Consider a type I linear-phase FIR lowpass filter with impulse response hLP[n] of length...

Consider a type I linear-phase FIR lowpass filter with impulse response h_LP[n] of length (M + 1) and frequency response

The system has the amplitude function A_e(e^jω) shown in Figure P7.46.

This amplitude function is the optimal (in the Parks–McClellan sense) approximation to unity in the band 0 ≤ ω ≤ ω_p, where ω_p = 0.27π, and the optimal approximation to zero in the band ω_s ≤ ω ≤ π, wherein ω_s = 0.4π.

(a) What is the value of M?

Suppose now that a highpass filter is derived from this lowpass filter by defining

(b) Show that the resulting frequency response is of the form

(c) Sketch B_e(e^jω) for 0 ≤ ω ≤ π.

(d) It is asserted that for the given value of M (as found in part (a)), the resulting highpass filter is the optimum approximation to zero in the band 0 ≤ ω ≤ 0.6π and to unity in the band 0.73π ≤ ω ≤ π. Is this assertion correct? Justify your answer.