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Odd Impulse Reponses

Note that odd impulse responses of the form $ h(n)=-h(-n)$ are closely related to zero-phase filters (even impulse responses). This is because another Fourier symmetry relation is that the DTFT of an odd sequence is purely imaginary [84]. In practice, Hilbert transform filters and differentiators are often implemented as odd FIR filters [68]. A purely imaginary frequency response can be divided by $ j$ to give a real frequency response. As a result, filter-design software for one case is easily adapted to the other [68].

Equivalently, an odd impulse response can be multiplied by $ j$ in the time domain to yield a purely imaginary impulse response that is Hermitian. Hermitian signals have real Fourier transforms [84]. Therefore, a Hermitian impulse response gives a filter having a phase response that is either zero or $ \pi $ at each frequency.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University