Canonical Form 



of Linear Noisy Netv^orks 



Lossless network transformations performed on a noisy network, in 

 such a way that the number of terminal pairs is unchanged, change the 

 impedance matrix as well as the noise spectra. However, these lossless 

 network transformations do not change the eigenvalues of the character- 

 istic noise matrix. Thus we know that each noisy network possesses 

 some essential noise characteristics, unalterable by those lossless network 

 transformations which preserve the number of terminal pairs. On this 

 basis, we expect to be able to find a fundamental form of the network 

 which places these characteristics directly in evidence. In this chapter, 

 we shall develop such a form of the network. This fundamental or "ca- 

 nonical" network form is, of course, attainable through lossless network 

 transformations performed on the original network. The existence of a 

 canonical form for every linear noisy network greatly clarifies its most 

 important noise characteristics and simplifies the study of fundamental 

 limits on its noise performance. Since the canonical network contains 

 not more than n real parameters for every w-terminal-pair network, its 

 existence also shows that an «-terminal-pair, linear noisy network does 

 not possess more than n (real) invariants with respect to lossless trans- 

 formations. 



4.1. Derivation of the Canonical Form 



In this section we shall prove the following theorem : 



At any particular frequency, every n-terminal-pair network can he re- 

 duced by lossless imbedding into a canonical form consisting of n separate 



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