Figure P-1 depicts a pulse amplitude modulation(PAM) system. In general, p(t) is a periodi...

Figure P-1 depicts a pulse amplitude modulation(PAM) system. In general, p(t) is a periodic pulse signal with fundamental frequency ωp = 2π/Tp. Therefore, it can be represented as a Fourier series as shown in Fig. P-1. For this problem, we assume that p(t)is the periodic square wave shown in Fig. P-2.

(a) Assume that x(t)= cos(200πt). Make a plot over the interval 0 ≤ t ≤ 0.02 of the signals x(t), p(t), and xp(t)= p(t)x(t). Assume that ωp = 500π in making your plots of p(t)and xp(t). Observe that multiplication of x(t)by the periodic square wave has the effect of switching x(t)on and off periodically with period Tp = 1/250 = 0.004 seconds.

(b) Show that the Fourier transform of xp(t)= p(t )x (t )is

(c) For the inpu t x(t)with Fourier transform X(jω)depicted in Fig. P-3, make a sketch of Xp(jω)when ωp = 2π/Tp = 500π.

(d) If the frequency response of the LTI system is

and ωp = 2π/Tp = 500π, use the result of part (b) to determine the gain G and the cutoff frequency ωco so that x1(t)= x(t)

(e) Determine the minimumvalue of ωp and the corresponding values of G and ωco such that xr(t)= x(t).

(f) This system would work in essentially the same way if p(t)was changed to almost any periodic signal with fundamental frequency ωp. However, one important condition must be satisfied by the Fourier series coefficient a0. What is that condition?