IE$2110$

$4.$ Consider an ideal lowpass filter with a cutoff frequency of $1\mathrm{.}2$ Hz $.$ The input to this filter is the periodic square wave shown in Figure $1.$

$($a$)$ Find the dc value of the square wave in Figure $1.$

$(3$ Marks$)$

$($b$)$ Find the complex$-$exponential Fourier series coefficients of the fundamental frequency component of the square wave in Figure $1,$ given that

$c_n=\frac{1}{T_0}{\displaystyle \underset{T_0}{\overset{\phantom{}}{}}}x(t)e^{-jn{}_{0}t}dt,$

where ${}_0=\frac{2}{T_0}=2f_0,$ and $f_0$ is the fundamental frequency of the square wave.

$(7$ Marks$)$

$($c$)$ Find the output of the filter $y(t)$ and sketch its two$-$sided magnitude spectrum.

$(6$ Marks$)$

$($d$)$ Is it possible to sample $x(t)$ and $y(t)$ without aliasing? Explain your answers. What is the Nyquist rate for each case?

$(5$ Marks$)$

$($e$)$ Consider $z(t)=x(t)+y(t).$ Explain why Fourier series exists for $z(t)$ and hence find the trigonometric Fourier series coefficients of its fundamental frequency component.

$(4$ Marks$)$

$4$