In Section 12.4.3, we discussed an efficient scheme for sampling a bandpass continuous-t...

In Section 12.4.3, we discussed an efficient scheme for sampling a bandpass continuous-time signal with Fourier transform such that

The bandpass sampling scheme is depicted in Figure 12.12. At the end of the section, a scheme for reconstructing the original sampled signal s_r [n] was given. The original continuous-time signal s_c(t) in Figure 12.12 can, of course, be reconstructed from s_r [n] by ideal bandlimited interpolation (ideal D/C conversion). Figure P12.34-1 shows a block diagram of the system for reconstructing a real continuous-time bandpass signal from a decimated complex signal. The complex bandpass filter H_i (e^jω) in the figure has a frequency response given by Eq. (12.79).

(a) Using the example depicted in Figure 12.13, show that the system of Figure P12.34-1 will reconstruct the original real bandpass signal (i.e., y_c(t) = s_c(t)) if the inputs to the reconstruction system are y_rd[n] = s_rd[n] and y_id[n] = s_id[n].

(b) Determine the impulse response h_i [n] = h_ri[n] + jh_ii[n] of the complex bandpass filter in Figure P12.34-1.

(c) Draw a more detailed block diagram of the system of Figure P12.34-1 in which only real operations are shown. Eliminate any parts of the diagram that are not necessary to compute the final output.

(d) Now consider placing a complex LTI system between the system of Figure 12.12 and the system of Figure P12.34-1.This is depicted in Figure P12.34-2, where the frequency response of the system is denoted H(e^jω). Determine how H(e^jω) should be chosen if it is desired that