PROPERTIES OF SINE WA VE PLUS NOISE 147 



In this case the power spectrum is, from (8.3), (8.5), and (8.6), 



Wn{J) = -47ro- f cos{uf/o) log (1 - e~"') du 



(8.7) 



3/2^-/2/ (4n(r2) 



o 3/2 Y^ 



IcTTT Z^ n e 



the series being obtained by expanding the logarithm and integrating term- 

 wise. When this equation was used for computation it was found conven- 

 ient to apply the Euler summation formula to sum the terms in the series 

 beyond the (N — l)st. Writing b for/V(4o-2), the series in (8.7) becomes 



{N - 1)- 



-3/2^-6/ (iV-l) 



+ {./bY"- erf [(b/A0"1 + N-'^^-"'" [l - J^ (- ^ + *) 

 ?_ (^_ 12? 1 1_5_5 ^ _ 21 ^ i^\ 1 



(8.8) 



When h is zero the sum^^ of the series is 2.61237 • • • . The values for p = 

 in Table 4 were computed by taking iY = 12 in (8.8). As 6 ^ co the domi- 

 nant term in (8.8) is seen to be the one containing erf (choose iV so that 

 h = N^^^). Hence as/— > =0 



^iv(/) '-' 47rVV/. ^ (8.9) 



When both noise and the sine wave are present it is convenient to split the 

 power spectrum into three parts. The first part, Wi{f), is proportional to 

 PFjv(/), the power spectrum with noise alone. The second part W2(f) is 

 proportional to the form W(f) assumes when rms /,v <C Q and the third 

 part W-s{f) is of the nature of a correction term. This procedure is suggested 

 when we subtract the leading terms in the expressions (7.26) and (7.27) 

 (corresponding to ^ = 1 and k = 0, respectively) from yi. Likewise we 

 subtract the leading term in ^2, (7.27), at ^ = but do not bother to do so 

 at the end k = 1 because u^y2 approaches zero there. We therefore write 



yi - u'y2 = bi + ^""log (1 - k') - Kl - e-y/p - u'y2 

 + u^Kl - e-'Y/p] -e-' log (1 - k^) + (1 - ^2)^(1 - e-y/p 



= Z{u) - e- log (1 - k') - ^.^li^^ (1 - e-^f 



OqP 



1^ "Theory and Application of Infinite Series," Knopp, (1928), page 561. 



(8.10) 



