Odd Impulse Reponses

Note that odd impulse responses of the form 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 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 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 at each frequency.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.