Let $x(t)$ and $y(t)$ be two independent baseband information signals of bandwidth $W.$

As depicted in Figure $3($a$),$ there are two users that transmit $x(t)$ and $y(t)$ signals, respectively, by using certain modulators, in which the cut$-$off frequency of the ideal LPF is $f_c,$ and the center frequency and the bandwidth of the ideal HPF are $f_c$ and $2W,$ respectively.

Further, $r(t)$ denotes the total received signal at the receiver, and it is given by $r(t)=x_c(t)+y_c(t).$

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Answer the following questions.

a$)$ Determine the modulation type of the signal $x_c(t),$ and obtain its time$-$domain and frequency domain expressions.

b$)$ Determine the modulation type of the signal $y_c(t),$ and obtain its time$-$domain and frequency domain expressions.

c$)$ If the frequency spectra $x(f)$ and $Y(f)$ of the signals $x(t)$ and $y(t)$ are shown, respectively, in Figure $3($b$).$ Find the time$-$domain expressions of $x(t)$ and $y(t),$ and determine the time$-$ and frequency$-$domain expressions of the signal $r(t).$ Plot the frequency spectrum of $r(t).$

Figure $3($b$).$ The frequency spectra of the signals $x(t)$ and $y(t),$ respectively.

d$)$ Draw the block diagram of the demodulator that will recover $x(t)$ and $y(t)$ at the receiver side. Explain briefly how the demodulation process takes place.