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 signal norm , this can be expressed more succinctly as
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 plane.
Proof: We allow the signals and filter impulse response to be complex. By Parseval's theorem  for the DTFT, we have,C.2 for any signal ,
Thus, Parseval's theorem enables us to restate the definition of losslessness in the frequency domain:
where because the filter is LTI. Thus, is lossless by definition if and only if
We have shown that every lossless filter is allpass. Conversely, every unity-gain allpass filter is lossless.