Channel Covariance Matrices Research Articles

Subspace-Based Pilot Decontamination in User-Centric Scalable Cell-Free Wireless Networks

We consider a cell-free wireless system operated in Time Division Duplex (TDD) mode with user-centric clusters of remote radio units (RUs). Since the uplink pilot dimensions per channel coherence slot is limited, co-pilot users might incur mutual pilot contamination. In the current literature, it is assumed that the long-term statistical knowledge of all user channels is available. This enables Minimum Mean-Square Error channel estimation or simplified dominant subspace projection, which achieves significant pilot decontamination under certain assumptions on the channel covariance matrices. However, estimating the channel covariance matrix or even just its dominant subspace at all RUs forming a user cluster is not an easy task. In fact, if not properly designed, a piloting scheme for such long-term statistics estimation will also be subject to the contamination problem. In this paper, we propose a new channel subspace estimation scheme explicitly designed for cell-free wireless networks. Our scheme is based on 1) a sounding reference signal (SRS) using latin squares wideband frequency hopping, and 2) a subspace estimation method based on robust Principal Component Analysis (R-PCA). The SRS hopping scheme ensures that for any user and any RU participating in its cluster, only a few pilot measurements will contain strong co-pilot interference. These few heavily contaminated measurements are (implicitly) eliminated by R-PCA, which is designed to regularize the estimation and discount the “outlier” measurements. Our simulation results show that the proposed scheme achieves almost perfect subspace knowledge, which in turns yields system performance very close to that with ideal channel state information, thus essentially solving the problem of pilot contamination in cell-free user-centric TDD wireless networks.

Two Stage Beamforming in Massive MIMO: A Combinatorial Multi-Armed Bandit Based Approach

In frequency division duplex (FDD) massive multi-input multi-output (MIMO), the two-stage beamforming (TSB) using channel covariance matrices (CCM) can significantly reduce the downlink training length (DTL) and channel feedback. However, the overhead to estimate the CCM is large. In this paper, a combinatorial multi-armed bandit (CMAB) based TSB scheme is proposed without requirement of CMM. Particularly, the problem of the pre-beamforming matrix design is transformed into a CMAB problem. We consider the pre-beamforming matrix design in each slot as the arm selection in the CMAB, and convert the problem of the arm selection into a 0-1 integer linear programming problem, which can be solved by the branch-and-bound method. During the training process, the maximum likelihood method is used to detect the power of angle spectrum, and the angle range of each user is determined adaptively. We prove that the regret grows logarithmically with time, such that the proposed scheme converges towards the optimal action. Finally, simulation results demonstrate that the proposed scheme can significantly improve the spectral efficiency.

Rate Splitting in FDD Massive MIMO Systems Based on the Second Order Statistics of Transmission Channels

In this work, we present new results for the application of rate splitting multiple access (RSMA) to the downlink (DL) of a massive multiple-input-multiple-output system (MIMO) operating in frequency-division-duplex (FDD) mode. We propose a statistical precoding design relying on the channels’ second-order information when one-layer RS is implemented. The advantage of the statistical precoding lies in simplifying the precoder design by basing the optimization objective on the slowly-varying channel covariance matrices instead of the fast-changing instantaneous channel estimates. To this end, we use the so-called bilinear precoder, which was shown in [1] and [2] to have limited performance in the high SNR regime due to the imperfect channel state information (CSI) available at the base station (BS). We formulate the DL throughput maximization problem based on a widely used lower bound on the achievable sum rate and propose an iterative approach to solve the underlying optimization problem. Numerical results demonstrate the benefit of implementing one-layer RS. Furthermore, the proposed iterative approach achieves excellent results in terms of spectral efficiency compared to the state-of-the-art techniques.

Massive MIMO Communication With Intelligent Reflecting Surface

This paper studies the feasibility of deploying intelligent reflecting surfaces (IRSs) in massive multiple input multiple-output (MIMO) systems to improve the performance of users in the service dead zone. One question of paramount importance is as follows: if the overhead of channel training and the computational complexity of algorithm design arising from the huge number of IRS reflecting elements and base station (BS) antennas have to be controlled, can we provide reasonable performance to the users with weak direct channels? This paper provides an affirm answer to this question. Specifically, to reduce the channel training overhead, we consider an appealing protocol for the uplink communication in the IRS-assisted massive MIMO systems. Under this protocol, the IRS reflection coefficients are optimized based on the channel covariance matrices, which are generally fixed for many coherence blocks, to boost the long-term performance. Then, given the IRS reflecting coefficients, the BS beamforming vectors are designed in each coherence block based on the effective channel of each user, which is the superposition of its direct and reflected user-IRS-BS channels, to improve the instantaneous performance. Since merely the user effective channels are estimated in each coherence block, the training overhead of this protocol is the same as that in the legacy wireless systems without IRSs. Moreover, in the asymptotic regime that the numbers of IRS elements and BS antennas both go to infinity with a fixed ratio, we manage to first characterize the minimum mean-squared error (MMSE) estimators of the user effective channels and then quantify the closed-form user achievable rates as functions of channel covariance matrices with channel training overhead and estimation error taken into account. Interestingly, it is shown that the properties of channel hardening and favorable propagation still hold for the user effective channels, and satisfactory user rates are thus achievable even if simple BS beamforming solutions, e.g., maximal-ratio combining, are employed. Finally, thanks to the rate characterization, we design a low-complexity algorithm to optimize the IRS reflection coefficients based on channel covariance matrices.

Low-complex Bayesian estimator for imperfect channels in massive muti-input multi-output system

<span lang="EN-US">Motivated by the fact that the complexity of the computations is one of the main challenges in large multiple input multiple output systems, known as massive multiple-input multiple-output (MIMO) systems, this article proposes a low-complex minimum mean squared error (MMSE) Bayesian channel estimator for uplink channels of such systems. First, we have discussed the necessity of the covariance information for the MMSE estimator and how their imperfection knowledge can affect its accuracy. Then, two reduction phases in dimension and floating-point operations have been suggested to reduce its complexity: in phase 1, eigenstructure reduction for channel covariance matrices is implemented based on some truncation rules, while in phase 2, arithmetic operations reduction for matrix multiplications in the MMSE equation is followed. The proposed procedure has significantly reduced the complexity of the MMSE estimator to the first order O(M), which is less than that required for the conventional MMSE with O(M<sup>3</sup>) in terms of matrix dimension. It has been shown that the estimated channels using our proposed procedure are asymptotically aligned and serve the same quality as the full-rank estimated channels. Our results are validated by averaging the normalized mean squared error (NMSE) over a length of 500 sample realizations through a Monte Carlo simulation using MATLAB R2020a.</span>

Downlink Precoding for DP-UPA FDD Massive MIMO via Multi-Dimensional Active Channel Sparsification

In this paper, we consider user selection and downlink precoding for an over-loaded single-cell massive multiple-input multiple-output (MIMO) system in frequency division duplexing (FDD) mode, where the base station is equipped with a dual-polarized uniform planar array (DP-UPA) and serves a large number of single-antenna users. Due to the absence of uplink-downlink channel reciprocity and the high-dimensionality of channel matrices, it is extremely challenging to design downlink precoders using closed-loop channel probing and feedback with limited spectrum resource. To address these issues, a novel methodology – active channel sparsification (ACS) – has been proposed recently in the literature for uniform linear array (ULA) to design sparsifying precoders, which substantially reduces channel feedback overhead. Pushing forward this line of research, we aim to facilitate the potential deployment of ACS in practical FDD massive MIMO systems, by extending it from ULA to DP-UPA with explicit user selection and making the current ACS implementation simplified. To this end, by leveraging Toeplitz matrix theory, we start with the spectral properties of channel covariance matrices from the lens of their matrix-valued spectral density function. Inspired by these properties, we extend the original ACS using scalar-weight bipartite graph representation to the matrix-weight counterpart. Building upon such matrix-weight bipartite graph representation, we propose a multi-dimensional ACS (MD-ACS) method, which is a generalization of original ACS formulation and is more suitable for DP-UPA antenna configurations. The nonlinear integer program formulation of MD-ACS can be classified as a generalized multi-assignment problem (GMAP), for which we propose a simple yet efficient greedy algorithm to solve it. Simulation results demonstrate the performance improvement of the proposed MD-ACS with greedy algorithm over the state-of-the-art methods based on the QuaDRiGa channel models.

A Computationally Efficient Gradient Algorithm for Downlink Training Sequence Optimization in FDD Massive MIMO Systems

Future wireless networks will require advance physical-layer techniques to meet the requirements of Internet of Everything (IoE) applications and massive communication systems. To this end, a massive MIMO (m-MIMO) system is to date considered one of the key technologies for future wireless networks. This is due to the capability of m-MIMO to bring a significant improvement in the spectral efficiency and energy efficiency. However, designing an efficient downlink (DL) training sequence for fast channel state information (CSI) estimation, i.e., with limited coherence time, in a frequency division duplex (FDD) m-MIMO system when users exhibit different correlation patterns, i.e., span distinct channel covariance matrices, is to date very challenging. Although advanced iterative algorithms have been developed to address this challenge, they exhibit slow convergence speed and thus deliver high latency and computational complexity. To overcome this challenge, we propose a computationally efficient conjugate gradient-descent (CGD) algorithm based on the Riemannian manifold in order to optimize the DL training sequence at base station (BS), while improving the convergence rate to provide a fast CSI estimation for an FDD m-MIMO system. To this end, the sum rate and the computational complexity performances of the proposed training solution are compared with the state-of-the-art iterative algorithms. The results show that the proposed training solution maximizes the achievable sum rate performance, while delivering a lower overall computational complexity owing to a faster convergence rate in comparison to the state-of-the-art iterative algorithms.

Fingerprint-Based Covariance Matrix Estimation for Cell-Free Distributed Massive MIMO Systems

Channel covariance matrix, or large-scale channel state information, is usually assumed to be perfectly known, which is not practical due to the quickly changing environment. This letter proposes a highly accurate channel covariance matrix estimation method based on fingerprint-based localization. In initial pilot assignment stage, we eliminate the impact of inference and noise by introducing random phase shift to estimate channel covariance matrices, which are regarded as fingerprints for estimating locations. Users are clustered based on locations and pilots are reassigned accordingly to improve estimation accuracy. Simulation results show that the proposed method achieves higher covariance matrix estimation accuracy.

Low-Complexity ADMM-Based Algorithm for Robust Multi-Group Multicast Beamforming in Large-Scale Systems

We design an efficient robust multi-group multicast beamforming scheme for massive multiple-input multiple-output (MIMO) systems. Assuming only estimates of the channel covariance matrices are available at the base station with a bounded error, we formulate the robust quality-of-service (QoS) problem, which is to minimize the transmit power subject to the worst-case minimum signal-to-interference-plus-noise-ratio (SINR) guarantee. We directly solve the worst-case SINR problem and convert the robust QoS constraint into a number of non-convex constraints. Based on the recent convergence result of the alternating direction method of multipliers (ADMM) for non-convex problems, we develop an ADMM-based fast algorithm to directly tackle the reformulated non-convex problem with a convergence guarantee. The algorithm contains two layers of ADMM procedures. We design the outer-layer ADMM to decompose the problem into three convex subproblems and solve them alternatingly. We further develop an inner-layer consensus-ADMM-based algorithm to efficiently solve one subproblem. By exploring each subproblem structure and developing the special optimization techniques, we obtain closed-form or semi-closed-form solutions to each subproblem. These results lead to a fast iterative algorithm, which is guaranteed to converge to a stationary point of the original robust QoS problem. Simulation shows that our proposed algorithm provides a favorable performance compared with existing alternative methods with magnitudes of computational complexity reduction.

Millimeter Wave Beamforming Codebook Design via Learning Channel Covariance Matrices Over Riemannian Manifolds

Millimeter Wave Beamforming Codebook Design via Learning Channel Covariance Matrices Over Riemannian Manifolds

Spatially-Correlated Massive MIMO Channel: An Iterative Estimation Algorithm

In this letter, we propose an iterative algorithm for the joint estimation of the channel covariance matrices and the channel vector for the spatially-correlated massive MIMO system with the following main advantages: low pilot overhead, improved mean square error (MSE), and low complexity. The results show that this scheme outperforms the conventional estimation schemes in terms of the required pilot overhead and the resulting MSE. The results also show that the achievable sum rate of the proposed scheme tends to that of the ideal case that the channel covariance matrices are assumed to be perfectly known at the base station.

Max–min fair robust beamforming design for underlay device‐to‐device communications in cellular networks

SummaryThis paper investigates the robust downlink beamforming designs based on max–min fairness and base station (BS) power consumption constraint for device‐to‐device (D2D) communications underlaying cellular network, under the assumption of imperfect channel state information (CSI) at the BS. Our objective is to maximize the minimum signal‐to‐interference‐plus‐noise ratio (SINR) or non‐outage probability of users while guaranteeing that the consumed power at the BS is less than a threshold. In particular, three max–min fairness scenarios are considered. In the first scenario, the worst‐case SINR of cellular users (CUs) is maximized where the D2D SINR is guaranteed to be above a specified predetermined threshold. In the second scenario, we extend the first scenario and maximized D2D SINR while maximizing CUs' SINR. In these two scenarios, it is assumed that the errors are upper bounded in their Frobenius norms. In the third scenario, the minimum non‐outage probability of users is maximized, while a probabilistic model is considered for the uncertainty of channel covariance matrices. Although such optimization problems are not convex, the semidefinite relaxation (SDR) approach is used to obtain the optimal beamforming matrix, which always complies the rank‐one constraint. Simulation results show that significant improvement in terms of minimum SINR of CUs, the minimum non‐outage probability of CUs, and the probability of feasibility and the fairness index can be achieved with our proposed algorithms in comparison with other beamforming schemes.

Coordinated regularized zero-forcing beamforming with channel statistics based adaptive feedback for cooperative massive MIMO networks

Abstract In cooperative massive multiple-input multiple-output (MIMO) networks, channel statistics based adaptive feedback can considerably reduce feedback overhead for channel state information (CSI) acquisition as well as backhaul overhead for CSI sharing. When regularized zero-forcing beamforming is considered to coordinate interference with the skewed codebook, average sum rate depends on not only regularization parameters, but also quantization error impacts of serving and interfering channels according to their channel covariance matrices. To improve the average sum rate by effectively controlling the desired signal strength and the interference cancellation, we propose joint optimization of regularization parameters and feedback bit allocation by leveraging adaptive feedback according to the channel covariance matrices.

Downlink Training Design for FDD Massive MIMO Systems in the Presence of Colored Noise

Massive multiple-input multiple-output (MaMi) systems have attracted much research attention during the last few years. This is because MaMi systems are able to achieve a remarkable improvement in data rate and thus meet the immensely ongoing traffic demands required by the future wireless networks. To date, the downlink training sequence (DTS) for the frequency division duplex (FDD) MaMi communications systems have been designed based on the idealistic assumption of white noise environments. However, it is essential and more practical to consider the colored noise environments when designing an efficient DTS for channel estimation. To this end, this paper proposes a new DTS design by exploring the joint use of spatial channel and noise covariance matrices, when the channel is not reciprocal but the coherence block length remains limited. We derive an analytical solution for the mean square error (MSE) based on the proposed training design with colored noise. In addition, this paper exploits the method of random matrix theory to provide an analytical solution for the downlink (DL) achievable sum rate of the regularized zero forcing beamforming (RZFBF) precoder. Numerical results demonstrate that using the proposed DTS design, the MSE of the channel estimate is significantly reduced compared with the conventional training designs with white noise. Furthermore, the results show that the proposed pilot design markedly improves the DL achievable SR over the conventional training designs, especially at relatively low signal-to-noise-ratio (SNR) levels. This enables FDD MaMi systems to operate under more practical scenarios of colored noise and limited coherence time environments.

Statistical Beamforming for Massive MIMO Systems with Distinct Spatial Correlations.

In this paper, we propose a novel statistical beamforming (SBF) method called the partial-nulling-based SBF (PN-SBF) to serve a number of users that are undergoing distinct degrees of spatial channel correlations in massive multiple-input multiple-output (MIMO) systems. We consider a massive MIMO system with two user groups. The first group experiences a low spatial channel correlation, whereas the second group has a high spatial channel correlation, which can happen in massive MIMO systems that are based on fifth-generation networks. By analyzing the statistical signal-to-interference-plus-noise ratio, it can be observed that the statistical beamforming vector for the low-correlation group should be designed as the orthogonal complement for the space spanned by the aggregated channel covariance matrices of the high-correlation group. Meanwhile, the spatial degrees of freedom for the high-correlation group should be preserved without cancelling the interference to the low-correlation group. Accordingly, a group-common pre-beamforming matrix is applied to the low-correlation group to cancel the interference to the high-correlation group. In addition, to deal with the intra-group interference in each group, the post-beamforming vector for each group is designed in the manner of maximizing the signal-to-leakage-and-noise ratio, which yields additional performance improvements for the PN-SBF. The simulation results verify that the proposed PN-SBF outperforms the conventional SBF schemes in terms of the ergodic sum rate for the massive MIMO systems with distinct spatial correlations, without the rate ceiling effect in the high signal-to-noise ratio region unlike conventional SBF schemes.

Achievable Capacity with Spatial Channel Correlation in Massive MIMO Multi-Cell Systems

Massive Multi-Input and Multi-Output (MIMO) antenna system provides unlimited capacity by the spatial multiplexing and array gain. Since the data rate has been limited by the coherence interference due to pilot contamination (PC). In this paper, we propose transmit combine and precoding schemes to achieve asymptotic capacity in multi-cell scenario, when the number of base station antennas tends to infinity. The impact of spatial channel correlation on channel capacity is explored by considering the co-variance matrices of the user –terminals (UT)s .To do this, we presented linear processing schemes such as MMSE,MRC, and ZF.Where MMSE achieves high capacity in the presence of large-scale fading and PC. Since the diagonals of the channel covariance matrices were designed with non- zero Eigen values and linearly independent. The results outperform and obtain asymptotic limit, when the co-variance of UTs are linearly independent. The results were simulated by using MATLAB 2018b.

Non-Iterative Downlink Training Sequence Design Based on Sum Rate Maximization in FDD Massive MIMO Systems

This paper considers the problem of downlink (DL) training sequence design with limited coherence time for frequency division duplex (FDD) massive MIMO systems in a general scenario of single-stage precoding and distinct spatial correlations between users. To this end, a computationally feasible solution for designing the DL training sequences is proposed using the principle of linear superposition of sequences constructed from the users' channel covariance matrices. Based on the non-iterative superposition training structure and the P-degrees of freedom (P-DoF) channel model, a novel closed-form solution for the optimum training sequence length that maximizes the DL achievable sum rate is provided for the eigenbeamforming (BF) precoder. Additionally, a simplified analysis that characterizes the sum rate performance of the BF and regularized zero forcing (RZF) precoders in closed-form is developed based on the method of random matrix theory and the P-DoF channel model. The results show that the superposition training sequences achieve almost the same rate performances as state-of-the-art training sequence designs. The analysis of the complexity results demonstrates that more than four orders-of-magnitude reduction in the computational complexity is achieved using the superposition training design, which signifies the feasibility of this approach for practical implementations compared with state-of-the-art iterative algorithms for DL training designs. Importantly, the results indicate that the analytical solution for the optimum training sequence length with the P-DoF channel model can be effectively used with high accuracy to predict the sum rate performance in the more realistic one ring (OR) channel model, and thus, near optimal solutions can be readily obtained without resorting to computationally intensive optimization techniques.

A Spatial–Temporal Subspace-Based Compressive Channel Estimation Technique in Unknown Interference MIMO Channels

Spatial–temporal (ST) subspace-based channel estimation techniques formulated with $\ell 2$ minimum mean square error (MMSE) criterion alleviate the multi-access interference (MAI) problem when the interested signals exhibit low-rank property. However, the conventional $\ell 2$ ST subspace-based methods suffer from mean squared error (MSE) deterioration in unknown interference channels, due to the difficulty to separate the interested signals from the channel covariance matrices (CCMs) contaminated with unknown interference. As a solution to the problem, we propose a new $\ell 1$ regularized ST channel estimation algorithm by applying the expectation-maximization (EM) algorithm to iteratively examine the signal subspace and the corresponding sparse-supports. The new algorithm updates the CCM independently of the slot-dependent $\ell 1$ regularization, which enables it to correctly perform the sparse-independent component analysis (ICA) with a reasonable complexity order. Simulation results shown in this paper verify that the proposed technique significantly improves MSE performance in unknown interference MIMO channels, and hence, solves the BER floor problems from which the conventional receivers suffer.

Secret Key Generation Scheme Based on the Channel Covariance Matrix Eigenvalues in FDD Systems

In this letter, an efficient secret key generation (SKG) scheme exploiting the eigenvalue reciprocity of uplink (UL) and downlink (DL) channel covariance matrices in frequency division duplex (FDD) systems is developed. Our proposed method achieves a better performance in terms of SKG rate and secret key disagreement (SKD) ratio compared with the loopback method in the previous literature, meanwhile free from the security issue of loopback strategies. Numerical simulation results are presented to verify the feasibility and effectiveness of the proposed scheme.

Beam Squint and Channel Estimation for Wideband mmWave Massive MIMO-OFDM Systems

With the increasing scale of antenna arrays in wideband millimeter-wave (mmWave) communications, the physical propagation delays of electromagnetic waves traveling across the whole array will become large and comparable to the time-domain sample period, which is known as the spatial-wideband effect. In this case, different subcarriers in an orthogonal frequency division multiplexing (OFDM) system will "see" distinct angles of arrival (AoAs) for the same path. This effect is known as beam squint, resulting from the spatial-wideband effect, and makes the approaches based on the conventional multiple-input multiple-output (MIMO) model, such as channel estimation and precoding, inapplicable. After discussing the relationship between beam squint and the spatial-wideband effect, we propose a channel estimation scheme for frequency-division duplex (FDD) mmWave massive MIMO-OFDM systems with hybrid analog/digital precoding, which takes the beam squint effect into consideration. A super-resolution compressed sensing approach is developed to extract the frequency-insensitive parameters of each uplink channel path, i.e., the AoA and the time delay, and the frequency-sensitive parameter, i.e., the complex channel gain. With the help of the reciprocity of these frequency-insensitive parameters in FDD systems, the downlink channel estimation can be greatly simplified, where only limited pilots are needed to obtain downlink complex gains and reconstruct downlink channels. Furthermore, the uplink and downlink channel covariance matrices can be constructed from these frequency-insensitive channel parameters rather than through a long-term average, which enables the minimum mean-squared error (MMSE) channel estimation to further enhance performance. Numerical results demonstrate the superiority of the proposed scheme over the conventional methods in mmWave communications.