电子通讯 - 第四章 维纳滤波器

武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 1 现代数字信号处理现代数字信号处理 孙洪 Modern Digital Signal Modern Digital Signal ProcessingProcessing Chap 4 Linear Optimum Filtering Chap 4 Linear Optimum Filtering -------- Wiener FiltersWiener Filters 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 2 Introduction to Wiener FiltersIntroduction to Wiener Filters l Norber Wiener (1949) – Least Square Error Estimation – Data-dependent linear filter – Desired signal [ ] min)(2 =neE ( ) ( )å -= -= 1 0 ˆ p k k knxwns Wiener Filters å -= --=-= 1 0 )()()(ˆ)()( p k k knxwnsnsndne Fig.1: WF 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 3 4.1 Normal Equations 4.1 Normal Equations ---------- WienerWiener--Hopf Hopf Equations Equations l Error-Performance Surface l Normal Equation [ ] å å å-= - = - = -+-= 1 0 1 0 1 0 2 )()(2)0()( p k p k p k xxjkxskss jkrwwkrwrneE ( ) ( ) mkmRwmR p k xxkxs "-= å - = , 1 0 Wiener Filters Fig.2: Surface Fig.3: Matrix 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 4 4.2 Block4.2 Block--Data Formulation of the Data Formulation of the Wiener Filter Wiener Filter –– FIR Wiener FilterFIR Wiener Filter Wiener Filters ( ) ( ) XwXwsXwXwsssXwsXwsee TTTTTTTT +--=--= Xws =ˆ Xwse -= xsxx rwR = xsxx rRw 1-= xsxx rYRs 1ˆ -= Fig.4: Matrix of e 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 5 4.3 Minimum Mean4.3 Minimum Mean--squared Error squared Error –– FIR Wiener FilterFIR Wiener Filter Wiener Filters xsxxopt rRw 1-= ( ) ( ) XwXwsXwXwsssXwsXwsee TTTTTTTT +--=--= [ ] [ ] [ ] [ ] wRwrwr wxxEwnxsEwnsEneE xx T xs T ss TTT +-= +-= 2)0( )(2)()( 22 [ ] xsTssxxTss rwrwRwrneE -=-= )0()0()(min 2 2 ˆ 22 sse sss -= 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 6 4.4 Interpretation of Wiener Filters 4.4 Interpretation of Wiener Filters as Projection in Vector Space as Projection in Vector Space –– Principle of Principle of OrthogonalityOrthogonality Wiener Filters [ ] 0)(2 = ¶ ¶ neE wk [ ] kknxneE "=- ,0)()( [ ] [ ] 0)()(2)()(2)(2 =--=ú û ù ê ë é ¶ ¶ = ¶ ¶ knxneE w neneEneE w kk å - = --= 1 0 )()()( p k k knxwnsne Orthogonal equations Fig.5: Vector Space 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 7 4.5 4.5 NoncausalNoncausal IIR Wiener FiltersIIR Wiener Filters Wiener Filters )()()( zRzHzR xxncxs = ( ) ( )å ¥ -¥= -= k k knxwnsˆ l Linear IIR Filter l Wiener-Hopf Equation l The z-transform l The Wiener Filter ( ) ( ) mkmRwmR k xxkxs "-= å ¥ -¥= , )()()( zRzRzH xxxsnc = 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 8 4.6 Causal IIR Wiener Filters4.6 Causal IIR Wiener Filters Wiener Filters ( )zA( )nw ( )zB ( )nv + ( )zB1 ( )zG ( )nsˆ( )ne ( )ne ( )nx ( )nx ( )ns Whitening filter Optimum causal filter for white input [ ] ú û ù ê ë é == + )/1( )(1)(1)( **22 zB zRzRzG xs s s s ss e )/1()()( **2 zBzBzR sxx s= Fig.6: Spectral decomposition 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 9 4.7 Some Applications of Wiener 4.7 Some Applications of Wiener FiltersFilters Wiener Filters l Additive Noise Reduction l Wiener Channel Equaliser l Time-Alignment of Signals in Multichannel/multisensor Systems Fig.7: WF in frequency domain – S/N Fig.8: WF Channel Equaliser Fig.9: WF for multisensor system { } 0,),()()( =+= nsEnvnsnx )()()()( zVzHzSzX += )()()()( zVzHzSzX iii += 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 10 4.8 Implementation of Wiener 4.8 Implementation of Wiener FiltersFilters Wiener Filters l Computation of Correlations l The choice of Wiener filter order – The ability of the filter to remove distortions and reduce the noise – The computational complexity of the filter – The numerical stability of the Wiener solution Fig.10: WF for speech enhance 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 11 DiscussionDiscussion l A linear least square error estimation for Gaussian signal l A decorrelator l Without distortion of estimation only if – The spectra of the signal and the noise are seperable by a linear filter – The signal component of the input is linearly transformable to d(n) – The filter length is sufficiently large Wiener Filters Fig.11: WF as a decorrelator 武汉大学 电子信息学院 研究生课程 © 2002 孙洪 © 2002 孙洪 12 Summary for Wiener FiltersSummary for Wiener Filters l A data-dependant linear least square error estimation l Wiener-Hopf equation - solutions l Orthogonal equation - decorrelation Wiener Filters