How Extracting Information from Data Highpass Filters its Additive Noise

Victor S. Reinhardt

Abstract:	This paper examines the characteristics of three types of random error measures in the presence of negative power law (neg-p) noise: (a) the observable residual error after removing an estimate of an information containing causal function from data, (b) the jitter, the residual error with additional highpass (HP) filtering, and (c) Mth-order difference (?) variances, such as the Allan variance (1st-order ?-variance of the fractional frequency error y(t)) and the Hadamard-Picinbono variance (2nd-order ?-variance of y(t)). Measures (b) and (c) are used to mitigate perceived divergence problems in the mean square (MS) of Measure (a) due to the presence of neg-p noise. This paper proves that this perception is wrong; it shows that the MS of Measure (a) converges in the presence of neg-p noise by demonstrating that extracting a statistically optimal estimate of the causal behavior from data HP filters the noise in the measure. It is further shown that the order of this noise HP filtering increases with the complexity of the model function used to estimate the causal behavior in the data. Thus, if one is free to choose the complexity of the model function, the MS observable residual error is guaranteed to converge for any negative power in the noise PSD. Because of this, it is shown that the jitter can be defined simply as the observable residual error without additional HP filtering, making the jitter and residual error the same error measure. This paper finally shows that an Mth-order ?-variance is also a measure of the MS of the observable residual error for any number of data samples when the model function is an (M-1)th-order polynomial. This completes the equivalence, showing that Measures (a), (b), and (c) all measure the same kind of error when the model function for the causal behavior is a polynomial. The consequences of this equivalence are then explored. Among these is a physical explanation for the fact that the Allan variance is sensitive to frequency drift, while the Hadamard-Picinbono variance is not.
Published in:	Proceedings of the 39th Annual Precise Time and Time Interval Meeting November 27 - 29, 2007 Hyatt Regency Long Beach Long Beach, California
Pages:	559 - 580
Cite this article:	Reinhardt, Victor S., "How Extracting Information from Data Highpass Filters its Additive Noise," Proceedings of the 39th Annual Precise Time and Time Interval Meeting, Long Beach, California, November 2007, pp. 559-580.
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