502 BELL SYSTEM TECHNICAL JOURNAL 



through h] and &2 in the co, h plane will have a slope r, say, and at 

 CO = a linear phase intercept &o which may have any value. Hence, 

 the transfer exponent may be expressed as a function of frequency at 

 these two frequencies by the relations 



a = a' = constant, 

 and (90) 



6 = TCO + &0- 



The received voltage across R will then be a periodic function which is 

 attenuated by an amount a' napiers and is 



v{t) = e-°'[sin (coi(/ - r) - &o) + sin [woit - t) - 6o)], 



= 2e-°' cos i(co2 - wi)(/ - t) sin (K^i + W2)(^ - r) - bo). 



How the transmitting property of this circuit for the two frequencies 

 depends upon the phase intercept can be seen from a comparison of 

 (91) with (89). In order that the received voltage may be a time- 

 function of identically the same shape as the impressed voltage, but 

 with a time-of-transmission over the circuit of r seconds, it is necessary 

 that bo = 2mr radians, where n is any positive or negative integer. 

 This would mean no distortion of the impressed steady-state signal 

 made up of the two frequency components. If bo = (2n db l)7r, there 

 would be an apparent distortion only of a reversal in sign. However, 

 if ^Q = (2w ± Dtp, there would be maximum distortion in the trans- 

 mitted voltage. These conclusions may be tabulated briefly as follows : 



If bo = 2n7r, no distortion; 



If bo = (2w ± l)7r, apparent distortion of sign reversal; 



If &o = (2w ± Dtt, maximum distortion. 



The above discussion considered the case of any two frequencies. 

 If now we assume that the circuit has the characteristics (90) for 

 several or a range of frequencies, then the conclusions above obviously 

 apply as well to the steady-state transmission of an impressed e.m.f. 

 which is made up of any of those frequencies. Thus, for distortionless 

 steady-state transmission {without change of signal shape), the transfer 

 exponent must have for the frequency components impressed not only a 

 uniform attenuation and a linear phase relation, but also a proper linear 

 phase intercept bo = 2mT. If, in a physical system, (90) is satisfied 

 over a frequency range which includes zero frequency, then r would 

 necessarily be positive and bo = or a multiple of lir. 



Proceeding next to the transmission of an e.m.f. impressed suddenly 



