116 A. Hatstatt and K. E. Tatsis where H(iω) is the transfer function and i denotes the imaginary unit. It should be noted that when the measurement noise v(t) is additionally considered, the expression of the spectral density of h(t) described above should be slightly adjusted. Due to the fact that the noise term has only an additive effect to the signal h(t), the spectral density of the latter should be added to the one of the measurement noise. Due to the relation between the spectral density and the covariance function, which under stationary conditions are connected through the Fourier transform, the spectral characteristics of the signal h(t) can be inversely defined by specifying the covariance kernel of the signal, which results in a specific spectral content Sh(ω). The latter in turn can be converted to a state-space representation, according to the factorization postulated by Eq. (11). It is shown in [11] that such a conversion is possible, when the spectral density of the signal is amenable to a rational form. Covariance functions Various classes of covariance kernels exist, each of which can attribute different features to the modeled signal. This work is based on the use of Matern and periodic kernels, whose spectral density has an analytical description and their corresponding state-space matrices are presented below. Matern kernel The family of Matern kernels are used to describe stationary covariance functions for different types of random processes, while having only a few tunable hyperparameters. One of the most typically employed kernels belonging to this family is the one for ν =3/2, whose parameters are l and α. The state-space matrices in this case are obtained as Fc = 0 1 λ2 −2λ , Lc = 0 1 , Hc = 1 0 withλ= √3 l andσw =12 √3α2 l3 . Accordingly, the Matern kernel withν =5/2, in which case it is again parametrized with respect tol and α, corresponds to the following state-space matrices Fc = 0 1 0 0 0 1 −λ3 −3λ2 −3λ , Lc = 0 0 1 , Hc = 1 0 0 where λ= √5 l andσw =400 √5α2 3l5. Periodic kernel The periodic kernel enables the modeling of signals which repeat themselves exactly, such as the harmonic signals induced by rotating machines. This type of kernel is entirely deterministic, which implies that the response in time domain does not depend on the noise term. The state-space matrices of a signal whose covariance function is described by a periodic kernel are Fc = 0 −ω ω 0 , Lc = 0 1 , Hc = 1 0 where ωdenotes the eigenfrequency, while σw =0. Harmonic oscillator An extension of the periodic kernel in terms of spectral features is offered by the kernel that represents a harmonic oscillator. This represented by the following matrices Fc = 0 1 −ω2 −2ξω , Lc = 0 1 , Hc = −ω2 −2ξω where ωdenotes the eigenfrequency and ξ is the corresponding damping ratio. The dynamics of such a model are driven by a white noise signal with spectrumσw.
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