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Up: Sava & Fomel: Spectral
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We can now extend the equations derived for real numbers to
polynomials of Z, with
, and obtain spectral
factorization algorithms similar to the Wilson-Burg method
Sava et al. (1998), as follows:
| ![\begin{displaymath}
\fbox {$ \displaystyle
X_{n+1}=\frac{S+X_{n} \bar{G}}{ \bar{X_n} + \bar{G}}
$}
\end{displaymath}](img24.gif) |
(8) |
If L represents the limit of the series in (8),
![\begin{displaymath}
L \bar{L} + L \bar{G} = S + L \bar{G} \end{displaymath}](img25.gif)
and so
![\begin{displaymath}
L \bar{L} = S\end{displaymath}](img26.gif)
Therefore, L represents the causal or anticausal part of the given
spectrum
.
Table 3 summarizes the spectral factorization
relationships equivalent to those established for real numbers in
Table 1.
Table 3:
Spectral factorization
General |
![$X_{n+1} =\frac{S+X_n \bar G }{ \bar X_n+\bar G }$](img28.gif) |
Muir |
![$X_{n+1} =\frac{S+X_n }{ \bar X_n+1 }$](img29.gif) |
Secant |
![$X_{n+1} =\frac{S+X_n \bar X_{n-1}}{ \bar X_n+\bar X_{n-1}}$](img30.gif) |
Newton |
![$X_{n+1} =\frac{S+X_n \bar X_n }{2\bar X_n }$](img31.gif) |
Ideal |
![$X_{n+1} =\frac{S+X_n\sqrt{S} }{ \bar X_n+\sqrt{S} }$](img32.gif) |
The convergence properties are similar to those derived for
real numbers. As shown above, the Newton-Raphson method should have
the fastest convergence.
Next: A comparison with the
Up: Sava & Fomel: Spectral
Previous: The convergence rate
Stanford Exploration Project
4/20/1999