Fraudulent white noise: Flat power spectra belie arbitrarily complex processes

Power spectral densities are a common, convenient, and powerful way to analyze signals, so much that they now broadly deployed across the sciences engineering—from quantum physics cosmology from crystallography neuroscience speech recognition. The features reveal not only identify prominent signal frequencies but also hint at mechanisms generate correlation lead resonance. Despite their near-centuries-long run of successes in analysis, here we show flat power spectra can be generated by highly complex processes, effectively hiding all inherent structure signals. Historically, this circumstance has been widely misinterpreted, being taken as renowned signature “structureless” white noise—the benchmark randomness. We argue, contrast, extent most real-world systems exhibit correlations beyond pairwise statistics structures evade other statistical measures. As concrete physical examples, demonstrate fraudulent noise hides predictable both entangled chaotic crystals. To make these words warning operational, present constructive results explore how situation comes about high toll it takes understanding mechanisms. First, give closed-form solution for spectrum very broad class structurally generators. Second, close relationship between eigenspectra evolution operators spectra. Third, characterize minimal generative implied any spectrum. Fourth, construct arbitrarily processes with Finally, leveraging diagnosis problem, point developing more incisive tools discovering signals.10 MoreReceived 6 July 2020Accepted 16 December 2020DOI:https://doi.org/10.1103/PhysRevResearch.3.013170Published American Physical Society under terms Creative Commons Attribution 4.0 International license. Further distribution work must maintain attribution author(s) published article's title, journal citation, DOI.Published SocietyPhysics Subject Headings (PhySH)Research AreasChaosComplex systemsFluctuations & noiseInformation communication theoryNoiseStochastic processesNonlinear DynamicsInterdisciplinary PhysicsStatistical PhysicsGeneral Physics