STEADY STATE DELAY AND APERIODIC SIGNALS Hi 



communication art, these delays have been called phase and envelope delay, 

 respectively. If the medium exhibits dispersion they vary with frequency. 

 Let us fix our attention on the conditions throughout the medium at a 

 particular instant during the transmission of a sinusoidal disturl^ance. We 

 may determine the total change of phase in passing from the input to the 

 output. This may be more than a single cycle. If now we divide this 

 phase shift by the frequency, expressed in the same angular units, we get 

 the time which will be required for the phase at the input to progress to 

 the output, or the phase delay. Also it may readily be shown that the 

 derivative of this phase shift with respect to frequency is equal to the 

 envelope delay as defined above in terms of the group velocity. The 

 simplest treatment of this is based on the consideration of two sinusoidal 

 waves of equal amplitude and slightly different frequencies. 



While these delays can be easily interpreted for most media, difficulties 

 arise in the case of those substances which exhibit anomalous dispersion. 

 Here, in the neighborhood of certain frequencies, the phase shift varies 

 rapidly with frequency, and often appears to be discontinuous. Actually 

 the apparent discontinuity is a region of very rapid decrease of phase with 

 frequency, which leads to a negative value of envelope delay. In the same 

 region the transmission varies rapidly with frequency, and selective reflec- 

 tion occurs at the entering boundary. This effect can be explained in terms 

 of resonance in the elements which make up the fine structure of the 

 medium. 



The next step was to dissociate the idea of delay from that of velocity 

 in a medium, and associate it with a steady state transfer characteristic 

 between any two points of a linear system. This would permit its appli- 

 cation to all sorts of complicated networks in which uniform propagation 

 cannot be readily visualized. Here two types of characteristic are to 

 be distinguished. One, which is associated with what might be called 

 "damped" systems, exhibits a reasonably gradual variation of both phase 

 shift and attenuation with frequency. This is the analog of a medium 

 having normal dispersion. The other, which is associated with "resonant" 

 systems, exhibits the phenomena associated with anomalous dispersion. 

 In the case of filters and hollow wave guides these resonances give rise to 

 regions of high attenuation and reactive impedance, which are the analogs 

 of the regions of high absorption and selective reflection at the boundary 

 of a medium. In applying the idea of delay to networks then, we can expect 

 the results to agree with our intuitions only so long as we keep away from 

 the critical frequencies of resonant systems. 



In computing or measuring the phase shift of a system, at a single fre- 

 quency, the result is indeterminate so far as the addition of multiples of 

 1-K is concerned. This does not affect the envelope delay, which depends 



