The optimal filter of Kalman is derived for a general class of linear distributed-parameter systems with Gaussian disturbances and measurement noise. The concept of characteristic functional, which fully describes a distributed infinite-dimensional random variable, is used. The input disturbance and the measurement noise are assumed to be white in time, but they are allowed to have any correlation in space. A numerical example illustrates the theory.
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Tzafestaş et al. (1968) studied this question.
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