224 BELL SYSTEM TECHNICAL JOURNAL 



only on the derivative, and so this type of delay can be generalized directly 

 to include the transfer characteristics of arbitrary networks. To give an 

 exact meaning to phase delay some convention would have to be adopted 

 for determining what, if any, multiple of lir is to be added to the computed 

 phase for the frequency in question. Apparently no such convention has 

 been agreed upon which is of general application. For damped networks 

 which transmit frequencies down to zero, it is customary to assume the 

 phase shift to be zero at zero frequency, and, for higher frequencies, to add 

 multiples of It so that the phase shift varies continuously with frequency. 

 If, then, B is the computed phase shift, between — tt and tt, we may repre- 

 sent the continuously varying phase shift by ^ + Imir, where m is the 

 number of discontinuities in B which have been eliminated in passing from 

 zero to the frequency in question. The phase delay may then be defined as 



D, = ^ + ^"" . (1) 



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Any similar convention for resonant systems would be less simple, and 

 since, as will appear below, phase delay has little bearing on aperiodic 

 signals, it seems unwise to attempt to formulate such a convention here. 



In contrast with steady state delay, let us now examine the delay of an 

 aperiodic signal. If the signal is transmitted without distortion the con- 

 cept of delay of the signal as a whole is simple. If, because of distortion, 

 the sent and received signals are different we may still agree upon some 

 recognizable feature of each as determining its time of occurrence. If the 

 distortion is considerable the delay may vary greatly with the distinguishing 

 characteristic chosen. For example, if it depends on the behavior of com- 

 ponents of high frequency the delay may be quite different from what it 

 is if it depends on those of low. In the first case the result would be little 

 affected if, before transmission, the signal were sent through a high-pass 

 filter and, in the second, if it went through a low-pass filter. In each case 

 we measure a delay associated with a disturbance which comprises only 

 those Fourier components of the signal which occupy a particular limited 

 range of frequency. We may carry this idea farther and make use of a 

 very narrow band-pass filter. By varying the mid-frequency of this band 

 we obtain a delay which is a function of frequency. Its value, at any 

 frequency, is the delay, as defined by our convention, of a disturbance 

 which corresponds to that part of the spectrum of the signal which is in 

 the immediate neighborhood of the frequency in question. Our problem 

 then is to find recognizable features of a disturbance of this kind such that, 

 when they are used as criteria of delay, the result can be related directly 

 to the phase or envelope delay as defined in terms of periodic disturbances. 



Compared with the pair of equal sinusoids used in the derivation of 



