656 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



The natural modes and frequencies of infinite loss are determined 

 from the zeros of the polynomial A^. By (21), each zero is either a {z- 

 plane) natural mode, z^ , or the negativ^e of an infinite loss point —z„ . 

 If gain variations are inconsequential, there is likely to be some latitude 

 in designating each zero as a 2,, , or as a —z^. 



A zero of N with a positive real part would make a non-physical natural 

 mode, and hence it must be a —Za, corresponding to an infinite loss 

 point. A zero of N with a negative real part can be a natural mode z^ , 

 but this may not be required. It may be either a z„ or a, —z„ , provided 

 conjugate zeros are assigned in the same way, and provided the total 

 number of — z^ does not exceed the total number of z, . The latter con- 

 dition requires: 



At least half the zeros of N 

 must have negative real parts. 



The continued fraction (108) shows how many zeros will have nega- 

 tive real parts, before any zeros have been calculated. The following 

 theorem makes this easy: 



The number of zeros of N which have negative real parts 

 is equal to the number of positive coefficients in the trun- 

 cation of the continued fraction (108) ivhich determined N. 



When gain is not to be disregarded, but is to be exactly zero, the 

 synthesis technique needs few changes. The phase of an "all-pass" net- 

 work is related to the natural modes z^ as follows: 



il3 = E C,T, , k odd 



E ^ = -log ) ^ (109) 



2 



n(i + 4, 



This may be regarded as a special case of (20) for a = (which makes 

 Ck = for k even, and also happens to require z^ = —z^). In functional 

 form however, it is more like il3 of (21). It differs in only two regards. 

 In the power series in z, each C^ is divided by two. In the rational frac- 

 tion in z, all the zeros correspond to natural modes, and the poles cor- 

 respond to frequencies of infinite loss; but the poles are also exactly the 

 negatives of the zeros, as in the il3 eciuations of (21). 



Accordingly, the phase synthesis technique which ignores gain varia- 

 tions may be applied to the zero gain form of the problem by cutting 



