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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'' $ 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 $ n=1$, 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 $ -1$ samples (radians per radians-per-sample), or $ -T$ 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 $ n=1/2$ samples, it is linear phase, and $ \Theta(\omega) =
-\omega T/2$, as derived in Chapter 1. The phase delay and group delay are both $ 1/2$ 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.

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Software for Linear-Phase Filter Design
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Elementary Zero-Phase Filter Examples