Investigating the Biases in Allan and Hadamard Variances as Measures of Mth Order Random Stability

Victor R. Reinhardt

Abstract:	Allan and Hadamard variances (AHVs) generated from data consisting of purely random noise are well-known as mean square measures of M^th order random stability (MORS) over the difference interval?. When data contain deterministic drift (intrinsic aging plus environmentally induced temporal changes) as well as noise, however, it is also well known that AHVs can be biased measures of MORS. In such cases, one generally minimizes this bias by “removing” the drift from the data in question using auxiliary data fitting methods and then generating the AHV using the drift-removed residuals. This paper investigates the nature of one aspect of the residual AHV bias that remains after such drift-removal: kernel or Kstat(f)-bias. This bias exhibits itself in the spectral integral representation of a drift-removed AHV as an alteration of the spectral kernel Kstat(f) that relates the AHV to S(f), the power spectral density of the random data component from the non-drift removed AHV. This Kstat(f)-bias occurs because a fitting process, by its very nature, removes some of the noise along with the drift in the fit residuals, especially when negative power law noise is present. In this paper, charts of Kstat(f)-bias are generated as a function of ?/T for various AHV statistics (overlapping, total, and modified), various drift-removal methods (Greenhall, various polynomial order least squares fits, and environmental drift removal), and various power law noise processes. Selected drift-removed Kstat(f) charts are also examined to provide intuitive insight into why drift-removal Kstat(f)-bias behaves as it does. To aid in the generation of the numerous drift-removed bias charts in this paper, an efficient numerical technique is introduced for numerically computing drift-removed Kstat(f) directly in the time-domain from simple t-domain definitions of AHV statistics and driftremoval methods. This technique avoids the need for computationally intensive phase randomization of a t-domain input in order to eliminate errors that occur when the t-domain input is not wide-sense stationary. Finally, the paper demonstrates two important results when higher-order negative power law noise is present: (a) that noise whitening (increasing the order of a polynomial drift-removal fit until the data residuals appear uncorrelated) greatly increases the Kstat(f)-bias in drift-removed AHVs and leads to misleadingly-low estimates of the MORS, and (b) that the removal of temporally complex environmental drift can also generate significant Kstat(f)-bias.
Published in:	Proceedings of the 43rd Annual Precise Time and Time Interval Systems and Applications Meeting November 14 - 17, 2011 Hyatt Regency Long Beach Long Beach, California
Pages:	81 - 94
Cite this article:	Reinhardt, Victor R., "Investigating the Biases in Allan and Hadamard Variances as Measures of Mth Order Random Stability," Proceedings of the 43rd Annual Precise Time and Time Interval Systems and Applications Meeting, Long Beach, California, November 2011, pp. 81-94.
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