248 BELL SYSTEM TECHNICAL JOURNAL 



In a similar manner more sections can be added and the resonant 

 frequencies determined in terms of the cutoff frequencies and the 

 position of the attenuation peaks. The most general section con- 

 sidered in this paper has a maximum of five equivalent sections. For 

 this case by applying the process described above the propagation 

 constant and critical frequencies are given by the equations 



, ,P^ A + C + E (1 - coVco^^)(l - coVa)3^)^(l - coVco5^)^ 



^^ 2 \ + B + D\{\ - 0)70)2^)2(1 - o;Vw4')'(l - a;Vo;B2) ' ^^^^ 



where 



I 



5 5 5 5 



T^ = ^ ^ Y^ ^ ni,uninmomp\ m 7^ n\ m 9^ 0; 



m=l H=l 0=1 p=l 



m ^ p\ n 9^ p; n 9^ o; 9^ p; 

 E = mim^mimim-a. ' (31) 



The resonant frequencies are given by the equations 



U = 2/./V(l+B+J) , (32) 



/• 2 _ ^7A7fiV-^n-x^-r^y .^^. 



fA\2 + 5 + ^W^^~W) + /b2(5 + 2Z) - V52 _ 42)) ' 



;, = .^^!'<::'+^+^^. , (34) 



Ja\2A + C + VC2 _ 4^£) +/«2(C + 2£ - VC2 - 4y4£) 



For any smaller number of sections the values can be obtained by 

 letting some of the w's go to zero. For example, for a three section 

 filter nii — m^ = 0. For low, high, and all-pass networks the values 

 can be obtained by letting /a^ -^ 0, /b^ ^ 00 or Ja^ -^ ; /s^ -^ <x> . 



