302 BELL SYSTEM TECHNICAL JOURNAL 



A corresponding diagram which illustrates this process is that of 

 Fig. 5. 



The transfer constants of these wave-filters are also given by formula 

 (8) which includes (9). 



The image impedances in Sequence 2 are 



. W,,{_\+a'{U,-\-iVu)'] (12) 



where a, a', a", etc., have the same values as in (10). 



1.6 Relations Between Sequence 1 and Sequence 2 



Carrying through operations for the determination of the structures 

 of the series and shunt impedances in these wave-filters, the following 

 results are found: 



a. Each pair of structures of the same order in the two sequences is a 



pair of inverse networks of impedance product F?. 

 That is, if the series Tlf-type has the series and shunt impedances 

 Sifc'(m) and z^kim), and the shunt il/-type Zv/'{m) and So/'(;;0, the 

 inverse network relations are 



z,,'(m)z2,"(m) = z,u"{m)z,u'{m) = R\ 



For the il/il/'-types, using similar notation, 



su'(w, m')z2k"{ni, m') = zn"{m, m')z2k'{m, m') = R', 



and so on for the higher order pairs. Consequently, one structure of 

 each pair might be obtained from the other as such an inverse network.^^ 



b. The transfer constants of both structures of a pair are the same. 



This result would come from the inverse network relations which give 



both structures the same ratio of series to shunt impedances, a ratio 



which determines the transfer constant. It has already been found in 



formula (8) where the value of g is the same for both structures of any 



order. 



11 The structures indicated or to be shown in detail in Sequence 1 and Sequence 2 

 can be generahzed as ladder type derivations from any initial prototype Zu s^. This 

 is done by a simple replacement of su- and Cji by Zi and So, respectively; of K- by the 

 product 3122; and by the omission of the subscripts, k, throughout. 



