(Solved): In this exercise, you will simulate the Continuous-Time Fourier Series (CTFS) of continuous-time pe ...

In this exercise, you will simulate the Continuous-Time Fourier Series (CTFS) of continuous-time periodic signals in Matlab using its symbolic toolbox. BACKGROUND: For $T=\frac{2?}{w_0}$ periodic continuous-time signals, the CTFS synthesis and analysis equations are given respectively as below. $x(t)={?}_{k=??}^{?}{a}_k{e}^{jk{w}_{0}t}{a}_k=\frac{1}{T}{?}_{T}x(t)e^{?jk{w}_{0}t}dt$ CTFS representation converges for all periodic signals which have finite energy over a single period. The energy of the approximation error approaches to zero as the number of used harmonically related complex exponentials is increased. It can be shown that by using a finite number of CTFS coefficients, synthesis gives the best (in mean-squared error (MSE) sense) approximation to the original signal. EXERCISE: 1. For the below continuous-time periodic signal with fundamental period $T=2$, calculate and plot the CTFS coefficients $?a_k?$ (magnitude spectrum) for $k=?2:2$ against angular frequency $\left(w_0\right)$ using the given symbolic CTFS simulation below (as a symbolic function of $t$ ). $x(t)=e^{??t?}\text{ for }?1?t?1$ syms $t$ of define the variable $t$ as symbolic $\mathrm{x}=\exp (?\mathrm{abs}(t))$; of define the signal $\mathrm{x}$ figure, fplot $\left(x,\left[\begin{array}{ll}?1& 1\end{array}\right]\right)$; \% ctfs analysis - CTES analysis equation $T=2;?$ ? fundamental period $\mathrm{K}=?2:2$; ? all values of integer $\mathrm{k}$ for $k=1:$ length $(K)$ of CTFS analysis $ak(k)=\mathrm{int}\left(x^{?}\exp (?j?2?pi?K(k)?t/T),t,?T/2,T/2\right)/T$; $w(k)=K(k)?2?pi/T;$ \% angular frequency end Apply above procedure for $k=?10:10$. Compare and comment on the quality of approximation signals. Repeat the same procedure for periodic $x(t)=\mathrm{sinc}(t),?5?t?5,T=10$. 2. Next, simulate the CTFS synthesis equation to reconstruct an approximation to the periodic continuous-time signal $x(t)$ using using the 5 CTFS coefficients calculated above. otfs synthesis - CTFS synthesis equation xr $=0;$ o initialize approximation signal for $\mathrm{k}=1:$ length(K) \% YOUR CODE HERE end figure, fplot $\left(\mathrm{xr},\left[\begin{array}{ll}?5& 5\end{array}\right]\right);$ q $c$ ctfs_synthesis - CTFS synthesis equation $\mathrm{xr}=0;$ o initialize approximation signal for $k=1:$ length $(K)$ \% YOUR CODE HERE end figure, fplot (xr, [-5 5]) ; 3. Apply above procedure for $k=?10:10$. Compare and comment on the quality of approximation signals. 4. Repeat the same procedure for periodic $x(t)=\sin c(t),?5?t?5,T=10$.