## 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)