Minimum Mean-Square Error (MMSE) Weights

We're deriving the optimal MMSE weights. Recall from the previous page: Multiplying the terms above, the MSE becomes                          (1) Because the weights are constant (don't vary with time), they can be pulled out of the expectation operator, and the first term in equation (1) simplifies to: We'll now declare some definitions using equation (1). We'll define a commonly used matrix in signal processing - the Autocorrelation Matrix, written as , which is defined to be: The second term in equation (1) is the average power in the desired signal, : We'll define the correlation between the signals on the antennas and the desired signal as: The third term can be written using the above definition as: Finally, the fourth term in eq. (1) is the conjugate of the third term: All these defintions are at first a bit cumbersome, but equation (1) can now be rewritten as: We're pretty close to finding the desired weights now. Note that the above equation is actually a quadratic function of the weight vector W. Hence, to minimize the mean-squared error, we can take the gradient (or vector-derivative) with respect to W, set the result equal to zero, and we have the optimal weights. The gradient of the above equation is: Setting this to zero and solving for W, the optimal MMSE-weights are: These are the optimal weights we have been looking for. The result is statistical - in terms of expected values of the desired signals. In the next section, we'll look at this in a little more depth and present an example that illustrates this weighting method.