The Recovery of Distorted Band-Limited Signals

In this paper we devise a method for recovering band limited signals which have been subjected to a distorting procedure called companding. A band limited signal is a function in L2 whose Fourier transform vanishes outside of a bounded interval. In practice most real signals are in this class since they have no indefinitely high frequency components. The operation of companding a signal f(t) consists of replacing it by p[f(t)], where F(X) is a given function, This operation does not in general preserve the band-limitedness of f(t). We reproduce an unpublished uniqueness theorem of A. Beurling which shows that the knowledge of the transform of the companded signal on the interval where the transform of the original signal does not vanish determines uniqueIy the original signal, provided the companding function is monotonic increasing. We then prove the following existence theorem. To each function in L2 defined only on a bounded interval there is one and only one band limited function with band contained in this interval, the Fourier transform of whose companded version coincides with the given function. Our method is constructive and proceeds via a stable iteration scheme (Picard iterations). The convergence of our method requires that the companding function have a derivative which is bounded away from zero and infinity.