NKTWOIIK SYXTHKSIS ()37 



give the required double zero of [1 + H{z)] at z' = zl . One value of J, 

 so determined, is real and of order I'zl . This is the appropriate solution. 

 Then ] ,/" "/'""^ I is "f order l/lo"^^ , when j z | ^ 1. This suggests the 

 following appi'oxiniation in place of (47): 



H{z) ^ Gz^'^^' f^l^ (48) 



1 — J/z- 



Tiie approximation is at least as good as 1/zl"^* , compared with 

 unity, both in the useful interval and in the neighborhood of the singu- 

 larities 2o , and 2ff . This means that the approximation can be used: in 

 estimating the error a — a (in the useful interval), in calculating ./ and 

 G, and in finding the roots z„ . 



In the useful interval, \ z \ = 1, and therefore 1/z = z*. Then 

 (1 — J/z') is (1 — Jz~)*; and their ratio has magnitude unity. Thus 

 1 H(z) I = I G 1 , in the useful interval, to order of l/zl"'^* compared 

 with unity f. With \ J \ < 1, phase H(z) varies over the useful interval 

 to the same extent as the phase of /""^"'.J Fig. 9 illustrates the difference 

 ill a — a, as determined by (42) and (48). These curves, however, are 

 for single values of n and Zo ; and the improvement obtained by using 

 (48) would be different with different values of n or Zo . 



The values of / and G, determined from (48), and the requirement 

 that [1 + H(z)] must have two zeros at z^ = zl , turn out to be: 



J = 



71 + 1 I 2 



1 + -4 + a/(i - _^Y 4- . .^^.... 



(49) 



G = - _^„^^ 



2o 1 JZo 



Xote that this J is in fact smaller than l/zo . 



15. GENERALIZATION 



The several sections preceding describe a quite specific example, as 

 an introduction to synthesis applications. The next several sections de- 

 scribe how the specific methods of the example may be generalized, in 

 several respects. 



First, the ideal gain, a, is generalized, so that it need not even have 

 the sort of functional form associated with finite networks. Then, the 



t The I H(z) I determined b}' (42) is constant only to order \/sl . 

 t This is the most we can expect, when we have n singularities, whicli can pre- 

 vent the dominance of only lower order terms, through z^". 



