Signal restoration using efficient constrained least squares techniques

The problem of restoring missing samples from a flnite-time or space-domain vector, given a portion of its discrete frequency spectrum (DFT) and known bounds on the missing samples, is investigated. A matrix-based solution technique is proposed as a numerically efficient alternative to iterative solution via alternating projections onto convex sets (POCS). The problem is formulated as a set of linear equations that relate the known time and frequency samples to the missing timedomain samples. Finitely convergent gradient-based search algorithms are employed to compute the constrained least squares solution to this set of linear equations. It is shown that if the set of linear equations is exactly or overdetermined, the POCS iteration will converge to the solution computed by the gradient search method. The numerical efficiency of the proposed technique is demonstrated by a 1-D slab profile inversion example.