We are given an FIR bandpass filter h[n] that is zero phase, i.e., h[n] = h[−n]. Its ass...

We are given an FIR bandpass filter h[n] that is zero phase, i.e., h[n] = h[−n]. Its associated DTFT H(e^jω) is shown in Figure P7.35.

The filter is known to have been designed using the Parks–McClellan algorithm. The input parameters to the Parks–McClellan algorithm are known to have been:

The value of the input parameter M+1, which represents the maximum number of nonzero impulse response values (equivalently the filter length), is not known.

It is claimed that there are two filters, each with a frequency response identical to that shown in Figure P7.35, but having different impulse response lengths M + 1 = 2L + 1.

• Filter 1: M = M₁ = 14

• Filter 2: M = M₂ ≠ M₁

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

(a) What are possible values for M₂?

(b) The alternation theorem guarantees “uniqueness of the rth-order polynomial.” Explain why the alternation theorem is not violated.