MATHEMATICAL ANALYSIS OF RANDOM NOISE 49 



This gives the probability density (3.1-3). Hence the two representations 

 lead to the same result in this case. Evidently, they will continue to lead 

 to identical results as long as the central limit theorem may be used. In the 

 future use of the representation (2.8-6) we shall merely assume that the 

 central limit theorem may be applied to show that a normal distribution 

 is approached. We shall omit the work corresponding to equations (3.1-4). 

 The characteristic function for the distribution of /(/) is 



ave. e = e.\p — ^ u (3.1-6) 



3.2 The Distribution of / (/) and 7 (/ + r) 



We require the two dimensional distribution in which the first variable 

 is the noise current 7(0 and the second variable is its value T(t + r) at some 

 later time r. It turns out that this distribution is normal" , as we might 

 expect from the analogy with section 3.1. The second moments of this 

 distribution are 



Pii = P{t) =io= f wU) df 

 Jo 



/i22 = ^0 



Hn = I{t)nt + r) 



(3.2-1) 



The expression for mi2 is in line with our definition (2.1-4) for the correla- 

 tion function: 



^Pr ^ V'(t) = Limit I f I{t)I{t + r) dt (2.1-4) 



In order to get the distribution from the representation (2.8-6) we write 



AT 



71 = I(t) = S Cn cos (Unt — (pn) 



1 



v 



72 = I{t + t) = X <^« COS (aj„ t — ifn -'- Wn t) 



1 



-* It seems that the first person to obtain this distribution in connection with noise was 

 H. Thiede, Elec. Nachr. Tek. 13 (1936J, 84-95. 



