COMMUNICATION IN PRESENCE OF NOISE 



79 



(5-29) 



respectively. It is convenient to introduce the notation 



7 = «-l, 5 = ^-1 



s = r{\ + tA"), t ^ (1 + r)(l + b/m). 



It is seen that for the restricted region in which | a \ and 1 18 | are less than 



/ — (2q log qY'-, I 7 I and | 8 \ are at most 



0(q''- logi/- q) = 0(wi/2 logi/2 m). 



Hence s, /, (1 + 4siy''\ vi differ at most from r, 1 + r, 1 + 2r, 1 + l/r, 

 respectively, by terms of order nr'^'- log^'- m. Similar considerations show 

 that 



(4Trmb2)-"' = {2Trqy'Di[l + 0(w-i/2 logi/2 m)] 



(5-30) 



The argument of the exponential function in (5-27) must be expanded in 

 powers of y and 5. It turns out that when y and 8 lie in the restricted region, 

 powers above the second may be neglected. For the sake of convenience we 

 rewrite (5-13) and introduce zii 



z = Xou = -im-st = 4r(l -\- r)(m + y)(tn -f 8) 



Gi = 4^(1 + r)m- (5-31) 



z- z, = 4;'(1 + r)[m(y + 5) + 7^] 



so that 3 — Si is 0(m^'- \og^'- m). Then 



(1 + Astyi-' = (1 + z/m'Y'-' 



= (1 -f z,/in^'i- + (s - si)(l -f Si/w2)-i/V(2w2) (5-32) 



- (g - 3i)-(l + z,/m-)-'i-/{Sni') + Ro 



where R-: is of the same order as (z — ZiY/m , or m~^'~ log^/- m. It follows 

 that 



(1 -i- 45/)' 



1 -f- 2^ L ^" w-_ 



2r-(l -F rf (7 -f 5)- 

 (1 + 2ry m' 



+ Q{m~^'- \og'- m) 



i\ = 



(1 + r) 



1 + 



r(l -j- y/m 



_ r\\ +r){y + 8f 

 m\\ + 2ry 



r Y y + 8 yf\ 

 \ -\- 2r\_ m wz-J 



(5-33) 



+ 0{nr^'~ log''' w) 



