The ideal Hilbert transformer (90-degree phase shifter) has frequency response (over one...

The ideal Hilbert transformer (90-degree phase shifter) has frequency response (over one period

Figure P12.28-1 shows H(e^jω), and Figure P12.28-2 shows the frequency response of an ideal lowpass filter H_lp(e^jω) with cutoff frequency ω_c = π/2. These frequency responses are clearly similar, each having discontinuities separated by π.

(a) Obtain a relationship that expresses H(e^jω) in terms of H_lp(e^jω). Solve this equation for H_lp(e^jω) in terms of H(e^jω).

(b) Use the relationships in part (a) to obtain expressions for h[n] in terms of h_lp[n] and for h_lp[n] in terms of h[n].

The relationships obtained in parts (a) and (b) were based on definitions of ideal systems with zero phase. However, similar relationships hold for nonideal systems with generalized linear phase.

(c) Use the results of part (b) to obtain a relationship between the impulse response of a causal FIR approximation to the Hilbert transformer and the impulse response of a causal FIR approximation to the lowpass filter, both of which are designed by (1) incorporating an appropriate linear phase, (2) determining the corresponding ideal impulse response, and (3) multiplying by the same window of length (M + 1) samples, i.e., by the window method discussed in Chapter 7. (If necessary, consider

the cases of M even and M odd separately.) (d) For the Hilbert transformer approximations of Example 12.4, sketch the magnitude of the frequency responses of the corresponding lowpass filters.