MATHEMATICAL ANALYSIS OF RANDOM NOISE 101 



and is a tabulated function. Thus (3.10-11) gives the probabiUty density 

 of the envelope R. 



The average value of R" may be obtained by multipl}-ing (3.10-11) by R" 

 and integrating from to -x . Expansion of the Bessel function and term- 

 wise integration gives 



^ = (2^.r«r(« + l).— .F,g + M;|J 



= (2^.)-rg + l),F.(-|;l;-^j (3.10-12) 



where iFi is a h\^ergeometric function. In going from the first line to 

 the second we have used Kummer's first transformation of this function. 

 A special case is 



R2== p^ ^ 2rPo (3.10-13) 



When only noise is present, P = and 



R = (2^0)^^^ r(|) = (^f^)"' 



\ 2 / (3.10-14) 



R^ = 2iAo 



Before going further with (3.10-11) it is convenient to make the following 

 change of notation 



^ = 7172 ' ^^ = 7T72 ' ^ = Tm (3.10-15) 



V'o 'Ao Wo 



"a" is the ratio (sine wave amplitude)/(r.m.s. noise current). 



Instead of the random variable R we now have the random variable v whose 



probability density is 



p(v) = V exp 



■"4^1 ^"^""^^ (3.10-16) 



Curves of p{v) versus v are plotted in Fig. 6 for the values 0, 1, 2, 3, 5 of a . 

 Curves showing the probabiUty that v is less than a stated amount, i.e., dis- 

 tribution curves for v, are given in Fig. 7. These curves were obtained by 

 integrating p(v) numerically. The following useful expression for this 

 probability has been given by W. R. Bennett in some unpublished work. 



jf" piu) du = exp f-'^lAn |; h\ i^^av) (3.10-17) 



^ Curves of this function are given in "Tables of Functions", Jahnke and Emde (1938), 

 p. 275, and some of its properties are stated in Appendix 4C. 



