More General Allpass Filters

We have so far seen two types of allpass filters:

- The series combination of feedback and feedforward comb-filters is allpass when their delay lines are the same length and their feedback and feedforward coefficents are the same. An example is shown in Fig.2.30.
- Any delay element in an allpass filter can be replaced by an allpass filter to obtain a new (typically higher order) allpass filter. The special case of nested first-order allpass filters yielded the lattice digital filter structure of Fig.2.32.

**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
, we must have

In terms of the signal norm , this can be expressed more succinctly as

Notice that only stable filters can be lossless since, otherwise,
is generally infinite, even when
is finite. We
further assume all filters are *causal*^{3.14} for
simplicity. It is straightforward to show the following:

It can be shown [452, Appendix C] that 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.

Thus, ``lossless'' and ``unity-gain allpass'' are synonymous. For an allpass filter with gain at each frequency, the energy gain of the filter is for every input signal . Since we can describe such a filter as an allpass times a constant gain, the term ``allpass'' will refer here to the case .

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University