COMMUNICA TION IN PRESENCE OF NOISE 61 



power spectrum is limited to the band (0, F) but the signal, regarded as a 

 function of time, is not exactly zero outside its allotted interval of length T. 

 It turns out that both schemes lead to the same mathematical problem 

 which may be stated as follows: Given two universes of random numbers 

 both distributed normally about zero with standard deviations a and v, 

 respectively. Let the first universe be called the a (signal) universe and 

 the second the v (noise) universe. Draw 2iV + 1 numbers A_!^, A-i^+i, • • • , 

 Ao^\ • ■ • , Ai?^ at random from the a universe. These 2N -\- \ numbers 

 may be regarded as the rectangular coordinates of a point Pn in 2.Y + 1- 

 dimensional space. Draw 2N + 1 numbers B-n, • ■ • , Bi), ■ ■ • , B^ at 

 random from the v universe and imagine a (hyper-) sphere S of radius .Vo 

 = Po(?, where 



1/2 



•To 



= Z 5; = p^, (1-2) 



centered on the point Q whose coordinates are .4„ + Bn, n = —N,---, 

 0, • • • , iV. Return to the a universe, draw out A' sets of 2N -\- 1 numbers 

 each, denote thekth set by .4i^^, • • • , /lo^"\ • • • , .4^^' and the associated 

 point by Pa-. 



What is the probability that none of the A' points Pi, • • • , Pk lie within 

 the sphere 5? In other words what is the probability, which will be denoted 

 by "Prob. (PiQ, • • • , PkQ > PoQ)," that the A distances P^Q, • ■ • , PkQ 

 will all exceed the radius PqQ? In terms of the ^„'s and ^„'s we ask for 

 the probability that all K of the numbers .ri, xo, ■ ■ ■ , Xk exceed .Vo where 



Xk 



= Z (Ai'' - Air - Bj' = P,Q^ (1-3) 



Expression (1-2) for .vo is seen to be a special case of (1-3). The relationship 

 between the points Po, Q, Pi, P2, • • • , P/,, • • • , Px is indicated in Fig. 1. 

 The answer to this problem is given by the rather complicated expression 

 (4-12) which, when written out, involves Bessel functions of imaginary 

 argument and of order N — 1/2. When N and A become very large the 

 work of Section 5 shows that the probability in question is given by 



Prob. {PiQ, ••■ ,PkQ> P,Q) 



= (1 + erf H)/2 -f 0(1/A) + 0(7V-i/2 log''" N) (1-4) 



where, with r = v'/a", 



H = (^-4^ ')''' [(^V + 1/2) log. (1 + \/r) - log. (A + 1) 



, 1 , 27rA^(l -f 2r)" 

 + 2^°^^ (1 + .)^ 



(1-5) 



