FIR数字滤波器零极点灵敏度分析及优化实现
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摘要
针对有限字长效应导致滤波器零极点的位置偏移问题,基于状态空间实现结构研究FIR数字滤波器零极点对系数误差的灵敏性。不同于IIR滤波器,FIR滤波器状态空间模型中的系统矩阵具有亏损性。引入亏损矩阵广义特征向量分析极点的灵敏性,导出灵敏度表达式,并依据相似变换理论找寻最佳变换矩阵,提出FIR滤波器零极点灵敏度的优化实现。理论推导及仿真实验表明,FIR滤波器极点对系数误差敏感度较高,且所提优化实现方案能够降低灵敏度。
Aiming at the deviation of pole and zero in filters which caused by the finite word length(FWL) effects,the sensitivity of pole and zero for FIR digital filters to coefficient errors was studied based on the state-space model.Unlike the IIR filter,the system matrix in state-space model of the FIR filter was defective.A set of generalized eigenvectors of defective matrix was introduced to analyze the pole sensitivity and derive the measure expression,and optimal realizations with respect to pole-zero sensitivity for FIR filters were proposed by finding optimal transformation matrices according to the similarity transformation theory.Theoretical analysis and simulation experiments show that the poles of a FIR filter are more sensitive to coefficient errors,and the proposed optimal realizations can reduce the sensitivity.
Aiming at the deviation of pole and zero in filters which caused by the finite word length(FWL) effects,the sensitivity of pole and zero for FIR digital filters to coefficient errors was studied based on the state-space model.Unlike the IIR filter,the system matrix in state-space model of the FIR filter was defective.A set of generalized eigenvectors of defective matrix was introduced to analyze the pole sensitivity and derive the measure expression,and optimal realizations with respect to pole-zero sensitivity for FIR filters were proposed by finding optimal transformation matrices according to the similarity transformation theory.Theoretical analysis and simulation experiments show that the poles of a FIR filter are more sensitive to coefficient errors,and the proposed optimal realizations can reduce the sensitivity.
引文
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[2]YAN W Y,MOORE J B.Onβ-sensitivity minimization of linear state-space systems[J].IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications,1992,39(8):641-648.
[3]LI G,GEVERS M,SUN Y X.Performance analysis of a new structure for digital filter implementation[J].IEEE Transactions on Circuits Systems I:Fundamental Theory and Applications,2000,47(4):474-482.
[4]WILLIAMSON D.Round off noise minimization and pole-zero sensitivity in fixed-point digital filters using residue feedback[J].IEEETransactions on Acoustics,Speech,and Signal Processing,1986,34(5):1210-1220.
[5]LI G.On pole and zero sensitivity of linear systems[J].IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications,1997,44(7):583-590.
[6]徐巍华,吴俊,褚健.有限字长数字控制器的一种极点灵敏度优化结构[J].控制理论与应用,2000,17(4):613-618.XU W H,WU J,CHU J.A class of pole sensitivity optimization structures of an FWL digital controller[J].Control Theory and Applications,2000,17(4):613-618.
[7]LI G,LIM Y C,HUANG C G.Very robust low complexity lattice filters[J].IEEE Transactions on Signal Processing,2010,58(12):6093-6104.
[8]于爱华,黄朝耿,李刚,等.一种新型低复杂度的IIR格型滤波器[J].电子学报,2013,41(9):1703-1709.YU A H,HUANG C G,LI G,et al.A new class of low complexity IIRlattice filters[J].Acta Electronica Sinica,2013,41(9):1703-1709.
[9]MANTEY P.Eigenvalue sensitivity and state-variable selection[J].IEEE Transactions on Automatic Control,1968,13(3):263-269.
[10]SKELTON R E,WAGIE D A.Minimal root sensitivity in linear systems[J].Journal of Guidance Control and Dynamics,1984,7(5):570-574.
[11]GEVERS M,LI G.Parameterizations in control,estimation and filtering problems:accuracy aspects[M].New York:Springer-Verlag,1993.
[12]HINAMOTO T,DOI A,LU W S.Weighted pole and zero sensitivity minimization for state-space digital filters[C]//2015 IEEE International Symposium on Circuits and Systems(ISCAS).2015:2193-2196.
[13]HINAMOTO T,DOI A,LU W S.Minimization of weighted pole and zero sensitivity for state-space digital filters[J].IEEE Transactions on Circuits and Systems I:Regular Papers,2016,63(1):103-113.
[14]徐涛,于澜,鞠伟,等.计算亏损系统模态灵敏度的逐层递推演算方法[J].力学学报,2008,40(2):281-288.XU T,YU L,JU W,et al.Recursive solution on layer after layer for sensitivity analysis of modes on defective linear vibration system[J].Chinese Journal of Theoretical and Applied Mechanics,2008,40(2):281-288.
[15]陈塑寰,徐涛,韩万芝.线性振动亏损系统的矩阵摄动理论[J].力学学报,1992,24(6):747-753.CHEN S H,XU T,HAN W Z.Matrix perturbation for linear vibration defective systems[J].Acta Mechanica Sinica,1992,24(6):747-753.