##
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

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:

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

and frequency response

- 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

**Previous Section:**

Linear-Phase Filters (Symmetric Impulse Responses)