Solving for the Minimum Mean-Squared Error Weights

The minimum mean-squared error (MMSE) criterion optimizes the filter weights based on the input signals. Here you will learn how to find the optimum weights. You will use the method of completing the square to write the mean-squared error as a perfect square in the weights, which allows you to identify the optimum weights by inspection. You will also learn how the MMSE problem is equivalent to a least-squares problem involving an overdetermined system of linear equations. This relationship connects the MMSE problem to a rich set of results and methods for solving systems of linear equations. Least squares problems are frequently also encountered in estimation and imaging applications of signal processing.