小波分析及其工程应用讲义_4

Reconstruction of Approximations and Details(2)
the coefficient vectors cA1 and cD1 -- half the length of the original signal -- 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. That is, there are several ways to reassemble the original signal:
The Discrete Wavelet Transform(3)
Example of the Discrete Wavelet Transform
Multiple Level decomposition
The decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is broken 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, you'll select a suitable number of levels based on the nature of the signal, or on a suitable criterion such as entropy
Wavelet Package Aanlysis
In wavelet packet analysis, the details as well as the approximations can be split. This yields more than 2 2 different ways to encode the signal. This is the wavelet packet decomposition tree.
n 1
For instance, 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。
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.m test_1_02.m
Multi-step Decomposition and Reconstrction
This process involves two aspects: breaking up a signal to obtain the wavelet coefficients, reassembling the signal from the coefficients.
Reconstruction Filters
ψ The 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 reconstruction phases that are closely related (but not identical), we can "cancel out" the effects of aliasing.
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) 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
Reconstruction of Approximations and Details(1)
Reconstruct our original signal from the coefficients of the approximations and details. Reconstruct the approximations and details themselves from their coefficient vectors. Reconstruct the first-level approximation A1 from the coefficient vector cA1 Reconstruct the first-level detail D1 from the coefficient vector cD1 S=A1+D1
Wavelet Reconstruction
The process of Assembling these components back into the original signal without loss of information is called reconstruction, or synthesis. The mathematical manipulation that effects synthesis is called the inverse discrete wavelet transform (IDWT)
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-frequency components.
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, wavelets. 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 convolving 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.