I want to find the formula for the Fourier series $f_T(t)$ of a particular rectangular pulse train with the following properties:
- Period = $P$, a positive integer
- Amplitude $A = 1$
- Pulse width $\omega = 1$
- Function is odd, with
$$f_T(t)=\begin{cases} 1 & 0\leq t\leq 1 \\ 0 & \text{else} \end{cases}, 0\leq t\leq P$$
I have watched a zillion how-tos and read many pages, but I just can't get the hang of it, which is made harder by the function being odd. I'd love to be able to play about with the resulting function. Can anyone show how to construct it and what the final result is?
(Ideally in exponential form, but trig is fine too.)
Assuming $f_T(t)$ is defined as
$$f_T(t)=\left\{\begin{array}{cc} 1 & 0<t\leq 1 \\ 0 & 1<t\leq P \\ f_T(t \bmod P) & \text{True} \\ \end{array}\right.\tag{1}$$
where $P$ is a positive integer greater than $1$ the Fourier series for $f_T(t)$ is:
$$\tilde{f}_T(t)=\frac{1}{P}+\frac{1}{\pi}\sum\limits_{k=1}^K \frac{1}{k}\left(\sin\left(\frac{2 \pi k}{P}\right) \cos\left(\frac{2 \pi k t}{P}\right)+2 \sin^2\left(\frac{\pi k}{P}\right) \sin\left(\frac{2 \pi k t}{P}\right)\right)\tag{2}$$
The following three figures illustrate the Fourier series $\tilde{f}_T(t)$ defined in formula (2) in orange overlaid on the blue reference function $f_T(t)$ defined in formula (1) for $P\in\{2,3,4\}$.
Figure (1): Illustration of formula (2) for $\tilde{f}_T(t)$ evaluated at $P=2$ and $K=8$ (orange)
Figure (2): Illustration of formula (2) for $\tilde{f}_T(t)$ evaluated at $P=3$ and $K=12$ (orange)
Figure (3): Illustration of formula (2) for $\tilde{f}_T(t)$ evaluated at $P=4$ and $K=16$ (orange)