In Section 8.8 we considered synchronous discrete-time modulation and demodulation with

Chapter 8, Problem 8.47

(choose chapter or problem)

In Section 8.8 we considered synchronous discrete-time modulation and demodulation with a sinusoidal carrier. In this problem we want to consider the effect of a loss in synchronization in phase and/or frequency. The modulation and demodulation systems are shown in Figure P8.47(a), where both a phase and frequency difference between the modulator and demodulator carriers is indicated. Let the frequency difference w d - w c be denoted as ~w and the phase difference () d - () c as ~e. (a) If the spectrum of x[n] is that shown in Figure P8.47(b), sketch the spectrum of w[n], assuming ~w = 0. (b) If ~w = 0, show that w can be chosen so that the output r[ n] is r[ n] = x[n] cos~(). In particular, what is r[n] if~() = n/2? (c) For~() = 0, and w = WM + ~w, show that the output r[n] = x[n] cos[~wn] (assume that ~w is small).