Research On The Eigenfunctions Theory Of The Generalized 4F System And Its Applications

The theoretic and applied research on classical fractional Fourier transform(CFRFT) and linear canonical transform (LCT) is a hot subject in information optics.Their optical introductions make people survey the problems of optical propagation,imaging and information processing by a new viewpoint, and provide a new tool to dealwith these problems. During the optical information processing by use of CFRFT andLCT, the generalized 4f system is the most classical optical information processing sys-tem. The eigenfunctions of optical systems are the complex functions which maintainunchanged except for a constant when passing through the systems. If the eigenfunctionsform an orthogonal basis for the input of a system and the system is linear, then the effectof the whole system can be simply and effectively characterized by its eigenfunctions.Therefore, the eigenfunctions theory analysis of optical systems has important meanings.This dissertation makes an intensive study of the eigenfunctions theory of the generalized4f system, discusses the properties and numerical calculation method of the eigenfunc-tions of the generalized 4f system, analyzes the generalized 4f system and optical signalsby use of eigenfunctions, so as to provide new tools,methods and theory basis for thefurther development of optical information processing.First, the fact that the eigenfunctions of the generalized 4f system are generalizedprolate spheroidal wave functions (GPSWFs) was proved. The reproducing kernel func-tion for the space of bandlimited signals in LCT domain was given, and two samplingbasis of the same space were also given by use of the special property of the kernel func-tion. As a result, the uniform and nonuniform sampling theorems for bandlimited signalsin LCT domain were obtained. Besides, by use of computer simulations, the correctnessand effectiveness of the proposed sampling theorem were verified.Then,the eigenfunctions theory of the generalized 4f system were studied by useof sampling theory. By translating continuous eigenproblem to equivalent discrete eigen-problem, the orthogonal basis property of the eigenfunctions of the generalized 4f systemover the space of bandlimited signals in LCT domain was given, and by operator theorythe orthogonal basis property of the eigenfunctions of the generalized 4f system overthe space of finite energy signals on finite interval was obtained. Besides, the sampling analysis also provides a numerical calculation method for the eigenfunctions of the gen-eralized 4f system and the positive real value property and approximate step property ofthe eigenvalues. The effectiveness of the proposed calculation method was verified bycomputer simulation results.The effect of the generalized 4f system on system input was characterized by useof the eigenfunctions of the generalized 4f system, the space-bandwidth product and theresolution of inverse problem for the system were given; the energy preservation ratioproblem of the generalized 4f system was discussed. It is shown that the zero-order GP-SWF and its space limited version have the minimum energy lost when passing throughthe generalized 4f system among the classes of bandlimited signals in LCT domain andof finite energy signals over finite interval, respectively. Their energy preservation ra-tios were also given. Besides, it is also shown that among all the bandlimited signals inLCT domain, the zero-order GPSWF has the maximum energy concentration in the spacedomain.Finally, the optical signals were analyzed by use of the eigenfunctions of the gen-eralized 4f system. The sampling theorem and extrapolation method based on GPSWFsfor bandlimited signals in LCT domain were given, and the correctness and effectivenessof the obtained results were verified by use of computer simulation results; the mini-mum mean-squared error reconstruction (MMSER) formula and its iterative algorithmfor bandlimited signals in LCT domain from finite nonuniform samples were given, andthe properties of the MMSER were discussed. It is shown that the MMSER can reproduceoriginal signal as closely as possible.