010 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



nearly the same for other pulse shapes, provided the frequency spectrum / 

 of the pulses can be regarded as approximately constant over the im- I 

 portant portion of the band of the resonant circuit. This approximation ; 

 is legitimate for resonant circuits with a loss constant Q and pulse shapes '■ 

 at the input of repeaters as considered here. 



4.1 Resonant Circuit Response to Steady State Coniponent 1 



The first component consists of an infinite sequence of impulses of ^ 

 amplitude ^ and all of the same polarity, at intervals T. This sequence ! 

 has a fundamental f requeue}^ / = 1/T. When it impinges on a resonant 

 circuit with resonant frequency /n = / — Af and loss constant Q, the < 

 response is of the form 1 



As{t) = cos \f/ cos (o)/. — \p), (4.1) 



and 



tan ,A = Q(///n - M) ^ 2Q ^. (4.2) ^ 



The response is thus a steady state sinusoidal wave of frequency / 

 displaced from the fundamental component of the input wave by the 

 phase shift xp and reduced in amplitude by cos \p. This is the phase 

 shift and amplitude reduction of the resonant circuit at the frequency f 

 when the resonant frequency is /n . 



.2 Resonant Circuit Response to Random Signal Component 



The second component consists of an infinite random sequence of im- 

 pulses of amplitude ±§, at intervals T. The response of the resonant 

 circuit to this component will be a randomly fluctuating wave Ar{t) of 

 mean value 0. The maximum positi^'e amplitude is obtained when all 

 impulses of the second component are positive and is Arit) = As . The 

 maximum negative amplitude is Ar{t) = — *4s . Owing to the presence 

 of this component the total output of the resonant circuit .4,. + Ar{t) 

 can thus fluctuate between the limits and 2.4s , but the actual fluctua- 

 tions of significant probability will be smaller. 



The above fluctuations can be resolved into a component in phase 

 with the steady state response given by (4.1) and another component at 

 quadrature with the steady state timing wave. The rms values of these 

 components taken in relation to the amplitude of the steady state wave 

 are 



(4.3) 

 cos ^ 



