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The multidimensional demodulation problem is considered from the point of view of statistical estimation theory and a posteriori most probable signal estimates are derived. Correlated signals and noises are treated. This formulation yields a set of two matrix integral equations which must be solved for the optimum estimates. For amplitude modulation, the problem reduces to that of finding a set of time varying filters which are, again, solutions to a matrix integral equation. Special cases such as two-receiver systems, quadrature modulation, and single-sideband have particularly simple representations and are considered in some detail.