While the trivial ``bypass filter''
(§10.2.2), the ``bypass filter with a unit delay,''
is linear phase. It is (trivially) symmetric
, and the frequency response is
is a pure linear phase term
having a slope
samples (radians per radians-per-sample), or
(radians per radians-per-second). The phase- and group-delays are
each 1 sample at every frequency.
The impulse response of the simplest lowpass filter studied in
Chapter 1 was
Since this impulse response is symmetric about time
it is linear phase, and
, as derived
in Chapter 1. The phase delay and group delay are both
each frequency. Note that even-length linear-phase filters cannot be
time-shifted (without interpolation) to create a corresponding
zero-phase filter. However, they can be shifted to make a
near-zero-phase filter that has a phase delay and group delay equal to
half a sample at all passband frequencies.