262 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



15.16 If /(p) has a pole at p = «> , that pole is simple and 



fip) = pr + flip), 

 where r > 0, and fi(p) is PR. 



15.17 We shall use all of these in the next section, save 15.13. Our aim 

 is to prove properties analogous to 15.11, • • • , 15.16 for PR matrices 

 and operators. 



The reader familiar with the Brune process for realization of a 2-pole 

 will remember the importance of the properties 15.11, • • • , 15.16 for the 

 success of that process. Correspondingly, we must establish the analogs 

 of these properties to implement the general Brune process for 2ri-poles. 



XVI. POSITIVE REAL OPERATORS 



16.0 An operator Z{p) from K to V will be called positive real (PR) if in 

 some real coordinate frame the matrix [Z(p)] is a PR matrix in the sense 

 of 1.1 — that is 



(i) [^(p)] has rational elements Zrs(p) 



(ii) Zrsip) = ZrsiP) 

 (iii) Zrsip) = Zsrip) 



(iv) For any real /ceK and any peT+ 



ReiZip)k, k) >0. 



We intend in this section to establish for PR operators the properties 

 listed below. By subtracting 0.9 from the designation of each property 

 one obtains the designation of the analogous property of a PR scalar 

 function, stated earlier. 



16.01 Zip) has no poles in r+ . 



16.02 If Re(Z(p)/c, k) = for some peT+ , then Zip)k = for all p. 



16.03 If it exists, Z~\p) = Yip) is PR. 



16.04 If Zip) has a pole at p = po , it has one at p = po . 



16.05 If Zip) has a pole at p = tcoo , that pole is simple* and 



2p 



Zip) = -T-x^ R + Ziip), 



p -r Wo 

 where R is real, symmetric, and semi-definite, not zero, and Ziip) is PR. 

 * i.e., of order one. 



