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 $ \alpha=0$. The real impulse response $ h(n)$ of a zero-phase filter is even.11.1 That is, it satisfies

$\displaystyle h(n) = h(-n), \quad n\in{\bf Z}
$

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 $ h(n)=g\delta(n)$). 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,

$\textstyle \parbox{0.8\textwidth}{\emph{a real, even impulse response corresponds to a real, \\
even frequency response.}}$
This follows immediately from writing the DTFT of $ h$ in terms of a cosine and sine transform:

$\displaystyle H(e^{j\omega T}) \eqsp$   DTFT$\displaystyle _{\omega T}(h)
\eqsp \sum_{n=-\infty}^\infty h(n) \cos(\omega nT)
- j \sum_{n=-\infty}^\infty h(n) \sin(\omega nT)
$

Since $ h$ 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

$\displaystyle H(e^{j\omega T}) \eqsp \sum_{n=-\infty}^\infty h(n) \cos(\omega nT)
$

for any real, even impulse-response $ h$. Thus, the frequency response $ H(e^{j\omega T})$ is a real, even function of $ \omega$.

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

$\textstyle \parbox{0.8\textwidth}{A filter is said to be \emph{zero phase} when...
... frequency
$\omega$, and when $H(e^{j\omega T})>0$\ in the filter passband(s).}$

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 $ \omega_c$ rad/s, the passband is $ \omega\in[-\omega_c,\omega_c]$.

$ \pi $-Phase Filters

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


Phase $ \pi $ in the Stopband

Practical zero-phase filters are zero-phase in their passbands, but may switch between 0 and $ \pi $ 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 $ \pi $ over various intervals within stopbands. Similarly, practical ``linear phase'' filters are typically truly linear phase across their passbands, but typically exhibit discontinuities by $ \pi $ radians in their stopband(s). As long as the stopbands are negligible, which is the goal by definition, the $ \pi $-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 design
The impulse response h is returned in linear-phase form, so it must be left-shifted $ (N-1)/2=5$ samples to make it zero phase.

Figure 10.1: Impulse response and frequency response of a length 11 zero-phase FIR lowpass filter. Note that the frequency response is real because the filter is zero phase. Also plotted (in dashed lines) are the desired passband and stopband gains.
\includegraphics[width=\twidth ]{eps/remezexa}

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 $ \pi $ radians, so that the phase alternates between zero and $ \pi $ in the stopband. This is typical of practical zero-phase filters.

Figure: Amplitude response and phase response of the length 11 zero-phase FIR lowpass filter in Fig.10.1.
\includegraphics[width=\twidth ]{eps/remezexb}


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 $ h(n)=\delta(n)$ has frequency response $ H(e^{j\omega T})=1$, which is zero phase for all $ \omega$.

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

    $\displaystyle h(n) \eqsp b_{1}\delta(n+1) + b_0\delta(n) + b_1 \delta(n-1),
$

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

    $\displaystyle H(z) \eqsp b_{1}z + b_0 + b_1 z^{-1}
$

    and the frequency response is

    $\displaystyle H(e^{j\omega T}) \eqsp b_{1}e^{j\omega T}+ b_0 + b_1 e^{-j\omega T}\eqsp b_0 + 2 b_1 \cos(\omega T)
$

    which is real for all $ \omega$.

  • Extending the previous example, every order $ 2N$ zero-phase real FIR filter has an impulse response of the form

    \begin{eqnarray*}
\lefteqn{
h(n) \eqsp
b_{N}\delta(n+N)
\;+\; \cdots
\;+\; b_{...
...ots
& & \;+\; b_1 \delta(n-1)
\;+\; \cdots
\;+\; b_N\delta(n-N)
\end{eqnarray*}

    and frequency response

    $\displaystyle H(e^{j\omega T}) \eqsp b_0 \;+\; 2 \sum_{k=1}^N b_k \cos(k\omega T)
$

    which is clearly real whenever the coefficients $ b_k$ 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 $ h(n)=\delta(n+1/2) + \delta(n-1/2)$. Since the bandlimited digital impulse signal $ \delta (n)$ is ideally interpolated using bandlimited interpolation [91,84], giving samples of sinc$ (n)\isdeftext \sin(\pi n)/(\pi n)$--the unit-amplitude sinc function having zero-crossings on the integers, we see that sampling $ h$ on the integers yields an IIR filter:

    $\displaystyle h(n) = \sum_{m=-\infty}^{\infty}$   sinc$\displaystyle (n-m-1/2) +$   sinc$\displaystyle (n-m+1/2)
$

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


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