PROPERTIES OF SINE WAVE PLUS NOISE 125 



integrating (4.5) with respect to 6 and 0' over their respective ranges. The 

 integration with respect to 6' may be performed at once giving 



-3/2 





. (4.6) 

 exp |- 2^^^ \B{F^ - IRQcos 6 + Q') -}- (boR' + b.Qsin 0)'] 



From (4.1) and (4.6) it follows that the expected number TVr of times per 

 second the envelope passes through R with positive slope is 



exp |- ^^ [B{R' - 2RQC0S 6 + Q') + {boR' + ^^iQsin 6)'] 



When the power spectrum w(J) of the noise current In is symmetrical about 

 the sine wave frequency /q, bi is zero and B is equal to 60^2. In this case the 

 integrations in (4.7) may be performed. We obtain 



«-m.-(sh{-'^) 



(b2Y'^ fProbability density ofl 

 \27r/ |_envelope at the value Rj 



(4.8) 



where the second Hne is obtained from expression (3.12). As will be seen 

 from its definition (A2-1), bo is equal to the mean square value ^0 of In 

 (and also of Ic and 7 s). 

 Introducing the notation 



V = Rbo'"" = R/rms In 



(4.9) 

 a = Abo^^"" = Q/rms In, 



which is the same as that of equations (3.10-15) of Reference A except that 

 there P denotes the ampHtude of the sine wave and plays the same role as 

 Q does here, enables us to write (4.8) as 



The function p{v) is plotted as a function of v for various values of a in Fig. 

 6, Section 3.10, of Reference A. 

 It is interesting to note that 



(bi/boY'^/Tr = Expected number of zeros per second of U (or of /«) (4.11) 



