小波分析及其工程应用讲义_4

1、WAVELT TRANSFORM AND ITS APPLICATIONS小波变换与工程应用WAVELT TRANSFORM AND ITS APPLICATIONS李艳 讲师自动化科学与工程学院The Discrete Wavelet Transform(1)In wavelet analysis, we often speak of approximations and details.The approximations are the high-scale, low-frequency components. The details are the low-scale, high-fr

2、equency components. The Discrete Wavelet Transform(2)These signals A and D are interesting, but we get 2000 values instead of the 1000 we had. By looking carefully at the computation, we may keep only one point out of two in each of the two 2000-length samples to get the complete information. This i

3、s the notion of down-sampling. We produce two sequences called cA and cD.The Discrete Wavelet Transform(3)Example of the Discrete Wavelet TransformMultiple Level decomposition The decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is br

4、oken down into many lower resolution components. This is called the wavelet Since the analysis process is iterative, in theory it can be continued indefinitely. In reality, the decomposition can proceed only until the individual details consist of a single sample or pixel. In practice, youll select

5、a suitable number of levels based on the nature of the signal, or on a suitable criterion such as entropy Wavelet ReconstructionThe process of Assembling these components back into the original signal without loss of information is called reconstruction, or synthesis. The mathematical manipulation t

6、hat effects synthesis is called the inverse discrete wavelet transform (IDWT)Reconstruction FiltersThe downsampling of the signal components performed during the decomposition phase introduces a distortion called aliasing. It turns out that by carefully choosing filters for the decomposition and rec

7、onstruction phases that are closely related (but not identical), we can cancel out the effects of aliasing. Reconstruction of Approximations and Details(1)Reconstruct our original signal from the coefficients of the approximations and details.Reconstruct the approximations and details themselves fro

8、m their coefficient vectors.Reconstruct the first-level approximation A1 from the coefficient vector cA1Reconstruct the first-level detail D1 from the coefficient vector cD1S=A1+D1Reconstruction of Approximations and Details(2)the coefficient vectors cA1 and cD1 - half the length of the original sig

9、nal- cannot directly be combined to reproduce the signal.It is necessary to reconstruct the approximations and details before combining them. Extending this technique to the components of a multilevel analysis, we find that similar relationships hold for all the reconstructed signal constituents. Th

10、at is, there are several ways to reassemble the original signal:Relationship of Filters to Wavelet shapes The wavelet function is determined by the high-pass filter, which also produces the details of the wavelet decomposition. There is an additional function associated with some, but not all, wavel

11、ets. This is the so-called scaling function, . The scaling function is very similar to the wavelet function. It is determined by the low-pass quadrature mirror filters, and thus is associated with the approximations of the wavelet decomposition.In the same way that iteratively upsampling and convolv

12、ing the high-pass filter produces a shape approximating the wavelet function, iteratively upsampling and convolving the low-pass filter produces a shape approximating the scaling function.Multi-step Decomposition and ReconstrctionThis process involves two aspects: breaking up a signal to obtain the

13、wavelet coefficients, reassembling the signal from the coefficients. Wavelet Package AanlysisIn wavelet packet analysis, the details as well as the approximations can be split. This yields more than different ways to encode the signal. This is the wavelet packet decomposition tree.122nFor instance,

14、wavelet packet analysis allows the signal S to be represented as A1 + 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。Introduce of Wavelet Function（1）Introduce of Wavelet Function（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)

15、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

16、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)n对平稳信号消噪一维小波分析用于用于信号消噪处理(3)n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