Investigation of matrix inversion methods for application in adaptive beamforming algorithms

The paper presents a study of covariance matrix inversion methods in adaptive beamforming for antenna arrays. Two signal processing paradigms are considered, namely spatial processing and space–time processing, for which the structure of the covariance matrix and its impact on the choice of inversion algorithms are analyzed. Wiener-optimal weight vectors, obtained as the solution of the mean square error minimization problem, are used as a reference solution. The Cholesky decomposition, a recursive Levinson-type algorithm, the Bareiss method, and an FFT-based approximation are compared in terms of the accuracy of reproducing the optimal weights, the resulting training mean square error, the shape of the radiation pattern and computational complexity. Numerical simulations are performed in MATLAB for different antenna array geometries under the same noise scenario. The article considers the relationship between the structure of the covariance matrix in spatial and space-time processing tasks, the choice of algorithms for its inversion, and their computational efficiency. It is shown that exact inversion methods provide results consistent with the Wiener-optimal solution, whereas approximate methods significantly reduce computational cost at the expense of a controlled increase in error. The obtained results confirm the practical relevance of structured covariance matrix inversion methods for space-time adaptive signal processing.

1. Glushankov Ye.I., Tsarik V.I. Direct Adaption Methods for Linear and Circular Antenna Arrays. Journal of the Russian Universities. Radioelectronics. 2023;26(1):6–16. (In Russ.). https://doi.org/10.32603/1993-8985-2023-26-1-6-16

2. Glushankov E.I., Tsarik I.V., Tsarik V.I. Spatially distributed technology of adaptive formation of the antenna array pattern. Scientific News. 2022;(29):165–169. (In Russ.).

3. Van Trees H.L. Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory. New York: John Wiley & Sons; 2002. 1472 p.

4. Golub G.H., Van Loan Ch.F. Matrix Computations. Baltimore: The Johns Hopkins University Press; 2013. 756 p.

5. Aubry A., Babu P., De Maio A., Rosamilia M. Advanced Methods for MLE of Toeplitz Structured Covariance Matrices With Applications to Radar Problems. IEEE Transactions on Information Theory. 2024;70(12):9277–9292. https://doi.org/10.1109/TIT.2024.3474977

6. Dzhigan V.I. Adaptive signal filtering: Theory and algorithms. Moscow: Tekhnosfera; 2013. 528 p. (In Russ.).

7. Sosulski Ja., Tangermann M. Introducing block-Toeplitz covariance matrices to remaster linear discriminant analysis for event-related potential brain-computer interfaces. Journal of Neural Engineering. 2022;19(6). https://doi.org/10.1088/1741-2552/ac9c98

8. Woolrich M., Hunt L., Groves A., Barnes G. MEG beamforming using Bayesian PCA for adaptive data covariance matrix regularization. NeuroImage. 2011;57(4):1466–1479. https://doi.org/10.1016/j.neuroimage.2011.04.041

9. Haykin S.S. Adaptive Filter Theory. Boston: Pearson; 2014. 912 p.

10. Glushankov E., Tsarik V. Quality Analysis of Space and Space-Frequency Adaptive Signal Processing Algorithms in Satellite Navigation Systems. Proceedings of Telecommunication Universities. 2022;8(3):37–43. (In Russ.). https://doi.org/10.31854/1813-324X-2022-8-3-37-43

11. Böttcher A., Grudsky S.M. Toeplitz matrices, asymptotic linear algebra and functional analysis. Gurgaon: Hindustan Book Agency; 2000. 126 p. https://doi.org/10.1007/978-93-86279-04-0

12. Akaike H. Block Toeplitz matrix inversion. SIAM Journal on Applied Mathematics. 1973;24(2):234–241. https://doi.org/10.1137/0124024

13. Cybenko G. The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations. SIAM Journal on Scientific and Statistical Computing. 1980;1(3):303–319. https://doi.org/10.1137/0901021

14. Chan T.F. An optimal circulant preconditioner for Toeplitz systems. SIAM Journal on Scientific and Statistical Computing. 1988;9(4):766–771. https://doi.org/10.1137/0909051

15. Gray R.M. Toeplitz and Circulant Matrices: A Review. Foundations and Trends in Communications and Information Theory. 2006;2(3):155–239.