Adaptive cross‐sensor beam forming with sparse and random planar arrays

Abstract

Let the quadrature field measured at sensor j of a planar array, located at (xj, yj), be Fj. The cross‐sensor field Gl is formed as the product Gl = FjFk*, where k represents sensor k and * denotes complex conjugate. The beam response B̃ of the array in look direction φ̂ is B̃(φ̂) = ΣlwlGφ,l*〈Gl〉, where {w̃l} are cross‐sensor weights, Gφ̂,l is the expected value of Gl for a single incident wave of unit amplitude and bearing φ̂, and 〈〉 denotes a time average. To be specific, Gφ̂,l = expi[h(yj − yk) cosφ̂ + h(xj − xk)sinφ̂], where h is the horizontal wave number. A simple adaptive algorithm can be used to remove the strongest signal in the beam response function. Let φ̂1 represent the bearing of the strongest signal and B̃(φ̂1) = A12 be the maximum value of B̃(φ̂). The values of 〈Gl are modified by subtraction 〈Gl〉 = 〈Gl〉 − A12 expi[h(yj − yk) cosφ̂1 + h(xj − xk) sinφ̂1]. A new beam response can now be calculated and the second largest signal then subtracted, etc. Examples are shown for both sparse and random planar arrays. [Work supported by NAVELEX 320.]