524 BELL SYSTEM TECHNICAL JOURNAL 



Theory Underlying Phase Distortion Measurement 



It should be understood that what is meant here by phase shift is 

 really the insertion phase shift; that is, the phase shift of a system is the 

 change in phase at the receiving terminal due to the insertion of the 

 system under consideration between the generator and the receiving 

 terminal. In the same way, when delays are mentioned, insertion 

 delays are understood unless otherwise specified. 



A certain amount of time is required for the transmission of any 

 signal from one place to another; and it has been found that, for a 

 natural reproduction of tone or speech, or the satisfactory transmission 

 of any signal, not only must the attenuation of the various component 

 frequencies be approximately equalized, but also the time of propaga- 

 tion of these same component parts must be nearly the same for dif- 

 ferent frequencies. This time of propagation is, of course, closely 

 related to the change in phase of the component frequencies during 

 transmission. 



In order to have no phase distortion it is necessary that the phase 

 shift be linear with frequency within the frequency range required for 

 transmission.^' "^ Graphically, this means that if the phase shift is 

 plotted as a function of frequency, the resulting graph will be a straight 

 line within the frequency range under consideration. It is evident 

 then that for such a condition the first derivative of the phase shift with 

 respect to frequency, or the slope of the phase shift-frequency curve, is 

 constant. 



The slope or first derivative is closely related to the delay of the 

 envelope. The following statement results from a mathematical 

 consideration of the building up of sinusoidal currents in systems similar 

 to those which we are considering here.^ The envelope of the oscillations 

 in response to an e.m.f. E cos wt applied at time t = reaches 50 per 

 cent of its ultimale steady value at time t — d^jdw arid its rate of building 

 up is inversely proportioytal to \ d~0'dur. Various assumptions are made 

 in arriving at this conclusion, but it does not seem necessary to discuss 

 these here e.xcept to mention the condition that the attenuation of 

 the system under consideration should be approximately equalized 

 over the frequency range in the neighborhood of the applied frequency. 



It is apparent then that this quantity d^jdbi plays a fundamental 

 role in determining the delay of a system. Moreover, the use of d^/dco 

 has an advantage over /3 in that it is constant for a distortionless sys- 

 tem, while (3 varies with frequency. The quantity dlS'du: will simply 

 be defined here as the "envelope delay" of a system in frequency 



'"Building Up of Sinusoidal Currents in Long Periodically Loaded Lines," 

 J. R. Carson, B. S. T. J., Vol. Ill, p. 558, 1924. 



