THE ELECTRIC WAVE-FILTER 23 



unity at the extreme frequencies zero and infinity. The four pass 

 bands have, of course, the same locations as in Fig. 7. 



Multiplying the ratio by a constant greater than unity introduces 

 stop ( — ) bands along with the stop ( + ) bands; multiplying it by 

 a constant less than unity removes all infinite attenuations; these 

 changes within the stop bands are made without altering the loca- 

 tions of the four pass bands. 



Wave-Filter Having Assigned Pass Bands 



In connection with practical applications we especially desire to 

 know what latitude is permitted in the preassignment of properties 

 for a wave-filter. If we consider first the ideal lattice wave-filter, 

 its limitations are those inherent in the form which its two inde- 

 pendent resistanceless one-point impedances^ Zi and Zo may assume. 

 The mathematical form of this impedance is shown by formula (7) 

 of the appendix, which may be expressed in words as follows: 



Within a constant factor the most general one-point reactance obtain- 

 able by means of a finite, pure reactance network is an odd rational 

 function of the frequency which is completely determined by assigning 

 the resonant and anti-resonant frequencies, subject to the condition that 

 they alternate and include both zero and infinity. 



The corresponding general expressions for the quotient and product 

 of the impedances Zi and Zj are shown by formulas (8) and (9). 

 Definite, realizable values for all of the 2n-f2 parameters and 

 2n + l optional signs occurring in these formulas may be deter- 

 mined in the following manner: 



(a) Assign the location of all n pass bands, which must be treated 

 as distinct bands even though two or more are confluent; this 

 fixes the values of the 2n roots pi . . . p2n which correspond 

 to the successive frequencies at the two ends of the bands. 



(b) Assign to the lower or upper end of each pass band propagation 

 without phase change from section to section; this fixes the 

 corresponding optional sign in formula (8) as + or — , respec- 

 tively. 



(c) Assign a value to the propagation constant at any one non- 

 critical frequency (that is, assign the attenuation constant in a 



* A one-point impedance of a network is the ratio of an impressed electromotive 

 force at a point to the resulting current at the same point — in contradistinction to 

 two-point impedances, where the ratio applies to an electromotive force and the 

 resulting current at two different points. 



