Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 312

j

Im( ) z

z 1

d T

a

Re( ) z

0

0 1

FIGURE 8.9

Frequency mapping from the analog domain to the digital domain.

Next, we examine frequency mapping between the s-plane and the z-plane. As illustrated in
Figure 8.9 , the analog frequency u a is marked on the ju -axis on the s-plane, whereas u d is the digital
frequency labeled on the unit circle in the z-plane.
We substitute s ¼ jw a and z ¼ e juT into the bilinear transformation in Equation (8.11) to get

e ju d T 1
e ju d T þ 1

2
T

ju a ¼ (8.13)

Simplifying Equation (8.13) leads to

tan u d 2

2
T

u a ¼ (8.14)

Equation (8.14) explores the relation between the analog frequency on the ju axis and the corre-
sponding digital frequency u d on the unit circle. We can also write its inverse as

tan 1 u a 2

2
T

u d ¼

(8.15)

The range of the digital frequency u d is from 0 radians per second to the folding frequency u s = 2
radians per second, where u s is the sampling frequency in terms of radians per second. We present
a plot of Equation (8.14) in Figure 8.10 .
From Figure 8.10 , when the digital frequency range 0 u d 0 : 25 u s is mapped to the analog
frequency range 0 u a 0 : 32 u s , the transformation appears to be linear; however, when the
digital frequency range 0 : 25 u s u d 0 : 5 u s is mapped to the analog frequency range for
u a > 0 : 32 u s , the transformation is nonlinear. The analog frequency range for u a > 0 : 32 u s is
compressed into the digital frequency range 0 : 25 u s u d 0 : 5 u s . This nonlinear frequency
mapping effect is called frequency warping . We must incorporate frequency warping into IIR filter
design.
The following example will illustrate the frequency warping effect in the BLT.