We have so far seen two types of allpass filters:
Definition:
A linear, time-invariant filter is said to be
lossless if it preserves signal
energy for every input signal. That is, if the input signal is
, and the output signal is
, we must have
Notice that only stable filters can be lossless since, otherwise,
is generally infinite, even when
is finite. We
further assume all filters are causal2.9 for
simplicity. It is straightforward to show the following:
It can be shown [426, Appendix D] that stable, linear,
time-invariant (LTI) filter transfer function is lossless if
and only if
Thus, ``lossless'' and ``unity-gain allpass'' are synonymous. For an
allpass filter with gain at each frequency, the energy gain of the
filter is
for every input signal
. Since we can describe
such a filter as an allpass times a constant gain, the term
``allpass'' will refer here to the case
.