Example Linear-Phase Filters

While the trivial ``bypass filter'' $h(n)=\delta(n)$ was zero-phase, the ``bypass filter with a unit delay,'' $h(n) = \delta(n-1)$ is linear phase. We know this because it is (trivially) symmetric about time $n=1$. Since the frequency response is $H(z) = e^{-j\omega T}$, 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 $h = \delta(n) + \delta(n-1)$ (or in matlab syntax, h=[1 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.