320 BELL SYSTEM TECHNICAL JOURNAL 



2.2 it is seen that our fundamental inversion formulae (2,1-5) and (2.1-6) 

 are unchanged. Thus, 



w(J) = A [ Ut) cos 2rfT dr (2.5-1) 



Jo 



^(t) = [ w(J) cos lirfr df (2.5-2) 



Jo 



where the bars indicate averages taken over the parameters with /or rheld 

 constant. 



The definitions of w and ^ appearing in these equations are Ukewise ob- 

 tained from (2.1-3) and (2.1-4) 



w 



and 



HI) = Limit ^JilZlL' (2.5-3) 



Ut) = Limit i f I{{)I{l -h r) dt (2.5-4) 



r— M i •'0 



The values of / and t are held fixed while averaging over the parameters. 

 In (2.5-3) S{f) is regarded as a function of the parameters obtained from 

 /(/) by 



S(f) = [ I(t)e-'''^'dt (2.1-2) 



Jo 



Similar expressions may be obtained for the average power spectrum for 

 d.c. and periodic components. All we need to do is to average the ex- 

 pression (2.2-11) 



Sometimes the average value of the product /(/)/(/ -t- t) in the definition 

 (2.5-4) of \}/(t) is independent of the time T. This enables us to perform 

 the integration at once and obtain 



1^(t) = /(/)/(/ -I- r) (2.5-5) 



This introduces a considerable simplification and it appears that the simplest 

 method of computing w'f) for an /(/) of this sort is first to compute iA(r), and 

 then use the inversion formula (2.5-1). 



2.6 First Example — The Shot Effect 



We first compute the average on the right of (2.5-5). By using the 

 method of averaging employed many times in part I, we have 



I{i)I{t -\- r) = i: p{K) UiOI^it -\- r) (2.6-1) 



