308 BELL SYSTEM TECHNICAL JOURNAL 



when V -^ oo , it is known that tlie real and iiraginan- parts of this sum are 

 random variables, which tend to become independent and normally dis- 

 tributed about zero. This suggests the manner in which the normal dis- 

 tribution of the coefBcients arises. Averaging over the 6k s in (1.7-3) gives 

 when n > 



dnK = hr.K = (1.7-5) 



Some further algebra gives 



T~ _ 7Y~ _ K 2 



QnK — OnK — ^ A^ 



2 (1.7-6) 



anKbuK = dnKOmK = b„KbmK = 



where n 9^ m and n, m > 0. 



So far, we have been considering the case of exactly K arrivals in our 

 interval of length T. Now we pass to the general case of any number of 

 arrivals by making use of formulas analogous to 



5 = Z PiK)ZK (1.7-7) 



A = 



as has been done in section 1.3. Thus, for w > 0, 

 dn = hn = 



an = On = — Rn = (Tn (1-7-8) 



Cnbn = anUm = bnb„, = 0, H 9^ M 



In the second Hne we have used (r„ to denote the standard deviation of c„ 

 and bn . ^ 

 by writing 



and bn . We may put the expression for (t„ in a somewhat different form 



/n = I = n^f, A/ = 1 (1.7-9) 



where /„ is the frequency of the «th component. Using (1.7-4), 



(tI = 2,/A/l [ F(/)e~"'^^"' 



dl 



(1.7-10) 



Thus, <Tn is proportional to v/T. 



The probability density function P{ai , • • ■ Cj^ , b] , • • • by) for the 27V co- 

 efficients, ai , • • ' as , bi , • • • by may be derived in much the same fashion 

 as was the probability density of the noise current in section 1.4. Here N 



