Odd Impulse ReponsesNote 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 . In practice, Hilbert transform filters and 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 that is Hermitian. Hermitian signals have real Fourier transforms . Therefore, a Hermitian impulse response gives a filter having a phase response that is either zero or at each frequency.
Symmetric Linear-Phase Filters
Zero-Phase Filters (Even Impulse Responses)