Implementing IIR/FIR Filters

φΩ ( ) tan 1 –
d
k
Ω/Ω
c
( Ω/Ω
c
) 2 N/2
= ∑ -----------------------------k
= 1
– 1
Eqn. 5-12
Eqn. 5-11, and Eqn. 5-12 describe the response
characteristics of an N th -order lowpass Butterworth
filter in the continuous frequency analog domain.Since
the quantity of interest is usually 20 log
G(Ω), Eqn. 5-11 can be transformed into a sum
(over the second-order sections) of 20 log (G k ),
where G k is the gain of the k th section. Similarly, the
total phase is just the sum of the phase contribution
by each section from Eqn. 5-12.
The bilinear transformation is used to convert the
continuous frequency domain transfer function into
the digital domain representation, where θ, the normalized
digital domain frequency equal to 2πf/f s ,
can be thought of as the ratio of frequency to sampling
frequency scaled by 2π. Substituting s from
Eqn. 2-13 and Ω c from Eqn. 2-14 into the k th section
of Eqn. 5-7 yields the digital domain form of the Butterworth
lowpass filter (for the k th second-order
section):
H
k
( z)
α
k
1 2z 1 –
z 2 –
( + + )
1
-- γ
2 k
z 1 –
β
k
z 2 –
=
-------------------------------------------------
– +
Eqn. 5-13
MOTOROLA 5-5