Page 259 - Analog and Digital Filter Design
P. 259
256 Analog and Digital Filter Design
described in this chapter (Passive First-Order Equalizers). The group delay is
greatest at low frequencies and is smoothly decaying as the frequency rises.
Second-order equalizers are described by four factors: Sigma, Omega, Q, and
B. Sigma and B are the real and imaginary coordinates of the pole-zero con-
stellation. Omega (or mR) is the peak delay frequency, and Q is the Q-factor of
this peak; these two factors are required to find component values in a second-
order equalizer.
Higher-order equalizers are described by a combination of first- and second-
order factors. Third-order equalizers use a first- and second-order section in
series, so values are given for Sigma 1, Sigma 2, Omega 2, Q2, and B2. Sigma
1 describes the first-order section and the other factors describe the second-order
section. Fourth-order equalizers use two second-order sections in series. Sigma
1, Omega 1, Ql, and B1 describe one second-order section, and Sigma 2, Omega
2, (22, and B2 describe the other.
Passive filter component values for these equalizers are also given in sepa-
rate tables (Table 9.2 and Tables 9.4-9.8). These are normalized for one-ohm
termination and a one radian per second cutoff frequency. Active equalizer com-
ponents are not given since there are many solutions, unlike the passive equal-
izer where the solution depends on both the impedance and the filter's cutoff
frequency.
Group Delay of Butterworth Filters
To find the goup delay of a Butterworth filter it is necessary to carry out the
following steps:
1. Find the denominator coefficients of the Butterworth transfer
function.
2. Multiply each coefficient by the Laplace variable (s) to the power of
the coefficient subscript.
3. Calculate the phase-shift function, using these coefficients and fre-
quency variables.
4. Differentiate the phase-shift function to find the group delay.
These steps have to be repeated for each filter-order required.
STEP ONE: The denominator coefficients can be found using an iterative
formula given by Herdsman.'
