$5\mathrm{.}8-1$ Consider a message signal that consists of three sinusoids m$($t$)=2$cos$(2$ $400$t$)+$cos$(2$ $800$t$)3$sin$(2$ $1200$t$)$

that lasts $0\mathrm{.}2$ second.

$($a$)$ Using a sampling frequency of $4000$ Hz$,$ illustrate the spectrum of the ideally sampled signal and compare with the spectrum of the original signal m$($t$).$

$($b$)$ Using an ideal LPF of bandwidth B$=2000$ Hz for reconstruction, illustrate the difference in time domain between the reconstructed signal and the original signal. What happens if we select B$=1500$ Hz instead?

$($c$)$ If we use a simple rectangular pulse of width Ts $($t$/$Ts$)$ to reconstruct the original signal from the quantized samples, design and implement the equalizer filter to recover the message as discussed in Sec. $5\mathrm{.}1\mathrm{.}2.$