COMMUNICATION IN PRESENCE OF NOISE 85 



P(xo, ti) = — r— . / z ^e "'" [ave. e"'] <^3 



Zirl •/— 00, abcvcO 



(Al-6) 



= 1- -^. s-V^'Mave. e"lJ2 



2tI J-oo. below 



where the subscripts "above 0" and "below 0" indicate that the path of 

 integration is indented so as to pass above or below, respectively, the pole 

 at 3 = 0. The value of ave. exp (izx) may be obtained by setting N + 1/2 

 = m in (4-5). The new notation 



xo = Ms = 2ms, u = 2ml, 2z = ^ (Al-7) 



enables us to write ^ 



1 r°° 



P{xo,u) = - ^ . r~' exp m[-is^ - log (1 - /f) 



Zirl J-oo.aboveO (A 1-8) 



- t + t(l- 7f)"'] d^. 

 The further change of variable 



I - it = V (A 1-9) 



carries (Al-8) into 



P{x,,u) = ~ f {{ - vr'exp[mF(v)]dv (Al-10) 



2Tri Jk 



where the path of integration K is the straight line in the complex v plane 

 running from l + Zxtol — joo with an indentation to the right of z' = 1, 

 and 



F{i') = sv - log V + t/v - s - f. (Al-11) 



The K used here should not be confused with the K denoting the number 

 of messages in the body of the paper. We have run out of suitable symbols. 

 An asymptotic expression for (Al-10) will now be obtained by the method 

 of "steepest descents." The saddle points are obtained by setting the 

 derivative 



F'{v) = s - \/v - l/v"- (Al-12) 



to zero and are at 



^1 = [1 + (1 + 4siy'V2s 



1,2 = [1 - (1 -f- 4sty'']/2s (Al-13) 



