Page 169 - Analog and Digital Filter Design

1 66 Analog and Digital Filter Design

The gain at DC is denoted by the symbol "A." The resistors labeled R and R'
can be any arbitrary value; a typical value may be in the range 1 kR to 100kQ
say 10 kR. The resistors labeled R have an effect on the input impedance of the
filter section.

Denormalizing State Variable or Biquad Designs

The simplest approach with state variable and biquad filters is to scale the poles
(and zeroes, in the biquad case) before using the design equations. Choose a
capacitor value, and then use the equations to find the resistor values. If the
resistor values are very small or very large, select a new capacitor value and try
again. Again, aim to keep the resistor values between 1 kR and 100kR.

Here's an example of denormalizing a biquad highpass filter design. Design a
second-order Inverse Chebyshev filter that gives 40 dB stopband attenuation.
This uses normalized lowpass pole locations 0.70705 f j0.71416 and a zero loca-
tion 10.04963. The passband cutoff frequency is l kHz. Let R = 10 kR and let
the reactance of C = 10 kR at 1 kHz (C = 1/[27rFc. 10'1). Thus, using the nearest
E6 preferred value, C = 150 nF.

First compute the pole locations from the normalized lowpass values:

Thus, oIlP = 0.700102 and wHP = 0.707142.

The frequency scaling factor at 1 kHz is 2nFc = 6283 radls. Multiplying the pole
locations by this factor gives:

O/ip = 0.700102.6283 = 4398.87
and wHp = 0.707142.6283 = 4443.10.

Now compute the section's frequency from the normalized highpass pole
location

W,,HP = 1/dHp' + 61,lp' = 0.995084.6283 = 6252.29.
Now compute the highpass zero location, using oZHp = l/oz and the nor-
malized lowpass zero location of 10.04963, which gives the highpass zero at