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.

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University