Yield-Optimized Superoscillations

• Published in: IEEE transactions on signal processing Vol. 61; no. 12; pp. 3113 - 3118
• Main Authors: Katzav, E., Schwartz, M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.06.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Constraints; Detection, estimation, filtering, equalization, prediction; Distortion; Eigenvalues; Eigenvalues and eigenfunctions; Energy use; Exact sciences and technology; Fourier analysis; Information, signal and communications theory; Optimization; quantum theory; Signal and communications theory; Signal processing; Signal, noise; Studies; supergain; superoscillations; superresolution; Telecommunications and information theory; time-frequency analysis; Transaction processing; Time-frequency analysis; Superresolution; superoscillations; Scientific application; Signal processing; Numerical method; Eigenvalue problem; Eigenvalues and eigenfunctions; Band limited signal; supergain; Quantum theory; Constrained optimization
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2013.2258018

Superoscillating signals are band-limited signals that oscillate in some region faster their largest Fourier component. While such signals have many scientific and technological applications, their actual use is hampered by the fact that an overwhelming proportion of the energy goes into that part of the signal, which is not superoscillating. In the present paper, we consider the problem of optimization of such signals. The optimization that we describe here is that of the superoscillation yield, the ratio of the energy in the superoscillations to the total energy of the signal, given the range and frequency of the superoscillations. The constrained optimization leads to a generalized eigenvalue problem, which is solved numerically. It is noteworthy that it is possible to increase further the superoscillation yield at the cost of slightly deforming the oscillatory part of the signal, while keeping the average frequency. We show, how this can be done gradually, which enables a tradeoff between the distortion and the yield. We show how to apply this approach to nontrivial domains, and explain how to generalize this to higher dimensions.