Simple Linear-Phase Filter Examples

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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'' $h(n)=\delta(n)$ is zero-phase (§10.2.2), the ``bypass filter with a unit delay,'' $h(n) = \delta(n-1)$ is linear phase. It is (trivially) symmetric about time , and the frequency response is $H(z) = e^{-j\omega T}$ , which is a pure linear phase term $\Theta(\omega)=-\omega T$ 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 $h = \delta(n) + \delta(n-1)$ [ $H(z)=1+z^{-1}$ ]. Since this impulse response is symmetric about time samples, it is linear phase, and $\Theta(\omega) = -\omega T/2$ , 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.

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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