Consider an FIR system whose impulse response is

This system is an example of a class of filters known as frequency-sampling filters, problem 6.37 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 first-and second-order IIR system.

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