8.69. Let x[n] be an N-point sequence such that x[n] = 0 for n < 0 and for n >...

8.69. Let x[n] be an N-point sequence such that x[n] = 0 for n < 0 and for n > N − 1. Let xˆ [n] be the 2N-point sequence obtained by repeating x[n]; i.e.,

Consider the implementation of a discrete-time filter shown in Figure P8.69. This system has an impulse response h[n] that is 2N points long; i.e., h[n] = 0 for n < 0 and for n > 2N − 1.

(a) In Figure P8.69-1, what is ˆX [k], the 2N-pointDFT of xˆ [n], in terms ofX [k], theN-point DFT of x[n]?

(b) The system shown in Figure P8.69-1 can be implemented using only N-point DFTs as indicated in Figure P8.69-2 for appropriate choices for System A and System B. Specify System A and System B so that yˆ [n] in Figure P8.69-1 and y[n] in Figure P8.69-2 are equal for 0 ≤ n ≤ 2N − 1. Note that h[n] and y[n] in Figure P8.69-2 are 2N-point sequences and w[n] and g[n] are N-point sequences.