Zero-Phase Filters

(Even Impulse Responses)

A *zero-phase filter* is a special case of a linear-phase filter
in which the phase slope is
. The real impulse response
of a zero-phase filter is *even*.^{11.1} That is, it satisfies

Note that every even signal is symmetric, but not every symmetric signal is even. To be even, it must be symmetric about time 0 .

A *zero-phase filter cannot be causal* (except in the trivial
case when the filter is a constant scale factor
).
However, in many ``off-line'' applications, such as when filtering a
sound file on a computer disk, causality is not a requirement, and
zero-phase filters are often preferred.

It is a well known Fourier symmetry that *real, even signals have
real, even Fourier transforms* [84]. Therefore,

This follows immediately from writing the DTFT of in terms of a cosine and sine transform:

DTFT

Since is even, cosine is even, and sine is odd; and since even times even is even, and even times odd is odd; and since the sum over an odd function is zero, we have that

for any real, even impulse-response . Thus, the frequency response is a real, even function of .

A real frequency response has phase zero when it is positive, and
phase
when it is negative. Therefore, we define
a *zero-phase filter* as follows:

Recall from §7.5.2 that a *passband* is defined as a
frequency band that is ``passed'' by the filter, *i.e.*, the filter is
not designed to minimize signal amplitude in the band. For example,
in a lowpass filter with cut-off frequency
rad/s, the
passband is
.

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