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Dimension-reduced generalized likelihood ratio detection based on sampling of normal modes in deep ocean

## Dimension-reduced generalized likelihood ratio detection based on sampling of normal modes in deep ocean

Kong De-Zhi, Sun Chao, Li Ming-Yang, Zhuo Jie, Liu Xiong-Hou
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• #### Abstract

In this paper, two generalized likelihood ratio (GLR) detectors are presented for the case of multiple snapshots of test data to detect the presence of an underwater acoustic source in the deep ocean. The two GLR detectors are termed the eigenvalue detector (EVD) and the constant false alarm rate eigenvalue detector (CFAR EVD), respectively. Theoretical analysis and numerical results show that for a given input signal-to-noise ratio (SNR) of the array, the GLR detectors achieve higher output SNRs when the spatial dimension of test data decreases. To further enhance the detection performances of the GLR detectors, we propose a dimension-reduced (DR-GLR) method based on array sampling of modal information. This DR-GLR method combines the characteristics of sound propagation and array receiving. According to normal mode theory, acoustic signals emitted from the acoustic source lie in the modal space spanned by the sampled modal information of the array. Resulting from the restriction of the array size, it often occurs in deep ocean when the dimension of " effective modal subspace” is less than that of the test data which is equivalent to the number of hydrophones. Based on this phenomenon, we reconstruct the modal information by merely retaining the " effective modal subspace” to formulate the dimension reduction matrix. The DR-GLR test statistics is deduced by employing the dimension reduction matrix when using the vertical linear array (VLA) and the horizontal linear array (HLA), respectively. The DR-GLR detectors when using an HLA require more computational amount than when using a VLA. Simulation experiments are conducted to analyze the detection performances of the two GLR detectors, and verify the performance improvement effects of DR-GLR detectors. The numerical results show that the CFAR EVD presents good robustness to the uncertainty of the noise power and the DR-GLR detectors outperform the GLR detectors in detection performance. It also turns out the acoustic signals received by the HLA lie in a lower-dimensional " effective modal subspace” than by the VLA, and thus when using an HLA the DR-GLR detectors present higher detection probabilities than using a VLA. Moreover, the smaller the dimension of the " effective modal subspace”, the better the performance improvement of the DR-GLR detectors will be. The dimension of the " effective modal subspace” increases with hydrophone spacing and/or the source frequency increasing.

#### References

 [1] 段睿 2016 博士学位论文 (西安: 西北工业大学) Duan R 2016 Ph. D. Dissertation (Xian: Northwestern Polytechnical University) (in Chinese) [2] 李启虎, 李敏, 杨秀庭 2008 声学学报 33 193 Li Q H, Li M, Yang X T 2008 Acta Acustica 33 193 [3] Gorodetskaya E Y, Malekhanov A I, Sazontov A G 2008 IEEE J. Oceanic Eng. 24 1109 [4] Sha L, Nolte L W 2006 IEEE J. Oceanic Eng. 31 5263 [5] Sha L, Nolte L W 2005 J. Acoust. Soc. Am. 117 5653 [6] 刘宗伟, 孙超, 易锋, 郭国强, 向龙凤 2014 声学学报 39 309 Liu Z W, Sun C, Yi F, Guo G Q, Xiang L F 2014 Acta Acustica 39 309 [7] 刘宗伟, 孙超, 吕连港 2015 声学学报 05 5949 Liu Z W, Sun C, Lv L G 2015 Acta Acustica 05 5949 [8] Hari V N, Anand G V 2013 Digital Signal Processing 23 1645 [9] 李明杨, 孙超, 邵炫 2014 物理学报 63 204302 Li M Y, Sun C, Shao X 2014 Acta Phys. Sin. 63 204302 [10] Li M Y, Sun C, Willett P 2017 IEEE J. Oceanic Eng. 43 131 [11] Scharf L L, Friedlander B 1994 IEEE trans. Signal Process. 42 2146 [12] Collison N E, Dosso S E 2000 J. Acoust. Soc. Am. 107 3089 [13] 刘宗伟, 孙超, 向龙凤, 易锋 2014 物理学报 63 034304 Liu Z W, Sun C, Xiang L F, Yi F 2014 Acta Phys. Sin. 63 034304 [14] 斯蒂芬 Kay 著 (罗鹏飞, 张文明 译) 2011 统计信号处理基础 (北京: 电子工业出版社) 第573—574页 Kay S M (translated by Luo P F, Zhang W M) 2011 Fundamentals of Statistical Signal Processing (Beijing: Pushlishing House of Electronics Industry) pp573−574 [15] Wang P, Fang J, Han N, Li H B 2010 IEEE Trans. Veh. Technol. 59 1791 [16] Hack D E, Rossler C W, Patton L K 2014 IEEE Signal Process. Lett. 21 1002 [17] Jin Y, Friedlander B 2004 IEEE Trans. Signal Process. 53 13 [18] Kong D Z, Sun C, Liu X H, Xie L, Jiang G Y 2017 Oceans 2017 Aberdeen, UK, June 19−22, 2017 p1 [19] Kong D Z, Sun C, Liu X H, Li M Y, Xie L 2018 Oceans 2018 Kobe, Japan, May 28−31, 2018 p1 [20] Morgan D R, Smith T M 1990 J. Acoust. Soc. Am. 87 737 [21] Tandra R, Sahai A 2008 IEEE J. Sel. Top. Signal Process. 2 4 [22] Haddadi F, Malek M M, Nayebi M M, Aref M R 2009 IEEE Trans. Signal Process. 58 452

#### Cited By

• 图 1  EVD的输出信噪比随接收数据快拍数和空间维度的变化曲线, ${\rm snr} = 1$　(a) 固定空间维度$N = 20$; (b) 固定快拍数$L = 100$

Figure 1.  The output SNR of EVD varying with various snapshot number and spatial dimension, ${\rm snr} = 1$: (a) spatial dimension $N = 20$; (b) snapshot number $L = 100$.

图 2  使用VLA时DR-GLR检测器的算法流程图

Figure 2.  The flow diagrams of the DR-GLR detectors when using a VLA

图 3  水平阵声源信号入射方位

Figure 3.  The arrival angle of acoustic signal on the HLA.

图 4  使用HLA时DR-GLR检测器的算法流程图

Figure 4.  The flow diagrams of the DR-GLR detectors when using a HLA.

图 5  深海波导及相关环境参数

Figure 5.  Deep-sea waveguide and environmental parameters

图 6  深海声速剖面

Figure 6.  Deep-sea sound speed profile.

图 7  不同信噪比下检测概率曲线比较, $P_{\rm FA}$ = 0.01　(a) $L = 20$; (b) $L = 40$

Figure 7.  Probability of detection curves with various SNRs, $P_{\rm FA}$ = 0.01: (a) $L = 20$; (b) $L = 40$.

图 8  噪声功率不确定, 不同信噪比下检测概率曲线比较　(a)$L = 20$; (b)$L = 40$

Figure 8.  Probability of detection curves with various SNRs when noise power is uncertain: (a) $L = 20$; (b) $L = 40$.

图 9  不同数据维度下的检测概率曲线对比, 快拍数$L = 40$　(a) EVD; (b) CEVD

Figure 9.  Probability of detection curves with various spatial dimension: (a) EVD; (b) CEVD.

图 10  阵列采样模态信息及相应模态矩阵的奇异值　(a) VLA采样的各阶模态; (b) VLA; 归一化的各阶奇异值分布; (c) HLA采样的各阶模态; (d) HLA, 归一化的各阶奇异值分布

Figure 10.  Modal information sampled on the array and singular values of corresponding mode matrices: (a) Various modes sampled on the VLA; (b) normalized singular values associated with the VLA; (c) various modes sampled on the HLA; (d) normalized singular values associated with the HLA.

图 11  不同信噪比下的检测概率曲线对比, 快拍数$L = 20$　(a) VLA； (b) HLA

Figure 11.  Probability of detection curves of different detectors, snapshot number $L = 20$: (a) VLA; (b) HLA.

图 12  噪声功率不确定, 不同信噪比下检测概率曲线对比, 快拍数$L = 40$　(a) VLA; (b) HLA

Figure 12.  Probability of detection curves of different detectors when noise power is uncertain, snapshot number $L = 40$: (a) VLA; (b) HLA.

图 13  声源激发的各阶简正波模态函数和水平波数　(a)各阶模态函数幅值随波导深度的变化; (b)各阶水平波数分布

Figure 13.  Modal functions and horizontal wavenumber of various normal modes excited by the acoustic source: (a) Modal functions along with various depths; (b) distribution of various horizontal wavenumbers.

图 14  阵列配置对降维程度的影响, $N = 40$　(a) 阵列深度100 m; (b) 阵元间距4 m

Figure 14.  The influence of array configuration on the degree of dimension reduction, $N = 40$: (a) Array depth of 100 m; (b) hydrophone spacing of 4 m.

图 15  降维系数随声源频率的变化曲线

Figure 15.  The dimension reduction coefficient varying with increasing frequency.

•  [1] 段睿 2016 博士学位论文 (西安: 西北工业大学) Duan R 2016 Ph. D. Dissertation (Xian: Northwestern Polytechnical University) (in Chinese) [2] 李启虎, 李敏, 杨秀庭 2008 声学学报 33 193 Li Q H, Li M, Yang X T 2008 Acta Acustica 33 193 [3] Gorodetskaya E Y, Malekhanov A I, Sazontov A G 2008 IEEE J. Oceanic Eng. 24 1109 [4] Sha L, Nolte L W 2006 IEEE J. Oceanic Eng. 31 5263 [5] Sha L, Nolte L W 2005 J. Acoust. Soc. Am. 117 5653 [6] 刘宗伟, 孙超, 易锋, 郭国强, 向龙凤 2014 声学学报 39 309 Liu Z W, Sun C, Yi F, Guo G Q, Xiang L F 2014 Acta Acustica 39 309 [7] 刘宗伟, 孙超, 吕连港 2015 声学学报 05 5949 Liu Z W, Sun C, Lv L G 2015 Acta Acustica 05 5949 [8] Hari V N, Anand G V 2013 Digital Signal Processing 23 1645 [9] 李明杨, 孙超, 邵炫 2014 物理学报 63 204302 Li M Y, Sun C, Shao X 2014 Acta Phys. Sin. 63 204302 [10] Li M Y, Sun C, Willett P 2017 IEEE J. Oceanic Eng. 43 131 [11] Scharf L L, Friedlander B 1994 IEEE trans. Signal Process. 42 2146 [12] Collison N E, Dosso S E 2000 J. Acoust. Soc. Am. 107 3089 [13] 刘宗伟, 孙超, 向龙凤, 易锋 2014 物理学报 63 034304 Liu Z W, Sun C, Xiang L F, Yi F 2014 Acta Phys. Sin. 63 034304 [14] 斯蒂芬 Kay 著 (罗鹏飞, 张文明 译) 2011 统计信号处理基础 (北京: 电子工业出版社) 第573—574页 Kay S M (translated by Luo P F, Zhang W M) 2011 Fundamentals of Statistical Signal Processing (Beijing: Pushlishing House of Electronics Industry) pp573−574 [15] Wang P, Fang J, Han N, Li H B 2010 IEEE Trans. Veh. Technol. 59 1791 [16] Hack D E, Rossler C W, Patton L K 2014 IEEE Signal Process. Lett. 21 1002 [17] Jin Y, Friedlander B 2004 IEEE Trans. Signal Process. 53 13 [18] Kong D Z, Sun C, Liu X H, Xie L, Jiang G Y 2017 Oceans 2017 Aberdeen, UK, June 19−22, 2017 p1 [19] Kong D Z, Sun C, Liu X H, Li M Y, Xie L 2018 Oceans 2018 Kobe, Japan, May 28−31, 2018 p1 [20] Morgan D R, Smith T M 1990 J. Acoust. Soc. Am. 87 737 [21] Tandra R, Sahai A 2008 IEEE J. Sel. Top. Signal Process. 2 4 [22] Haddadi F, Malek M M, Nayebi M M, Aref M R 2009 IEEE Trans. Signal Process. 58 452
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•  Citation:
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• Abstract views:  192
• Cited By: 0
##### Publishing process
• Received Date:  13 January 2019
• Accepted Date:  30 May 2019
• Available Online:  26 November 2019
• Published Online:  01 September 2019

## Dimension-reduced generalized likelihood ratio detection based on sampling of normal modes in deep ocean

###### Corresponding author: Sun Chao, csun@nwpu.edu.cn;
• 1. School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China
• 2. Key Laboratory of Ocean Acoustics and Sensing Ministry of Industry and Information Technology, Xi’an 710072, China
• 3. State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China

Abstract: In this paper, two generalized likelihood ratio (GLR) detectors are presented for the case of multiple snapshots of test data to detect the presence of an underwater acoustic source in the deep ocean. The two GLR detectors are termed the eigenvalue detector (EVD) and the constant false alarm rate eigenvalue detector (CFAR EVD), respectively. Theoretical analysis and numerical results show that for a given input signal-to-noise ratio (SNR) of the array, the GLR detectors achieve higher output SNRs when the spatial dimension of test data decreases. To further enhance the detection performances of the GLR detectors, we propose a dimension-reduced (DR-GLR) method based on array sampling of modal information. This DR-GLR method combines the characteristics of sound propagation and array receiving. According to normal mode theory, acoustic signals emitted from the acoustic source lie in the modal space spanned by the sampled modal information of the array. Resulting from the restriction of the array size, it often occurs in deep ocean when the dimension of " effective modal subspace” is less than that of the test data which is equivalent to the number of hydrophones. Based on this phenomenon, we reconstruct the modal information by merely retaining the " effective modal subspace” to formulate the dimension reduction matrix. The DR-GLR test statistics is deduced by employing the dimension reduction matrix when using the vertical linear array (VLA) and the horizontal linear array (HLA), respectively. The DR-GLR detectors when using an HLA require more computational amount than when using a VLA. Simulation experiments are conducted to analyze the detection performances of the two GLR detectors, and verify the performance improvement effects of DR-GLR detectors. The numerical results show that the CFAR EVD presents good robustness to the uncertainty of the noise power and the DR-GLR detectors outperform the GLR detectors in detection performance. It also turns out the acoustic signals received by the HLA lie in a lower-dimensional " effective modal subspace” than by the VLA, and thus when using an HLA the DR-GLR detectors present higher detection probabilities than using a VLA. Moreover, the smaller the dimension of the " effective modal subspace”, the better the performance improvement of the DR-GLR detectors will be. The dimension of the " effective modal subspace” increases with hydrophone spacing and/or the source frequency increasing.

Reference (22)

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