Problem $01.$ In DSB$-$SC$,$ the modulation and demodulation processes are shown in block diagrams in Figure $1.$ Assume that $\backslash ($ m$($t$)=10\backslash$cos $500$ t $\backslash )$ and $\backslash ($ c$($t$)=\backslash$cos $5000$ t $\backslash ).$

Figure $1.$ DSB$-$SC modulator and demodulator.

$($a$)($points $7)$ Derive the expressions in frequency$-$domain, and sketch the corresponding spectrum, of signals at points $\backslash (\backslash$mathbf$\{1,2,3,4,5,6\}\backslash ),$ and $\backslash (\backslash$mathbf$\{7\}\backslash ).$ Assume that the transmitted signal is received at the receiver with no channel effects.

$($b$)($points $2)$ Find the bandwidth of the modulating and modulated signals.

$($c$)($points $3)$ What modifications do the above DSB$-$SC circuit need to go through so that the resultant modified circuit will operate as an AM circuit? Draw the equivalent AM circuit.

$($d$)($points $3)$ Do you see any advantage of using AM over DSB$-$SC circuits?

$($e$)($points $2)$ Mention an application of AM over DSB$.$

$($f$)($points $3)$ Find the bandwidths of both AM and DSB modulated waves.