Odd Impulse Reponses

Note that *odd impulse responses* of the form
are
closely related to zero-phase filters (even impulse responses). This
is because, by Fourier symmetry, the DTFT of an odd sequence is purely
*imaginary* [83]. In
practice, *Hilbert transform filters* and *FIR
differentiators* are often implemented as odd FIR filters
[65]. 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
[65].

Equivalently, an odd impulse response can be multiplied by in the
time domain to yield a purely imaginary impulse response which is
*Hermitian*, and Hermitian signals have real, even Fourier
transforms, as needed for the ``zero-phase'' property.

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