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BELL SYSTEM TECHNICAL JOURNAL 



where the bar denotes an average with respect to / and g is a function of r. 

 From (A2-3) the correlation function for Is is —f where the double prime 

 on g denotes the second derivative with respect to r. 



Attention is sometimes fixed upon the variation in distance between suc- 

 cessive zeros of /. The time between two successive zeros of / at, say, /o and h 

 is the time taken for qt + B, as appearing in R cos {qt + ^), to increase by tt. 

 This assumes that the envelope R does not vanish in the interval. For the 

 moment we write B as B{t) in order to indicate its dependence on the time /. 

 Then /o and h must satisfy the relation 



qh + B{ti) - qh - B{to) = t 

 Since B{t) is a relatively slowly varying function we write 



B{t,) - B{h) = {h - h)B'{h) + (/i - toW{to)/2 + 



(3.9) 



Table 1 

 Power Spectra of ly , Ic , Ig , and I's 



where the primes denote differentiation with respect to /. When this is 

 placed in (3.9) and terms of order (/i — toY neglected, we obtain 



2{h^^)-h'^2^^'^^'^ 



(3.10) 



which relates the interval between successive zeros to &'. 



The expression on the right hand side of (3.10) may be defined as the in- 

 stantaneous frequency: 



Instantaneous frequency = /« + r- 



27r at 



(3.11) 



This definition is suggested when cos 27r// is compared with cos (qt + B) 

 and also by (3.10) when we note that the period of the instantaneous fre- 



