Q $1)$

Consider a message signal $m(t)$ with the spectrum shown in Figure P$2\mathrm{.}8.$ The message bandwidth $W=2kHz.$ This signal is applied to a product modulator, together with a carrier wave $A_{c}cos(2f_{c}t),$ producing the DSB$-$SC modulated signal $s(t).$ The modulated signal is next applied to a coherent detector. Assuming perfect synchronism between the carrier waves in the modulator and detector, determine the spectrum of the detector output when: $($a$)$ the carrier frequency $f_c=12\mathrm{.}5kHz$ and $($b$)$ the carrier frequency $f_c=7\mathrm{.}5kHz.$ What is the lowest carrier frequency for which each component of the modulated signal $s(t)$ is uniquely determined by $m(t)?$

Q $2)$

The AM signal

$s(t)=A_c[1+m^{5n}(t)]cos(2f_{c}t)$

is applied to the system shown in Figure P$2\mathrm{.}7.$ Assuming that $|k_{d}m(t)|<mn>1</mn>$ for all $t$ and the message signal $m(t)$ is limited to the interval $-WfW$ and that the carriet frequency $f_e>2W$ show that $m(t)$ can be obtained from the square$-$rooter output $v_3(t).$

Q$3)$

Figure P$2\mathrm{.}2$ shows the circuit diagram of a square$-$law modulator. The signal applied to the nonlinear device is relatively weak, such that it can be represented by a square law:

$($:$v_2(t)=a_1{v}_1(t)+a_2{v}_1^{3}t$