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The characterization and optimization of nonlinear systems is considered from the function space point of view. The design of nonlinear systems is regarded as the problem of mapping the function space of the past of the input onto a line that corresponds to the amplitude of the filter output. An orthogonal functional representation for a system is shown to result from any mapping which partitions this space into nonoverlapping cells. The complication of solving for optimum systems in terms of measured higher order statistics is circumvented by formulating the problem so that particular statistical measurements directly yield the optimum systems.