## 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.

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Antisymmetric Linear-Phase Filters

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Odd Impulse Reponses