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Kolmogoroff cross-spectral factorization, therefore, provides a tool
to factor the helical 1-D filter of length 2*N*_{x} + 1 into
minimum-phase causal and anti-causal filters of length *N*_{x} +1.
Fortunately, filter coefficients drop away rapidly from either end.
In practice, small-valued coefficients can be safely discarded,
without violating the minimum-phase requirement; so for a
given grid-size, the cost of the matrix inversion scales linearly with
the size of the grid.
The unitary form of equation (3) can be maintained
by factoring the right-hand-side matrix,
in equation (6), with Kolmogoroff before
applying it to .

| |
(11) |

| (12) |

** Next:** Impulse response
** Up:** Helical boundary conditions
** Previous:** Cross-spectral factorization
Stanford Exploration Project

7/5/1998