PROPERTIES OF SINE WA VE PLUS NOISE 131 



The probability density p{d') of 6' and its cumulative distribution, ob- 

 tained by numberical integration, are shown in Figs. 5 and 6. 



The probability that 0' exceeds a given d'l is equal to the probability that 

 z exceeds Si, where Si denotes (6o/W^^*"^i, and both probabilities are equal to 



^£(1 +.V'%Fxg; l;p(l + zY^'^dz (5.12) 



The probability that 6' > d[ becomes e~'/(4:zl) as 01 -^ oo . 



When Q » rms In the leading term in the asymptotic expansion of the 

 1^1 in (5.9) gives 



P(e') ~ ^e-""''"\ a^ = VQ' (5.13) 



when it is assumed that s- « 1. This expression holds only for the central 

 portion of the curve for p{d'). Far out on the curve, p{d') still varies as 

 d'~^. Equation (5.13) may be obtained directly by using the approximation 

 (3.6) that 6' is nearly equal to Ig/Q and noticing that bz is the mean square 

 value of li. 



If the sine wave is absent, p is zero and 



P(e') = hibo/b2y'hi + zT''' (5.14) 



which is consistent with the results given between equations (3.4-10) and 

 (3.4-11) of Reference A. In this case (5.12) becomes 



^-|'(l + z?)-'« (5.15) 



Although the standard deviation of 6' is infinite an idea of the spread of 

 the distribution may be obtained from the average value of \ 6' \. Setting 

 bi equal to zero in (4.5) in order to obtain the case in which the noise band is 

 symmetrical about the sine-wave frequency leads to 



[V\ = — 4t I dR del dR' dd'B'R' 

 4:Tr''Oo02 "'o 'J-TT J- 00 Jo 



exp i 1-(R:' - 2QR cos d + Q'')/bo - (R'' -f R^d'')/b2^ 



The integrals in R', B' cause no difficulty and the integral in B is proportional 

 to the Bessel function Io{QR/bo). When the resulting integral in R is 

 evaluated^^ we obtain 



ib2/boy^h-'"lo{p/2) (5.16) 



where p = Q'/{2bo). 



13 See, for example, G. N. Watson, "Theory of Bessel Functions," Cambridge (1944), 

 p. 394, equation (5). 



