Year-end Sale % OFF 
Abstract
Power spectral densities are a common, convenient, and powerful way to analyze signals, so much so that they are now broadly deployed across the sciences and engineering—from quantum physics to cosmology and from crystallography to neuroscience to speech recognition. The features they reveal not only identify prominent signal frequencies but also hint at mechanisms that generate correlation and lead to resonance. Despite their near-centuries-long run of successes in signal analysis, here we show that flat power spectra can be generated by highly complex processes, effectively hiding all inherent structure in complex signals. Historically, this circumstance has been widely misinterpreted, being taken as the renowned signature of “structureless” white noise—the benchmark of randomness. We argue, in contrast, to the extent that most real-world complex systems exhibit correlations beyond pairwise statistics their structures evade power spectra and other pairwise statistical measures. As concrete physical examples, we demonstrate that fraudulent white noise hides the predictable structure of both entangled quantum systems and chaotic crystals. To make these words of warning operational, we present constructive results that explore how this situation comes about and the high toll it takes in understanding complex mechanisms. First, we give the closed-form solution for the power spectrum of a very broad class of structurally complex signal generators. Second, we demonstrate the close relationship between eigenspectra of evolution operators and power spectra. Third, we characterize the minimal generative structure implied by any power spectrum. Fourth, we show how to construct arbitrarily complex processes with flat power spectra. Finally, leveraging this diagnosis of the problem, we point the way to developing more incisive tools for discovering structure in complex signals.10 MoreReceived 6 July 2020Accepted 16 December 2020DOI:https://doi.org/10.1103/PhysRevResearch.3.013170Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasChaosComplex systemsFluctuations & noiseInformation & communication theoryNoiseStochastic processesNonlinear DynamicsInterdisciplinary PhysicsStatistical PhysicsGeneral Physics
Highlights
The challenge of discovering structure in noisy signals is compounded manifold, as we demonstrate in the following, when our chosen observables hide arbitrary amounts of inprinciple-predictable structure behind a familiar signature of white noise—the flat power spectrum
We show that fraudulent white noise arises in measurements of entangled quantum systems
Though, none of the structure of the conditional probability density functions {p(X |s)}s matters for the power spectrum, except for the average value observed in each state
Summary
Success in discovering the mechanisms that underlie the systems we seek to understand, though, requires distinguishing structure from noise Often, this distinction falls to discretion: Structure is that part of a signal we can predict, while noise stands in as a catch-all for everything else. This conundrum holds especially in the analysis of signals from truly complex systems, as when analyzing data from multielectrode arrays in brain tissue [1] or social experiments [2] These systems are often said to be “noisy” even though the so-called noise may be entirely functionally relevant but in an unknown way [3]. It introduces a general condition for fraudulent white noise processes—structured processes with a flat power spectrum—which applies very broadly, including to inputdependent processes with nonstationary high-order statistics. Appendices present detailed derivations, as well as several generalizations, of the main results
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
