In the system shown in Figure P7.45, the input Xc(t) is band limited with Xc(jw) = 0

Chapter 7, Problem 7.45

(choose chapter or problem)

In the system shown in Figure P7.45, the input Xc(t) is band limited with Xc(jw) = 0, lwl > 27T X 104 . The digital filter h[n] is described by the input-output relation HUw) l ~ Conversion x[n] = Xc(nT) y[n] Conversion Yp(t) _ill_ Yc(t) sequence to a h[n] impulse to an train ~ -~ 1T T T +oo p(t) = L o(t-nT) n = -x Figure P7 .45 II y[n] = T L x[k]. (P7.45-l) k= -'X (a) What is the maximum value of T allowed if aliasing is to be avoided in the transformation from Xc(t) to Xp(t). (b) With the discrete-time LTI system specified through eq. (P7.45-l), determine its impulse response h[n]. (c) Determine whether there is any value ofT for which ,!~ y[ n] = )~II! L x,.( T) d T. (P7 .45-2) If so, determine the maximum value. If not, explain and specify how T would be chosen so that the equality in eq. (P7.45-2) is best approximated. (Think carefully about this part; it is easy to jump to the wrong conclusion!)