MATHEMATICAL ANALYSIS OF RANDOM NOISE 287 



5. There are two representations of a random noise current which are 

 especially useful. The first one is 



N 



I{t) = S (^n cos unt -{• bn Sin a}„ /) (2.8-1) 



n-1 



where c„ and &„ are independent random variables which are distributed 

 normally about zero with the standard deviation ■\/w{fn)Af and where 



OJn = 27r/„ , /„ = wA/ 

 The second one is 



AT 



^(0 = Jl Cn cos (co„/ - <pn) (2.8-6) 



n=l 



where v'n is a random phase angle distributed uniformly over the range 

 (0, 27r) and 



c„ = [2w(/„)Afr 



At an appropriate point in the analysis N and A/ are made to approach 

 infinity and zero, respectively, in such a manner that the entire frequency 

 band is covered by the summations (which then become integrations). 



6. The normal distribution in several variables and the central hmit 

 theorem are discussed in sections 2.9 and 2.10. 



Part in — Statistical Properties of Noise Current 



1. The noise current is distributed normally. This has already been 

 discussed in section 1.6 for the shot-efi'ect. It is discussed again in section 

 3.1 using the concepts introduced in Part II, and the assumption, used 

 throughout Part III, that the average value of the noise current /(/) is zero. 

 The probabiUty that I{t) lies between I and / + ^/ is 



where \{/o is the value of the correlation function, i/'(t), of /(/) at r = 



^0 = ^(0) = ( w{f) df, (3.1-2) 



w(f) being the power spectrum of /(/). \J/o is the mean square value of 

 /(/), i.e., the r.m.s. value of /(/) is xpo'^. 



The characteristic function (ch. f.) of this distribution is 



ave.e'"^^'^ = exp-^°«^ (3.1-6) 



