Page 355 - Schaum's Outline of Theory and Problems of Signals and Systems

342 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

6.46. Consider designing a discrete-time LTI system with system function HJz) obtained by
applying the bilinear transformation to a continuous-time LTI system with rational
system function H,(s). That is,

Show that a stable, causal continuous-time system will always lead to a stable, causal
discrete-time system.
Consider the bilinear transformation of Eq. (6.183)

Solving Eq. (6.188) for z, we obtain

Setting s = jw in Eq. (6.1891, we get

Thus, we see that the jw-axis of the s-plane is transformed into the unit circle of the z-plane.
Let

z =re'" and s =a + jo
Then from Eq. (6.188)

r2- 1 2r sin $2
1 +r2+2rcosR +'1+r2+2rcos~

Hence,

2 2r sin R
w=-
T, 1 +r2+2rcosR

From Eq. (6.191a) we see that if r < 1, then a < 0, and if r > 1, then cr > 0. Consequently, the
left-hand plane (LHP) in s maps into the inside of the unit circle in the z-plane, and the
right-hand plane (RHP) in s maps into the outside of the unit circle (Fig. 6-32). Thus, we
conclude that a stable, causal continuous-time system will lead to a stable, causal discrete-time
system with a bilinear transformation (see Sec. 3.6B and Sec. 4.6B). When r = 1, then o = 0
and
2 sin 0 2 R
w=- = -tan-
T, I + cos R T, 2