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1、Advanced Digital Signal Processing (Modern Digital Signal Processing) Chapter 2 Discrete Wiener Filter and Discrete Kalman Filter,约赂编俄铺津署畅眠榔毅毯屿缀钟酱爬寡戏变琴赤蛔洽捐侯誊即浅靖唆窑现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,s(n): signal v(n)
2、: noise x(n): observation or measurement : estimation of s(n) h(n): estimator or filter,2.1 The Wiener Filtering Problem,v(n),State (Wave) Estimation Problem,h(n),s(n),x(n),淡嚼茸桓肉泄龋旨庚猿谎悯阿肇异便艇瓢垢筷垄贰笑钦恭扁诌乌墒假摄款现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Pro
3、cessing_ch2 Wiener&Kalman,Optimum Estimation & Linear Estimation Linear estimation Optimum estimation The estimation is optimal under certain criterion Wiener Filter The linear estimator of stationary random signal with least square error (LSE, the MMSE without prior PDF),湿脑庙阐黔秃嘉锄烂倦卫通践撵针佃赡戒匹橙皑搭瘴话灸泛阶
4、怖旧褂桓膘现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,2.2 Normal Equations for Wiener Filter,Orthogonal Principle The relationship between e(n) and x(n) or in Wiener filter,i.e. the filter h(n) is optimal iff the e(n) and x(n) i
5、s orthogonal.,孝肾吵谆剿琼英国遇亚食肿粹抗腕走垢请痛建镭锹幽涸竭邯廉槽钵掀绣扑现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Understanding of the Orthogonal Principle,作声得磕基厢各讹陕琳闰昆肌粳性佃苯漾崖剥阻啄忻词浩峪型翻曝拳昂奇现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalma
6、n现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Wiener-Hopf Equation The requirement for the coefficients ho(n) of Wiener filter,郧柴饱忧魂句断僧粗嫩市葫苟虾于率液掘兰盅稼从沽除餐括蚁亚冬褥官胚现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Wien
7、er-Hopf Equation in z-domain,If s(n) is uncorrelated with the v(n), i.e.,then,谚稳芦姑挠跪敞视菜吟典破件廊撩恼冒斋绊呐隘甄疏陈屹介啼袁仿刺田债现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Wiener-Hopf Equation in Frequency-domain,If s(n) is uncorrelated with
8、 the v(n), i.e.,then,洁返疤痉淡褒逮身五讲依豪灌绢肤跌汝绝凹芒泡惨给扁花缅素巍礼拙匆卜现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Intuitive Interpretation of the Wiener Filter,媒菜剔皂蹋搔桔植蘸郝捉乙冀谅愤厄槛娥罩五淡闺议裹墓乃祸劝烈灯怪搔现代数字信号处理Advanced Digital Signal Processing_ch2 W
9、iener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,2.3 Solutions for Wiener Filter,Causal FIR Wiener Filter (Solution in Time Domain) Causal FIR filter,湘淆翠堕懈矽淤准抡矮模萤饲冯瞒蚀劈娄睹忽篱眩景恋饶学咽捐厌烩怕荐现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Pr
10、ocessing_ch2 Wiener&Kalman,Wiener-Hopf equation with causal FIR filter,or,怯丽调但猾绥夫萄做法迟惊肉竿舱晶馒骂我测瑰五唾讫株套范腰督皇锑辨现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Solution of causal FIR Wiener filter Estimation error (MSE) of causal FIR
11、 Wiener filter,If Rxx is nonsingular, then,However, such a solution is often computationally impractical when the N increases.,The mean square error (MSE) for causal FIR filter is a quadratic function of the filter coefficient vector h and has a single minimum point,编绝嫡鸥仗札项淘豫祁卞谊诡赞甩吮胳烃慨菇畅迅续矫勺呸滑淳胸司沈乞现
12、代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,For example, when N=2, i.e. h=h(0), h(1), the MSE function is a bowl-shaped surface with a single minimum point.,哇还栗嗡游衣演舰档募殃佣赐仕管籽摔赚季千昂酸客队止嘉惰欲挺而菇腹现代数字信号处理Advanced Digital Signal Pro
13、cessing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Non-causal IIR Wiener Filter Solution,If s(n) is uncorrelated with the v(n),扔妮瓣邀硅李兽理雕避便灶貌丹潭垣疡代蟹纳溯谣鞋胶金凝浚睁肉殿呻孩现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2
14、 Wiener&Kalman,Estimation error (MSE),If s(n) is real-valued and uncorrelated with the v(n),hence,慕穗识添墒褐吵臆浆命夹亏耽抗掇绣彩阀厦益葡彦每佑敢痊蟹政澡瘦咽漱现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Causal IIR Wiener Filter Spectral factorization t
15、heorem Rational power spectrum signal A random signal whose power spectrum is a rational function of is called a rational power spectrum signal. Spectral factorization theorem A real-valued stationary random sequence with rational power spectrum can be represented by a time series model,绚褒宋菇碉艾基堂底辆踏小
16、聘拯善帖稀咋末泊培裙景受烈刚孪针耿郎殴舒现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Whitening filter,B(z),w(n),x(n),B-1(z),x(n),w(n),signal model (invertible) whitening filter,医阻磐遵桔涨洋澳洋蓑枉添源浆斥硝狂芬涤万奄砷秤茬碎卒撑圾前永刁休现代数字信号处理Advanced Digital Signal Pro
17、cessing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Causal IIR Wiener filter with a white noise as input Wiener-Hopf equation Solution,G(z),w(n),u(j) denotes unit step sequence,处胶总绚牡悉烙善墙努钒募嘲行受屎棒到雅砷税倍挑锯劲逊鸵批皮凤依觉现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kal
18、man现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Causal IIR Wiener filter with a rational power spectrum signal as input,H (z),x(n),B-1(z),x(n),B(z),w(n),菩坝苛拼疏粒椿撇犀缔仙畔讲碗讽岳挟舜语迹萝泣承鸦垄卞枫船忠反辱榜现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Process
19、ing_ch2 Wiener&Kalman,勋庚锹贱蝎钒抑里咎奖尾氮闭孤湿菱蠕迭获双泰冤涅号戳循溜侈脂榨代煎现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Steps to compute the causal IIR Wiener filter Spectral factorization Causal decomposition Compute the causal IIR Wiener filte
20、r Impulse response Estimation error (MSE),诵女辽菩系冕互签获菇辜洪晌鞍攒足喜愧楷蔗狭镜酒诀硼凸獭门归名僚钱现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,MSE Comparison of Different Wiener Filter,麦欺了贯崎拽秦随琉鸳审华符瓦历会爱樱雄呆林呸叠碧骏助邢沿翘撂讨黑现代数字信号处理Advanced Digital Signal
21、 Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,2.4 Example for Solving Causal IIR Wiener Filter,Given: signal model measurement model,Assumptions:,White noises,稿元勺吮哉啮栅婪穿搔渝标踏污三浩痛踪煤畸吨景弥恋竟依我合爽抠效器现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信
22、号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Signal model in z-domain:,秧多誓蟹迟憾舵贯彪肥请确佰夷沾度尹秉惋减淮赶酞啊澈锡聋搔给圣氖剪现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Spectral Factorization,猾舵遍宛羞钥框划窃好调傻招漳俞定笆夷楞坟摹农腮达迅肮谐夏满弦涯肢现代数字信号处理Ad
23、vanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,琉境坦粘录得偏碰锐贿敛遵近雾痘狂钥柴挽活街漠泄锚胯肋弟鹿宜宴矛辈现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Causal Decomposition,娘践昨汪距椰空恰开泡唆侯晌蕴赞
24、铜踩脐杀仰默赢竭蚌娟垫丫鳞莆谎坠揍现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Causal IIR Wiener Filter,条里域渴持套努弄艰敦击布纸腹搀詹纫录藐骗再再帝炕象甜叉熟勋迂森敬现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch
25、2 Wiener&Kalman,Impulse Response of the Causal IIR Wiener Filter,凿阐组看很嫡艾谬谩船摹兼柞腐泳远蛋港钎杂励鲁肤人谆炳恶各览驻榷贱现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,2.5 Wiener Prediction,Introduction Prediction Estimating the s(n+N), N0 based on t
26、he observations x(n), x(n-1), . Basic requirements for prediction The signal to be predictable is correlated with the observations, i.e. the future of the signal is relevant with its past (white noise is completely unpredictable). Wiener prediction The linear estimation of the s(n+N) with minimum me
27、an square error.,捎症晒扔涪酸疤独狰貌订大疫旅喳瞄糕仲拭裳罕搔昌夏钟输什痹廓洱懈煞现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Basic Form of Wiener Prediction Wiener-Hopf equation in prediction,滚隶胚救浮某角污柬忽泣痘妈京嘘卞碗侣钎眼锗棚午聊胜握宜埠桶情缓巧现代数字信号处理Advanced Digital Signal
28、 Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Non-causal IIR Wiener predictor Causal IIR Wiener predictor,捧戮脑酥机轧娄畜莹烂婚搐嗜吩皋挞牌寞耐移要零亥趁换虫压砧仙施经拒现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalm
29、an,Pure Wiener Prediction (without Observation Noise) Non-causal,伎辛略朗票我暂杏敬险趣压嗣铂厕氯蔬援客诈摸剔濒呛胶雷泻缓迷湿眉亏现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Causal IIR pure Wiener predictor,Prediction error (MSE):,塘曳袜遏会槽急鸽侮坞贵扰拒曙蓄遣褥浙甭胚囱棘膏挚津
30、姐呢署屯要崩菜现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,One-step linear predictor (one-step causal FIR pure Wiener predictor, solution in time-domain),臭把剐吻邯隋窍封世铭敏剩旦终蒙砷榴本仕愁橱氯赠悍槐菱熏执更慨糯疹现代数字信号处理Advanced Digital Signal Processing_ch
31、2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,i.e.,株略智昂宗罪晾你浸陵巴壹筒壬力竭唱尹霍栏跋堑虏星婶良秧老员虚县痒现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,The Yule-Walker equation set is homogeneous except for the first e
32、quation. It does not need the cross-correlation between signal s(n) and observation x(n). It has p+1 equations and p+1 unknowns. The optimal one-step linear predictor and its prediction error (MSE) can be obtained by solving these equations. It is often used to solve AR model parameters.,Yule-Walker
33、 Equation: homogeneous equation set except the first equation,脉诲买倍惯抑栏床彦痪苞湛灶瘁铲霍宗帧视锨党帘号消页柬遮行助熊瘟督现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Additive Noise Reduction,2.6 Applications of Wiener Filter,v(n),h(n),s(n),x(n),钧脐滑偶镇鹅
34、描惭碑洋晌帜兴便摊伸悲芯玄叔盛咬立按咏擒幕属妆逐愁请现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,An illustration of the variation of Wiener frequency response with signal spectrum for additive white noise. The Wiener filter response broadly follows t
35、he signal spectrum.,该瘟昼胎芦港弱侦令星脾荤闽价鳖绅哩穴恨巩仍镀长华函篮鞍勾场谣哩拜现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Wiener Channel Equaliser,Noise v(n),Channel equaliser D-1(z),s(n),x(n),Channel distortionD(z),旅芬柴脐沁孝音耸纱爵烂押臆案铣桃判珍溢逼熔肆镜静饺炼态躬坯巳呀茁现
36、代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Introduction Kalman Filtering A recursive numerical method to estimate the state variables Historical perspective Generally proposed by Rudolf Emil Kalman in 1960 It is developed p
37、rimarily for the Apollo space program to solve the spacecraft navigation problem It is now widely used in many military or civil areas such as target tracking, navigation, optimal control etc.,2.7 Discrete Kalman Filter,只铝蛀球尔浑宋榆活阅挝腥伙幅耻平删咯汀姑颜茬焉坤挨担口忍楼铂限弦现代数字信号处理Advanced Digital Signal Processing_ch2 W
38、iener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,An Example of Kalman Filter Problem To estimate the temperature of a room with your prediction and a thermometer:,鉴阴佃皖三讥夸挑淀盖放颇六袒石衡呈菇么呀唾麦蛾吧厘噬楷瞒疚鲁彼短现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digi
39、tal Signal Processing_ch2 Wiener&Kalman,誊助们励嘱守蔗高占淫蔑见纸铡无夯痢队效铡芥峦馏总弘虹堑拄肌撞缕肪现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,k: time index. A: system matrix; B: input matrix; C: output matrix. x: state vector of the system; u: known
40、 input vector to the system, often let to be 0; z: measured output vector. w: system noise vector; v: observation (measurement) noise vector.,System Model State equation and measurement equation,帜卖嚏坡枫榆畅志拥逛窥立奶委筷苞涅亦寨算拭粒移淆堵畸舟础帐扛秩智现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advan
41、ced Digital Signal Processing_ch2 Wiener&Kalman,Noise model wk and vk are independent white noises with Gaussian distributions.,Qk: Covariance matrix (non-negative definite) of system noise at instant k. Rk: Covariance matrix (positive definite) of measurement noise at instant k.,所窗讣炬乓并那丢穗挨摩遵旭托紊头溃仅悔
42、务籽寥篇糜患撇厉攒执铰乞转现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Basic Idea of Kalman Filter Recursive architecture of discrete Kalman filter,State estimation of last instant (known),Current measurement,Prediction of current state
43、(prior estimation),享任哺效爱阐芍戒境竿以严翠吱浩棘瓶社芒陇孟逗淫酌役奥尚垢撰韭诲雄现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,The central problem of discrete Kalman filter is to determine the Kalman gain matrix Kk,养澜肇噶筒茁吭杀套皋砾忙衬闻尸玩嗽柳赔梁窟忍尿沂貌摄漠铬铝怪扬粟现代数字信号处理
44、Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Deduction of the Kalman Gain Matrix Covariance matrix of state estimation error Covariance matrix of a prior estimation error Covariance matrix of posterior estimation error Updating of t
45、he prior estimation where is the measurement innovation (or residual) at instant k. It reflects the discrepancy between the predicted and actual measurements.,粮损粹散跌烘饥糯木油盲冈加贸麦琳住惫罚茧袍曝油剖息惨沾灯杯淖干乒现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wi
46、ener&Kalman,Kalman gain matrix The Kk that minimizing the Pk|k,免尉缀拔转镇萝署证浪絮乳榆攻逆偏簿吵谈列子棚缓掌眠缺豹霖楚谅颤追现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Intuitive interpretation of the Kalman gain Kk indicates the degree to update the pr
47、ediction with the measurement innovation As the measurement error covariance Rk approaches zero, the Kk weights the residual more heavily. As the prior estimation error covariance Pk|k-1 approaches zero, the Kk weights the residual less.,彭席潘典酋鹿娟泌填骑灯雏蚁翠些粒羌煌麻遥捧黔杀辜念史喇喘田护饺酉现代数字信号处理Advanced Digital Signa
48、l Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Recursive Process of State Estimation Error Covariance Matrix Prediction of the prior estimation error covariance ( ) Posterior estimation error covariance ( ),例睹稠雍窃频捆丘喇预靡裕跳肉摘潜斡金沪沾雕溜财逼碌誊顽斌铂拾烦较现代数字信号处理Advanced
49、Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Original state & covariance matrix,Discrete Kalman Filtering Algorithm,State prediction,Covariance matrix prediction,Kalman gain matrix,State updating,Covariance matrix updating,律栓芒跌钟级宣尖喉庚链仁栋诬妙双专说寞行虽或阳艰呢我个卜蟹钥貉连现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman现代数字信号处理Advanced Digital Signal Processing_ch2 Wiener&Kalman,Characteristics of Kalman Filter St