Consider the analysis–synthesis system shown in Figure P4.59-1. The lowpass filter h0[n]...

Consider the analysis–synthesis system shown in Figure P4.59-1. The lowpass filter h₀[n] is identical in the analyzer and synthesizer, and the highpass filter h₁[n] is identical in the analyzer and synthesizer. The Fourier transforms of h₀[n] and h₁[n] are related by

(a) If X(e^jω) and H₀(e^jω) are as shown in Figure P4.59-2, sketch (to within a scale factor) X₀(e^jω), G₀(e^jω), and Y₀(e^jω).

(b) Write a general expression for G₀(e^jω) in terms of X(e^jω) and H₀(e^jω). Do not assume that X(e^jω) and H₀(e^jω) are as shown in Figure P4.59-2.

(c) Determine a set of conditions on H₀(e^jω) that is as general as possible and that will guarantee that |Y(e^jω)| is proportional to |X(e^jω)| for any stable input x[n].

Note: Analyzer–synthesizer filter banks of the form developed in this problem are very similar to quadrature mirror filter banks. (For further reading, see Crochiere and Rabiner (1983), pp. 378–392.)