TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 23 



VI. Random Interference 



We have hitherto confined attention to the transient phenomena 

 when the form of the applied voltage was explicitly given. In the 

 problem of the behavior of wave-filters and selective circuits in general 

 to such disturbances as "static" in radio transmission and "noise" 

 in wire transmission this is not the case, and the applied force is 

 usually more or less completely random. By this it is meant that 

 the interfering disturbance, which may be supposed to originate in 

 a large number of unrelated sources, varies in an irregular, uncon- 

 trollable manner, and is characterized statistically by no predominant 

 frequency. Consequently the wave form of the applied force at any 

 particular instant is entirely indeterminate. This fact makes it 

 necessary to treat the problem as a statistical one, and deal with 

 mean values. In the following we shall derive formulas for the mean 

 energy absorbed from random interference; and then define and 

 discuss the selective figure of merit of networks with respect to random 

 interference. 



The mathematical treatment of the problem will be based on 

 formulas VI to VIII of section II. To apply these formulas to the 

 problem of random disturbances and their effect on selective net- 

 works, consider a long interval of time, or epoch, say from to T. 

 During this epoch we suppose that the network is subjected to a 

 large number of individual impressed forces f\{t),fi{t) . . .f„(t), 

 which are unrelated and vary in intensity and wave form in an irreg- 

 ular, indeterminate manner, and thus constitute what will be called 

 random interference. If we write 



]T(/)=/ 1 (o+/ 2 (o+...+/»(o, 



then by VI, y(0 is representable as a Fourier integral, thus: 

 V (/) = —J | F(w) | cos [at + (w) ] do> 



while, in accordance with formula VIII, the energy absorbed by the 

 selective network from this random interference is measured by 18 



l r" \FM\' 

 w =7j e lzMF rfw - 



18 It should be clearly understood that Z(ico) is the transfer impedance of the 

 receiving with respect to the driving branch of the network, and that W is the 

 energy absorbed by a unit resistance located in the former. 



