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Chapter 2:
Canonical Expansions
Abstract

Just as an appropriate representation of a deterministic function can facilitate an application, such as with trigonometric Fourier series, decomposition of a random function can make it more manageable. Specifically, given a random function X(t), where the variable t can either be vector or scalar, we desire a representation of the form in Eq. (2.1), where x1(t), x2(t),... are deterministic functions, Z1,Z2, ... are uncorrelated zero-mean random variables, the sum may be finite or infinite, and some convergence criterion is given. Equation 2.1 is said to provide a canonical expansion (representation) for X(t). The terms Zk, xk(t), and Zkxk(t) are called coefficients, coordinate functions, and elementary functions, respectively. {Zk} is a discrete white-noise process such that the sum is an expansion of the centered process X(t) minus[1] μX(t) in terms of white noise. Consequently, it is called a discrete white-noise representation.