$5\mathrm{.}7-1$ Consider a message signal that consists of three sinusoids

$$

$($

$$

$)$

$=$

$2$

$$

cos

$$

$($

$2$

$$

$$

$400$

$$

$)$

$+$

cos

$$

$($

$2$

$$

$$

$800$

$$

$)$

$$

$3$

$$

sin

$$

$($

$2$

$$

$$

$1200$

$$

$)$

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

$$

$=$

$2000$

$$Hz

for reconstruction, illustrate the difference in time domain between the reconstructed signal and the original signal. What happens if we select

$$

$=$

$1500$

$$Hz

instead?

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