualizing a graph of the sum of infinite Dirac functions
I'm currently trying to solve a problem on my "Signals and Systems" class and I'm stumped.The question has 2 parts:
1-) The function T.I can be defined as:
$$
T\cdot I(t) = \sum_{n=-\infty}^{...
0
votes
0
answers
21
views
Convolution of f(-x) with delta
i'm struggling with the following convolution:
$\ f(-x,-y)* \delta(x,y-a)$
I don't know if it should be solved that way:
$\ f(-x,-(y-a))=f(-x,-y+a) $
or: (just the shifting)
$\ f(-x, -y-a) $
- 41
4
votes
1
answer
106
views
Is the convolution of the Heaviside distribution with itself well defined?
During a lecture, we were told that for two distributions $S,T \in \mathcal{D'}(\mathbb{R})$ the convolution $S* T$ is defined if either $S$ or $T$ had a compact support, or, if both of them were ...
- 197
0
votes
0
answers
127
views
Convolution of shifted function with Delta Dirac Function
We know that $f(t)*δ(t-a)=f(t-a)$. How about $f(t-a)*δ(t-a)$? (where * is convolution and $δ(t)$ is dirac delta function)
I think it's $f(t-a)*δ(t-a) = f(t-a)$ because convolution with delta function ...
0
votes
1
answer
76
views
Asymptotic expansion of a convolution integral
Suppose I have a convolution integral
$$\tilde{F}(x; g)=\int dy F(x-y) K(y;g),$$
with a kernel
$$K(x;g)\equiv\frac{1}{\pi}\frac{g}{x^2+g^2}.$$
The result of the integral depends on the parameter $g$ ...
- 391
2
votes
1
answer
202
views
Convolution of discrete measures
Exercise (Billingsley's Probability and Measure (3e), p.271, Exr.20.13): Given discrete measures, $\mu$ and $\nu$, consisting of masses $\alpha_n$ and $\beta_n$ for $n = 0, 1,2,\ldots$ Show that $\mu *...
- 43
1
vote
1
answer
167
views
What is the convolution of $1/t$ and $\operatorname{rect}(t)$?
I'm interested in calculating:
$$ \mathscr{F}\{\theta \operatorname{sinc}(x)\}$$
Where $\theta$ is the Heaviside function. I attempted to solve that by transforming it into:
$$ \mathscr{F}\{\theta\}*\...
- 596
0
votes
0
answers
258
views
Convolution between a sinc and a Dirac comb
I am not an expert on distributions, but I would like to know if there is a meaningful way to define the convolution
$$ sinc * P_T $$
between $ sinc(x)=\frac{sin(x)}{x} $ and $ P_T = \sum_{k=-\infty}^{...
- 1,072
1
vote
2
answers
93
views
Is my derivation of the derivative of a product of the Heaviside function and a function correct? (A follow-up question)
That's a follow-up question to the accepted answer to this post.
After some thinking, I ended up deriving the derivative differently. I'm wondering if the dear Math stack exchange community can tell ...
0
votes
0
answers
51
views
Simplifying Expression with Infinite Sum of Delta Functions convolved with Exponential Function
I need some help with simplifying the following expression:
$$\sum_{n= -\infty }^\infty \delta(t-2n) \ast e^{-2t}u(t)$$
Using the convolution integral and taking the unit step into account, I get:
$$\...
- 1
0
votes
0
answers
105
views
Is the diract delta function a unique identity under convolution?
I just thought about how amazing of a coincidence it is that this limit object as an offshoot of the Weierstrass approximation theorem and proofs of Fourier series ended up also being the identity ...
- 579
0
votes
1
answer
275
views
Using Inverse Laplace to find the frequency response of a transfer function - Help needed!
The frequency response is the inverse Laplace transform of a transfer function. I am tasked to apply the inverse Laplace on the transfer function below in order to convert it into the time domain.
$$...
1
vote
0
answers
47
views
Convolution of converging measure.
For any $N>0$, let $(x_{i,N})_{1\leq i \leq N}$ be a set of points in the compact $K \subset \mathbb{R}^d$. To lighten the notations, we will note $x_{i}=x_{i,N}$.
Assume that the following weak ...
- 2,190
1
vote
0
answers
53
views
Fourier transform and convolution theorem
I am trying to convert the spectral density $S_{XX}$ of an arbitrary time-dependent signal $X(t)$ into the Fourier domain. In particular, I have the following definitions for Fourier/Inverse Fourier ...
- 333