Orthogonal sets of data windows constructed from trigonometric polynomials

In contrast to common practise in many other physical sciences, the statistical analysis of PTTI data is often based directly on time domain techniques rather than on frequency domain (spectral analysis) techniques. The predominant analysis technique in the PTTI community, namely, the two-sample (or Allan) variance, is often used to indirectly infer frequency domain properties under the assumption of a power-law spectrum. Here we argue that direct use and estimation of the spectrum of PTTI data have a number of potential advantages. First, spectral estimators are typically scaled independent chisquare random variables with a known number of degrees of freedom. These properties allow easy computation of the variance of estimators of various quantities that are direct functions of the spectrum. Second, the effect of detrending data can be quantified more easily in the frequency domain than in the time domain. Third, the variance of estinaators of the two-sample variance can be expressed in terms of readily estimated spectral density functions. This allows one to generate confidence intervals for the two-sample variance without explicitly assuming a statistical model. Fourth, there exist tractable statistical techniques for estimating the spectrum from data sampled on an unequally spaced grid or from data corrupted by a small proportion of additive outliers. The two-sample variance cannot be readily generalized to these situations.