Consider a type I linear-phase FIR lowpass filter with impulse response hLP[n] of length (M + 1) and frequency response
The system has the amplitude function Ae(ejω) shown in Figure P7.46.
This amplitude function is the optimal (in the Parks–McClellan sense) approximation to unity in the band 0 ≤ ω ≤ ωp, where ωp = 0.27π, and the optimal approximation to zero in the band ωs ≤ ω ≤ π, wherein ωs = 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 Be(ejω) 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.
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