Antisymmetric Linear-Phase Filters

In the same way that odd impulse responses are related to even impulse responses, linear-phase filters are closely related to antisymmetric impulse responses of the form , . An antisymmetric impulse response is simply a delayed odd impulse response (usually delayed enough to make it causal). The corresponding frequency response is not strictly linear phase, but the phase is instead linear with a constant offset (by ). Since an affine function is any function of the form , where and are constants, an antisymmetric impulse response can be called an affine-phase filter. These same remarks apply to any linear-phase filter that can be expressed as a time-shift of a -phase filter (i.e., it is inverting in some passband). However, in practice, all such filters may be loosely called ``linear-phase'' filters, because they are designed and implemented in essentially the same way [68].

Note that truly linear-phase filters have both a constant phase delay and a constant group delay. Affine-phase filters, on the other hand, have a constant group delay, but not a constant phase delay.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.