Mathematical Analysis of Random Noise 

 BY s. o. RICE 



{Concluded from Jidy 1944 issue) 



PART III 

 STATISTICAL PROPERTIES OF RANDOM XOISE CURRENTS 



3.0 IXTRODUCTIOX 



In this section we use the representations of the noise currents given in 

 section 2.8 to derive some statistical properties of /(/). The first six sec- 

 tions are concerned with the probabihty distribution of /(/) and of its zeros 

 and maxima. Sections 3.7 and 3.8 are concerned with the statistical prop- 

 erties of the envelope of /(/). Fluctuations of integrals involving /'(/) 

 are discussed in section 3.9. The probability distribution of a sine wave 

 plus a noise current is given in 3.10 and in 3.11 an alternative method of 

 deriving the results of Part III is mentioned. Prof. Uhlenbeck has pointed 

 out that much of the material in this Part is closely connected with the 

 theory of Markoff processes. Also S. Chandrasekhar has written a review 

 of a class of physical problems which is related, in a general way, to the 

 present subject." 



3.1 The Distribution of the Noise Cureent^^ 



In section 1.4 it has been shown that the distribution of a shot effect 

 current approaches a normal law as the e.xpected number of events per 

 second, v, increases without limit. 



In Une with the spirit of this Part, Part III, we shall use the representation 



.V 



HO = Z) («" cos CO,,/ -\- bn sin aj„0 (2.8-1) 



to show that /(/) is distributed according to a normal law. This is obtained 

 at once when the procedure outlined in section 2.8 is followed. Since <7„ 

 and bn are distributed normally, so are a„ cos ooj and bn sin a)„/ when / is 

 regarded as fixed. /(/) is thus the sum of 2X independent normal variates 

 and consequently is itself distributed normally. 



22 Stochastic Problems in Physics and Astronomy, Rei\ of Mod. Phys., Vol. 15, pp. 

 1-89 (1943j. 



23 An interesting discussion of this subject by V. D. Landon and K. A. Norton is given 

 in the I.R.E. Proc, 30 (Sept. 1942j pp. 425-429. 



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