MATHEMATICAL ANALYSIS OF RANDOM NOISE 285 



possible use of this result is to determine whether a noise due to random in- 

 dependent events occuring at the rate of v per second may be regarded as 

 "random noise" in the sense of this work. 



4. \Vlien /(/), as given by (1.5-1), is analyzed as a Fourier series over an 

 interval of length T a set of Fourier coefficients is obtained. By taking 

 many different intervals, all of length T, many sets of coefficients are 

 obtained. If v is sufficiently large these coefficients tend to be distributed 

 normally and independently. A discussion of this is given in section 1.7. 



Part II — Power Spectra and Correlation Functions 



1. Suppose we have a curve, such as an oscillogram of a noise current, 

 which extends from / = to / = <x> . Let this curve be denoted by I{t). 

 The correlation function of /(/) is i/'(t) which is defined as 



^(r) = Limit I [ /(/)/(/ + r) dl (2.1-4) 



r-»oo 1 •'0 



where the limit is assumed to exist. This function is closely connected 

 with another function, the power spectrum, w(/), of /(/). /(/) may be 

 regarded as composed of many sinusoidal components. If I(i) were a 

 noise current and if it were to flow through a resistance of one ohm the 

 average power dissipated by those components whose frequencies lie be- 

 tween/and/ + df would be w(f) df. 

 The relation between w(f) and \I/(t) is 



w{f) = ^ I rPir) cos It/t dr (2.1-5) 



Jo 



^(t) = [ wif) COS IttJt df (2.1-6) 



Jo 



When /(/) has no d.c. or periodic components, 



w(f) = Limit ^'y^l' (2.1-3) 



where 



"-'"^' dt. 



su) = f me-'' 



Jo 



Jo 

 The correlation function for 



/(O = ^ + C cos (27r/o^ - <p) 

 is 



^(r) =^ A' + ^ cos 27r/oT (2.2-3) 



These results are discussed in sections 2.1 to 2.4 inclusive. 



