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*.^{12.1}That is, it satisfies

Note that 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 usually preferred.

It is a well known Fourier symmetry that real, even signals have real, even Fourier transforms. 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
A real frequency response has phase zero when it is positive, and a
phase of radians when it is negative. Therefore, a ``zero-phase
filter,'' as we have defined [
] may actually have a
phase response of 0 or at each frequency. In practice, the
filter is usually precisely zero-phase in all ``pass bands'', while it
switches between 0 and in the stop bands (frequency intervals
where the gain is desired to be zero). A better name, in general,
would be *piecewise constant-phase filter*. For any real filter,
the constant phase can be either 0 or . Similarly, the term
``linear phase'' could be sharpened to ``linear-phase with
discontinuities by any multiple of radians''.

Figure 11.1 shows the impulse response and frequency response
of a length 11 zero-phase FIR lowpass filter designed using the Remez
exchange algorithm.^{12.2} The Matlab code for designing this filter is as
follows:

N = 11; % filter length - must be odd b = [0 0.1 0.2 0.5]*2; % band edges M = [1 1 0 0 ]; % desired band values h = remez(N-1,b,M); % Remez multiple exchange designThe impulse response

Figure 11.2 shows the amplitude and phase responses of the FIR
filter. We see that each zero-crossing in the frequency response
results in a phase jump of radians. The phase is zero
throughout the pass-band, but in the *stop-band*, the phase
alternates between zero and . In practice, very few ``zero-phase
filters'' have a truly zero phase response at all frequencies.
Instead, the phase typically alternates between zero and , as
shown in Fig.11.2. However, the phase values typically
occur only in the stop-band of the filter, *i.e.*, at frequencies where
the amplitude response is so small that it can be neglected. At
frequencies for which the phase response is , the filter may be
said to be ``inverting'', *i.e.*, it negates the frequency components at
such frequencies.

In view of the foregoing, the term ``zero-phase filter'' is only
approximately descriptive. More precise terms would be
*0-or--phase filter*, *even-impulse-response
filter*, or *real-frequency-response filter*.

[How to cite this work] [Order a printed hardcopy]

Copyright ©

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

[Automatic-links disclaimer]