Abstract
This chapter provides a spectral representation of stationary, discrete-time random signals and determines that its autocovariance function is a positive-definite sequence cumulative power spectrum is also introduced. Stochastic integrals with respect to signals with uncorrelated increments are developed permitting introduction of computer algorithms to simulate stationary signals with a given spectral density.
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Notes
- 1.
See Sect. 4.1 or, e.g., M. Denker and W.A. Woyczyński, Introductory Statistics and Random Phenomena. Uncertainty, Complexity and Chaotic Behavior in Engineering and Science, Birkhäuser-Boston 1998.
- 2.
See, e.g., G.B. Folland, Real Analysis, J. Wiley, New York 1984.
- 3.
A step proving the existence of an infinite such sequence requires an application of the so-called Kolmogorov Extension Theorem, see, e.g., P. Billingsley, Probability and Measure, Wiley, New York, 1986.
- 4.
Recall that the indicator function 1 A(w) is defined as being equal to 1 for w belonging to set A, and being 0 for w outside A.
- 5.
See, e.g., G. B. Folland, Real Analysis, W. Wiley, New York 1984.
- 6.
In the sense that it preserves the norms: the standard deviation is the norm in space \(L^2_0({\mathbf {P}})\), and \(\|a\|=(\int _0^1|a(f)|{ }^2 \,d\mathcal {S}_X(f))^{1/2}\), for an a(f) in \(L^2(d\mathcal {S}_X(f))\).
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Woyczyński, W.A. (2019). Spectral Representation of Discrete-Time Stationary Signals and Their Computer Simulations. In: A First Course in Statistics for Signal Analysis. Statistics for Industry, Technology, and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-20908-7_10
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DOI: https://doi.org/10.1007/978-3-030-20908-7_10
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