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Comparative analysis of two-sided linear prediction (TSP) and one-sided linear prediction (OSP) in terms of linear-phase filter design, prediction error, spectral estimation and use in time series interpolation shows that the error processes of TSP and OSP are nonwhite and white, respectively, and that the mean square error of TSP is smaller than that of OSP. The two-sided AR modelling problems are different and similar in TSP and OSP, respectively. During spectral estimation TSP corresponds to the spectral peaks while OSP to the spectral nulls.

A new data-adaptive estimator for determining the spectrum of nonstationary signals employs the data on the signal components and the minimum mean-squared error estimate of the time-varying amplitude of the sinusoidally-represented signal. Repetition of the process yields a power distribution similar to the Wold-Cramer evolutionary spectrum. Simulation examples demonstrate the high performance of the estimator with respect to positivity and time-frequency resolution.

Two orthonormal tapers, the sinusoidal taper and the minimum bias (MB) taper, facilitate multitaper spectral analysis. Both sinusoidal and MB tapers do not involve an additional spectral bandwidth parameter, with the sinusoidal tapers converging to MB tapers as sequence length tends to infinity. Addition or deletion of tapers allows the local adjustment of the multitaper spectral bandwidth estimate. The orthonormal taper-based spectral estimation approach has a performance superior to that of Slepian tapers.

The estimation of the spectra of nonstationary signals is performed by a newly developed, evolutionary periodogram method. The expression of time varying amplitude at various frequencies as orthonormal function expansion is the basis for this method. The standard periodogram estimator occurs in the evolutionary periodogram as a special case.

A dependable estimator of the stage of cross bispectrum was identified and the precise distribution of the stage of an intricate Gaussian sample bispectrum was obtained. In a majority of instances, a Gaussian distribution can be used to estimate the precise distribution. The variance of the time delay estimate calculated from the sample cross bispectrum of a signal in additive correlated noise is obtained by using this Gaussian estimate.