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BELL SYSTEM TECHNICAL JOURNAL 



— (e) ... . , 



have the objection that the variance/zj is infinite; however, this will 

 cause us no trouble. 



With Eq. (5.25) instead of (5.22) the spectral of density Ri takes the form 



- 

 1 



eii' 



k dk 



+ (^ k"' C02 + Z)2 k^ 



Q(y) = 



= {x/'^Ts")-0a'R'Arl/o:-Q(y), 

 y(y - -\ogy) 



1 -\-f 



y = ro}/D. 



(5.26) 



In obtaining the above equation we have made the usual assumption that 

 the frequencies of interest are larger than ojo = ^ir'^D/D, and have replaced 

 the original sum by an integral. The function Q(y) has the following proper- 

 ties: 



Q(y) a^ —- y log y for y <K 1 



IT 



Qiy) ~ 1 for y » 1 



(5.27) 



Hence for w <<C D/^, S(u) ^ log w, the integral of which converges as a; — > 0; 

 whereas, for w ^ D/(^, 5(co) differs negligibly from that given by the unre- 

 fined theory (Eq. (5.24))^ 



The self-covariance Ri{l) R\{l -f u) now exists for all non- vanishing u and 

 is given by 



R,{t)R,{t +u) = (x/2Ts")-^a'R*A, • j j 



-,2 —Duk'^kdk 



I e 



JO 1 + P ^2 

 = (x/47r5'0 '^a' R'A2-e'""^\-Ei(-Du/f)] J 



(5.28) 



where 



— Ei{—x) = I e " dv/v, 



J X 



<^ —log yx for X <K 1, 



c^ — for X » 1, 



X 



7 = 0.5772. 



Thus for u « ^yZ), Ri{t)Ri(t + u) a - log {jDu/f') and for u » ^/D, 

 Ri(l)Riit + u) a 1/u. 



