# Compressive Sensing Based Design of Sparse Tripole Arrays

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Design Methods

#### 2.1. Array Model

#### 2.2. Compressive Sensing Based Design of Sparse Tripole Arrays

## 3. Design Examples

#### 3.1. Broadside Design Examples

Example | Aperture$/\lambda $ | Number of Tripoles | $\overline{\Delta d}/\lambda $ | Min($\Delta d$) | Computation Time (s) | ${\widehat{\theta}}_{ML}$ | PSL (dB) |
---|---|---|---|---|---|---|---|

Non-reweighted | |||||||

M= 101 | 10 | 16 | 0.67 | 0.15 | 5.61 | ${90}^{\xb0}$ | −32.28 |

Reweighted | |||||||

M= 101 | 5.20 | 8 | 0.74 | 0.60 | 25.92 | ${90}^{\xb0}$ | −23.25 |

Non-reweighted | |||||||

M= 301 | 10 | 14 | 0.77 | 0.57 | 23.86 | ${90}^{\xb0}$ | −32.53 |

Reweighted | |||||||

M= 301 | 5.13 | 8 | 0.73 | 0.60 | 88.88 | ${90}^{\xb0}$ | −22.94 |

Non-reweighted | |||||||

M= 501 | 10 | 14 | 0.77 | 0.58 | 49.90 | ${90}^{\xb0}$ | −32.69 |

Reweighted | |||||||

M= 501 | 5.12 | 8 | 0.73 | 0.60 | 168.52 | ${90}^{\xb0}$ | −22.89 |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${d}_{n}/\lambda $ | 0 | 0.80 | 1.62 | 2.42 | 3.25 | 4.02 | 4.72 | 5.28 | 5.98 | 6.75 | 7.58 | 8.38 | 9.20 | 10 |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

${d}_{n}/\lambda $ | 2.43 | 3.23 | 4.03 | 4.70 | 5.30 | 5.97 | 6.77 | 7.57 |

#### 3.2. Off-Broadside Design Examples

${\theta}_{ML}$ | Aperture$/\lambda $ | Number of Tripoles | $\overline{\Delta d}/\lambda $ | Min($\Delta d$) | Computation Time (s) | ${\widehat{\theta}}_{ML}$ | PSL (dB) |
---|---|---|---|---|---|---|---|

Non-reweighted ${10}^{\xb0}$ | 10 | 15 | 0.71 | 0.68 | 22.17 | ${10}^{\xb0}$ | −35.18 |

Reweighted ${10}^{\xb0}$ | 5.73 | 9 | 0.72 | 0.70 | 71.17 | ${10}^{\xb0}$ | −21.27 |

Non-reweighted ${30}^{\xb0}$ | 10 | 18 | 0.59 | 0.55 | 23.40 | ${30}^{\xb0}$ | −30.92 |

Reweighted ${30}^{\xb0}$ | 6.53 | 12 | 0.59 | 0.57 | 69.05 | ${29}^{\xb0}$ | −23.27 |

Non-reweighted ${50}^{\xb0}$ | 10 | 23 | 0.45 | 0.35 | 24.88 | ${50}^{\xb0}$ | −24.69 |

Reweighted ${50}^{\xb0}$ | 8.20 | 6 | 1.64 | 1.57 | 144.89 | ${49}^{\xb0}$ | −18.12 |

Non-reweighted ${70}^{\xb0}$ | 10 | 24 | 0.43 | 0.17 | 21.18 | ${69}^{\xb0}$ | −19.61 |

Reweighted ${70}^{\xb0}$ | 10 | 8 | 1.43 | 0.17 | 115.22 | ${70}^{\xb0}$ | −17.51 |

${\theta}_{ML}$ | Aperture$/\lambda $ | Number of Tripoles | $\overline{\Delta d}/\lambda $ | Min($\Delta d$) | Computation Time (s) | ${\widehat{\theta}}_{ML}$ | PSL (dB) |
---|---|---|---|---|---|---|---|

Non-reweighted ${50}^{\xb0}$ | 10 | 18 | 0.59 | 0.48 | 25.25 | ${50}^{\xb0}$ | −15.23 |

Non-reweighted ${70}^{\xb0}$ | 10 | 20 | 0.53 | 0.33 | 25.44 | ${70}^{\xb0}$ | −7.59 |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${d}_{n}/\lambda $ | 0 | 0.68 | 1.40 | 2.12 | 2.85 | 3.55 | 4.28 | 5 | 5.72 | 6.45 | 7.15 | 7.88 | 8.60 | 9.32 | 10 |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

${d}_{n}/\lambda $ | 2.13 | 2.83 | 3.57 | 4.27 | 5 | 5.73 | 6.43 | 7.17 | 7.87 |

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Van Trees, H.L. Optimum Array Processing, Part IV of Detection, Estimation, and Modulation Theory; John Wiley & Sons, Inc.: New York, NY, USA, 2002. [Google Scholar]
- Rodger, J.A. Toward reducing failure risk in an integrated vehicle health maintenance system: A fuzzy multi-sensor data fusion kalman filter approach for IVHMS. Expert Syst. Appl.
**2012**, 39, 9821–9836. [Google Scholar] [CrossRef] - Jarske, P.; Saramaki, T.; Mitra, S.K.; Neuvo, Y. On properties and design of nonuniformly spaced linear arrays. IEEE Trans. Acoust. Speech Signal Process.
**1988**, 36, 372–380. [Google Scholar] [CrossRef] - Haupt, R.L. Thinned arrays using genetic algorithms. IEEE Trans. Antennas Propag.
**1994**, 42, 993–999. [Google Scholar] [CrossRef] - Yan, K.K.; Lu, Y. Sidelobe reduction in array-pattern synthesis using genetic algorithm. IEEE Trans. Antennas Propag.
**1997**, 45, 1117–1122. [Google Scholar] - Chen, K.; He, Z.; Han, C. Design of 2-dimensional sparse arrays using an improved genetic algorithm. In Proceedings of the IEEE Workshop on Sensor Array and Multichannel Processing, Waltham, MA, USA, 12–14 July 2006; pp. 209–213.
- Cen, L.; Ser, W.; Yu, Z.L.; Rahardja, S. An improved genetic algorithm for aperiodic array synthesis. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Las Vagas, NV, USA, 31 March–4 April 2008; pp. 2465–2468.
- Trucco, A.; Murino, V. Stochastic optimization of linear sparse arrays. IEEE J. Ocean. Eng.
**1999**, 24, 291–299. [Google Scholar] [CrossRef] - Repetto, S.; Trucco, A. Designing superdirective microphone arrays with a frequency-invariant beam pattern. IEEE Sens. J.
**2006**, 6, 737–747. [Google Scholar] [CrossRef] - Caorsi, S.; Lommi, A.; Massa, A.; Pastorino, M. Peak sidelobe level reduction with a hybrid approach based on gas and difference sets. IEEE Trans. Antennas Propag.
**2004**, 52, 1116–1121. [Google Scholar] [CrossRef] - Oliveri, G.; Massa, A. Genetic algorithm (GA)-enhanced almost difference set (ADS)-based approach for array thinning. IET Microwav. Antennas Propag.
**2011**, 5, 305–315. [Google Scholar] [CrossRef] - Candes, E.; Romberg, J.; Tao, T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory
**2006**, 52, 489–509. [Google Scholar] [CrossRef] - Ji, S.; Xue, Y.; Carin, L. Bayesian compressive sensing. IEEE Trans. Signal Process.
**2008**, 56, 2346–2356. [Google Scholar] [CrossRef] - Wang, W.; Wu, R. High resolution direction of arrival (DOA) estimation based on improved orthogonal matching pursuit (OMP) algorithm by iterative local searching. Sensors
**2013**, 13, 11167–11183. [Google Scholar] [CrossRef] [PubMed] - Li, L.; Zhang, W.; Li, F. The design of sparse antenna array. CoRR
**2008**, 11, 110–117. [Google Scholar] - Carin, L. On the relationship between compressive sensing and random sensor arrays. IEEE Antennas Propag. Mag.
**2009**, 51, 72–81. [Google Scholar] [CrossRef] - Cen, L.; Ser, W.; Cen, W.; Yu, Z.L. Linear sparse array synthesis via convex optimization. In Proceedings of the IEEE International Symposium on Circuits and Systems, Paris, France, 30 May–2 June 2010; pp. 4233–4236.
- Prisco, G.; D’Urso, M. Exploiting compressive sensing theory in the design of sparse arrays. In Proceedings of the IEEE Radar Conference, Kansas City, MO, USA, 23–27 May 2011; pp. 865–867.
- Oliveri, G.; Carlin, M.; Massa, A. Complex-weight sparse linear array synthesis by bayesian compressive sampling. IEEE Trans. Antennas Propag.
**2012**, 60, 2309–2326. [Google Scholar] [CrossRef] - Fuchs, B. Synthesis of sparse arrays with focused or shaped beampattern via sequential convex optimizations. IEEE Trans. Antennas Propag.
**2012**, 60, 3499–3503. [Google Scholar] [CrossRef] - Prisco, G.; D’Urso, M. Maximally sparse arrays via sequential convex optimizations. IEEE Antennas Wirel. Propag. Lett.
**2012**, 11, 192–195. [Google Scholar] [CrossRef] - Candes, E.J.; Wakin, M.B.; Boyd, S.P. Enhancing sparsity by reweighted l
_{1}minimization. J. Fourier Anal. Appl.**2008**, 14, 877–905. [Google Scholar] [CrossRef] - Williams, D.B.; Madisetti, V. (Eds.) Digital Signal Processing Handbook, 1st ed.; CRC Press, Inc.: Boca Raton, FL, USA, 1997.
- Zhong, X.; Premkumar, A.; Wang, H. Multiple wideband acoustic source tracking in 3-D space using a distributed acoustic vector sensor array. IEEE Sens. J.
**2014**, 14, 2502–2513. [Google Scholar] [CrossRef] - Zheng, G. A novel spatially spread electromagnetic vector sensor for high-accuracy 2-D DOA estimation. Multidimens. Syst. Signal Process.
**2015**, 16, 1–26. [Google Scholar] [CrossRef] - Wang, Y.; Xia, W.; He, Z.; Sun, Y. Polarimetric detection for vector-sensor processing in quaternion proper gaussian noises. Multidimens. Syst. Signal Process.
**2015**, 1–22. [Google Scholar] [CrossRef] - Compton, R.T. The tripole antenna: An adaptive array with full polarization flexibility. IEEE Trans. Antennas Propag.
**1981**, 29, 944–952. [Google Scholar] [CrossRef] - Leprettre, B.; Martin, N.; Glangeaud, F.; Navarre, J.P. Three-component signal recognition using time, time-frequency, and polarization information-application to seismic detection of avalanches. IEEE Trans. Signal Process.
**1998**, 46, 83–102. [Google Scholar] [CrossRef] - Bansal, R. Tripole to the rescue [antennas]. IEEE Antennas Propag. Mag.
**2001**, 43, 106–107. [Google Scholar] [CrossRef] - Wong, K. Direction finding/polarization estimation-dipole and/or loop triad(s). IEEE Trans. Aerosp. Electron. Syst.
**2001**, 37, 679–684. [Google Scholar] [CrossRef] - Wong, K. Blind beamforming/geolocation for wideband-FFHs with unknown hop-sequences. IEEE Trans. Aerospace Electron. Syst.
**2001**, 37, 65–76. [Google Scholar] [CrossRef] - Lundback, J.; Nordebo, S. Analysis of a tripole array for polarization and direction of arrival estimation. In Proceedings of the IEEE Workshop on Sensor Array and Multichannel Signal Processing, Barcelona, Spain, 18–21 July 2004; pp. 284–288.
- Au-Yeung, C.K.; Wong, K. CRB: sinusoid-sources’ estimation using collocated dipoles/loops. IEEE Trans. Aerospace Electron. Syst.
**2009**, 45, 94–109. [Google Scholar] [CrossRef] - Yuan, X.; Wong, K.; Xu, Z.; Agrawal, K. Various compositions to form a triad of collocated dipoles/loops, for direction finding and polarization estimation. IEEE Sens. J.
**2012**, 12, 1763–1771. [Google Scholar] [CrossRef] - Getu, B. Theoretical capacity of tripole array MIMO systems within a fixed area. In Proceedings of the Wireless and Mobile Networking Conference (WMNC), Dubai, UAE, 23–25 April 2013; pp. 1–4.
- Zoltowski, M.; Wong, K. Closed-form eigenstructure-based direction finding using arbitrary but identical subarrays on a sparse uniform cartesian array grid. IEEE Trans. Signal Process.
**2000**, 48, 2205–2210. [Google Scholar] [CrossRef] - Zoltowski, M.; Wong, K. ESPRIT-based 2-D direction finding with a sparse uniform array of electromagnetic vector sensors. IEEE Trans. Signal Process.
**2000**, 48, 2195–2204. [Google Scholar] [CrossRef] - Wong, K.; Zoltowski, M. Closed-form direction finding and polarization estimation with arbitrarily spaced electromagnetic vector-sensors at unknown locations. IEEE Trans. Antennas Propag.
**2000**, 48, 671–681. [Google Scholar] [CrossRef] - Wong, K.; Li, L.; Zoltowski, M. Root-MUSIC-based direction-finding and polarization estimation using diversely polarized possibly collocated antennas. IEEE Antennas Wirel. Propag. Lett.
**2004**, 3, 129–132. [Google Scholar] [CrossRef] - Wong, K.; Yuan, X. Vector cross-product direction-finding with an electromagnetic vector-sensor of six orthogonally oriented but spatially noncollocating dipoles/loops. IEEE Trans. Signal Process.
**2011**, 59, 160–171. [Google Scholar] [CrossRef] - He, J.; Ahmad, M.O.; Swamy, M. Extended-aperture angle-range estimation of multiplefresnel-region sources with a linear tripole array using cumulants. Signal Process.
**2012**, 92, 939–953. [Google Scholar] [CrossRef] - Zheng, G.; Wu, B.; Ma, Y.; Chen, B. Direction of arrival estimation with a sparse uniform array of orthogonally oriented and spatially separated dipole-triads. IET Radar Sonar Navig.
**2014**, 8, 885–894. [Google Scholar] [CrossRef] - Song, Y.; Yuan, X.; Wong, K. Corrections to vector cross-product direction-finding with an electromagnetic vector-sensor of six orthogonally oriented but spatially noncollocating dipoles/loops [jan 11 160-171]. IEEE Trans. Signal Process.
**2014**, 62, 1028–1030. [Google Scholar] [CrossRef] - Hawes, M.; Liu, W. Location optimization of robust sparse antenna arrays with physical size constraint. IEEE Antennas Wirel. Propag. Lett.
**2012**, 11, 1303–1306. [Google Scholar] [CrossRef] - Hawes, M.; Liu, W. Sparse array design for wideband beamforming with reduced complexity in tapped delay-lines. IEEE Trans. Acoust. Speech Lang. Process.
**2014**, 22, 1236–1247. [Google Scholar] [CrossRef] - Hawes, M.; Liu, W. Compressive sensing-based approach to the design of linear robust sparse antenna arrays with physical size constraint. IET Microwav. Antennas Propag.
**2014**, 8, 736–746. [Google Scholar] [CrossRef][Green Version] - Hawes, M.; Liu, W. Design of fixed beamformers based on vector sensor arrays. Int. J. Antennas Propag.
**2015**, 2015, 1–9. [Google Scholar] [CrossRef] - Zhang, X.; Liu, Z.; Liu, W.; Xu, Y. Quasi-vector-cross-product based direction finding algorithm with a spatially stretched tripole. In Proceedings of the 2013 IEEE International Conference of IEEE Region 10 (TENCON13), Xi’an, China, 22–25 October 2013; pp. 1–4.
- Research, C. CVX: Matlab Software for Disciplined Convex Programming, Version 2.0 Beta. Available online: http://cvxr.com/cvx (accessed on 20 September 2015).
- Grant, M.; Boyd, S. Graph implementations for nonsmooth convex programs. In Recent Advances in Learning and Control (a tribute to M. Vidyasagar; Blondel, V., Boyd, S., Kimura, H., Eds.; Springer-Verlag Limited: Berlin Heidelberg, Germany, 2008; pp. 95–110. [Google Scholar]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hawes, M.; Liu, W.; Mihaylova, L. Compressive Sensing Based Design of Sparse Tripole Arrays. *Sensors* **2015**, *15*, 31056-31068.
https://doi.org/10.3390/s151229849

**AMA Style**

Hawes M, Liu W, Mihaylova L. Compressive Sensing Based Design of Sparse Tripole Arrays. *Sensors*. 2015; 15(12):31056-31068.
https://doi.org/10.3390/s151229849

**Chicago/Turabian Style**

Hawes, Matthew, Wei Liu, and Lyudmila Mihaylova. 2015. "Compressive Sensing Based Design of Sparse Tripole Arrays" *Sensors* 15, no. 12: 31056-31068.
https://doi.org/10.3390/s151229849