286 BELL SYSTEM TECHNICAL JOURNAL 



2. So far we have supposed /(/) to be some definite function for which a 

 curve may be drawn. Now consider /(/) to be given by a mathematical 

 expression into which, besides /, a number of parameters enter. w(/) and 

 i/'(t) are now obtained by averaging the integrals over the possible values 

 of the parameters. This is discussed in section 2.5. 



3. The correlation function for the shot effect current of (1.2-1) is 



^(t) = V j_ F{l)F{t + T)dt-\-\v j F{t) dt\ (2.6-2) 



The distributed portion of the power spectrum is 



w,{f) = 2v I 5(/) f 

 where 



/-|-00 

 F{t)e-''"^' dt (2.6-5) 



The complete power spectrum has in addition to wi(/) an impulse func- 

 tion representing the d.c. component /(/). 



In the formulas above for the shot effect it was assumed that the expected 

 number, v, of electrons per second did not vary with time. A case in which 

 V does vary with time is briefly discussed near the end of Section 2.6. 



4. Random telegraph signal. Let /(/) be equal to either a or — c so that 

 it is of the form of a flat top wave, and let the lengths of the tops and bot- 

 toms be distributed independently and exponentially. The correlation 

 function and power spectrum of / are 



^(t) = a'e-'"'^' (2.7-4) 



wif) = 2,2 _r 2 (2.7-5) 



where /x is the expected number of changes of sign per second. 



Another type of random telegraph signal may be formed as follows : Divide 

 the time scale into intervals of equal length h. In an interval selected at 

 random the value of /(/) is independent of the value in the other intervals 

 and is equally likely to be -fo or —a. The correlation function of /(/) is 

 zero for I T I > A and is 



•■('-'i') 



for < I T I < A and the power spectrum is 



»W=^"C-^7 (2.7-9) 



