In the second-order case, we have, for the analog prototype,
where, from Eq.(I.2),
(from Eq.(I.9)), where, as discussed in §I.3, we set
to obtain a digital cut-off frequency at
and the digital filter transfer function is
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(I.4) |
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(I.5) | |
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(I.6) | |
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(I.7) |
It is also immediately verified that
In the analog prototype,
the cut-off frequency is
rad/sec, where,
from Eq.(I.1), the amplitude response
is
. Since we mapped the cut-off frequency precisely
under the bilinear transform, we expect the digital filter to have
precisely this gain.
The digital frequency response at one-fourth the sampling rate is
Note from Eq.(I.8) that the phase at cut-off is exactly -90 degrees
in the digital filter. This can be verified against the pole-zero
diagram in the
plane, which has two zeros at
, each
contributing +45 degrees, and two poles at
, each contributing -90
degrees. Thus, the calculated phase-response at the cut-off frequency
agrees with what we expect from the digital pole-zero diagram.
In the
plane, it is not as easy to use the pole-zero diagram
to calculate the phase at
, but using Eq.(I.3), we
quickly obtain
and exact agreement with
A related example appears in §9.2.4.