## 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.

**Next Section:**

Symmetric Linear-Phase Filters

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Zero-Phase Filters (Even Impulse Responses)