290 BELL SYSTEM TECHNICAL JOURNAL 



where ^0 is the mean square value of /(/). The probability that /?(/) lies 

 between Ri and Ri + dRi and at the same time R(t + t) lies between 

 R2 and Ri + dRi when / is chosen at random is obtained by multiplying 

 (3.7-13) by dRi dRi . For an ideal band-pass filter, the expected number 

 of maxima of the envelope in one second is 



Mm(/,-fa) (3.8-15) 



When R is large, say y > 2.5 where 



1/2 

 y = Tm , ^0 = r.m.s. value of /(/), 



the probability that a maximum of the envelope, selected at random from 

 the universe of such maxima, lies between R and R -^ dR is approximately 



, 9 . .,2,9 dR 

 1.13(/ - l)e-''" -TTi 

 Wo 



A curve for the corresponding probability density is shown for the range 

 < y < 4. Curves which compare the distribution function of the maxima 

 of R with other distribution functions of the same type are also given. 



8. In section 3.9 some information is given regarding the statistical 

 behavior of the random variable : 



rh+T 



E = \ l\t) dt (3.9-1) 



where h is chosen at random and /(/) is a noise current with the power 

 spectrum wif) and the correlation function i/'(t). The average value 

 niT of E is T\pQ and its standard deviation cjt is given by (3.9-9). For a 

 relatively narrow band-pass fi.lter 



(Tt t 



niT VT{fb — fa) 



when T(fb — /a) ^ 1. This follows from equation (3.9-10). An ex- 

 pression which is believed to approximate the distribution of E is given by 

 (3.9-20). 



9. In section 3.10 the distribution of a noise current plus one or more 

 sinusoidal currents is discussed. For example, if / consists of two sine waves 

 plus noise: 



I = P cos pt -\- Q cos qt -\- In, (3.10-20) 



where p and q are incommensurable and the r.m.s. value of the noise cur- 

 rent In is ypl ^, the probabihty density of the envelope R is 



R f rJo{Rr)MPr)MQr)e-'^°'''" dr (3.10-21) 



Jo 



where /o( ) is a Bessel function. 



