UploadFile_6417

Analog-to-Digital Filter Transformations 435
2. The entire left half-plane maps into the inside of the unit circle. Hence
this is a stable transformation.
3. The imaginary axis maps onto the unit circle in a one-to-one fashion.
Hence there is no aliasing in the frequency domain.
Substituting σ =0in (8.66), we obtain
z = 1+j ΩT 2
1 − j ΩT 2
= e jω
since the magnitude is 1. Solving for ω as a function of Ω, we obtain
( ) ΩT
ω =2tan −1 or Ω = 2 ( ω
)
2
T tan (8.67)
2
This shows that Ω is nonlinearly related to (or warped into) ω but that
there is no aliasing. Hence in (8.67) we will say that ω is prewarped into Ω.
into a digital filter using the bilinear transfor-
□ EXAMPLE 8.15
s +1
Transform H a(s) =
s 2 +5s +6
mation. Choose T =1.
Solution
Using (8.65), we obtain
H(z) =H a
(
2
T
∣ ) ( )
1 − z −1 ∣∣∣
= H
1+z −1 a 2 1 − z−1
1+z −1 T =1
=
(
2 1 − z−1
+1
1+z−1 )
2 1 − 2 ( )
z−1
+5 2 1 − z−1
+6
1+z −1 1+z −1
Simplifying,
H(z) = 3+2z−1 − z −2
= 0.15 + 0.1z−1 − 0.05z −2
20+4z −1 1+0.2z −1
MATLAB provides a function called bilinear to implement this
mapping. Its invocation is similar to the imp invr function, but it also
takes several forms for different input-output quantities. The SP toolbox
manual should be consulted for more details. Its use is shown in the
following example.
□
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.