This appendix addresses the general problem of characterizing all digital allpass filters, including multi-input, multi-output (MIMO) allpass filters. As a result of including the MIMO case, the mathematical level is a little higher than usual for this book. The reader in need of more background is referred to [84,37,98].
Our first task is to show that losslessness implies allpass.
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
, then we have
In terms of the
Notice that only stable filters can be lossless, since otherwise
can be infinite while
is finite. We further
assume all filters are causalC.1 for
simplicity. It is straightforward to show the following:
Theorem: A 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
Proof: We allow the signals
and filter impulse response
to be complex. By Parseval's theorem
[84] for the DTFT, we have,C.2 for any signal
,
i.e.,
Thus, Parseval's theorem enables us to restate the definition of losslessness in the frequency domain:
where
We have shown that every lossless filter is allpass. Conversely, every unity-gain allpass filter is lossless.