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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 generally taken to include 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. However, in practice, such a filter may be called a linear-phase filter, since it is designed and implemented in essentially the same way [65].

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, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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