294 BELL SYSTEM TECHNICAL JOURNAL 



7. A method discovered independently by Van Vleck and North shows 

 that the correlation function ^(t) of the output current for an unbiased 

 linear rectifier is 



*M = |' + |%F.[-i-J;J;|] (4.7-6) 



where the input voltage is Vn • The correlation function )^(t) of Vjf is 

 denoted by ^t and the mean square value of Fa- is xpo . The power spectrum 

 W{f) of / may be obtained from 



W{f) = 4 f ^(r) cos 2irfT dr (4.6-1) 



Jo 



by expanding the hypergeometric function and integrating termwise using 



Gn{f) = I 4^r COS 2tvJt dr. (4C-1) 



Jo 



Appendix 4C is devoted to the problem of evaluating the integral for G„(/). 



8. Another method of obtaining the correlation function i/'(t) of /, termed 

 the "characteristic function method," is explained in section 4.8. It is 

 illustrated in section 4.9 where formulas for ^(r) and W{f) are developed 

 when the voltage P cos pt + Vn is applied to a general non-Unear device. 



9. Several miscellaneous results are given in section 4.10. The char- 

 acteristic function method is used to obtain the correlation function for a 

 square-law device. The general formulas of section 4.9 are applied to the 

 case of a v^^ law rectifier when the input noise spectrum has a normal law 

 distribution. Some remarks are also made concerning the audio-frequency 

 output of a linear rectifier when the input voltage V is 



Q{\ + r cos pt) cos qt -{- Vn . 



10. A discussion of the hypergeometric function iFi{a; c; x), which often 

 occurs in problems concerning a sine wave plus noise, is given in 

 Appendix 4B. 



TART I 



THE SHOT EFFECT 



The shot effect in vacuum tubes is a typical example of noise. It is due 

 to fluctuations in the intensity of the stream of electrons flowing from the 

 cathode to the anode. Here we analyze a simpUfied form of the shot effect. 



