[en] In this paper, we investigate symbol-level precoding (SLP) and efficient decoding techniques for downlink transmission, where we focus on scenarios where the base station (BS) transmits multiple quadrature amplitude modulation (QAM) constellation streams to users equipped with multiple receive antennas. We begin by formulating a symbol-level joint design scheme aimed at collaboratively optimizing the transmit precoding and receive combining matrices. This coupled problem is addressed by employing the alternating optimization (AO) method, and closed-form solutions are derived by analyzing the obtained two subproblems. Furthermore, to address the dependence of the receive combining matrix on the transmit signals, we switch to maximum likelihood detection (MLD) method for decoding. Notably, we have demonstrated that the smallest singular value of the precoding matrix significantly impacts the performance of MLD method. Specifically, a lower value of the smallest singular value results in degraded detection performance. Additionally, we show that the traditional SLP matrix is rank-one, making it infeasible to directly apply MLD at the receiver end. To circumvent this limitation, we propose a novel symbol-level smallest singular value maximization problem, termed SSVMP, to enable SLP in systems where users employ the MLD decoding approach. Moreover, to reduce the number of variables to be optimized, we further derive a more generic semidefinite programming (SDP)-based optimization problem. Numerical results validate the effectiveness of our proposed schemes and demonstrate that they significantly outperform the traditional block diagonalization (BD)-based method.
Disciplines :
Computer science
Author, co-author :
Tong, Xiao ; Xi’an Jiaotong University, School of Information and Communications Engineering, Faculty of Electronic and Information Engineering, Xi’an, China
Li, Ang ; Xi’an Jiaotong University, School of Information and Communications Engineering, Faculty of Electronic and Information Engineering, Xi’an, China ; Southeast University, National Mobile Communications Research Laboratory, Nanjing, China
LEI, Lei ; University of Luxembourg > Interdisciplinary Centre for Security, Reliability and Trust > SigCom > Team Symeon CHATZINOTAS ; Xi’an Jiaotong University, School of Information and Communications Engineering, Faculty of Electronic and Information Engineering, Xi’an, China ; Southeast University, National Mobile Communications Research Laboratory, Nanjing, China
Hu, Xiaoyan ; Xi’an Jiaotong University, School of Information and Communications Engineering, Faculty of Electronic and Information Engineering, Xi’an, China
Dong, Fuwang ; Harbin Engineering University, College of Intelligent Systems Science and Engineering, Harbin, China
CHATZINOTAS, Symeon ; University of Luxembourg > Interdisciplinary Centre for Security, Reliability and Trust (SNT) > SigCom
Masouros, Christos ; University College London, Department of Electronic and Electrical Engineering, London, United Kingdom
External co-authors :
yes
Language :
English
Title :
MU-MIMO Symbol-Level Precoding for QAM Constellations with Maximum Likelihood Receivers
Publication date :
October 2025
Journal title :
IEEE Transactions on Communications
ISSN :
0090-6778
eISSN :
1558-0857
Publisher :
Institute of Electrical and Electronics Engineers Inc.
Manuscript received March 3, 2025; revised May 13, 2025 and August 18, 2025; accepted September 24, 2025. The work of Xiao Tong was supported by the Free Exploration Student Project of Xi\u2019an Jiaotong University under Grant xzy022024055. The work of Ang Li was supported in part by the National Natural Science Foundation of China under Grant 62101422, 62371386, in part by the Science and Technology Program of Shaanxi Province under Grant 2024JC-JCQN-59, 2025RS-CXTD-008, in part by the open research fund of National Mobile Communications Research Laboratory, Southeast University (No. 2024D01) and in part by the Xiaomi Young Scholars Program. The work of Lei Lei was supported in part by the NSFC project under Grant 62471375, in part by the Qin Chuang Yuan High-Level Innovation and Entrepreneurship Talent Program under Grant QCYRCXM-2023-049, and in part by the Key Research and Development Program of Shaanxi under Grant 2024GX-YBXM-065. The work of Xiaoyan Hu was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 62471380 and Grant 62201449. (Corresponding authors: Lei Lei and Ang Li.) X. Tong and X. Hu are with the School of Information and Communications Engineering, Faculty of Electronic and Information Engineering, Xi\u2019an Jiaotong University, Xi\u2019an, Shaanxi 710049, China (e-mail: xiao.tong.2023@stu.xjtu.edu.cn; xiaoyanhu@xjtu.edu.cn).
H. Lu et al., “A tutorial on near-field XL-MIMO communications towards 6G,” IEEE Commun. Surveys Tuts., Early Access, 2024.
F. Dong, F. Liu, Y. Cui, W. Wang, K. Han and Z. Wang, “Sensing as a service in 6G perceptive networks: A unified framework for ISAC resource allocation,” IEEE Trans. Wireless Commun., vol. 22, no. 5, pp. 3522-3536, May 2023.
F. Dong et al., “Communication-Assisted Sensing in 6G Networks,” IEEE J. Sel. Areas Commun., vol. 43, no. 4, pp. 1371-1386, 2025.
Z. Wang et al., “A tutorial on extremely large-scale MIMO for 6G: Fundamentals, signal processing, and applications,” IEEE Commun. Surveys Tuts., vol. 26, no. 3, pp. 1560-1605, 2024.
A. Wiesel, Y. C. Eldar, and S. Shamai (Shitz), “Zero-forcing precoding and generalized inverses,” IEEE Trans. Signal Process., vol. 56, no. 9,pp. 4409–4418, Sep. 2008.
C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst, “A vector perturbation technique for near-capacity multiantenna multiuser communication—Part I: Channel inversion and regularization,” IEEE Trans. Commun., vol. 53, no. 1, pp. 195–202, Jan. 2005.
Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 461–471, Feb. 2004.
K. Zu, R. C. de Lamare, and M. Haardt, “Generalized design of low-complexity block diagonalization type precoding algorithms for multiuser MIMO systems,” IEEE Trans. Commun., vol. 61, no. 10,pp. 4232–4242, Oct. 2013.
S. Lu et al., “Random ISAC signals deserve dedicated precoding,” IEEE Trans. Signal Process., vol. 72, pp. 3453-3469, 2024.
M. Alodeh et al., “SSymbol-level and multicast precoding for multiuser multiantenna downlink: A state-of-the-art, classification, and challenges,” IEEE Commun. Surveys Tuts., vol. 20, no. 3, pp. 1733-1757, 2018.
A. Li et al., “A tutorial on interference exploitation via symbol-level precoding: Overview, state-of-the-art and future directions,” IEEE Commun. Surveys Tuts., vol. 22, no. 2, pp. 796–839, 3rd Quart., 2020.
C. Masouros and E. Alsusa, “A novel transmitter-based selective-precoding technique for DS/CDMA systems,” in 2007 IEEE International Conference on Communications, Glasgow, UK, pp. 2829-2834, 2007.
C. Masouros and E. Alsusa, “Dynamic linear precoding for the exploitation of known interference in MIMO broadcast systems,” IEEE Trans. Wireless Commun., vol. 8, no. 3, pp. 1396–1404, Mar. 2009.
A. Li and C. Masouros, “Interference exploitation precoding made practical: Optimal closed-form solutions for PSK modulations,” IEEE Trans. Wireless Commun., vol. 17, no. 11, pp. 7661–7676, 2018.
A. Li, C. Masouros, B. Vucetic, Y. Li, and A. L. Swindlehurst, “Interference exploitation precoding for multi-level modulations: Closed-form solutions,” IEEE Trans. Commun., vol. 69, no. 1, pp. 291–308, 2021.
,,,,, [16] L. Liu, C. Masouros and A. L. Swindlehurst, “Robust symbol level precoding for overlay cognitive radio networks,” IEEE Trans. Wireless Commun., vol. 23, no. 2, pp. 1403-1415, 2024.
G. Zhang, C. Shen, B. Ai and Z. Zhong, “Robust symbol-level precoding and passive beamforming for IRS-aided communications,” IEEE Trans. Wireless Commun., vol. 21, no. 7, pp. 5486-5499, 2022.
Z. Xiao, R. Liu, M. Li, Y. Liu, and Q. Liu, “Low-complexity designs of symbol-level precoding for MU-MISO systems,” IEEE Trans. Commun., vol. 70, no. 7, pp. 4624–4639, Jul. 2022.
Y. Xu, F. Fang, S. Wang and D. Cai, “Energy efficiency maximization for downlink NOMA designs via symbol-level precoding,” IEEE Trans. Veh. Technol., vol. 73, no. 10, pp. 14974-14987, Oct. 2024.
S. Zhang, B. Di and H. Zhang, “Energy efficient symbol level precoding design in a RIS assisted communication system,” IEEE Commun. Lett., vol. 28, no. 5, pp. 1141-1145, May 2024.
Z. Liao and F. Liu, “Symbol-level precoding for integrated sensing and communications: A faster-than-Nyquist approach,” IEEE Commun. Lett., vol. 27, no. 12, pp. 3210-3214, Dec. 2023.
N. Babu, A. Kosasih, C. Masouros and E. Björnson, “Symbol-level precoding for near-field ISAC,” IEEE Commun. Lett., vol. 28, no. 9, pp. 2041-2045, Sept. 2024.
Y. Chen, F. Liu, Z. Liao and F. Dong, “Symbol-level precoding for MIMO ISAC transmission based on interference exploitation,” IEEE Commun. Lett., vol. 28, no. 2, pp. 283-287, Feb. 2024.
E. S. P. Lopes, L. T. N. Landau and A. Mezghani, “Minimum symbol error probability discrete symbol level precoding for MU-MIMO systems with PSK modulation,” IEEE Trans. Commun., vol. 71, no. 10, pp. 5935-5949, Oct. 2023.
Y. Wang, H. Hou, W. Wang and X. Yi, “Symbol-level precoding for average SER minimization in multiuser MISO systems,” IEEE Wireless Commun. Lett., vol. 13, no. 4, pp. 1103-1107, April 2024.
Z. Wei, C. Masouros and T. Xu, “Constructive interference based joint combiner and precoder design in multiuser MIMO systems,” in 2021 IEEE International Conference on Communications Workshops (ICC Workshops), Montreal, QC, Canada, 2021.
X. Tong, A. Li, L. Lei, F. Liu and F. Dong, “Symbol-level precoding for MU-MIMO system with RIRC receiver,” IEEE Trans. Commun., vol. 72, no. 5, pp. 2820-2834, May 2024.
S. Cai, T.-H. Chang, and H. Zhu, “Joint symbol level precoding and receive beamforming for multiuser MIMO downlink,” in Proc. IEEE 20th Int. Workshop Signal Process. Adv. Wireless Commun. (SPAWC), Cannes, France, Jul. 2019, pp. 1–5.
S. Cai, H. Zhu, C. Shen, and T.-H. Chang, “Joint symbol level precoding and receive beamforming optimization for multiuser MIMO downlink,” IEEE Trans. Signal Process., vol. 70, pp. 6185–6199, 2022.
S. Yang and L. Hanzo, “Fifty years of MIMO detection: The road to large-scale MIMOs,” IEEE Commun. Surveys Tuts., vol. 17, no. 4, pp. 1941-1988, Fourthquarter 2015.
K. Kato, K. Fukawa, R. Yamada, H. Suzuki and S. Suyama, “Low-complexity MIMO signal detection employing multistream constrained search,” IEEE Trans. Veh. Technol., vol. 67, no. 2, pp. 1217-1230, Feb 2018.
M. Matthaiou, N. D. Chatzidiamantis, G. K. Karagiannidis and J. A. Nossek, “ZF detectors over correlated K fading MIMO channels,” IEEE Trans. Commun., vol. 59, no. 6, pp. 1591-1603, June 2011.
Y. Jiang, M. Varanasi, and J. Li, “Performance analysis of ZF and MMSE equalizers for MIMO systems: An in-depth study of the high SNR regime,” IEEE Trans. Inf. Theory., vol. 57, no. 4, pp. 2008–2026, Apr 2011.
R. Lupas and S. Verdú, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inf. Theory., vol. 35, no. 1, pp. 123–136, Jan 1989.
X. Zhu and R. D. Murch, “Performance analysis of maximum likelihood detection in a MIMO antenna system,” IEEE Trans. Commun., vol. 50, no. 2, pp. 187–191, Feb 2002.
M. O. Damen, H. E. Gamal, and G. Caire, “On maximum-likelihood detection and the search for the closest lattice point,” IEEE Trans. Inf. Theory., vol. 49, no. 10, pp. 2389–2402, Oct 2003.
K. J. Kim and J. Yue, “Joint channel estimation and data detection algorithms for MIMO-OFDM systems,” in Proc. 36th Asilomar Conf. Signals Syst. Comput., Pacific Grove, USA, pp. 1857–1861, Nov 2002.
K. Higuchi, H. Kawai, N. Maeda, H. Taoka and M. Sawahashi, “Experiments on real-time 1-Gb/s packet transmission using MLD-based signal detection in MIMO-OFDM broadband radio access,” IEEE J. Sel. Areas Commun., vol. 24, no. 6, pp. 1141-1153, June 2006.
Z. Xiao, R. Liu, M. Li, Y. Liu and Q. Liu, “Low-complexity designs of symbol-level precoding for MU-MISO systems,” IEEE Trans. Commun., vol. 70, no. 7, pp. 4624-4639, July 2022.
C. Masouros and G. Zheng, “Exploiting known interference as green signal power for downlink beamforming optimization,” IEEE Trans. Signal Process., vol. 63, no. 14, pp. 3628–3640, Jul. 2015.
M. Alodeh, S. Chatzinotas and B. Ottersten, “Symbol-level multiuser MISO precoding for multi-level adaptive modulation,” IEEE Trans. Wireless Commun., vol. 16, no. 8, pp. 5511-5524, Aug. 2017.
