Implementing IIR/FIR Filters

pass Butterworth filter (three second-order sections)
in both the analog domain and digital domain. Note
that the gain of the first section (k = 1) is greater than
unity near the cutoff frequency but that the overall
composite response never exceeds unity. This fact
allows for easy implementation of the Butterworth filter
in cascaded direct form (i.e., scaling of sections
is not needed as long as the sections are implemented
in the order of decreasing k). Overflow at the
output of any section is then guaranteed not to occur
(the gain of the filter never exceeds unity). Note that
the digital response (see Figure 5-28) is identical to
the analog response but warped from the right along
the frequency axis. Imagine the zero at plus infinity
in the analog response mapping into the zero at f s /2
in the digital case. Also note that, because of this
mapping, the digital response falls off faster than the
-12 dB/octave of the analog filter when the cutoff is
near f s /2.
The previous analysis is nearly identical to the case
of the highpass filter except the coefficients (see
Figure 3-24) have slightly different values. Since
the bandstop case is just the sum of a lowpass and
highpass case, it can be analyzed by these techniques.
The bandpass case, however, is more
difficult and requires considerably more work (see
Reference 14) because the center frequency of
each section is now different and the formula for
calculating these frequencies is not as simple as
the formulas for the previous filter types. In addition,
to complicate matters further, scaling between
sections becomes more of a problem since the off-
5-8 MOTOROLA