COMMUNICATION IN PRESENCE OF NOISE 75 



Since p^^iii, Xn) is the joint probability density of ti and .Vo, the integration 

 with respect to .Vo in the first part of (5-7) yields the probability density of 

 u, and the integration with respect to ii in the second part gives the prob- 

 ability density /Jci(.Vo) (stated in (4-10)) of .Vu. Thus (8) 



i 



dxopoiu, .Vo) = 



[«/2(l + rWe 



q -«/2(l + r) 



'o "■—"••- 2(l + r)r(g-f 1) 



(5-8) 

 e 



I 



dupo(n, .To) 



:.vo/2H'r"'°''^ 



'o "" ' 2rT(q + 1) 



Setting (S-S) in (5-7) and putting u = 2(1 -f r)y and .Vo = 2ry in the two 

 parts of (5-7) reduces them to the same form. Thus (5-7) is equal to 



2 - 



r(<7 + 



T)Ly'^~''y (5-^) 



with / = (2q log qY^-. In order to show that (5-9) is 0(1 '9) we use the ex- 

 pansion 



-y -f g log y = -q -\- qlogq - (y - q)-/(2q) + (y - qY/{3q-) 



- (v - qYqlq + (y - q)d\-'/4: 



where ^ 6 ^ 1. Let v represent the sum of the (y — qY and (y — qY terms, 

 and expand exp r as 1 + v plus a remainder term. The integral of exp — 

 (y — q)-/(2q), taken between the hmits q ±: (, can be shown to be of the 

 form 1 — 0(1 'q) by integrating by parts as in obtaining the asymptotic 

 expansion for the error function. The term in (y — qY vanishes upon integra- 

 tion and the remainder terms may be shown to be of 0(1/ q). In all of this 

 work a square root of q comes in through the fact that 



1 > i2irqY'-q''e-''/r(q -f 1) > exp [- \/il2q)] (5-10) 



We have just shown that the error introduced by restricting the region of 

 integration as indicated by (5-6) introduces an error of order 1 'q which 

 vanishes as 5 ^ ^c . The normal law approximation to po(ii, Xo) predicted by 

 the central limit theorem holds over this restricted region. However, instead 

 of appealing to the central limit theorem to determine the accuracy of the 

 approximation, we prefer to deal directly with the functions involved. 



Consideration of (5-4) and the behavior of p(,{u, .Vu) suggests the substitu- 

 tion 



.To = 2r{q -I- a) 



u = 2(1 -f r){q -f /3) 



(5-11) 



