Page 261 - Analog and Digital Filter Design

258 Analog and Digital Filter Design

denominator. Conversion from the S-plane to find the complex frequency vari-
able (jw) is needed. The phase-shift function is the negative arc tangent of this
resultant equation, with the denominator multiplied by j.

phase = -arc tan[(j3m3 + 2jw)/j(2j2m2 + l)]

There are complex j multipliers to consider. Squaring this function gives minus
one. So j2 + -1 and j’ + -j.
phase = -arc tan[(-jw3 + 2jw)/j(-2m2 + l)]

The complex factor j cancels, leaving:

phase = -arc tan[(-& + 2w)/(-2w2 + l)]

STEP FOUR: Finally, the group delay is the differentiation of the phase-shift
function. The result of this differentiation is the rate of change of the function,
which is the group delay. Differentiation is a complex subject; however. for
this purpose, it is sufficient to know that the differentiation of arc tan (x) is
l/(x’ + 1). I do not propose to go into this further here, but the resultant equa-
tion has only even powers of w, the highest power being in the denominator and
equal to (1 + w)”’, where n is the filter-order. Examples of calculation for up to
third-order Butterworth filters are given by Helszajn.’

Having obtained the equations for the group delay of Butterworth filters, up to
twelfth-order, MATHCAD6 was used to optimize the equalizer. An example of
this is given in Figure 9.13; here a seventh-order design is equalized by a second-
order equalizer section. The frequency variable (0) is steps in increments of
MOO radians per second. The tuned frequency of the second-order equalizer
(%) is symbolized by Q. The resultant group delay has an equi-ripple charac-
teristic. Equi-ripple is the state where all the peaks are equal in amplitude and
all the troughs are equal in amplitude, although not necessarily at an equal fre-
quency spacing.