Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
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Digital filter
specifications
Digital filter
H(z)
Bilinear
transform
Bilinear
transform
w
w
Analog filter
specifications
Analog filter
H(s)
Aanlog filter
design
Figure 6.10 Digital IIR filter design using the bilinear transform
The bilinear transform is a mapping or transformation that relates points on the s-
and z-planes. It is defined as
s
2
T
z À 1
z 1
2
T
1 À z
À1
1 z
À1
,
6:3:16
or equivalently,
z
1 T=2s
1 À T=2s
:
6:3:17
This is called the bilinear transform because of the linear functions of z in both the
numerator and denominator of (6.3.16).
As discussed in Section 6.1.2, the jV-axis of the s-plane (s 0) maps onto the unit
circle in the z-plane. The left (s < 0) and right (s > 0) halves of the s-plane map into the
inside and outside of the unit circle, respectively. Because the jV-axis maps onto the unit
circle (jzj 1), there is a direct relationship between the s-plane frequency V and the z-
plane frequency !. Substituting s jV and z e
j!
into (6.3.16), we have
jV
2
T
e
j!
À 1
e
j!
1
:
6:3:18
It can be easily shown that the corresponding mapping of frequencies is obtained as
V
2
T
tan
!
2
,
6:3:19
or equivalently,
! 2 tan
À1
VT
2
:
6:3:20
Thus the entire jV-axis is compressed into the interval Àp=T, p=T for ! in a one-to-
one manner. The range 0 3 I portion in the s-plane is mapped onto the 0 3 p portion
of the unit circle in the z-plane, while the 0 3 ÀI portion in the s-plane is mapped onto
260
DESIGN AND IMPLEMENTATION OF IIR FILTERS
Summary :
It is defined as s 2 T z À 1 z 1 2 T 1 À z À1 1 z À1 , 6:3:16 or equivalently, z 1 T=2s 1 À T=2s : 6:3:17 This is called the bilinear transform because of the linear functions of z in both the numerator and denominator of (6.3.16).
Tags :
filter,bilinear,splane,transform,unit,onto,circle,into,portion,design,jaxis,digital,6316