In this problem, we develop a technique for designing discrete-time filters with minimum...

In this problem, we develop a technique for designing discrete-time filters with minimum phase. Such filters have all their poles and zeros inside (or on) the unit circle. (We will allow zeros on the unit circle.) Let us first consider the problem of converting a type I linearphase FIR equiripple lowpass filter to a minimum-phase system. If H(e^jω) is the frequency response of a type I linear-phase filter, then

1. The corresponding impulse response

is real and M is an even integer.

2. It follows from part 1 that H(e^jω) = A_e(e^jω)e ^−jωn0 , where A_e(e^jω) is real and n₀ = M/2 is an integer.

3. The passband ripple is δ1; 6; i.e., in the passband, A_e(e^jω) oscillates between (1 + δ₁) and (1 − δ₁). (See Figure P7.57-1.)

4. The stopband ripple is δ₂; i.e., in the stopband, −δ₂ ≤ A_e(e^jω) ≤ δ₂, and A_e(e^jω) oscillates between −δ₂ and +δ₂. (See Figure P7.57-1.)

The following technique was proposed by Herrmann and Schüssler (1970a) for converting this linear-phase system into a minimum-phase system that has a system function H_min(z) and unit sample response h_min[n] (in this problem, we assume that minimum-phase systems can have zeros on the unit circle):

Step 1. Create a new sequence

Step 2. Recognize that H₁(z) can be expressed in the form

for some H₂(z), where H₂(z) has all its poles and zeros inside or on the unit circle and h₂[n] is real.

Step 3. Define

The denominator constant where normalizes the passband so that the resulting frequency response H_min(e^jω) will oscillate about a value of unity.

(a) Show that if h₁[n] is chosen as in Step 1, then H₁(e^jω) can be written as

where H₃(e^jω) is real and nonnegative for all values of ω.

(b) If H₃(e^jω) ≥ 0, as was shown in part (a), show that there exists an H₂(z) such that

where H₂(z) is a minimum-phase system function and h₂[n] is real (i.e., justify Step 2).

(c) Demonstrate that the new filter H_min(e^jω) is an equiripple lowpass filter (i.e., that its magnitude characteristic is of the form shown in Figure P7.57-2) by evaluating δ’₁ and δ’₂.What is the length of the new impulse response h_min[n]?

(d) In parts (a), (b), and (c), we assumed that we started with a type I FIR linear-phase filter. Will this technique work if we remove the linear-phase constraint? Will it work if we use a type II FIR linear-phase system?