6 INTRODUCTION [Ch. 1 



The proof of this criterion will be developed through the concept ot 

 noise measure. Inasmuch as the general criterion involves (at least) 

 arbitrary lossless interconnections of amplifiers, including feedback, 

 input mismatch, and so forth, a rather general approach to the noise 

 measure is required. In particular, we must show that the noise measure 

 has a real significance of its own which is quite different from and much 

 deeper than the one suggested by its appearance in Eq. 1.2. There it 

 appears only as an algebraic combination of noise figure and available 

 gain that happens to be convenient for describing amplifier cascades. 

 Here the properties of M with regard to lossless transformations are be- 

 coming involved. 



Consideration of these properties brings us into the entire general 

 subject of external network transformations of noisy Hnear networks. 

 Among these, lossless transformations form a group in the mathematical 

 sense. The quantities invariant under the group transformations must 

 have a physical significance. Investigation of these invariants forms a 

 substantial part of the present study. To be sure, for the special case of 

 a two-terminal-pair amplifier, the optimum noise performance, through 

 its related noise measure, turns out to be one of the invariants; but several 

 other interpretations of the invariants prove equally interesting, and 

 the development of the entire subject is simplified by presenting them 

 first. 



The simplest formulation and interpretation of the invariants of a 

 linear noisy network result from its impedance representation. The 

 following chapter is therefore devoted to a discussion of network trans- 

 formations, or "imbeddings," in terms of the impedance-matrix repre- 

 sentation. The concept of exchangeable power as an extension of 

 available power is then introduced. 



In Chapter 3, the n invariants of a Hnear noisy w-terminal-pair network 

 are found as extrema of its exchangeable power, with respect to var- 

 iations of a lossless w-to-one-terminal-pair network transformation. It 

 is found that an w-terminal-pair network possesses not more than these 

 n invariants with respect to lossless w-to-w-terminal-pair transformations. 

 These n invariants are then exhibited in a particularly appealing way in 

 the canonical form of the network, achievable by lossless transformations 

 and characterized by exactly n parameters. This form is introduced in 

 Chapter 4. 



Through Chapter 4, the invariants are interpreted only in terms of the 

 extrema of the exchangeable power. New interpretations are considered 

 next. They are best introduced by using other than the impedance- 

 matrix description. Accordingly, in Chapter 5, general matrix representa- 

 tions are studied, where it is pointed out that usually a different 



