62 NETWORK REALIZATION OF OPTIMUM PERFORMANCE [Ch. 7 

 The characteristic-noise matrix of this amplifier can be computed to be 



N = J(P - TPT+)-W 



2 w 



1 



4(W^ - 1) 



-Eni 



2 w^ - 1 



+ En\ In\ 



\Enl\ + 



-Snl Inl 



2|wP - 1 



-1- \T 2 



En\Inl + 



In\ 



2 kr - 1 



(7.8) 



(6.25) 



2|w|2 - 1 

 The amplifier noise measure in matrix form is given by Eq. 6.25 



yWy 1 



' y+(P - TVT^)y IUTq ^f 



where y is a column vector fulfilling the condition: 



yi 



in which Zs is the source impedance. 



The noise measure of Eq. 6.25 reaches its least positive value when 

 the vector y is equal to that eigenvector y^^^ of N which pertains to the 

 positive eigenvalue of N. The vector y is adjusted by an adjustment of 

 the source impedance Zs. Hence, the noise measure can be optimized by 

 a lossless impedance-matching network at the input of the amplifier if 

 the actual source impedance with a positive real part can be transformed 

 into the value Zs'^^ prescribed by the eigenvector y^^\ 



«»-©■ 



(7.9) 



Thus, if the noise measure is to be optimized by a lossless mismatching 

 network, it is necessary and sufficient that 



Re 





Re {Zs^^n > 



(7.10) 



The proof that the inequality (Eq. 7.10) is fulfilled for \u\ > 1 will now 

 be carried out. 



The proof is greatly facilitated if we use a wave formalism rather than 

 the voltage-current formalism. We assume that transmission lines of 

 1-ohm characteristic impedance are connected to the amplifiers. The 

 incident waves ax and a^ and the reflected waves hi and 62 on these 



