COMPAKISON OF SIGNALLING ALI'IIABKTS ,") 1 7 



bets. All alphabet is considered poor il' its point on the ethcicncy j^rapli 

 Hes far above the ideal curve R/W = C/W = log (1 + Y). 



4. The alphabets 



The alphabets which appear on the efficiency graph arc the following: 



excess three (XSS): tiie t(Mi sequences of 4 binary digits whicli I'cpre- 

 , sent 3, 4, • • • , and 12 in binary notation; 



two out of five: tlie ten setincnccs of five binary digits which contain 

 exactly two ones; 



pulse position (PPIO): the ten sequences of ten binarj^ digits which 

 contain exactly one one; 



2" binary: all of 2 sequences of D binary digits. 



pulse amplitude (PAn) : the 2n + 1 sequences of length 1 consisting 

 of — n, — w + 1, • • • , n. This alphabet gives rise to a sort of quantized 

 amplitude modulation. 



pulse length (PLn) : the ?i + 1 sequences of n binary digits of the form 

 11 • • • 10 • • • 0, i.e., a run of ones followed by a run of zeros. 



Minimizing alphabets (K, D) : The above alphabets are taken from 

 actual practice. They are convenient because, aside from PAn, they 

 require a signal generator with only two amplitude levels. If we ignore 

 ease of generating the signals as a factor, a great many geometric ar- 

 rangements of points suggest themselves as possible good alphabets. 

 The principle by which one arrives at good alphabets may be described 

 as follows. When a D and K have been determined which give the desired 

 information rate R [by Equation (16)] try to arrange the K letter points 

 in D dimensional space in such a way that the distances between pairs 

 of points are all greater than some fixed distance and that the average 

 of the K sc[uared distances to the origin is minimized. By Equations 

 (9) and (13) it is seen that, apart from the small influence of the factor 

 A^, this process must minimize the signal to noise ratio Y required. 



Ordinarily it is difficult to prove that a configuration is a minimizing 

 one. Even to recognize a configuration which leads to a relative minimum 

 {i.e. a minimum over all nearby configurations) is not always easy. The 

 eight vertices of a cube, for example, do not give a relative minimum. 

 Consequently, most of the alphabets to be described are only conjectured 

 to be "best possible." Each of them satisfies one necessary requirement 

 of minimizing alphabets that the centroid of the point configuration 

 (assuming a unit mass at each letter point) lies at the origin. That this 

 condition is necessary follows from the easily derived identity 



r2 = n - Ro 



