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 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

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

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

. 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 $\omega\in[-\omega_c,\omega_c]$ .

Subsections

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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See Also

Introduction to Digital Filters
    Filters Preserving Phase

             -Phase Filters