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.