0
votes
0answers
13 views

What is the formula for single frequency generation function obtained from FFT?

What is the correct formula of a function that generates specific tone from fourier transform? I thought that having: transformata - an array with FFT of a source sample. v = transformata[freq] - ...
1
vote
1answer
55 views

Fourier Series Approximations of Functions

From a few examples of smooth functions, discontinuous functions and continuous functions which have a 'kink' (i.e. $|x|$ where left and right limits disagree)... I've seen that the fourier series ...
-1
votes
1answer
29 views

How to extend a given function to an odd function with period 2 (Fourier Series)

This is the question. f(x) = 7x^2 + 4 I want to know how to extend the above fuction to become a odd function with period 2.
2
votes
2answers
59 views

Fourier Series of what appears to be a sawtooth series

Find the Fourier series of \begin{equation} f(x)=\begin{cases} x-[x] \quad &\text{if $x$ is not an integer} \\ \frac{1}{2} \quad &\text{if $x$ is an integer} \end{cases} \end{equation} ...
2
votes
0answers
44 views

Representing series $f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$ as a Dirac comb function.

Consider the function $$f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$$ where $\omega_n= \sqrt{(\frac{n \pi c}{l})^2-(\frac{r_0}{2})^2}.$ If we neglect the term ...
1
vote
0answers
35 views

Relations between complex functions satisfying a specific condition

What is the relation between the following two complex functions: $$g(\theta)=\sum_n x[n]\ y[n]\ e^{in\theta}$$ and $$f(\theta)=\sum_n \left(x[n]\pm i\sqrt{1-x[n]^2}\right)\ y[n]\ e^{in\theta}$$ ...
6
votes
0answers
245 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
9
votes
1answer
190 views

Fourier Series involving the Jacobi Symbol

We know that the Fourier Series $$s(x)=\sum_{k\neq0}\frac{1}{k}\exp\left(2\pi ik x\right)$$ corresponds to the sawtooth function, $s(x)=\left\{x\right\} -\frac{1}{2}$. Suppose that ...
7
votes
4answers
316 views

The leap to infinite dimensions

Extending this question, page 447 of Gilbert Strang's Algebra book says What does it mean for a vector to have infinitely many components? There are two different answers, both good: 1) The ...