  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.
