## Symmetric Linear-Phase Filters

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

*zero-phase*filter. Thus, every odd-length linear-phase filter can be expressed as a delay of some zero-phase filter,

*z*transforms (§6.3), the transfer function of a linear-phase filter is

*linear phase term*times which is real. Since can go negative, the phase response is

### 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:

`firls`are implemented in the free, open-source, Octave Forge collection as well.

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.

**Next Section:**

Antisymmetric Linear-Phase Filters

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Odd Impulse Reponses