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.