Singular value decomposition utilizing parallel algorithms on graphical processors

One of the current challenges in underwater acoustic array signal processing is the detection of quiet targets in the presence of noise. In order to enable robust detection, one of the key processing steps requires data and replica whitening. This, in turn, involves the eigen-decomposition of the sample spectral matrix, C_x = 1/K σ_x^KX(k)X ^H(k) where X(k) denotes a single frequency snapshot with an element for each element of the array. By employing the singular value decomposition (SVD) method, the eigenvectors and eigenvalues can be determined directly from the data without computing the sample covariance matrix, reducing the computational requirements for a given level of accuracy (van Trees, Optimum Array Processing). (Recall that the SVD of a complex matrix A involves determining V, σ, and U such that A = UσV^H where U and V are orthonormal and σ is a positive, real, diagonal matrix containing the singular values of A. U and V are the eigenvectors of AA^H and A ^HA, respectively, while the singular values are the square roots of the eigenvalues of AA^H.) Because it is desirable to be able to compute these quantities in real time, an efficient technique for computing the SVD is vital. In addition, emerging multicore processors like graphical processing units (GPUs) are bringing parallel processing capabilities to an ever increasing number of users. Since the computational tasks involved in array signal processing are well suited for parallelization, it is expected that these computations will be implemented using GPUs as soon as users have the necessary computational tools available to them. Thus, it is important to have an SVD algorithm that is suitable for these processors.