Non-iterative implementation of a class of iterative signal restoration algorithms

In this paper, we show that a class of iterative signal restoration algorithms, which includes as a special case the discrete Gerchberg-Papoulis algorithm, can always be implemented directly (i.e., non-iteratively). In the exactly- and over-determined cases, the iterative algorithm always converges to a unique least squares solution. In the under-determined case, it is shown that the iterative algorithm always converges to the sum of a unique minimum norm solution and a term dependent on initial conditions. For the purposes of early termination, it is shown that the output of the iterative algorithm at the r^th iteration can be computed directly using a singular value decomposition-based algorithm. The computational requirements of various iterative and non-iterative implementations are discussed, and the effect of the relaxation parameter on the regularization capability of the iterative algorithm is investigated.