COMPARISON' OF SIGXALLIXG ALPIIABI:TS 513 



/(/) of the form (7), or in other words into a seciueiice of .suinplc \iihies 

 f{in/2\V). To do this, we construct a new alphabet containing K letters 

 which are different sequences of real iuuul)ers of some fixed length, 

 say D places. When we let the letters of the new alphabet correspond to 

 letters of the old one the message is translated into a sequence of real 

 numbers which we use for the sequence /(m/21F). 

 If the 7v letters of the sequence alphabet are 



Si'. On , • • • , OiD 

 S2' Q21 , ' ' ' , (f2D 



the expression (8) for the average power of the function /(/) becomes 



P = ^{d\ + d\+ ■■■ -{-dl) (9) 



where 



D 



di\ = S « ii • 



If the D numbers in the sequence 8i are regarded as coordinates of a 

 point in Euclidean D dimensional space, d1 represents the square of the 

 distance from the point representing Si to the origin. 



AVhen/(0 is transmitted, the received signal will be/(0 + n{t) where 

 n{t) is some (unknown) white Gaussian noise signal. The noise signals 

 n{t) are characterized by the fact that their sample values n{m/2W) 

 are independently distributed according to Gaussian laws. That is, 



P,„b(„(^.)<x) = ^ £.--... (H)) 



The variance a^ of the distribution of noise samples is, by an application 

 of (8), the power of this ensemble of noise signals. 



At the receiving end of the channel, there is a detector which observes 

 each block of D sample values /(m/2T^) + n(m/2W) and tries to decide 

 which one of the A' letters Si , ■ ■ ■ , Sk was sent. In terms of the geo- 

 metric picture, the detector divides all of I) dimensional space into A' 

 non-o\'ei-lapping regions Ui , • ■ • , Uk with the property that, if the J) 

 received sample values are represented by a point in Ui , the detector 



