(a) Figure P7.40-1 shows the frequency response Ae(ejω) of a lowpass Type I Parks– McCle...

(a) Figure P7.40-1 shows the frequency response A_e(e^jω) of a lowpass Type I Parks– McClellan filter based on the following specifications. Consequently it satisfies the alternation theorem.

Passband edge: ω_p = 0.45π

Stopband edge: ω_s = 0.50π

Desired passband magnitude: 1

Desired stopband magnitude: 0

The weighting function used in both the passband and the stopband is W (ω) = 1.

What can you conclude about the maximum possible number of nonzero values in the impulse response of the filter?

(b) Figure P7.40-2 shows another frequency response B_e(e^jω) for a Type I FIR filter. B_e(e^jω) is obtained from A_e(e^jω) from part (a) as follows:

where k₁ and k= are constants. Observe that B_e(e^jω) displays equiripple behavior, with different maximum error in the passband and stopband.

Does this filter satisfy the alternation theorem with the passband and stopband edge frequencies indicated and with passband ripple and stopband ripple indicated by the dashed lines?