
Frequency response of the resulting z-domain transfer function
^
H(z) and the continuous-time
transfer function H(s) will agree at the selected critical frequency, that is,
H(s)
j
s¼jv
0
¼
^
H(z)
z¼e
jv
0
T
(8:462)
The prewarped crit ical freque ncy ^v
0
is obtained from (Jacquot 1981)
^v
0
¼
2
T
tan
v
0
T
2
(8:463)
The following example illustrates the process of prewarping a second-order continuous-time filter
transfer function to force agreement in the frequency response functions at the natural frequency of
the filter.
Example 8.13
An analog filter is described by
H(s) ¼
s þ v
2
n
s
2
þ 2zv
n
s þ v
2
n
(z ¼ 0:25, v
n
¼ 1000 rad=s) (8:464)
(a) Find H(z) using the bilinear transform with ...