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BELL SYSTEM TECIIXICAL JOURNAL 



the rectangular wave, the half-cycle sine wave, and the rectangular 

 wave extending through part of the dot provided the reference time 

 / = is properly chosen. 



Returning to proposition II, let us write 



5,-(/ + 7'/2) =A'{t + To+rj)rfj(x) (Ix, 



the subscript j indirating the particular t\pc of impressed dot signal, 

 and To the \alue of t for any type of signal, taken as reference. Then 



Sj(l+T,2) = [.-1 '(/ + roi + ^A " (/ + ro) + . . 7^jJ{j{x)dx (2a) 



Now, the condition that proposition I shall hold to a good degree of 

 approximation is that the expansion (2a) shall converge rapidK'. 

 Since the maximum possible value of tj is T 2 and since in practise 

 it is much smaller than T/2, the required convergence obtains for 

 much larger values of T, that is, slower speeds of signaling than that 

 required in the expansion (1). Furthermore, for the three types of 

 signals specifically under consideration t\, to and tz differ from one 

 another by quantities \cr\- much smaller than T '1 in all actual trans- 

 mission systems. 



If the conditions of proposition III are introduced, the approximation 

 is still closer and proposition I is valid for still lower signaling speeds. 



In order to arrive at quantitative ideas of the minimum signaling 

 speeds at which the foregoing proposition is valid, it is necessary, 

 of course, to specify the arrival cur\e of the transmission system 

 under consideration. An application of the foregoing analysis to 

 representative transmission systems both with and without a "cut-off" 

 frequency has shown that it is \alid to a \er\' good degree of approxi- 

 mation for speeds considerabK- Inwir tiian the highest attainable 

 under practical conditions. 



.XIM'F.NDIX B 



Use has been made of tiu- liniiuiia 



ir = A' log ni 

 where 11'= the speed nf ti-.inMnission <>{ intelliijeiue 



A" =a constant 

 and w =thc niunber of current \alues emplo\ed. 



The assumptions which imderlie this formula and its derivation will 

 now be given. 



Let us assume a code whose characters are .ill of liu- sanu- duration. 

 This is usually the case in jirinter codes, if ;/ is the number of signal 



