70 BELL SYSTEM TECHNICAL JCURXAL 



where the v universe from which the B's are drawn has, as before, standard 

 deviation v given by (.V + 1 '2)v- = Ws. When the signal /o(/) is sent, the 

 input to the receiver is /,)(/) + J{l) and the process of reception consists of 

 selecting the smallest of the A' + 1 .v^'s 



x, = 2T-' f [IdO - /o(/) - J{t)?dl (3-4) 



•'-00 



= i: (at - Air -bs- 



n=—S 



The second expression for .v/, is the same as the one given by (2-7) for the 

 first encoding scheme, and the discussion in Section 2 following (2-7) may 

 also be applied to the second encoding scheme. In particular, the probabiUty 

 of obtaining no error in transmitting a signal through noise is the same in 

 both systems of encoding, and is given by the Prob. {P\Q, • • • , PkQ > PoQ) 

 of the mathematical problem of Section 1. 



4. Solution of the Mathematical Problem 



We shall simplify the work of solving the mathematical problem stated 

 in Section 1 by taking a = 1 and v-/(r- = r. First regard the 4X + 2 numbers 

 An , Bn, n = — .Y, • • • , N as fixed or given beforehand. Geometrically, this 

 corresponds to having the points Pq and Q given. Select a typical set of 

 random variables A,, , w = —N,---, N, k > and consider the associated 

 set of variables 



y„ = ^f - A'y - 5„ = .4i" -f y„. (4-1) 



y„ is a random variable distributed normally about its average value 



% = -Ai'^ - Bn (4-2) 



with standard deviation a = \. The quantity .ya-, defined by (1-3) and repre- 

 senting the square of the distance between Pk and Q, may be written as 



N 



n=—N 



Thus Xk is the sum of the squares of 2.V + 1 independent and normally 

 distributed variates, having the same standard deviation but different 

 average values. The probability density of such a sum is remarkable in 

 that it does not depend upon the y„'s individually but only on the smu of their 

 squares which we denote by 



«= z ill = i; u\r -^ Bnf 



n=-S n=-\ 



(4-4) 

 _ 1 fEnergv' in sent signal + Encrgyl 



|_in noise J 



