$($b$)$ The block diagram in part $($a$)$ provides a nice implementation of a modulator because it may be easier to build a squarer than to build a multiplier. Based on the successful use of a squaring operation in the modulator, we decide to use the same squaring operation in the demodulator. Let

$x(t)=A_{c}m(t)\sqrt[\phantom{2}]{2}cos(2f_{c}t)$

where $m(t){}^{F-1}{?}^{F}M(f)$ is bandlimited to $B,$ i$.$e$.,|M(f)|=0$ for $|f|>B.$ Again, assume $f_{c}B$ Consider the block diagram shown in Figure $6\mathrm{.}3.$

Figure $6\mathrm{.}3$: Block diagram for Problem $2$b

Use

$H_{LP}(f)=\{\begin{array}{cc}1,& |f|B\\ 0,& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}\text{.}& \end{array}$

Find $y^{\text{I}}(t).$ Does this block diagram work as a demodulator; that is$,$ is $y^{\text{I}}(t)$ proportional to $m(t)?$