Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 348

Selecting the poles located in the left-half s-plane, we obtain

[ − 0 . 4468 + j1 . 1614 , − 0 . 4468 + j1 . 1614 , − 0 . 8935] .

The poles of the normalized Type II Chebyshev filter are located at the inverse
of the above locations and are given by

[ − 0 . 2885 − j0 . 7501 , − 0 . 2885 + j0 . 7501 , − 1 . 1192] .

The zeros of the normalized Chebyshev Type II filter are computed using
Eq. (7.60) and are given by

[ − j1 . 1547 , + j1 . 1547 , ∞ ] .

The zero at s =∞ is neglected. The transfer function for the normalized
Type II Chebyshev filter is given by
H ( S ) = K ( S + j1 . 1547)( S − j1 . 1547)

( S + 0 . 2885 + j0 . 7501)( S + 0 . 2885 − j0 . 7501)( S + 1 . 1192) ,
which simplifies to
H ( S ) = K ( S 2 + 1 . 3333)

S 3 + 1 . 6962 S 2 + 1 . 2917 S + 0 . 7229 .
Since H ( ω ) at ω = 0is1 . 3333 / 0 . 7229 = 1 . 8444 , K is set to 1 / 1 . 8444 =
0 . 5422 to make the dc gain equal to unity. The new transfer function with unity
gain at ω = 0isgivenby
H ( S ) = 0 . 5422( S 2 + 1 . 3333)

S 3 + 1 . 6962 S 2 + 1 . 2917 S + 0 . 7229 .
Step 4 normalizes H ( S ) based on the following transformation:

0 . 5422(( s / 100) 2 + 1 . 3333)
( s / 100) 3 + 1 . 6962( s / 100) 2 + 1 . 2917( s / 100) + 0 . 7229 ,

H ( s ) = H ( S ) S = s / 100 =

which simplifies to

54 . 22( s 2 + 1 . 3333 10 4 )
s 3 + 1 . 6962 10 2 s 2 + 1 . 2917 10 4 s + 0 . 7229 10 6 .

H ( s ) =

Step 5 plots the magnitude spectrum, which is shown in Fig. 7.10. As expected,
the frequency characteristics in Fig. 7.10 have a monotonic gain within the
pass band and ripples within the stop band. Also, it is noted that the magnitude
spectrum H ( ω ) = 0 between the frequencies of ω = 100 and ω = 150 radians/s.
This zero value corresponds to the location of the complex zeros in H ( s ). Setting

0.891 1

0.6

0.4

Fig. 7.10. Magnitude spectrum
of the Type II Chebyshev lowpass
filter designed in Example 7.8.

0.1778

0

0 50 100 150 200 250