Prove the downsampling equation (12.8.1) by using the property X’(f)= D(f)X(f) where D(f ) is the…

Prove the downsampling equation (12.8.1) by using the property X’(f)= D(f)X(f) where D(f ) is the ideal interpolator defined by Eq. (12.2.24), and using the results of Problem 12.11. Why can’t we write X(f )= X’(f )/D(f )?

Problem 12.11

Consider the sampling of an analog signal xa(t) at the two sampling rates fs and fs’ = Lfs. The corresponding signal samples are x(n)= xa(nT) and x (n’)= xa(n’T’). Because T = LT’ , it follows that x(n) will be the downsampled version of x’(n’) in the sense of Eq. (12.5.1), that is, x(n)= xa(nT)= xa(nLT’)= x’(nL). The spectra of x(n) and x’(n’) are given by the Poisson summation formulas:

Using the change of variables k = k’L+m, where m = 0, 1, . . . , L−1, show that the spectrum of the downsampled signal is given by the discrete-time version of the Poisson summation formula:

Why is the factor L needed? Show that the same equation can be expressed in terms of the normalized digital frequencies ω = 2πf /fs and ω’ = 2πf /fs ‘as