Question $5$

$($a$)$ Show that the DSB$-$SC receiver, shown in Fig. $3,$ has a post$-$detection signal$-$to$-$noise ratio given

by

$SN{R}_{\text{p}\text{o}\text{s}\text{t}}=\frac{P_m}{2B}co{s}^2$

$,$ where $P_m$ is the modulating signal power.

Fig. $3.$ DSB$-$SC receiver.

$($b$)$ A white noise voltage of power spectral density $\frac{}{2}$ is input to an integrator having transfer

function $H(f)=\frac{j}{2f}[exp(-j2fT)-1].T$ is the integration time interval and $$ is the

integrator time constant. Show that the normalised power of the integrator output noise voltage

is given by $N=\frac{T}{2{}^2}.$

Note: ${\displaystyle \underset{0}{\overset{}{}}}(sin\frac{x}{x}{)}^{2}dx=\frac{}{2}$