140 BELL SYSTEM TECHNICAL JOURNAL 



The characteristic function for I cu ^c2, I si, Is2, Id, hi, I si, Is2 is, from 

 (A2-2) and (A2-3) of Appendix II and Section 2.9 of Reference A, 



ave. exp i[zil ci + 22/02 + 23/* 1 + 24/^2 + 25/ci + 25/02 + 27/si + 28/52] 



= exp (- §) [bo{z\ + 22 + 23 + 24) + ^>2(25 + 26 + 2? + zl) 



-\-2bi{ZiZi + Z228 — 2325 — 2426) (7.5) 



+ 2g{ZiZ2 + 2324) + 2g'(SiZ6 - 2225 + 2328 - 242?) 

 — 2g"(Z626 + 2728) + 2h{ziZA — Z2 23) 



-\-2h'(ziZ8 + S227 — 23Z6 — 2425) — 2h'^{z5Zs — z^zj)]. 



Since we have included bi, h, h' , h" this holds when/g is not necessarily at 

 the center of the noise band. However, henceforth we return to our assump- 

 tion that fq is placed at the center of a symmetrical noise band and take 

 ii, h, h' , h" to be zero. 



The probability density oi Id, • • • Is2 which is to be placed in (7.4) is 

 the eight-fold integral 



dzi • ' • / dzs 



J-co (7.5) 



exp [—izilci— • • • — izdU X [ch.f.] 



where "ch.f." denotes the characteristic function obtained by setting bi, 

 hj h', h" equal to zero on the right hand side of (7.5). 

 The integral (7.4) for 12(r) may be written as 



12(t) = /i - /2 - /3 + /4 (7.7) 



where Ji is the 16-fold integral 



+00 f, +00 /. +00 



dzs 



/-rw /•-t-oo /.-1-00 /.- 



did"- dl[^{2TV dz,--- 



00 »'— 00 «/— 00 •/— ( 



exp [-izilci - . • . - izsl's2] (7.8) 



(Q + Icl){Q + 1 02) In 1^2 



X [ch.f.] 



m + iciY + ihm + ic2y + /a 



and /2, J I, Ji are obtained from Ji by replacing the product (Q + Id) 



(Q + /e2)/:i/:2 by (Q + id)idiX2, Isjuq + /c2)/:2, /.i/:i/.2/:2 



respectively. 



The integration with respect to /ci and /«i in (7.8) may be performed at 

 once. We replace (> + / e i and I dhy x and y, respectively, and use 



[ dx [ dy _£_ e-"'-»f " = Z^. (7.9) 



