A discrete-time system with the pole-zero pattern shown in Figure P10.50(0 referred to a...

A discrete-time system with the pole-zero pattern shown in Figure P10.50(0 referred to as a first-order all-pass system, since the magnitude of the frequency response is constant regardless of frequency.

(a) Demonstrate algebraically that is constant

To demonstrate the same property geometrically, consider the vector a-gram in Figure P10.50(b). We wish to show that the length of is proportional to the length of independently of the frequency ω.

(b) Express the length of using the law of cosines and the fact that is one leg of a triangle for which the other two legs are the unit vector and a vector of length a.

(c) In a manner similar to that in part (13), determine the length of and show that it is proportional in length to independently of ω.