550 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



construction of a finite passive network which, as a 2w-pole, has the im- 

 pedance matrix Z(p) — i.e. is a 2/?-pole equivalent to N. We call such 

 a network a (physical) realization of N, or of Z(p). The dual probkun, 

 that of realizing a PR admittance matrix, can be handled dually. 



Let Zo(p) = Z{p), No = N, no = n. We describe an inductive proce- 

 dure which, given a 2nr-pole N^ , r > 0, either 

 (i) Constructs a physical realization of N^ , or 



(ii) Constructs a 2nr+i-pole Nr+i such that if N^+i is physically real- 

 izable, then Nr is. 



To show that this induction actually gives a realization of any PR 

 matrix Zo{p) we must demonstrate that, first, it is effective — i.e. that 

 at any stage N^ at least one of (i) and (ii) is possible. Second, we must 

 show that the process terminates with the construction of a finite net- 

 work. The details of these demonstrations are given in the paragraphs 

 4.1 et seq. of this section. In the paragraphs 4.01 to 4.07 we describe the 

 logical pattern into which these details are to be fit when they are 

 established. 



4.01 There are nine basic operations by which the networks N^ are con- 

 structed. We name the operations here, in order to give a clearer picture 

 of the logic of the process, but their mathematical treatment is deferred 

 to later paragraphs. 



IP: A PR impedance matrix Zr(p) which has poles on p = ioi is 

 represented as 



1 2o 



P k P -T (^k 



where Zr+\{p) is PR and has no poles on p = ico. 

 AP: A PR admittance matrix Yr{p) is represented dually: 



1 2v 



P k P -V <J^k 



ID: A PR impedance matrix Zr{p) is represented as W'Zr+i{p)W, 



where Zf+i{p) is a non-singular Zr+i(p) bordered by zeros. 

 AD: Dual to ID. 

 Res: A PR matrix Zr(p) is represented as 



Zrip) = aS + Zr+l(p), 



where S is real, constant, symmetric, and positive definite, 

 and a > is the largest a for which Zr+\{p) is PR. 

 Con : The dual to Res. 



