﻿ 基于小波包分析的时反聚焦算法改进
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 应用科技  2018, Vol. 45 Issue (6): 47-52  DOI: 10.11991/yykj.201712006 0

### 引用本文

KANG Weixin, LI Haifeng. Improvement of time reversal focusing algorithm based on wavelet packet analysis[J]. Applied Science and Technology, 2018, 45(6), 47-52. DOI: 10.11991/yykj.201712006.

### 文章历史

Improvement of time reversal focusing algorithm based on wavelet packet analysis
KANG Weixin, LI Haifeng
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: In the detection of pile foundation, the interference of multipath effect leads to that the damage echo signal with small scattering degree will be flooded in the multipath signals, and it is hard to distinguish the damage signal. If the signal is mixed with noise, the damaged signal will be flooded, and the signal to noise ratio of the detection will decrease, degrading quality of the signal detection system. In order to solve the above problems, the improved time-reversal focusing algorithm based on wavelet analysis is proposed in this paper. Namely, denoise the signal wavelet generated from the signal measured on the spot, and then analyze the wavelet packet for the denoised signal, so as to achieve focusing and noise suppression at the dammage point of the signal. Experimental results show that the algorithm has better focusing and noise suppression effects.
Keywords: pile foundation    multi-path effect    noise    signal to noise ratio    wavelet denoise    wavelet packet decomposition    time reversal    focus

1 小波去噪

 $f\left( t \right) = s\left( t \right) + n\left( t \right)$

 ${w_f}\left( {j,k} \right) = {2^{\frac{j}{2}}}\sum\limits_{n = 0}^{N - 1} {f\left( n \right)} \psi \left( {{2^j}n - k} \right)$

 $\begin{array}{*{20}{c}} {{s_f}\left( {j + 1,k} \right) = {s_f}\left( {j,k} \right) {\cdot} h\left( {j,k} \right)} \\ {{w_f}\left( {j + 1,k} \right) = {s_f}\left( {j,k} \right) {\cdot} g\left( {j,k} \right)} \end{array}$

 $\begin{gathered} {S_f}\left( {j - 1,k} \right) = {S_f}\left( {j,k} \right) {\cdot} {\overset{\frown} h} \left( {j,k} \right) + {w_f}\left( {j,k} \right){\cdot}{\overset{\frown} g} \left( {j,k} \right) \\ \end{gathered}$

 ${\omega _{j,k}} = {u_{j,k}} + {v_{j,k}}$

 $T = \sigma \sqrt {2\ln n} /\exp (\left( {l - 1} \right)/2)$

 ${{\overset{\frown} \omega} _{j,k}} = \left\{ {\begin{array}{*{20}{c}} {{\omega _{j,k}} + T - \displaystyle\frac{{2T}}{{\exp \left( {\left| {\displaystyle\frac{{{\omega _{j,k}}}}{T}} \right| - 1} \right) + 1 + n}}},\;\;{{\omega _{j,k}} \leqslant - T} \\ {\displaystyle\frac{{2\operatorname{sgn} \left( {{\omega _{j,k}}} \right){{\left| {{\omega _{j,k}}} \right|}^{n + 1}}}}{{\left[ {\exp \left( {\left| {\displaystyle\frac{{{\omega _{j,k}}}}{T}} \right| - 1} \right) + 1 + n} \right]{T^n}}}},\;\;{\left| {{\omega _{j,k}}} \right| < T} \quad\quad\quad\\ {{\omega _{j,k}} - T + \displaystyle\frac{{2T}}{{\exp \left( {\left| {\displaystyle\frac{{{\omega _{j,k}}}}{T}} \right| - 1} \right) + 1 + n}}},\;\;{{\omega _{j,k}} \geqslant T} \end{array}} \right.$

1）利用小波变换分解含噪信号 $f\left( k \right)$ ，可得到一组小波系数 ${\omega _{j,k}}$

2）用阈值处理小波系数 ${\omega _{j,k}}$ ，确定小波系数的估计值，使 $\left\| {{{{\overset{\frown} \omega} }_{j,k}} - {\omega _{j,k}}} \right\|$ 最小；

3）用小波逆变换对 ${{\overset{\frown} \omega} _{j,k}}$ 进行重构，得到估计的损伤信号 ${\overset{\frown} f} \left( k \right)$ ，即为去噪后的信号。

2 小波包分解

${L^2}\left( R \right) = \mathop \oplus \limits_{j \in Z} {W_j}$ ${L^2}\left( R \right)$ 按照不同空间尺度j下分解而成的子空间 ${W_j}\left( {j \in Z} \right)$ 的正交和。

 $\left\{ {\begin{array}{*{20}{c}} \begin{gathered} M_j^0 = {V_j} \\ M_j^1 = {W_j} \\ \end{gathered} &({j \in {\bf Z}} )\end{array}} \right.$

 $\begin{array}{*{20}{c}} {M_{j + 1}^0 = M_j^0 \oplus M_j^1}&({j \in {\bf Z}}) \end{array}$

$M_j^n$ ${m_n}\left( t \right)$ 的闭包空间，则 ${m_n}\left( t \right)$ 满足双尺度方程：

 $\left\{ {\begin{array}{*{20}{c}} {{m_{2n}}\left( t \right) = \sqrt 2 \displaystyle\sum\limits_{k \in Z} {h\left( k \right)} {m_n}\left( {2t - k} \right)} \\ {{m_{2n + 1}}\left( t \right) = \sqrt 2 \displaystyle\sum\limits_{k \in Z} {g\left( k \right)} {m_n}\left( {2t - k} \right)} \end{array}} \right.$ (1)

 $\left\{ {\begin{array}{*{20}{c}} {{m_0}\left( t \right) = \displaystyle\sum\limits_{k \in Z} {{h_k}} {m_0}\left( {2t - k} \right)} \\ {{m_1}\left( t \right) = \sqrt 2 \displaystyle\sum\limits_{k \in Z} {{g_k}} {m_0}\left( {2t - k} \right)} \end{array}} \right.$ (2)

 $\left\{ {\begin{array}{*{20}{c}} {\varphi \left( t \right) = \displaystyle\sum\limits_{k \in Z} {{h_k}\varphi \left( {2t - k} \right),{{\left\{ {{h_k}} \right\}}_{k \in Z}}} \in {l^2}} \\ {\psi \left( t \right) = \displaystyle\sum\limits_{k \in Z} {{g_k}\varphi \left( {2t - k} \right),{{\left\{ {{g_k}} \right\}}_{k \in Z}}} \in {l^2}} \end{array}} \right.$ (3)

 $M_{j + 1}^n = M_j^n \oplus M_j^{2n + 1}(j \in {\bf Z},n \in {{\bf Z}_ + })$

 $g_j^n\left( t \right) \in M_j^n,g_j^n\left( t \right) = \sum\limits_l {d_l^{j,n}} {m_n}\left( {{2^j}t - l} \right)。$
3 时反聚焦

 $h\left( t \right) = \sum\limits_{i = 1}^\infty {{A_i}} \left( {t - {\tau _i}} \right)$

 $y\left( t \right) = {r_c}s\left( t \right) \text{·} h\left( t \right) + n\left( t \right)$ (4)

 $y\left( t \right) = {r_c}s\left( t \right)\cdot h\left( t \right) \cdot h\left( t \right) + n\left( t \right)$ (5)

 $y\left( t \right) = {r_c}y\left( { - t} \right) \cdot h\left( t \right) \cdot h\left( t \right) + n\left( t \right)$ (6)

 \begin{aligned} z\left( \omega \right) =& {r_c}{Y^ * }\left( \omega \right)H\left( \omega \right)H\left( \omega \right) + N\left( \omega \right) = \\ & r_c^2 \cdot X \cdot \left( \omega \right){\left| {H\left( \omega \right)} \right|^4} + r_c^2{N^ * }\left( \omega \right){\left| {H\left( \omega \right)} \right|^2} + N\left( \omega \right) \\ \end{aligned} (7)

4 改进算法

1）在symN小波基上对检测信号 $y\left( t \right)$ 进行小波分解；对分解后的小波系数进行阈值去噪处理；

2）小波重构，获得去噪信号 $s\left( t \right)$

3）对去噪后的信号 $s\left( t \right)$ 进行小波包分解，从分解后的信号中得到M个奇异点，截取时宽为 ${T_{\rm{i}}} = {t_{i2}} - {t_{i1}}$ $i = 1, 2, \cdots, M$ $i$ 为第 $i$ 个奇异点；

4）桩基应力波检测信号数据和激励信号的函数表达都是已经确定的，分别为 $y\left( t \right)$ $x\left( t \right)$ 。将 $y\left( t \right)$ $x\left( t \right)$ 变换到频域，变换后的信号分别为 $Y\left( \omega \right)$ $X\left( \omega \right)$ ，则桩基检测信号的信道冲击响应函数为

 $H\left( \omega \right) = Y\left( \omega \right)/X\left( \omega \right)$

5）用宽度为 ${T_{{i}}}$ 的时间窗对信号进行截取，窗内乘一，窗外置零，得到M个截取信号，时域和频域分别为为 ${s_{ic}}\left( t \right)$ ${S_{ic}}\left( \omega \right)$ 的信号；

6）将每个截取信号 ${s_{ic}}\left( t \right)$ 在整个信号时间T内进行时间反转，时反后得到的M个时域为 ${t_i}\left( t \right) = {s_{cr}}\left( {T - t} \right)$ ${r_i}\left( t \right) = {S_{cr}}\left( \omega \right) \cdot \exp \left( { - {\rm i}{ \omega t}} \right)$ 的时反信号。

5 实验与仿真

 $s\left( t \right) = A \cdot \sin \left( {\omega t} \right)$

${y_s}\left( t \right)$ 进行3层小波阈值去噪，之后进行小波包分解，分解后的回波信号如图10所示。

6 结论

1）本文基于小波分析的时间反转改进算法，可以在含有噪声的情况下，识别出损伤点；

2）可以对损伤信号在损伤点处进行聚焦，并具有更强的抑制噪声能力，对损伤信号的信噪比平均提升了9.4 dB；

3）对每个损伤点都具有聚焦、抑制噪声和提高信噪比的能力，改进算法可以对一维构件实测中损伤的模式识别与定位以及信号的处理有一定的益处。

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