A Method for Power Spectrum Parameter Estimation

An asymptotic analysis is carried out for an approximate method of estimating the parameters of the power spectrum of a zero-mean stationary Gaussian random process from an observed realization of limited duration. Maximum likelihood estimates are obtained with the approximation that the coefficients of the Fourier series expansion of the realization are uncorrelated. This is equivalent to other approximation techniques which assume a periodic covariance function. The dispersion of the estimates is evaluated in terms of a quantity called the differential variance. It is shown that with this quantity as a criterion, the approximate estimates are as good, asymptotically, as the exact maximum likelihood estimates. An approximate expression for the differential variance in terms of the power spectrum is given and it is shown that this expression asymptotically approaches its exact value. These results follow from a general expression, obtained by means of a converse to the Schwarz inequality, which compares the differential variance of the approximate estimates with that of the maximum likelihood estimates. This expression is evaluated for the power spectrum parameter estimation problem in terms of the covariance matrix of the Fourier coefficients. The asymptotic behavior of these covariances is bounded so that the convergence of the elements of the inverse covariance matrix can be demonstrated. The results on the differential variance of the approximate estimates are then established by matrix methods. Accepted for the Air Force Stanley J. Wisniewski Lt Colonel, USAF Chief, Lincoln Laboratory Office