【作者】 赵辉；

【导师】 冉启文；

【作者基本信息】 哈尔滨工业大学 ， 物理电子学， 2011， 博士

【摘要】 经典分数傅里叶变换和线性正则变换理论及应用的研究是信息光学的一个热点研究方向,二者的光学引入使得人们可以用一个新的观点去审视光的传播、成像、信息处理等问题,并提供了新的工具去处理这些问题。在以经典分数傅里叶变换和线性正则变换为工具进行光学信息处理的过程中,广义4??系统是最典型的一种光学信息处理系统。光学系统的本征函数是通过该系统后除了一个常数增益外保持不变的复函数。如果系统的本征函数构成系统输入的正交基且此系统是线性的,则整个系统的作用就可以通过其本征函数简单而有效地描述。因此,光学系统的本征函数理论分析具有非常重要的意义。本文深入研究广义4??系统本征函数理论,讨论广义4??系统本征函数及相应本征值的性质和数值计算方法,借助本征函数分析广义4??系统和光学信号,为光学信息处理的进一步发展提供新的工具、方法及理论基础。本文首先证明了广义4??系统的本征函数为广义扁长椭球波函数。给出了线性正则变换域带限信号空间的再生核函数并利用其特殊性质给出了此空间的两组采样基,从而得到了线性正则变换域带限信号的均匀和非均匀采样定理。此外,借助计算机仿真模拟验证了所给采样定理的正确性与有效性。随后,借助采样理论研究了广义4??系统本征函数理论。通过将连续本征问题转化为等价的离散本征问题,得到了广义4??系统本征函数系在线性正则变换域带限信号空间上的正交基性质;利用算子理论给出了广义4??系统本征函数系在有限区间内能量有限信号空间上的正交基性质。此外,采样理论分析还给出了广义4??系统本征函数的数值计算方法及本征值的正实数性和近似阶梯性。计算机仿真模拟结果证实了所提计算方法的有效性。借助广义4??系统的本征函数描述了广义4??系统对系统输入的作用,给出了系统的空间带宽积和逆问题的求解方法;讨论了广义4??系统的能量保持率问题,证明了零阶广义扁长椭球波函数及其空限形式分别是线性正则变换域带限信号空间和能量有限信号空间中通过广义4??系统后能量损失最小的信号,并分别给出了其能量保持率。此外,还指出零阶广义扁长椭球波函数是所有线性正则变换域带限信号中在空域具有最大能量聚集性的信号。最后,借助广义4??系统的本征函数分析了光学信号。给出了线性正则变换域带限信号基于广义扁长椭球波函数的采样定理和外推方法,并借助计算机仿真模拟结果证实了所得结果的正确性和有效性;给出了线性正则变换域带限信号基于有限非均匀采样的最小均方误差重构公式及迭代算法,讨论了最小均方误差重构的性质,结果表明:通过最小均方误差重构可以尽可能近似地恢复原连续信号。更多还原

【Abstract】 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.更多还原