Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 361

Note that the transfer function for the bandpass Butterworth filter is the same
as that derived by hand in Example 7.9.

7.4.3 Lowpass to bandstop filter

The transformation to convert a lowpass filter with the transfer function H ( S )
into a bandstop filter with transfer function H ( s ) is given by the following
expression:

= s ( ξ p2

− ξ p1 )
s 2 + ξ p1 ξ p2 ,

S (7.74)

where S = σ + j ω represents the lowpass domain and s = γ + j ξ represents
the bandstop domain. The frequency ξ = ξ p1 and ξ p2 represents the two pass-
band corner frequencies for the bandpass filter with ξ p2 >ξ p1 . Note that the
transformation in Eq. (7.74) is the inverse of the lowpass to bandpass transfor-
mation specified in Eq. (7.69).
In terms of the CTFT domain, Eq. (7.74) can be expressed as follows:
ω = ξ ( ξ p2 − ξ p1 )

ξ p1 ξ p2 − ξ 2 ,

(7.75)

which can be used to confirm that Eq. (7.74) is indeed a lowpass to bandstop
transformation.
As for the bandpass filter, Eq. (7.75) leads to two different values of the
stop-band frequencies,

ξ s1 ( ξ p2 − ξ p1 )
ξ p1 ξ p2 − ξ s1

ξ s2 ( ξ p2 − ξ p1 )
ξ p1 ξ p2 − ξ s2

ω s1 = and ω s2 = (7.76)

for the lowpass filter. The smaller of the two values is selected as the stop-band
corner frequency for the normalized lowpass filter. Example 7.12 designs a
bandstop filter.

Example 7.12
Design a bandstop Butterworth filter with the following specifications:

pass band I (0 ≤ξ ≤ 100 radians/s)

− 2dB ≤ 20 log 10 H ( ξ ) ≤ 0;

stop band (150 ≤ξ ≤ 250 radians/s) 20 log 10 H ( ξ ) ≤− 20 dB;

pass band II ( ξ ≥ 370 radians/s) − 2dB ≤ 20 log 10 H ( ξ ) ≤ 0 .

Solution
For ξ p1

= 100 radians/s and ξ p2 = 370 radians/s, Eq. (7.70) becomes
ω = 270 ξ

3 . 7 10 4 − ξ 2 ,
to transform the specifications from the domain s

= γ + j ξ of the bandstop

filter to the domain S = σ + j ω of the lowpass filter. The specifications for the