Spectral Density Estimation of Stochastic Vector Processes

A spectral density matrix estimator for stationary stochastic vector processes is studied. As the duration of the analyzed data tends to infinity, the probability distribution for this estimator at each frequency approaches a complex Wishart distribution with mean equal to an aliased version of the power spectral density at that frequency. It is shown that the spectral density matrix estimators corresponding to different frequencies are asymptotically statistically independent. These properties hold for general stationary vector processes, not only Gaussian processes, and they allow efficient calculation of updated probabilities when formulating a Bayesian model updating problem in the frequency domain using response data. A three-degree-of-freedom Duffing oscillator is used to verify the results.