50 BELL SYSTEM TECHNICAL JOURNAL 



From the central limit theorem for two dimensions it follows that I\ and /g 

 are distributed normally. As in (3.1) 



Mn = /i = E cl-h -^ / w{J) df = ;Ao 



1 JO 



M22 = h = I'l — 4^0 (3.1-2) 



.V 



Ml2 = /i /2 = X/ C n a-^'C. { cos (a)„ i — (/:,t) COS (w„ / — (p„ + aj„ t) } 

 1 



Now the quantity within the parenthesis is 



cos (cOni ~ <Pn) COS a)„r — COS (cOni — ^n) sin (a)„/ — (^„) sin OJnT" 



and when we take the average with respect to <^„ the second term drops 

 out, giving 



Ml 



2 = X) c'„-| COS aj„r ^ / w{f) COS 2x/r c?/ = \}/r (3.2-3) 



1 JQ 



where we have used co„ = 27r/„ and the relation (2.1-6) between w{j) and \p{r). 

 The probability densit}^ function for /i and Ii may be stated. From the 

 discussion of the normal law in 2.9 it is 



2x 



exp 



— i/'o/i — 



2(^1 - ^l) J ^'-'-'^ 



For a band pass filter whose range extends from /a to/b we have 



rib 

 ^T = I Wo cos 27r/r (// 



Sm OJbT — Sm COaT ,- - _s 



= Wo {3.2-s) 



Ztt 



= — sin TT(Jb — fa) cos irrifh + /a) 



TTT 

 lAo = Wo(/6 — /a) 



where wo is the constant value of u'(/) in the pass band and 



uh = 2Tvfh (3.2-6) 



OJo = 27r/a 



According to our formula (3.2-4), /i and 1 2 are independent when \pr 

 is zero. For the r's which make i/'r zero, a knowledge of /i does not add to 

 our knowledge of lo . For example, suppose we have a narrow filter. Then 



,Ar = Owhenr = [2(ft + fa)]'' 



\{/r is nearly — i/'o when t = [fb -\- faT 



