Mathematical Analysis of Random Noise 



By S, O. RICE 



Introduction 



THIS paper deals with the mathematical analysis of noise obtained by 

 passing random noise through physical devices. The random noise 

 considered is that which arises from shot effect in vacuum tubes or from 

 thermal agitation of electrons in resistors. Our main interest is in the sta- 

 tistical properties of such noise and we leave to one side many physical 

 results of which Nyquist's law may be given as an example. 



About half of the work given here is believed to be new, the bulk of the 

 new results appearing in Parts III and IV. In order to provide a suitable 

 introduction to these results and also to bring out their relation to the work 

 of others, this paper is written as an exposition of the subject indicated in 

 the title. 



When a broad band of random noise is applied to some ph5'sical device, 

 such as an electrical network, the statistical properties of the output are 

 often of interest. For example, when the noise is due to shot efifect, its 

 mean and standard deviations are given by Campbell's theorem (Part I) 

 when the physical device is hnear. Additional information of this sort 

 is given by the (auto) correlation function which is a rough measure of the 

 dependence of values of the output separated by a fixed time interval. 



The paper consists of four main parts. The first part is concerned with 

 shot effect. The shot effect is important not only in its own right but 

 also because it is a typical source of noise. The Fourier series representa- 

 tion of a noise current, which is used extensively in the following parts, may 

 be obtained from the relatively simple concepts inherent in the shot efifect. 



The second part is devoted principally to the fundamental result that the 

 power spectrum of a noise current is the Fourier transform of its correlation 

 function. This result is used again and again in Parts III and IV. 



A rather thorough discussion of the statistics of random noise currents 

 is given in Part III. Probability distributions associated with the maxima 

 of the current and the maxima of its envelope are developed. Formulas 

 for the expected number of zeros and maxima per second are given, and a 

 start is made towards obtaining the probability distribution of the zeros. 



When a noise voltage or a noise voltage plus a signal is applied to a non- 



^ An account of this field is given by E. B. Moullin, "Spontaneous Fluctuations of 

 Voltage," Oxford (1938). 



282 



