Sec. 7.2] OPTIMIZATION, INDEFINITE CASE 63 



transmission lines are related to the terminal voltages and currents by 



ai =i(Fi+7i); b2 = HV^ -\- h) 

 b, =i(^i --^i); cL2 = i{V2-h) 



These transformations are conveniently summarized in matrix form. We 

 define the matrix 



R=5 



^[1 -1] (-^) 



We then have 



The general-circuit-parameter representation of the network is 



V - Tu = 8 (5.2) 



The matrix equation for the new choice of variables v' and u' has the 

 general form 



v' - T'u' = 8' (7.14) 



The relations between the matrices T' and 8', on the one hand, and T 

 and 8, on the other hand, are easily derived by using Eqs. 7.12 and 7.13 

 in Eq. 5.2 above. We obtain 



T' = RTR-i (7.15) 



5' = R8 = i 



2 



'8i + 62 

 _5i — §2 



] = B:'] (^-^^^ 



Let us now rewrite the noise measure, Eq. 6.25, in terms of the matrices 

 T' and 8'. For this purpose we note that, from the definition of Eq. 7.11 

 for R, we have 



R = R+ = jR-i (7.17) 



Using Eqs. 7.16 and 7.17, we can write the noise matrix 88^ as 



88^ = R-^8^R-i = 4R8^R^ (7.18) 



Furthermore, we have from Eqs, 7.15 and 7.17 



P - TPT^ = P - R-iT'RPR+T'^(R-^)+ 



= 2R[2RPR+ - T'(2RPR^)T't]R^ 

 = 2R(P' - T'P'T'+)R+ 



