136 BELL SYSTEM TECHNICAL JOURNAL 



where the function F{iu) and the path of integration C are chosen to fit the 

 device. Examples of such devices are given in Appendix 4A. The corre- 

 lation function ^(r) of I{t) is given by 



dt 



dv 



^(r) = Limit 1 \ I{t)I{t + t) 



= Limit j\- [ dt f F{iu)e'"''^'^ du f F{iv)e''"'^'^'^ 



T-*oo 4:ir^ I JQ Jc Jc 



= -^ f F(iu) du [ F{iv) dv (4.8-6) 



47r~ J c J c 



1 r"" 



Limit - / exp [iuV{i) + ivV{t + t)] di 



= — -, / F{iu) du I F{iv)g{u, v, t) dv 

 At" Jc J c 



This is the fundamental formula of the ch. f. method. 



\Mien ]'{t) is the sum of a noise voltage and a regular voltage, as in 

 (4.8-2), (4.8-6) becomes 



^(r) = — [ Fiiu)e-^'^°"^''' du f F{iv)e-^'^''-'''' 



Att-Jc Jc (4,8-7) 



e-'^r^" g,(u, V, t) dv 



where gs(u, v, r) is the ch. f. of Vs{t) and Vs(t + r) given by (4.8-5). This 

 is a definite expression for ^(t). All that follows is devoted to the evalua- 

 tion of this integral and to the evaluation of 



W{f) = 4 /" ^(t) cos IwfT dr (4.6-1) 



Jo 



for the power spectrum of /. 



Quite often /(/) will contain dc and periodic components. It seems con- 

 venient to deal with these separately since they correspond to terms in 

 ^(t) which cause the integral (4.6-1) for W{f) to diverge. In fact, from 

 Section 2.2 it follows that a correlation function of the form 



,2 . C 



A' + ~ cos 27r/oT (2.2-3) 



corresponds to a current 



^ -f C cos (2x/o/ - if) (2.2-2) 



