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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.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2023-09-17 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University