小波分析及其工程应用讲义_4

WAVELTTRANSFORMANDITSAPPLICATIONS小波变换与工程应用WAVELTTRANSFORMANDITSAPPLICATIONS李艳讲师自动化科学与工程学院TheDiscreteWaveletTransform(1)Inwaveletanalysis,weoftenspeakofapproximationsanddetails.Theapproximationsarethehigh-scale,low-frequencycomponents.Thedetailsarethelow-scale,high-frequencycomponents.TheDiscreteWaveletTransform(2)ThesesignalsAandDareinteresting,butweget2000valuesinsteadofthe1000wehad.Bylookingcarefullyatthecomputation,wemaykeeponlyonepointoutoftwoineachofthetwo2000-lengthsamplestogetthecompleteinformation.Thisisthenotionofdown-sampling.WeproducetwosequencescalledcAandcD.TheDiscreteWaveletTransform(3)ExampleoftheDiscreteWaveletTransformMultipleLeveldecompositionThedecompositionprocesscanbeiterated,withsuccessiveapproximationsbeingdecomposedinturn,sothatonesignalisbrokendownintomanylowerresolutioncomponents.ThisiscalledthewaveletSincetheanalysisprocessisiterative,intheoryitcanbecontinuedindefinitely.Inreality,thedecompositioncanproceedonlyuntiltheindividualdetailsconsistofasinglesampleorpixel.Inpractice,you'llselectasuitablenumberoflevelsbasedonthenatureofthesignal,oronasuitablecriterionsuchasentropyWaveletReconstructionTheprocessofAssemblingthesecomponentsbackintotheoriginalsignalwithoutlossofinformationiscalledreconstruction,orsynthesis.Themathematicalmanipulationthateffectssynthesisiscalledtheinversediscretewavelettransform(IDWT)ReconstructionFiltersThedownsamplingofthesignalcomponentsperformedduringthedecompositionphaseintroducesadistortioncalledaliasing.Itturnsoutthatbycarefullychoosingfiltersforthedecompositionandreconstructionphasesthatarecloselyrelated(butnotidentical),wecancancelouttheeffectsofaliasing.ReconstructionofApproximationsandDetails(1)Reconstructouroriginalsignalfromthecoefficientsoftheapproximationsanddetails.Reconstructtheapproximationsanddetailsthemselvesfromtheircoefficientvectors.Reconstructthefirst-levelapproximationA1fromthecoefficientvectorcA1Reconstructthefirst-leveldetailD1fromthecoefficientvectorcD1S=A1+D1ReconstructionofApproximationsandDetails(2)thecoefficientvectorscA1andcD1--halfthelengthoftheoriginalsignal--cannotdirectlybecombinedtoreproducethesignal.Itisnecessarytoreconstructtheapproximationsanddetailsbeforecombiningthem.Extendingthistechniquetothecomponentsofamultilevelanalysis,wefindthatsimilarrelationshipsholdforallthereconstructedsignalconstituents.Thatis,thereareseveralwaystoreassembletheoriginalsignal:RelationshipofFilterstoWaveletshapesThewaveletfunctionisdeterminedbythehigh-passfilter,whichalsoproducesthedetailsofthewaveletdecomposition.Thereisanadditionalfunctionassociatedwithsome,butnotall,wavelets.Thisistheso-calledscalingfunction,.Thescalingfunctionisverysimilartothewaveletfunction.Itisdeterminedbythelow-passquadraturemirrorfilters,andthusisassociatedwiththeapproximationsofthewaveletdecomposition.Inthesamewaythatiterativelyupsamplingandconvolvingthehigh-passfilterproducesashapeapproximatingthewaveletfunction,iterativelyupsamplingandconvolvingthelow-passfilterproducesashapeapproximatingthescalingfunction.Multi-stepDecompositionandReconstrctionThisprocessinvolvestwoaspects:breakingupasignaltoobtainthewaveletcoefficients,reassemblingthesignalfromthecoefficients.WaveletPackageAanlysisInwaveletpacketanalysis,thedetailsaswellastheapproximationscanbesplit.Thisyieldsmorethandifferentwaystoencodethesignal.Thisisthewaveletpacketdecompositiontree.122nForinstance,waveletpacketanalysisallowsthesignalStoberepresentedasA1+AAD3+DAD3+DD2.小波包函数除了尺度和平移两个参数外，增加了一个频率参数，克服了小波时间分辨率高时频率分辨率低的缺陷。IntroduceofWaveletFunction（1）IntroduceofWaveletFunction（2）根据不同的标准，小波函数具有不同的类型(1)小波函数和尺度函数及其傅立叶变换的支撑长度。即当时间或频率趋向无穷大时，函数从一个有限值收敛到0的速度；(2)对称性。在图像处理中用于避免移相；(3)消失矩阶数。有利于数据压缩；(4)正则性。有利于信号或图像的重构获得较好的平滑效果。在MATLAB命令行输入：waveinfo(‘’)命令可以查看函数简要说明例如：waveinfo(‘db’)在MATLAB命令行输入：wavemenu，打开小波工具箱GUI可以查看详细帮助参考文献：故障信号检测的小波基选择方法.PDF小波函数的性质及其应用研究.PDFApplications一维小波分析用于信号奇异性检测一维小波分析用于用于信号消噪处理一维小波分析用于识别含噪信号的有用信号发展趋势二维小波分析用于图像压缩二维小波分析用于图像消噪二维小波分析用于图像增强二维小波分析用于图像融合利用小波包进行特征提取利用小波包进行信号消噪处理利用小波包进行图像压缩一维小波分析用于信号奇异性检测(1)信号中的奇异点及不规则突变部分经常带有比较重要的信息，例如在故障诊断中，故障通常表现为输出信号发生突变。在这些奇异信号中,信号的奇异程度是不同的,根据研究的需要,常将其分为剧变奇异信号和缓变奇异信号。剧变奇异信号是指信号本身具有突变,缓变奇异信号则指信号本身是连续的,但其某阶导数具有间断或奇变。对信号进行多尺度分析，在信号出现突变时，小波变换后的系数具有模值极大值，可以通过对极大值点的检测确定故障发生的时间。小波的选择，需要注意具有良好的正则性。例程：test_1_01.mtest_1_02.m一维小波分析用于信号奇异性检测(2)Test_1_01.m第一类间断点一维小波分析用于信号奇异性检测(3)Test_1_02.m一维小波分析用于用于信号消噪处理(1)一维小波分析用于用于信号消噪处理(2)对平稳信号消噪一维小波分析用于用于信号消噪处理(3)对非平稳信号消噪一维小波分析用于识别含噪信号的有用信号发展趋势（1）一维小波分析用于识别含噪信号的有用信号发展趋势（2）利用小波包进行特征提取（1）利用小波包分析建立一个能够表征系统状态的特征向量，以便利用模式识别的方法对该系统进行故障检测与定位算法：1对采样信号进行小波包分解2对小波包分解系数重构，提取各频带范围的信号3求各频带信号的总能量4构造特征向量，以能量为元素构造特征向量，进行归一化处理5确定在正常与各种故障状态下，特征向量的特征值及容差范围利用小波包进行特征提取（2）利用小波包进行信号消噪处理1信号的小波包分解2计算最佳树（确定最佳小波包基）3小波包分解系数的阈值量化4小波包重构