
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.
Next Section: Forward-Backward FilteringPrevious Section: Symmetric Linear-Phase Filters