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Nonstationary fluctuations in stationary time-series

In making an estimate of the power spectrum of a process from a short sample the stationarity assumption used to formulate the spectrum estimate is often violated, so that reliable assessment of the error of the estimate is difficult. Here we examine the problems of detecting narrow-band non-stationarity in short data samples and the influence of such non-stationarity on the variance of spectrum estimates. We decompose the covariance matrix of the eigencoefficients used in multiple-window spectrum estimation methods into a series of known basis matrices with scalar coefficients. For a given bandwidth and sample size, we describe simultaneous orthogonal expansions for both the power (time) function and for the eigencoefficient covariance matrix. The limiting power basis functions are eigenfunctions of a narrow band sinc2 kernel while the corresponding basis matrices are trace-orthogonal so that the observable non-stationary is described by a series of quadratic forms.