## 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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Forward-Backward Filtering

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Symmetric Linear-Phase Filters