MATHEMATICAL AXALYSIS OF RANDOM XOISE 105 



the noise to correspond to a very large number of very small random dis- 

 placements. 



Another way of deriving (3.10-21) is to assume (p — co,,.)/, (q — Wm)t, 

 <Pi , ifi , • ■ ■ are independent random angles. The characteristic function 

 of -4, 5 is 



ave. e*"^+*'''^ = /o(PV^"^M^-)/o((2Vi<H^')e"^'^°'''^"'^''^ 

 The probability density oi A, B is 



(ly I""" du f^ dv e--^--'« ave. e'""^^"'' 



\Mien the change of variables 



A = R cos d u = r cos ip 



B = R sin 6 v = r sin ip 



is made the integration with respect to (p may be performed. The double 

 integral becomes 



^ I rMPr)MQr)MRr)i 

 Zir Jo 



-i'l'oli)'-^ 



'" dr 



This leads directly to (3.10-21) when we observe that dAdB = RdRdd. 

 Incidentally, if 



I = Q(l -\- k cos pt) cos qt + 7jv 



in which p << q, similar considerations show that the probability density 

 of i? is 



- [ da [ rJo{Rr)Jo[Qr{l + k cos a)]e-^'^'"'''dr 



T Jo Jo 



when Wm is taken to be q. The integration with respect to r may be per- 

 formed. This relation is closely connected with (3.10-11). 



Returning now to the case in which / is the sum of two sine waves plus 

 noise, we may show from (3.10-21) and 



,ri+l 



rjTl-Ti -p 



(■ * i) 



[ R''-''Jo{Rr)dR= ^ . 



.-r(-l) 



