COMPARISON OF SIGNALLING ALPHABETS 515 



111 the cases to be computed Qa is a rapidly decreasing function of 

 A',7 and the only terms worth keeping in (12) are the ones for which 

 Rij is the smallest of the nunihers /i?,! , • • • , RiK ■ Moreover since 

 till' Qij are all small' enough so that the upper and lower bounds differ 

 only by a few per cent, the ui)per bound is a good approximation to p,- . 

 Then a simple approximate formula for the average letter error prob- 

 ability /> = 0>i + • • • + Vk)/K is 



'V'ZTr J Tola 



where 2/-o is the smallest of the /\ (7v — 1) '2 distances Ra and .V is the 

 average o\-er all l(>ttei-s in the ali)iiabet of the numb(n- of lettei' points 

 which are a distance 2/o away. 



3. Efflciency graph 



The efficiency graph to be described was constructed originally to 

 comj)are alphabets for signalling telephone numbers of length equal to 

 t(Mi decimal digits. It was desired that on the average only one telephone 

 number in 10^ should be received incorrectly. As described in Part I 

 section 2, if the telephone numbers are encoded into sequences of letters 

 from an alphabet of K letters, we must reciuire that the average prob- 

 ability of error in any letter be 



p = 10"' logio K (U) 



or smaller. 



Given an alphabet, one can compute with the help of (13) and (14) 

 and a table of the error integral the largest value of the noise power <x~ 

 which can be tolerated. The average power of the transmitted signal is 

 P given by Equation (9). Hence we can compute the smallest signal to 

 noise ratio 



Y = P/a' (15) 



which will be satisfactory. 



A letter containing log K bits of information is transmitted during 

 an interval of D 2W seconds. Hence the rate at which information is 

 received is 



R = '^KME (Hi) 



bits per second. Again Ec}uation (16) ignores a term representing in- 



