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
