Plot the pole-zero constellation as well as the magnitude response of the transfer function of Exercise 4.20 for \(M=6,7,8\) and comment on the results.

Exercise 4.20

Some FIR filters present a rational transfer function:

(a) Show that the transfer function

\[H(z)=\frac{\left(r^{-1} z\right)^{-(M+1)}-1}{r e^{\mathrm{j} 2 \pi /(M+1)} z^{-1}-1}\]

corresponds to an FIR filter (Lyons, 2007).

(b) Determine the operation performed by such a filter.

(c) Discuss the general case when a rational transfer function corresponds to an FIR filter.