48 BELL SYSTEM TECHNICAL JOURNAL 



that (2.8-6) expresses I{t) as the sum of a large number of independent ran- 

 dom variables 



/(O = Xl + X2 -\- ••' + Xn 

 Xn = Cn COS {wj — (Pn) 



and hence that as N — > x I{t) becomes distributed according to a normal 

 law. In order to make the limiting process definite we first choose .Y and 

 A/such^that iVA/ = F where 



r wU) df <e\ wU) df 



J F Jo 



where e is some arbitrarily chosen small positive quantity. We now let 

 N —^ °o and A/ — ^ in such a way that XAf remains equal to F. Then 



. ^' 



A = Xl + xl + "• + xl = Yl 2W(/„)A/C0S2 {cOnt — (Pn) 



1 •'0 



.V 



B =\^\^+ '" + 1^ = E (2w(/„)^y)''Vos(a;„^- s^„) |-^ 



<4(A/)^'^ rw/)^^ 



Jo 



where the bars denote averages with respect to the (^'s, t being held constant. 

 If we assume that the integrals are proper, the ratio BA~^ ^ — > as .V -^ =c , 

 and consequently the central limit theorem* may be used if wif) = for 

 f > F. Since we may make F as large as we please by choosing e small 

 enough, we may cover as large a frequency range as we wish. For this 

 reason we write =o in place of F. 



Now that the central hmit theorem has told us that the distribution of 

 I(t), as given by (2.8-6), approaches a normal law, there remains only the 

 problem of finding the average and the standard deviation: 



.V 



1(0 = Zl <^n cos (cOn/ — <pn) = 

 1 



A' 



P(f) = Z) cl C0S2 (w„/ - <pn) (3.1-5) 



1 



-^ / w{f) df = 4^0 

 Jo 



* Section 2.10. 



