Abstract
This thesis studies some of the problems arising in the analysis of random signals. The digital computer simulations of the first and second order Gaussian processes are employed for the problems requiring empirical investigations. Moreover some exact autocorrelation functions are also used for further demonstrations. The required filter characteristics, for generation of the first and second order processes with prescribed autocorrelation functions, are designed and the equations for the digital computer simulations are derived. The Gaussian data are then generated for a variety of simulation studies being undertaken. The statistical errors in the digital estimates of the probability density functions are considered. The sampling properties of the autocorrelation estimates from uniformly sampled data are also studied; the theoretical and empirical estimate errors are compared and a simplification of the complicated expression, giving the expected error magnitudes, is examined. The maximum determinant method of autocorrelation function extrapolation is studied empirically. The reliability test and the extrapolation errors are examined and the best choice of the truncation point is deduced. The equivalence of the maximum determinant and maximum entropy approaches is shown analytically. Some simulation examples of the maximum entropy spectra and their transformations to the autocorrelation domain are also reported. A problem arising in certain situations is that the zero lag coefficient may be known and followed by a number of unknown coefficients and then a knowledge of the remaining portion of the autocorrelation function. A method of estimating the missing initial coefficients has been introduced in Stone (1978), where further research has also been suggested on it, regarding the selection of the estimates. This and further studies of the method are reported in this thesis. The problem of aliasing is analysed and demonstrated. The effects of data interpolation on the spectral estimates are then investigated. In particular, the application of linear and cubic spline interpolation methods, to the autocorrelation function and the sampled data, are considered. Finally, the thesis studies the sequential sampling scheme. Its contribution in minimizing the problem of aliasing, when the sampling interval is restricted to a minimum allowable value, is proved and demonstrated. The methods of estimating the autocorrelation function and spectra, in sequential sampling, are discussed and presented.