An orthonormal Laguerre expansion yielding Rice´s envelope density function for two sine waves in noise

Through an orthonormal Laguerre expansion, expressions are derived for a lesser known Rician probability distribution-the probability density function (PDF) of the envelope of two fixed-amplitude randomly phased sine waves in narrowband Gaussian noise-and for the integral of the density, the cumulative distribution function (CDF). The principal formula derived has been checked analytically, numerically, and (approximately) graphically. Analytically, the moment-generating function for the PDF of the square of the envelope has been found to be a three-term product of elementary functions times an I₀ Bessel function (and thus to be in closed form); in confirmation, the same result has been secured via another, more direct route