MATHEMATICAL ANALYSIS OF RANDOM NOISE 307 



dilators. Some argument arose as to whether the coefficients in (1.7-1) 

 were statistically independent or not. It was finally decided that the}' 

 are independent. 



The shot effect current has been represented in this way by Schottky. 

 The Fourier series representation has been discussed by H. Nyquist and 

 also by Goudsmit and Weiss. Remarks made by A. Schuster are equiv- 

 alent to the statement that a„ and b,, are distributed normally. 



In view of this wealth of information on the subject it may appear super- 

 fluous to say anything about it. However, for the sake of completeness, 

 we shall outline the thoughts which lead to (1.7-1). 



In Une with our usual approach to the shot effect, we suppose that exactly 

 A' electrons arrive during the interval (0, T), so that the noise current for 

 the interval is 



Ui) = E F(/ - 4) (1.7-2) 



k=i 



The coefficients in the Fourier series expansion of /«:(/) over the interval 

 (0, T) are a„K and b„K where 



OnK - ibnK = ^ I] j F {t - 4) exp ~^ -y- r^ 



-It [y^t) exp [-i ^' (/ + 4)]^/ 



= Rne-'^-^te''"'" (1.7-3) 



fr=i 



In this expression 



ft - 2^^^* 



RnC-''" = C„ - iSn = ^ £" F{t)e-'''"'"Ut 



(1.7-4) 



In the earlier sections the arrival times /i , /2 , • ■ ■ Ik were regarded as K 

 independent random variable each distributed uniformly over the interval 

 (0, T). Hence the ^a's may be regarded as random variables distributed 

 uniformly over the interval to Itt. 



Incidentally, it will be noted that in (1.7-3) there occurs the sum of K 

 randomly oriented unit vectors. When A' becomes very large, as it does 



Mkm. d. Physik, 57 (1918) pp. 541-567. 



' Unpublished Memorandum, "Fluctuations in Vacuum Tube Noise and the Like," 

 March 17, 1932. 



1" Investigation of Hidden Periodicities, Terrestrial Magnetism, 3 (1898), pp. 13-41. 

 See especially propositions 1 and 2 on page 26 of Schuster's paper. 



