Allpass Filters

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 *causal*^{C.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
[84] 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

Since this must hold for all , we must have for all , except possibly for a set of measure zero (

We have shown that every lossless filter is allpass. Conversely, every unity-gain allpass filter is lossless.

- Allpass Examples
- Paraunitary Filters
- MIMO Allpass Filters

- Allpass Problems

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