COMMUNICATION IN PRESENCE OF NOISE 81 



by (5-38)). Then, since A exp S{a, 0) is positive, it follows from (5-40) that 



L(Ai) > / > ^(Xo) (5-41) 



Also since exp [— ^X exp S(a, /3)] lies between and 1 for all real values of 

 a and ^ it may be shown from (5-24) that i>(X) is equal to /(X) -\- Q{q~^'~) 

 where 



da / d^ A exp [- Qia, /3) - ^Xe^^"'^^] (5-42) 



■CO •'—00 



Here X is a constant and Q{a, 0), A, S{a, (3) are defined by (5-19) and (5-37). 

 From (5-39) and (5-41) we obtain 



Prob. {PiQ, •■■ , PkQ > PoQ) = /(I) + e[/(Xi) - /(I)] (5-43) 

 + (1 - d)[Ji\,) - /(I)] + 0(1/A0 + 0(5-1/2 log3/2 q) 



where < < 1. It will be shown later that /(Xi) and /(X2) differ from /(I) 

 by terms which are certainly not larger than 0{q~^'^). 



The problem now is to evaluate the integral (5-42) for /(X). It turns out 

 that exp [— ^X exp S(a, (3)] acts somewhat like a discontinuous factor which 

 is unity when S{a, 13) + log ^X is negative and zero when it is positive. In 

 order to investigate this behavior we make the change of variable 



a — (3 — w a — y — rw 



{I -{- r)a - rl3 = y (3 = y - (1 -{- r)w (5-44) 



da dl3 = dw dy 



From (5-19), (5-37), and (5-42) 



Q{a, (3) = [f- + (1 + 2rmD = fD + ^^~/2q 



S{a, /3) = w - y^Z) (5-45) 



/oo »00 



dy / dw Dx exp [- fD - ^'/2q - A\e"'~"'''] 

 ■00 •'—00 



Here and in the following work j3 is to be regarded as a function of iv and y. 

 Split the interval of integration with respect to w into the two subintervals 



(— 00 , Wo) and (wo, ^) where 



li'o = f-D - log ^X (5-46) 



and y is temporarily regarded as constant. In the first interval 



/wo 

 exp[- ^'/2q- g"""'"] Jw 



(5-47) 

 e-^'"" dw - (1 - exp [- ^-"'0])^-^^/=" dw 



