Abstract
The interpolation concept can fill the holes in the difference coarray of nonuniform linear arrays (NLAs), thus ensuring effective enhancement of the degrees of freedom and expansion of the array aperture. However, compared to interpolation-based direction of arrival (DOA) estimation techniques, the application of interpolation to beamforming approaches has not been universally investigated. In this study, two robust adaptive beamforming algorithms are proposed. First, the observed data is interpolated and the signal of interest (SOI) is subtracted. Second, the vector completion problem of the received data associated with the derived virtual array is formulated, which can be implemented with the utilization of atomic norm minimization. Unlike previous algorithms, the proposed method obtains the interference plus noise covariance matrix (INCM) and the DOA of the SOI by solving the vector completion problem rather than matrix completion. Thereafter, this problem is solved iteratively in two convex steps, whose closed-form expressions for the alternating direction method of multipliers are derived to enhance computational efficiency. The first proposed algorithm calculates the weight vector using rows corresponding to the physical sensor locations of the INCM. The second one fills the gaps of NLAs using the virtual filling technique to form a consecutive array and directly applies the resulting INCM to compute the weight vector. Numerical simulations demonstrate the superior overall performance of the two proposed methods under nonideal scenarios.
Similar content being viewed by others
Data Availability
All data generated or analysed during this study are included in this published article.
References
D.A. Abraham, Underwater acoustic signal processing (Springer, Berlin, 2019), pp. 621–741. https://doi.org/doi.org/10.1007/978-3-319-92983-5_9.
M. Bekrani, A. H. T. Nguyen, A. W. H. Khong, An adaptive non-linear process for under-determined virtual microphone beamforming, 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp 4495–4499 (2021).
B.N. Bhaskar, G. Tang, B. Recht, Atomic norm denoising with applications to line pectral estimation. IEEE Trans. Signal Process. 61(23), 5987–5999 (2013)
J. Capon, High-resolution frequency-wavenumber spectrum analysis. Proc. IEEE 57(8), 1408–1418 (1969)
P. Chen, Z. Chen, P. Miao, et al., RIS-ADMM: an ADMM-based passive and sparse sensing method with interference removal, arXiv preprint arXiv:2206.06172 (2022)
P. Chen, Y. Yang, Y. Wang et al., Adaptive beamforming with sensor position errors using covariance matrix construction based on subspace bases transition. IEEE Signal Process. Lett. 26(1), 19–23 (2019)
Y. Cheng, X. Zhang, T. Liu et al., Coprime array-adaptive beamforming via atomic-norm-based sparse recovery. IET Radar Sonar Navig. 15(11), 1494–1507 (2021)
Y. Chu, Z. Wei, Z. Yang, New reweighted atomic norm minimization approach for line spectral estimation. Signal Process. 206, 108897 (2023)
Y. Du, H. Xu, W. Cui et al., Adaptive beamforming algorithm for coprime array based on interference and noise covariance matrix reconstruction. IET Radar Sonar Navig. 16(4), 668–677 (2022)
B. Feng, D.C. Jenn, Grating lobe suppression for distributed digital subarrays using virtual filling. IEEE Antennas Wirel. Propag. Lett. 12, 1323–1326 (2013)
Q. Ge, Y. Zhang, Z. Feng et al., Novel robust adaptive beamformer in the presence of gain-phase errors. Circuits Syst. Signal Process. 40, 1926–1947 (2021)
M. Grant, S. Boyd, Y. Ye, CVX: Matlab software for disciplined convex program-ming, version 2.1 [Online]. Available: http://cvxr.com/cvx, (2017).
Y. Gu, C. Zhou, N. A. Goodman, et al., Coprime array adaptive beamforming based on compressive sensing virtual array signal, in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2981–2985 (2016).
J. He, T. Shu, V. Dakulagi et al., Simultaneous interference localization and array calibration for robust adaptive beamforming with partly calibrated arrays. IEEE Trans. Aerosp. Electron. Syst. 57(5), 2850–2863 (2021)
F. Huang, W. Sheng, X. Ma, Modified projection approach for robust adaptive array beamforming. Signal Process. 92(7), 1758–1763 (2012)
Y. Huang, M. Zhou, S.A. Vorobyov, New designs on MVDR robust adaptive beamforming based on optimal steering vector estimation. IEEE Trans. Signal Process. 67(14), 3624–3638 (2019)
Y. Ke, C. Zheng, R. Peng et al., Robust adaptive beamforming using noise reduction preprocessing-based fully automatic diagonal and steering vector estimation. IEEE Access 5, 12974–12987 (2017)
Y. Li, Y. Chi, Off-the-grid line spectrum denoising and estimation with multiple measurement vectors. IEEE Trans. Signal Process. 64(5), 1257–1269 (2016)
K. Liu, Y.D. Zhang, Coprime array-based robust beamforming using covariance matrix reconstruction technique. IET Commun. 12(17), 2206–2212 (2018)
S. Liu, J. Zhao, Y. Zhang et al., 2D DOA estimation algorithm by nested acoustic vector-sensor array. Circuits Syst. Signal Process. 41, 1115–1130 (2021)
J. Lu, J. Yang, B. Hou et al., Virtual reconstruction-based robust adaptive beamforming for distributed digital subarray antennas. Int. J. Antennas Propag. 2021, 6649439 (2021)
Y. Lv, F. Cao, F. Wu et al., Robust adaptive beamforming based on covariance matrix reconstruction using Gauss-Legendre quadrature and steering vector estimation. EURASIP J. Adv. Signal Process. 9, 1–20 (2023)
Y. Lv, C. He, F. Cao et al., Interleaved arrays with second-order difference coarray generation for mutual coupling reduction. Circuits Syst. Signal Process. 2023, 1–29 (2023)
H. Qiao, P. Pal, Gridless line spectrum estimation and low-rank Toeplitz matrix compression using structured samplers: A regularization-free approach. IEEE Trans. Signal Process. 65(9), 2221–2236 (2017)
G. Tang, B.N. Bhaskar, P. Shah et al., Compressed sensing off the grid. IEEE Trans. Information Theory 59(11), 7465–7490 (2013)
P.P. Vaidyanathan, P. Pal, Sparse sensing with co-prime samplers and arrays. IEEE Trans. Signal Process. 59(2), 573–586 (2011)
S.A. Vorobyov, A.B. Gershman, Z. Luo, Robust adaptive beamforming using worst-case performance optimisation: a solution to the signal mismatch problem. IEEE Trans. Signal Process. 51(2), 313–324 (2003)
J. Xu, G. Liao, L. Huang et al., Robust adaptive beamforming for fast-moving target detection with FDA-STAP radar. IEEE Trans. Signal Process. 65(4), 973–984 (2017)
J. Xu, G. Liao, S. Zhu et al., Response vector constrained robust LCMV beamforming based on semidefinite programming. IEEE Trans. Signal Process. 63(21), 5720–5732 (2015)
X. Yang, Y. Li, F. Liu et al., Robust adaptive beamforming based on covariance matrix reconstruction with annular uncertainty set and vector space projection. IEEE Antennas Wirel. Propag. Lett. 20(2), 130–134 (2020)
J. Yang, G. Liao, J. Li, Robust adaptive beamforming in nested array. Signal Process. 114, 143–149 (2015)
L. Yu, Y. Wei, W. Liu, Adaptive beamforming based on nonuniform linear arrays with enhanced degrees of freedom, TENCON-2015 IEEE Region 10 Conference, 1–5 (2015).
X. Yuan, L. Gan, Robust adaptive beamforming via a novel subspace method for interference covariance matrix reconstruction. Signal Process. 130, 233–242 (2017)
X. Zhang, W. Jiang, K. Huo et al., Robust adaptive beamforming based on linearly modified atomic-norm minimization with target contaminated data. IEEE Trans. Signal Process. 68, 5138–5151 (2020)
Z. Zheng, Y. Huang, W. Wang et al., Augmented covariance matrix reconstruction for DOA estimation using difference coarray. IEEE Trans. Signal Process. 69, 5345–5358 (2021)
Z. Zheng, T. Yang, D. Jiang et al., Robust and efficient adaptive beamforming using nested subarray principles. IEEE Access 8, 4076–4085 (2019)
Z. Zheng, T. Yang, W. Wang et al., Robust adaptive beamforming via coprime coarray interpolation. Signal Process. 169, 107382 (2020)
Z. Zheng, Y. Zheng, W. Wang et al., Covariance matrix reconstruction with interference steering vector and power estimation for robust adaptive beamforming. IEEE Trans. Veh. Technol. 67(9), 8495–8503 (2018)
C. Zhou, Y. Gu, X. Fan et al., Direction-of-arrival estimation for coprime array via virtual array interpolation. IEEE Trans. Signal Process. 66(22), 5956–5971 (2018)
C. Zhou, Y. Gu, S. He et al., A robust and efficient algorithm for coprime array adaptive beamforming. IEEE Trans. Veh. Technol. 67(2), 1099–1112 (2018)
C. Zhou, Z. Shi, Y. Gu, Coprime array adaptive beamforming with enhanced degrees-of-freedom capability, in Proc. IEEE Radar Conference (RadarConf), 1357–1361 (2017).
Acknowledgements
The work has been supported partly by the National Natural Science Foundation of China (Grant nos. 62071481, 61903375, and 61773389), Natural Science Foundation of Shaanxi Province (2021KJXX-22, 2020JQ-298), Post-doctoral Science Foundation of China (2019M663635), and Special Support Plan for High-level Talents in Shaanxi Province (TZ0328).
Author information
Authors and Affiliations
Contributions
YL: Writing and methodology. FC: Supervision and software. CH: Review. WZ: Review. JX: Review. JY: Review.
Corresponding author
Ethics declarations
Conflict of interest
All data generated or analysed during this study are included in this published article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A
First, we substitute \({\mathbf{y}} - p_{0} {\mathbf{b}}_{0}\) into the left side of (18), it gains
Analyzing the \(q\)-th term in (40), we get
Therefore, the following formula is derived:
This completes the proof.
Appendix B
For \({\mathbf{y}}^{i + 1}\), let
where \({\mathbf{Z}}^{i} = \left[ {\begin{array}{*{20}l} {{\mathbf{Z}}_{11}^{i} } \hfill & {\quad {\mathbf{z}}_{12}^{i} } \hfill \\ {{\mathbf{z}}_{21}^{i} } \hfill & {\quad z_{22}^{i} } \hfill \\ \end{array} } \right]\), \({\mathbf{z}}_{12}^{i} = {\mathbf{z}}_{21}^{{i,{\text{H}} }}\), \({\mathbf{F}}^{i} = \left[ {\begin{array}{*{20}c} {{\mathbf{F}}_{11}^{i} } & {\quad {\mathbf{f}}_{12}^{i} } \\ {{\mathbf{f}}_{21}^{i} } & {\quad f_{22}^{i} } \\ \end{array} } \right]\), and \({\mathbf{f}}_{12}^{i} = {\mathbf{f}}_{21}^{{i,{\text{H}} }}\). Subsequently, the update form for \({\mathbf{y}}^{i + 1}\) is calculated as
In the case of \(p_{0}^{i + 1}\), we make
The closed form for \(p_{0}^{i + 1}\) is yielded:
For \({\mathbf{u}}^{i + 1}\), let
The l-th entry of \({\mathbf{u}}^{i + 1}\) is derived as
Here, \({\text{Tr}} \left( {{\mathbf{X}},l} \right) = \sum\nolimits_{\chi } {X_{l + \chi - 1, \, \chi } } ,\chi = 1,2, \ldots ,L - l + 1\) and \({\mathbf{e}}_{1}\) stands for a zero vector, except that its first entry is 1.
In the case of \(t^{i + 1}\), we make
Therefore, we update \(t^{i + 1}\) as
For \({\mathbf{Z}}^{i + 1}\), it can be formulated as
Based on the eigenvalue decomposition, we obtain
Then, \({\mathbf{Z}}^{i + 1} = {\mathbf{D}}{\varvec{\Lambda}}^{\!+}{\mathbf{D}}^{\text{H}}\) is obtained, where \({{\varvec{\Lambda}}}^{ + }\) is gained by substituting the negative entries in \({{\varvec{\Lambda}}}\) with 0.
In the case of \({\mathbf{F}}^{i + 1}\), we have
Thereafter, \({\mathbf{F}}^{i + 1}\) is updated as
ADMM is completed once the maximum number of iterations, that is, \(i_{\max }\), is achieved or the stopping criterion is met.
Appendix C
The deviation between \({\hat{\mathbf{y}}}\) and \({\mathbf{r}}_{I}\), excluding their interpolated elements, can be expressed as
According to (55), we get
Introducing the Cauchy–Schwarz inequality into the last term of (56) yields
Based on (23), (57), and \(\left\| {{\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\|_{2} \le \left\| {\tau ({\mathbf{y}}^{{\mathbf{*}}} ) \oplus {\mathbf{G}} - {\mathbf{R}}_{I} } \right\|_{\text{F}}\), (56) can be transformed into [35]
where \({\mathbf{G}} = \tau ({\mathbf{g}})\), and \({\mathbf{R}}_{I} = \tau ({\mathbf{r}}_{I} )\) stands for the covariance matrix of the derived virtual array [39]. Subsequently, the off-the-shelf conclusion in Lemma 1 [24] is exploited, described as
Lemma 1
If \({\mathbf{x}}(k),k = 1,2, \cdots ,K\) is a Gaussian random vector following the distribution \({\mathbf{x}}(k)\sim CN(0,{\mathbf{R}}_{x} )\),
Here, \({\text{P}} \left\{ \cdot \right\}\) represents the probability. Without loss of generality, generalizing (59) to the virtual domain yields [35, 39]
with a probability of, at least, \(1 - 2{\text{e}}^{ - 2c\sqrt K }\). In (60), we consider \({\mathbf{u}}^{{\mathbf{*}}}\) to be a precise estimate of \({\mathbf{y}}_{q}^{{\mathbf{*}}}\). Then, defining \(\mu_{2} = \tfrac{\mu }{2}\left\{ {\tfrac{1}{L}{\text{Tr}} \left( {\tau ({\mathbf{u}}^{{\mathbf{*}}} - {\hat{\mathbf{u}}})} \right) + t^{{\mathbf{*}}} - \hat{t}} \right\}\) and denoting \(\left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}}} \right\|_{2}\) by \(y_{g}\), (58) is expressed in the following general form:
Thus, we get \(y_{g} - \mu_{1} - \sqrt {\mu_{1}^{2} + \mu_{2} } \le 0\), which means
This completes the proof.
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
Lv, Y., Cao, F., He, C. et al. Robust Adaptive Beamforming via Virtual Interpolation-Based Atomic Norm Minimization. Circuits Syst Signal Process 42, 7377–7403 (2023). https://doi.org/10.1007/s00034-023-02450-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-023-02450-y