﻿ 能量法估计全息面信噪比研究
 舰船科学技术  2017, Vol. 39 Issue (3): 150-154 PDF

Estimation of the signal to noise ration in the planar acoustical holography based on energy method
DAI Zhen, HE Qi-wei, WAN Hai-bo
Naval Engineering University, Wuhan 430033, China
Abstract: The signal to noise ratio (SNR) is a very important parameter in theplanar acoustical holography. However, it can’t be determined directly. To solve the problem, a formula in ‘-space’ is given based on theenergy method. According to the formula, the sampling interval has a great influence on the SNR estimation value. To determine the accuracy of the estimation value, the measurement data on the different areas of the holography surface is chosen to estimate. When the result is not accurate, the estimation value can be improved by decreasing the sampling interval. The maximum estimation value is given in different estimation precision and different sampling interval to provide a reference for practical applications.
Key words: planar near-field acoustic holography     signal to noise ratio     reconstruction error
0 引　言

1 平面 NAH 基本原理[8]

 $\begin{split}\\[-12pt]{P_H}(x,y,{z_H}) = & \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{P_s}({{x',y',}}{{{z}}_s})} } {g_D}\times\\[8pt]& {{(x - x',y - y',}}{{{z}}_H}{\rm{ - }}{{{z}}_s}{{){\rm d}x'{\rm d}y'}}{\text{，}}\end{split}$ (1)

 ${g_D}(x,y,z) = \frac{{z(1 - ikr)}}{{2\pi {r^3}}}{\text{，}}$ (2)

 $F({k_x},{k_y}) = \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {f(x,y){e^{ - i({k_x}x + {k_y}y)}}{\rm d}} } x{\rm d}y\text{，}$ (3)

 ${P_H}({k_x},{k_y},{z_H}) = {P_S}({k_x},{k_y},{z_S}){G_D}({k_x},{k_y},{z_H} - {z_S})\text{，}$ (4)

 ${G_D}({k_x},{k_y},z) = {e^{i{k_z}z}}\text{，}$ (5)

$k_x^2 + k_y^2 \leqslant {k^2}$ 时，

 ${k_z} = \sqrt {{k^2} - (k_x^2 + k_y^2)}\text{；}$ (6)

$k_x^2 + k_y^2 > {k^2}$ 时，

 ${k_z} = i\sqrt {(k_x^2 + k_y^2) - {k^2}}\text{，}$ (7)

 ${P_H}({k_x},{k_y},{z_H}) = {P_S}({k_x},{k_y},{z_S}){e^{i{k_z}({z_H} - {z_S})}}\text{。}$ (8)

 ${P_S}({k_x},{k_y},{z_S}) = {P_H}({k_x},{k_y},{z_H}){e^{ - i{k_z}({z_H} - {z_S})}}\text{。}$ (9)
2 信噪比定义与噪声能量计算

 $SNR = 10\lg \frac{{{E_S}}}{{{E_N}}}{\rm{ = }}10\lg \frac{{E - {E_N}}}{{{E_N}}}\text{。}$ (10)

 $\left\{ \begin{array}{l}E[{p_N}(x,y,{z_H})] = \displaystyle\frac{1}{{MN}}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{p_N}({x_i},{y_j}) = 0{\text{，}}} } \\[5pt]D[{p_N}(x,y,{z_H})] = \displaystyle\frac{1}{{MN}}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {\{ {p_N}({x_i},{y_j}) - } } \\[5pt]\quad \quad E[{p_N}(x,y,{z_H})]{\} ^2} \!\!=\!\! \displaystyle\frac{1}{{MN}}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{p_N}{{({x_i},{y_j})}^2{\text{。}}}} } \end{array} \right.$ (11)

 ${E_N} = MND[{p_N}(x,y,{z_H})]\text{。}$ (12)

 图 1 波数域分布划分 Fig. 1 Division of the wavenumber region

 $\begin{split}\\[-12pt]D[{P_N}({k_x},{k_y},{z_H})] = \frac{1}{{MN}}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {} }\text{，} \\{p_N}{({k_x}_i,{k_{yj}},{z_H})^2} \approx D[{P_N}({\Omega _3})]\end{split}\text{。}$ (13)

 ${E_N}\!\!\! =\!\!\! \sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{p_N}{{({x_i},{y_j})}^2}} }\!\!\! =\!\!\! \frac{1}{{MN}}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{p_N}{{({k_x}_i,{k_{yj}},{z_H})}^2}} }\text{，} \!\!\!\!$ (14)

 ${E_N} \approx D[{P_N}({\varOmega _3})] = \frac{1}{b}{({\left\| {{P_N}({\varOmega _3})} \right\|_2})^2}\text{，}$ (15)

 $\begin{split}\\[-12pt]E = \sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {p{{({x_i},{y_j})}^2}} } = & \frac{1}{{MN}}\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {p{{({k_x}_i,{k_{yj}},{z_H})}^2}} }=\\& \frac{1}{{MN}}{({\left\| {p({k_x},{k_y},{z_H})} \right\|_2})^2}\text{，}\end{split}$ (16)

 $SNR \approx {\kern 1pt} 10\lg \frac{{b{{({{\left\| {p({k_x},{k_y},{z_H})} \right\|}_2})}^2} - MN{{({{\left\| {{P_N}({\Omega _3})} \right\|}_2})}^2}}}{{MN{{({{\left\| {{P_N}({\Omega _3})} \right\|}_2})}^2}}}\text{。}$ (17)
3 信噪比估值判别

 $\varepsilon {\rm{ = }}\frac{{\left| {SNR' - SNR} \right|}}{{SNR}}\text{。}$ (18)

 ${L_i} = L - 2(i - 1)\Delta {\text{，}}i = 1,2 \cdots 9{\text{。}}$ (19)

 图 2 全息面不同区域下信噪比估值 Fig. 2 The estimation for the SNR in different areas of the holography surface

 $\varepsilon ' = \frac{{\left| {SN{R_{\max }} - SN{R_{\min }}} \right|}}{{SN{R_{\max }}}}{\text{。}}$ (20)

 图 3 信噪比估值误差对比 Fig. 3 Comparison of the SNR estimation errors

1）保持采样间隔 $\Delta$ 不变，以全息面坐标原点为中心，选择合适的n 值（可以令 ${L_n} = L/2$ ），从全息面中选择n 个正方形区域 ${L_i} \times {L_i}(i = 1,2 \cdots n)$

2）对所取区域的声压数据分别进行信噪比估计，得到一系列估值 $SN{R'_i}$ ，找到其最大估值 $SN{R_{\max }}$ 和最小估值 $SN{R_{\min }}$ ，并计算 $\varepsilon '$

3）根据实际测量情况，选择合适的 ${\varepsilon _r}$ ，如果 $\varepsilon ' < {\varepsilon _r}$ ，认为估值精确，并将 $SN{R_{\max }}$ 作为理论信噪比值。

4 减小信噪比估值误差的方法

 图 4 不同采样间隔下的信噪比估值误差 Fig. 4 TheSNR estimation errors in different sampling interval

5 结　语

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