24 BELL SYSTEM TECHNICAL JOURNAL 



We now introduce the function R(o>) which will be termed the 

 energy spectrum of the random interference, and which is defined by 

 the equation 



R{u) = ~\F{u)\\ (10) 



Dividing both sides by T and writing W'/T = e, formula VIII becomes 



RM , 



7Tc/o 



|Z(*w)| 



(11) 



Both € and i?(co) become independent of T provided the epoch is 

 made sufficiently great, and e measures the mean energy absorbed per 

 unit time from the random interference. The practical significance 

 of this formula is contained in the statement that the required function 

 of the selective network, as regards random interference, is to minimize 

 the ratio of e to the signal energy. Consequently this ratio furnishes an 

 index of the merit of the network. 



In order to rigorously evaluate the integral of formula (11) the 

 energy spectrum R(u) of the interference must be completely specified 

 over the entire interval of integration. Obviously this information 

 cannot be deduced without imposing some restrictions on the char- 

 acter of the interference, or making some hypothesis regarding the 

 mechanism in which it originates. On the other hand if the forces 

 /i(0. /2W • • • /n(0 are absolutely random in a strict mathematical 

 sense, it would appear that all frequencies are equally probable in the 

 spectrum R(u) and that, consequently, the most probable energy dis- 

 tribution is that which makes i?(co) a constant, independent of 00. 

 This inference, however, has not been theoretically established; 

 indeed, the problem does not appear to admit of satisfactory solution 

 by the calculus of probabilities. Furthermore, deductions based on 

 the assumption that the interference is random in a strict mathemati- 

 cal sense might well be inadequate for the applications contemplated, 

 and the "most probable" spectrum in serious disagreement with the 

 spectrum of the actual interference 19 to which we wish to apply the 

 results of the present study. 



Fortunately, in view of these difficulties, a complete specification 

 of R(ao) is not at all necessary for a practical solution of the problem. 

 This is a consequence of the following facts: 



19 For example, the spectrum of the interference presented to the terminals of the 

 selective network will be modified by the characteristics of the "transducer," over 

 which the disturbances are transmitted. Thus both in radio and wire systems, 

 the greater attenuation suffered in transmission by high frequencies, will reduce the 

 relative intensity of the high frequency part of the spectrum. 



