Kajplan 



butlon is given by 



p(£ <x) = 1 - e'"" (129) 



In addition to information on the probability density, additional 

 statistical properties of the drift force are provided in terms of the 

 autocorrelation and power spectral density functions for that force. 

 The relationship between the statistical characteristics of an input 

 to the nonlinear form of drift force representation, to the output 

 characteristics, as defined by the autocorrelation and spectral 

 density, is a useful description which can be applied in further 

 analyses and for simulation studies. The problem of a square -law 

 detector has been treated in the available literature, e.g. [ 20] and 

 the results can be applied to the present case. If a general narrow 

 band Gaussian random process, represented by the variable x(t) , 

 is the input to a square-law device, and the output is defined as r , 

 the autocorrelation function Rr2(T) is given by 



R^2(t) = 40- j + 4rJ(t) 



= (7-) +4rJ{t) (130) 



where 



72 = 2(rJ (131) 



is the mean value of the square of the envelope, with tr^ the mean 

 square value of the input function x(t). The relations between the 

 autocorrelation and power spectral .densities of the input function 

 are given by 



SM = I J Rx('^)e"'''* dr, R,(T) = ^y^S»e'"* dc (132) 



where 



.00 



Rx(0) = ^x = -iy Sx(co) dco (133) 



'0 



thus being in conformity with the relations described by the Neumann 

 wave spectrum and all linear functions derived from that spectral 

 formulation. 



1072 



