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Because prediction filters are finite-impulse-response filters, they
can be characterized by the zeros of their *z*-transform.
From equation (2), we know
that the zeros of the prediction filter
are .Therefore, we can express this filter as follows:

| |
(10) |

If we scan the amplitude spectrum of this filter over the *s* plane,
we can find *L* notches at

| |
(11) |

that locate all the zeros of the *z*-transform of this filter.
Similarly, we can express the prediction filter as follows:

| |
(12) |

where denotes the phases of the *M*th order complex roots of
the unity. Now, if we scan the amplitude spectrum of over
the *s* plane, we can
find notches at

| |
(13) |

*M* times as many notches as that of .Comparing equation (13) with equation (11),
it is apparent that these two equations become identical
when is equal to zero. Thus, *L* out of zeros
of are the zeros of .Our goal is to identify these *L* zeros when is known.
If the component of data at frequency is not spatially aliased, then
has *L* zeros between two vertical lines
and , which are *L* zeros of . However,
if the component of data at frequency is spatially aliased,
the task of identifying the zeros becomes complicated and requires
sophisticated algorithms.

** Next:** Dealiasing prediction filters with
** Up:** DEALIASING THE PREDICTION FILTERS
** Previous:** DEALIASING THE PREDICTION FILTERS
Stanford Exploration Project

11/18/1997