6 BELL SYSTEM TECHNICAL JOURNAL 



1. "Constant k" Wave- Filter Having Any Preassigned 

 Transmitting and Attenuating Bands 



The ''constant k" wave-filter belonging to any class 1 is defined as 

 that ladder type wave-filter whose product of series and shunt impedances, 

 and therefore characteristic impedance, k, of the corresponding smooth 

 line, is constant independent of frequency. 



The reasons for seeking the "constant k" wave-filter of any class 

 are, briefly: 



(1) Its physical structure is readily found which will give any 

 preassigned transmitting and attenuating bands. 



(2) Each of its two mid-point characteristic impedances passes 

 thru the same values, different in the two cases, in all transmitting 

 bands. 



(3) Its design is preliminary to and furnishes a logical basis for the 

 derivation of general wave-filters possessing desirable attenuation 

 and impedance characteristics. 



Letting the two impedances of the "constant k" wave-filter be 

 denoted with extra suffixes as Z\\ and z 2 [, we have seen that if there 

 is a relation between these impedances such that 



%ik z 2 k = k 2 = Constant, (5) 



the series and shunt impedances of the "constant k" wave-filter must be 

 inverse networks to each other. Only one of them, say z ik , need then be 



temporarily considered, the ratio — — , becoming ( -^) which by (3) 



4z2k \2k/ 



shows that free transmission occurs wherever the series impedance 



passes with increasing frequency thru the values from z ik = —ilk 



to z lk = o, and Zi k = o to z u . = -\-i2k. At each critical frequency 



separating a transmitting from an attenuating band z ik has the 



value Zi k = ±i2k. By (4) an attenuating band includes a frequency 



at which Z\ k is anti-resonant. Hence, in a "constant k" wave-filter 



the transmitting and attenuating bands include the frequencies at which 



the series impedance is resonant and anti-resonant, respectively. The 



1 The class of a wave-filter, as defined in the present paper, is determined by the 

 number of its transmitting bands and their general locations on the frequency scale; 

 the type by its general structure. Thus, the low-band-and-high pass class transmits 

 in a band including zero frequency, in one internal band, and in a band including 

 infinite frequency. A class is complementary to another if its transmitting and 

 attenuating bands correspond in order to the attenuating and transmitting bands, 

 respectively, of the other. One class is higher, or lower, than another if it has in 

 addition to those of the latter at least one more, or one less, transmitting or attenu- 

 ating band. 



