稀疏信号处理简介

稀疏信号处理简介“Signal&informationprocessingis···anArt”—PetreStoica电子科技大学电子工程学院万群2020/12/242一所大学，两个战场科学：一个是和其他世界一流大学共同面对的国际学术前沿战场技术：另一个是为我们国家经济、社会、国防、发展战略需要服务的战场UESTC2020/12/243内容从几个问题开始稀疏重建理论几个例子阵列信号处理的例子实孔径超分辨、阵列稀疏布阵无线定位的例子MDS、MC信道估计的例子2020/12/244一、从几个问题开始高斯分布凭什么无所不在？MMSE是最优的？吝啬原则：免费的午餐？分辨率受孔径限制？机器学习：支持向量是稀疏的？什么是多维标度问题?2020/12/245高斯分布：AnequationisforeternityThefundamentalnatureofthisdistributionanditsmainpropertieswerederivedbyLaplace(1781)whenGausswassixyearsoldThedistributionitselfhadbeenfoundbydeMoivre(1733)beforeLaplacewasborn2020/12/246Gauss’sQuestion(1809)Whatwouldbeadistributiondensityf(x;θ)forwhichthemaximumlikelihoodestimateofθisthesamplemeanweusethemodernterminologyadoptedbythescientificcommunitymorethanacenturylater（themethodofmaximumlikelihoodwasproposedbyFisherin1921）2020/12/247DERIVATIONOFGAUSS(1809)Usingi.i.d.observations,themaximumlikelihoodestimateofparameteroflocation2020/12/248Derivationanyrealnumbercanbearbitrarilyaccuratelyapproximatedbyrationalnumbers2020/12/249ResultGaussassumedthesamplemeanduetoitscomputationalconvenienceandderivedtheGaussianlaw.ThislineofreasoningisquitetheoppositetothemodernexpositionintextbooksonstatisticsandsignalprocessingwheretheLSmethodisderivedfromtheassumedGaussianity.2020/12/2410为什么要折衷？性能最优计算最简单跑题了？2020/12/24111.1高斯分布凭什么无所不在？TheroleofGaussianmodelsinsignalprocessingisbasedontheoptimalpropertyoftheGaussiandistributionminimizingFisherinformationovertheclassofdistributionswithaboundedvariance.Thecentrallimittheorem(CLT)isnotonlyauniquereasonbutperhapsitisevennotthemainreason2020/12/2412Fisherinformation2020/12/24131.2MMSE是最优的？Ifhisknowntobesparse,canwedoevenbetterthantheMMSEestimate?Andifso,howmuchbettercanwedo?有偏估计！2020/12/2414NP-Hard？现代最小二乘(P0)0minxsubjectto2bAx(P1)1minxsubjectto2bAx2020/12/24151.3吝啬原则：免费的午餐？多成分混合（合成，正问题）分离各个成分（感知，反问题）1()Mkkkxa2()()Hsax2020/12/2416贪婪的谱估计=滤波：2()()Hswx211121121121111/22()()()()()1()()()()()()()()()()()()1()()()()()MHCBFkkkMVDRHMHkkkHPPvARHHHPPPHnnnMUSICHHHnnkkPwasaqRawsaRaaqqqawsaqqaaqUUawsaUUaaq1M2020/12/24171.4分辨率受孔径限制？DFTOOOOOOOO=OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOX2020/12/2418BWE2020/12/24191GHz(S)+1GHz(X)=10GHz？Lband1to2GHzSband2to4GHzCband4to8GHzXband8to12GHzKuband12to18GHz2020/12/2420超分辨是一个欠定问题在线测量+先验模型稀疏2020/12/24211.5机器学习：支持向量是稀疏的？thetrainingsamplehyperplanethatdoestheseparation-2020/12/2422primalformulationoftheproblem2020/12/2423convexquadraticprogrammingproblem2020/12/2424dualformulation2020/12/2425GreatwatershedinoptimizationItisnotbetweenlinearityandnonlinearity,butconvexityandnon-convexity—R.Rockafellar,SIAMReview1993220xaybxycxdye2020/12/24261.6什么是多维标度问题?测距定位2020/12/2427倒行逆施：解的表示计算最简单性能最优2020/12/2428子空间分析2020/12/2429矩阵完整性分析Rank=2,3节点之间无测量节点之间测量误差很大计算最简单性能最优所需测量不多！2020/12/2430He-WenWei,RongPeng,QunWan,Zhang-XinChen,andShang-FuYe,MultidimensionalScalingAnalysisforPassiveMovingTargetLocalizationwithTDOAandFDOAMeasurements,IEEETransactionsonSignalProcessing,vol.58,no.3,pp.1677-1688,2010S.Qin,Q.Wan,Z.X.Chen,AFastMultidimensionalScalingAnalysisforMobilePositioning,IETSignalProcessing.Zhang-XinChen,He-WenWei,QunWan,Shang-FuYeandWan-LinYang，ASupplementtoMultidimensionalScalingFrameworkforMobileLocation:AUnifiedView，IEEETransactionsonSignalProcessing,vol.57,no.5,pp.2230-2234,May2009HewenWei,QunWan,ShangfuYe,ANovelWeightedMultidimensionalScalingAnalysisforTime-of-Arrival-BasedMobileLocation,IEEETransactionsonSignalProcessing,Vol.56,No.7,July2008,pp.3018-3022HewenWei,QunWan,ShangfuYe,Multidimensionalscalingbasedpassiveemitterlocalizationfromrange-differencemeasurements,IETSignalProcessing,Volume2,Issue4,December2008Page(s):415-423Zhang-XinChen,QunWan,He-WenWeiandWan-LinYang，ANovelSubspaceApproachforHyperbolicMobileLocation,ChineseJournalofElectronics，2009年第3期,pp.569-573HuangJiYan,WanQun,Commentson‘TheCramer-RaoBoundsofHybridTOA/RSSandTDOA/RSSLocationEstimationSchemes’,IEEEComm.Letters,Vol.11,Issue11,Nov.2007,pp.848-8492020/12/2431二、稀疏重建理论基追踪：BasisPursuit，贪婪算法稀疏重建条件：RIP字典确定型随机型结构+随机型计算最简单性能最优所需测量最少2020/12/24322020/12/2433CVX:convexoptimizationMarch3,2008，[e-mail]l1_lslarge-scalel1-regularizedleast-squaresl1_logreglarge-scalel1-regularizedlogisticregressionGGPLABgeometricprogrammingL1-MAGICconvexoptimizationtoCompressedSensingSparseLabsparsesolutionstolinearequations,particularlyunderdeterminedsystemsCurrentsoftware2020/12/24342020/12/24352020/12/24362020/12/24372020/12/24382020/12/2439三、几个例子阵列信号处理的例子实孔径超分辨阵列稀疏布阵无线定位的例子MDSMC信道估计的例子SVR2020/12/2440稀疏布阵：同阵元数，优化5.7dB00.8751.8752.753.754.6255.6256.57.58.3759.37510.2513.125-101x(wavelength)yOurarraycompare_ieee_trans_sp_1988_vol.36_no.3_pp372-80-60-40-20020406080-30-25-20-15-10-50Direction(degree)NormalizedpatternMaximumsidelobelevelis-19.01dB2020/12/2441稀疏信道估计r=Sh+ecvx_beginvariablesh;minimize(norm(S*h-r,2)+0.5*norm(h,1));cvx_end1ms10kbps:101Gbps:100万2020/12/244200.050.10.150.20.250.3010203