Symmetric Linear-Phase Filters
Simple Linear-Phase Filter Examples
- The example of §10.2.1 was in fact a linear-phase FIR
filter design example. The resulting causal finite impulse response
was left-shifted (``advanced'' in time) to make it zero phase.
- While the trivial ``bypass filter''
(§10.2.2), the ``bypass filter with a unit delay,''
is linear phase. It is (trivially) symmetric
about time , and the frequency response is
is a pure linear phase term
having a slope
of samples (radians per radians-per-sample), or seconds
(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 samples,
it is linear phase, and
, as derived
in Chapter 1. The phase delay and group delay are both sample at
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.
Software for Linear-Phase Filter Design
The Matlab Signal Processing Toolbox covers many applications with the following functions:
Methods for FIR filter design are discussed in the fourth book of the music signal processing series , and classic references include [64,68]. There is also quite a large research literature on this subject.
Antisymmetric Linear-Phase Filters
Odd Impulse Reponses