Zero-Phase Filters Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

## Zero-PhaseFilters(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.12.1That 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.

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 for a zero-phase filter . This is a real and even function of .

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 design

The impulse response h is returned in linear-phase form, so it must be left-shifted samples to make it zero phase. 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.

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