## Allpass Filter Sections

The *allpass filter* passes all frequencies with equal gain. This
is in contrast with a lowpass filter, which passes only low
frequencies, a highpass which passes high-frequencies, and a bandpass
filter which passes an interval of frequencies. An allpass filter may
have any phase response. The only requirement is that its amplitude
response be constant. Normally, this constant is
.

From a physical modeling point of view, a unity-gain allpass filter
models a *lossless system* in the sense
that it *preserves signal energy*. Specifically, if
denotes the input to an allpass filter , and if denotes
its output, then we have

This equation says that the total energy out equals the total energy in. No energy was created or destroyed by the filter. All an allpass filter can do is delay the sinusoidal components of a signal by differing amounts.

Appendix C proves that Eq.(B.9) holds if and only if

### The Biquad Allpass Section

The general biquad transfer function was given in Eq.(B.8) to be

^{B.3}

In terms of the poles and zeros of a filter , an allpass filter must have a zero at for each pole at . That is if the denominator satisfies , then the numerator polynomial must satisfy . (Show this in the one-pole case.) Therefore, defining takes care of this property for all roots of (all poles). However, since we prefer that be a polynomial in , we define , where is the order of (the number of poles). is then the flip of .

For further discussion and examples of allpass filters (including muli-input, multi-output allpass filters), see Appendix C. Analog allpass filters are defined and discussed in §E.8.

### Allpass Filter Design

There is a fairly large literature thread on the topic of
*allpass filter design*. Generally, they fall into two main
categories: *parametric* and *nonparametric* methods.
Parametric methods can produce allpass filters with optimal
group-delay characteristics [42,41].
Nonparametric methods, while suboptimal, can design very large-order
allpass filters, and errors can usually be made arbitrarily small by
increasing the order [100,70,1],
[78, pp. 60,172]. In music applications, it is usually the
case that the ``optimality'' criterion is unknown because it depends
on aspects of sound perception (see, for example,
[35,72]). As a result, perceptually
weighted nonparametric methods can often outperform optimal parametric
methods in terms of cost/performance. For a nonparametric method that
can design very high-order allpass filters according to highly
flexible criteria, see [1].

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