MATHEMATICAL ANALYSIS OF RANDOM NOISE 283 



linear device, such as a square-law or linear rectifier, the output will also 

 contain noise. The methods which are available for computing the amount 

 of noise and its spectral distribution are discussed in Part IV. 



Acknowledgement 



I wish to thank my friends for many helpful suggestions and discussions 

 regarding the subject of this paper. Although it has been convenient to 

 acknowledge some of this assistance in the text, I appreciate no less sincerely 

 the considerable amount which is not mentioned. In particular, I am in- 

 debted to Miss Darville for computing the curves in Parts III and IV. 



Summary of Results 



Before proceeding to the main body of the paper, we shall state some of 

 the principal results. It is hoped that this summary will give the casual 

 reader an over-all view of the material covered and at the same time guide 

 the reader who is interested in obtaining some particular item of informa- 

 tion to those portions of the paper which may possibly contain it. 



Part I— Shot Effect 



Shot effect noise results from the superposition of a great number of 

 disturbances which occur at random. A large class of noise generators 

 produce noise in this way. 



Suppose that the arrival of an electron at the anode of the vacuum tube 

 at time / = produces an effect F{t) at some point in the output circuit. 

 If the output circuit is such that the effects of the various electrons add 

 linearly, the total effect at time / due to all the electrons is 



/(/) = z nt- h) (1.2-1) 



where the k^ electron arrives at tk and the series is assumed to converge. 

 Although the terminology suggests that /(/) is a current, and it will be 

 spoken of as a noise current, it may be any quantity expressible in the form 

 (1.2-1). 

 1. Campbell's theorem: The average value of /(/) is 



I(J) = V \ Fit) dt (1.2-2) 



•Loo 



and the mean square value of the fluctuation about this average is 



ave. [/(/) - 7(0]' = V £ F^{t) dt (1.2-3) 



