92 BELL SYSTEM TECHNICAL JOURNAL 



mally, approximately. Hence when T is very large the probability that E 

 lies in dE is 



dE (E — Wr)- 



exp - .2 (3.9-15) 



where 



^«r = r [ W(f) df 

 Jo 



al = T f w\f) df 

 Jo 



(3.9-16) 



the second relation being obtained by letting T -^ °o in (3.9-9). The 

 analogy with Campbell's theorem, section 1.2, is evident. WTien we deal 

 with a band pass filter we may use (3.9-10) and (3.9-11). 



Consider a relatively narrow band pass filter such that we may find a T 

 for which Tfa >> It but T{fb —fa) < < -64. Thus several cycles of fre- 

 quency /„ are contained in T but, from (3.8-15), the envelope does not change 

 appreciably during this interval. Thus throughout this interval /(/) may 

 be considered to be a sine wave of amplitude R. The corresponding value 

 of E is approximately 



2 



where the distribution of the envelope R is given by (3.7-10). From this 

 it follows that the probabihty of E lying in dE is 



dE E dE -Elmr fin 1'7\ 



-— exp - -— = — e ^ (3.9-17) 



when E is small but not too small. 



When we look at (3.9-14) and (3.9-17) we observe that they are of the 

 form 



n+l pn 



^ ^ -"^ dE (3.9-18) 



T{n + 1) 



Moreover, the normal law (3.9-15), may be obtained from this by letting n 

 become large. This suggests that an approximate expression for the dis- 

 tribution of E is given by (3.9-18) when a and n are selected so as to give 

 the values of Wr and ctt obtained from (3.9-3) and (3.9-9). This gives 



a = ^4^ «+l=^ (3.9-19) 



