# A Sparse-Based Off-Grid DOA Estimation Method for Coprime Arrays

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Signal Model

## 3. Proposed Sparse-Based Off-Grid Direction of Arrival Estimation Method

#### 3.1. Coarse Estimation Process

#### 3.2. Fine Estimation Process

#### 3.3. Summary of the Steps about the Proposed Method

**Step****1:**- Compute the sample covariance matrix ${\widehat{\mathit{R}}}_{xx}$ according to Equation (4);
**Step****2:****Step****3:****Step****4:**- Normalize the sample covariance matrix estimation error according to Equations (9), (12), (17) and (18). Then, solve the optimization problem (20) to get $\overline{\mathit{p}}$. The grid points corresponding to K maximum values of $\overline{\mathit{p}}$ are the coarse DOAs ${\mathbf{\theta}}_{coarse}$;
**Step****5:**- Get modified sample covariance matrix $\tilde{\mathit{R}}$ according to Equation (26);
**Step****6:**- Vectorize $\tilde{\mathit{R}}$ and construct the off-grid model according to Equation (27);
**Step****7:**- Fix the grid bias vector ${\mathbf{\delta}}^{\left(j\right)}$ and solve the optimization problem (29) to get ${\tilde{\mathit{p}}}^{\left(\right)}$;
**Step****8:****Step****9:**- Terminate the iterative process if convergence criteria $\left(\right)open="\parallel "\; close="\parallel ">{\mathbf{\delta}}^{\left(\right)}-{\mathbf{\delta}}^{\left(j\right)}/\left(\right)open="\parallel "\; close="\parallel ">{\mathbf{\delta}}^{\left(j\right)}$ is satisfied or the number of iterations exceeds the maximum one. Otherwise, return to
**Step 7**and continue the iteration process. **Output:**- The final grid bias vector ${\mathbf{\delta}}_{final}$ and DOA estimation results $\widehat{\mathbf{\theta}}\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}{\mathbf{\theta}}_{coarse}+{\mathbf{\delta}}_{final}$.

## 4. Numerical Simulations

#### 4.1. Detection Performance

#### 4.2. Resolution Ability

#### 4.3. Estimation Accuracy

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- 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] - Bekkerman, I.; Tabrikian, J. Target detection and localization using MIMO radars and sonars. IEEE Trans. Signal Process.
**2006**, 54, 3873–3883. [Google Scholar] [CrossRef] - So, H.C. Source localization: algorithms and analysis. In Handbook of Position Location. Theory, Practice, and Advances; Zekavat, R., Buehrer, R.M., Eds.; Wiley-IEEE Press: Piscataway, NJ, USA, 2011; pp. 25–66. [Google Scholar]
- Wan, L.; Kong, X.; Xia, F. Joint range-doppler-angle estimation for intelligent tracking of moving aerial targets. IEEE Internet Things J.
**2018**, 5, 1625–1636. [Google Scholar] [CrossRef] - Schmidt, R. Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag.
**1986**, 34, 276–280. [Google Scholar] [CrossRef] - Roy, R.; Kailath, T. ESPRIT-estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoust. Speech Signal Process.
**1989**, 37, 984–995. [Google Scholar] [CrossRef] - Dai, J.; Xu, W.; Zhao, D. Real-valued DOA estimation for uniform linear array with unknown mutual coupling. Signal Process.
**2012**, 92, 2056–2065. [Google Scholar] [CrossRef] - Heidenreich, P.; Zoubir, A.M.; Rubsamen, M. Joint 2-D DOA estimation and phase calibration for uniform rectangular arrays. IEEE Trans. Signal Process.
**2012**, 60, 4683–4693. [Google Scholar] [CrossRef] - Goossens, R.; Rogier, H. A hybrid UCA-RARE/Root-MUSIC approach for 2-D direction of arrival estimation in uniform circular arrays in the presence of mutual coupling. IEEE Trans. Antennas Propag.
**2007**, 55, 841–849. [Google Scholar] [CrossRef] - Moffet, A. Minimum-redundancy linear arrays. IEEE Trans. Antennas Propag.
**1968**, 16, 172–175. [Google Scholar] [CrossRef] [Green Version] - Pal, P.; Vaidyanathan, P.P. Nested arrays: A novel approach to array processing with enhanced degrees of freedom. IEEE Trans. Signal Process.
**2010**, 58, 4167–4181. [Google Scholar] [CrossRef] - Si, W.; Peng, Z.; Hou, C.; Zeng, F. Two-dimensional DOA estimation for three-parallel nested subarrays via sparse representation. Sensors
**2018**, 18, 1861. [Google Scholar] [CrossRef] [PubMed] - Vaidyanathan, P.P.; Pal, P. Sparse sensing with co-prime samplers and arrays. IEEE Trans. Signal Process.
**2011**, 59, 573–586. [Google Scholar] [CrossRef] - Qin, S.; Zhang, Y.D.; Amin, M.G. Generalized coprime array configurations for direction-of-arrival estimation. IEEE Trans. Signal Process.
**2015**, 63, 1377–1390. [Google Scholar] [CrossRef] - Weng, Z.; Djurić, P.M. A search-free DOA estimation algorithm for coprime arrays. Digit. Signal Process.
**2014**, 24, 27–33. [Google Scholar] [CrossRef] - Pal, P.; Vaidyanathan, P.P. Coprime sampling and the MUSIC algorithm. In Proceedings of the 2011 IEEE Digital Signal Processing Workshop and IEEE Signal Processing Education Workshop (DSP/SPE), Sedona, AZ, USA, 4–7 January 2011; pp. 289–294. [Google Scholar]
- Pal, P.; Vaidyanathan, P.P. A grid-less approach to underdetermined direction of arrival estimation via low rank matrix denoising. IEEE Signal Process Lett.
**2014**, 21, 737–741. [Google Scholar] [CrossRef] - Zhou, C.; Zhou, J. Direction-of-Arrival Estimation with Coarray ESPRIT for Coprime Array. Sensors
**2017**, 17, 1779. [Google Scholar] [CrossRef] [PubMed] - Shan, T.J.; Wax, M.; Kailath, T. On spatial smoothing for direction-of-arrival estimation of coherent signals. IEEE Trans Acoust. Speech Signal Process.
**1985**, 33, 806–811. [Google Scholar] [CrossRef] - Tan, Z.; Nehorai, A. Sparse Direction of Arrival Estimation Using Co-Prime Arrays with Off-Grid Targets. IEEE Signal Process. Lett.
**2014**, 21, 26–29. [Google Scholar] [CrossRef] - Tan, Z.; Eldar, Y.C.; Nehorai, A. Direction of arrival estimation using co-prime arrays: A super resolution viewpoint. IEEE Trans. Signal Process.
**2014**, 62, 5565–5576. [Google Scholar] [CrossRef] - Zhou, C.; Shi, Z.; Gu, Y.; Goodman, N.A. DOA estimation by covariance matrix sparse reconstruction of coprime array. In Proceedings of the 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brisbane, Australia, 19–24 April 2015; pp. 2369–2373. [Google Scholar]
- Zhang, Y.D.; Amin, M.G.; Himed, B. Sparsity-based DOA estimation using co-prime arrays. In Proceedings of the 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Vancouver, BC, Canada, 26–31 May 2013; pp. 3967–3971. [Google Scholar]
- Zhang, Y.; Ye, Z.; Xu, X.; Hu, N. Off-grid DOA estimation using array covariance matrix and block-sparse Bayesian learning. Signal Process.
**2014**, 98, 197–201. [Google Scholar] [CrossRef] - Tan, Z.; Yang, P.; Nehorai, A. Joint sparse recovery method for compressed sensing with structured dictionary mismatches. IEEE Trans. Signal Process.
**2014**, 62, 4997–5008. [Google Scholar] [CrossRef] - Jagannath, R.; Hari, K. Block sparse estimator for grid matching in single snapshot DOA estimation. IEEE Signal Process. Lett.
**2013**, 20, 1040–1043. [Google Scholar] [CrossRef] - Yang, Z.; Xie, L. Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization. IEEE Trans. Signal Process.
**2016**, 64, 995–1006. [Google Scholar] [CrossRef] - Yang, Z.; Xie, L.; Zhang, C. Off-grid direction of arrival estimation using sparse Bayesian inference. IEEE Trans. Signal Process.
**2013**, 61, 38–43. [Google Scholar] [CrossRef] - Dai, J.; Bao, X.; Xu, W.; Chang, C. Root sparse Bayesian learning for off-grid DOA estimation. IEEE Signal Process. Lett.
**2017**, 24, 46–50. [Google Scholar] [CrossRef] - Wang, Q.; Zhao, Z.; Chen, Z.; Nie, Z. Grid evolution method for DOA estimation. IEEE Trans. Signal Process.
**2018**, 66, 2374–2383. [Google Scholar] [CrossRef] - Wu, X.; Zhu, W.P.; Yan, J.; Zhang, Z. Two sparse-based methods for off-grid direction-of-arrival estimation. Signal Process.
**2018**, 142, 87–95. [Google Scholar] [CrossRef] - Wang, Y.; Yang, M.L.; Chen, B.X.; Xiang, Z. Improved DOA estimation based on real-valued array covariance using sparse Bayesian learning. Signal Process.
**2016**, 129, 183–189. [Google Scholar] [CrossRef] - Yang, J.; Yang, Y.X.; Liao, G.S.; Lei, B. A super-resolution direction of arrival estimation algorithm for coprime array via sparse Bayesian learning inference. Circuits Syst. Signal Process.
**2018**, 37, 1907–1934. [Google Scholar] [CrossRef] - Sun, F.; Wu, Q.; Sun, Y.; Ding, G.; Lan, P. An iterative approach for sparse direction-of-arrival estimation in co-prime arrays with off-grid targets. Digit. Signal Process.
**2017**, 61, 35–42. [Google Scholar] [CrossRef] - Shaghaghi, M.; Vorobyov, S.A. Subspace leakage analysis and improved DOA estimation with small sample size. IEEE Trans. Signal Process.
**2015**, 63, 3251–3265. [Google Scholar] [CrossRef] - Liu, Z.; Huang, Z.; Zhou, Y. Sparsity-inducing direction finding for narrowband and wideband signals based on array covariance vectors. IEEE Trans. Wirel. Commun.
**2013**, 12, 1–12. [Google Scholar] [CrossRef] - Ottersten, B.; Stoica, P.; Roy, R. Covariance matching estimation techniques for array signal processing applications. Digit. Signal Process.
**1998**, 8, 185–210. [Google Scholar] [CrossRef] - Tibshirani, R. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. B
**1996**, 58, 267–288. [Google Scholar] - Grant, M.; Boyd, S. CVX: MATLAB Software for Disciplined Convex Programming, Version 2.1. March 2014. Available online: http://cvxr.com/cvx (accessed on 10 July 2018).
- Liu, C.L.; Vaidyanathan, P.P. Cramér–rao bounds for coprime and other sparse arrays, which find more sources than sensors. Digit. Signal Process.
**2017**, 61, 43–61. [Google Scholar] [CrossRef] - Shi, J.; Hu, G.; Zhang, X.; Sun, F.; Zhou, H. Sparsity-based two-dimensional doa estimation for coprime array: From sum—Difference coarray viewpoint. IEEE Trans. Signal Process.
**2017**, 65, 5591–5604. [Google Scholar] [CrossRef]

**Figure 2.**Spatial spectrums of the four algorithms with 17 sources. (

**a**) Off-grid sparse Bayesian inference (OGSBI); (

**b**) Low rank matrix denoising (LRD); (

**c**) Least absolute shrinkage and selection operator (LASSO); (

**d**) Proposed method.

**Figure 3.**Resolution ability comparision for proposed method and other three algorithms with two closely spaced sources. (

**a**) OGSBI; (

**b**) LRD; (

**c**) LASSO; (

**d**) Proposed method.

**Figure 5.**RMSE as a function of snapshots with $\mathrm{SNR}=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}$.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Si, W.; Zeng, F.; Hou, C.; Peng, Z.
A Sparse-Based Off-Grid DOA Estimation Method for Coprime Arrays. *Sensors* **2018**, *18*, 3025.
https://doi.org/10.3390/s18093025

**AMA Style**

Si W, Zeng F, Hou C, Peng Z.
A Sparse-Based Off-Grid DOA Estimation Method for Coprime Arrays. *Sensors*. 2018; 18(9):3025.
https://doi.org/10.3390/s18093025

**Chicago/Turabian Style**

Si, Weijian, Fuhong Zeng, Changbo Hou, and Zhanli Peng.
2018. "A Sparse-Based Off-Grid DOA Estimation Method for Coprime Arrays" *Sensors* 18, no. 9: 3025.
https://doi.org/10.3390/s18093025