COMMUNICATION IN PRESENCE OF NOISE 87 



shows. When < Vi < \, i.e., oo > xo> x, the two paths may still be made 

 to coincide but it is necessary to add the contribution of the pole dX v = 1 

 as K is pulled over it. This is equivalent to passing from the first to the 

 second of equations (Al-6). The path 6 = which makes Imag. F{v) of 

 (Al-15) zero turns out to be the curve of "steepest ascent" and hence need 

 not be considered. As (Al-13) shows, the saddle point V2 does not enter into 

 our considerations because it lies on the negative real v axis and the path 

 of integration K in (Al-10) cannot be made to pass through it without 

 trouble from the singularity of F{v) at i) = 0. 



We now suppose Xq < x so that 5 and / are such as to make z'l > 1. In 

 order to remove the factor {\ — v) from the denominator of the integrand 

 in (Al-10), we change the variable of integration from v to w: 



V — \ = e"^, (1 — v)~Hv = —dw 

 P(xo, u) = — ;r—. I exp [mF(l + e")] dw 



(Al-19) 



As :; comes in along the path of steepest descent, the path of integration L 

 for w comes in from w = 'x> -{- iw and dips down towards the real w axis 

 as arg v decreases from ir. L crosses the real w axis perpendicularly at the 

 point 



wi = log {vi - 1) (Al-20) 



and then runs out to w = oo — iw along a curve which tends to become 

 parallel to the real w axis, wi may be either positive or negative. When xq 

 is almost as large as £-, wi is large and negative. 



Since F{v) is real along the path of steepest descent, F{\ + e"") is real 

 along L. This real value is — oo at the ends of L and attains its maximum 

 value F{v-^, given by (Al-16), at w = Wi. Wi is a saddle point in the complex 

 w plane because 



-^ F{\ + en = F'il + e^e"' = F'We" (Al-21) 



dw 



vanishes at w = wi. 



Instead of F(l + e^) itself we shall be concerned with 



T = F(l + e"^) - F{1 + e") (Al-22) 



so that (Al-19) may be written as 



_ exp|mF(l + .")] f ^-„„ ^^ (^j.23) 



2in Jl 



The variable t is real on the path of integration L, is zero at wi, and in- 

 creases to -f 00 as we follow L out to w = =© zt iir. It is convenient to split 



