The Toeplitz matrix in the analysis of the adaptive algorithms of the uniformly spaced adaptive array

The LMS algorithm uses an estimate of the gradient vector, whereas one of its modifications, the Newton algorithm that steers the weight-vector increment and generates a straighter path to the optimum weight, must contain an estimate of the inverse of the input signal correlation matrix. The initial (a priori) estimate of the inverse matrix is essential in the implementation of this combined LMS-Newton method with fewer iterations of the weight conversion process. It is shown that advantage can be taken of the fact that for the uniformly spaced adaptive array of identical elements the structure of the input correlation matrix is Toeplitz, which eliminates the necessity of direct inversion and opens the way for implementation of the readily available rapid inversion algorithms at the design and performance analysis stage.<>