Problem

In this problem, we consider a method of obtaining an implementation of a variable-cutof...

In this problem, we consider a method of obtaining an implementation of a variable-cutoff linear-phase filter. Assume that we are given a zero-phase filter designed by the Parks– McClellan method. The frequency response of this filter can be represented as

and its system function can therefore be represented as

with e^j^θ = Z. (We use Z for the original system and z for the system to be obtained by transformation of the original system.)

(a) Using the preceding expression for the system function, draw a block diagram or flow graph of an implementation of the system that utilizes multiplications by the coefficients a_k, additions, and elemental systems having system function (Z +Z −1)/2.

(b) What is the length of the impulse response of the system? The overall system can be made causal by cascading the system with a delay of L samples. Distribute this delay as unit delays so that all parts of the network will be causal.

(c) Suppose that we obtain a new system function from Ae(Z) by the substitution

Using the flow graph obtained in part (a), draw the flow graph of a system that implements the system function Be(z).What is the length of the impulse response of this system? Modify the network as in part (b) to make the overall system and all parts of the network causal.

(d) If Ae(e^j^θ ) is the frequency response of the original filter and Be(e^j^ω) is the frequency response of the transformed filter, determine the relationship between θ and ω.

(e) The frequency response of the original optimal filter is shown in Figure P7.59. For the case α1 = 1 − α0 and 0 ≤ α0 < 1, describe how the frequency response Be(e^j^ω) changes as α₀ varies. Hint: Plot Ae(e^j^θ ) and Be(e^j^ω) as functions of cos θ and cos ω. Are the resulting transformed filters also optimal in the sense of having the minimum maximum weighted approximation errors in the transformed passband and stopband?