Page 438 - Analog and Digital Filter Design

Design Equations 435
~

Pole to origin is = m,,. Real part of pole coordinate = CF

Another way of expressing these is:

+
ct), =2Qo and 2Q= ii'(s)? 1.

Notice that these equations show that Q depends on the ratio of w/a so, as the
pole-zero diagram is scaled for a higher cutoff frequency, the value of Q remains
unchanged. The natural frequency m,,,, is dependent upon a, and this changes in
proportion to the scaling of the diagram. Zero locations are scaled in a similar
way, moving away from the origin (s = 0) and along the imaginary axis.

Digital Filter Equations

Finding FIR Filter Zero Coefficient Using L'Hopital's Rule
The maximum coefficient value is at iz = 0, but this cannot be calculated because
we would be dividing by zero. The value 401 is calculated by differentiating the
numerator and denominator separately, and then letting iz = 0. This is known as
L'Hopital's rule, named after a French mathematician.

Let us look at a sinc function where the first zero coefficient is at rz = 5. The
sampled sinc(s) function has values given by ~[II].

The value of h[O] for this equation can be found using L'Hopital's rule:

:
h[o] =-.-cos - = - = 0.2
lr5

The next value, 1z[ 11, is simply h[l] = = 0.187 1. Likewise, by substituting
TT
values for n, other values are calculated.