650 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



It is still true that 1 + H{z) will have the same number of zeros as 

 poles in the region \z\ < 1, so long as a — 5 is reasonably small in the 

 interval | 2 | = 1. In equation (87), however, the poles of 1 + H(z) 

 include the zeros z„ of D (the arbitrary infinite loss points), as well as 

 the poles of R{z)R{ — z). When the coefficients of D are to be chosen as 

 in the previous section, the contour rule merely says that any z, and z^ 

 in the region | 2 | < 1 will occur in like numbers. 



Fortunately, the frequent occurrence of | z„ | < 1 is softened by the 

 following curious circumstance. Almost always, any | z, \ and ] z^ | < 1 

 are so nearly identical that factors {z — z„) and {z — z„) may be can- 

 celled out without any important effect on H{z), or a — a. Cancellations 

 of this sort were encountered a number of times before an explanation 

 was discovered. Actually the explanation is quite simple. 



At any zero of 1 + H{z), H{z) = — 1. On the other hand, H{z) is 

 small when \z\ = 1. Generally, it is much smaller in most of the inter- 

 val \z\ < 1. For instance, when C-ik = Cok through k = m, H{z) is pro- 

 portional to Z™'''^, in the neighborhood oi z = 0. As a result, | H{z) \ 

 rarely becomes as large as 1, in the region | 2; 1 < 1, except in the very 

 close proximity of a pole. In other words, in the region \z\ < 1, any 

 zero Za is usually very close to a pole Za — usually so close that the 

 corresponding factors z — z„ may be cancleled out without significant 

 effect on a — a. 



The occurrence of non-physical natural modes {Rez^ = 0) is the same 

 as before; but adjustments to correct for these, in an efficient manner, 

 are much more complicated. In addition, there may be non-physical 

 infinite loss points, z„ . To correct for non-physical singularities, the 

 simplest procedure would be to change one or both of the highest order 

 coefficients in N and D of (82), that is Kn" and Kn' . This would be 

 inefficient, however, for it would spoil the match of Cn" to Cn" , or C„' 

 to Cn' . The unmodified design, defined by (84), can match terms through 

 order n" + n', and it is desirable to change only the highest order terms 

 in adjusting the design. 



More efficient adjustments are in fact feasible. They sometimes re- 

 quire an increased element of art; but the art may be based on specific 

 principles. Some particularly useful principles are described in the next 

 section. These apply to various other corrections besides correction of 

 non-physical zeros and poles. Examples are reduction in phase to make 

 two-terminal realization possible, and increase in shunt capacity in 

 two-terminal designs. In general, they offer a means of making 

 m < n' -{■ n" in (84), and using the remaining degrees of freedom to 

 meet other conditions. 



