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 | Maximum entropy spectral analysis |  |
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The following derivation of the relationship between entropy and power spectrum is essentially
the same as that given by ().
The derivation is included here for completeness.
The entropy of
random variables
,
,
is given by
![$\displaystyle H_N = - \int P(X_1,\ldots,X_N) \ln[a^{2N}P(X_1,\ldots,X_N)]d^NX = -2N\ln a - \int P\ln Pd^NX,$](img176.png) |
(A-1) |
where
is the joint probability density and
is a constant with the same units as
. The power
spectrum
computed from the autocorrelation values
depends only on the
second-order statistics of the time series
. Therefore, the given time series
cannot be distinguished from a normal (Gaussian) process.
The joint probability density for a normal process with
variables of zero mean is
(using matrix notation, where
is the transpose of
)
![$\displaystyle P(X_1,\ldots,X_N) = \left[\left(2\pi e\right)^N\det T_{N-1}\right]^{-\frac{1}{2}} \exp\left(-\frac{1}{2}X^T\cdot T^{-1}_{N-1}\cdot X\right),$](img180.png) |
(A-2) |
where
is the
Toeplitz matrix [see ()]
given by Equation (11) and
is the
-vector
determined by
.
Substituting (A-2) into (A-1), we find
![$\displaystyle H_N = \frac{1}{2}\ln\left[(2\pi e)^N\det T_{N-1}\right]-2N\ln a.$](img182.png) |
(A-3) |
Setting the arbitrary constant
for convenience, Equation (A-3)
then becomes
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(A-4) |
Since (A-4) necessarily diverges as
, a better measure
of the information content of the series is the average entropy per variable
given by
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(A-5) |
The eigenvalues
of
are real and nonnegative since
is Hermitian and nonnegative
definite. Furthermore,
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(A-6) |
so
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(A-7) |
The Szëgo theorem (, ,)
states that, if
is any continuous function, then
![$\displaystyle \lim_{N\to \infty} \frac{1}{N}\sum_{i=1}^N F(\lambda_i) = \frac{1}{2W}\int_{-W}^W F[2WP(f)] \,df,$](img190.png) |
(A-8) |
where, as before,
is the Nyquist frequency,
is the power spectrum, and the
's are the
eigenvalues of
.
Combining Equations (A-7) and (A-8), we find
![$\displaystyle h = \frac{1}{4W}\int^W_{-W} \ln[2WP(f)]\,df,$](img192.png) |
(A-9) |
which is the sought after result.
 |
 |
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 | Maximum entropy spectral analysis |  |
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Next: APPENDIX B
Up: Berryman: MESA
Previous: acknowledgments
2009-04-13