$1.($a$)$ A double sideband suppressed carrier amplitude modulation $($DSBSC$-$AM$)$ system is used to transmit a message signal $\backslash ($ m$($t$)=3\backslash$sin $\backslash$left$(200\backslash$pi t$+\backslash$frac$\{\backslash$pi$\}\{4\}\backslash$right$)+4\backslash$cos $(200\backslash$pi t$+\backslash )\backslash (\backslash$frac$\{\backslash$pi$\}\{4\}\backslash ))$ using a cosine carrier signal with amplitude $5$ Volts and frequency of $5$ kHz with zero phase.

$($i$)$ Express the message signal $\backslash ($ m$($t$)\backslash )$ in a single sine form.

$($ii$)$ Determine the time and frequency domain expressions of DSBSC$-$AM signal and sketch clearly its amplitude spectrum with proper labelling of amplitude and frequency.

$(13$ Marks$)$

Note: Question No$.1$ continues on page $2.$

IE$3012$

$($b$)$ The suppressed carrier AM signal in part $($a$)$ is coherently demodulated using a product modulator with the local oscillator of frequency $\backslash ($ f$\_\{0\}\backslash ).$ This is followed by an ideal low pass filter $($LPF$)$ to get the demodulated output.

$($i$)$ Sketch clearly the amplitude spectrum of the signal before and after LPF$.$

$($ii$)$ Determine LPF bandwidth required for the demodulator output to contain as little distortion as possible.

$($iii$)$ How much distortion occurs at the demodulated output?