## Symmetric Linear-Phase Filters

As stated at the beginning of this chapter, the impulse response of every causal, linear-phase, FIR filter is symmetric:  Assume that is odd. Then the filter is a zero-phase filter. Thus, every odd-length linear-phase filter can be expressed as a delay of some zero-phase filter, By the shift theorem for z transforms (§6.3), the transfer function of a linear-phase filter is and the frequency response is which is a linear phase term times which is real. Since can go negative, the phase response is For frequencies at which is nonnegative, the phase delay and group delay of a linear-phase filter are simply half its length: ### 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-PhaseFilter Design

The Matlab Signal Processing Toolbox covers many applications with the following functions: All of these functions except 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 , 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