Sec. 7.2] 



OPTIMIZATION, INDEFINITE CASE 



65 



The positive eigenvalue of N' is 



x.=^ 



wr-1 



"1 "2 



I // Is /|2 Is /|2\ 1/1 r*" /I? /|2h; /|2 |s /? /* 2\ 



(7.26) 



From the definition of the eigenvector, the matrix equation results: 

 of which the second component is 



-b^'H^W^ - WVW2^'^ = \,W2^'^ 



and thus 





U) 



-25/*52 



/*x / 



«P-1 



? '72 I Ts'T2 I 



\0l +02 + 



,WP-1 



, ^ , ^\2 



Isj / 2 J Is / 2 I 

 lOl I +|02 I ) -' 



wr-1 



|5iV*r 



(7.27) 



But |w| > 1, and the eigenvalue Xi is real. Therefore, from Eq. 7.26 



|w| - 

 which, in Eq. 7.27 yields 



Oi 02 



< 



(112 \ 



1^1 Is /|2 I Is '121 



.12 _ 1 I5l I + 1^2 1 J 



^2 



(1) 



Wl 



(1) 



< 1 



(7.28) 



The eigenvector w^^^ for which the noise measure is optimized thus 

 corresponds to a reflection coefficient less than unity, that is, to a passive 

 source impedance. The condition in Eq. 7.28 is equivalent to the condi- 

 tion, Eq. 7.10, in the general-circuit-matrix notation. We have thus 

 proved that the noise measure of a unilateral amplifier can be optimized 

 with a lossless mismatching network between the source (having an 

 impedance with a positive real part) and the amplifier. The output im- 

 pedance of this amplifier always has a positive real part. It follows that 

 any number of optimized amplifiers, with det (P — TPT'^) < 0, can be 

 cascaded with appropriate lossless mismatching networks between suc- 

 cessive stages so as to achieve an arbitrarily high gain. The excess-noise 



