Spectra for the Product of Gaussian Noises

Products of Gaussian noises often emerge as the result of non-linear detection techniques or as parasitic effects, and their proper handling is important in many practical applications, including 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 2 W , and it is zero for frequencies greater than 2 W . 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 are corroborated by computer simulations.