38 



OTHER REPRESENTATIONS 



[Ch.5 



We have 



Comparison with Eq. 5.15 shows that 



Qt = R^QzR 



(5.18) 



where R is the matrix that transforms the general-excitation vector 



into the voltage and current vector 



, according to Eq. 5.7. 



We are now ready to set up the general characteristic-noise matrix N 

 for any matrix representation, Eq. 5.6. 



5.4. The General Characteristic-Noise Matrix 



We have introduced the most general matrix representation of a linear 

 network in Eq. 5.6. We have defined the associated power matrix Qt in 

 Eq. 5.15. With these two we may define a general characteristic-noise 

 matrix Nr corresponding to this matrix representation. The require- 

 ments are that this matrix Ny : 



1. Should reduce exactly to the form of Eq. 3.15 when the network is 

 described on the impedance basis. 



2. Should be related to Eq. 3.15 by a similarity transformation when 

 the same network is described on other than impedance basis (for example, 

 admittance, scattering, and so forth). 



Under these conditions, Ny will contain the n network invariants as its 

 eigenvalues. 



Here we shall follow the simple expedient of defining Nr and then 

 proving its relationship to the matrix defined by Eq. 3.15. Thus, let 



Ny = {[l I -t] Qt-' [l I -t]}~'88T (5.19) 



For the impedance-matrix representation, using Eqs. 5.1, 5.6, and 

 5.17, we obtain 



[l ; -Z] Qz-' [l ; -Zj = -2(Z + Z+) 

 Introducing Eq. 5.20 into Eq. 5.19, we have 



Nz = -J(z + z+)-^eeT 



(5.20) 

 (5.21) 



