MATHEMATICAL AX ALTS IS OF RANDOM NOISE 111 



we may refer all terms to the mid-band frequenc}' /^ — o),n/2Tr, as is done 

 in equations (3.7-2) to (3.7-4). 

 In this way we obtain 



]' = A cos o:J - B sin wj = R cos (coj + 6), (4.1-8) 



where .1 and B are relatively slowly varying functions of / given by 



A = P cos {P — W,n)t + Q cos [q — W,,,)/ + ^ Cn cos (C0„^ — CO,,,/ — <^n), 



n 



5 = P sin (/> — avj/ + Q sin (</ — co™)/ + XI c„ sin (a;,,/ — oj,,,/ — <^„) 



and 



R^ = A' -{- B\ R > 



(4.1-9) 



tan^ = B/A. 



This delinition of R has also been given in equations (3.10-22, 23, 24). 

 The envelope of V is R and the output current is 



I = aR: \ + \ cos (2co„J + 2^) (4.1-10) 



Since i? is a slowly varying function of time, so is R\ The power spectrum 

 of R' is confined to frequencies much lower than 2fm and consequently the 

 power spectrum of R' cos {2w„4 + 26) is clustered around 2/,,, . Thus the 

 only term in / contributing to the low frequency output is aR'/2 which is 

 what we wished to show. 



We now return to the statistical properties of Iti . First, consider the 

 case in which T' consists of noise only, T' = Vn , so that the probabihty 

 density of the envelope R is 



R ^-fi^/^if-o 



where 

 Hence 



^ e-'"'"' (3.7-10) 



xPo = [rms V^■f = Vl (4.1-11) 



7 ai?2 



^0 2 i^-u 



aR' R .-^2/2^0 



arpo 



4^0 (4.1-12) 



a\J/Q 



