Use of the Autocorrelation Matrix in Spectral Analysis.

This thesis derives and discusses the properties of the autocorrelation matrix and its relationship to the power spectrum. It then considers how this matrix maybe used as an aid in minimising two particular problems, that can arise in spectral analysis. The problems considered, are those encountered when only a knowledge of certain portions of the autocorrelation function can be obtained. The first problem is that of truncation’, where the latter portion of an autocorrelation function is unknown and the second, is a problem that can arise when a time series is randomly or sequentially sampled and there exists a restriction on the minimum allowable sample time. In this case the zero lag coefficient is known, then there are a number of unknown coefficients, followed by knowledge of the remaining portion of the autocorrelation function. In both these situations problems can arise in analysing the resulting power spectrum, unless plausible estimates can be given to the unknown coefficients. To aid the first problem this thesis proposes an 'extrapolation method’ and for the second an ’interpolation method’. Both these methods yield estimates for the unknown coefficients, ensuring that the known coefficients retain their original values and that the properties of the autocorrelation matrix are maintained. With both estimation methods an allowable range often exists, from which a value must be chosen for a particular estimate. The selection of this value is discussed together with its physical and theoretical effects on the resulting power spectrum. The reliability of the estimation methods is discussed and results are given of applying both methods, to theoretical and experimental autocorrelation functions. Finally the thesis considers the relationship between the autocorrelation matrix and the power spectrum and investigates the possibility of producing spectral estimates, from eigenvalue and eigenvector analysis of the autocorrelation matrix