**Search Introduction to Digital Filters**

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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

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

*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
,

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

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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