146 BELL SYSTEM TECHNICAL JOURNAL 



Is is of the normal law type (our method could be applied to other types 

 but ^ and g" would be more complicated functions of r and Table 3 would 

 have to be extended to negative values of ^, if they should occur). Second, 

 we resort to numerical integration to obtain a portion of W{J). Because 

 of the second item our results are either tabulated or are given as curves, 

 shown in Figs. 8 and 9, except when Q = (noise only) in which case the 

 power spectrum of B' is given by the series (8.7). 

 The power spectrum of In is assumed to be 



^(/) = ^^ .-</-A'^'<-^) (8.1) 



The mean square value of In is equal to that of a noise current whose power 

 spectrum has the constant value of ^o/(o'a/2x) over a band of width /& — fa 

 = o-\/27r = 0-2.507. The value of w{j) is one quarter of its mid-band value 

 at the points/ — fq = zhay/l loge 4 = ±0-1.665 (the 6 db points) and the 

 distance between these points is ?>3?>()(j. Integration of (8.1) shows that the 

 mean square value of In is xf/o in accordance with our customary notation. 

 The mid-band value of w{f) is 4^o/(or\/2w). 



Assuming fq^ a and evaluating the integrals (A2-1) of Appendix II 

 defining bo and g gives 



Oq = ^0, g = \f/oe = \poe 



g'/g = -uu' = -iTau, g^g = -(27r(r)'(l - u') ^^ 2) 



-^ = - (2to-) , k = g/bo = e 



s 



where we have set 



u = Ittct, li' = lira (S.3) 



and the primes on g and u denote differentiation with respect to r. The cor- 

 relation function is accordingly, from (7.16). 



12(r) = 27rV2(yi - u^y^) (8.4) 



If 6\t) be regarded as a noise current its power spectrum is 



W{f) = 4 f 12(r) cos lirfr dr 

 Jo 



(8.5) 



When noise alone is present, p is zero and (7.25) yields 



fi(r) = -27rV2 log. (1 - k^) = -27rV log, (1 - e""') (8.6) 



