Problem $2$:

$($a$)$ Derive the Fourier series coefficients $\backslash ($ a$\_\{$k$\}\backslash )$ and $\backslash ($ b$\_\{$k$\}\backslash )$ of a periodic square wave $\backslash ($ x$($t$)\backslash )($signal toggling between $-1$ and $+1).$

$($b$)$ Write a MATLAB script that computes the Fourier series coefficients of the square wave from a$),$ and that plots the Fourier series. Do not forget to set the figure title and label the axis using the MATLAB commands 'title', 'xlabel' and 'ylabel', respectively. As an additional requirement, the MATLAB script has to allow setting the following parameters:

$\backslash ($ N $\backslash ).$ Number of Fourier series coefficients to be computed

$\backslash ($ f$\_\{0\}\backslash )$ Frequency of the square wave $[$in Hz $]$

$\backslash ($ t $\backslash ).$ Vector defining discrete time steps at which the Fourier series is to be computed and plotted

$($c$)$ Vary the parameters $\backslash ($ N $\backslash )$ and $\backslash ($ f$\_\{0\}\backslash ).$ What effects can you observe?