138 BELL SYSTEM TECHNICAL JOURNAL 



7. Correlation Function for — 



at 



In this section we shall compute the correlation function 12(t) of d'{t). 

 We are primarily interested in 12 (r) because it is, according to a fundamental 

 result due to Wiener, the Fourier transform^^ of the power spectrum W{j) 

 of ^(/). 



We shall first consider the case in which the sine wave power is very large 

 compared with the noise power so that, from (3.6), 6 is approximately 

 IJQ and 6' approximately I's/Q. Then using (A2-3) and (A2-1) 



o(t) = e'{t)e\t + r) - Qr-n's{t)i[{t + r) 



= -g"Q~" = 4^V r^(/)(/ -A)' cos 2Hf-f,)Tdf 

 Jo 



(7.1) 



When wif) is effectively zero outside a relatively narrow band in the neigh- 

 borhood oifqj as it is in the cases with which we shall deal, (7.1) leads to the 

 relation (divide the interval (0, oo) into (0, /g) and (fg, oo), introduce new 

 variables of integration fi = fq — f,f2 = f — fq in the respective intervals, 

 replace the upper limit /g of the first integral by go , combine the integrals, 

 and compare with (2.1-6) of Reference A) 

 Power spectrum of d'{t) = W(f) 



= ^T'fQ-'lw(Jq + /) + w(fq - /)] (7.2) 



This form is closely related to results customarily used in frequency modula- 

 tion studies. It should be remembered that in (7.2) it is assumed that 

 < / «/« and rms In « Q. 



Additional terms in the approximation for Q(t) may be obtained by 

 expanding 



d = arc tan ' 



Q + lc 



in descending powers of Q, multiplying two such series (one for time t and 

 the other for time / + r) together, and averaging over /. If Id, I si and 

 /c2, 1*2 denote the values of 7^, /« at times t and / + r respectively, the 

 average values of the products of the /'s may be obtained by expanding the 

 characteristic function (obtainable from equation (7.5) given below by 

 setting 2& = z« = 27 = 28 = 0) of the four random variables Id, I si, Ici, I si- 

 This method is explained in Section 4.10 of Reference A. When w{}) is 

 symmetrical about/, it is found that 



" The form which we shall use is given by equation (2.1-5) of Reference A. 



