Dae Gwan Lee

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.

Superoscillations of Prescribed Amplitude and Derivative

Superoscillations occur when a bandlimited signal oscillates at a rate higher than its maximum frequency. We show that it is possible to construct superoscillations by constraining not only the value of the signal but also that of its derivative. This allows a better control of the shape of the superoscillations. We find that for any given bandwidth, no matter how small, there exists a unique signal of minimum energy that satisfies any combination of amplitude and derivative constraints, on a sampling grid as fine as desired. We determine the energy of the signal, for any grid, regular or irregular. When the set of derivative constraints is empty the results reduce to minimum energy interpolation. In the absence of amplitude constraints, we obtain pure derivative-constrained extremals. The flexibility gained by having two different types of constraints makes it possible to design superoscillations based only on amplitudes, based only on derivatives, or based on both. In the last case, the amplitude and derivative sampling grids can be interleaved or aligned. We explore this flexibility to build superoscillations that cost less energy. Illustrating examples are given.

Superoscillations with Optimal Numerical Stability

A bandlimited signal can oscillate at a rate faster than its bandlimit. This phenomenon, called “superoscillation”, has applications e.g. in superresolution and superdirectivity. The synthesis of superoscillations is a numerically difficult problem. We introduce time translation σ as a design parameter and give an explicit closed formula for the condition number of the matrix of the problem, as a function of σ. This enables us to determine the best possible condition number, which is several orders of magnitude better than otherwise achievable.

Superoscillations With Optimum Energy Concentration

Oscillations of a bandlimited signal at a rate faster than the bandlimit are called “superoscillations” and have applications e.g. in superresolution and superdirectivity. The synthesis of superoscillating signals is a numerically difficult problem. Minimum energy superoscillatory signals seem attractive for applications because (i) the minimum-energy solution is unique (ii) it has the smallest energy cost (iii) it may yield a signal of the smallest possible amplitude. On the negative side, superoscillating functions of minimum-energy depend heavily on cancellation and give rise to expressions that have very large coefficients. Furthermore, these coefficients have to be found by solving equations that are very ill-conditioned. Surprisingly, we show that by dropping the minimum energy requirement practicality can be gained rather than lost. We give a method of constructing superoscillating signals that leads to coefficients and condition numbers that are smaller by several orders of magnitude than the minimum-energy solution, yet yields energies close to the minimum. In contrast with the minimum-energy method, which builds superoscillations by linearly combining functions with an ill-conditioned Gram matrix, our method combines orthonormal functions, the Gram matrix of which is obviously the identity. Another feature of the method is that it yields the superoscillatory signal that maximises the energy concentration in a given set, which may or may not include the superoscillatory segment.