A Note on Linear Least Squares Filtering

This paper is concerned with a rigorous, yet intuitively appealing, justification of the optimum mean-square-error linear filter design procedure under the constraint of physical realizability. By use of the formulation and solution of an equivalent finite dimensional geometric problem, the following two (intuitively) difficult questions are answered. 1) Why can't the physically realizable optimum design be implemented by simply taking the physically real portion of the unconstrained optimum? 2) What is the significance of the "whitening" filter operation which must be implemented before the physical realizability constraint can be introduced? Finally, the problem is shown to be closely related to the principal axis transformation problem for quadratic surfaces and a three-dimensional example is provided to complete the analogy.