Showing papers on "Biorthogonal system published in 1972"

Solutions of the fokker--planck equation in detailed balance.

Abstract: The solutions of the Fokker-Planck equation in detailed balance are investigated. Firstly the necessary and sufficient conditions obtained by Graham and Haken are derived by an alternative method. An equivalent form of these conditions in terms of an operator equation for the Fokker-Planck Liouville operator is given. Next, the transition probability is expanded in terms of an biorthogonal set of eigenfunctions of a certain operatorL. The necessary and sufficient conditions for detailed balance leads to a simple operator equation forL. This operator equation guarantees that on!y half of the biorthogonal set needs to be calculated. Finally the dependence of the eigenvalues on the reversible and irreversible drift coefficient is discussed.

L-Orthogonal Signal Transmission and Detection

Abstract: This paper investigates the detail capabilities and performance characterization of systems that employ L -orthogonal signaling techniques. L -orthogonal signals represent a unified set of signals wherein the polyphase and orthogonal (biorthogonal) signal sets are included as special cases. This fact is important since orthogonal (biorthogonal) and polyphase signaling sets represent opposing forces as far as tradeoffs between error probability, energy-to-noise ratio, and bandwidth requirements are concerned. Bounds on the performance of the optimum receiver and the performance of various suboptimum (practical) receiver structures are given. Coherent and differentially encoded signals are also pursued. Various comparisons and tradeoffs are made by means of numerical evaluation of the error-probability expressions.

Mixed Conjugate Finite-Element Approximations of Linear Operators

Abstract: The concept of conjugate approximation functions is generalized to mixed formulations of linear boundary value problems. Biorthogonal basis functions are generated using two distinct finite-element models of the domain of a given function. Various special cases are considered. It is shown that four distinct models can be developed corresponding to a single linear operator. A comparison of the concepts of the hypercircle and conjugate approximation is given.