78 BELL SYSTEM TECHNICAL JOURNAL 



where we have written Ii , h , h , h for Id , Tsi , Ic2 , I$2 . We now make, 

 the transformation 



/i = Ri cos ^1 Is = R2 cos 02 



I2 = Ri sin 61 Ii = R2 sin 62 



and average the resulting probabiUty density over di and 62 in order to get 

 the probability that Ri and R2 lie in dRi and dR2 . It is 



R\ R2 dR\ dRo 



f ddi \ dd2 exp 

 •'o 



4:TV- A Jo 



{^oR\ + lAoi?' - 2(jiizRiR2 cos {do - di) - 2fxuRiR2 sin (^o - di)] 



lA 



Since the integrand is a periodic function of Bo we may integrate from 

 Q2 = 61 to 62 = 61 -\- 27r instead of from to lir. This integration gives the 

 Bessel function, /o , of the first kind with imaginary argument. The result- 

 ing probabiUty density for i?i and Ro is 



R1R2 r 1R1R2 r 2 , 2 il/2\ "Ao /T32 , „2n /, - . ,x 



[mi3 + M14J 1 exp - —- {Ri + R2) (3./-13) 



^^'\-A 



A \ A ^" '^ / ^ 2.4 



where, from (3.7-12), 



.,222 

 A = \f/o — His — fJLu 



His and nu are given by (3.7-11). Of course, Ri and R2 are always positive. 

 For an ideal band pass filter with cut-offs at/a and/s we set 



fm = ^^^, -^(f) = ^^0 for fa<f < /6 



and obtain 



h = "^oifb — fa) 



'■^^ o / /• r N ji- '^0 sin x(/6 — /a)r 



13 = / Wo COS 27r(/ — fm)T df = 



''fa 

 r/b 



14 = / Wo sin 27r(/ — /,„)r df ^ 



-'fa 



The /o term in (3.7-13), which furnishes the correlation between Ri and i?o , 

 becomes 



sin X 

 R1R2 ~T~ 



_ sin^ X 



