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 [83,37,98].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
Notice that only stable filters can be lossless, since otherwise can be infinite while is finite. We further assume all filters are causalD.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
Proof: We allow the signals and filter impulse response to be complex. By Parseval's theorem  for the DTFT, we have,D.2 for any signal ,
We have shown that every lossless filter is allpass. Conversely, every unity-gain allpass filter is lossless.