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 



—• ^ " J!. Z, T^ r 1 — cos /cor 1 . , , , , , 

 £' = 4 E E E 71 ■ WY-2 — e^'-^'rn-tk) 



I H{x)H{y)bm{x)bk{y)dxdy 



x=uc *-'V = Uc 



+ 2l I I G{v,x)H(y)8m(x)8k{y)dxdyd'n 

 I G{r^,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 8m(x)8m(y) = 0, unless x = y. 



7n=l 



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 



E/' ^ 2dfl f H^iuc)no(uc)duc 



I I G(t7, Uc)G{z, Uc)noiUc)ducdzdr] 



,1=0 '^z=0 ^Ue= ^a(n-yia') 



I I G{r}, Uc)G{z, Uc)no{uc)ducdzdri \ . (44) 



ij=0 ♦^2=1) '^«c=VaU-V) 



In terms of the variable y = V^m', the average rate of emission may 

 readily be shown to be 



noiuc)dUc = — ye~v^dy. 



