$\pi$ -Phase Filters

Under our definition, a zero-phase filter always has a real, even impulse response [ $h(n) = h(-n)$ ], but not every real, even, impulse response is a zero-phase filter. For example, if $h(n)$ is zero phase, then $-h(n)$ is not; however, we could call $-h(n)$ a ``$\pi$ -phase filter'' if we like (a zero-phase filter in series with a sign inversion).