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 .

-Phase Filters

Under our definition, a zero-phase filter always has a real, even impulse response [ ], but not every real, even, impulse response is a zero-phase filter. For example, if is zero phase, then is not; however, we could call a -phase filter'' if we like (a zero-phase filter in series with a sign inversion).

Phase in the Stopband

Practical zero-phase filters are zero-phase in their passbands, but may switch between 0 and in their stopbands (as illustrated in the upcoming example of Fig.10.2). Thus, typical zero-phase filters are more precisely described as piecewise constant-phase filters, where the constant phase is 0 in all passbands, and over various intervals within stopbands. Similarly, practical linear phase'' filters are typically truly linear phase across their passbands, but typically exhibit discontinuities by radians in their stopband(s). As long as the stopbands are negligible, which is the goal by definition, the -phase regions can be neglected completely.

Example Zero-PhaseFilter Design

Figure 10.1 shows the impulse response and frequency response of a length 11 zero-phase FIR lowpass filter designed using the Remez exchange algorithm.11.2 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 10.2 shows the amplitude and phase responses of the FIR filter designed by remez. The phase response is zero throughout the passband and transition band. However, each zero-crossing in the stopband results in a phase jump of radians, so that the phase alternates between zero and in the stopband. This is typical of practical zero-phase filters.

Elementary Zero-Phase Filter Examples

A practical zero-phase filter was illustrated in Figures 10.1 and 10.2. Some simple general cases are as follows:

• The trivial (non-)filter has frequency response , which is zero phase for all .

• Every second-order zero-phase FIR filter has an impulse response of the form

where the coefficients are assumed real. The transfer function of the general, second-order, real, zero-phase filter is

and the frequency response is

which is real for all .

• Extending the previous example, every order zero-phase real FIR filter has an impulse response of the form

and frequency response

which is clearly real whenever the coefficients are real.

• There is no first-order (length 2) zero-phase filter, because, to be even, its impulse response would have to be proportional to . Since the bandlimited digital impulse signal is ideally interpolated using bandlimited interpolation [91,84], giving samples of sinc--the unit-amplitude sinc function having zero-crossings on the integers, we see that sampling on the integers yields an IIR filter:

sinc   sinc

• Similarly, there are no odd-order (even-length) zero-phase filters.

Next Section:
Odd Impulse Reponses
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Linear-Phase Filters (Symmetric Impulse Responses)