Symmetric Linear-Phase FiltersAs 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 DesignThe Matlab Signal Processing Toolbox covers many applications with the following functions:
Antisymmetric Linear-Phase Filters
Odd Impulse Reponses