I- 1 ITERS lOR C'.l RRIRR SYSTEMS 213 



The filter described here actually consists of two filters connected in 

 tandem, each derived from a different power ratio. This step was taken 

 because of the relatively low dissipation factor realizable with coils of reason- 

 able size. By dividing the total attenuation between two power ratios, 

 lower overall distortion due to dissipation was achieved. The distortion 

 represented by the non-dissipative ripj^le "ap" was minimized by so assign- 

 ing the frequencies of infinite attenuation to the two functions that phasing 

 in of the ripples was avoided as far as possible. 



The two power ratios selected are: 



/«i = 1 + (e2«P _ Dcosh'e,^ 



For these the peak frequencies were assigned on an alternate basis as 

 follows : 



To 0/, : nil , W3 , W6 and im 



To di2 : Wi , '"i; , ffh and m^ 



with the value of "m" decreasing from Wi to W7 and ttii = 1. The param- 

 eter "m" has the same meaning as in image filter theory. 



The next step in the process is the finding of the roots of the two power 

 ratios. These may be obtained from the following expansions: 



For e"^ representing a reference filter of 3^ sections: 



(W3 -f xfims + x)\m, + .v)-(l + :v) 



+ I - — r— i ) (ws - .v)"(w6 - xfinii - xfil - x) = 



which is expressed in the form 



Ki[x + Ci.v + 02 X + Qsx -f fl4-^" + ObX -f ae-v + ^7 = 0] 

 For e'"' representing a reference filter of 4 sections: 

 (;«2 -f .v)(w, -f .x:)(w6 -f a;)(l -f x) 



/.Up 1 

 J c^p -j- 1 



which is expressed by 



Ki[x^ + a^x^ -f c^x^ -f (710.V + an = 0] 



^ = 4A+^ 



In the above expressions x — \/ \ +75 where p = iw and a^ for the filter 



