Figure P8.24 shows a system to be used for sinusoidal amplitude modulation, where x(t)

Chapter 8, Problem 8.24

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Figure P8.24 shows a system to be used for sinusoidal amplitude modulation, where x(t) is band limited with maximum frequency WM, so that X(jw) = 0, lwl > WM. As indicated, the signal s(t) is a periodic impulse train with period T and with an offset from t = 0 of d. The system H(jw) is a bandpass filter. (a) With Li = 0, WM = 7ri2T, w1 = 7r!T, and wh = 37r/T, show that y(t) is proportional to x(t) cos wet, where We = 27r!T. (b) If WM, w1, and wh are the same as given in part (a), but dis not necessarily zero, show that y(t) is proportional to x(t) cos(wct + Oc), and determine We and 0 c as a function of T and d. (c) Determine the maximum allowable value of WM relative toT such that y(t) is proportional to x(t) cos( wet + 0 c).