20 



CHARACTERISTIC-NOISE MATRIX 



[Ch.3 



where the (real) column matrix | has every element zero except the i\h, 

 which is 1 : 



^i =0, j r^i 



^i = 1 



(3.2) 



Matrix | can be visualized as a double-pole w-throw selector switch which 

 chooses electrically only one of the n terminal pairs according to its sole 

 nonzero element. 



The variation of the lossless network in Fig. 2.4 now corresponds to 

 variation of the transformation network Zt in Fig. 2.2 through all possible 

 lossless forms. We wish to find the stationary values of Pe,i correspond- 

 ing to variation of Zy, To render explicit this variation, E' is first 

 expressed in terms of the original E and Z, using Eq. 2.10. Accordingly, 



where 



E'E'+ = Z,a(Z + Z„„)-lEEt[(Z + Zaa)-^]^Z,/ = T^EE+T (3.3) 



(3.4) 



T+ = Z6a(Z + TaaY^ 



Second, expressing Z' in terms of Z by means of Eq. 2.9 yields 



Z -f Z ''" = — Z6a(Z + Zoo)'~'^Za6 + T^hh + Z^ft 

 -Z„6^[(Z + Z„a)-1]+Z,/ 



(3.5) 



The conditions of Eq. 2.5, guaranteeing that the transformation network 

 is lossless, convert the foregoing relation to 



TJ 4- Z't = Z6a[(Z -f Zaa)-^ + (Z+ ^ Za^T'^Zj 



= Z6a(Z + Zaa)-'[{Z^ + ZaJ) + (Z + Zaa)] (Z^ + ZaJ)-'Zj 

 = T^(Z + Zt)T (3.6) 



It follows that 



p . = 



1 (iV)EEt(Tl) 

 2(|V)(Z + Z+)(t|) 



(3.7) 



in which matrix t (not |) is to be varied through all possible values 

 consistent with the lossless requirements upon the transformation 

 network. 



