MATHEMATICAL ANALYSIS OF RANDOM NOISE 311 



The correlation function, which turns cut to be a very useful tool, appar- 

 ently was introduced by G. I. Taylor.'^ Recently it has been used by quite 

 a few writers in the mathematical theory of turbulence. 



In section 2.1 the power spectrum and correlation function of a specific 

 function, such as one given by a curve extending to / = oo , are defined by 

 equations (2.1-3) and (2.1-4) respectively. That they are related by the 

 Fourier inversion formulae (2.1-5) and (2.1-6) is merely stated; the cUs- 

 cussion of the method of proof being delayed until sections 2.3 and 2.4. In 

 section 2.3 a discussion based on Fourier series is given and in section 2.4 a 

 parallel treatment starting with Parseval's integral theorem is set forth. 

 The results as given in section 2.1 have to be supplemented when the func- 

 tion being analyzed contains a d.c. or periodic components. This is taken 

 up in section 2.2. 



The first four sections deal with the analysis of a specific function of /. 

 However, most of the applications are made to functions which behave as 

 though they are more or less random in character. In the mathematical 

 analysis this randomness is introduced by assuming the function of / to be 

 also a function of suitable parameters, and then letting these parameters be 

 random variables. This question is taken up in section 2.5. In section 2.6 

 the results of 2.5 are applied to determine the average power spectrum and 

 the average correlation function of the shot effect current. The same thing 

 is done in 2.7 for a flat top wave, the tops (and bottoms) being of random 

 length. The case in which the intervals are of equal length but the sign 

 of the wave is random is also discussed in 2.7. The representation of the 

 noise current as a trigonometrical series with random variable coefficients 

 is taken up in 2.8, The last two sections 2,9 and 2,10 are devoted to prob- 

 ability theory. The normal law and the central Hmit theorem, respectively, 

 are discussed. 



2.1 Some Results of Generalized Harmonic Analysis 



We shall first state the results which we need, and then show that they are 

 plausible by methods which are heuristic rather than rigorous. Suppose 

 that /(/) is one of the functions mentioned above. We may think of it as 

 being specified by a curve extending from t = — oo to / = oo. /(/) may 

 be regarded as ccmposed of a great number of sinusoidal components whose 

 frequencies range from to + oo . It does not necessarily have to be a noise 

 current, but if we think of it as such, then, in flowing through a resistance of 

 one ohm it will dissipate a certain average amount of power, say p watts. 



1* Diffusion by Continuous Movements, Proc. Land. Math. Soc, Ser. 2, 20, pp. 196- 

 212 (1920). 



" See the text "Modern Developments in Fluid Dynamics" edited by S. Goldstein, 

 Oxford (1938). 



