The bilinear transformation is a mapping for obtaining a rational z-transform Hd(Z) from

Chapter 10, Problem 10.65

(choose chapter or problem)

The bilinear transformation is a mapping for obtaining a rational z-transform Hd(Z) from a rational Laplace transform Hc(s). This mapping has two important properties:

1. If Hc(s) is the Laplace transform of a causal and stable LTI system, then Hd(Z) is the z-transform of a causal and stable LTI system.

2. Certain important characteristics of \(\left|H_{c}(j \omega)\right|\) are preserved in \(\left|H_{d}\left(e^{j \omega}\right)\right|\). In this problem, we illustrate the second of these properties for the case of all-pass filters.

(a) Let

\(H_{c}(s)=\frac{a-s}{s+a} \text {, }\)

 where a is real and positive. Show that

\(\left|H_{c}(j \omega)\right|=1\)

(b) Let us now apply the bilinear transformation to Hc(s) in order to obtain Hd(Z). That is,

\(H_{d}(z)=\left.H_{c}(s)\right|_{s=\frac{1-z^{-1}}{1+z^{-1}}}\)

Show that Hd(Z) has one pole (which is inside the unit circle) and one zero (which is outside the unit circle).

(c) For the system function Hd(Z) derived in part (b), show that \(\left|H_{d}\left(e^{j \omega}\right)\right|=1\).