Split versions of the Levinson-like and Schur-like fast algorithms for solving block-slanted-Toeplitz systems of equations

In Joshi and Yagle (1998) the Fredholm equations of one-dimensional (1-D) inverse scattering and LLS estimation were transformed via the orthonormal wavelet transform into a series of symmetric “block-slanted-Toeplitz” (BST) systems of equations. Levinson-like and Schur-like fast algorithms were presented for solving the BST systems. Here, we present split versions of the Levinson-like and Schur-like fast algorithms. The significance of these split algorithms is as follows. Although the Levinson-like and Schur-like fast algorithms reduce the complexity of solving the BST systems from O(n³) to O(n²), there still exists an inherent redundancy in these algorithms in the case where the BST system matrices have centrosymmetric blocks. This situation arises when a symmetric wavelet basis function (like the Littlewood-Paley) is used in the problem transformation. This redundancy is exploited here to derive the split Levinson-like and split Schur-like fast algorithms. These split algorithms reduce the number of multiplications required at each iteration by a factor of two, as compared with the Levinson-like and Schur-like algorithms