6 BELL SYSTEM TECHNICAL JOURNAL 



finite number of maxima and minima, it is representable by the 

 Fourier integral 



f(t)=-f |F(w)| cos[co/ + e(co)]</oo, 



7T t/0 



VI 



rhere 



|^(co)| 2 = [" f f(t) cos at d/1 + ["r /(/) sin to/ <fc~| . VII 



Let this force be applied to a network in branch 1 and let the 

 resultant current I n {t) be measured in branch n. Let the steady- 

 state transfer impedance at frequency oo/2ir be denoted by Zi t ,(iu) 

 and let z n (iu) and cos n denote the impedance and power factor of 

 branch n at frequency w/2ir. It may then be shown that 



r 70 1 r M lFfco)i 2 



JO IT J \Zi H (iU)\ 2 



and, as special cases, 



and 



f T U(t)\Ht= - f" I F{<S)\Hu. 



i T Jo 



X 



The total energy W, absorbed by branch n from the applied force 

 is given by 



W=- /' W J F( " )l 'i2 l 2 »(^)l cosG„-Jco. Villa 



T Jo \Z\ n (tco)l 2 



Comparison of the formulas for W and W shows that, if the branch 

 n is a simple series combination of impedance elements, W M /fo 

 energy absorbed by a unit resistance element in branch n from the applied 

 force /(/). 



In the subsequent discussion of the behavior of selective circuits to 

 random interference and applied forces of finite duration, W of 

 formula VIII will be taken, therefore, as a measure of the energy 

 absorbed by the receiving branch or element. Similarly formula IX 

 measures the energy absorbed when the applied force is impulsive. 

 The application of formula VIII rather than Villa, when they differ 

 except for a constant, is justified because we are concerned with the 

 energy absorbed by a receiving element proper, which can be repre- 

 sented by a simple resistance. 



The advantage of formula VIII, in addition to its simplicity, resides 

 in the fact that the right hand side is usually quite easily computed, 



