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BELL SYSTEM TECHNICAL JOURNAL 



The symbol 0{N-'^'^ log''^ N) stands for a term of order N'^''^ log3/2 N, i.e., 

 a positive constant C and a value No can be found such that the absolute 

 value of the term in question is less than CN~'^'~ log^'^ N' when N > .Yi,. 

 In order to obtain actual numerical values for C and A^o, considerably 

 more work than is given here would be required. The term 0(1/A') is of 

 the same nature. The "order of" terms have been carried along in the work 

 of Section 5 in order to guard against error in the many approximations 

 which are made in the derivation of (1-4). 



is the origin of coordinates 

 a'-^Iv-Ao^V-A^n^ of point Pk 

 in space in 2m + t dimensions 



Fig. 1 — Diagram indicating relalionshi]) between points Po, Q, and Pk corresponding 

 to signal, signal plus noise, and k^^ signal not sent (k > 0), respectively. 



The last term within the bracket in (1-5) has been retained even though 

 it gives terms of order A'^"*'' log A^ when (1-5) is j^ut in (1-4) and could 

 thus be included in 0(.V""'' log^'- N). As shown by the table in the next 

 paragraph, inclusion of this term considerably imjjroves the agreement 

 between (1-4) and values of I^rob. (PiQ, • ■ ■ , PkQ > P^Q) obtained by 

 integrating the exact expression (4-12) numerically. This suggests that 

 the term ()(A' "- log^'- A') in (1-4) is unnecessarily large. 



Although the "order of" terms in (1-4) give us some idea of the accuracy 

 of the apj)roximati()n expressed by (1-4) and (1-5), a better one is desirable. 

 With this in mind the lengthy task of computiiTg the exact ex[)ression (4-12) 

 for Prob. (PiQ, ■ • • , PkQ > PoQ) by numerical integration was undertaken. 



