Implementing IIR/FIR Filters

sections. The second-order section of a Butterworth
filter can be derived from the simple RCL network of
Figure 1-1. However, the Butterworth damping factors
are predetermined values, which can be shown to
yield a maximally flat passband response (see Reference
14). The Butterworth damping coefficients are
given by the following equation:
d k
( 2k – 1)π
= 2sin----------------------
2N
Eqn. 5-8
Eqn. 5-8 is the characteristic equation that determines
a Butterworth filter response. Note that for a
single-section (k = 1) second-order (N = 2) lowpass
filter, dk = 2 sin (π/4) = 2 as expected for a maximally
flat response. For a fourth-order filter with two
second-order sections, d1 = 2 sin (π/8), where k = 1
and N = 4, and d2 = 2 sin (3π/8),
where k= 2 and N = 4.
Eqn. 5-7 represents an all-pole response (the only
zeros are at plus and minus infinity in the analog sdomain).
The poles of a second-order section are
the roots of the quadratic denominator as given by:
p
k1
2 1/2
= – d
k
/2–
j1 ⎛ – d
k
/4⎞
⎝ ⎠
and
p
k2
d
k
/2 j⎛ 2
1 – d
k
/4⎞
⎝ ⎠
1/2
=
+
MOTOROLA 5-3
(a)
(b)
Eqn. 5-9