﻿ 统计最优柱面近场声全息参数选取方法研究
 舰船科学技术  2018, Vol. 40 Issue (3): 120-127 PDF

Research on parameter selection for the statistically optimal cylindrical near-field acoustical holography
LI Biao, LI Xi-you, WANG Zhi-qiang, LIU Xing-hua
Dalian Scientific Test and Control Technology Institute, Dalian 116013, China
Abstract: Aiming at the application of acoustic source localization technique based on cylindrical statistic optimal near-field acoustical holography, study on the parameter selection of near-field acoustical holography. According to the scanning measurement method of the complex sound pressure of holographic plane and cylindrical statistic optimal near-field acoustical holography algorithm mathematical model, this paper analyzes the near-filed acoustic holography measurement system parameters influencing the source localization accuracy, and develops the system parameters and information processing parameters selection method. In addition, this paper establishes cylindrical statistically optimal near-field acoustical holography practical information processing technology. Through numerical simulation and model test ,the validity of the parameter selection method is verified, which provides a practical basis for parameters selection for submarine the near-field acoustical holography.
Key words: statistical optimum     near-field acoustical holography     sound source localization
0 引　言

1 柱面统计最优近场声全息算法 1.1 测量模型

 图 1 柱面全息面及声源重构面网格剖分 Fig. 1 Mesh reconstruction of cylindrical holographic surface and sound source

 ${k_{z,N}} = \frac{1}{2}\left( {\frac{{2\pi }}{\Delta }} \right)\text{。}$ (1)

 \begin{aligned}&{k_{s,N}} = \frac{1}{2}{\left. {\left( {\frac{{2\pi }}{{\Delta s}}} \right)} \right|_{\Delta s = \frac{{2\pi {r_h}}}{{P - 1}}}} = \frac{{P - 1}}{{2{r_h}}}\text{，}\\ &{k_{z,N}} = \frac{1}{2}{\left. {\left( {\frac{{2\pi }}{{\Delta z}}} \right)} \right|_{\Delta z = \frac{{2L}}{{Q - 1}}}} = \frac{{\pi (Q - 1)}}{{2L}}\text{。}\end{aligned} (2b)

1.2 多参考全息面复声压

 ${\left[ {\begin{array}{*{20}{c}}{{y_{1k}}} \!\!&\!\! {{y_{2k}}}\!\! &\!\! \cdots & {{y_{Jk}}}\end{array}} \right]^{{T}} } = {r_{1k}}{\left[ {\begin{array}{*{20}{c}}{{h_{1k}}} \!\! & \!\! {{h_{2k}}} \!\! & \!\! \cdots \!\! &\!\! {{h_{Jk}}}\end{array}} \right]^{{T}}\text{，} }$

 ${{{Y}}_k} = {R_k}{{{H}}_k}\text{，}$ (3b)

${\overline{\overline Y} _{jk}}$ 为全息面上第j个阵元的第k次复声压值， ${C_{(.)(.)}}$ 为互谱算子，则根据方程（3）建立单参考点全息面复声压场为

 ${\overline{\overline Y} _{jk}} = \sqrt {{C_{{y_{jk}}{y_{jk}}}}} {e^{j\angle {C_{{y_j}_k{r_{1k}}}}}}\text{。}$ (4)

 $\left[ {\begin{array}{*{20}{c}}{{y_1}} \\{{y_2}} \\\vdots \\{{y_N}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{h_{11}}} & {{h_{12}}} & \cdots & {{h_{1M}}} \\{{h_{21}}} & {{h_{22}}} & \cdots & {{h_{2M}}} \\\vdots & \vdots & \ddots & \vdots \\{{h_{N1}}} & {{h_{N2}}} & \cdots & {{h_{NM}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{r_1}} \\{{r_2}} \\\vdots \\{{r_M}}\end{array}} \right]\text{，}$

 ${{y}} = {{{H}}_{{yr}}}{{r}}\text{，}$ (5b)

 ${{{H}}_{{yr}}} = {{E}} \{ {{y}}{{{r}}^{{H}} }\} \cdot {{{E}} ^{ - 1}}\{ {{r}}{{{r}}^{{H}} }\} = {{{C}}_{{yr}}}{{C}}_{{rr}}^{ - 1}\text{，}$ (6)

 ${{{C}}_{{rr}}} = {{U\Sigma }}{{{V}}^{{H}} } = {{U\Sigma }}{{{U}}^{{H}}\text{，} }$ (7)

 ${{Y}} = {{{H}}_{{yr}}}{{U}}{{{\Sigma }}^{1/2}} = {{{C}}_{{yr}}}{{U}}{{{\Sigma }}^{ - 1/2}}\text{。}$ (8)

1.3 柱面统计最优数学模型

 $\left( {{\nabla ^2} - \frac{1}{{{c^2}}}\frac{{{\partial ^2}}}{{\partial {t^2}}}} \right)p = 0\text{，}$ (9)

 ${\nabla ^2} = \frac{{{\partial ^2}}}{{\partial {r^2}}} + \frac{1}{r}\frac{\partial }{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} + \frac{{{\partial ^2}}}{{\partial {z^2}}}\text{，}$

Helmholtz方程（9）的自由场行波解

 $\begin{array}{l}p(r,\varphi ,z) = \sum\limits_{n = - \infty }^{ + \infty } {{e^{jn\varphi }}\frac{1}{{2\pi }}\int_{ - \infty }^{ + \infty } {{ d}{k_z}{P_n}({r_s},{k_z})} } \text{，}\\{e^{j{k_z}z}}\left\{ {\begin{array}{*{20}{c}}{\frac{{{{H}}_n^{(1)}({k_r}r)}}{{{{H}}_n^{(1)}({k_r}{r_s})}},{k_r} = \sqrt {{k^2} - k_z^2} }\text{，}\\[5pt]{\frac{{{{{K}}_n}({k_r}^\prime r)}}{{{{{K}}_n}({k_r}^\prime {r_h})}},{k_r}^\prime = \sqrt {k_z^2 - {k^2}} }\text{，}\end{array}} \right.\end{array}$ (10)

 ${\varPhi _{n,{k_z}}}(r,\phi ,z) = \left\{ {\begin{array}{*{20}{c}}{\frac{{{\mathop{ H}\nolimits} _n^{(1)}({k_r}r)}}{{{\mathop{ H}\nolimits} _n^{(1)}({k_r}{r_s})}}{e^{jn\varphi }}{e^{j{k_z}z}},\;\;{k_r} = \sqrt {{k^2} - k_z^2} }\\{\frac{{{{\mathop{ K}\nolimits} _n}({k_r}'r)}}{{{{\mathop{ K}\nolimits} _n}({k_r}'{r_s})}}{e^{jn\varphi }}{e^{j{k_z}z}},\;\;{k_r}' = \sqrt {k_z^2 - {k^2}} }\end{array}} \right. \!\! \text{，}r \geqslant {r_s}\text{。}$ (11)

 $p(r,\varphi ,z) \! \approx \! \frac{1}{{2\text{π} }}\sum\limits_{n = 0}^N {\sum\limits_{m = 0}^M {{P_n}({r_s},m \cdot \Delta {k_z}){\varPhi _{n,m}}(r,\varphi ,z)} } \Delta {k_z}\left| {_{{k_z} = m \cdot \Delta {k_z}}} \right.\text{，}$ (12)

 $({r_t},{\varphi _p},{z_q})\xrightarrow{{l = (q - 1)P + p}}\left( {{{\vec r}_{t,j}}} \right)\text{，}$ (13)

 $p({\vec r_{t,j}}) = \sum\limits_{l = 1}^{PQ} {{c_l}({{\vec r}_{h,j}})p({{\vec r}_{h,l}})}\text{。}$ (14)

 ${\varPhi _{n,m}}({\vec r_{t,j}}) = \sum\limits_{l = 1}^{PQ} {{c_l}({{\vec r}_{h,j}}){\varPhi _{n,m}}({{\vec r}_{h,l}})} \left| {_{{k_z} = m \cdot \Delta {k_z}}} \right.\text{，}$ (15)

 ${{b}} = {{Ac}}\text{，}$ (16)

 ${ b} \!=\! {\left[ {{b_{kl}}} \right]_{MN \times PQ}}\mathop \equiv \limits^{l = (q - 1)P + p,k = (n - 1)M + m} {\left[ {{\varPhi _{n,m}}(r,{\varphi _p},{z_q})} \right]_{MN \times PQ}}\text{，}$
 ${{A}} \!=\! {\left[ {{a_{kl}}} \right]_{MN \times PQ}}\mathop \equiv \limits^{l = (q - 1)P + p,k = (n - 1)M + m} {\left[ {{\varPhi _{n,m}}({r_h},{\varphi _p},{z_q})} \right]_{MN \times PQ}}\text{，}$
 ${{c}} \!=\!\! {\left[ {{c_{kl}}} \right]_{PQ \times PQ}}\mathop \equiv \limits^{l = (q - 1)P + p,k = (n - 1)M + m} {\left[ {{c_l}(r,{\varphi _p},{z_q})} \right]_{PQ \times PQ}}\text{。}$

 ${{c}} = {({{{A}}^{{H}} }{{A}} + \Theta {{I}})^{ - 1}}{{{A}}^{{H}} }{{b}}\text{，}$ (17)

 $\begin{split}&{\left[ {{p_{r(j)}}} \right]_{1 \times PQ}} = {\left[ {{p_h}_{(j)}} \right]_{1 \times PQ}}{\left[ {{c_{lk}}} \right]_{PQ \times PQ}}=\\& {{p}}_h^{\mathop{ T}\nolimits} {({{{A}}^{\mathop{ H}\nolimits} }{{A}} + \Theta {{I}})^{ - 1}}{{{A}}^{\mathop{ H}\nolimits} }{{b}}\text{。}\end{split}$ (18)

2 定位精度影响因素分析

2.1 信息处理参数

 $\left| {{k_{z,c}}} \right| < \frac{{\pi D}}{{27.3({r_h} - {r_s})}}\text{，}$ (19)

 $\left| {{k_{s,c}}} \right| = \frac{n}{{{r_s}}} < \frac{{\pi D}}{{27.3({r_h} - {r_s})}}\text{，}$ (19b)

 ${W_{{{k - space}}}} = \left\{ \begin{array}{l}1 - \displaystyle\frac{1}{2}{e^{ - \left( {1 - \left| {{k_t}} \right|/{k_c}} \right)/{\alpha _k}}}\;\;\text{，}\;\;\left| {{k_t}} \right| < {k_c}\text{，}\\\displaystyle\frac{1}{2}{e^{\left( {1 - \left| {{k_t}} \right|/{k_c}} \right)/{\alpha _k}}}\;\;\;\;\text{，}\;\;\;\;\;\left| {{k_t}} \right| > {k_c}\text{。}\end{array} \right.$ (20)

 图 2 正则化因子对重构声压沿轴向分布影响 Fig. 2 Influence of regularization factor on the distribution of reconstructive sound pressure along the axial diretion

 图 3 倏逝波信噪比对重构声压沿轴向分布影响 Fig. 3 Influence of evanescent wave sing-to-noise ratio on the distribution of reconstructive sound pressure along the axial diretion

 图 4 窗型参数对重构声压沿轴向分布影响 Fig. 4 Influence of window parameter on the distribution of reconstructive sound pressure along the axial direction
2.2 测量阵参数

Tukey窗函数

 ${W_{{{Tukey}}}} = \left\{ {\begin{array}{*{20}{c}}\begin{array}{l}\displaystyle\frac{1}{2} + \displaystyle\frac{1}{2}\cos \left[ {\frac{{2\pi }}{\alpha }\left( {z - {z_R}/2} \right)/{L_{w,T}} - \pi } \right]\text{，}\\{z_R}/2 - {L_{w,T}} \leqslant z \leqslant {z_R}/2\text{，}\\1\quad\text{，} z < {z_R}/2 - {L_{w,T}}\text{，}\\0\quad\text{，} z > {z_R}/2\text{。}\end{array}\end{array}} \right.$

 图 5 阵元间距对定位精度影响 Fig. 5 Influence of array element spacing on positioning accuracy

 图 6 全息面尺寸对定位精度影响 Fig. 6 Influence of holographic size on positioning accuracy

 图 7 测量距离对定位精度影响 Fig. 7 Influence of measuring distance on positioning accuracy

 图 8 重构面上声压沿轴向分布曲线 Fig. 8 The distribution of sound pressure along the axial direction of the reconstructed surface

3 数值仿真

 图 9 多点源全息面声压幅值谱 Fig. 9 Sound pressure amplitude spectrum of multi sound holographic surface

 图 10 多点源重构面声压幅值谱 Fig. 10 Sound pressure amplitude spectrum of multi point source reconstruction surface
4 试验验证

 图 11 复声压测量结果 Fig. 11 Complex sound pressure measurement results

 图 12 声压重构结果 Fig. 12 Sound pressure reconstruction results

 图 13 壳体振动测量结果 Fig. 13 Shell vibration measurement results
5 结　语

1）利用Tukey窗函数应用于轴向滤波，可有效消除轴向有效测量孔径效应，窗函数参数 $\alpha$ 值选取应根据声源分辨率要求进行优化选取；

2）算法实现过程，正则化因子应采用较低的信噪比参数，倏逝波信噪比参数应采用较高值，同时考虑有效测量孔径效应要求进行优化选取；

3）实际测量时，测量系统阵元间距小于1/3倍声源波长、轴向测量孔径大于3倍声源波长、测量距离小于声源1/5倍波长。

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