PROPERTIES OF SINE WAVE PLUS NOtSE 111 



our work we denote the power spectrum of In by w{f) and its corr'elat'iotf' 

 function by i/'W- Themeansquare value of /at is denoted by ^o- ^ ^' * 



The study of the probabiUty distribution of / is essentially a study of the 

 integraP 



1 r ri - Qcosei 



where 



<fi(x) = 77^. e-'" (1.3) 



and p{I) is the probabiUty density of /, i.e. p{l)dl is the probability that a 

 value of current selected at random will lie in the interval /, / + dl. An- 

 other expression for p{I) is given by equation (3.10-6) of Reference A, 

 namely 



Pil) = i- f e-'"-*"'" UQz) dz (1.4) 



where JoiQz) denotes the Bessel function of order zero. 

 The substitutions 



enable us to write (1.2) as 



Pi(y) = V^opii) = - f Ay -acose)de, (1.6) 



TT Jo 



where pi(y) denotes the probabiUty density of y. This is the expression 

 actually studied. Curves showing pi(y) and the cumulative distribution 

 function 



f p{h)dh = f Pi{yi)dyi 

 J— 00 •'—00 



= - I (p-i(y — 0' cos 0) ddy 



IT Jo 



(1.7) 



where 



<P-iioc) = r <p{x,)dx, = i + ierf (x/V2) (1.8) 



J— 00 



3 W. R. Bennett, "Response of a Linear Rectifier to Signal and Noise," Jour. Aeons. 

 Soc. Amer. Vol. 15 (1944), 164-172, and B.S.T.J. Vol. 23 (1944), 97-113. 



