Discrete estimators of covariance components of vectorial periodically nonstationary random processes

The properties of estimators for invariants of covariance tensor-function of vectorial periodically correlated random processes, calculated on the base of discrete data, are analyzed. It is shown that aliasing effect of the first kind leads to incorrect estimation of the mean function Fourier coefficients and the second kind leads to decreasing a convergence of covariance components estimator. The conditions of avoidance of the aliasing effect of the first and the second kinds are obtained. Formulas for the estimator variance and bias, which allow comparing efficiency of the discrete and the continuous estimators, are derived. The consistency of estimators is proved. Dependences of the estimators variances and biases on realization length and signal parameters are found.

LSM-harmonic analysis of bi-periodic nonstationary vibration signals

The estimators of parameters of bi-periodic nonstationary vibration signal deterministic part, obtained with using the least squares method (LSM), are analyzed. LSM estimation allows avoiding aliasing effects. The formulas for estimators of variance and bias, which describe their dependences on realization length and signal covariance components, are derived. The results are specified for the quadrature model of the signal. LSM has shown its efficiency for separation of harmonics with close frequencies, so it should be considered as the main method for vibration signals analysis. It is shown that its usage allows one to obtain unbiased estimators of bi-periodic nonstationary vibration signal deterministic part regardless of realization length and harmonic frequencies.