STEADY STATE DELAY AND APERIODIC SIGNALS 233 



sensibly constant. When r — ±ai , it is reduced to ~~7~ times its maximum. 



For r- > > ar, it varies inversely as | t | . 

 To investigate the oscillating factor of (5) we note that now 



7 = ±5t ± ^, 



where the sign of 8t depends on that of ai and that of - does not. The 

 oscillating factor then is 



cos [(coi + 8)t — (di — coid'i) - 77], 

 where 



rj = arctan — ± - . (8) 



T 2 



The frequency, (wi + 5), is that of the edge of the segment of the spectrum 



where the amplitude is relatively very large. The phase differs from that 



for small values of ai by a quantity -q which is an ambiguous function of the 



time T. This ambiguity may be removed if we assume that the phase 



varies continuously and that, for very small values of r, the amplitude has 



the same sign as the spectrum component corresponding to an infinitesimal 



/ 



(Xi 



value of 5. As t increases through zero, arctan — changes discontinuously 



r 



from + 7: to ±~ according as ai ^ 0. To avoid a similar discontinuity, 



in 7j we say that the sign of - in (8) is to be taken opposite for positive and 



negative values of r. If we make it ± for r < 0, and + for t > 0, according 

 as ai ^ 0, then 77 is zero in the neighborhood of t = 0. Since the amplitude 

 factor is always positive, this corresponds to a spectral component of positive 



amplitude. If we make the sign of - + for t < 0, and ± for r > 0, i? 



becomes ± tt, which is the equivalent of a negative amplitude. Hence a 

 knowledge of the spectral component of frequency ooi enables us to determine 



TV 



the sign in (8). For large values of (t), rj reduces to ±-. 



Here we have assumed the amplitude of the input signal to be independent 

 of frequency. If this is not the case the same conditions hold at the input 

 as have just been discussed for the output of a resonant system. 



The main conclusion to be drawn from the foregoing is that when the 

 amplitude is changing rapidly with frequency, the component of an aperiodic 



