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THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



in agreement with (14). The spectral density can now be obtained from 

 (11) or (12); it is 



w(f) = 



46/1 



1 + 47r2/2/i2- 



This is the same as would be obtained for a Markov process that alter- 

 nately assumed the values + \/ah, — y/ah at the Poisson rate of (2/1)"^ 

 changes of sign per sec. (Cf . Rice p. 325.) 



VI EXAMPLE 2. HOLDING-TIME DISTRIBUTED UNIFORMLY IN {a,0) 



Let h{u) be constantly equal to (|S — a)~^ in the interval (a, /3), and 

 constantly elsewhere. Then by (12), 



w{f) = 



a 

 a 



TT*-;- 



1 f 



1 — / cos 2irft dt 



1 - 



Now we see that 



J{ij) = \ h{u) du 



sin 27r//3 — sin 27r fa 

 27r/(/3 - a) . 



1 for y ^ oc 

 ^ - y 



^ - a 



[0 for 7/^/3 



for a ^ y ^ 



so that 



Rir) = 



a 



a 



+ 



13 - a 



< r < 



a 



g (^ - tY 



2 - a 

 



a 



^ r ^ /i 



(15) 



T ^ l3 



is the covariance function of the process N{t) when holding-time is dis- 

 tributed uniformly in (a, (3). 



If, formally, we let (/3 - a) approach while keeping §(a + /3) fixed, 

 then the holding-times become concentrated in the neighborhood of the 

 mean, h; in the hmit, as h{i() tends to a singular normal distribution ^ 

 with mean, h, and variance zero, we obtain ' 



a 



w{f) = -^, [1 - cos 2irfh] 



(16) 



