On Distribution of Rice–Middleton Model

The probability density function in Rice–Middleton model, which describes the behavior of the single sinusoidal random signal combined with Gaussian noise is expressed in three mutually independent ways: firstly, with the aid of an integral representation of the modified Bessel function of the first kind of integer order; secondly, by a hyperbolic cosine differential operator and thirdly, applying the Grünwald–Letnikov fractional derivative. The cumulative distribution functions are also described in all these cases, and also using the Nuttall Q–function. An associated, seemingly new, probability distribution is introduced which cumulative distribution function and the raw moments of general real order are obtained whilst the characteristic function’s power series form is inferred. The exposition ends with a discussion in which by–product summations are given for the considered Neumann series of the second type built by modified Bessel functions of the second kind having integer order.