MATHEMATICAL ANALYSIS OF RANDOM NOISE 81 



In line with the earher work we set 



Xi = Ic = R cos 6 Xi = Is = R sin 6 

 X2 = l[ = R' sin ^ + i? cos dd' 

 x^ = l[ = R' cosd - R sin 66' 

 .x-3 = I'J = R" cos 6 - 2R' sin 66' 



- R cos 66'- - R sin 66" 

 X6 = I's = R" sin 6 + 2R' cos 66' 



- R sin 66'^ + R cos 00" 



The angle 6 varies from to lir and 6' and 6" vary from — oo to + oc . By 

 forming the Jacobian it may be shown that 



dxi dx2 • • • dxe = R^ dR dR' dR" dd d6' d6" 



Also, the quantities in (3.8-4) are 



xl + xl = R^ 0:1X3 + XiXs = RR" - R^d'^ 



X1X2 — XiX^ = R'6' X2 + xl = R'' + R''6'' 



X2Xz - x^xg = RR"6' - 2R'~6' - R'R6" - Ri'6'^ 



xl + xl = R'" - 2RR"6'- + AR'V + 4RR'6'd" 



+ i?'0'* + R^d"^ 



The expression for p(R, 0, R") is obtained when we set these values of the 

 x's in (3.8-4) and integrate the resulting probability density over the ranges 

 of 6, 6', 6": 



^(^' "' ^") = 8^ i '' L "' L """ ^'-'-'^ 



exp -^^[BqR^ + 2BiR:6' - 2B.XRR" - R^6'^) 



+ B22R-6'- - 2B3R6'{R" - R6'-) 



+ B,(R"' - 2RR"6'^ + R'6" + R'd'")] 



The integrations with respect to 6 and 6" may be performed at once leaving 

 p(R, 0, R") expressed as a single integral which, unfortunately, appears to 

 be flifficult to handle. For this reason we assume that w(f) is symmetrical 

 about the mid-band frequency /« . From (3.8-2), Z»i and bs are zero and 

 from (3.8-3), Bi and B3 are zero. 



