96 BELL SYSTEM TECHNICAL JOURNAL 



WTien we make use of the fact that \p{y) is an even function of y we see, from 

 (3.9-26), that the mean square fluctuation of A{t) is 



{A{t) - Af = Yif) - A' =- [ e-""xly\y) dy (3.9-28) 



a Jo 



'^(t) may be expressed in terms of integrals involving the power spectrum 

 wif) of I{t). The work starts with (3.9-24) and is much the same as in 

 going from (3.9-8) to (3.9-9). The result is 



^(r) = A' + [ df, [ dfow{fi)wif2) 

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r cos 2x(/i + /2)r cos 2ir{fi - fijr 1 



la' + [2x(/x + f2)f "^ «2 + [2x(/i - /2)PJ 



It is convenient to define 'w(—f) for negative frequencies to be equal to 

 ■w(f). The integration with respect to /2 may then be taken from — »: to 

 + oc and we get 



Jo J-« a- -\- [lTr{ji — J2)\ 



The power spectrum W(f) of ^4 (/) may be obtained by integrating "^(r) : 



Wif) = 4 [ ^(t) cos 27r/r dr 

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Let us concern ourselves with the fluctuating portion A{t) — A oi A{t). 

 Its power spectrum Wdf) is 



Wcif) = 4 / (\^(t) - A') cos Itt/t dr 

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The integration is simplified by using Fourier's integral formula in the form 



/ dr / #2F(/2) cos 27r(« -/2)t = |F(w) 

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We get 



Wcif) = 2 ,\ 2,2 [ df,[wif,)wif+f,) +w(/x)w(-/ + /0] 



(3.9-30) 



= aN^^I« ^(/0-(/-/i)^/: 



The simplicity of this result suggests that a simpler derivation may be 

 found. If we attempt to use the result 



wif) = Limit 2MZli (2.5-3) 



r-»oo -/ 



