COMMUNICATION IN PRESENCE OF NOISE 9l 



When (A2-1) is substituted in Bessel's differential equation, which we 

 write as 



^' dy"- ^'^^-^^ '^ ^dy ^"'"^^^ ~ ^'^^ "*" ^'^^"^^^^ ^ ^' 

 we obtain a differential equation for V: 



V" = (4 - y)w-V4 - {2qu' + w-^)y-W - V'"- (A2-4) 



Here the primes denote differentiation with respect to y. The constants of 

 integration associated with (A2-4) are to be chosen so that 



y _^ 3,2/(4^ + 4) as y -> 0. (A2-5) 



This condition is obtained by comparing the limiting form of (A2-1), in 

 which w -^ 1 + >'V2, with 



Condition (A2-5) completely determines V since substitution of the 

 assumed solution 



F = 4->(5 + i)-y + ciy + C2/ + . • . 



in (A2-4) leads to relations which determine Ci, Ci, • • • successively. 

 Let V = V. Then (A2-4) becomes 



v' = c - 2bv - v^ (A2-6) 



where c and b are known functions of y defined by 



c = (4 - y'~)w-'/4, b = (qw-\- w-^/2)y-' (A2-7) 



From (A2-5), v -^ y/{2q + 2) as y -^ and therefore 



V 



I vdy (A2-8) 



•'0 



We first show that \v\ < l/(2q — 1) when q > 1. The (y, v) plane may 

 be divided into regions according to the sign of v'. The equations of the 

 dividing lines between these regions are obtained by setting ii' = in (A2-6). 

 Thus, for a given value of y, v' is positive if V2 < v < vi and negative if 

 V > viOT V < Vi where 



v,= -b+ (b' + cy = c/[b + (62 + cyi^] 



v^= -b+ (62 + cyi^ (A2-9) 



When y > we have b ^ q. A plot of c versus y shows that | c | ^ 1. Hence, 



