The system shown in Figure P10.29 is proposed as a spectrum analyzer. The basic operatio...

The system shown in Figure P10.29 is proposed as a spectrum analyzer. The basic operation is as follows: The spectrum of the sampled input is frequency-shifted; the lowpass filter selects the lowpass band of frequencies; the downsampler “spreads” the selected frequency band back over the entire range −π < ω <π; and the DFT samples that frequency band uniformly at N frequencies.

Assume that the input is bandlimited so that = 0 for ≥ π/T . The LTI system with frequency response H(e^jω) is an ideal lowpass filter with gain of one and cutoff frequency π/M. Furthermore, assume that 0 < ω₁ < π and the data window w[n] is a rectangular window of length N.

(a) Plot the DTFTs, X(e^jω), Y(e^jω), R(e^jω), and R_d (e^jω) for the given and for ω₁ = π/2 and M = 4. Give the relationship between the input and output Fourier transforms for each stage of the process; e.g., in the fourth plot, you would indicate

R(e ^{j ω} ) = H(e^jω)Y (e^jω).

(b) Using your result in part (a), generalize to determine the band of continuous-time frequencies in that falls within the passband of the lowpass discrete-time filter. Your answer will depend on M, ω₁ and T. For the specific case of ω₁ = π/2 and

M = 4, indicate this band of frequencies on the plot of given for part (a).

(c) (i) What continuous-time frequencies in are associated with the DFT values V [k] for 0 ≤ k ≤ N/2?

(ii) What continuous-time frequencies in do the values for N/2 < k ≤ N −1 correspond to? In each case, give a formula for the frequencies .