In this chapter stochastic models are analyzed in various ways. The following issues will be treated.
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Linear filtering and its effect on the spectrum. Complex-valued signals will be allowed (Section 4.2).
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Spectral factorization (Section 4.3), permitting complex-valued signals.
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A review of linear continuous-time models and their properties (Section 4.4), and sampling of such models (Section 4.5).
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Some aspects on the positive real part of the spectrum (Section 4.6).
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Properties of the bispectrum when stochastic signals are filtered (Section 4.7).
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Algorithms for covariance calculations based on finite-order models (Section 4.8).
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Bibliography
There is a rich literature on continuous-time stochastic systems. An excellent book with profound mathematics combined with engineering application is Åström, K.J., 1970. Introduction to Stochastic Control Theory. Academic Press, New York.
See also Jazwinski, A.H., 1970. Stochastic Processes and Filtering Theory, Academic Press, New York.
Another book dealing with stochastic dynamic models is the classical text Box. G.E.P., Jenkins, G.W., 1976. Time Series Analysis: Forecasting and Control, second ed. Holden-Day, San Francisco.
A comprehensive treatment is also given in Brown, R.G., 1983. Introduction to Random Signal Analysis and Kaiman Filtering. John Wiley & Sons, New York.
The Kalman-Yakubovich lemma is important in several system theoretic contexts. For example, see Anderson, B.D.O., Vongpanitlerd, S., 1973.: Network Analysis and Synthesis. Prentice-Hall, Englewood Cliffs, NJ.
Narendra, K.S., Taylor, J.H., 1973. Frequency Domain Criteria for Absolute Stability. Academic Press, New York.
Algorithms similar to the one presented in Section 4.8.2 are given in Demeure, C.J., Mullis, C.T., 1989. The Euclid algorithm and fast computation of cross-covariance and autocovariance sequences. IEEE Transactions on Acoustics, Speech and Signal Processing. ASSP-37, 545–552.
The algorithms presented in Section 4.8.4 is based on Ježek, J., 1990. Conversion of the polynomial continuous-time model to the delta discrete-time and vice versa. Proc. 11th IFAC World Congress, Tallinn, 1990.
For an efficient algorithm for spectral factorization in the polynomial approach, see Kučera, V., 1979. Discrete Linear Control. John Wiley & Sons, Chichester.
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Söderström, T. (2002). Analysis. In: Discrete-time Stochastic Systems. Advanced Textbooks in Control and Signal Processing. Springer, London. https://doi.org/10.1007/978-1-4471-0101-7_4
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