104 BELL SYSTEM TECHXICAL JOURNAL 



WTien I consists of two sine waves plus noise 



I = Pccspt-{-Q sin qt + /.v , (3.10-20) 



where the radian frequencies p and q are incommensurable, the probability 

 density of the envelope R is 



R / rJo{Rr)Jo{Pr)Jo{Qr)e-'^'''" dr (3.10-21) 



where xpo is I'\ . \Mien Q is zero the integral may be evaluated to give 

 (3.10-11). When both P and Q are zero the probabiUty density for R 

 when only noise is present is obtained. If there are three sine waves instead 

 of two then another Bessel function must be placed in the integrand, and 

 so on. To define R it is convenient to think of the noise as being confined 

 to a relatively narrow band and the frequencies of the sine waves lying 

 within, or close to, this band. As in equations (3.7-2) to (3.7-4), we refer 

 all terms to a representative mid-band frequency /„» = Wm/27r by using 

 equations of the t}'pe 



cos pt = cos [(p — 03,,,)/ + 0}J] 



= cos (p — Oim)t cos OJmt — Sin (p — (jim)t Sln 0)mt. 



In this way we obtain 



V = A cos o^a - B sin wj = R cos {wj + 6) (3.10-22) 



where A and B are relatively slowly var}-ing functions of t given by 

 A = P cos {p — Um)i + Q cos {q — 0^,n)t 



+ Z^ C„ cos (a3„/ — Wmt — (pn) 



B = P sin {p — o}m)t + Q sin (g — Urn)i 



+ 2_/ ^n sin {uni — Wmt — cpn) 



(3.10-23) 



and 



R^ = A^ + B'-, R > 



tan d = B/A 



(3.10-24) 



As might be expected, (3.10-21) is closely associated with the problem 

 of random flights and may be obtained from Kluyver's result by assuming 



39 G. X. Watson, "Theory of Bessel Functions" (Cambridge, 1922), p. 420. 



