Spectral density matrix

The first term in the bracket stands for the mean, and the second for the spectral density matrix. For the continuous formulation, the covariances for the model and observation errors are given as... 

Here, we briefly describe MFDA. Considering the usual situations where the trajectory is sampled with discrete time steps, we focus on discrete time and continuous frequency spaces. For simplicity, we set the sampling time interval of the trajectory data. At, to unity. We define the spectral density matrix S(f) as the inverse Fourier transform of a lagged variance-covariance matrix C(r) = (v(f)v (f+x)). 

For the estimator of the spectral density matrix S(f) under the finite sampling step T, we adopt the multitaper method (MTM) [26]. The estimator of MTM is given by averaging K independent estimates. 

For a given set of model parameters, the response x is a zero-mean Gaussian process and the (/, / ) element of its power spectral density matrix function S is given by [249] ... 

Consider a discrete stochastic vector process y and a finite number of discrete data points = y , n = 1,2,..., A. Based on D a discrete estimator of the spectral density matrix of the stochastic process y is introduced ... 

Statistical Properties of the Spectral Density Matrix Estimator... 

Next, the statistical properties of the spectral density matrix estimator Sy j cok) are investigated. Denote by Vn((Ok) and licok) the real and imaginary part, respectively, of (cok) so ... 

The modal forcing f is a linear combination of the components of g so it is also a zero-mean Gaussian vector process. It is stationary with the spectral density matrix function ... 

The forcing parameters defining the spectral density matrix function Sg and the modulating function A. 

The Bayesian spectral density approach approximates the spectral density matrix estimators as Wishart distributed random matrices. This is the consequence of the special structure of the covariance matrix of the real and imaginary parts of the discrete Fourier transforms in Equation (3.53) [295]. Another approximation is made on the independency of the spectral density matrix estimators at different frequencies. These two approximations were verified to be accurate at the frequencies around the peaks of the spectmm. The spectral density estimators in the frequency range with small spectral values will become dependent since aliasing and leakage effects have a greater impact on their values. Therefore, the likelihood function is constructed to include the spectral density estimators in a limited bandwidth only. In particular, the loss of information due to the exclusion of some of the frequencies affects the estimation of the prediction-error variance but not the parameters that govern the time-frequency structure of the response, e.g., the modal frequencies or stiffness of a structure. 

Or alternatively by the corresponding cross- form spectral density matrix given by... 

Clearly, the proper definition of the power-spectral density matrix is the cmcial step to address for the stochastic modeling of the... 

The terms of the evolutionary spectmm-compatible power-spectral density matrix are then given by the following equation ... 

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