Allpass Filter SectionsThe 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
Allpass Filter DesignThere 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 .
Elementary Filter Sections