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In this article, the definition on the observe ability and its relation tothe signal detecting performance are studied systematically for the lineardiscrete-time(LDT) systems. Firstly, to define and analyze the observe abilityfor the practical systems with the measured noise, six kinds of bounded noisemodels are classified. For the noise energy bounded case, the observabilityellipsoid and the image observability ellipsoid are defined by the stateobserved error and then a novel concept on the LDT systems, called as theobserve ability, is proposed. Based on that, some theorems and properties aboutthe observe ability and the signal detecting performances are given and proven,and then the reason that to maximize the observe ability is to optimize thesignal detecting performances is established. Secondly, a dual relation betweenthe observability ellipsoid and the controllability ellipsoid, which volumesand radii are respectively with some inverse relations, is stated and proven.Accordingly, the analytical computing equations for the volume of the twoobservability ellipsoids are got and some analytical shape factors of theseellipsoids are deconstructed. Based on these effective compting for thevolumes, radii, and shape factors, analyzing and optimizing for the observeability can be carried out. Thirdly, to compare rationally the state observeability between the different systems or different system parameters, thenormalization of the output variables, the state variables, and the systemmodels are discussed. Finally, some numerical experiments and their resultsshow the effectiveness of the computing and comparing methods for the observeability.

Definition and Analytical Expression on State Observe Ability for Linear Discrete-time Systems with the Bounded Noise Energy