A Kronecker product CLMS algorithm for adaptive beamforming

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Abstract

In this paper, an adaptive algorithm is derived by considering that the beamforming vector can be decomposed as a Kronecker product of two smaller vectors. Such a decomposition leads to a joint optimization problem, which is then solved by using an alternating optimization strategy along with the steepest-descent method. The resulting algorithm, termed here Kronecker product constrained least-mean-square (KCLMS) algorithm, exhibits (in comparison to the well-known CLMS) improved convergence speed and reduced computational complexity; especially, for arrays with a large number of antennas. Simulation results are shown aiming to confirm the robustness of the proposed algorithm under different operating conditions.

Introduction

Adaptive beamforming has emerged as an effective technique for meeting the growing demand for high capacity in mobile communication systems. Such a technique is used to dynamically adjust the radiation pattern of an antenna array in order to suppress interference and strengthen the signal-of-interest (SOI). As a result, an enhanced signal-to-interference-plus-noise ratio (SINR) is obtained at the array output [1], [2], [3], [4], allowing to increase both spectral and energy efficiency of the system [5], [6]. In this sense, adaptive beamforming is being considered along with massive MIMO (multiple-input/multiple-output) systems to improve the capacity of 5G networks [7], [8], [9], since substantial SINR gains can be achieved with large antenna arrays [10]. Among the adaptive beamforming algorithms available in the open literature, the constrained least-mean-square (CLMS) [11] has been widely used to adjust the beamforming vector (other approaches are discussed in [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]), due to its low computational complexity and dependence only on knowledge of the angle-of-arrival (AOA) of the SOI [23]. This algorithm emerges as a solution for the linearly constrained minimum variance (LCMV) problem, which aims to minimize the beamforming output power subject to a set of linear constraints. However, as discussed in [24], the performance of this algorithm is significantly affected by the number of antennas of the array; specifically for large antenna arrays, the algorithm exhibits slow convergence speed and demands high computational resources. Therefore, the use of the CLMS algorithm in modern beamforming applications involving arrays with a large number of antennas can be prohibitive [25].

Aiming to overcome the aforementioned aspects, [25] recently introduced the idea of representing the beamforming vector as a Kronecker product of two smaller vectors. Thereby, the beamforming design can be broken into smaller optimization problems, leading to the development of more efficient algorithms (in terms of convergence speed, numerical stability, computational cost, required memory, and introduced delay). In this context, focusing on adaptive beamforming applications involving arrays with a large number of antennas (such as in massive MIMO systems [10], [26]), the present research work has the following goals:

  • i)

    to reformulate the LCMV optimization problem by expressing the beamforming vector as a Kronecker product of two smaller vectors;

  • ii)

    to derive an adaptive beamforming algorithm that solves the reformulated optimization problem based on an alternating optimization strategy and steepest-descent method;

  • iii)

    to explain the proposed algorithm using a two-dimensional geometric representation;

  • iv)

    to discuss about the computational complexity of the proposed algorithm in comparison to the well-known CLMS algorithm; and

  • v)

    to assess the robustness of the proposed algorithm and provide performance comparisons for different operating conditions.

The remainder of this paper is organized as follows. Section 2 revisits the signal model and the LCMV problem. Section 3 introduces the Kronecker product decomposition, establishes the alternating optimization strategy, presents the derivation of the proposed algorithm, provides an explanation of the algorithm behavior using a geometric representation, and discusses its computational complexity. Section 4 presents simulation results for assessing the performance of the proposed algorithm. Lastly, Section 5 presents concluding remarks.

Throughout this paper, the adopted mathematical notation follows the standard practice of using lower-case boldface letters for vectors, upper-case boldface letters for matrices, and both italic Roman and Greek letters for scalar quantities. Superscripts H and ⁎ represent the Hermitian transpose of a matrix and the complex conjugate, respectively, E() denotes the expected value, and ⊗ stands for the Kronecker product.

Section snippets

Signal model and optimization problem

The scenario considered here consists of K single-antenna mobile terminals (co-channel users) and a base station equipped with an array of M antennas (as described in [27]). Thus, the baseband input vector can be expressed as [23], [28]x(n)=k=1Ksk(n)+z(n) with sk(n) and z(n)CM×1 representing, respectively, the baseband signal vector corresponding to the kth user and the complex additive white Gaussian noise with power σz2 present at each antenna of the array. Then, assuming L multipath

Proposed approach

Here, the beamforming problem is first reformulated by assuming that the beamforming vector can be expressed as a Kronecker product of two smaller beamforming vectors, thus giving rise to a joint optimization problem. Next, an alternating optimization strategy is described in order to solve the joint optimization problem. Based on such a strategy, the Kronecker product CLMS (KCLMS) algorithm is derived by using the steepest-descent method. Then, a geometric interpretation of the algorithm

Simulation results

Here, Monte Carlo (MC) simulation results (50 independent runs) are shown to assess the performance of the proposed KCLMS algorithm in comparison to the CLMS algorithm [11]. To this end, we consider a base station equipped with a ULA having M omnidirectional antennas uniformly spaced by a half wavelength. The scenario is formed by a SOI located at 30o and six interfering signals located at {60o, 50o, 30o, 0o, 40o, 70o} for Case i), or at {60o, 50o, 30o, 0o, 33o, 70o} for Case ii). Note

Concluding remarks

In this paper, assuming that the beamforming vector can be expressed as a Kronecker product of two smaller vectors, the Kronecker product constrained least-mean-square (KCLMS) algorithm was proposed. Such an adaptive beamforming algorithm outperforms the constrained least-mean-square (CLMS) algorithm, presenting better tradeoff between convergence speed and steady-state SINR as well as lower computational load (especially, as the number of antennas in the array increases). Simulation results

CRediT authorship contribution statement

Eduardo Vinicius Kuhn: Investigation, Methodology, Software, Writing – original draft, Writing – review & editing. Ciro André Pitz: Conceptualization, Methodology, Software, Writing – original draft. Marcos Vinicius Matsuo: Methodology, Software, Writing – original draft, Writing – review & editing. Khaled Jamal Bakri: Validation, Writing – original draft, Writing – review & editing. Rui Seara: Resources, Supervision, Writing – review & editing. Jacob Benesty: Conceptualization, Methodology,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This research work was supported in part by the Brazilian National Council for Scientific and Technological Development (CNPq).

The authors are thankful to the Handling Editor and the anonymous reviewers for their valuable and constructive comments and suggestions. Thanks are also due to Mr. Guilherme Martignago Zilli for his helpful comments on an early draft of this paper.

Eduardo Vinicius Kuhn received the B.S. degree in Telecommunications Engineering from the Assis Gurgacz University, Brazil, in 2010, and the M.Sc. and the Ph.D. degrees in Electrical Engineering from the Federal University of Santa Catarina, Brazil, in 2012 and 2015, respectively. He also was a postdoctoral researcher at the LINSE–Circuits and Signal Processing Laboratory of the Federal University of Santa Catarina, from July 2015 to June 2017 and from July 2019 to June 2021, and at the

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  • Cited by (0)

    Eduardo Vinicius Kuhn received the B.S. degree in Telecommunications Engineering from the Assis Gurgacz University, Brazil, in 2010, and the M.Sc. and the Ph.D. degrees in Electrical Engineering from the Federal University of Santa Catarina, Brazil, in 2012 and 2015, respectively. He also was a postdoctoral researcher at the LINSE–Circuits and Signal Processing Laboratory of the Federal University of Santa Catarina, from July 2015 to June 2017 and from July 2019 to June 2021, and at the INRS-EMT: National Institute of Scientific Research–Energy, Materials, and Telecommunications Research Centre, Canada, from January to April 2020. Since 2015, he is a Professor of the Electronics Engineering Department at the Federal University of Technology–Paraná, Brazil. His research interests include adaptive filters and algorithms, stochastic modeling, speech processing, biomedical signal processing, and communications systems.

    Ciro André Pitz received the B.S. degree in Telecommunication Engineering and the M.Sc. degree in Electrical Engineering from the Regional University of Blumenau, Brazil, in 2008 and 2010, respectively. In 2015, he received the Ph.D. degree in Electrical Engineering from the Federal University of Santa Catarina, Brazil. From 2015 to 2017, he was a postdoctoral researcher at the LINSE–Circuits and Signal Processing Laboratory of the Federal University of Santa Catarina. In 2018, he joined the Department of Control, Automation and Computational Engineering at the Federal University of Santa Catarina, where he is currently a Professor. His research interests include adaptive signal processing theory and its application in communications systems.

    Marcos Vinicius Matsuo received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from the Federal University of Santa Catarina, Brazil, in 2010, 2012, and 2016, respectively. From 2016 to 2018, he was a Professor of the Department of Electronics at the Federal Institute of Santa Catarina, Brazil. In 2018, he joined the Department of Control, Automation, and Computation Engineering at the Federal University of Santa Catarina, where he is currently a Professor of Control and Automation Engineering. His research interests include adaptive signal processing, communications systems, and machine learning.

    Khaled Jamal Bakri received the B.S. degree in Electronics Engineering from the Federal University of Technology–Paraná, Brazil, in 2018, and the M.Sc. degree in Electrical Engineering from the Federal University of Santa Catarina, Brazil, in 2020. Currently, he is working towards the Ph.D. degree in Electrical Engineering at the Federal University of Santa Catarina. His research interests include adaptive signal processing and communications systems.

    Rui Seara received the B.S. and M.Sc. degrees in Electrical Engineering from the Federal University of Santa Catarina, Brazil, in 1975 and 1980, respectively. In 1984, he received the Doctoral degree in Electrical Engineering from the Paris-Sud University, Paris, France. He joined the Electrical Engineering Department at the Federal University of Santa Catarina, Brazil, in 1976, where he is currently a Professor of Electrical Engineering, and Director of LINSE–Circuits and Signal Processing Laboratory. His research interests include digital and analog filtering, adaptive signal processing algorithms, image and speech processing, and digital communications.

    Jacob Benesty received the M.Sc degree in Microwaves from Pierre & Marie Curie University, France, in 1987, and Ph.D. degree in Control and Signal Processing from Orsay University, France, in April 1991. During his Ph.D. (from Nov. 1989 to April 1991), he worked on adaptive filters and fast algorithms at the Centre National d'Etudes des Telecommunications (CNET), Paris, France. From January 1994 to July 1995, he worked at Telecom Paris University on multichannel adaptive filters and acoustic echo cancellation. From October 1995 to May 2003, he was first a Consultant and then a Member of the Technical Staff at Bell Laboratories, Murray Hill, NJ, USA. In May 2003, he joined the University of Quebec, INRS-EMT, in Montreal, Quebec, Canada, as a Professor. He is also an Adjunct Professor with Aalborg University, Denmark, a Guest Professor with Northwestern Polytechnical University, Xi'an, China, and a Visiting Professor with the Technion, Haifa, Israel.

    His research interests include signal processing, acoustic signal processing, and multimedia communications. He is the inventor of many important technologies. In particular, he was the lead researcher at Bell Labs who conceived and designed the world-first real-time hands-free full-duplex stereophonic teleconferencing system. Also, he conceived and designed the world-first PC-based multi-party hands-free full-duplex stereo conferencing system over IP networks.

    He is the editor of the book series Springer Topics in Signal Processing. He was the general chair and technical chair of many international conferences and a member of several IEEE technical committees. Four of his journal papers were awarded by the IEEE Signal processing Society and in 2010 he received the Gheorghe Cartianu Award from the Romanian Academy. He has co-authored and co-edited/co-authored numerous books in the area of acoustic signal processing.

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