The causal system is more practical then the noncausal system in the world. Causality implies only the past input can effect the future output. As a consequence, noncausal system is seldom investigation. The purpose of this thesis is to study the signal recury for a noncausal system. The principle of signal estimation is based upon the Wiener-Hopf equation. Therefore, the correlation computation is very important. By transforming the noncausal dynamic equations to a causal equation, we achieve a partial recursive computation structure for correlation computation. However the current input is not independent of the past signal in the noncausal system. Hence, the Mason Rule is applied to solved this problem to make the above recursive structure complete. Furthermore, a recursive computation of Mason Rule for stage propagation is developed in this thesis to accelerating the processing speed. Our algorithm is applied to image restoration. We first segment the image to find the required generating input ponen for each correlated region. Secondly, we extend our 1-D algorithms to 2-D algorithm to restore the image. Our method is compared with the method developed base upon the Gaussian Markov model. The experiments results demonstrate the advantage of method in both visual quailty and numerical results.

In this thesis, the asymptotic behavior and oscillation of solutions of dynamic equations on time scales are studied. In the first part of the thesis, asymptotic equivalence and asymptotic equilibrium of dynamic systems are investigated. Sufficient conditions are established for the asymptotic equivalence of linear systems and linear and quasilinear systems, respectively, and for the asymptotic equilibrium of quasilinear systems by unifying and extending some known results for differential systems and difference systems to dynamic systems on arbitrary time scales. In particular, for the asymptotic equivalence of differential systems, the well-known theorems of Levinson and Yakubovich are improved and the well-known theorem of Wintner for the asymptotic equilibrium of linear differential systems is generalized to arbitrary time scales. Some of our results for asymptotic equilibrium are new even for difference systems. In the second part, the oscillation of solutions of a particular class of second order nonlinear delay dynamic equations and, more generally, two-dimensional nonlinear dynamic systems, including delay-dynamic systems, are discussed. Necessary and sufficient conditions are derived for the oscillation of solutions of nonlinear delay dynamic equations by extending some continuous results. Specifically, the classical theorems of Atkinson and Belohorec are generalized. Sufficient conditions are established for the oscillation of solutions of nonlinear dynamic systems by unifying and extending the corresponding continuous and discrete results. Particularly, the oscillation criteria of Atkinson, Belohorec, Waltman, and Hooker and Patula are generalized.

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在這篇論文中，我們討論的是一階非線性衝擊動態方程的週期邊界值問題。利用Schaefer定理及Banach固定點定理，我們得到一些解的存在性結果。 / In this thesis, we are concernd with nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales. By using Schaefer’s theorem and Banach’s fixed point theorem we acquire some new existence results.