Q1)
Consider a message signal $m(t)$ with the spectrum shown in Figure P2.8. The message bandwidth $W = 2$ kHz. This signal is applied to a product modulator, together with a carrier wave $A_c \cos(2\pi f_c t)$, 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 $f_c = 12.5$ kHz and (b) the carrier frequency $f_c = 7.5$ kHz. What is the lowest carrier frequency for which each component of the modulated signal $s(t)$ is uniquely determined by $m(t)$?
Q2)
The AM signal
$s(t) = A_c[1 + m_m(t)] \cos(2\pi f_c t)$
is applied to the system shown in Figure P2.7. Assuming that $|k_m m(t)| < 1$ for all $t$ and the message signal $m(t)$ is limited to the interval $-W \le f \le W$ and that the carrier frequency $f_c > 2W$ show that $m(t)$ can be obtained from the square-rooter output $v_o(t)$.
Q3)
Figure P2.2 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:
$v_o(t) = a_1 v_i(t) + a_2 v_i^2(t)$
where $a_1$ and $a_2$ are constants, $v_i(t)$ is the input voltage, and $v_o(t)$ is the output voltage. The input voltage is defined by
$v_i(t) = A_c \cos(2\pi f_c t) + m(t)$
where $m(t)$ is a message signal and $A_c \cos(2\pi f_c t)$ is the carrier wave.
(a) Evaluate the output voltage $v_o(t)$.
(b) Specify the frequency response that the tuned circuit in Figure P2.2 must satisfy in order to generate an AM signal with $f_c$ as the carrier frequency.
(c) What is the amplitude sensitivity of this AM signal?