COMMUNICATION IN PRESENCE OF NOISE 65 



(1-8), is shown in the following table. The probability of no error is de- 

 noted by p and the terms are given in the same order as on the right of 

 (1-8) in order to show their relative importance. The ratio M/Mi{=R/Ri) 

 for r = 0.1 is shown as a function of M in Fig. 2. 



For r = Ws/Ws = 0.1 

 Mi bits M for p = .5 M for p = .99 M for p = .99999 



102 ^^ _ + 3.75 Ml - 24.3 + 3.75 Mj - 44.6 + 3.75 



W " '' + 7.07 " - 243+ 7.07 " - 446+ 7.07 



106 u "4-10.38 " -2430+10.38 " -4460+10.38 



For r = W^/Ws = 1 



102 Mj-O-i- 4.44 M7-33.4+ 4.44 Af7-61.2+ 4.44 

 10^ *' "+ 7.76 " - 334+ 7.76 " - 612+ 7.76 

 10' " "+11.08 " -3340+11.08 " -6120 + 11.08 



There may be some question as to the accuracy of the values for p = .99999, 

 especially for Mi = 100, since this corresponds to points on the tail of 

 the probability distribution where the "order of" terms in (1-4) become 

 relatively important. 



Of course, for a given bandwidth, the ideal rate of signaling Ri (given by 

 (1-1)) for r = .1 exceeds that for r = 1 in the ratio (logo ll)/(log2 2) = 

 3.46. 



The above results agree with the statement that, by efficient encoding, 

 the rate of signaling R can be made to approach the ideal rate Ri = Mi/T 

 given by (1-1). As applied to our two schemes, the term "efficient encoding" 

 means using a very large value oi FT or N. To see this, divide both sides of 

 (1-8) by Ml and rearrange the terms: 



1 - M/Mi = aH M'"' + 0(M7' log Mi) (1-10) 



When Mi is replaced by RiT in M/Mj, the fraction M/T occurs. We shall 

 set R = M/T and call R the rate of signaling corresponding to some fixed 

 probability of error (which determines H). Thus, when (1-7) and the defini- 

 tion (1-9) for a are used, (1-10) goes into 



^^^ ~ ^^ = ^ A- 0((\o<y FT) /FT) (1-11) 



Ri [(1 + r)FTY'Uoge(l + 1/r) + ^^lo, /^ i |, /^ i ; U ii; 



Equation (1-11) shows that when r and H are fixed (i.e. when the noise 

 [)Ower/signal power and the probability of error are fixed) R/Ri approaches 

 unity as FT — » 00 . This is shown in Fig. 2 for the case r = 0.1. S'mceR/Ri => 

 M/Mi, M/Mj must approach unity and consequently M as well as Mi in- 



