基于Delta算子的自适应滤波及应用
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摘要
Delta算子方法是一种新的离散化方法，不仅避免了在高速采样时常规的移位算子方法引起的病态条件问题，而且当采样周期趋于零时，Delta算子离散模型趋近于原来的连续模型，从而使得传统的连续域设计方法可直接用于离散域设计，在高速信号处理与数字采样控制方面具有重要的应用价值。
本论文是作者在参加河南省自然科学基金《基于Delta算子的自适应逆控制系统的鲁棒设计》和河南省高校青年骨干教师资助计划《自适应逆控制的Delta算子方法及性能研究》等项目的研究工作的总结。首先综述自适应逆控制的研究进展，然后重点研究基于Delta算子的自适应滤波方法及应用问题。取得主要结果如下：
(1)研究在自适应逆控制设计中常采用的X-滤波和ε-滤波LMS自适应算法，分析这两种算法的性能，通过仿真验证这两种算法明显优于传统LMS算法；然后提出了Delta算子描述LMS算法的另一种矩阵表示方法，并给出X-滤波和ε-滤波LMS自适应算法的Delta算子模型描述形式。
(2)研究基于Delta算子的QR分解LMS(Delta-QR-LMS)滤波问题，推导出Delta-QR-LMS算法的递推步骤；分析该算法的收敛性和稳定性，仿真结果表明Delta-QR-LMS算法具有较快的收敛速度和较好的误差跟踪性能。
(3)将Delta算子引入到自适应均衡器的设计，提出Delta算子描述的改进型LMS算法的判决反馈均衡器模型。分析表明，基于Delta算子的均衡器在收敛性和稳定性方面明显优于q算子描述的均衡器。
Delta operator is a new discretization method that plays the important role in high-speed signal processing and digital sampled control. Delta operator approach can not only avoid the ill-conditioned problems caused by the conventional shift operator based theory when fast sampling, but also the delta operator formulated model approaches the corresponding continuous time model as the sampling interval goes to zero, thus the design method for continuous time systems can be directly applied to discrete time systems.
The thesis is supported by the Natural Science Foundation of Henan Province entitled "Robust design for adaptive inverse control systems using delta operator" and Foundation for University Key Teacher by Henan Province entitled "Delta operator approach to adaptive inverse control design and performance analysis". Development of adaptive inverse control is firstly surveyed, then the problems of the delta operator formulated adaptive filtering theory and applications are studied. The major results of the thesis are as following.
(1) The filtered-X and filtered- ? LMS algorithms are considered which are widely used in adaptive inverse control system. The performance analysis of the above two algorithms is given. Then the simpler matrix descriptions of LMS algorithm using delta operator are presented, the delta operator formulated filtered-X and filtered- ?LMS algorithms are also derived.
(2) The filtering problem of the delta operator formulated LMS algorithm with QR decomposition (Delta-QR-LMS) is studied. The recursive formulae of the Delta-QR-LMS is developed, both the convergence and the stability of this algorithm are analyzed. Simulation results indicate Delta-QR-LMS algorithm has the advantages compared with Delta-LMS.
(3) Delta operator model is introduced to the adaptive equalizer design. Updating-parameter expressions for decision feedback equalizer (DFE) based on delta operator are deduced. And the simulation results confirm the high effectiveness of proposed equalizer.
本论文是作者在参加河南省自然科学基金《基于Delta算子的自适应逆控制系统的鲁棒设计》和河南省高校青年骨干教师资助计划《自适应逆控制的Delta算子方法及性能研究》等项目的研究工作的总结。首先综述自适应逆控制的研究进展，然后重点研究基于Delta算子的自适应滤波方法及应用问题。取得主要结果如下：
(1)研究在自适应逆控制设计中常采用的X-滤波和ε-滤波LMS自适应算法，分析这两种算法的性能，通过仿真验证这两种算法明显优于传统LMS算法；然后提出了Delta算子描述LMS算法的另一种矩阵表示方法，并给出X-滤波和ε-滤波LMS自适应算法的Delta算子模型描述形式。
(2)研究基于Delta算子的QR分解LMS(Delta-QR-LMS)滤波问题，推导出Delta-QR-LMS算法的递推步骤；分析该算法的收敛性和稳定性，仿真结果表明Delta-QR-LMS算法具有较快的收敛速度和较好的误差跟踪性能。
(3)将Delta算子引入到自适应均衡器的设计，提出Delta算子描述的改进型LMS算法的判决反馈均衡器模型。分析表明，基于Delta算子的均衡器在收敛性和稳定性方面明显优于q算子描述的均衡器。
Delta operator is a new discretization method that plays the important role in high-speed signal processing and digital sampled control. Delta operator approach can not only avoid the ill-conditioned problems caused by the conventional shift operator based theory when fast sampling, but also the delta operator formulated model approaches the corresponding continuous time model as the sampling interval goes to zero, thus the design method for continuous time systems can be directly applied to discrete time systems.
The thesis is supported by the Natural Science Foundation of Henan Province entitled "Robust design for adaptive inverse control systems using delta operator" and Foundation for University Key Teacher by Henan Province entitled "Delta operator approach to adaptive inverse control design and performance analysis". Development of adaptive inverse control is firstly surveyed, then the problems of the delta operator formulated adaptive filtering theory and applications are studied. The major results of the thesis are as following.
(1) The filtered-X and filtered- ? LMS algorithms are considered which are widely used in adaptive inverse control system. The performance analysis of the above two algorithms is given. Then the simpler matrix descriptions of LMS algorithm using delta operator are presented, the delta operator formulated filtered-X and filtered- ?LMS algorithms are also derived.
(2) The filtering problem of the delta operator formulated LMS algorithm with QR decomposition (Delta-QR-LMS) is studied. The recursive formulae of the Delta-QR-LMS is developed, both the convergence and the stability of this algorithm are analyzed. Simulation results indicate Delta-QR-LMS algorithm has the advantages compared with Delta-LMS.
(3) Delta operator model is introduced to the adaptive equalizer design. Updating-parameter expressions for decision feedback equalizer (DFE) based on delta operator are deduced. And the simulation results confirm the high effectiveness of proposed equalizer.
引文
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[4] 覃景繁，欧阳景正．一种新的变步长自适应滤波算法．数据采集与处理，1997,12(3)：171-194．
[5] Widrow B, Plett G L. Adaptive inverse control. Proc. the 1993 Int. Symposium on Intelligent Control, Chicago, 1993, 1-6.
[6] Plett G L. Adaptive inverse control of unmodeled stable SISO and MIMO linear systems. Int. J. Adaptive Control and Signal Processing, 2002, 16(4): 243-272.
[7] Liu G P, Kadir K V, Bilings S A. On-line identification of nonlinear systems using Volterra polynomial basis function neural networks. Neural Networks, 1998,11(9): 1645-1657.
[8] Zheng Qingsheng, Zafiriou E.A local form of small gain theorem and analysis of feedback Volterra systems. IEEE Trans Automatic Control, 1999, 44(3): 635-640.
[9] 党映农，韩崇昭．基于Volterra级数模型的非线性系统自适应控制稳定性研究．控制理论与应用，2002，19(1)：80-84．
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[15] 高鹰，谢胜利．一种变步长LMS自适应滤波算法及分析．电子学报，2001，29(8)：1094-1097．
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[17] Krstajie B, Stankovie L J, Uskokovie Z. An approach to variable step-size LMS algorithms. Electronics Letters, 2002, 38(16): 927-928.
[18] Kwong R H, Johnson E W. A variable step size LMS algorithm. IEEE Trans Signal Processing, 1992, 40(7): 1633-1642.
[19] Mathews V J, Xie Z. A stochastic gradient adaptive filter with gradient adaptive step size. IEEE Trans Signal Processing, 1993, 41: 2075-2087.
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[21] 胡春风，高立，张筱华，等．一种改进的频域LMS自适应滤波器算法．通信学报，1998，19(9)：75-79．
[22] Costa M H, Bermudez J C M. Stochastic analysis of the filtered-X LMS algorithm in systems with nonlinear secondary paths. IEEE Trans Signal Processing, 2002, 50(6): 1327-1342.
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