MATHEMATICAL ANALYSIS OF RANDOM NOISE 



289 



where 



n . /-. TA + fn)' 



O = V 3 J- : 



jb — Ja 



ri = 



/6+/a 



fb and/o being the upper and lower cut-off frequencies. 



5. In section 3.5 several multiple integrals which occur in the work of 

 Part III are discussed. 



6. The distribution of the maxima of /(/) is discussed in section 3.6. The 

 expected number of maxima per second is 



- aOO -|1;2 



fw{f)df 

 Jo 



2I ^;' J 



f fMDdf 

 Jo 



(3.6-6) 



For a band-pass filter the expected number of maxima per second is 



'3/1 -AT 



15 ft- flJ 



(3.6-7) 



For a low-pass filter where /a = this number is 0.775/6 . 

 The expected number of maxima per second lying above the hne /(/) = /i 

 is approximately, when /i is large, 



g-fi/2iAo ^ i[the expected number of zeros of / per second] (3.6-11) 



where \{/o is the mean square value of /(/). 



For a low-pass filter the probability that a maximum chosen at random 

 from the universe of maxima lies between / and I -{- dl is approximately, 

 when I is large, 



V^ -y2,2 dl 



ye 



1-2 



(3.6-9) 



where 



y = ,7T72 



7. When we pass noise through a relatively narrow band-pass filter one 

 of the most noticeable features of an oscillogram of the output current is 

 its fluctuating envelope. In sections 3.7 and 3.8 some statistical properties 

 of this envelope, denoted by R or Rit), are derived. 



The probability that the envelope lies between R and R -\- dRis 



R ^-R-il2^o 



4^0 



dR 



(3.7-10) 



