Foundations for Stochastic Spectral Analysis

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Abstract
Introduction In the previous chapter we produced representations for various deterministic functions and sequences in terms of linear combinations of sinusoids with different frequencies (for mathematical convenience we actually used complex exponentials instead of sinusoids directly). These representations allow us to easily define various energy and power spectra and to attach a physical meaning to them. For example, subject to square integrability conditions, we found that periodic functions are representable (in the mean square sense) by sums of sinusoids over a discrete set of frequency components, while nonperiodic functions are representable (also in the mean square sense) by an integral of sinusoids over a continuous range of frequencies. For periodic functions, the energy from ∞ to ∞ is infinite, so we can define their spectral properties in terms of distributions of power over a discrete set of frequencies. For nonperiodic functions, the energy from –∞ to ∞ is finite, so we can define their properties in terms of an energy distribution over a continuous range of frequencies. We now want to find some way of representing a stationary process in terms of a ‘sum’ of sinusoids so that we can meaningfully define an appropriate spectrum for it; i.e., we want to be able to directly relate our representation for a stationary process to its spectrum in much the same way we did for deterministic functions. Now a stationary process has associated with it an ensemble of realizations that describe the possible outcomes of a random experiment.

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