620 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



know what further conditions must be imposed upon the Za , so that 

 the corresponding pa will meet the special conditions necessary for 

 physical networks. A simple analysis of the definition (8) of Za , and of 

 the well known restrictions on the p^ , leads to the following assertion; 



The physical restrictions on z„ are 

 exactly the same as those on pa . 



It is ob\dous, for example, that conjugate complex z, are necessary 

 for conjugate complex p^ . Also, because \za\ > 1, z„ dominates —l/z„ . 

 Then the sign of Re pa is the same as that of the Re z^ , and pa with 

 negative real parts require Za with negative real parts, and so on. 



Thus the direct choice of z„ is restricted in exactly the same way as- 

 the choice of pa , except for the additional general requirement \ Za \ > 1. 

 The last condition imposes no important restriction on the corresponding 

 Pt, . Initially, it was adopted to make z^ unique for any p„ (not at a 

 useful real frec[uency); but this condition does also play an essential 

 role in the z-plane formulation of the interpolation problem. 



6. NETWORK GAIN AND PHASE IN TERMS OF Z 



A first step in the 2-plane formulation of the interpolation problem is' 

 the formulation of the network gain and phase functions, (1) and (2), in 

 terms of z. This is most usefully examined as a transformation of func- 

 tional form, rather than as a conformal mapping. 



The gain and phase function (1) transforms as follows: The analytic 

 relation between p and z is regular in the neighborhood of the singular- 

 ities Pa of the network function. Therefore, there will be similar singu- 

 larities of the transformed function at the ^-plane mappings of p„ ^ 

 which are z, and — l/z, . These singularities, and also suitable behavior 

 at infinity, are exhibited by the following expression for a -{- f/3 as a 

 function of z. 



XT is used here to designate a product of factors of the type following it.f 

 t The expression is readily confirmed in the following very elementary man- 

 ner: For every factor ( 1 — — j , in (9), there is also a factor (l + — ) • The 

 product of the two maj' be expanded as follows: 



.--in + i =i 



'^ ' '•■ (io> 



