Canonical Form of Linear Noisy Networks

At any single frequency, every
n
-terminal-pair noisy linear network has at most n real parameters that are invariant with respect to all lossless "imbeddings" of that network. Such an "imbedding" is defined by constructing an arbitrary lossless
2n
-terminal-pair network,
n
of whose terminal pairs are connected to those of the original network, and the remaining
n
of which form a new set of
n
terminal pairs. Moreover, by a suitable choice of this imbedding structure, the original network can always be reduced to a canonical form which places clearly in evidence its
n
invariants. The canonical form consists of
n
isolated one-terminal-pair networks each of which comprises a (negative or positive) resistance in series with a noise voltage generator, and these various noise generators are mutually uncorrelated. The
n
exchangeable powers from the
n
isolated terminal pairs are the
n
invariants of the original network. The invariants have other physical meanings. Each meaning is best brought out by a corresponding particular matrix description of the network. Transformations between matrix descriptions are studied and applied to show that the invariants are interpretable as the
n
stationary values of the exchangeable power obtainable from any one of the new terminal pairs created by a lossless imbedding, as the imbedding network is varied through all lossless forms. Finally, the two invariants of a two-terminal-pair network are shown to fix the extrema of its noise measure, one of which is known to represent, for an amplifier, the minimum excess noise figure achievable at high gain.