Sec. 7.2] OPTIMIZATION, INDEFINITE CASE 61 



impedance matrix 



Z = 



I (^-3) 



where r is real, we obtain, for the unilateral gain (Eq. 7.1) of the com- 

 bined network, 



[Re(^)+rf + Im^(z^) . 



-1 +r2 + 2rRe(w) ^^^ 



In Eq. 7.4, U can always be made positive and greater than unity by 

 choosing 



r > + \/[Re (w)]2 + 1 - Re (w) 



Thus, the network of Fig. 7.16 can always be given a unilateral gain that 

 is positive and greater than unity. Then, according to Mason's work,^ 

 it can be unilateralized and brought into the form of Fig. 7.1a. 

 The amplifier in Fig. 7.1c, however, has 



det (P - TPT+) > (7.5) 



and cannot, under any terminal condition, absorb power. Obviously it 

 cannot be reduced to any of the other forms by any lossless transforma- 

 tion whatsoever. It is a "negative-resistance" amplifier. 



Recalling that the optimum noise measure is not changed by any loss- 

 less transformation, we conclude that noise-performance optimization of 

 amplifiers need only be carried out on networks of the specific forms 

 Figs. 7.1a and 7.1c, and such a procedure will be sufficient to include all 

 nonpassive cases. 



7.2. Optimization of Amplifier, Indefinite Case 



We imagine that the given amplifier with det (P — TPT^) < is 

 initially in, or is reduced to, the form of Fig. 7.1a. We shall show that 

 -^e.opt can be realized for this circuit by suitable input mismatch, retain- 

 ing positive source impedance. 



The general circuit matrix of the amplifier in Fig, 7.1a is 



and the noise column matrix (elements not shown in Fig. 7.1a) is 



6 = [t:] (") 



