# Multiple Signal Classification Algorithm Based Electric Dipole Source Localization Method in an Underwater Environment

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## Abstract

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## 1. Introduction

## 2. Localization of Electric Dipole Source in Finite Region

#### 2.1. Localization Model of Electric Dipole Source

#### 2.2. Localization Based on the Multiple Signal Classification Algorithm

- Step 1: Discretize the boundary of the locating area and calculate the boundary matrix ${\mathbf{Z}}_{\mathrm{s}}^{\mathrm{s}}$ according to the prior information about the boundary. By inverting the boundary matrix, we have ${\left({\mathbf{Z}}_{\mathrm{s}}^{\mathrm{s}}\right)}^{{}^{-1}}$.
- Step 2: By measuring the voltage of each channel of the receiving antenna array, the matrix ${\mathbf{\Psi}}_{t}$ is formed with the size of $K\times N$.
- Step 3: According to (23), the covariance matrix ${R}_{\mathbf{\Psi}}$ can be constructed.
- Step 4: Obtain the required signal subspace ${\mathbf{U}}_{\mathrm{S}}$ and noise subspace ${\mathbf{U}}_{\mathrm{N}}$ via the eigendecomposition of the constructed matrix ${R}_{\mathbf{\Psi}}$.
- Step 5: Mesh the locating area with a set of spatial points ${\mathit{r}}_{\mathrm{p}\left(1\right)},{\mathit{r}}_{\mathrm{p}\left(2\right)},\cdots ,{\mathit{r}}_{\mathrm{p}\left(\mathrm{M}\right)}$.
- Step 6: Calculate the matrix $\mathbf{A}$ according to (17) with the estimated dipole source position ${\mathit{r}}_{\mathrm{p}\left(i\right)}$, $i=1,2,\cdots ,\mathrm{M}$.
- Step 7: Obtain the eigenvalues ${\lambda}_{1\left(i\right)}$, ${\lambda}_{2\left(i\right)}$ and ${\lambda}_{3\left(i\right)}$ via the generalized eigendecomposition ${\lambda}_{\mathrm{GEIG}}({\mathbf{A}}^{\mathrm{H}}{\mathbf{U}}_{\mathrm{N}}{\mathbf{U}}_{\mathrm{N}}^{\mathrm{H}}\mathbf{A},{\mathbf{A}}^{\mathrm{H}}\mathbf{A})$, where ${\lambda}_{1\left(i\right)}\le {\lambda}_{2\left(i\right)}\le {\lambda}_{3\left(i\right)}$.
- Step 8: Find the global minima of ${\lambda}_{min}={\lambda}_{1\left(j\right)}$. The dipole source position is estimated by ${\mathit{r}}_{\left(j\right)}$.

- Step 1: Scan the entire locating region with the interval of $\u25b3={\u25b3}_{\mathrm{g}}$ by the use of point-by-point scan method, and output the estimation position ${r}_{est}$.
- Step 2: Scan the local region near the estimation position ${r}_{est}$ with the interval of $\u25b3={\u25b3}_{m}$, where $m=1,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}3,\phantom{\rule{0.277778em}{0ex}}\cdots ,\phantom{\rule{0.277778em}{0ex}}M$.
- Step 3: If m equals to M, go to Step 4. Otherwise, update the estimation position ${r}_{est}$, update the interval by $m=m+1$, reduce the searching range and go to Step 2.
- Step 4: Estimate the position by using the CG method and output the final estimation position.

## 3. Numerical Examples

## 4. Experiment

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**The upper plot shows the response of the receiving antenna array to the simulated noiseless source with 100 sample points. The lower plot shows the response of the receiving antenna array for the signal plus noise data such that the signal-to-noise ratio (SNR) is 20 dB.

**Figure 4.**Imaging the spatial spectrum $P\left({\mathit{r}}_{\mathrm{p}}\right)$ on the plane z. (

**a**) $z=0.309$ m for estimating the position of electric dipole source at ${p}_{1}$; (

**b**) $z=0.312$ m for estimating the position of electric dipole source at ${p}_{2}$; (

**c**) $z=0.324$ m for estimating the position of electric dipole source at ${p}_{3}$. The highlight blue spot indicates the true location.

**Figure 9.**The electric dipole source and uniform circular array receiving antenna (UCARA) , left the physical size of the electric dipole source, right side the size of UCARA.

**Figure 11.**Spatial spectra based on the measured data. (

**a**) is the estimation for ${p}_{1}$; (

**b**) is the estimation for ${p}_{2}$; (

**c**) is the estimation for ${p}_{3}$; (

**d**) is the estimation for ${p}_{4}$; and (

**e**) is the estimation for ${p}_{5}$.

Electrode Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

x (m) | 0.4 | 0.5 | 0.6 | 0.4 | 0.5 | 0.7 | 0.6 | 0.34 | 0.5 |

y (m) | 0.6 | 0.4 | 0.6 | 0.4 | 0.5 | 0.3 | 0.5 | 0.55 | 0.5 |

z (m) | 0.5 | 0.6 | 0.6 | 0.4 | 0.7 | 0.6 | 0.4 | 0.55 | 0.5 |

Electrode Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

x (m) | 40 | 50 | 50 | 50 | 40 | 60 | 50 | 50 | 50 |

y (m) | 80 | 140 | 100 | 100 | 100 | 100 | 80 | 120 | 100 |

z (m) | −30 | −30 | −40 | −20 | −30 | −30 | −30 | −30 | −30 |

Electrode Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

x (m) | −0.099 | 0 | 0.099 | 0.14 | 0.099 | 0 | $-0.099$ | $-0.14$ | 0 |

y (m) | 0.099 | 0.14 | 0.099 | 0 | $-0.099$ | $-0.14$ | $-0.099$ | 0 | 0 |

z (m) | −0.3 | −0.3 | −0.3 | −0.3 | $-0.3$ | $-0.3$ | $-0.3$ | $-0.3$ | $-0.3$ |

Dipole Source | Actual Position | Estimated Position | Maximum Error (m) | Minimum Error (m) | RMS Error (m) |
---|---|---|---|---|---|

${p}_{1}$ | $[0.0,0.3,-0.3]$ | $[0.02,\mathrm{0,32},-0.30]$ | 0.035 | 0.024 | 0.026 |

${p}_{2}$ | $[0.1,0.3,-0.3]$ | $[0.11,0.31,-0.30]$ | 0.009 | 0.009 | 0.009 |

${p}_{3}$ | $[0.2,0.3,-0.3]$ | $[0.20,0.29,-0.29]$ | 0.076 | 0.023 | 0.046 |

${p}_{4}$ | $[0.3,0.3,-0.3]$ | $[0.30,0.31,-0.28]$ | 0.026 | 0.017 | 0.020 |

${p}_{5}$ | $[0.4,0.3,-0.3]$ | $[0.41,0.30,-0.28]$ | 0.033 | 0.028 | 0.030 |

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**MDPI and ACS Style**

Xu, Y.; Xue, W.; Li, Y.; Guo, L.; Shang, W.
Multiple Signal Classification Algorithm Based Electric Dipole Source Localization Method in an Underwater Environment. *Symmetry* **2017**, *9*, 231.
https://doi.org/10.3390/sym9100231

**AMA Style**

Xu Y, Xue W, Li Y, Guo L, Shang W.
Multiple Signal Classification Algorithm Based Electric Dipole Source Localization Method in an Underwater Environment. *Symmetry*. 2017; 9(10):231.
https://doi.org/10.3390/sym9100231

**Chicago/Turabian Style**

Xu, Yidong, Wei Xue, Yingsong Li, Lili Guo, and Wenjing Shang.
2017. "Multiple Signal Classification Algorithm Based Electric Dipole Source Localization Method in an Underwater Environment" *Symmetry* 9, no. 10: 231.
https://doi.org/10.3390/sym9100231