294 BELL SYSTEM TECHNICAL JOURNAL 



This operation gives a new image impedance Wi{s) which is more 

 general than the corresponding one of the prototype. 



The derived structures represented by formulas (2) and (4) as well 

 as their common prototype are given in Fig. 1. A comparison of 

 formulas (1) to (5) shows that for the same value of the parameter 5 

 both derived networks have the same transfer constant T{s) and that 



z,'{s)z^"{s) = z,"{s)z.2'{s) = WiWo = Wi{s)Wo(s) = z^z.. 



Thus the series and shunt impedances of one derived structure are 

 inverse networks of impedance product ZiZo of the shunt and series 

 impedances, respectively, of the other one derived from the same 

 prototype, Si, Zo- vSimilarly, the pair of image impedances Wi and W2 

 and the pair Wi{s) and W^is) are inverse impedances of this same 

 product. In fact, either infinite structure might have been obtained 

 from the other as such an inverse network; the transfer constants of the 

 two would then necessarily be identical for the ratio of series to shunt 

 impedance would be the same in both. 



1.3 "Constant k'' Wave-Filter, The Initial Prototype 

 The "constant k" wave-filter of any class, that is, having any 

 preassigned transmitting and attenuating bands, is a reactance 

 network of ladder type whose product of series and shunt impedances, 

 and therefore iterative impedance k of the corresponding uniform line, 

 is a constant independent of frequency. Putting k equal to the 

 resistance R of the line or impedance with which the wave-filter is 

 normally to be associated, we have 



ZikZ2k = k^ = R^ = a. constant. 



Here and in what follows the additional subscript k implies a relation 

 to the "constant ^" wave-filter. 



When there is dissipation in the reactance elements, the above 

 relation is strictly satisfied by requiring that the coil dissipation 

 constant, (/, and the condenser dissipation constant, d', be equal for 

 each pair of inverse network elements. For example, when d = d' 



{d + i)2irfLik _ Lik ^ D2 



{d' + i)27rfC2k C.U 



There are several reasons for choosing the "constant ^" wave-filter 

 as the initial prototype. 

 1. Its structure and method of design for any class is definitely known. ^ 



7 See footnote 1. Also U. S. Patent No. 1,509,184 to O. J. Zobel, dated September 

 23, 1924. 



