Direct Construction of Superoscillations

Oscillations of a bandlimited signal at a rate faster than its maximum frequency are called “superoscillations” and have been found useful e.g., in connection with superresolution and superdirectivity. We consider signals of fixed bandwidth and with a finite or infinite number of samples at the Nyquist rate, which are regarded as the adjustable signal parameters. We show that this class of signals can be made to superoscillate by prescribing its values on an arbitrarily fine and possibly nonuniform grid. The superoscillations can be made to occur at a large distance from the nonzero samples of the signal. We give necessary and sufficient conditions for the problem to have a solution, in terms of the nature of the two sets involved in the problem. Since the number of constraints can in general be different from the number of signal parameters, the problem can be exactly determined, underdetermined or overdetermined. We describe the solutions in each of these situations. The connection with oversampling and variational formulations is also discussed.