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 . In practice, Hilbert transform filters and FIR differentiators are often implemented as odd FIR filters . 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 .
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.