Antisymmetric Linear-Phase Filters

In the same way that odd impulse responses are related to even impulse
responses, linear-phase filters are generally taken to include
*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*. However, in practice, such a filter
may be called a linear-phase filter,
since it is designed and implemented in essentially the same way
[65].

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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