514 THE BELL SYSTEM TECHNICAL JOTTRNAL, MAY 1952 



decides that Si was sent. By Equation (10), the probabihty that the de- 

 tector picks the wi'ong letter when Si is sent is 



Pi = 7^3^7y3 J j_ • ■ ■ J c~"''''^''^ diji ■ ■• dyo (11) 



where Ui is the set of all points not in Ui and r, is the distance from 

 (Ui , • • • , Vd) to the point representing Si . 



For any given alphabet the best possible detector (in the sense that it 

 minimizes the average probability of making an error in guesssing a 

 letter) is called a maximum likelihood detector. The I'egion lU for a maxi- 

 mum likelihood detector consists of all points (i/i , • ■ ■ , iju) which are 

 closer to the point Si than to any other letter point »S>(/',: < Vj for all 

 j 7^ i). To prove that this choice of U i is best possible consider any other 

 detector such that Ui contains a set V of points in which I'i > Tj . A 

 direct calculation shows that the detector obtained by removing V from 

 Ui and making F part of U j has a smaller probability of error per letter. 

 The set of points equidistant from two given points is a hyperplane. The 

 region Ui of a maximum likelihood detector is a convex region bounded 

 by segments of the hyperplanes 



Ti = n , Ti = ro , • • • . 



To compare signalling alphabets under the most favorable possible 

 circumstances, we always compute letter error probabilities assuming 

 that the detector is a maximum likelihood detector. 



2. Computation of error probabilities 



Exact evaluation of the letter error probability integral (11) is im- 

 possible except in a few special cases. Fortunately we are only interested 

 in (11) when a- is small enough in comparison to the size of Ui to make the 

 integral small. Then fairly accurate approximate formulas can be de- 

 rived. 



Theorem 3. Let Rij be the distance between letter points *S, and Sj . Then 



11(1 - Qi,) < Pi <ZQiJ (12) 



where 



1 r°° 



Qij = —y= / e ""^^ dx. 



The proof of Theorem 3 follows from the fact that Qij is the prob- 

 ability that, when »S\ is transmitted, the recei\'ed sequence will be closer 

 to Sj than to Si . 



