COMMUNICATION IN PRESENCE OF NOISE 83 



poses, it seems worthwhile to set down approximate expressions for the 

 terms which have been dismissed as 0(g~^'^). From the above work, 



J(\) = (1 + erf B)/2 + A C dy r^'^//"" T"''"' exp [- e"'"'^"] dw 



•'-00 l.*'tt'0 



/Wo 

 e-^"%l - exp[-e'"-^''])dw 

 00 



/"'" 2 1 



log A\ j (5-53) 



^ (1 + erf B)/2 + A T dy T*''' { -.577.. + y'D}e-^\"' 



= (l+erf5)/2+(l^^y'[-.577...+ 



4-1(1 + r)-^l + (2 + 4r)52}]e-^' 



where /3i = y + (1 + r) log A\ and we have made use of the fact that 

 jSy^? changes relatively slowly in comparison with w when q is large. 



Since J(\) differs from (1 + erf B)/2 by 0(5-1/2), and since the three B's 

 for X equal to Xi, 1, and X2 differ by not more than 0(q~^/^ log (X2/X1)) = 

 0(5-1 log^/2 gj^ fj-om (5-50) and (5-38), it follows that the terms involving 

 /(Xi) and /(X2) in (5-43) may be included in the term 0(q-^'^ log'/^ ^) jj^ 

 using our result it is more convenient to deal with N and K -{- 1 instead of 

 q = N — 1/2 and K. Hence instead of B we deal with H defined by 



_ 1 (1 + rY" (K -^ \)(l + l/rr-\l + r) 



2 (5 + 1/2)1/2 «S [27r(5 + 1/2)(1 + 2r)]i/2 ' ^^'^^^ 



The difference B — H, with X = 1 and H finite, may be shown to be (with 

 considerable margin) 0{l/K) -f 0(5-1/2). From (5-43), as amended by the 

 first sentence in this paragraph, it follows that 



Prob. (PiQ, ■■■ ,PkQ> PoQ) = (1 + erf H)/2 + 0(1/A') + 0(5-1/2 log3/2 5) 



(1-4) 



where the difference between erf B and erf H has been absorbed by the 

 "order of" terms. When 5 + 1/2 is replaced by N in (5-54) the result is ex- 

 pression (1-5) for H. 



