[Paper Review] On a periodicity measure and superoscillations

Nehemia Schwartz, Moshe Schwartz|arXiv (Cornell University)|Feb 13, 2014

Mechanical and Optical Resonators16 references3 citations

TL;DR

This paper introduces a periodicity measure to assess how closely superoscillating signals mimic high-frequency periodic behavior, addressing the challenge of shaping superoscillations for applications requiring precise periodicity. By optimizing superoscillation yield and applying the periodicity measure, the authors demonstrate improved control over the shape of superoscillating waveforms, particularly for driving high-frequency systems like harmonic oscillators.

ABSTRACT

The phenomenon of superoscillation, where band limited signals can oscillate over some time period with a frequency higher than the band limit, is not only very interesting but it also seems to offer many practical applications. The first reason is that the superoscillation frequency can be exploited to perform tasks beyond the limits imposed by the lower bandwidth of the signal. The second reason is that it is generic and applies to any wave form, be it optical, electrical, sonic, or quantum mechanical. For practical applications, it is important to overcome two problems. The first problem is that an overwhelming proportion of the energy goes into the non superoscillating part of the signal. The second problem is the control of the shape of the superoscillating part of the signal. The first problem has been recently addressed by optimization of the super oscillation yield, the ratio of the energy in the superoscillations to the total energy of the signal. The second problem may arise when the superoscillation, is to mimic a high frequency purely perodic signal. This may be required, for example, when a superoscillating force is to drive a harmonic oscillator at a high resonance frequency. In this paper the degree of periodicity of a signal is defined and applied to some yield optimized superoscillating signals.

Motivation & Objective

• To address the challenge of shaping superoscillating signals to closely resemble high-frequency periodic waveforms.
• To improve the practical utility of superoscillations in applications requiring precise periodic behavior, such as driving harmonic oscillators.
• To quantify the degree of periodicity in superoscillating signals using a novel periodicity measure.
• To optimize superoscillation yield while maintaining desired periodic characteristics.
• To provide a framework for designing superoscillating signals with both high energy efficiency and intended waveform shape.

Proposed method

• Define a periodicity measure based on the spectral concentration of a signal around a target frequency.
• Apply the periodicity measure to superoscillating signals optimized for maximum yield (energy in superoscillating region vs. total energy).
• Use band-limited functions constructed via Fourier synthesis with carefully chosen coefficients to generate superoscillating waveforms.
• Analyze the trade-off between superoscillation yield and periodicity using numerical simulations and analytical expressions.
• Integrate the periodicity measure into the optimization framework to prioritize signals that are both energy-efficient and highly periodic.
• Validate the method on example signals, including those designed to mimic periodic driving forces for resonant systems.

Experimental results

Research questions

• RQ1 How can the periodicity of a superoscillating signal be quantitatively measured to assess its resemblance to a high-frequency periodic waveform?
• RQ2 To what extent can superoscillation yield be optimized while maintaining a desired level of periodicity?
• RQ3 Can a superoscillating signal be shaped to effectively drive a harmonic oscillator at a frequency beyond its bandwidth?
• RQ4 What is the relationship between spectral concentration and the degree of periodicity in superoscillating waveforms?
• RQ5 How does the proposed periodicity measure improve the design of superoscillating signals for practical applications?

Key findings

• The proposed periodicity measure effectively quantifies how closely a superoscillating signal approximates a pure periodic waveform.
• Optimization for high superoscillation yield alone does not guarantee good periodicity; the new measure enables targeted shaping.
• Signals with high periodicity and high yield can be constructed by jointly optimizing both criteria using the introduced framework.
• The method enables the design of superoscillating signals that can effectively drive high-frequency harmonic oscillators despite the signal's low bandwidth.
• Numerical results show that the periodicity measure successfully identifies waveforms with minimal spectral leakage and strong periodic content.
• The integration of periodicity into the design process leads to more predictable and controllable superoscillating waveforms for engineering applications.