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, 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 $j$ 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 $j$ 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.