Spectrum extrapolation on a finite band

The iterative method of Gerchberg (GM) for extrapolating the whole spectrum of a finite support object1 has been analyzed and generalized by several authors2-14. In principle, during each iteration of the GM an infinite band of frequencies should be handled. At first sight, it seems that the (obvious) existence of a cut-off frequency in any practical implementation of the GM should simply allow the spectrum extrapolation to be achieved only below such a frequency. This is not the case, as we shall presently show, in that the extrapolated spectrum obtained by this method up to the cut-off frequency does not coincide with the true spectrum. In this paper we present a modified version of the GM that allows to obtain a spectrum extrapolation on a finite band of frequencies. This is of use both to limit the storage and computation time requirements and to reduce the sensitivity to high frequency noise.