932 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1 9 O? 



With these approximations 



1 _ e-oV-o^'^'^^Iil + iH (17) 



where 



i^-'^Q, (18) 



CO 



will be recognized as the phase shift of the resonant current at the fre- 

 (|iiency w, as obtained from (3) with w = coo + Aco. 



A further approximation that can be introduced in (12) is 





C:^ g»"'o 



(19) 



since to < T and Auto and coo^o/2Q « 1 . 



With the above approximations (12) becomes 



^4/ =. -^ ^ e't"'o-^i cos ^. (20) 



2 TT 



The real part of this expression is 



A, = ^9. cos (c/,, - rp) cos ,A, (21) 



2 TT 



which is the response to the steady state component of the pulse train. 



3 Response to Random Com'ponent of Impulse Train 



Let a secjuence of impulses of amplitude ^ and randomly positive and 

 negative polarity impinge on the resonant circuit at intervals T. The 

 response is then, 



Ar(t) = ^ Z ± cos coit - nT)e-^^'-"''''''. (22) 



2 ,i=o 



This expression for the random component differs from (9) for the sys- 

 tematic component in that the impulses have random ± polarity. If 

 all signs are chosen the same, the values of Ar(t) will be either —As(,t) 

 or -\-Asit). The resultant response of the resonant circuit, i.e. As(t) + 

 Ar{t), can thus vary between the limit and 2As{t). Ar(t) represents a 

 random fluctuation about A^it) as a mean value. In the following the 

 rms value of this fluctuation is evaluated. 



In order to determine the components of Ar{t) in phase and at quadra- 



