IDEAL FILTERS 



219 



Fig. 1 — The symmetrical lattice. 



The relation (1) requires for transmission, i.e., for 6 imaginary, that 

 Zx and Zy differ in sign; for attenuation, i.e., for 9 real, that Zx and Z„ 

 be alike in sign. In the case of the low-pass filter, this amounts to 

 requiring correspondence of zeros (resonances) in one arm to poles 

 (anti-resonances) in the other for / < fc, and of zeros to zeros and 

 poles to poles for f > fc, where fc, the cut-off, is a critical frequency 

 which appears in one arm only.^'' If we denote these critical fre- 

 quencies by /i, fi, • • • , fr in the range below fc, and by fi, fi, *•',// 

 in the range above fc, and if we make use of a well-known theorem ^^ 

 we readily find that Zx and Zy have forms similar to ^^ 



iK^f 



aia^ 



CLr—\(lc(l2 



a s-1 



aias 



flrfll • • • CLs 



Zj y 



iKyaias • • • a^a^ • • • a's-i 

 f a2Cii ' ' ' ar-idi • • • aj 



(3) 



1" In the basic theory given by Dr. Campbell, in the paper just referred to, it is 

 shown that in general a lattice having many natural frequencies is a milti-band-pass 

 filter. The extension of the theory in the manner shown above, in which separate 

 parameters for the control of the transfer constant and image impedance are obtained 

 by imposing special conditions on the natural frequencies, thus rendering many bands 

 confluent, was discovered and exploited independently by W. Cauer and one of the 

 present writers (see Cauer, "Siebschaltungen" and later papers; or H. W. Bode, U. S. 

 Patent No. 1828454, also "A General Theory of Electric Wave Filters," loc. cit.). 

 The published work by Dr. Cauer gives a particularly complete discussion. It ap- 

 pears from a recent informal communication from Dr. Campbell to the authors, how- 

 ever, that this extension was also considered by him and was described briefly in the 

 Yale-Harvard Lectures on Wave Filters delivered in 1923. The lectures have unfortu- 

 nately not been published. Their content, however, is similar to that given in the 

 discussion above. 



" R. M. Foster, "A Reactance Theorem," this Journal, Vol. Ill, No. 2, April, 1924, 

 p. 259. 



'2 The cut-off factor Oc may appear in either the numerator or denominator of 

 either form, and the line- and cross-arms may be interchanged. Otherwise the ex- 

 pression is general. 



