64 BELL SYSTEM TECHNICAL JOURNAL 



Both of the encoding schemes are concerned with sending, in an interval 

 of duration T, one of A' + 1 different messages. According to communica- 

 tion theory (1, 2, 3) this corresponds to sending at the rate of T~^^ loga 

 {K + 1) bits per second. However, instead of discussing the rate of trans- 

 mission, it is more convenient, from the standpoint of (1-4), to deal with 

 the total number of bits of information sent in time T. Thus, selecting and 

 sending one of the A' + 1 possible messages is equivalent to sending 



M = logo(A + 1) (1-6) 



bits of information. M, or one of the adjacent integers if M is not an integer, 

 is the number of "yes or no" questions required to select the sent message 

 from the A + 1 possible messages (divide the A -|- 1 messages into two 

 equal, or nearly equal, groups; select the group containing the sent message 

 by asking the person who knows, "Is the sent message in the first group?"; 

 proceed in this way until the last subgroup consists of only the sent mes- 

 sage). The amount of information which would be sent in time T at the 

 ideal rate Ri defined by (1-1) is 



Mj = TRr = FT log2 (1 + l/r)= (N -\- 1/2) logs (1 + 1/r) (1-7) 



where use has been made of Wf^/Ws = v'/tr- = r, and the relation N < 

 FT < iV^ -t- 1 (which turns out to be common to both encoding schemes) 

 has been approximated by A^ + 1/2 = FT. 



When (1-6) and (1-7) are used to eliminate N and A from (1-5) the 

 result is an expression for the actual amount M of information sent (in 

 time T) in terms of (1) the amount Mi which is sent by transmitting at 

 the ideal rate (1-1) for a time T, (2) the ratio r of the noise power to the 

 signal power, and (3) the probability of no error in sending M bits of in- 

 formation in time T, this probability being given as (1 -j- erf H)/2: 



M = Ml- qMY'H + b (1-8) 



where 

 a = 2 



r iog2 e Y 



L(l + r) log. (1 + l/r)J ' 



, 1, r 27 r(l + 2r)Mj ] 

 ' - 2 ^°^^ L(l + OMog.(l + l/.)J 



(1-9) 



Here the "order of" terms in (1-4) have been neglected together with 

 similar terms which arise when N -f 1/2 is used for A'' in com])uting a and 

 b. The term b is usually small compared to aM,II. 



The more slowly we send, the less chance there is of error. The relation- 

 shij) between M, Mi and the ])r()l)ability of no error, as cominited from 



