COMMUNICATION IN PRESENCE OF NOISE 63 



The values obtained in this way are Usted in the second column of the 

 following table. The values of Prob. (PiQ, • • ■ , PkQ > PoQ) obtained 

 from (1-4) (in which the "order of" terms are ignored) and (1-5) are given 

 in the third column. Column IV lists values obtained from (1-4) and a 

 simplified form of (1-5) obtained by omitting the last term in (1-5). These 

 values are less accurate than those in the third column. The values in 

 Column V are computed from (1-5) and a modified form of (1-4) obtained 

 by adding the correction term shown in equation (5-53) (with B — H). 

 The values in Column V are presumably the best that can be done with the 

 approximations made in Section V of this paper, although the first entry 

 renders this a little doubtful. 



Prob. {PiQ, ■■■ , PkQ > PoQ) for N = 99.5 & r = 1 



to the ideal rate of signaling. The non-integer value of 99.5 for N is ex- 

 plained by the fact that the calculations were started before the present 

 version of the theory was worked out. It will be noticed that for A' -f- 1 = 

 9100^-30 ^^ Qf ^]r^Q approximate values exceed the .994 obtained by numerical 

 integration. I am in doubt as to whether the major part of the discrepancy 

 is due to errors in numerical integration (due to the considerable difficulty 

 encountered) or to errors in the approximations. 



In both encoding schemes, the point Po corresponds to the transmitted 

 signal, Q to the transmitted signal plus noise, and Pi, P^, • • • Pk to K 

 other possible signals. The average signal power turns out to be (A^ + 1/2)<t- 

 and the average noise power to be (^V + l/2)j'-. Furthermore, 



.vo = twice the average power in the noise. 



Xk = " " " " " " " plus the ^th signal. 



Prob. (PiQ, ■ ■ ■ PkQ > PoQ) = Probability that none of the K other 



signals will be mistaken for the signal sent, 

 i.e., the probability of no error. 



The random numbers A „ are taken to be distributed normally instead 

 of some other way because this choice makes the encoding signals (in our 

 two schemes) resemble random noise, a condition which seems to be neces- 

 sary for efficient encoding (1, 2). 



