[5] Let the quadrature functions “*b*” and “*c*” be Gaussian and characterized by standard deviations σ_{b} and σ_{c}, respectively; and let “*g*” have a standard deviation *σ*. It is generally argued [*Steinberg*, 1976] that the variances must be the same, for otherwise the standard deviation of “*g*” will be dependent on the choice of coordinates for the quadrature. It then follows that

Use of this, together with the fact that “*b*” and “*c*” are uncorrelated, it can be shown that the joint probability density function (PDF) w is given by

where as usual 〈X〉 is used to denote expectation value. Once the above quadrature components are converted into polar via *b* = ∣*g*∣ cos(θ), and *c* = ∣*g*∣ sin(θ), and upon performing the angular integration, and using “*A*” to denote the random amplitude function of “*g*,” we obtain the PDF of the amplitude “*a*,” which is given by [*Steinberg*, 1976; *Davenport and Root*, 1958]:

where

and *I*_{0} is the modified Bessel function of zero order [*Abramowitz and Stegun*, 1964]. Equation (3), the Rice distribution [*Davenport and Root*, 1958], is commonplace in random processes, such as in the problem of multiple reception [*Lange*, 1967]. The PDF of phase can be obtained by similar means. Upon introduction of the angular function B, defined as

and performing radial integration, it follows that [*Davenport and Root*, 1958]:

valid for all *B*, and where erf(x) stands for the standard error function [*Abramowitz and Stegun*, 1964].

[6] The PDF corresponding to power can be obtained via use of an identity in Probability Theory [*Davenport and Root*, 1958, p.309] which relates the density functions of two arbitrary random processes Y and X, for *Y* = *X*^{2}:

Since power *p* is the square of the amplitude, the above, coupled with (3) results in the PDF of power being given by

Although the individual quadrature components of “*g*” are Gaussian, its amplitude, phase and power are not. For the case of Gaussian components, the situation is reminiscent of a Gaussian distribution, which when viewed in the complex plane which defines the quadrature components, exhibits an elliptical cross section (see Figure 1). When observed in polar coordinates, two relative maxima will be found as the radial variable is kept fixed and the azimuth is varied from 0 to 2π. Thus this simplistic view results in a bimodal phase distribution.