PROPERTIES OF SINE WAVE PLUS NOISE 127 



enables us to write (4.7) as 



•'- T -^ 7 sin 



acO8 0-x2/2 - (^'^^^ 



(4.14) 



Part of the integrand may be integrated with respect to x and the remaining 

 portion integrated by parts with respect to 6. The double integral in the 

 second Hne of (4.13) then becomes 



''-r •'-xL''7sine J 



= 1^1 + tV cos ^)g«cos(>-(.sin^)V2 ^^ 



= 2x1: % (- ^T [/„(«) + 7^a-'/„«(a)l. 



n=0 W! \ «/ 



The series is obtained by expanding exp [—(7 sin Sy/l] in the second 

 equation in powers of sin 6 and integrating termwise. 



5. Probability Density of — 



dt 



As was pointed out in Section 3 the time derivative 6' of the phase angle 

 associated with the envelope is closely related to the instantaneous fre- 

 quency. The probabiHty density p{d') of d' may be expressed in terms of 

 modified Bessel functions as shown by equation (5.4). Curves for p{d') are 

 given when the sine wave frequency fq lies at the middle of a symmetrical 

 band of noise. Although the expressions for p{d') are rather complicated, 

 those for the averages B' and | d' \ given by equations (5.7) and (5.16) are 

 relatively simple. 



The probability density p{d') may be obtained by integrating the expres- 

 sion (4.5) for p(R,R',d, 6') with respect to R, R\ 6. The integration with 

 respect to R', the limits being — 00 and + =0 , gives the probability density 

 fori?, ^, ^': 



piR, e, 6') = —^ ( -?^ ) exp l-aR' + 2bR cos 6 + c sin'^ 



47r2 \boB/ ^^ j^ 



-b^'/{2B)] 



