Nearly everyone has some notion of random or noiselike signals. One has only to tune a radio away from a station, turn up the volume, and the result is static, or noise. If one were to look at a recording of such a signal, it would appear to wander on aimlessly with no apparent order in its amplitude pattern, as shown in Fig. 1.1. Signals of this type cannot be described with explicit mathematical functions such as sine waves, step functions, and the like. Their description must be put in probabilistic terms. Early investigators recognized that random signals could be described loosely in terms of their spectral content, but a rigorous mathematical description of such signals was not formulated until the 1940s, most notably with the work of Wiener and Rice (1, 2).

Probability plays a key role in the description of noiselike or random signals. It is especially important in Kalman filtering, which is also sometimes referred to as statistical filtering. This is a bit of a misnomer though, because Kalman filtering is based on probabilistic descriptors of the signals and noise, and these are assumed to be known at the beginning of the filtering problem. Recall that in probability we assume that we have some a priori knowledge about the likelihood of certain elemental random outcomes. Then we wish to predict the theoretical relative frequency of occurrence of combinations of these outcomes. In statistics we do just the reverse. ...

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