MATHEMATICAL AX A LYSIS OF KAXDO.]f NOISE 119 



where 



I Jo 



Wl + W2 , ^ fh + th r rj. , , . rj.. 



n = 2 — + « - 7y — = joT -V {n - j^T) 



(4.3-5) 



We choose the band so narrow that 



n. - wi « TJT or ^t « 1 (4.3-6) 



This enables us to write approximately 



In - ibn = Z e--^^(("/^)-/0)-| r e-''-'''l,{l) 



r=l i J <,r-\)T 



dt 



Ti = T/t, T being chosen to make n an integer. Suppose we do this for 

 a large number of intervals of length T. Then /jv(0 will differ from interval 

 to interval. The set of integrals for r = \ gives us an array of values which 

 we regard as defining the distribution of a complex random variable, say 

 xi . Similarly the set of integrals for r = 2 defines the distribution of a 

 second random variable ;V2 , and so on to aVi . Because we have chosen t 

 so large that /a'(/) in any one integral is practically independent of its values 

 in the other integrals we may say that Xi , ^2 , • • • Xr^ are independent. 



We have 



~i2iT(,(nlT)-/o)TT ^ 



e 



/, _ ^-A — V ^-'2T((ni+l/T)-/o)rr 



— tb„„_ = £ e" 



j2?r((no/r)-/o)rT 



and if no — ni « ri , as was assumed in (4.3-6), we may apply the central 

 limit theorem to show that c„i , b,n , On^+i , • • ■ fln., , b,,., tend to become in- 

 dependent and normally distributed about zero as we let the band width 

 j8 ^ and T —^ <x> (and hence ri — > ^■-■^ ) in such a way as to keep n^ — Hi 

 fixed. In this work we make use of the fact that Isit) is such that the real 

 and imaginary parts of xi, X2, • • • Xr all have the same average and standard 

 deviation. It is convenient to assume /oT' is an integer. 



Thus as the band width ^ approaches zero the band output Jn given by 

 (4.3-4) may be represented in the same way, namely as (2.8-1), as was the 

 random noise current studied in Part III. Hence Jn tends to have the 



