Canonical Form of Linear Noisy Networks

Abstract

At any single frequency, every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n</tex> -terminal-pair network, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> of whose terminal pairs are connected to those of the original network, and the remaining <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> of which form a new set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> invariants. The canonical form consists of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> exchangeable powers from the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> isolated terminal pairs are the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> 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.