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A Wavelet Perspective on the Allan Variance

机译：小波视角的艾伦方差

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摘要

The origins of the Allan variance trace back 50 years ago to two seminal papers, one by Allan (1966) and the other by Barnes (1966). Since then, the Allan variance has played a leading role in the characterization of high-performance time and frequency standards. Wavelets first arose in the early 1980s in the geophysical literature, and the discrete wavelet transform (DWT) became prominent in the late 1980s in the signal processing literature. Flandrin (1992) briefly documented a connection between the Allan variance and a wavelet transform based upon the Haar wavelet. Percival and Guttorp (1994) noted that one popular estimator of the Allan variance—the maximal overlap estimator—can be interpreted in terms of a version of the DWT now widely referred to as the maximal overlap DWT (MODWT). In particular, when the MODWT is based on the Haar wavelet, the variance of the resulting wavelet coefficients—the wavelet variance—is identical to the Allan variance when the latter is multiplied by one-half. The theory behind the wavelet variance can thus deepen our understanding of the Allan variance. In this paper, we review basic wavelet variance theory with an emphasis on the Haar-based wavelet variance and its connection to the Allan variance. We then note that estimation theory for the wavelet variance offers a means of constructing asymptotically correct confidence intervals (CIs) for the Allan variance without reverting to the common practice of specifying a power-law noise type . We also review recent work on specialized estimators of the wavelet variance that are of interest when some observations are missing (gappy data) or in the presence of contamination (rogue observations or outliers). It is a simple matter to adapt these estimators to become estimators of the Allan variance. Finally we note that wavelet variances based upon wavelets other than the Haar offer interesting generalizations of the Allan variance.
机译：艾伦方差的起源可追溯到50年前，两篇开创性的论文，一份是艾伦（1966），另一份是巴恩斯（1966）。从那时起，艾伦方差在表征高性能时间和频率标准方面一直发挥着领导作用。小波首先出现在1980年代初的地球物理文献中，而离散小波变换（DWT）在1980年代后期的信号处理文献中变得十分突出。 Flandrin（1992）简要记录了Allan方差和基于Haar小波的小波变换之间的联系。 Percival and Guttorp（1994）指出，可以用DWT的一种形式来解释Allan方差的一种流行的估算器-最大重叠估算器，现在该模型被广泛称为最大重叠DWT（MODWT）。特别是，当MODWT基于Haar小波时，当小波系数乘以二分之一时，所得小波系数的方差（小波方差）与艾伦方差相同。因此，小波方差背后的理论可以加深我们对Allan方差的理解。在本文中，我们回顾了基本的小波方差理论，重点是基于Haar的小波方差及其与Allan方差的联系。然后我们注意到，小波方差的估计理论提供了一种构造Allan方差的渐近正确置信区间（CI）的方法，而无需恢复指定幂律噪声类型的常规做法。我们还回顾了有关小波方差的专门估计器的最新工作，当缺少某些观测值（盖比数据）或存在污染（无赖观测值或离群值）时，这是有意义的。使这些估计量适应成为Allan方差的估计量是一件简单的事情。最后，我们注意到基于非Haar小波的小波方差提供了Allan方差的有趣概括。

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