54 NOISE MEASURE [Ch. 6 



noise measure. Large values of \Ge\ must correspond to large values of 

 conventional available gain. Therefore, the excess-noise figure at large 

 conventional available gain is limited to values greater than, or equal to, 

 \i/kTo A/ under the most general lossless external network operations on 

 the amplifier. These include, for example, lossless feedback, input mis- 

 match, and so forth. On the supposition that the noise figure at large 

 conventional available gain is a meaningful measure of the quality of 

 amplifier noise performance, the minimum positive value of the noise 

 measure, 



is a significant noise parameter of the amplifier. The significance of 

 ■^e.opt will be further enhanced by the proofs, given in the remaining 

 sections, of the following statements : 



1. The lower bound Me, opt on the noise measure of an amplifier can 

 actually be achieved by appropriate imbedding. Moreover, this is ac- 

 complished in such a way that subsequent cascading of identical units 

 realizes Me,opt as the excess-noise figure at arbitrarily high gain. 



2. An arbitrary passive interconnection of independently noisy ampU- 

 fiers with different values of Me, opt cannot yield a new two-terminal-pair 

 amplifier with an excess-noise figure at large conventional available gain 

 lower than Me, opt of the best component amplifier (the one with the 

 smallest value of Me, opt) • 



3. The use of passive dissipative imbedding networks for a given 

 amplifier driven with a positive source impedance cannot achieve a posi- 

 tive noise measure less than its Me, opt- 



We shall take up statements 2 and 3 first. 



6.5. Arbitrary Passive Interconnection of Amplifiers 



To prove statements 2 and 3 of Sec. 6.4, we begin by considering a 

 general lossless interconnection of n independently noisy amplifiers, as 

 shown in Fig. 6.7. A 2w-terminal-pair network results. By open- 

 circuiting all but two of the resulting terminal pairs, we obtain the most 

 general two-terminal-pair amplifier obtainable from the original ones by 

 lossless interconnection. In Sec. 4.2 we have developed the general 

 theory of such an imbedding and reduction of terminal pairs. Indeed, 

 Fig. 4.3 includes the situation of Fig. 6.7. We know that the eigen- 

 values of the characteristic-noise matrix of the network lie between the 

 most extreme eigenvalues of the characteristic-noise matrices of the 

 original n amplifiers. Therefore the lowest positive eigenvalue of the 



