Convergence of iterative nonexpansive signal reconstruction algorithms

Iterative algorithms for signal reconstruction from partial time- and frequency-domain knowledge have proven useful in a number of application areas. In this paper, a general convergence proof, applicable to a general class of such iterative reconstruction algorithms, is presented. The proof relies on the concept of a nonexpansive mapping in both the time and frequency domains. Two examples studied in detail are time-limited extrapolation (equivalently, band-limited extrapolation) and phase-only signal reconstruction. The proof of convergence for the phase-only iteration is a new result obtained by this method of proof. The generality of the approach allows the incorporation of nonlinear constraints such as time- (or space-) domain positivity or minimum and maximum value constraints. Finally, the underrelaxed form of these iterations is also shown to converge even when the solution is not guaranteed to be unique.