Abstract
The present DOA estimation approach based on sparse Bayesian learning has two major shortcomings: high algorithm complexity and large estimate errors. These two flaws are mostly the result of an excessive number of complex-valued matrix inversion operations and incorrect grid partitioning in the EM stage of sparse Bayesian learning (SBL). To overcome the aforementioned issues, we propose a three-procedure, low-complexity DOA estimate technique based on SBL. First, the roots of the estimated covariance are used to complete the real-valued conversion and generate the received signal matrix, which includes the virtual array steering vector. Second, a novel iterative method for immobile spots is devised using the probability distribution function of the noise variance. Finally, iterations are completed utilizing dynamic grid approaches to improve DOA estimation accuracy. The simulation findings reveal that the proposed technique significantly speeds up DOA estimation and, to a lesser extent, improves estimation accuracy.






Similar content being viewed by others
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
References
M. Agatonovic, Z. Stankovic, N. Doncov, Application of artificial neural networks for efficient high-resolution 2D DOA estimation. Radio Eng. 21(4), 1178–1186 (2012)
S.D. Babacan, R. Molina, A.K. Katsaggelos, Bayesian compressive sensing using laplace priors. IEEE Trans. Image Process. 19(1), 53–63 (2010)
Z. Cao, J. Dai, W. Xu, C. Chang, Fast variational bayesian inference for temporally correlated sparse signal recovery. IEEE Signal Process. Lett. 28, 214–218 (2021)
P. Chen, Z. Cao, Z. Chen, X. Wang, Off-grid DOA estimation using sparse Bayesian learning in MIMO radar with unknown mutual coupling. IEEE Trans. Signal Process. 67(1), 208–220 (2019)
F. Chen, J. Dai, N. Hu, Z. Ye, Sparse Bayesian learning for off-grid DOA estimation with nested arrays. Digital Signal Process. 82, 187–193 (2018)
Z. Chen, W. Ma, P. Chen, Z. Cao, A robust sparse Bayesian learning-based DOA estimation method with phase calibration. IEEE Access. 8, 141511–141522 (2020)
J. Dai, X. Bao, W. Xu, C. Chang, Root sparse Bayesian learning for off-grid DOA estimation. IEEE Signal Process. Lett. 24(1), 46–50 (2017)
J. Dai, H.C. So, Sparse Bayesian learning approach for outlier-resistant direction-of-arrival estimation. IEEE Trans. Signal Process. 66(3), 744–756 (2018)
J. Dai, H.C. So, Real-valued sparse Bayesian learning for DOA estimation with arbitrary linear arrays. IEEE Trans. Signal Process. 69, 4977–4990 (2021)
M. Delbari, A. Javaheri, H. Zayyani, F. Marvasti, Non-coherent DOA estimation via majorization-minimization using sign information. IEEE Signal Process. Lett. 29, 892–896 (2022)
P. Gerstoft, C.F. Mecklenbräuker, A. Xenaki, S. Nannuru, Multisnapshot sparse Bayesian learning for DOA. IEEE Signal Process. Lett. 23(10), 1469–1473 (2016)
N.R. Goodman, Statistical analysis based on a certain multivariate complex gaussian distribution (anintroduction). Ann. Math. Stat. 34(1), 152–177 (1963)
M. Haardt, J.A. Nossek, Unitary ESPRIT: how to obtain increased estimation accuracy with a reduced computational burden. IEEE Trans. Signal Process. 43(5), 1232–1242 (1995)
K.-C. Huarng, C.-C. Yeh, A unitary transformation method for angle of arrival estimation. IEEE Trans. Signal Process. 39(4), 975–977 (1991)
S. Ji, Y. Xue, L. Carin, Bayesian compressive sensing. IEEE Trans. Signal Process. 56(6), 2346–2356 (2008)
C.-L. Liu, P.P. Vaidyanathan, Correlation subspaces: generalizations and connection to difference coarrays. IEEE Trans. Signal Process. 65(19), 5006–5020 (2017)
W. Liu. Super Resolution DOA Estimation based on Deep Neural Network. Scientific Reports, 10(1), (2020).
Z.-M. Liu, Z.-T. Huang, Y.-Y. Zhou, An efficient maximum likelihood method for direction-of-arrival estimation via sparse bayesian learning. IEEE Trans. Wireless Commun. 11(10), 1–11 (2012)
D. Malioutov, M. Cetin, A.S. Willsky, A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Trans. Signal Process. 53(8), 3010–3022 (2005)
B. Ottersten, P. Stoica, R. Roy, Covariance matching estimation techniques for array signal processing applications. Digital Signal Process. 8(3), 185–210 (1998)
A. Paulraj, V. U. Reddy, T. J. Shan and T. Kailath, Performance Analysis of the Music Algorithm with Spatial Smoothing in the Presence of Coherent Sources. in MILCOM 1986 - IEEE Military Communications Conference: Communications-Computers: Teamed for the 90's,(Monterey, CA, USA, 1986),pp. 41.5.1–41.5.5.
H. M. Pour, Z. Atlasbaf, M. Hakkak , Performance of Neural Network Trained with Genetic Algorithm for Direction of Arrival Estimation, in Mobile Computing & Wireless Communication International Conference (Amman, Jordan, 2007), pp. 197–202.
R. Roy, T. Kailath, ESPRIT-estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoust. Speech Signal Process. 37(7), 984–995 (1989)
R. Schmidt, Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag. 34(3), 276–280 (1986)
P. Stoica, E.G. Larsson, A.B. Gershman, The stochastic CRB for array processing: a textbook derivation. IEEE Signal Process. Lett. 8(5), 148–150 (2001)
M.E. Tipping, Sparse Bayesian learning and the relevance vector machine. J. Mach. Learn. Res. 1, 211–244 (2001)
Z. Wan and W. Liu, A Fast Group Sparsity Based Phase Retrieval Algorithm for Non-Coherent DOA Estimation. in 2020 54th Asilomar Conference on Signals, Systems, and Computers (Pacific Grove, CA, USA, 2020), pp. 220–224.
Z. Wan and W. Liu, Non-Coherent DOA Estimation of Off-Grid Signals With Uniform Circular Arrays. in ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)(Toronto, Canada, 2021), pp. 4370–4374.
Z. Wan, W. Liu, Non-coherent DOA estimation via proximal gradient based on a dual-array structure. IEEE Access. 9, 26792–26801 (2021)
Q. Wang, H. Yu, J. Li, F. Ji, Y. Chen, Sparse Bayesian learning based algorithm for DOA estimation of closely spaced signals. J. Electron. Inf. Technol. 43(3), 708–716 (2021)
Q. Wang, H. Yu, J. Li, F. Ji, F. Chen, Sparse Bayesian learning using generalized double Pareto prior for DOA ESTIMATION. IEEE Signal Process. Lett. 28, 1744–1748 (2021)
Q. Wang, Z. Zhao, Z. Chen, Z. Nie, Grid evolution method for DOA estimation. IEEE Trans. Signal Process. 66(9), 2374–2383 (2018)
Z. Yang, L. Xie, C. Zhang, Off-grid direction of arrival estimation using sparse bayesian inference. IEEE Trans. Signal Process. 61(1), 38–43 (2013)
H. Zamani, H. Zayyani, F. Marvasti, An iterative dictionary learning-based algorithm for DOA estimation. IEEE Commun. Lett. 20(9), 1784–1787 (2016)
H. Zayyani, M. Babaie-Zadeh, C. Jutten, An iterative Bayesian algorithm for sparse component analysis in presence of noise. IEEE Trans. Signal Process. 57(11), 4378–4390 (2009)
Y. Zou, B. Li, C.H. Ritz, Multi-source doa estimation using an acoustic vector sensor array under a spatial sparse representation framework. Circuits Syst. Signal Process. 34, 1–28 (2015)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
Authors have no conflict of interest to declare.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A
Conclusion 1 proof.\({\tilde{\mathbf{R}}}_{y}^{1/2}\) is the Hermitian square root of \({\tilde{\mathbf{R}}}_{y}\), so \({\tilde{\mathbf{W}}}\) can be split into the following equation
Based on \(({\mathbf{AC}} \otimes {\mathbf{BD}}) = ({\mathbf{A}} \otimes {\mathbf{B}})({\mathbf{C}} \otimes {\mathbf{D}})\), derive \({\tilde{\mathbf{W}}}^{{ - {1/2}}}\) as follows
According to \(vec\left\{ {{\mathbf{ABC}}} \right\} = ({\mathbf{C}}^{{\varvec{T}}} \otimes {\mathbf{A}})vec\left\{ {\mathbf{B}} \right\}\), we get.
\({\mathbf{r}}_{{\varvec{W}}} = \sqrt T vec\left\{ {{\mathbf{I}}_{M} } \right\}\).
Conclusion 2 proof. \(i,j \in \left\{ {1,2,...,L} \right\}\),\(L\) is the number of grids and the DOA estimation range \([ - {\varvec{\pi}}/2,{\varvec{\pi}}/2]\) is divided evenly into \(L\) \(\theta\).
Based on \({\tilde{\mathbf{W}}}^{{ - {1/2}}} = \sqrt T \left( {({\tilde{\mathbf{R}}}_{y}^{1/2} )^{{\varvec{T}}} \otimes {\tilde{\mathbf{R}}}_{y}^{1/2} } \right)^{ - 1}\), it follows that
According to \(({\mathbf{A}} \otimes {\mathbf{B}})^{{\varvec{H}}} = {\mathbf{A}}^{{\varvec{H}}} \otimes {\mathbf{B}}^{{\varvec{H}}}\), it follows that
Appendix B
The results derived from Appendix A \({\mathbf{r}}_{{\varvec{W}}} = \sqrt T vec\left\{ {{\mathbf{I}}_{M} } \right\}\),\(({\tilde{\mathbf{W}}}^{{ - {1/2}}} {\mathbf{A}}_{\theta }^{{}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}}\) are further deduced as follows
According to \(vec\left\{ {{\mathbf{ABC}}} \right\} = ({\mathbf{C}}^{{\varvec{T}}} \otimes {\mathbf{A}})vec\left\{ {\mathbf{B}} \right\}\), we get
Since the previous equation demonstrates that \(({\tilde{\mathbf{W}}}^{{ - {1/2}}} {\mathbf{A}}_{\theta }^{{}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}}\) is real and that \({\mathbf{V}}\) and \(\Lambda\) are likewise real, it is clear from the following equation that \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{r} }}\) is real. We can also demonstrate that \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Psi } }}\) and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varepsilon }\) are true values.
\(({\tilde{\mathbf{W}}}^{{ - {1/2}}} {\mathbf{A}}_{\theta }^{{}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}} = ({\mathbf{U}}\Lambda {\mathbf{V}}^{{\varvec{T}}} )^{{\varvec{H}}} {\mathbf{r}}_{{\varvec{W}}} = {\mathbf{V}}\Lambda^{{\varvec{T}}} {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{r} }}\).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wang, G., Kang, Y. & Wang, H. Low-Complexity DOA Estimation Algorithm based on Real-Valued Sparse Bayesian Learning. Circuits Syst Signal Process 43, 4319–4338 (2024). https://doi.org/10.1007/s00034-024-02649-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-024-02649-7