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

In that discussion, it was assumed that the signal was initially sampled with sampling frequency The bandpass sampling scheme is depicted in Figure 12.12. After we form a complex bandpass discrete-time signal s[n] with one-sided Fourier transform S(e^jω), the complex signal is decimated by a factor M, which is assumed to be the largest integer less than or equal to

(a) By carrying through an example such as the one depicted in Figure 12.13, show that if the quantity is not an integer for the initial sampling rate chosen, then the resulting decimated signal s_d[n] will have regions of nonzero length where its Fourier transform S_d(e^jω) is identically zero.

(b) How should the initial sampling frequency 2π/T be chosen so that a decimation factor M can be found such that the decimated sequence s_d[n] in the system of Figure 12.12 will have a Fourier transform S_d (e^jω) that is not aliased yet has no regions where it is zero over an interval of nonzero length?