Antisymmetric Linear-Phase Filters

In the same way that odd impulse responses are related to even impulse responses, linear-phase filters are closely related to antisymmetric impulse responses of the form $h(n)=-h(N-1-n)$ , $n=0:N-1$ . An antisymmetric impulse response is simply a delayed odd impulse response (usually delayed enough to make it causal). The corresponding frequency response is not strictly linear phase, but the phase is instead linear with a constant offset (by $\pm\pi/2$ ). Since an affine function is any function of the form $f(\omega)=\alpha \omega + \beta$ , where $\alpha$ and $\beta$ are constants, an antisymmetric impulse response can be called an affine-phase filter. These same remarks apply to any linear-phase filter that can be expressed as a time-shift of a $\pi$ -phase filter (i.e., it is inverting in some passband). However, in practice, all such filters may be loosely called ``linear-phase'' filters, because they are designed and implemented in essentially the same way [68].

Note that truly linear-phase filters have both a constant phase delay and a constant group delay. Affine-phase filters, on the other hand, have a constant group delay, but not a constant phase delay.