Nonstationarity: patching |

Since signal and noise are uncorrelated, the spectrum of data is the spectrum of the signal plus that of the noise. An equation for this idea is

This says resonances in the signal and resonances in the noise will both be found in the data. When we are given and it seems a simple matter to subtract to get . Actually it can be very tricky. We are never given and ; we must estimate them. Further, they can be a function of frequency, wave number, or dip, and these can be changing during measurements. We could easily find ourselves with a negative estimate for which would ruin any attempt to segregate signal from noise. An idea of Simon Spitz can help here.

Let us reexpress equation (15) with prediction-error filters.

(16) |

Inverting

(17) |

The essential feature of a PEF is its zeros. Where a PEF approaches zero, its inverse is large and resonating. When we are concerned with the zeros of a mathematical function we tend to focus on numerators and ignore denominators. The zeros in compound with the zeros in to make the zeros in . This motivates the ``Spitz Approximation.''

(18) |

It usually happens that we can find a patch of data where no signal is present. That's a good place to estimate the noise PEF . It is usually much harder to find a patch of data where no noise is present. This motivates the Spitz approximation which by saying tells us that the hard-to-estimate is the ratio of two easy-to-estimate PEFs.

It would be computationally convenient if we had expressed not as a ratio. For this, form the signal by applying the noise PEF to the data . The spectral relation is

(19) |

Inverting this expression and using the Spitz approximation we see that a PEF estimate on is the required in numerator form because

(20) |

Nonstationarity: patching |

2013-07-26