Symmetric Linear-Phase Filters
As stated at the beginning of this chapter, the impulse response of every causal, linear-phase, FIR filter is symmetric:












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''
is zero-phase (§10.2.2), the ``bypass filter with a unit delay,''
is linear phase. It is (trivially) symmetric about time
, and the frequency response is
, which 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:
![$\displaystyle \begin{tabular}{rl}
\texttt{remez()} & (optimal Chebyshev linear-...
...or general FIR (or IIR)\\
& filter design \cite[page 50]{JOST}).
\end{tabular}$](http://www.dsprelated.com/josimages_new/filters/img1217.png)
Methods for FIR filter design are discussed in the fourth book of the music signal processing series [87], and classic references include [64,68]. There is also quite a large research literature on this subject.
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Antisymmetric Linear-Phase Filters
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Odd Impulse Reponses