A zero-phase FIR filter h[n] has associated DTFT H(ejω), shown in Figure P7.34....

A zero-phase FIR filter h[n] has associated DTFT H(e^jω), shown in Figure P7.34.

The filter is known to have been designed using the Parks–McClellan (PM) algorithm. The

input parameters to the PM algorithm are known to have been:

• Passband edge: ω_p = 0.4π

• Stopband edge: ω_s = 0.6π

• Ideal passband gain: G_p = 1

• Ideal stopband gain: G_s = 0

• Error weighting function W(ω) = 1

The length of the impulse response h[n], is M + 1 = 2L + 1 and

h[n] = 0 for |n| > L.

The value of L is not known.

It is claimed that there are two filters, each with frequency response identical to that shown in Figure P7.34, and each having been designed by the Parks–McClellan algorithm with different values for the input parameter L.

• Filter 1: L = L₁

• Filter 2: L = L₂ > L₁.

Both filters were designed using exactly the same Parks–McClellan algorithm and input parameters, except for the value of L.

(a) What are possible values for L₁?

(b) What are possible values for L₂ > L₁?

(c) Are the impulse responses h₁[n] and h₂[n] of the two filters identical?

(d) The alternation theorem guarantees “uniqueness of the r^th-order polynomial.” If your answer to (c) is yes, explain why the alternation theorem is not violated. If your answer is no, show how the two filters, h₁[n] and h₂[n], are related.