Approximation of phase and envelope distributions of Gaussian random amplitude-modulated signals

In this paper the probability density functions (pdf) for the phase and the envelope of an amplitude-modulated (AM) signal, where the modulation is modeled as a random Gaussian function, in the presence of narrowband Gaussian noise are calculated. An exact, closed-form expression is obtained for the phase pdf, while approximate results, accurate for small and large carrier-to-noise ratios (CNR), are derived for the envelope pdf.