604 BELL SYSTEM TECHNICAL JOURNAL 



The mean square of this equation for the noise generator voltage 

 may be obtained by finding the average of the square of the expression 

 over a very long period of time; that is 



_. « ^ ^ t2 r 1 



£^ = 4 E E E f^ - 



1=1 k=l m=l ^ L 



cos /cot 



ilo,(.tm-tk) 



x\ ^ f H{x)H(y)8,n(x)8,(y)dxdy 



I I G{y],x)II{y)b„,{x)bk{y)dxdydri 



I ( G{v,x)G{z,y) 



X 8m{x)8k(y)dxdydzdr] \. (43) 



In the above equation, the contribution to the mean square noise 

 generator voltage from the summation with respect to m, for a fixed 

 value of k, is zero unless m = k, since the long time average of the 

 emission deviation must be zero. Furthermore, since the electrons 

 are emitted independently of one another 



p 

 E 8mix)8„,{y) = 0, unless x = y. 



From these considerations, the contribution to the mean square 

 generator voltage from the second integral in (43) is zero since x and y 

 have no common value. The contribution from the last integral is a 

 bit more difficult to obtain. However, from (41), the contribution to 

 the mean square noise generator voltage from the frequencies between 

 / and f -\- df can be shown to be 



£7 = 2(f/| r HHu,:)no{tic)duc 



I I G(ri, Uc)G{z, Uc)no{uc)dUcdzdr] 



G{r], Uc)G{z, Uc)no(uc)dUcdzdr] \ . (44) 



In terms of the variable y — ^au , the average rate of emission may 

 readily be shown to be 



/ \7 ^J- a _2j 



no{Uc)dUc = — yt ^ ay. 



