【作者】 刘庆；

【导师】 戴青云； 凌永权(Bingo Wing-Kuen Ling)；

【作者基本信息】 广东工业大学 ， 信息与通信工程， 2020， 博士

【摘要】 滤波理论在现代信息技术领域占据着举足轻重的地位,信号的获取、传输,以及处理和存储,都离不开滤波技术。通过优化方法来设计多速率滤波器组,很大程度上改善了滤波性能,为最优时频分析等相关领域提供了全新的思路。但是滤波器组优化设计问题是一个无限约束条件优化问题。如何提出快速有效的优化算法来设计滤波器组,并保证这些无限个约束条件获得满足是该领域公认的核心关键问题。本文主要针对多速率滤波器组优化设计的若干个问题进行研究。包括多速率滤波器的结构设计、优化问题求解和应用验证。整篇论文包括三个方面内容:1、针对常规的最大抽取M通道仿酉镜像线性相位滤波器组的仿酉条件定义在频域中无法获得最优解的难题,重新定义仿酉条件,并通过优化方法求得最优解;2、针对常规采样的非兼容非均匀滤波器组无法实现完美重构问题,采用基于块采样的方式实现非兼容非均匀滤波器组最优设计,并在多路复用器中应用验证;3、针对目前多种滤波器组采样结构是对称的无法完美重构的问题,研究了一种新的基于周期窗函数的滤波器组的结构设计,将多种结构滤波器组设计进行推广。论文的主要贡献总结如下:1)提出了一种最大抽取M通道仿酉镜像线性相位FIR滤波器组的最优化设计方法。由于常规最大抽样M通道仿酉镜像线性相位FIR滤波器组设计的仿酉条件定义在频域中,频域是一个连续集,滤波器的频率响应被表示为一组三角函数的高阶多项式,很难找到优化问题目标函数的梯度,因此,最优滤波器组的设计问题是不可追踪的,这给滤波器优化设计带来计算量和频率选择性等方面的困扰。为此,本文提出了一种新的解决思路。首先将这一条件表示为有限个离散的关于采样频率点处的函数方程,把查找最优旋转角度的问题转换成滤波器的最优频率响应采样问题,要求对滤波器的幅度响应采样的点数大于滤波器长度,将滤波器组的设计问题转化为一个仿酉镜像总误差在L1范数下最小化的优化问题。但是这个优化问题是一个非凸问题,求解困难。为此,采用范数松弛的序列二次规划方法寻找其局部最优解,通过使用不同的初始条件迭代上述过程,得到了一个近似全局最优解。计算机数值模拟结果表明,该设计方案优于现有设计方案。2)提出了一种基于块采样的单输入单输出线性时不变非均匀多路复用器的最优设计方法,该方法能够实现近似完美重构。基于常规采样的非兼容非均匀滤波器组无法实现完美重构,只能通过多输入多输出线性时不变系统或单输入单输出线性时变系统来设计,但这种设计方法难以在实现代价和完美重构误差之间取得平衡。为此,本文阐述了单输入单输出线性时不变系统和基于块采样的非均匀多路复用器的完美重构条件,并基于此条件把滤波器组的设计问题转化为一个完美重构误差在L1范数下最小化的优化问题,通过求解优化问题得到基于块采样的单输入单输出线性时不变非均匀多路复用器的设计方案。计算机数值模拟结果表明,所设计的基于块采样的非均匀多路复用器对信道噪声具有较强的鲁棒性,重构误差较小。3)提出了一种周期窗函数的滤波器组结构设计。针对块采样非均匀分析滤波器组采样结构和块采样非均匀综合滤波器组的采样结构是对称的,无法实现精确完美重构。首先,本文提出了一种窗滤波器组,它由一组分析滤波器、一组分析周期窗、一组综合滤波器和一组综合周期窗组成。当综合周期窗函数为恒定时,窗滤波器组成为常规采样非均匀滤波器组和块采样非均匀滤波器组的推广。当综合周期函数为时变函数时,综合周期窗函数滤波器组是非对称结构块非均匀滤波器组的推广。从而使信号能够用更灵活的时频分解方法生成子带系数。其次,研究了恒定综合周期窗函数和时变综合周期窗函数的滤波器组,推导了其精确完美重构条件。并在此基础上,进一步对各类结构的滤波器组的仿酉条件进行了研究,并分别给出简单实例验证。综上所述,本文基于多速率滤波器组设计的若干问题,结合最优化方法展开研究,其内容是信息科学与数理科学交叉性的创新研究,相信对于发现特性和规律,并解决实际应用问题具有重要的研究意义。更多还原

【Abstract】 Filtering theory plays an important role in the field of modern information technology.It is an indispensable technology for the acquisition,transmission,processing and storage of signals.Through the optimization algorithm to design the filter,the actual demand index and filter performance are improved to a great extent,which provides a new idea for the optimal time-frequency analysis and other related fields.However,the optimal design of filter banks is an optimization problem with infinite constraints.How to propose a fast and effective optimization algorithm to design digital filters and filter banks,and ensure that these infinite constraints are satisfied is the core and key problem in this field.In this paper,several problems on the optimal design of filter banks are studied.It includes the structure design,solving the optimization problem and the application of the filter.The whole paper includes three aspects:(1)Aiming at the problem that the traditional maximum decimation M-channel mirror paraunitary filter bank can’t get the optimal solution in the frequency domain,the paraunitary condition is redefined and the optimal solution is obtained by the optimization method;(2)The block sampling method is adopted to solve the problem that the traditional sampling incompatible non-uniform filter banks cannot achieve perfect reconstruction.Further more,the optimal design of incompatible non-uniform filter banks is realized and verified in multiplexers.(3)Aiming at solve the problem that the structures of many filter banks are symmetrical and can not be reconstructed completely,a new structure design of filter banks based on periodic window function is studied,whichextends the design of multiple structure filter banks.The main contributions of this dissertation are summarized as follows.1)An optimal design method of maximum decimated M-channel mirror paraunitary matrix linear phase FIR filter banks is proposed.The traditional maximum sampling M-channel mirror paraunitary matrix linear phase FIR filter banks are defined in the frequency domain.Since the frequency domain is a continuous set,and the frequency response of the filter is expressed as a set of high-order polynomials of a set of trigonometric functions,it is difficult to find the gradient of the objective function of the optimization problem.Therefore,the problem of optimal filter bank design is not traceable,which brings difficulties to filter optimization design in terms of computational complexity and frequency selectivity.To solve these problems,we propose a new solution.Firstly,this condition is expressed as a finite number of discrete function equations at the sampling frequency points.The problem of finding the optimal rotation angle is transformed into the problem of optimal frequency response sampling of the filter.The number of sampling points for the amplitude response of the filter is required to be greater than the filter length.The filter design problem is transformed into an optimization problem of minimizing the total mirror paraunitary moment position error under L1 norm.However,this optimization problem is a non convex problem,which is difficult to solve.Therefore,the norm relaxed sequential quadratic programming method is used to find the local optimal solution.By iterating the above process with different initial conditions,an approximate global optimal solution is obtained.The numerical simulation results show that the design scheme is better than the existing state-of-art methods.2)An optimal design method of single input single output linear time invariant nonuniform multiplexing filter banks based on block sampling is presented.This method can achieve approximately perfect reconstruction.The traditional sampling based incompatible nonuniform filter banks can not achieve perfect reconstruction.They can only be designed by multiple inputs multiple outputs linear time invariant filters or single input single output linear time-varying systems.However,this design method is difficult to balance the implementation cost and the perfect reconstruction error.In this dissertation,the perfect reconstruction conditions of single input single output linear time invariant filter and non-uniform multiplexing filter based on block sampling are described,and a perfect reconstruction error under L1 norm is established based on this condition.By solving this optimization problem,the design scheme of the non-uniform multiplexer with single input and single output linear time invariant filter based on block sampling is obtained.The numerical simulation results show that the designed non-uniform multiplexer based on block sampling has strong robustness to channel noise and small reconstruction error.3)A filter bank structure design with periodic window function is proposed to solve the problem that the block sampling nonuniformity analysis filter bank and the block sampling nonuniformity synthesis filter bank can not achieve accurate and perfect reconstruction.That is because their sampling structures are symmetrical.Firstly,a window filter bank is proposed,which consists of a group of analysis filters,a group of analysis periodic windows,a group of comprehensive filters and a group of comprehensive periodic windows.When the comprehensive periodic window function is constant,the window filter is composed of the traditional sampling non-uniform filter bank and the block sampling non-uniform filter bank.When the periodic function is time-varying,the integrated periodic window function filter bank is a generalization of asymmetric block nonuniform filter banks.Thus,the signal can be decomposed into subband coefficients by a more flexible time-frequency decomposition method。Secondly,the filter banks with constant comprehensive periodic window function and time-varying comprehensive periodic window function are studied,respectively,and their exact perfect reconstruction conditions are derived.Further more,the paraunitary conditions of periodic window functions are further studied,and some representative examples are given to verify them.In summary,this dissertation studied several key problems of multirate filter bank design,through the optimization method to carry out the research.These contents are the innovative research on the intersection of information science and mathematical science.It is believed that it has important research significance for discovering characteristics and laws and solving practical application problems.更多还原