Extreme massive multiple-input multiple-output (MIMO) in the upper mid-band spectrum (7-20 GHz) is a key enabler for 6G. In orthogonal frequency division multiplexing (OFDM) systems, two-stage beamforming is essential for practical deployment, but most existing works rely on simplified channels and do not consider real-time processing constraints. Current methods require full-dimensional covariance computation and/or eigenvalue decomposition (EVD), resulting in prohibitive memory use and computational complexity. We propose a block Krylov-Nyström framework for beamspace beamforming that avoids full covariance storage and reduces complexity compared to EVD counterparts. The proposed scheme reconstructs full beamforming weights from low-rank channel information using the block Krylov-Nyström subspace, ensuring numerical stability and efficiency. To capture real-time constraints, we further introduce two evaluation methodologies: a complexity-to-delay conversion integrated with channel aging, and a spectral efficiency-to-complexity ratio (SECR) metric. Simulations with 3GPP 38.901 channel models demonstrate that the proposed scheme achieves benchmark spectral efficiency with significantly lower latency and complexity, offering up to 15-30 higher SECR than baseline methods. These results establish the block Krylov-Nyström framework as a practical and robust solution for real-time massive MIMO-OFDM beamforming in 6G networks.

The increasing demand for high-capacity and energy-efficient wireless communications has made extreme massive Multiple Input Multiple Output (MIMO) a critical technology for 6G networks. Hybrid beamforming is essential to trade-off spectral efficiency and power consumption, yet existing solutions are typically array-specific. Additionally, most work focuses on single-polarized antenna systems, whereas dual-polarized arrays are widely used in practical deployments. This paper proposes a universal shifted codebook set-based hybrid beamforming framework for dual-polarized massive MIMO. The key lies in decoupling the array architecture impact from the hybrid beamforming optimization, transferring it into a designed shifted codebook set. This enables seamless adaptation across different hybrid array architectures without modifying core algorithm modules. Based on this framework, we develop two efficient algorithms: shifted-codebook orthogonal matching pursuit and shifted-codebook greedy grid-of-beam for joint and two-stage hybrid beamforming, respectively. Simulations using 3GPP dual-polarized channels show that our algorithms outperform the state-of-the-art solution, offering higher spectral efficiency and improved energy efficiency while maintaining flexibility for product evolution and MIMO hardware upgrades.

The upper mid-band, spanning from 7 GHz to 20 GHz, emerges as a promising candidate spectrum for 6G, offering extensive coverage and remarkable capacity. It drives massive Multiple Input Multiple Output (MIMO) to adopt extremely large antenna arrays and the so-called extreme massive MIMO will play an essential role. However, as the number of antennas at base stations increases, the demands on signal processing intensify significantly, imposing substantial limitations on product implementation. Reduced-rank beamforming such as beamspace beamforming can reduce the complexities associated with channel estimation and precoding. This paper introduces a low-complexity block Krylov-beamspace beamforming, targeting the implementation of two-stage long-term/short-term fully digital rank-reduction beamforming in 6G extreme massive MIMO at upper mid-band. The key lies in the concept of using the derived multiple signature vectors to calculate the beamspace beamforming matrix that spans the block Krylov subspace. An adaptive and fast-converging beamspace determination scheme is also developed for the proposed algorithm. We further investigate its implementation in the practical cross-polarized antenna systems. The proposed beamforming is able to provide an eigen-based performance but with a significant complexity reduction, exhibiting a great potential in the evolution of extreme massive MIMO for 6G.

Angular reciprocity based downlink (DL) channel covariance matrix (CCM) reconstruction has the significant superiority of requiring no uplink feedback in frequency division duplex multi-antenna systems. In this realm, maximal angle of arrival estimation-based schemes show superior performance due to its noise-robust merit for environment with a dominant scattering cluster. To extend the scheme to multiple scattering clusters environments, we propose to decompose the DL CCM into the covariance matrix of each scattering cluster and then reconstruct each DL sub-covariance matrix utilizing the estimated angular reciprocal parameters. Simulation results verify the validity of the proposed scheme in a 3GPP standard based multiple scattering clusters channel model.

Beamspace beamforming, which is advantageous in reducing channel estimation efforts and the subsequent precoding complexity, has great potential for massive Multiple Input multiple Output (MIMO) systems with extremely large antenna arrays. Sub-6 GHz will still play an essential role for a wide coverage and a reliable connectivity in future wireless communications. It is thus of prime interest to investigate efficient beamspace beamforming technologies for Frequency Division Duplex (FDD) massive MIMO. Standard beamspace beamforming schemes can be realized by either downlink grid-of-beam or uplink sounding, where beams are selected from the Discrete Fourier Transformation (DFT) codebook or calculated by Eigen-Value Decomposition (EVD) from the measured channel covariance matrix. However, in FDD when channel reciprocity does not hold, it is difficult to design efficient beamspace beamforming algorithms that not only delivers high performance but also has low complexity and low induced overhead. This paper proposes a novel Krylov-beamspace beamforming scheme for FDD massive MIMO. Taking advantages of angular reciprocity in FDD, we develop a generalized algorithm to compensate the uplink channel covariance matrix for the downlink use, which is more robust against scattered channels and imperfect reciprocity than the conventional counterpart. The beamspace beamforming matrix is designed by the derived beam signature vector and the compensated channel covariance matrix, resulting in the so called “beam”-Krylov subspace. Two concepts utilizing limited downlink feedback and/or uplink sounding are proposed, which can provide enough flexibilities in implementing the new scheme. The proposed algorithm does not need any complicated EVD calculations and even outperforms the baseline method, which is low-complexity/low-overhead and quite applicable for FDD massive MIMO.

For the prominent superiority in supporting high-data-rate applications with reduced hardware complexity and energy consumption, hybrid precoding in millimeter-wave (mmWave) massive multi-input multi-output (MIMO) system has attracted considerable attentions recently. To adequately utilize the available antenna resources and reduce power consumption, we investigate energy efficient hybrid precoding for an adaptive partially-connected structure in mmWave massive MIMO system with antenna group overlapping to achieve array gain. Specifically, the energy efficient hybrid precoding is firstly formulated as an energy efficiency (EE) maximizing problem, where the number of active phase shifters, the connection relationship between radio frequency chains and antennas, and precoding matrix are jointly optimized. Then, utilizing the diagonal characteristic of digital precoding in a partially-connected structure, we propose a decomposition-based low-complexity approach to tackle the large-scale mix-integer non-convex EE optimization problem in the fully adaptive hybrid precoding scheme. Besides, to provide further insights on how the EE-oriented design affects the system performance, a partially adaptive scheme is also introduced. Finally, simulation results verify the viability and effectiveness of the proposed schemes.

The cell-free massive Multiple Input Multiple Output (MIMO) architecture is considered for millimeter Wave (mmWave) bands to overcome propagation challenges at higher frequencies such as significant pathloss, easy blockage, and shadowing. Instead of using a single and co-located extreme large antenna array, multiple large arrays with a smaller dimension in a hybrid architecture can be efficiently deployed over multiple Access Points (APs) in a distributed way. We propose a distributed hybrid beamforming concept for mmWave cell-free massive MIMO systems, which supports multiantenna or even hybrid array based User Equipments (UEs) and multi-stream transmissions. Analog beamforming at both APs and UEs can be obtained in a joint and multi-phase manner to maximize the total effective array gain in the cell-free system. We also develop an interference aware digital precoding method to deal with multi-user/multi-stream transmissions, achieving a high multiplexing gain. Simulation results show a good potential of using the distributed hybrid beamforming in cell-free systems.

Although Terahertz communication systems can provide high data rates, it needs high directional beamforming at transmitters and receivers to achieve such rates over a long distance. Therefore, an efficient beam training method is vital to accelerate the link establishment. In this study, we propose a low-complexity beam training scheme of terahertz communication system which uses a low-cost small-scale hybrid architecture to assist a large-scale array for data transmission. The proposed scheme includes two key stages: (1) coarse AoAs/AoDs estimation for beam subset optimization in auxiliary array stage, and (2) accurate AoAs/AoDs estimation by exploiting channel sparsity in data transmission array stage. The analysis shows that the complexity of the scheme is linear with the number of main paths, and thus greatly reduces the complexity of beam training. Simulation results have verified the better performance in spectral efficiency of the proposed scheme than that of the related work.

Massive Multiple Input Multiple Output (MIMO) is able to boost the system throughput. A key challenge is a large overhead of Channel State Information (CSI) feedback with the increased number of antenna ports in Frequency Division Duplexing (FDD) massive MIMO systems. Conventional methods apply either compressed sensing or beamformed reference signal to reduce the CSI overhead. However, there are still other probems such as additional overhead, user’s implementation complexity, or performance limitation. We propose a machine learning enhanced CSI acquisition and training solution for FDD massive MIMO. It can efficiently recover the CSI with more ports than those of the CSI feedback. Furthermore, a practical training strategy is developed, which shows the feasibility of using uplink dataset to train the neural network for the downlink use in FDD.

Massive Multiple Input Multiple Output (MIMO) at millimeter wave bands is able to boost the system throughput. A key challenge for the hybrid beamforming design in massive MIMO systems is the acquisition of the full channel state information, since the number of radio frequency chains is much smaller than that of the antennas. Conventional methods require a longer measurement time, a large overhead, or costly signal processing efforts. Therefore, we propose an efficient and adaptable deep neural network based low-rank channel recovery scheme for a hybrid array based massive MIMO system. The proposed neural network architecture includes a common feature extraction module and the adaptable recovery module. The feature extraction, built on the convolutional neural network with residual learning functionality, can efficiently learn the essential features from the low-rank measurements. The adaptable key recovery module maps the essential features to the full channel information. The proposed architecture enables an efficient learning procedure and can be easily adapted to different cases. Simulation results are carried out and compared with existing solutions, showing the potential of applying deep learning concepts in millimeter wave massive MIMO systems.

Applying relays in the multi-user massive Multiple Input Multiple Output (MIMO) system at Millimeter Wave (mmWave) bands can greatly help to overcome the large propagation loss at higher frequencies and boost the communication link. We propose an efficient hybrid beamforming technique for the relay assisted mmWave multi-user massive MIMO system. The design includes the analog transmit-receive coordinated beam alignment procedure and non-linear precoding based digital beamforming. The analog beam alignment can be flexibly applied to different hybrid array architectures. We propose to jointly design digital beamforming cross the base station, the relay, and the multi-antenna users to enhance the system performance. It is based on the Geometric Mean Decomposition Tomlinson Harashima Precoding (GMD-THP) scheme, and only linear processing is applied at the relay for a cost-effective reason. We evaluate the performance of the proposed technique and compare it with its linear and fully digital counterparts.

Non-linear precoding gained attention recently in 3GPP (the third generation partnership project) as a possible technique to improve the performance of multi-user multiple input multiple output (MIMO). In this paper, research and standardization activities related non-linear precoding for 3GPP are described. Implementation, performance evaluation of nonlinear precoding and analysis of 3GPP specification impact are presented in this paper. Both system and UE throughput evaluation results, conducted at both 5GHz and 30GHz, demonstrate that non-linear precoding yield superior performance compared to linear precoding. In addition, two novel non-linear precoding based transmission schemes which require lower computational complexities than conventional non-linear precoding schemes are proposed. The first scheme separates UEs into linearly precoded and nonlinearly precoded UEs, reducing computational complexities required by non-linear precoding. The second scheme is a precoding scheme which intentionally introduces inter-UE interference through linear precoding and implements nonlinear precoding to remove inter-UE interference; multi-user diversity gain and reduction in computational complexity are obtained simultaneously. Finally, new abstraction performance metrics for the proposed schemes are presented. The simulation results demonstrate that the proposed methods outperform linear precoding techniques and offer a tradeoff between computational complexity and throughput performance.

Multi-beam mobile satellite systems have a great potential in providing broadband mobile services over a large area to achieve a high system throughput. On ground beamforming techniques are quite promising but require a large feeder link bandwidth to deliver all the feeds??? signals. There are two potential solutions to solve this problem. 1) Hybrid space-ground beamforming is able to reduce the feeder link bandwidth and save spectral resources, which exhibits a good trade-off between the performance and the space/ground complexity. 2) The groundbased beamforming using multiple distributed gateways divides the whole spectrum of the feeder link into several sub-sets, relaxing the bandwidth and processing requirements at each gateway. In this paper, we propose a distributed non-linear precoding based hybrid space-ground beamforming technique, where on-board beamforming aims at suppressing interference among gateways and on-ground beamforming is implemented distributively at each gateway. The proposed technique takes advantages of both the hybrid scheme and the distributed multiple gateway architecture, and does not require any cooperation or any exchange of channel state information among gateways. We show the benefits of the proposed technique over the fully onground beamforming schemes as well as its linear counterparts.

To assist hybrid beamforming in multi-user massive Multiple Input Multiple Output (MIMO) systems at millimeter wave frequencies, we propose Auxiliary Processing Network (APN) based multi-phase channel alignment techniques. The APN, which is realized in parallel to the main beamforming network, have two implementation architectures, namely the hybrid analog-digital APN based on antenna switching as well as the fully digital APN with low-resolution Analog-to-Digital Converters (ADCs). For different APN architectures, we design various channel estimation algorithms for the channel alignment procedure and evaluate the corresponding performance.

Multi-beam mobile satellite systems aim at providing broadband and high speed mobile services over a large area to achieve a high system throughput, where hybrid space-ground beamforming is one of the most promising candidates for ground-based beamforming techniques. It not only reduces the feeder link bandwidth to save spectral resources, but also takes advantages of both the on-ground and on-board processing, exhibiting a good trade-off of the performance and the space/ground complexity. In this paper, we propose an efficient hybrid space-ground precoding technique for multi-beam mobile satellite systems. It consists of coarse on-board beamforming and reduced-rank on-ground beamforming based on the feed selection of the phased array antenna. The advantages of the proposed hybrid precoding are shown as compared to the fully on-ground beamforming as well as the existing solutions.

For massive multiple input multiple output systems at millimeter wave (mm Wave) bands, we consider an efficient hybrid array architecture, namely, overlapped subarray (OSA), and develop a Unified Low Rank Sparse (ULoRaS) recovery algorithm for hybrid beamforming in downlink multiuser scenarios. The ULoRaS scheme takes advantage of the transmit-receive coordinated beamforming procedure to achieve large array gains. It has no dimensionality constraint and can be applied to the generalized OSA architecture, including both the fully connected and the widely discussed non-OSA cases. It is shown that the proposed ULoRaS algorithm for the novel OSA design is a good compromise of the performance and the required hardware complexity.

Targeting massive Multiple Input Multiple Output (MIMO) systems, we consider an efficient multi- panel array architecture and propose hybrid beamforming techniques for millimeter wave frequencies. Taking advantages of the multi-panel array, hybrid two-stage processing including coarse and fine beamforming can be carried out in a distributed fashion. Panel-coordinated analog beamforming as well as decoupling only requires channels measured at each panel itself. Non- linear precoding is then applied on a per-panel basis. It is shown that the proposed schemes relax the antenna dimensionality constraint and exhibit a good performance even in an overloaded case, which are practically feasible in both the single-/multi-cell scenarios.

We propose a subspace-based Widely Linear (WL) blind channel estimation scheme based on the iterative power method for the WL constrained minimum variance Code Division Multiple Access (CDMA) receiver. The novel technique approximates the noise subspace by using a matrix power and the WL processing fully exploits the second-order non-circularity of the signal. Two adaptive recursive least squares algorithms are developed using power iterations, which completely avoid the computationally intensive singular value decomposition. Simulation results show an improved performance of the proposed algorithms in terms of convergence and complexity as compared to their linear counterparts.

Massive Millimeter Wave (mmWave) Multiple Input Multiple Output (MIMO) systems utilize hybrid beamforming techniques to alleviate the implementation complexity of combining a large number of antennas. To solve the optimization problem in the hybrid design supporting multiple users and multiple data streams per user, we first propose a coordinated Radio Frequency (RF) beamforming technique based on the Generalized Low Rank Approximation of Matrices (GLRAM) approach and then develop an efficient Modified GLRAM (MGLRAM) algorithm. The proposed scheme only requires the information of the composite channel, instead of the complete physical channel matrix which is assumed to be known in the existing literature. It takes advantage of the coordination between the base station and users to achieve a maximal array gain and has no dimensionality constraint. The multiplexing gain is then exploited by applying the Block Diagonalization (BD) technique. It is shown that our proposed scheme is a practical and competing solution, which can be easily applied to both the Time Division Duplex (TDD) and Frequency Division Duplex (FDD) systems.

The objective of this paper is to review state-of-the-art techniques of beamforming in mobile satellite systems and evaluate the potential benefits/drawbacks of on-ground beamforming compared with on-board beamforming approach. The paper also provides a short analysis of beamforming error sources in on-ground beamforming such as propagation effects at feeder link level, on-board degradations at payload level, differential atmospheric perturbations, and Doppler shift effect. An investigation of signal processing techniques is also performed to provide a preliminary assessment of the interest for employing adaptive beamforming and precoding techniques in multi-spots mobile satellite systems. Copyright © 2013 John Wiley & Sons, Ltd.