Advanced-Digital-Signal-Processing(color).ppt

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1、Chapter 2 Wiener Filtering and Kalman Filter(1) Wiener Filtering Problem ns nvObserved data or Measured dataSignal Noise or InterferenceAdditive combination 2.1 Normal equations of Wiener filter nx nhneEninxihnhnxnsimin2 nsnsneLinear estimationOptimum estimation(minimum mean-square error)Estimation

2、errorSmoothing Filtering Prediction niixinhns0 11npniixinhnsP-order forward one step linear prediction (LPC) 10Niixinhns 121003210012000100001210NxxxxhNhNhNhhhhhhhNssss 1, 1, 0,0NninxihnsniCausality: Low diagonal matrix Wiener Filter(2) Orthogonal equations jjnxneEjjnxneEjhneneEjhn,0, 1, 0,022(3) no

3、rmal equations (Wiener-Hopf equations) nxnsmnxnsEmRnxmnxinxEimRmimRihmRsxxxixxsx and of sequencen correlatio-cross of sequenceation autocorrel where, The Normal Equations of Causal Wiener Filter 0,0mimRihmRixxsx TTNnxnxnxnNhhh11110 xh2.2 Solution of Wiener-Hopf Equations2.2.1 FIR (Finite Impulse Res

4、ponse) Wiener Filter 1210032130122101121012101, 1, 0,10NhhhhRNRNRNRNRRRRNRRRRNRRRRNRRRRNmimRihmRxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxsxsxsxsxNixxsx nnsENRRRRsxsxsxsxxP 1210 nnERNRNRNRNRRRRNRRRRNRRRRTxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxR 0321301221011210RhPRhPTTor Solution: nnnEnnsEnnnsxxxxxRPxhPRhTT1TT1opt

5、12.2.2 Non-causal IIR (Infinite Impulse Response) Wiener FilterThe normal equations : dzzzHjnhzSzSzHzSzHzSnoptoptxxsxoptxxoptsx121 mimRihmRixxsx,2.2.3 Causal IIR Wiener FilterThe normal equations : 0,0mimRihmRixxsxMinimum Phase Sequence A stable and causal sequence that has a rational z-transform wi

6、th all of its zeros (and poles ) inside theunit is said to be a minimum phase sequence. For example, the finite duration sequence Maaa,10is a minimum phase sequence if 1,11100iMiiMiiizzzazazAMinimum phase polynomial , Minimum phase systemWhy does a sequence with its all zeros inside the unit circle

7、have minimum phase lag?Suppose is a order polynomial having only one zero inside the unit circle zFzzzA111where is a order polynomial with its all zeros outside unit circle. zA zFM1MThe zero:1zz conjugate and reciprocal relation zFzzzB11The zero:11 zz111111 1 1zzzzzz zFzzzA111 zFzzzB11 BAzBzAzzzzzzz

8、zzzzzz 11111111111(1) Amplitude characteristic(2) Phase lag characteristicPhase lag: AeAeAAAjAjAarg,arg BeBeBBBjBjBarg,argThe phase difference : jjjjjjjeeezezeezezBA211111 ABABBA0 , 022 2 2argarg11arg argzezejjSuppose is a order polynomial having its allzero inside the unit circle. zAMMaximum phase

9、sequenceMaximum phase PolynomialMaximum phase systemM2sequences having the same amplitude characteristic but different phases characteristics.2. Spectral Factorization Theorem The rational power spectrum of a real stationary random signal can be factored into a product of theform3. Model of Wide-Sen

10、se Stationary Random SignalsynthesisanalysisThe whitening filter s.polynomial phase minimum areboth and , where12zDzNzDzNzBzBzBzSxxProof for the spectral factorization theorem:Symmetry of the power spectrum of a real stationary random signal: 1zSzSxxxxReal zero:Complex zero:iz1iz 1izizReal zeroCompl

11、ex zeroIn order to that is stable and causal , is also stable and causal, and both must be minimum phase polynomials. zB zDzNzB zNzDzB1Causal and stable zB1All poles ( all zeros of ) are inside the unit circle zDAll poles ( all zeros of )are inside the unit circle zN zD zNAll zeros of and are inside

12、 the unit circle. zN zD(1) Any regular process may be realized as the output of a causal and stable filter that is driven by white noise having a variance of . This is known as the innovations representation of the process. (model)(2) If is filtered with the inverse filter , then the output is white

13、 noise with a variance of . The formation of this white noise process is called the innovations process .(3) Since and are related by an invertible transformation, either process may be derived from each other. Therefore, they both contain the same information.2 nx zB12 n nx 0,0mimRihmRixxsx 0iinxih

14、nhnxns nnx2 ng zSzGnunRngmmgimigmRmmRmimRigmRssisis2220220110, where0,a white noise with variance A causal IIR Wiener filter with impulse response 4. causal IIR Wiener filter (with a white noise as input)Stable, Causal. The poles inside the unit circle.5. A input with rational power spectrum polynom

15、ils phase minimum areboth and ,12zDzNzDzNzBzBzBzSxxSpectral factorized theoremis causal and optimal zG zSzGs21 iinxifn zGzBzHc1 121zBzSzBzHsxc 12111 zBzSzGzBzSzSzFzSmRmfimRifinxifmnsEnmnsEmRsxsxsxssxisxisComputational steps Product factorization (spectral factorization theorem)(2) Sum factorization

16、of 111zBzSzBzSzBzSsxsxsx(3) Compute the system 121zBzSzBzHsxc(4) Compute the impulse response(5) Compute the minimum mean-square error 12zBzBzSxx2.3 Mean-Square Error of Wiener Filter 0 2minesRnsneEnsnsneEneEn zSzHzSzSzSzSzSdzzxSjRndzzzSjmRxsoptsss ssssssescuesesmceses 210 211.min1 dzzzSzHzSjncuxsop

17、tss1.min 21Why is Wiener filter optimal? vvssssoptvvssssxxssoptSSSHzSzSzSzSzSzH 0 01 0optvvoptssoptvvHSHSHS2.4 Computation of Causal Wiener FilterGiven: signal model measurement model 1 ,1anwnasns 1 ,cnvncsnxAssumptions: 00, 0, 1 iwnvEisnvERivnvEininQiwnwEninini RzSQzSvvwwWhite noisesSolution: Razaz

18、QczSzSczSzSazazcQzcSzSzcSzSzSazazQzSnvncsnxnwnasnsvvssvcsxxsssvssvcsssxss11 11 11 , 1122112(1) Product factorization (spectral factorization theorem) 1,11, 111111 1111121221222212fazfzzBzBzBazazzRaQcaRzRaQcaRRaQcRazazQczSxxzffzffffzfzzRaQcaRzRaQcaRRaQc212221222122221111111111RaQcaRffRacQf22222211110

19、,22PPcRRPaPQRicatti Equation(2) Sum factorization fzazcQfzazazazcQzBzSzGsx111111111 methodexpansion fraction partial (2) residueby method formulainversion (1) ?zGPCRPaQcaRf22221222or vfaRaf(1) Inversion formula method 1111azfacQzG(2) Partial fraction methodfzzafcQf11fzBz1 1111azfacQzG circle)unit th

20、einside (poles 0,1 ,11sRe21.111nafacQazfzazcQdzzzGjngncunn 111111 111azafcQazAfzazcQzG(3) Compute the system function of the causal IIR filter gain Wiener the111111 111111 1211211122facQGfzGfzfacQazfacQazfzzGzBzHcAnother method for computing causal IIR Wiener filter 122221 )3(1or )2(solution positiv

21、e ) 1 (fzGzHcGafPcRaRfPcRcPGPPcRRPaPQcPcRaRf2afPPcRRPaPQ122PcRcPcPfacQG2221PcR22cGaPcRPacaPcRaRf1222 PcRcPG2cGaf1222or vfaRaf2.5 Scale Kalman Filter 1 ,1 ,nnxnnsnnsns nxGnnsfnnsn11 1111 nnsacnxGnnsannsnThe first prediction:The second prediction:Innovation:111nnsnnsa1111nnsacnnscnnx 111nnsacnxnnxnxnc

22、Gaf1The optimal gain (Kalman gain) 1min 1212222nnsnsneneEnPcnPRcnPcRncPGGnnsnsEneEnnnPrediction error powerPrediction errorThe minimum mean square error nPnPcGnPcGnPnnsnsEneEnnn1 22Prediction error power QnanP122wQComputational steps QnanP12 nPcRncPGn2 nPcGnn 1 1111 nnsacnxGnnsannsnInitiation 11,110

23、,001sGPs2.6 Vector Kalmin Filter2.6.1 Signal Vector and Data Vector qinwnsansiiii, 2, 1 , 1 nnnaaanwnwnwnnsnsnsnqTqTqwAssAws1000000 212121 nnnbanwnnsnsnnsnsnwnbsnasnsnsnsnsnsnsnwnbsnasnswAssAws101 0 11111 ,212112211121 nnncccnvnvnvnnxnxnxnqkkinvnscnxkTkTkiiiivCsxCvx000000000)( , 2, 1 2121212.6.2 Derivation of Vector Kalman Filter nnnnnnvAsxwAss1TTTBAABAAAABBAbabaaabba122Scale arithmetic matrix arithmetic 111111nnnnnnnnnnnnnnnnnnnnnnnnnnnnnsCAxGsAsPCGIRCPCCPGQAAP1TTT