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.