TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 25 



(a) In the case of efficient selective networks, the important con- 

 tributions to the integral (11) are confined to a finite continuous range 

 of co which includes, but is not greatly in excess of, the range which 

 the network is designed to select. 20 This fact is a consequence of the 

 impedance characteristics of selective networks and of the following 

 properties of the spectrum R(u>). 



(b) R{u>) is a continuous, finite function of co which converges to 

 zero at infinity and is everywhere positive. It possesses no sharp 

 maxima or minima, 21 and its variation with respect to co, where it 

 exists, is slow. These properties of R(w) are believed to be evident 

 from physical considerations, and will not be elaborated. 



Now referring to formula (11), since the numerator and denominator 

 of the integrand are everywhere positive, it follows that a value 

 u m of co exists, such that 



= -R(o: m ) f 



dco 



Now suppose that the network is designed to select frequencies in the 

 range coifSco^co2. Then from the properties of the network and of 

 the spectrum i?(co) discussed above, it follows that u m lies close to, or 

 within, the range coi^co^coo. In any case, if the band co 2 — an is 

 made so narrow that the curvature of R(u>) over the interval is negligi- 

 ble, then with negligible error w m may be taken as 2ir times the "mid- 

 frequency" of the band. That is to say, with negligible error, co w 

 may be defined either as (coi+co 2 )/2 or as -v/coico?. 



The foregoing argument may be summarized in the following 

 proposition : 



The mean energy e absorbed per unit time from random interference by 

 a selective network designed to select the band of frequencies corresponding 

 to coi^jco^coo is measured by the formula 



e=*-pR(u m ), (12) 



7T 



where p denotes the infinite integral 



dw 





lz(ico)| 2 



20 This statement excludes from present consideration networks, which, like the 

 high pass filter, select an infinite band of frequencies. This limitation, however, is 

 of no practical consequence, because such networks are quite useless as regards 

 random interference. This question will be briefly discussed later. 



21 The existence of sharp maxima and minima would indicate the presence of 

 systematic interference, which should not be regarded as part of the random inter- 

 ference. 



