Singular value decomposition in adaptive beamforming

Karhunen-Loeve (K-L) expansions are fundamental to analysis of optimum array processors. It is shown how vector K-L expansion can be obtained by a generalized Fourier transform of the array output vector and an eigenvalue decomposition. In actual implementation of adaptive array processor singular value decomposition (SVD) of a matrix formed from transformed data is used instead of eigenvalue decomposition. It is also shown how K-L orthonormal system can be calculated from another orthonormal system by an eigenvector transformation that diagonalizes the covariance matrix of the original orthonormal expansion coefficients. Thus we have a computationally viable method for construction K-L orthonormal systems.