Sec. 4.1] DERIVATION OF THE CANONICAL FORM 29 



{possibly negative) resistances in series with uncorrelated noise voltage 

 generators. 



We note first that a lossless imbedding of the w-terminal-pair network 

 transforms the two matrices Z + Z^ and EE^ in identical colinear manner, 

 as shown in Eqs. 3.3 and 3.6. We have also noted that the matrix t which 

 appears in the transformation is entirely unrestricted by the conditions 

 (Eqs. 2.5) of losslessness for the imbedding network. 



It is always possible to diagonalize simultaneously two Hermitian 

 matrices, one of which (eW) is positive definite (or, as a limiting case, 

 semidefinite), by one and the same colinear transformation. Thus, 

 suppose that both Z' + Z'^ and E'E'^ have been diagonalized by a 

 proper imbedding of the original network (see Fig. 2.2). This means 

 that the impedance matrix Z' of the resulting network is of the form 



Z' = diag (Ru R2,--, Rn) + Zrem (4.1) 



where Zrem fulfills the condition of the impedance matrix of a lossless 



network Zrem = -(Zrem)^- 



Suppose, finally, that a lossless (and therefore noise-free) network with 

 the impedance matrix —Zrem is connected in series with our network 

 (Z'j E') , as shown in Fig. 4.1. The result is a network with the impedance 

 matrix 



Z" = Z' - Zrem = diag (i?i, R2, --■,Rn) (4.2) 



The open-circuit noise voltages remain unaffected when a lossless source- 

 free network is connected in series with the original network. Thus, 



E'' = E' or E"E''+ = E'E'"^ (4.3) 



Consequently, the two operations lead to a network with the diagonal 

 impedance matrix Z" of Eq. 4.2 and a diagonal noise matrix E"E''^. 

 This canonical form of the network consists of n separate resistances in 

 series with uncorrelated noise voltage generators, as shown in Fig. 4.2. 



Noting that the series connection of a lossless network is a special case 

 of a lossless imbedding, we have proved the theorem stated at the begin- 

 ning of this section. 



A lossless imbedding leaves the eigenvalues of the characteristic-noise 

 matrix invariant. Thus, the eigenvalues of the characteristic-noise 

 matrix N'' of the canonical form of the original network are equal to those 

 of N of the original network. But, the eigenvalues X„ of N" are clearly 

 its n diagonal elements 



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