218 BELL SYSTEM TECHNICAL JOURNAL 



is proportional to the derivative of the phase characteristic. The 

 realization of a linear phase shift in the transmission band therefore 

 automatically carries with it the satisfaction of the requirement of 

 uniform loss, in this range. ^ It can also be shown that the other 

 characteristics of the network will not be appreciably affected by slight 

 uniform dissipation. 



Moreover, it is well known that the image impedance and transfer 

 constant of a lattice structure are controlled by independent param- 

 eters.^ We can, therefore, dissociate the problem of providing the 

 required constant image impedance in the transmission band from that 

 of providing the required loss and phase characteristics.'' For the 

 moment we shall fix our attention on the transfer constant. 



With these simplifications our problem reduces to that of con- 

 structing a filter whose transfer constant on a non-dissipative basis 

 represents a linear phase shift in the transmission band and an infinite 

 loss in the attenuation bands, these being separated by narrow "transi- 

 tion intervals" in the neighborhood of the cut-offs. These transition 

 intervals may be taken small at pleasure, but must be assigned in 

 advance to insure the physical realizability of the network. 



Formulation of Requirements — Low-Pass Filters * 



If the impedances of the arms of a lattice are Z^ and Zy, Fig. 1, it is 

 well known that the image transfer constant and the image impedance 

 are given by the expressions ^ 



/Z. ^^^ 



Zi = VZ^. (2) 



* Strictly speaking, a slight qualification should be placed upon this statement. 

 Our process of approximating the ideal characteristics will lead to a phase shift which 

 ripples about the desired linear characteristic, the number of ripples depending upon 

 the number of elements used. As the number of elements is increased indefinitely, 

 the linear characteristic is approximated more and more closely, but it is evidently 

 not a necessary consequence of this that the slope of the ripples should approach 

 constancy. We shall be able to show, however, that with the actual process used, the 

 amplitude of the ripples decreases so rapidly that dB/dw approaches constancy as B 

 approaches linearity. 



* This follows at once from equations (4) and (5), p. 220. 



^ A method of choosing the lattice parameters to give a substantially constant 

 impedance in the transmission band has in fact already been obtained by W. Cauer, 

 "Siebschaltungen," V. D. I. Verlag, Berlin, 1931; or "Ein Interpolationsproblem 

 mit Funktionen mit Positivem Realteil," Math. Zeit., November, 1933, p. i. An 

 alternative method will eventually be developed as a by-product of the present 

 analysis. 



* The extension to filters of other types is given on p. 225. 



8 G. A. Campbell, "Phvsical Theory of the Electric Wave Filter," this Journal, 

 Vol. I, No. 2, November, 1922, p. 1. 



