A low-complexity algorithm to digitally uncouple the mutual coupling effect in antenna arrays via symmetric Toeplitz matrices - Perera, Sirani M.; Lingsch, Levi; Madanayake, Arjuna; Belostotski, Leonid (Journal of computational and applied mathematics, 2026)
In this paper, we solve systems of linear equations having an n×n coefficient matrix as a symmetric Toeplitz matrix having elements found via the measured mutual coupling effects of electromagnetic fields caused by antenna array elements. This coefficient matrix is called the mutual coupling matrix. In general, these mutual coupling matrices are characterized as dense matrices. However,building on our prior work, we have introduced a symmetric Toeplitz structure, defining its elements through the self- and mutual coupling effects of antenna array elements. Thus, in this paper, we propose an algorithm to uncouple the mutual coupling effect of antenna arrays using O(nlog(n)) as opposed to O(n3) complexity while defining the mutual coupling matrix as a matrix defined by the structure, i.e., a symmetric Toeplitz matrix. The proposed mutually coupled systems will be solved using a sparse factorization of the uncoupling matrices consisting of diagonal and butterfly matrices. The proposed algorithm has low arithmetic complexity compared to brute-force computations in solving systems of linear equations associated with mutual coupling matrices. The proposed factorization also leads to an alternative method to solve the system of linear equations having symmetric Toeplitz matrices as coefficient matrices with O(nlog(n)) as opposed to the O(n3) complexity algorithm. To evaluate the accuracy and efficiency of the proposed Toeplitz solver, we have benchmarked our algorithm against highly optimized libraries such as SciPy, NumPy, and PyTorch, specifically focusing on operations involving Toeplitz system solvers and inversion. We show that the proposed Toeplitz solver achieves exceptional efficiency, especially when utilizing GPU acceleration in PyTorch, all while maintaining accuracy. For the demonstration of numerical results based on the proposed digital uncoupling algorithm and the effect of attenuation, we use S-parameters at 1.4 GHz of an 8-element sub-array and a 16-element sub-array. We show that the diagonal elements of the uncoupling matrices steadily decrease as one moves away from the main diagonal, highlighting the diminishing effect of mutual coupling and the predominance of self-coupling over mutual coupling. Finally, an 8-element signal flow graph will be presented to show the uncoupling of mutual coupling effects of antenna arrays in digital signal processing perspective.