CN103971021A - Method for quantitative wavelet basis selection based on maximum common information entropy - Google Patents

Abstract

Translated fromChinese

本发明公开了一种基于最大共信息熵的定量小波基选择方法，包括以下步骤：采用小波基库中预选出的各小波基作为小波包变换的小波基函数，对采集到的信号进行小波包变换，获得各个频带的小波包系数；根据获得的各个频带的小波包系数，对小波包节点[4，0]的小波包系数进行重构；把原始信号看作数据序列X，把数据序列X经4层小波包变换得到的小波包节点[4，0]的重构信号看作数据序列Y，分别计算采用各小波基对信号进行分解时两个数据序列的共信息熵值；通过对各个共信息熵值的比较，选择共信息熵值最大时所采用的小波基作为最适合分析该类信号的小波基，实现定量化小波基的选择。本发明实现了定量化小波基的选择，大大提高了选择小波基的准确率。

The invention discloses a method for selecting a quantitative wavelet base based on maximum common information entropy, which comprises the following steps: using each wavelet base preselected in a wavelet base library as a wavelet base function for wavelet packet transformation, and performing wavelet packet processing on the collected signal Transform to obtain the wavelet packet coefficients of each frequency band; according to the obtained wavelet packet coefficients of each frequency band, reconstruct the wavelet packet coefficients of the wavelet packet node [4, 0]; regard the original signal as a data sequence X, and regard the data sequence X The reconstructed signal of the wavelet packet node [4, 0] obtained by the 4-layer wavelet packet transformation is regarded as the data sequence Y, and the common information entropy value of the two data sequences when the signal is decomposed by each wavelet base is calculated respectively; For the comparison of the common information entropy value, the wavelet base used when the common information entropy value is the largest is selected as the most suitable wavelet base for analyzing this type of signal, so as to realize the selection of quantitative wavelet base. The invention realizes the selection of the quantitative wavelet base, and greatly improves the accuracy rate of selecting the wavelet base.

Description

Translated fromChinese

基于最大共信息熵的定量小波基选择方法Quantitative wavelet base selection method based on maximum common information entropy

技术领域technical field

本发明涉及一种以信息论中的共信息熵来开展信号分析的定量小波基选择方法。The invention relates to a quantitative wavelet base selection method for carrying out signal analysis with common information entropy in information theory.

背景技术Background technique

近年来，小波理论在非平稳信号分析中得到了广泛的应用，通过小波变换分析处理信号时，能否选择合适的小波基函数直接关系到特征提取的成败。然而目前，在应用小波变换进行信号分解时，绝大多数研究者采用了定性的方法根据小波形状去选择小波，这种方法依赖于研究者的主观判断，准确率较低。如何通过研究原始信号和小波包系数之间的关系从多种类型的小波基函数中定量地选出适合于特定应用的小波基进行有效的信号特征提取，当前没有公开的解决办法。In recent years, wavelet theory has been widely used in the analysis of non-stationary signals. When analyzing and processing signals through wavelet transform, whether the appropriate wavelet basis function can be selected is directly related to the success or failure of feature extraction. However, at present, when using wavelet transform to decompose signals, most researchers use a qualitative method to select wavelets according to wavelet shapes. This method relies on the subjective judgment of researchers, and the accuracy is low. How to quantitatively select a wavelet basis suitable for a specific application from various types of wavelet basis functions by studying the relationship between the original signal and the wavelet packet coefficients for effective signal feature extraction, there is currently no public solution.

发明内容Contents of the invention

发明目的：为了克服现有技术中存在的不足，本发明提供一种基于最大共信息熵的定量小波基选择方法。Purpose of the invention: In order to overcome the deficiencies in the prior art, the present invention provides a quantitative wavelet base selection method based on maximum common information entropy.

技术方案：为解决上述技术问题，本发明的一种基于最大共信息熵的定量小波基选择方法，包括以下步骤：Technical solution: In order to solve the above-mentioned technical problems, a kind of quantitative wavelet base selection method based on maximum common information entropy of the present invention comprises the following steps:

(1)根据采集到的信号，得到信号序列x(i)，i＝1，2，...，N，i表示等时间间隔采样时间点，N为信号长度；(1) According to the collected signal, obtain the signal sequence x(i), i=1, 2, ..., N, i represent equal time interval sampling time points, and N is the signal length;

(2)采用小波基库中预选出的各小波基作为小波包变换的小波基函数，按照公式(a)对采集到的信号x(i)进行4层小波包变换；(2) Using each wavelet base pre-selected in the wavelet base library as the wavelet base function of wavelet packet transformation, according to the formula (a), carry out 4 layers of wavelet packet transformation to the collected signal x(i);

${\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>+</span>+</span>\mathrm{<span><span>1,2</span>1,2</span>}\mathrm{<span><span>k</span>k</span>}}<span><span>=</span>=</span>\underset{\mathrm{<span><span>m</span>m</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>h</span>h</span>}<span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>,</span>,</span>\mathrm{<span><span>k</span>k</span>}}$

${\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>+</span>+</span>\mathrm{<span><span>1,2</span>1,2</span>}\mathrm{<span><span>k</span>k</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\underset{\mathrm{<span><span>m</span>m</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>g</span>g</span>}<span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>,</span>,</span>\mathrm{<span><span>k</span>k</span>}}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>a</span>a</span>}<span><span>)</span>)</span>$

获得各个频带的小波包系数α_j，k,m,其中，$j=\mathrm{0,1,2,3},k=\mathrm{0,1},...{,2}^j-1,m=\mathrm{0,1},...,2^{n_0-j}-1,$n₀＝log₂N≥J，J为最大分解层数，n为滤波器阶数，h和g分别为小波包分解的低通滤波器和高通滤波器组系数，且满足公式g(n)＝(-1)ⁿh(1-n)，α_j，k＝{α_j，k,1,α_j，k,2,...，α_j，k，m}表示小波包分解中第j层、第k个子空间的小波包系数，m为该子空间小波包系数的个数；Obtain the wavelet packet coefficients α_j,k,m of each frequency band, where,$j=\mathrm{0,1,2,3},k=\mathrm{0,1},...{,2}^j-1,m=\mathrm{0,1},...,2^{{\mathrm{no}}_0-j}-1,$ n₀ ＝log₂ N≥J, J is the maximum number of decomposition layers, n is the filter order, h and g are the low-pass filter and high-pass filter bank coefficients of wavelet packet decomposition respectively, and satisfy the formula g(n) ＝(-1)ⁿ h(1-n), α_{j, k} ＝{α_{j, k, 1} , α_{j, k, 2} ,..., α_{j, k, m} } represent the first The wavelet packet coefficients of layer j and the kth subspace, m is the number of wavelet packet coefficients of the subspace;

(3)根据所述步骤(2)获得的各个频带的小波包系数，按照公式(b)对小波包节点[4，0]的小波包系数进行重构；(3) according to the wavelet packet coefficients of each frequency band obtained in the step (2), the wavelet packet coefficients of the wavelet packet node [4,0] are reconstructed according to formula (b);

${\mathrm{<span><span>y</span>the\; y</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>,</span>,</span>\mathrm{<span><span>k</span>k</span>}}<span><span>=</span>=</span>\underset{\mathrm{<span><span>n</span>no</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>+</span>+</span>\mathrm{<span><span>1,2</span>1,2</span>}\mathrm{<span><span>k</span>k</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>g</span>g</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>+</span>+</span>\mathrm{<span><span>1,2</span>1,2</span>}\mathrm{<span><span>k</span>k</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>b</span>b</span>}<span><span>)</span>)</span>$

得到重构信号y，其中，h1和g1分别为小波包重构的低通滤波器和高通滤波器组系数。A reconstructed signal y is obtained, where h1 and g1 are the low-pass filter and high-pass filter bank coefficients of wavelet packet reconstruction respectively.

(4)把原始信号x看作数据序列X，把数据序列X经4层小波包变换得到的小波包节点[4，0]的重构信号y看作数据序列Y，根据公式(c)(d)分别计算数据序列X和数据序列Y的香农熵H(X)和H(Y)，根据公式(e)计算两个数据序列X、Y的联合熵H（X，Y)；(4) The original signal x is regarded as data sequence X, and the reconstructed signal y of wavelet packet node [4, 0] obtained by data sequence X through 4-layer wavelet packet transformation is regarded as data sequence Y, according to the formula (c)( d) Calculate the Shannon entropy H(X) and H(Y) of the data sequence X and the data sequence Y respectively, and calculate the joint entropy H(X, Y) of the two data sequences X and Y according to the formula (e);

$\mathrm{<span><span>H</span>h</span>}<span><span>(</span>(</span>\mathrm{<span><span>X</span>x</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span>\underset{\mathrm{<span><span>x</span>x</span>}<span><span>\&Element;</span>\&Element;</span>\mathrm{<span><span>X</span>x</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span>\mathrm{<span><span>log</span>log</span>}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>c</span>c</span>}<span><span>)</span>)</span>$

$\mathrm{<span><span>H</span>h</span>}<span><span>(</span>(</span>\mathrm{<span><span>Y</span>Y</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span>\underset{\mathrm{<span><span>x</span>x</span>}<span><span>\&Element;</span>\&Element;</span>\mathrm{<span><span>Y</span>Y</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>y</span>the\; y</span>}<span><span>)</span>)</span>\mathrm{<span><span>log</span>log</span>}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>y</span>the\; y</span>}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>d</span>d</span>}<span><span>)</span>)</span>$

$\mathrm{<span><span>H</span>h</span>}<span><span>(</span>(</span>\mathrm{<span><span>X</span>x</span>}<span><span>,</span>,</span>\mathrm{<span><span>Y</span>Y</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span>\underset{\mathrm{<span><span>x</span>x</span>}<span><span>\&Element;</span>\&Element;</span>\mathrm{<span><span>X</span>x</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\underset{\mathrm{<span><span>y</span>the\; y</span>}<span><span>\&Element;</span>\&Element;</span>\mathrm{<span><span>Y</span>Y</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>,</span>,</span>\mathrm{<span><span>y</span>the\; y</span>}<span><span>)</span>)</span>\mathrm{<span><span>log</span>log</span>}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>,</span>,</span>\mathrm{<span><span>y</span>the\; y</span>}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>e</span>e</span>}<span><span>)</span>)</span>$

其中，p(x)和p(y)则分别代表数据序列X和数据序列Y的概率密度，p(x，y)表示数据序列X和数据序列Y的联合概率密度。Among them, p(x) and p(y) represent the probability density of data sequence X and data sequence Y respectively, and p(x, y) represents the joint probability density of data sequence X and data sequence Y.

(5)按照公式(f)分别计算采用各小波基函数对信号进行分解时两个数据序列的共信息熵值；(5) Calculate the total information entropy values of the two data sequences when each wavelet basis function is used to decompose the signal according to formula (f);

I(X；Y)＝-H(X，Y)+H(X)+H(Y) (f)I(X;Y)=-H(X,Y)+H(X)+H(Y) (f)

(6)选择共信息熵值最大时所采用的小波基作为最适合分析该类信号的小波基。(6) Select the wavelet base that is used when the value of common information entropy is the largest as the wavelet base that is most suitable for analyzing this type of signal.

进一步地，所述预选出的小波基函数为Haar小波、Db2小波、Db4小波、Db6小波、Db8小波、Db10小波、Coif1小波、Coif2小波、Coif3小波、Coif4小波、Coif5小波、Sym2小波、Sym3小波、Sym4小波、Sym5小波、Sym6小波、Sym7小波、Sym8小波、Bior1.3小波、Bior2.4小波、Bior2.6小波、Bior4.4小波、Bior5.5小波、Bior6.8小波、rBio1.3小波、rBio2.4小波、rBio2.6小波、rBio4.4小波、rBio5.5小波、rBio6.8小波。Further, the pre-selected wavelet basis functions are Haar wavelet, Db2 wavelet, Db4 wavelet, Db6 wavelet, Db8 wavelet, Db10 wavelet, Coif1 wavelet, Coif2 wavelet, Coif3 wavelet, Coif4 wavelet, Coif5 wavelet, Sym2 wavelet, Sym3 wavelet , Sym4 wavelet, Sym5 wavelet, Sym6 wavelet, Sym7 wavelet, Sym8 wavelet, Bior1.3 wavelet, Bior2.4 wavelet, Bior2.6 wavelet, Bior4.4 wavelet, Bior5.5 wavelet, Bior6.8 wavelet, rBio1.3 wavelet , rBio2.4 wavelet, rBio2.6 wavelet, rBio4.4 wavelet, rBio5.5 wavelet, rBio6.8 wavelet.

有益效果：本发明相对于现有技术而言，通过原始信号和小波包系数之间的关系，基于最大共信息熵的定量小波基选择方法，实现了定量化小波基的选择，大大提高了选择小波基的准确率。Beneficial effects: Compared with the prior art, the present invention realizes the selection of quantitative wavelet bases through the relationship between the original signal and wavelet packet coefficients and the quantitative wavelet base selection method based on the maximum common information entropy, greatly improving the selection Accuracy of wavelet basis.

附图说明Description of drawings

图1为Db10小波信号加入信噪比SNR＝20dB的高斯白噪声后的样本信号。Fig. 1 is a sample signal after adding Gaussian white noise with a signal-to-noise ratio SNR=20dB to the Db10 wavelet signal.

图2为节点[4，0]的小波包系数。Figure 2 shows the wavelet packet coefficients of node [4,0].

图3为数据序列X经4层小波包变换得到的小波包节点[4，0]的重构信号小波包节点[4，0]的重构信号的时域图。Fig. 3 is a time-domain diagram of the reconstructed signal of the wavelet packet node [4, 0] obtained by transforming the data sequence X through the 4-layer wavelet packet.

图4为两组数据序列的共信息熵、联合熵和香农熵之间的关系。Figure 4 shows the relationship among the common information entropy, joint entropy and Shannon entropy of two sets of data sequences.

具体实施方式Detailed ways

下面结合附图对本发明作更进一步的说明。The present invention will be further described below in conjunction with the accompanying drawings.

一种基于最大共信息熵的定量小波基选择方法，包括以下步骤：A quantitative wavelet base selection method based on maximum common information entropy, comprising the following steps:

(1)根据采集到的信号，得到信号序列x(i)，i＝1，2，...，N，i表示等时间间隔采样时间点，N为信号长度；(1) According to the collected signal, obtain the signal sequence x(i), i=1, 2, ..., N, i represent equal time interval sampling time points, and N is the signal length;

(2)采用小波基库中预选出的各小波基作为小波包变换的小波基函数，所述预选出的小波基函数为Haar小波、Db2小波、Db4小波、Db6小波、Db8小波、Db10小波、Coif1小波、Coif2小波、Coif3小波、Coif4小波、Coif5小波、Sym2小波、Sym3小波、Sym4小波、Sym5小波、Sym6小波、Sym7小波、Sym8小波、Bior1.3小波、Bior2.4小波、Bior2.6小波、Bior4.4小波、Bior5.5小波、Bior6.8小波、rBio1.3小波、rBio2.4小波、rBio2.6小波、rBio4.4小波、rBio5.5小波、rBio6.8小波。按照公式(a)对采集到的信号x(i)进行4层小波包变换；(2) Each wavelet basis pre-selected in the wavelet base library is used as the wavelet basis function of wavelet packet transformation, and the pre-selected wavelet basis functions are Haar wavelet, Db2 wavelet, Db4 wavelet, Db6 wavelet, Db8 wavelet, Db10 wavelet, Coif1 wavelet, Coif2 wavelet, Coif3 wavelet, Coif4 wavelet, Coif5 wavelet, Sym2 wavelet, Sym3 wavelet, Sym4 wavelet, Sym5 wavelet, Sym6 wavelet, Sym7 wavelet, Sym8 wavelet, Bior1.3 wavelet, Bior2.4 wavelet, Bior2.6 wavelet , Bior4.4 wavelet, Bior5.5 wavelet, Bior6.8 wavelet, rBio1.3 wavelet, rBio2.4 wavelet, rBio2.6 wavelet, rBio4.4 wavelet, rBio5.5 wavelet, rBio6.8 wavelet. Carry out 4 layers of wavelet packet transformation to the collected signal x(i) according to the formula (a);

${\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>+</span>+</span>\mathrm{<span><span>1,2</span>1,2</span>}\mathrm{<span><span>k</span>k</span>}}<span><span>=</span>=</span>\underset{\mathrm{<span><span>m</span>m</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>h</span>h</span>}<span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>,</span>,</span>\mathrm{<span><span>k</span>k</span>}}$

${\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>+</span>+</span>\mathrm{<span><span>1,2</span>1,2</span>}\mathrm{<span><span>k</span>k</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\underset{\mathrm{<span><span>m</span>m</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>g</span>g</span>}<span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>,</span>,</span>\mathrm{<span><span>k</span>k</span>}}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>a</span>a</span>}<span><span>)</span>)</span>$

获得各个频带的小波包系数α_j，k,m，其中，$j=\mathrm{0,1,2,3},k=\mathrm{0,1},...{,2}^j-1,m=\mathrm{0,1},...,2^{n_0-j}-1,$n₀＝log₂N≥J，J为最大分解层数，n为滤波器阶数，h和g分别为小波包分解的低通滤波器和高通滤波器组系数，且满足公式g(n)＝(-i)ⁿh(1-n)，α_j，k＝{α_j，k,1，α_j，k,2，...，α_j，k,m}表示小波包分解中第j层、第k个子空间的小波包系数，m为该子空间小波包系数的个数；Obtain the wavelet packet coefficients α_j,k,m of each frequency band, where,$j=\mathrm{0,1,2,3},k=\mathrm{0,1},...{,2}^j-1,m=\mathrm{0,1},...,2^{{\mathrm{no}}_0-j}-1,$ n₀ ＝log₂ N≥J, J is the maximum number of decomposition layers, n is the filter order, h and g are the low-pass filter and high-pass filter bank coefficients of wavelet packet decomposition respectively, and satisfy the formula g(n) ＝(-i)ⁿ h(1-n), α_{j, k} ＝{α_{j, k,1} , α_{j, k,2} ,..., α_{j, k,m} } represent the first The wavelet packet coefficients of layer j and the kth subspace, m is the number of wavelet packet coefficients of the subspace;

(3)根据所述步骤(2)获得的各个频带的小波包系数，按照公式(b)对小波包节点[4，0]的小波包系数进行重构；(3) according to the wavelet packet coefficients of each frequency band obtained in the step (2), the wavelet packet coefficients of the wavelet packet node [4,0] are reconstructed according to formula (b);

${\mathrm{<span><span>y</span>the\; y</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>,</span>,</span>\mathrm{<span><span>k</span>k</span>}}<span><span>=</span>=</span>\underset{\mathrm{<span><span>n</span>no</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}<span><span>(</span>(</span>{\mathrm{<span><span>h</span>h</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>+</span>+</span>\mathrm{<span><span>1,2</span>1,2</span>}\mathrm{<span><span>k</span>k</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>g</span>g</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}\mathrm{<span><span>n</span>no</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&alpha;</span>\&alpha;</span>}}_{\mathrm{<span><span>j</span>j</span>}<span><span>+</span>+</span>\mathrm{<span><span>1,2</span>1,2</span>}\mathrm{<span><span>k</span>k</span>}<span><span>+</span>+</span>\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>b</span>b</span>}<span><span>)</span>)</span>$

得到重构信号y，其中，h₁和g₁分别为小波包重构的低通滤波器和高通滤波器组系数。A reconstructed signal y is obtained, where h₁ and g₁ are the low-pass filter and high-pass filter bank coefficients of wavelet packet reconstruction respectively.

(4)把原始信号x看做数据序列X，把数据序列X经4层小波包变换得到的小波包节点[4，0]的重构信号y看做数据序列Y，根据公式(c)(d)分别计算数据序列X和数据序列Y的香农熵H(X)和H(Y)，根据公式(e)计算两个数据序列X、Y的联合熵H(X，Y)；(4) The original signal x is regarded as the data sequence X, and the reconstructed signal y of the wavelet packet node [4, 0] obtained by the data sequence X through 4-layer wavelet packet transformation is regarded as the data sequence Y, according to the formula (c)( d) Calculate the Shannon entropy H(X) and H(Y) of the data sequence X and the data sequence Y respectively, and calculate the joint entropy H(X, Y) of the two data sequences X and Y according to the formula (e);

$\mathrm{<span><span>H</span>h</span>}<span><span>(</span>(</span>\mathrm{<span><span>X</span>x</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span>\underset{\mathrm{<span><span>x</span>x</span>}<span><span>\&Element;</span>\&Element;</span>\mathrm{<span><span>X</span>x</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span>\mathrm{<span><span>log</span>log</span>}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>c</span>c</span>}<span><span>)</span>)</span>$

$\mathrm{<span><span>H</span>h</span>}<span><span>(</span>(</span>\mathrm{<span><span>Y</span>Y</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span>\underset{\mathrm{<span><span>x</span>x</span>}<span><span>\&Element;</span>\&Element;</span>\mathrm{<span><span>Y</span>Y</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>y</span>the\; y</span>}<span><span>)</span>)</span>\mathrm{<span><span>log</span>log</span>}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>y</span>the\; y</span>}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>d</span>d</span>}<span><span>)</span>)</span>$

$\mathrm{<span><span>H</span>h</span>}<span><span>(</span>(</span>\mathrm{<span><span>X</span>x</span>}<span><span>,</span>,</span>\mathrm{<span><span>Y</span>Y</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>-</span>-</span>\underset{\mathrm{<span><span>x</span>x</span>}<span><span>\&Element;</span>\&Element;</span>\mathrm{<span><span>X</span>x</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\underset{\mathrm{<span><span>y</span>the\; y</span>}<span><span>\&Element;</span>\&Element;</span>\mathrm{<span><span>Y</span>Y</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>,</span>,</span>\mathrm{<span><span>y</span>the\; y</span>}<span><span>)</span>)</span>\mathrm{<span><span>log</span>log</span>}\mathrm{<span><span>p</span>p</span>}<span><span>(</span>(</span>\mathrm{<span><span>x</span>x</span>}<span><span>,</span>,</span>\mathrm{<span><span>y</span>the\; y</span>}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>e</span>e</span>}<span><span>)</span>)</span>$

其中，p(x)和p(y)则分别代表数据序列X和数据序列Y的概率密度，p(x，y)表示数据序列X和数据序列Y的联合概率密度。Among them, p(x) and p(y) represent the probability density of data sequence X and data sequence Y respectively, and p(x, y) represents the joint probability density of data sequence X and data sequence Y.

(5)按照公式(f)分别计算采用各小波基函数对信号进行分解时两个数据序列的共信息熵值；(5) Calculate the total information entropy values of the two data sequences when each wavelet basis function is used to decompose the signal according to formula (f);

I(X；Y)＝-H(X，Y)+H(X)+H(Y) (f)I(X;Y)=-H(X,Y)+H(X)+H(Y) (f)

(6)选择共信息熵值最大时所采用的小波基作为最适合分析该类信号的小波基。(6) Select the wavelet base that is used when the value of common information entropy is the largest as the wavelet base that is most suitable for analyzing this type of signal.

利用本发明对Db10小波信号加入信噪比SNR＝20dB的高斯白噪声后的样本信号进行定量小波基的选取。The present invention is used to select the quantitative wavelet basis for the sample signal after adding Gaussian white noise with a signal-to-noise ratio SNR=20dB to the Db10 wavelet signal.

如图1，样本信号由Db10小波信号加入信噪比SNR＝20dB的高斯白噪声生成，对样本信号进行等时间间隔采样，得到序列x(i)，i＝1，2，…，N。其中，横坐标代表采样点数N＝20000，纵坐标代表样本信号的幅值。As shown in Figure 1, the sample signal is generated by adding Gaussian white noise with SNR=20dB to the Db10 wavelet signal. The sample signal is sampled at equal time intervals to obtain the sequence x(i), i=1, 2,...,N. Wherein, the abscissa represents the number of sampling points N=20000, and the ordinate represents the amplitude of the sample signal.

如图2，采用小波基库中预选出的各小波基作为小波包变换的小波基函数，按照公式(a)对样本信号x(i)进行4层小波包变换，获得节点[4，0]的小波包系数α_4，0。其中，横坐标代表小波包节点[4，0]的小波包系数的个数，纵坐标代表小波包节点[4，0]的小波包系数值。As shown in Figure 2, each wavelet basis pre-selected in the wavelet basis library is used as the wavelet basis function of wavelet packet transformation, and the sample signal x(i) is subjected to 4-layer wavelet packet transformation according to the formula (a), and the node [4, 0] is obtained The wavelet packet coefficient α_4,0 . Wherein, the abscissa represents the number of wavelet packet coefficients of the wavelet packet node [4, 0], and the ordinate represents the value of the wavelet packet coefficient of the wavelet packet node [4, 0].

如图3，根据获得的小波包系数，利用公式(b)对小波包节点[4，0]的小波包系数进行重构，得到重构信号y。其中，横坐标代表重构后的样本信号长度，纵坐标代表重构信号y的幅值。As shown in Figure 3, according to the obtained wavelet packet coefficients, use the formula (b) to reconstruct the wavelet packet coefficients of the wavelet packet node [4, 0] to obtain the reconstructed signal y. Wherein, the abscissa represents the length of the reconstructed sample signal, and the ordinate represents the amplitude of the reconstructed signal y.

如图4，横坐标分别代表数据序列X和数据序列Y的数据长度，纵坐标分别代表数据序列X和数据序列Y的幅值，H(X)和H(Y)分别代表数据序列X和数据序列Y的香农熵，H(X，Y)代表两个数据序列X、Y的联合熵H(X，Y)。如图4可知，共信息熵I(X；Y)，描述了一组数据序列中所含有的另一组数据序列的信息多少，其数值的大小往往反映了两组数据序列的相似程度，共信息熵越大则两组数据序列就越相似。As shown in Figure 4, the abscissa represents the data length of data sequence X and data sequence Y respectively, the ordinate represents the amplitude of data sequence X and data sequence Y respectively, and H(X) and H(Y) represent data sequence X and data sequence Y respectively. The Shannon entropy of sequence Y, H(X, Y) represents the joint entropy H(X, Y) of two data sequences X, Y. As shown in Figure 4, the total information entropy I(X; Y) describes how much information is contained in another set of data sequences in one set of data sequences, and its numerical value often reflects the degree of similarity between the two sets of data sequences. The greater the information entropy, the more similar the two sets of data sequences are.

表1Table 1

小波基wavelet共信息熵total information entropy小波基wavelet共信息熵total information entropy小波基wavelet共信息熵total information entropyHaarHaar20.3220.32Coif5Coif524.3024.30Bior2.6Bior2.624.2824.28Db2Db224.2624.26Sym2Sym224.2824.28Bior4.4Bior4.424.2424.24Db4Db424.2924.29Sym3Sym324.2724.27Bior5.5Bior5.524.2724.27Db6Db624.3024.30Sym4Sym424.2524.25Bior6.8Bior6.824.2624.26

Db8Db824.2924.29Sym5Sym524.2724.27rBio1.3rBio1.324.2724.27Db10Db1024.3124.31Sym6Sym624.2724.27rBio2.4rBio2.424.2524.25Coif1Coif124.2624.26Sym7Sym724.2824.28rBio2.6rBio2.624.2724.27Coif2Coif224.2424.24Sym8Sym824.2824.28rBio4.4rBio4.424.2524.25Coif3Coif324.2824.28Bior1.3Bior1.320.3120.31rBio5.5rBio5.524.2824.28Coif4Coif424.2624.26Bior2.4Bior2.424.2824.28rBio6.8rBio6.824.2924.29

参见表1，把原始样本信号x(i)看做数据序列X，把数据序列X经4层小波包变换得到的小波包节点[4，0]的重构信号y看做数据序列Y，根据公式(c)(d)分别计算数据序列X和数据序列Y的香农熵H(X)和H(Y)，根据公式(e)计算两个数据序列X、Y的联合熵H(X，Y），再按照公式(f)分别计算采用各小波基函数对信号进行分解时两个数据序列的共信息熵值。Referring to Table 1, the original sample signal x(i) is regarded as the data sequence X, and the reconstructed signal y of the wavelet packet node [4, 0] obtained by the data sequence X through 4-layer wavelet packet transformation is regarded as the data sequence Y, according to Formulas (c) and (d) calculate the Shannon entropy H(X) and H(Y) of data sequence X and data sequence Y respectively, and calculate the joint entropy H(X, Y) of two data sequences X and Y according to formula (e) ), and then calculate the co-information entropy values of the two data sequences when using each wavelet basis function to decompose the signal according to the formula (f).

由于小波变换的本质是利用伸缩移位后的小波基与待分析信号之间的相关性操作，小波基与待分析信号越相似，其特征提取的能力越强。从表1中可以看出，采用小波基库中的Db10小波进行信号分析后的重构信号与样本信号的共信息熵值最大，根据本发明所述的定量小波基选择方法，选择Db10小波作为最为适合分析样本信号的小波基。又己知样本信号由Db10小波信号加入信噪比SNR＝20dB的高斯白噪声生成，所述样本信号与Db10小波基最为相似，从而，验证了本发明提出的基于最大共信息熵的定量小波基选择方法的有效性。Because the essence of wavelet transform is to use the correlation operation between the stretched and shifted wavelet base and the signal to be analyzed, the more similar the wavelet base is to the signal to be analyzed, the stronger the ability of feature extraction. As can be seen from Table 1, the joint information entropy value of the reconstructed signal and the sample signal after signal analysis using the Db10 wavelet in the wavelet base library is the largest, and according to the quantitative wavelet base selection method of the present invention, the Db10 wavelet is selected as The wavelet basis most suitable for analyzing sample signals. Known sample signal is added the Gaussian white noise generation of SNR=20dB by Db10 wavelet signal again, described sample signal and Db10 wavelet base are the most similar, thereby, verified the quantitative wavelet base based on maximum common information entropy that the present invention proposes The validity of the selection method.

以上所述仅是本发明的优选实施方式，应当指出：对于本技术领域的普通技术人员来说，在不脱离本发明原理的前提下，还可以做出若干改进和润饰，这些改进和润饰也应视为本发明的保护范围。The above is only a preferred embodiment of the present invention, it should be pointed out that for those of ordinary skill in the art, without departing from the principle of the present invention, some improvements and modifications can also be made, and these improvements and modifications are also possible. It should be regarded as the protection scope of the present invention.

Claims (2)

1. the quantitative Selection of Wavelet Basis method based on maximum co-information entropy, is characterized in that: comprise the following steps:

(1) according to the signal collecting, obtain burst x (i), i=1,2 ..., N, i represents constant duration sampling time point, N is signal length;

(2) each wavelet basis of selecting in advance in employing wavelet basis storehouse, as the wavelet basis function of wavelet package transforms, carries out 4 layers of wavelet package transforms according to formula (a) to the signal x (i) collecting;

${\mathrm{\&alpha;}}_{j+\mathrm{1,2}k}=\underset{m}{\mathrm{\&Sigma;}}h(m-2n){\mathrm{\&alpha;}}_{j,k}$

${\mathrm{\&alpha;}}_{j+\mathrm{1,2}k+1}=\underset{m}{\mathrm{\&Sigma;}}g(m-2n){\mathrm{\&alpha;}}_{j,k}---\left(a\right)$

Obtain the wavelet packet coefficient α of each frequency band_{j, k, m}, wherein,$j=\mathrm{0,1,2,3},k=\mathrm{0,1},...{,2}^j-1,m=\mathrm{0,1},...,2^{n_0-j}-1,$N₀=log₂n>=J, J is maximum decomposition level number, and n is filter order, and h and g are respectively low-pass filter and the Hi-pass filter group coefficient of WAVELET PACKET DECOMPOSITION, and meet formula g (n)=(1)ⁿh (1-n), α_{j, k}={ α_{j, k, 1}, α_{j, k, 2}..., α_{j, k, m}representing the wavelet packet coefficient of j layer in WAVELET PACKET DECOMPOSITION, k sub spaces, m is the number of this subspace wavelet packet coefficient;

(3) wavelet packet coefficient of each frequency band obtaining according to described step (2), is reconstructed the wavelet packet coefficient of wavelet packet node [4,0] according to formula (b);

$y_{j,k}=\underset{n}{\mathrm{\&Sigma;}}(h_1(m-2n){\mathrm{\&alpha;}}_{j+\mathrm{1,2}k}+g_1(m-2n){\mathrm{\&alpha;}}_{j+\mathrm{1,2}k+1})---\left(b\right)$

Obtain reconstruction signal y, wherein, h₁and g₁be respectively low-pass filter and the Hi-pass filter group coefficient of wavelet package reconstruction.(4) original signal x is regarded as to data sequence X, the wavelet packet node [4 that data sequence X is obtained through 4 layers of wavelet package transforms, 0] reconstruction signal y regards data sequence Y as, according to (d) respectively Shannon entropy H (X) and the H (Y) of computational data sequence X and data sequence Y of formula (c), calculate the combination entropy H (X, Y) of two data sequence X, Y according to formula (e);

$H\left(X\right)=-\underset{x\&Element;X}{\mathrm{\&Sigma;}}p\left(x\right)\log p\left(x\right)---\left(c\right)$

$H\left(Y\right)=-\underset{x\&Element;Y}{\mathrm{\&Sigma;}}p\left(y\right)\log p\left(y\right)---\left(d\right)$

$H(X,Y)=-\underset{x\&Element;X}{\mathrm{\&Sigma;}}\underset{y\&Element;Y}{\mathrm{\&Sigma;}}p(x,y)\log p(x,y)---\left(e\right)$

Wherein, p (x) and p (y) distinguish the probability density of representative data sequence X and data sequence Y, and p (x, y) represents the joint probability density of data sequence X and data sequence Y.

(5) calculate respectively according to formula (f) the co-information entropy that adopts each wavelet basis function two data sequences when signal is decomposed;

I(X；Y)＝-H(X，Y)+H(X)+H(Y) (f)

(6) wavelet basis adopting while selecting co-information entropy maximum is as the wavelet basis of the most applicable such signal of analysis.

2. the quantitative Selection of Wavelet Basis method based on maximum co-information entropy according to claim 1, it is characterized in that: the wavelet basis function of selecting in advance in described step (2) is Haar small echo, Db2 small echo, Db4 small echo, Db6 small echo, Db8 small echo, Db10 small echo, Coif1 small echo, Coif2 small echo, Coif3 small echo, Coif4 small echo, Coif5 small echo, Sym2 small echo, Sym3 small echo, Sym4 small echo, Sym5 small echo, Sym6 small echo, Sym7 small echo, Sym8 small echo, Bior1.3 small echo, Bior2.4 small echo, Bior2.6 small echo, Bior4.4 small echo, Bior5.5 small echo, Bior6.8 small echo, rBio1.3 small echo, rBio2.4 small echo, rBio2.6 small echo, rBio4.4 small echo, rBio5.5 small echo, rBio6.8 small echo.