PROPERTIES OF SINE WAVE PLUS NOISE 139 



0102 = Q2 + Q4 + 3^6 + • • • 

 . 72 



Q(r) = 0[02 = - T^ ^1^2 (7.3) 



(IT- 



= - J - |i (??" + «") - |e (/«" + 2«'^«) + • • • 



From the exact expression for 12 (r) obtained below it is seen that the last 

 equation in (7.3) is really asymptotic in character and the series does not 

 converge. We infer that this is also true for the first equation of (7.3). 



We shall now obtain the exact expression for the correlation function 12(t) 

 of 0'{t) when fq is at the center of a symmetrical band of noise. At first sight 

 it would appear that the easiest procedure is to calculate the correlation 

 function for 0(t) and then obtain 12(r) by differentiating twice. However, 

 difficulties present themselves in getting outside the range — tt, tt since ^ 

 enters the expressions only as the argument of trigonometrical functions. 

 Because I could not see any way to overcome this difficulty I was forced to 

 deal with 0' directly. Unfortunately this increases the complexity since 

 now the distribution of the time derivatives of Ic and Is also must be con- 

 sidered. 



We have 



sec'^ = 1 + 



\Q + J 



,, _ (Q + Ic)i: - IsK ^ (Q + Ic)l's - Isl'c 



sec2 e(Q + icY (Q + icY + n 



(7.4) 



and the value of 0'{t)0'{t + r) is the eight-fold integral 



Q(t) = f dh, "• J dIs2p{IcU -'- , Is2) 



{Q + Icl)l'sl - IsJc l ^ (Q + Ic2)ls2 - Is2lc2 



(Q + iciY + III (Q + ic^y + ih 



where p{I c\, ' • • , l's2) is an eight-dimensional normal probabiUty density. 

 As before, the subscripts 1 and 2 refer to times / and / + r, respectively. 

 The most direct way of evaluating the integral (7.4) is to insert the expres- 

 sion for ^(/ci, • • • , 1^2) and then proceed with the integration. Indeed, 

 this method was used the first time the integral (7.4) was evaluated. Later 

 it was found that the algebra could be simplified by representing p{I c\, • • • , 

 Is2) as the Fourier transform of its characteristic function. The second 

 procedure will be followed here. 



