Formulation for Single-channel Output Measurements

This section considers the response measurement of a single-degree-of-freedom system or a single-channel measurement of a multi-degree-of-freedom system. Discrete data is sampled with a time step Ar and y denotes the measured response at time t = nAt. The measurement is different from the model response  

the stochastic process x represents the response of a single-degree-of-freedom (SDOF) system or the response of a particular degree of freedom of a multi-degree-of-lfeedom (MDOF) system. The prediction error is modeled as a zero-mean discrete (band-limited) white noise process e with variance and spectral intensity  

Consider the scaled discrete Fourier transform of the stochastic process x  

Bayesian Methods for Structural Dynamics and Civil Engineering 

the spectral density estimator of y can be computed in analogy to Equation (3.23) as follows  

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