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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\mathbb{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 its frequency
response $H(e^{j\omega T})$\ is a real and even function of radian 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]$ .

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University