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BELL SYSTEM TECHNICAL JOURNAL 



2. The probability that /(/) lies between h and /i + dl, and /(/ + r) 

 lies between 1% and h + dli when / is chosen at random is 



hl^l - ^IV' ^^ e.p \zMLzJ^A±JMlb\ (3.2-4) 

 where i/'^ is the correlation function ^ij) of /(/): 



^(r) = / w{j) cos 27r/r <// 

 Jo 



(3.2-3) 



The ch. f. for this distribution is 



ave, c 



iuno-^i.ni^r^ ^ ^^.p I _^o ^^. _j. 



I -y («' + v') - rA,«J (3.2-7) 



3. The expected number of zeros per second of /(/) is 



-ll/2 



1 r_^(o)' ''"'" ' 



ttL Hi 



\^(0) J 



1/2 





(3.3-11) 



assuming convergence of the integrals. The primes denote differentiation 

 with respect to r: 



dr- 



For an ideal band-pass filter whose pass band extends from/a to fb the ex- 

 pected number of zeros per second is 



uf» - /J 



(3.3-12) 



When fa is zero this becomes 1.155/6 and when/a is very nearly equal to 

 fb it approaches fb-\-fa. 



4. The problem of determining the distribution function for the length 

 of the interval between two successive zeros of /(/) seems to be quite diffi- 

 cult. In section 3.4 some related results are given which lead, in seme 

 circumstances, to approximations to the distribution. For example, for 

 an ideal narrow band-pass filter the probability that the distance between 

 two successive zeros lies between t and t -{- dr is approximately 



2 [1 + a\T - T^)'] 



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