26 BELL SYSTEM TECHNICAL JOURNAL 



and R(oj,„) is the spectral energy level of the interference at frequency 

 w m /2x. co m lies close to or within the band coi <oo <co 2 , and when this band 

 is sufficiently small with respect to the curvature of R(<a), w m /2ir may be 

 aken as the mid-frequency of the band. 



Formula (12) is of very considerable practical and theoretical im- 

 portance. It furnishes a basis for the experimental determination 

 of the energy spectrum R(oj), and this determination, for any given 

 epoch, can be made as accurate as desired by employing a band filter 

 which selects a sufficiently narrow band of frequencies. It also leads 

 immediately to the following important proposition. 



// a selective network is required to select the band of frequencies cor- 

 responding to coi^w^co 2 , the mean energy absorbed per unit time by 

 the network from random interference is necessarily greater than 



1 /"» R(u) 1 p dco 



This formula, therefore, determines the theoretical limit, beyond which 

 it is not possible to discriminate against random interference. 



We are now prepared to introduce a formula which defines the 

 figure of merit of a selective network with respect to random interference. 

 This formula gives the signal-to-random-interference energy ratio of 

 the network as compared with the corresponding ratio in an ideal refer- 

 ence circuit (defined below). 



Let the network, as above, be designed to select frequencies in the 

 band coi^co^co 2 . Then the energy absorbed per unit time from 

 steady-state forces in this frequency range is proportional to 



'- —f 



r co, \Z(iu))\ r 



The corresponding mean energy absorbed from random interference 

 is proportional to 



r*° dco 

 P ~J fZ(ico)| 2 



when the energy level of the interference is corrected to unity. 



The ratio S = <r/p defines the selective figure of merit of the network 

 with respect to random interference. 



Stated in words, the selective figure of merit of a network with respect 

 to random interference is equal to the statistical signal-to-random-inter- 

 ference energy ratio, divided by the corresponding ratio in an idea' band 

 filter which transmits without loss all frequencies in a "unit" band 

 (a>2 — «i = 1), and absolutely extinguishes all frequencies outside this band. 



