In this paper derivations are presented for exact asymptotic formulas for the means, variances, and covariances of four averaged estimators of physical parameters: two power spectra, the cospectrum and the quadspectrum, at any available discrete positive frequency. The raw estimators are averaged across q statistically independent replications of the experiment, not across nearby discrete frequencies. All these formulas together imply approximations to the mean and variance of estimated ordinary coherence, which then provide guidance in the choice ofq for reliable estimation.
I. INTRODUCTION
In the simplest kind of cross-spectral analysis one assumes that {(x(t),y(t)): -oo < t < oo} is a two-dimensional, weakly stationary, band-limited, Gaussian random process, with random real components x(t) and y(t), zero means, and absolutely integrable correlation functions.
At any available discrete positive frequency f, digital cross-spectral analysis is concerned with four real-valued physical parameters" G~(f) = power spectrum of x; Cxr(f) = cospectrum of x and y; Gy(f) = power spectrum of y; Q~,r(f) = quadspectrum ofx and y.
In general, the true values of these four physical parameters are unknown. Thus it is useful to define an averaged estimator of each parameter that is computable from experimental data. Before the observations are taken, these four estimators are random variables, with a joint probability distribution. That distribution may be summarized, incompletely but nonetheless meaningfully, by 14 statistical parameters: four means, four variances and six covariances.
First, in this paper, exact asymptotic formulas are derived for the means, variances and covariances of these four estimators. Second, these formulas are applied to derive the approximate mean and variance of the estimator of ordinary coherence. These approximations provide useful guidance for the design of experiments to achieve adequate reliability of estimation in cross-spectral analysis.
2. RELEVANT FORMULAS
2.1. STATISTICAL PARAMETERS In this paper the usual formulas are assumed for the means, variances, covariances and correlations of a joint probability density function, as found in any introductory text in mathematical statistics. The Isserlis Theorem [1, p. 27] also is used for real-valued, normally 39