Spectra for the product of Gaussian noises

TL;DR

This paper analytically derives the power density spectrum of the product of Gaussian noises using Rice's formalism, revealing a linear decrease from zero to twice the bandwidth and confirming results through simulations.

Contribution

It provides a novel analytical calculation of the noise spectrum for Gaussian noise products, including correlation effects, validated by simulations.

Findings

Spectrum decreases linearly from zero to 2W

Spectrum is zero beyond 2W

Correlation doubles the variance of the squared noise

Abstract

Products of Gaussian noises often emerge as the result of non-linear detection techniques or as a parasitic effect, and their proper handling is important in many practical applications, including in fluctuation-enhanced sensing, indoor air or environmental quality monitoring, etc. We use Rice's random phase oscillator formalism to calculate the power density spectra variance for the product of two Gaussian band-limited white noises with zero-mean and the same bandwidth W. The ensuing noise spectrum is found to decrease linearly from zero frequency to 2W, and it is zero for frequencies greater than 2W. Analogous calculations performed for the square of a single Gaussian noise confirm earlier results. The spectrum at non-zero frequencies, and the variance of the square of a noise, is amplified by a factor two as a consequence of correlation effects between frequency products. Our analytic results is corroborated by computer simulations.