MATHEMATICAL ANALYSIS OF RANDOM NOISE 



77 



. We now study the correlation between R at time t and its value at some 

 later time / + r. Let the subscrij)ts 1 and 2 refer to the times t and t -\- t, 

 respectively. Then from (3.7-3) and the central limit theorem it follows 

 that the four random variables In , /*i , Ic2 , Is2 have a four dimensional 

 normal distribution. This distribution is determined by the second^ mo- 

 ments 



Id = Isl = Ic2 = Is2 = ^0 = Mil 



Ic\Ia\ — I dial — 



Icihi = /»i/«2 = t; Z^ c„ cos (w„r — WmX) 



2. n=l 



\ w{f) COS 27r(/ - /Jt df = mis 



(3.7-11) 



J AT 



Iclls2 — —IcilsX = T^^ZI Cn siu (w„ T — aj„. t) 



Z 71=1 



/ w{f) sin 27r(/ - /Jt df = nu 

 Jo 



M = 



The moment matrix for the variables in the order Id , /«: , la , I»i is 



"Ao M13 M14 



l/'o — M14 M13 



Mi3 — Mi4 iAo 



.Ml4 Ml3 l/'o _ 



and from this it follows that the cofactors of the determinant | M \ are 

 Mil = Mio = Mzz = Mu = }p(i{ypl — Mi3 — mh) 



= l/'o^, A = l/'o — Ml3 — Ml4 



Mi2 = .¥34 = 



Ml3 = M24 = — Ml3^ 



Mi4 = — lf23 = —H14A 



\M\ = A" 

 The probability density of the four random variables is therefore 



^^^v-^{Ui\ + il + il + il) 



(3.7-12) 



2m13(/i/3 + 12/4) - 2m14(/i/4 - 12/3)] 



