Page 186 - Analog and Digital Filter Design

183
Bandpass Filters

and
The passband center frequency is wo = Je BW is the bandwidth,
given by the difference between the upper and lower passband frequencies.
wL - 0,. This is not particularly easy to evaluate. However, Williams' has pub-
lished equations for finding the Q and resonant frequency, fR. of each stage of
a bandpass filter from a lowpass model. These are all that are needed to design
active bandpass filters. I have manipulated Williams' equations slightly, to be
consistent with those used to design bandstop filters. Bandstop filter equations
will be given in the next chapter.

To start with you need to know the Q of bandpass filter, QBp, and the real and
imaginary parts of the lowpass prototype pole location, oand w. The pole posi-
tions can be found by using the formulae or referring to tables given in Chapter
3. The bandpass Q is the center frequency,J;,, divided by the bandwidth.

o
r71 = -
QBP

The required Q =
8nz
This gives the frequency scaling factor, R' = @n + 4-

And the frequencies are fRI = -r;i and fR? = ~ifo.
w
These are the pole transformation equations. Now the zero locations are needed,
and, in an all-pole filter such as Chebyshev or Butterworth response, these are
at the S-plane origin and at infinity. In Cauer and Inverse Chebyshev filters the
zero locations have to be calculated, as follows:

w,
k = -
QBP
k'
12=-+1
2
The zero scaling factor can now be found, -7 = Jif +

A7
The bandpass zero frequencies are then f-,, =- 7 and SI-.: = &.
-
What does the S-plane diagram look like now? An example of a fourth-order
lowpass filter was given in Chapter 4, Figure 4.11. This had a Butterworth