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
Appendix C proves that Eq.(B.9) holds if and only if
That is, a filter is lossless if and only if it is an allpass filter having a gain of at every frequency .