Statistical properties of N random sinusoidal waves in additive Gaussian noise

The first order probability density functions of the sums of N independent sinusoidal waves having random amplitudes and phases in additive Gaussian noise are studied for the cases where N is fixed and where N in Poisson distributed. The conditional moments about the origin are obtained in closed form or both situations. The corresponding probability density functions for the envelope are also studied. The even conditional moments about the origin are also obtained in closed form. Representative numerical results are presented.