184 BELL SYSTEM TECHNICAL JOURNAL 



necessary property of constant resistance networks of this type, of a 

 three db loss at the cross-over frequency. 



If X is taken of the form i~-T 7— f t"he networks become band-pass 



and band-elimination filters, respectively. By taking other functions 

 for X multiple band structures may be designed, subject always to the 

 limitation that the combined bands of both filters must extend over 

 the whole frequency range, with a three db loss at each cross-over 

 point. 



The evaluation of the elements is easily done from equations (10). 

 The impedance denoted by aiX, for example, in the low-pass filter 

 would have the value i(f/fo) sin (wjln), which is an inductance of a 

 value (l/27r/o) [sin (7r/2w) J. For a terminating resistance dififerent from 

 unity the value of the first inductance is Li = (i?o/27r/'o)Csin (7r/2«)] 

 or in general any inductance is Lm = dmLo where Zo = Ro/lirfo. 

 Similarly, any capacitance is Cm = OmCo where Co = l/lirfoRo. The 

 corresponding formulas for the second network are C,„ = Co/a„t and 

 Lm = Lo/am- The same formulas hold for n even; in that case the 

 networks of Fig. 3 and Fig. 4 would terminate on the right in a shunt 

 arm with impedances of 1/aiX and X/ai, respectively. This is illus- 

 trated by Fig, 2 for n = 2. 



Filters with Characteristics Similar to Those of the 

 "M -derived" Type 



The networks shown in Fig. 3 and Fig. 4 have the same configuration 

 and similar characteristics to constant K filters. They are subject to 

 the same objection of a relatively slow rate of cut-off and an excessive 

 loss at frequencies remote from the cut-off. A type of characteristic 

 similar to that obtained with M-derived filters, with points of infinite 

 loss at finite frequencies, is necessary for an economical design in the 

 majority of cases. 



The loss characteristic of the network is of course fixed by the 

 function F(\), a ratio of two polynomials in X. It may be written 



tKK) - A,K ^^ 5iX2 + • • • + ^„X2"-2 • 



Now the first filter will have infinite loss points when the denominator 

 is zero, and the second filter when the numerator is zero. If these 



