NETWORK SYN'THESIS 



619 



O. Z-PLAXE MAl'PIiVGS OF NETWORK SINGULARITIES 



2-plane mappings of network singularities are also an essential part 

 of synthesis applications. The mapping z^ of a typical zero or pole p„ is 

 illustrated in Fig. 4. From (5), the analytic relation must be: 



Po = 



Wr 



(7) 



By its quadratic nature, there must be exactly two values of Zo , corre- 

 sponding to one Pa . The relation is such that replacing z^ by — l/z, 

 leaves p, unchanged; and hence the two values of z„ must be negative 

 reciprocals, each of the other. Thus, one mapping of pa falls outside the 

 unit 2-plane circle, and the other inside. 



A unique definition of z„ may be obtainetl by rcMiuiring that Za must 

 be the mapping outside the unit circle. Then | 2„ | > 1 by definition, and 

 the complete definition of Zc may be : 



We 



p. = 



Za\ > I 



(8) 



This definition is unique provided network singularities pa are excluded 

 from that very special line segment of the real frequency axis which 

 corresponds to the useful frequency interval, — Wc < co < +Wc (where 

 I Za 1 would be exactly 1). 



We are going to solve the interpolation problem by choosing the z^ 

 first, instead of the p-plane singularities p„ , after formulating the inter- 

 polation problem in suitable 2;-plane terms. For this, however, we must 



---tlCJc 



Pcr^ 



p-PLANE 



REAL p 



-LWc 



Fig. 4 — Mappings of a network singularity. 



