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BELL SYSTEM TECHNICAL JOURNAL 



creases linearly with FT. Thus, for efficient encoding M is large and, from 

 (1-6), so is A'. 



It should be remembered that equation (1-8) has been established only 

 for the two encoding schemes of this article. The question of how much 

 faster M/T approaches Ri for the more efficient encoding schemes mentioned 

 at the end of Section 2 still remains unanswered. 



0.6 



10^ lO'* 



M = BITS OF INFORMATION IN MESSAGE 



105 



10® 



Fig. 2 — Curves showing the approach of /?//?/ (= M I Mi) to unity as the message 

 length increases and the probabihty of no error remains tixed. R is the rate of signaHng 

 at which the probabihty of no error is p and Rj is the ideal rate. 



It gives me pleasure to acknowledge the help I have received in the prepa- 

 ration of this memorandum from conversations with Messrs. H. Xyquist, 

 John Riordan, C. E. Shannon, and M. K. Zinn. I am also indebted to Miss 

 M. Darville for comj)uting the tables shown above and for checking a num- 

 ber of the equations numerically. 



2. The First Encoding Scheme 



Suppose that we have A' + 1 different messages any one of which is to 

 be transmitted over a uniform frequency band extending from zero to the 

 nominal cut-off frequency F in a time interval of length T. The adjective 

 "nominal" is used because the sudden starting and stopping of the signals 

 given by the first encoding scheme produces frequency components higher 



