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A random function is a complicated mathematical entity. For some applications, one need only consider second-order characteristics and not be concerned with the most delicate mathematical aspects; for others, it is beneficial to find convenient representations to facilitate the use of random functions.
If an appropriate canonical representation can be found, then dealing with a family of random variables defined over the domain of t is reduced to considering a discrete family of random variables. Equally as important is that, whereas there may be a high degree of correlation among the random variables composing the random function, the random variables in a canonical expansion are uncorrelated. The uncorrelatedness of the Z k is useful in eliminating redundant information and designing optimal linear filters.
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