PROPERTIES OF SINE WAVE PLUS NOISE 121 



length and inclination of QT will change due to the random fluctuations of 

 In. Thus the point T will wander around on the plane of the figure. If 

 rms In is much less than Q, T will be close to the point Q most of the time. 

 In this case 



6 = tan '-^ — 



Q + Ic Q (3.6) 



dt^ dtQ Q 



and a number of statistical properties of R and d may be obtained from th^ 

 corresponding properties of noise alone when we note that I^ Is, and Ig 

 behave like noise currents whose power spectra are concentrated in the 

 lower portion of the frequency spectrum. 



os = Q + ic =R cos e T 



Q S 



Fig. 4 — Graphical representation of / = ^cos qt + In - 



By squaring both sides of equations (3.1) and {2>3) and then averaging 

 with respect to / and the <^n's we may show that I^Is, and In all have the 

 same rms value, namely 4'\''^. 



It may be seen from {33) that the power spectra of Ic and Is are both 

 given by 



^(/. + /) + ^(A-/) (^^-7) 



where it is assumed that < / «/q. Likewise the power spectrum of the 

 time derivative I's oi I s is 



47r2/X/,+/) + Z£;(/,-/)] (3.8) 



This follows from the representation of /« obtained by differentiating the 

 expression (3.3) for 7^ with respect to /, the procedure being the same as in 

 the derivation of equation (7.2) in Section 7. The power spectra shown in 

 Table 1 were computed from equations (3.7) and (3.8). 



The correlation function for 7^, and hence also for 7^, is, from equations 

 (A2-1) and (A2-3) of Appendix II, 



7,(/)7,(/ + r) = g = [ ^(/) cos lirif - f,)T df 

 Jq 



