The power spectrum is the magnitude squared of the frequency distribution. That is:

Ps=Ys Ys*

Where Ps is the power spectrum, Ys is the frequency spectrum and Ys* is the complex conjugate of the frequency spectrum. Taking the inverse Fourier transform of the power spectrum we get the autocorrelation function.

Rs(Tau)=F^-1{Ps} where Rs is the auto correlation function Tau is the time delay

F^-1 is the inverse Fourier transform

The correlation matrix can be written from the auto correlation function as follows:

RR=

[Rs(t1) Rs(t2) … Rs(Tn)]

[Rs(t2) Rs(t1) … Rs(Tn-1)]

…

[[Rs(tn) Rs(t1) … Rs(T1})]

Now that we have the correlation matrix we can whiten the linear equation

Y=Ax

By multiplying both sides of the equation by the square root of the inverse of the correlation matrix. The inverse of the correlation matrix is known as the information matrix.

]]>Perhaps we should weight our information in the frequency components as:

1/sqrt(f*T) Where T is the data length in time and f is the frequency. From this frequency weighting we should be able to devise a whitening filter.

]]>Mark

]]>“Figure 2. Top – archived version. Middle – 3 series version pre-stabilization. Bottom – 3-series version, post-stabilization.” ]]>

Mark

]]>I saw the Stone book listed on Amazon, and may buy it at some point, maybe even the HKO book. I have access to both of those papers via IEEE as well. Thank you for the pointers. Odd that the new Karhunen would be unrelated, yet doing nearly identical work with the same name…

In the end, this all has to result in a dissertation topic, and, preferably, an actual dissertation! ðŸ™‚

Mark

]]>1) P. Comon, Independent Component Analysis, a new concept ?, Signal Processing, 36(3):287–314, April 1994.

2) J-F. Cardoso, Blind signal separation: statistical principles, Proceedings of the IEEE, 90(8):2009–2026, October 1998.

re #38: Juha Karhunen has no relation to Karhunen-Loeve, and to my knowledge is not an offspring either.

]]>A number of lawsuits have been filed by proponents of AGW to force the Federal Government to impose CO2 emission controls. The City of Boulder has filed suit. Calling Mann or any other scientist, both believer or skeptic, would be an option.

I don’t know the status of these suits but the AGW community should be careful what they wish for. The skeptics will be able to present their entire case under oath and on the record. To quote President Bush, “Bring it on!”.

]]>mark

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