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 . Therefore,
This follows immediately from writing the DTFT of in terms of a cosine and sine transform:
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 .