Formulation for Multiple-channel Output Measurements

Assume that discrete response data are available for No N ) instrumented DOFs and use Ar to denote the sampling time step. Because of measurement noise and modeling error, referred to hereafter jointly as prediction error, the measured response y e at time t = nAt differs 

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  

Since x and e are assumed statistically independent, q and e are also independent. By using Equations (3.8) and (3.44), taking expectation of Equation (3.46), yields  

Bayesian Methods for Structural Dynamics and Civil Engineering 

Grouping together the terms with the same value of n - n, the following expression can be obtained by taking expectation of Equation (3.50)  

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