Consider an FIR system whose impulse response is . This system is an examp...

Consider an FIR system whose impulse response is

.

This system is an example of a class of filters known as frequency-sampling filters. Problem 6.51 discusses these filters in detail. In this problem, we consider just one specific case.

(a) Sketch the impulse response of the system for the cases n₀ = 0 and n₀ = 15/2.

(b) Show that the system function of the system can be expressed as

(c) Show that if n₀ = 15/2, the frequency response of the system can be expressed as

Use this expression to sketch the magnitude of the frequency response of the system for n₀ = 15/2. Obtain a similar expression for n₀ = 0. Sketch the magnitude response for n₀ = 0. For which choices of n₀ does the system have a generalized linear phase?

(d) Draw a signal flow graph of an implementation of the system as a cascade of an FIR system whose system function is 1−z ⁻¹⁵ and a parallel combination of a 1st- and 2nd-order IIR system.