Minimum Mean-Square Error (MMSE) Weights

Weighting Methods
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We're deriving the optimal MMSE weights. Recall from the previous page:

mean squared error

Multiplying the terms above, the MSE becomes

             rewriting the MSE for phased array            (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:

first term of MSE

We'll now declare some definitions using equation (1). We'll define a commonly used matrix in signal processing - the Autocorrelation Matrix, written as autocorrelation matrix is statistical, which is defined to be:

Rxx or autocorellation Matrix

The second term in equation (1) is the average power in the desired signal, :

sigma squared represents received power level

We'll define the correlation between the signals on the antennas and the desired signal as:

correlation between desired signal and received signals

The third term can be written using the above definition as:

part of the MMSE weight derivation

Finally, the fourth term in eq. (1) is the conjugate of the third term:

final term in MSE weight derivation

All these defintions are at first a bit cumbersome, but equation (1) can now be rewritten as:

final MSE

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:

gradient of error term

Setting this to zero and solving for W, the optimal MMSE-weights are:

solving for MMSE weights in phased array

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.

Next: MMSE (Part 3)

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