Equivalent transformation for a partially adaptive array

A partially adaptive array is one in which elements of a phased array are controlled or adaptively weighted in groups or in which certain elements, called auxiliary elements, are made controllable. Mathematically, this type of array is formed by transforming all of the elements of an array by a nonsquare matrix such that the resulting output vector has a length less than the number of array elements. It is shown that there is an equivalent matrix transform that can effectively be utilized in analyzing the partially adaptive array's performance when a small number of external jammers are present. Processor implementation and convergence rate considerations lead to the desirability of reducing the dimensionality of the cancellation processor while maintaining good sidelobe interference protection. A meaningful measure of canceller performance is to compute the optimal output signal-to-noise ratio. This expression is a function of the jammer, direction-of-arrival vectors (DOAVs), jammer powers, the array steering vector, and internal noise. It is shown that if this expression is computed for the fully adaptive array then it is easily computed for the partially adaptive array by transforming the jammer DOAVs and the steering vector by the orthogonal projection matrix defined by the rows of the subarray transformation matrix and substituting these vectors back into the original expression for the fully adaptive array