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More General Allpass Filters

We have so far seen two types of allpass filters:

We now discuss allpass filters more generally in the SISO case. (See Appendix D of [452] for the MIMO case.)

Definition: A linear, time-invariant filter $ H(z)$ is said to be lossless if it preserves signal energy for every input signal. That is, if the input signal is $ x(n)$ , and the output signal is $ y(n) = (h\ast x)(n)$ , we must have

$\displaystyle \sum_{n=-\infty}^{\infty} \left\vert y(n)\right\vert^2 =
\sum_{n=-\infty}^{\infty} \left\vert x(n)\right\vert^2.

In terms of the $ \ensuremath{L_2}$ signal norm $ \left\Vert\,\,\cdot\,\,\right\Vert _2$ , this can be expressed more succinctly as

$\displaystyle \left\Vert\,y\,\right\Vert _2^2 = \left\Vert\,x\,\right\Vert _2^2.

Notice that only stable filters can be lossless since, otherwise, $ \left\Vert\,y\,\right\Vert$ is generally infinite, even when $ \left\Vert\,x\,\right\Vert$ is finite. We further assume all filters are causal3.14 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 $ H(z)$ is lossless if and only if

$\displaystyle \left\vert H(e^{j\omega})\right\vert = 1, \quad \forall \omega.

That is, the frequency response must have magnitude 1 everywhere over the unit circle in the complex $ z$ plane.

Thus, ``lossless'' and ``unity-gain allpass'' are synonymous. For an allpass filter with gain $ g$ at each frequency, the energy gain of the filter is $ g^2$ for every input signal $ x$ . Since we can describe such a filter as an allpass times a constant gain, the term ``allpass'' will refer here to the case $ g=1$ .

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-02-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University