PROPERTIES OF SINE WAVE PLUS NOISE 135 



where we have set 



r = R{2bo)-''- X = rboOyh 



a = Qilbo)-''' = p'^ 7=1 + b,e''/b2 = 1+2' (6.12) 



(1 + u)D 



r being regarded as a constant when the variable of integration is changed 

 from Q" to x. 



The double integral in Q and x occurring in (6.11) is of the same form as 

 (A3-1) of Appendix III and hence may be transformed into (A3-3). Here 

 a = r a^c = — So:-, c + 6- = 0. The diameter of the path of integration C 

 may be chosen so large that the order of integration may be interchanged 

 and the integration wdth respect to r performed. The result is again an 

 integral of the form (A3-3) in which a- = ^. When this is reduced to (A 3-6) 

 it becomes 



,Y,, = e-\l'KiY^bT{b,hY^^'' [e-''i^h{hp/2) 



(6.13) 

 + (1 + t5)(1 + 75/2)-V/Ve {76(2 + yb)-\ p/y +5p/2!] 



where we have used Ie{—k, x) = Ie{k, x) which follows from the definition 

 (A 1-1) given in Appendix I. 

 When there is no sine wave present, p is zero and (6.13) becomes 





1/2 /i/ F - r + 40'' 



Ne' = ^{y-^ = ' ") "\ . (6.14) 



This gives a partial check on some of the above analysis since (6.14) may be 

 obtained immediately by setting a equal to zero in (6.11). Another check 

 may be obtained by letting p ^> ^c and using Ie{k, cc ) = (1 — k-)-^'-. 

 (6.13) becomes 



Ne' - (27r)-KVW^^-^-'"' (6.15) 



which agrees with the result obtained from 6' ^ L/Q- 



For the case in which the power spectrum w(J) of the noise is equal to the 

 constant value Wq over the frequency band extending from fq — ^/2 to 



bo = /3wo, b2 = ir-^'wo/S = tt^^^^o/S, b^ = ir'^'^Wo/o = ir'lS'bo/S (6.16) 



