PROPERTIES OF SINE WAVE PLUS NOISE 119 



of (1.10), Replacing the quantity within the brackets by the series shown in 

 the last equation of (2.5) ,'and integrating termwise leads to 



Nil, = iV«(./2)- ± ^^^ {fj" ... (-i; n + V, -f) (2.7) 



The series (2.7) converges for all values of a, y, and b. This follows from the 

 inequahty (1.11) which may be applied to <^^^"^(r), and from the fact that 

 the iFi is less than exp {h^/2) as may be seen by comparing corresponding 

 terms in their expansions. 



The expected number of zeros, per second, of I is 2iV(0) where we have set 

 /i, and hence y, equal to zero. In this case the integral in (2.6) may be 

 simplified somewhat and we obtain 



2iV(0) = iVo [e-«/o(/3) + ^ ^^ (^ , «)] (2.8) 



where /o(iS) is the Bessel function of order zero and imaginary argument and 



2 , ,2 2 72 



Ie{k, x) = j e "/o(^w) 



^0 



4 

 du. 



The integral Ie(k, x) is tabulated in Appendix I. 



I have been unable to obtain a simple derivation of (2.8). It was orig- 

 inally obtained from the following integral 



iV(/i) = — ^ /* de <p(y - a cos 6) f X(p{x + b sin 0) dx (2.9) 

 2 J-T •'0 



which may be derived from the second equation of (2.4) and the first of 

 (2.6). Setting /i and y equal to zero and writing out the ^'s gives 



2iV(0) =^' r dd I dx 

 27r J-v Jo 



X exp [-Ux + 2bx sin 6 + a cos' d + b^ sin' 6)]. 



Equation (2.8) was obtained by applying the method of Appendix III to 

 this expression. 



3. Definitions and Simple Properties of R and 



The remaining portion of this paper is concerned with the envelope R and 

 the corresponding phase angle 6. These quantities are introduced and some 

 of their simpler properties discussed in this section. 



Suppose that the frequency band associated with In is relatively narrow 



