A Comparison of Signalling Alphabets 



By E. N. GILBERT 



(Manuscript received March 24, 1952) 



Two channels are considered; a discrete channel which can transmit se- 

 quences of binary digits, and a continuous channel which can transmit hand 

 limited signals. The performance of a large number of simple signalling 

 alphabets is computed and it is concluded that one cannot signal at rates 

 near the channel capacity without using very complicated alphabets. 



INTRODUCTION 



C. E. Shannon's encoding theorems associate with the channel of a 

 communications system a capacity C. These theorems show that the 

 output of a message source can be encoded for transmission over the 

 channel in such a way that the rate at which errors are made at the re- 

 ceiving end of the system is arbitrarily small provided only that the 

 message source produces information at a rate less than C bits per second. 

 C is the largest rate with this property. 



Although these theorems cover a wide class of channels there are two 

 channels which can serve as models for most of the channels one meets 

 in practice. These are: 



1. The binary channel 



This channel can transmit only sequences of binary digits and 1 

 (which might represent hole and no hole in a punched tape; open-line 

 and closed line; pulse and no pulse; etc.) at some definite rate, say one 

 digit per second. There is a probability p (because of noise, or occasional 

 equipment failure) that a transmitted is received as 1 or that a trans- 

 mitted 1 is received as 0. The noise is supposed to affect different digits 

 independently. The cpacity of this channel is 



C = 1 + p\ogp+ (1 - p) log (1 - p) (1) 



bits per digit. The log appearing in Equation (1) is log to the base 2; 

 this convention will be used throughout the rest of this paper. 



1 C. E. Shannon, "A Mathematical Theory of Communication," Bell System 

 Tech. J., 27, p. 379-423 and pp. 623-656, 1948, theorems 9, 11, and 16 in particular. 



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