272 BELL SYSTEM TECIIXIC.IL JOrRXAL 



efficienc\- with which it excludes encrg>- outside the signaling range. 

 It will therefore be termed the figure of merit of the selective circuit and 

 denoted by S, thus 



0)2 — oil p 0)0 — COiJj^i /C[lu))- Jo \jC(tU)\' 



Staled in words, the figure of merit of a selective circuit with respect 

 to random interference is equal to the ratio of the mean square signal and 

 interference currents in the receiver, divided by the corresponding ratio 

 in an ideal band filter which transmits without loss all currents in a 

 "unit" band (ojo- o)i = l) and absolutely extinguishes currents outside 

 this band. 



Ill 



Before taking up practical applications of the foregoing formulas 

 further consideration will be given to the h\[)othesis, fundamental 

 to the argument, that over the frequency range whicii includes the 



important contributions to the integral / . „ . . ^ ,. , the spectrum i?(a)) 



^0 I Z(ia)) l" 



has negligible fluctuations so that the integral 



r 



— dw 

 Z(lcc) (■ 



ma\ , without ap|>reciable error, be replaced by 



Ju I Z(toi) \i 



where Um 2 ir is the "niifl-frec)uenc\" of the selective circuit. 



The original arguniciU in support nf this lu'pothesis was to ilie 

 effect that, since tlie interference is made uj) of a large number of 

 unrelated elementar\- disturbances distributed at random in time, 

 any sharp maxima or minima in the spectrum of the indi\idual di- 

 turbances would be smoothed out in the spectrum of the aggregate 

 disturbance. This argument is still bclicNcd to be (|iiite sound: the 

 importance of the question, ho\vi\cr. ctrlainK calls tor the more 

 detailed anaK'sis which follows; 



:V 



Lei <\'{l)=^(t:r(t-tr) (15) 



1 



where /, denotes the time of incidence of the r"' disturbance (t>r (/)• The 

 elementary disturbances 0i, <t>2 . . . (t>s are all i)erfectly arbitrary, so 



