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
In terms of the
Notice that only stable filters can be lossless since, otherwise,
is generally infinite, even when
is finite. We
further assume all filters are causal3.14 for
simplicity. It is straightforward to show the following:
It can be shown [452, Appendix C] that stable, linear,
time-invariant (LTI) filter transfer function
is lossless if
and only if
That is, the frequency response must have magnitude 1 everywhere over the unit circle in the complex
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
.