Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 310

The BLT method is a mapping or transformation of points on the s-plane to the z-plane. Equation
(8.11) can be alternatively written as

1 þ sT= 2
1 sT= 2

z ¼ (8.12)

The general mapping properties are summarized as following:
1. The left-half s-plane is mapped onto the inside of the unit circle of the z-plane.
2. The right-half s-plane is mapped into the outside of the unit circle of the z-plane.
3. The positive ju axis portion in the s-plane is mapped onto the positive half circle (the dashed line
arrow in Figure 8.8 ) on the unit circle, while the negative ju axis is mapped onto the negative half
circle (the dotted line arrow in Figure 8.8 ) on the unit circle.

To verify these features, let us look at the following illustrative example.

j

Im( ) z

Stable Region

Re( ) z

0

0 1

Stable Region

FIGURE 8.8

Mapping between the s-plane and the z-plane by the bilinear transformation.

EXAMPLE 8.3
Assume that T ¼ 2 seconds in Equation (8.12) , and that the following points are given:
1. s ¼1 þ j, on the left half of the s-plane
2. s ¼ 1 j, on the right half of the s-plane
3. s ¼ j, on the positive ju on the s-plane
4. s ¼j, on the negative ju on the s-plane

Convert each of the points in the s-plane to the z-plane, and verify mapping properties (1) to (3).
Solution:
Substituting T ¼ 2 into Equation (8.12) leads to

z ¼ 1 þ s
1 s

We can carry out mapping for each point as follows:

1 : 90

z ¼ 1 þð1 þ j Þ

j
2 j ¼

p : 26:57 ¼ 0:4472 : 116:57

1.

1 ð1 þ jÞ ¼

,