212 BELL SYSTEM TECHNICAL JOURNAL 



ends and the sets obtained by open circuiting one or both ends. The 

 same normal coordinate solutions also furnish convenient parameters 

 in terms of which general expressions for the image parameters of 

 the structure, can be built up. For a band-pass filter, for example, the 

 typical result is 



Tanhg = y^,^'--^"^^;;-/''-^^-S 



O'n ' ' ' Up 



ai • • • am-l^a'^Ciag+l • • - (Ir 

 CLl ' ' ' dm dq ' • ' dr-l 



Zi = ikij 



where a,- symbolizes a frequency factor of the form 1—7-^ and 



— /l — /2 — • • • — fm — fci — fn — fn+1 — • • • 



The formulae are almost exactly similar to those familiar in the theory 

 of the lattice, except that the quantities /i • • • /« which are now natural 

 frequencies of the network as determined under the previously de- 

 scribed conditions have a different significance. As in the lattice, 

 however, they fall into three groups : fi - • • fm and fq--' fr, which 

 affect Z I only;/„ - • - fp, which affect 9 only; and the cut-offs, fc^ and 

 /fj, which enter into both expressions. The formulae can be extended 

 to low-pass, high-pass and all-pass structures by allowing the cut-oflfs 

 to assume the limiting values zero and infinity respectively. 



Certain further restrictions upon the image impedance and transfer 

 constant of physically realizable filters may be obtained from the con- 

 sideration of another system of parameters, ri • • • r„, defined as the 

 roots of the equation tanh 9=1. They are usually of either single 

 or double multiplicity. The importance of the roots depends upon 

 the fact that in combination with the cut-offs they are sufficient to 

 determine 9 at all frequencies. The restrictions to which they lead 

 may be divided into two sets. The first affects the transfer constant 

 alone and is expressed in terms of limitations on the allowable positions 

 and multiplicities of the roots. The second is concerned with the 

 restrictions which must be placed upon the relation between the 

 transfer constant and the two image impedances. It may be expressed 

 by the statement that when the transfer constant and one image 

 impedance have been chosen as functions of frequency, the second 

 image impedance is determined as a function of frequency to within a 

 constant multiplier. The differences between the two image im- 

 pedances depend only upon the roots of single multiplicity, so that if 

 only double roots are involved the structure is necessarily symmetrical. 



