# Research and Analysis on the Localization of a 3-D Single Source in Lossy Medium Using Uniform Circular Array

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

**X**and

**N**are $M\times 1$ dimensional signal and noise matrix.

**A**is the steering vector.

**S**is $1\times K$ dimensional signal sequence at reference point with signal power of ${\sigma}_{s}^{2}$.

## 3. Proposed Method

**X**can be calculated:

#### 3.1. Phase Method

**R**to estimate the source location. We can thus obtain

#### 3.2. Amplitude Method

**R**,

#### 3.3. Synthesis Method

**R**adequately, we put the phase and amplitude information together to detect the source location in the conductive medium ($\alpha =\beta =\sqrt{\pi f\mu \sigma}$).

#### 3.4. Applicability Analysis

## 4. Numerical Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Lee, J.H.; Park, D.H.; Park, G.T.; Lee, K.K. Algebraic path following algorithm for localizing 3-D near-field sources in uniform circular array. Electron. Lett.
**2003**, 39, 1283–1285. [Google Scholar] [CrossRef] - Ko, Y.H.; Kim, Y.J.; Yoo, H.I.; Yang, W.Y.; Cho, Y.S. 2-D Do A Estimation with Cell Searching for a Mobile Relay Station with Uniform Circular Array. IEEE Trans. Commun.
**2010**, 58, 2805–2809. [Google Scholar] [CrossRef] - Krim, H.; Viberg, M. Two decades of array signal processing research: The parametric approach. IEEE Signal Process. Mag.
**1996**, 13, 67–94. [Google Scholar] [CrossRef] - Schmidt, R.O. Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag.
**1986**, 34, 276–280. [Google Scholar] [CrossRef] - Rao, B.D.; Hari, K.V.S. Performance analysis of root-MUSIC. IEEE Trans. Acoust. Speech Signal Process.
**1989**, 37, 1939–1949. [Google Scholar] [CrossRef] - Pesavento, M.; Gershman, A.B.; Haardt, M. Unitary root-MUSIC with a real-valued eigendecomposition: A theoretical and experimental performance study. IEEE Trans. Signal Process.
**2000**, 48, 1306–1314. [Google Scholar] [CrossRef] - Rot, R.; Kailath, T. Esprit-estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoust. Speech Signal Process.
**1989**, 37, 984–995. [Google Scholar] - Liu, G.; Sun, X. Two-Stage Matrix Differencing Algorithm for Mixed Far-Field and Near-Field Sources Classification and Localization. IEEE Sens. J.
**2014**, 14, 1957–1965. [Google Scholar] - Jung, T.; Lee, K. Closed-Form Algorithm for 3-D Single-Source Localization with Uniform Circular Array. IEEE Antennas Wirel. Propag. Lett.
**2014**, 13, 1096–1099. [Google Scholar] [CrossRef] - Wu, Y.; So, H.C. Simple and accurate two-dimensional angle estimation for a single source with uniform circular array. IEEE Antennas Wirel. Propag. Lett.
**2008**, 7, 78–80. [Google Scholar] - Liao, B.; Wu, Y.; Chan, S. A generalized algorithm for fast two dimensional angle estimation of a single source with uniform circular array. IEEE Antennas Wirel. Propag. Lett.
**2012**, 11, 984–986. [Google Scholar] [CrossRef] - Liu, Y.; Zhang, X.; Shao, J.; Zhang, B. Estimation of ship’s extremely low frequency electromagnetic signature based on fuzzy fusion for target detection. In Proceedings of the IEEE China Summit & International Conference on Signal and Information Processing, Xi’an, China, 9–13 July 2014; Volume 158, pp. 253–261. [Google Scholar]
- Tao, H.; Xin, J.; Wang, J.; Zheng, N.; Sano, A. Two-dimensional direction estimation for a mixture of noncoherent and coherent signals. IEEE Trans. Signal Process.
**2015**, 63, 318–333. [Google Scholar] [CrossRef] - Wang, G.; Xin, J.; Zheng, N.; Sano, A. Computationally efficient subspace-based method for two-dimensional direction estimation with L-shaped array. IEEE Trans. Signal Process.
**2011**, 59, 3197–3212. [Google Scholar] [CrossRef] - Xue, B.; Fang, G.; Ji, Y. Passive localization of mixed far-field and near-field sources using uniform circular array. Electron. Lett.
**2016**, 52, 1690–1692. [Google Scholar] [CrossRef] - Xue, B.; Fang, G.; Ji, Y. Effective algorithm mixed far-field and near-field sources localization using uniform circular array. Prog. Electromagn. Res. M
**2016**, 51, 139–146. [Google Scholar] [CrossRef]

**Figure 2.**Root mean square errors (RMSEs) of location estimations for single-source in the lossless medium versus SNRs. $\left(r,\theta ,\phi \right)=\left(1200,\frac{\pi}{6},\frac{\pi}{3}\right)$, and the snapshot number is 1000 with 500 independent trials.

**Figure 3.**RMSEs of location estimations for single-source in the lossless medium versus snapshot. $\left(r,\theta ,\phi \right)=\left(1200,\frac{\pi}{6},\frac{\pi}{3}\right)$, and the signal-to-noise ratio (SNR) is 10 dB with 500 independent trials.

**Figure 4.**RMSEs of location estimations for single-source in the weak lossy medium versus SNRs. $\left(r,\theta ,\phi \right)=\left(1200,\frac{\pi}{6},\frac{\pi}{3}\right)$, and the snapshot number is 1000 with 500 independent trials.

**Figure 5.**RMSEs of location estimations for single-source in the weak lossy medium versus snapshot. $\left(r,\theta ,\phi \right)=\left(1200,\frac{\pi}{6},\frac{\pi}{3}\right)$, and the SNR is 10 dB with 500 independent trials.

**Figure 6.**RMSEs of location estimations for single-source in the conductive medium versus SNRs. $\left(r,\theta ,\phi \right)=\left(120,\frac{\pi}{6},\frac{\pi}{3}\right)$, and the snapshot number is 1000 with 500 independent trials.

**Figure 7.**RMSEs of location estimations for single-source in the conductive medium versus snapshot. $\left(r,\theta ,\phi \right)=\left(120,\frac{\pi}{6},\frac{\pi}{3}\right)$, and the SNR is 10 dB with 500 independent trials.

**Figure 8.**RMSEs of location estimations for single-source in the lossy medium versus snapshot. $\left(r,\theta ,\phi \right)=\left(1200,\frac{\pi}{6},\frac{\pi}{3}\right)$, $\sigma $ is unknown, the SNR is 10 dB, and the snapshot number is 1000 with 500 independent trials.

**Figure 9.**RMSEs of location estimations for single-source in the lossy medium versus snapshot. $\left(r,\theta ,\phi \right)=\left(1200,\frac{\pi}{6},\frac{\pi}{3}\right)$, $\sigma $ is known, the SNR is 10 dB, and the snapshot number is 1000 with 500 independent trials.

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**MDPI and ACS Style**

Xue, B.; Qu, X.; Fang, G.; Ji, Y. Research and Analysis on the Localization of a 3-D Single Source in Lossy Medium Using Uniform Circular Array. *Sensors* **2017**, *17*, 1274.
https://doi.org/10.3390/s17061274

**AMA Style**

Xue B, Qu X, Fang G, Ji Y. Research and Analysis on the Localization of a 3-D Single Source in Lossy Medium Using Uniform Circular Array. *Sensors*. 2017; 17(6):1274.
https://doi.org/10.3390/s17061274

**Chicago/Turabian Style**

Xue, Bing, Xiaodong Qu, Guangyou Fang, and Yicai Ji. 2017. "Research and Analysis on the Localization of a 3-D Single Source in Lossy Medium Using Uniform Circular Array" *Sensors* 17, no. 6: 1274.
https://doi.org/10.3390/s17061274