154 BELL SYSTEM TECHNICAL JOURNAL 



Equation (4C-7) suggests that we may write the expression for G2(f) as 



G2{f) =\ f w{x)G,{f - x) dx (4C-8) 



This is seen to be true from (4C-2) and (4C-6). In fact it appears that 



GrW =1 [ Mf- x)Gr.-i(x) dx (4C-9) 



Z J— CO 



might be used for a step by step computation of Gn(f). 



We now consider GrXf) for the case of relatively narrow band pass filters. 

 As examples we take filters whose characteristics give the following wifYs 

 and >P(t)'s 



Table 1 



We shall refer to these filters as Filter a, Filter b, and Filter c, respectively. 

 All have /o as the mid-frequency of the pass band. The constants have 

 been chosen so that they all pass the same average power when a wide band 

 voltage is applied: 



\po = \ "^(f) df = mean square value of /(/) or V{t) 



and it is assumed that /o ^ o-, /o ^ a, /o ^ |3 so that the pass bands are 

 relatively narrow. 



Expressions for Gn(f) corresponding to several values of n are given in 

 Table 2. \Mien w = 1, Gi{f) is simply w(/)/4. G2(f) is obtained by set- 

 ting n= 2 in the definition (4C-1) for G„(f), squaring the i/'(r)'s of Table 1, 

 and using 



cos 27r/or =2 + 2 cos 47r/or 



