Fraudulent White Noise: Flat power spectra belie arbitrarily complex processes

TL;DR

This paper reveals that flat power spectra can be produced by highly complex processes, misleadingly suggesting randomness, and demonstrates how to identify hidden structures in such signals across various scientific fields.

Contribution

It provides a theoretical framework and constructive methods to understand how complex signals can exhibit flat spectra, challenging the assumption that flat spectra imply simple or random processes.

Findings

Flat power spectra can originate from complex, structured processes.

Eigen-spectra of evolution operators relate closely to power spectra.

Constructive methods to generate arbitrarily complex signals with flat spectra.

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 eigen-spectra 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.