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.

[1]  Moshe Schwartz,et al.  Yield-Optimized Superoscillations , 2012, IEEE Transactions on Signal Processing.

[2]  Wang Qiao,et al.  A simple model of Aharonov - Berry's superoscillations , 1996 .

[3]  G. V. Eleftheriades,et al.  Temporal Pulse Compression Beyond the Fourier Transform Limit , 2011, IEEE Transactions on Microwave Theory and Techniques.

[4]  Mark R. Dennis,et al.  A super-oscillatory lens optical microscope for subwavelength imaging. , 2012, Nature materials.

[5]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[6]  Michael V Berry,et al.  Exact nonparaxial transmission of subwavelength detail using superoscillations , 2013 .

[7]  G. Eleftheriades,et al.  Sub-Wavelength Focusing at the Multi-Wavelength Range Using Superoscillations: An Experimental Demonstration , 2011, IEEE Transactions on Antennas and Propagation.

[8]  S. Lang.  Complex Analysis , 1977 .

[9]  L. Levi,et al.  Fitting a bandlimited signal to given points , 1965, IEEE Trans. Inf. Theory.

[10]  C. Cahn,et al.  On the sampling the zeros of bandwidth limited signals , 1958, IRE Trans. Inf. Theory.

[11]  Paulo Jorge S. G. Ferreira,et al.  The energy expense of superoscillations , 2002, 2002 11th European Signal Processing Conference.

[12]  Nikolay I Zheludev,et al.  What diffraction limit? , 2008, Nature materials.

[14]  Yakir Aharonov,et al.  Quantum Paradoxes: Quantum Theory for the Perplexed , 2005 .

[15]  A. Requicha,et al.  The zeros of entire functions: Theory and engineering applications , 1980, Proceedings of the IEEE.

[16]  Achim Kempf,et al.  Unusual properties of superoscillating particles , 2004 .

[17]  Henry J. Landau,et al.  Extrapolating a band-limited function from its samples taken in a finite interval , 1986, IEEE Trans. Inf. Theory.

[18]  Nikolay I. Zheludev,et al.  Optical super-resolution through super-oscillations , 2007 .

[19]  Dae Gwan Lee,et al.  Direct Construction of Superoscillations , 2014, IEEE Transactions on Signal Processing.

[20]  Nikolay I Zheludev,et al.  Super-resolution without evanescent waves. , 2008, Nano letters.

[21]  Daniele C. Struppa,et al.  Some mathematical properties of superoscillations , 2011 .

[22]  Sandu Popescu,et al.  Evolution of quantum superoscillations, and optical superresolution without evanescent waves , 2006 .

[23]  A. Kempf.  Black Holes, Bandwidths and Beethoven , 1999 .

[24]  P.J.S.G. Ferreira,et al.  Superoscillations: Faster Than the Nyquist Rate , 2006, IEEE Transactions on Signal Processing.

[25]  Vaidman,et al.  How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. , 1988, Physical review letters.

[26]  George V Eleftheriades,et al.  Adaptation of Schelkunoff's Superdirective Antenna Theory for the Realization of Superoscillatory Antenna Arrays , 2010, IEEE Antennas and Wireless Propagation Letters.

[27]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[28]  Dae Gwan Lee,et al.  Superoscillations of Prescribed Amplitude and Derivative , 2014, IEEE Transactions on Signal Processing.

[29]  Yunfei Chen,et al.  On secrecy outage of MISO SWIPT systems in the presence of imperfect CSI , 2016, 2016 24th European Signal Processing Conference (EUSIPCO).

[30]  J. Anandan,et al.  Quantum Coherenece and Reality, In Celebration of the 60th Birthday of Yakir Aharonov , 1995 .