DISTORTION CORRECTION ■ 503 



at time / = 0, we note that since the e.m.f. can be expressed in terms 

 of a Fourier integral representation from /= — Qoto/= + °o we 

 may regard it as made up of a distribution of steady-state components. 

 For example, let the e.m.f. of (89) be impressed on a circuit at time 

 / = 0. Then 



E{t) = ( 7^ + - I -dy ] (sin coit + sin u^t), 



\2 ttJo y I 



= Ksm Wit + sm bill) -\ — I — 5 5 + „ — 7 cos /waco. 



(92) 



This represents the impressed voltage for negative as well as positive 

 values of t since in the first equation the factor of the sinusoids repre- 

 sents a function which is zero for all negative and unity for all positive 

 values of /. We may then interpret the last equation as giving for all 

 values of time the frequency distribution of steady-state components 

 of all frequencies which give the same result as the sinusoids of (89) 

 impressed suddenly at / = 0. This distribution extends over the 

 entire frequency range and has the largest amplitudes about co = wi 

 and CO = C02. 



Hence, if the initial part of the impressed e.m.f., as well as the final 

 steady state, is to be transmitted without distortion, the circuit transfer 

 voltage must have a characteristic which is distortionless not only with 

 respect to wi and co2 but also to all angular frequencies about them as 

 obtained from the analysis. That is, since the steady state is only 

 the limiting case of the transient state, an ideal circuit characteristic 

 for its distortionless transmission is only a part of and is included in 

 that for the transient state. Or, vice versa, ideal circuit characteristics 

 for the steady state are at least the same as for the transient state. 



These results are useful in studying a circuit whose attenuation is 

 constant and whose phase characteristic is approximately linear over 

 an internal frequency band. An extrapolation of this linear phase 

 characteristic to zero frequency may give a phase intercept which is 

 not ideal for preservation of wave-shape even in the steady state of 

 frequencies within the band, as we have seen. Increasing the fre- 

 quency range over which an ideal phase relation holds obviously 

 improves the transmission of transient voltages. Practically, good 

 results are obtained in a circuit wherein the attenuation is approxi- 

 mately constant and the phase is approximately proportional to 

 frequency over the required internal band of frequencies; then the 

 phase intercept, 6o, is zero and the time-of-phase-transmission, 

 Tp = &/co, is approximately constant and represents the time-of- 

 transmission of the circuit for those frequencies. 

 22> 



