Page 213 - Analog and Digital Filter Design

2 1 0 Analog and Digital Filter Design

lae or tables given in Chapter 3. The bandstop Q is the center frequency, fo,
divided by the width of the stopband.
wo2 = 02 + w‘

A=----- c3
wo .Qss
w
B=-
wo .Yss
f = B’- A’+ 4
g= j/+:/”B?

(Does this remind you of a well-known quadratic solving equation?
Try CI = 1, b = -f? and c = A’B’.)
AB
A=-
g

this gives T/I’=OSd(A+h)’ +(B+g)’
hl
and the frequencies are fRI = - and fR1 = Wh.
it7
The pole’s Q factor is given by Q = -. TV
A+h
Real poles have a Q factor of Q = oQss and a resonant frequency at fo.

Now to find the zero locations. In a prototype lowpass all-pole filter such as
Chebyshev or Butterworth response, zeroes are on the imaginary axis in the S-
plane, at infinity. During transformation into a bandstop response they move to
the center of the stopband. In a prototype lowpass Cauer and Inverse Cheby-
shev response, zeroes are just outside the passband. When transformed into a
bandstop response the zero locations move into the stopband, placed symmet-
rically above and below the center frequency. Their locations have to be calcu-
lated, as follows:
1
J=- where Z is the normalized lowpass zero frequency.
QssZ’

The zero frequencies are f-,, = f. [J - -1 and f-.> = [J + m].
2 2
So, what does the S-plane diagram look like now? In Chapter 4 an example of
a fourth-order lowpass filter was given (Figure 4.11). This had a Butterworth
response, with poles on a unit circle at -0.9239 k j0.3827 and -0.3827 2 j0.9239.