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