MATHEMATICAL ANALYSIS OF RANDOM NOISE 47 



The average value of /(/) as given by (2.8-1) is zero since dn = bn - 0: 



l{t) = (3.1-1) 



The mean square value of /(/) is 



X 



J'iO — X^ {'^'n COS" COn/ + b'n Sin C0„ /) 

 n = l 



V 



= E ^K/JA/ (3.1-2) 





w{f) df = ^^(0) = ^ 



In writing down (3.1-2) we have made use of the fact that all the c's and i's 

 are independent and consequently the average of any cross product is zero. 

 We have also made use of 



which were given in 2.8. \P(t) is the correlation function of /(/) and is 

 related to iv(f) by 



v., ^ ,A(t) = [ w(f) cos 27r/T df (2.1-6) 



as is explained in section 2.1. In this part we shall write the argument of 

 \P(t) as a subscript in order to save space. 



Since we know that I(t) is normal and since we also know that its average 

 IS zero and its mean square value is \po , we may write down its probability 

 density function at once. Thus, the probability of /(/) being in the 

 range I, I -j- dl is 



This IS the probabihty ot finding the current between 7 and I -\- dl a.i a. 

 time selected at random. Another way of saying the same thing is to state 

 that (3.1-3) is the traction of time the current spends in the range /, / + dl. 



In many cases it is more convenient to use the representation (2.8-0} 



1(0 = L Cn COS (O^nt - <Pn), c'n = 2w(fn)Af (2.8-6) 



n=l 



in which ipi , ■ ■ ■ ipn are independent random phase angles. In order to 

 deduce the normal distribution from this representation we first observe 



