Regularized Channel Inversion Research Articles

Optimized Power Control Design for Over-the-Air Federated Edge Learning

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Over-the-air federated edge learning</i> (Air-FEEL) has emerged as a communication-efficient solution to enable distributed machine learning over edge devices by using their data locally to preserve the privacy. By exploiting the waveform superposition property of wireless channels, Air-FEEL allows the “one-shot” over-the-air aggregation of gradient-updates to enhance the communication efficiency, but at the cost of a compromised learning performance due to the aggregation errors caused by channel fading and noise. This paper investigates the transmission power control to combat against such aggregation errors in Air-FEEL. Different from conventional power control designs (e.g., to minimize the individual <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mean squared error</i> (MSE) of the over-the-air aggregation at each round), we consider a new power control design aiming at directly maximizing the convergence speed. Towards this end, we first analyze the convergence behavior of Air-FEEL (in terms of the optimality gap) subject to aggregation errors at different communication rounds. It is revealed that if the aggregation estimates are unbiased, then the training algorithm would converge exactly to the optimal point with mild conditions; while if they are biased, then the algorithm would converge with an error floor determined by the accumulated estimate bias over communication rounds. Next, building upon the convergence results, we optimize the power control to directly minimize the derived optimality gaps under the cases without and with unbiased aggregation constraints, subject to a set of average and maximum power constraints at individual edge devices. We transform both problems into convex forms, and obtain their structured optimal solutions, both appearing in a form of regularized channel inversion, by using the Lagrangian duality method. Finally, numerical results show that the proposed power control policies achieve significantly faster convergence for Air-FEEL, as compared with benchmark policies with fixed power transmission or conventional MSE minimization.

Analog MIMO Communication for One-Shot Distributed Principal Component Analysis

A fundamental algorithm for data analytics at the edge of wireless networks is distributed principal component analysis (DPCA), which finds the most important information embedded in a distributed high-dimensional dataset by distributed computation of a reduced-dimension data subspace, called principal components (PCs). In this paper, to support one-shot DPCA in wireless systems, we propose a framework of analog MIMO transmission featuring the uncoded analog transmission of local PCs for estimating the global PCs. To cope with channel distortion and noise, two maximum-likelihood (global) PC estimators are presented corresponding to the cases with and without receive channel state information (CSI). The first design, termed coherent PC estimator, is derived by solving a Procrustes problem and reveals the form of regularized channel inversion where the regulation attempts to alleviate the effects of both receiver noise and data noise. The second one, termed blind PC estimator, is designed based on the subspace channel-rotation-invariance property and computes a centroid of received local PCs on a Grassmann manifold. Using the manifold-perturbation theory, tight bounds on the mean square subspace distance (MSSD) of both estimators are derived for performance evaluation. The results reveal simple scaling laws of MSSD concerning device population, data and channel signal-to-noise ratios (SNRs), and array sizes. More importantly, both estimators are found to have identical scaling laws, suggesting the dispensability of CSI to accelerate DPCA. Simulation results validate the derived results and demonstrate the promising latency performance of the proposed analog MIMO

Hybrid Beamforming for Terahertz Multi-Carrier Systems Over Frequency Selective Fading

We propose novel hybrid beamforming schemes for the terahertz (THz) wireless system where a multi-antenna base station (BS) communicates with a multi-antenna user over frequency selective fading. Here, we assume that the BS employs sub-connected hybrid beamforming and multi-carrier modulation to deliver ultra high data rate. We consider a three-dimensional wideband THz channel by incorporating the joint effect of molecular absorption, high sparsity, and multi-path fading, and consider the carrier frequency offset in multi-carrier systems. With this model, we first propose a two-stage wideband hybrid beamforming scheme which includes a beamsteering codebook searching algorithm for analog beamforming and a regularized channel inversion method for digital beamforming. We then propose a novel wideband hybrid beamforming scheme with two digital beamformers. In this scheme, an additional digital beamformer is developed to compensate for the performance loss caused by the constant-amplitude hardware constraints and the difference of channel matrices among subcarriers. Furthermore, we consider imperfect channel state information (CSI) and propose a probabilistic robust hybrid beamforming scheme to combat channel estimation errors. Numerical results demonstrate the benefits of our proposed schemes for the sake of practical implementation, especially considering its high spectral efficiency, low complexity, and robustness against imperfect CSI.

Optimized Power Control for Over-the-Air Computation in Fading Channels

Over-the-air computation (AirComp) of a function (e.g., averaging) has recently emerged as an efficient multiple-access scheme for fast aggregation of distributed data at mobile devices (e.g., sensors) at a fusion center (FC) over wireless channels. To realize reliable AirComp in practice, it is crucial to adaptively control the devices' transmit power for coping with channel distortion to achieve the desired magnitude alignment of simultaneous signals. In this paper, we solve the power control problem. Our objective is to minimize the computation error by jointly optimizing the transmit power at devices and a signal scaling factor (called denoising factor) at the FC, subject to individual average power constraints at devices. The problem is generally non-convex due to the coupling of the transmit powers at devices and denoising factor at the FC. To tackle the challenge, we first consider the special case with static channels, for which we derive the optimal solution in closed form. The derived power control exhibits a threshold-based structure: if the product of the channel quality and power budget for each device, called quality indicator, exceeds an optimized threshold, this device applies channel-inversion power control; otherwise, it performs full power transmission. We proceed to consider the general case with time-varying channels. To solve the more challenging non-convex power control problem, we use the Lagrange-duality method via exploiting its “time-sharing” property. The derived power control exhibits a regularized channel inversion structure, where the regularization balances the tradeoff between the signal-magnitude alignment and noise suppression. Moreover, for the special case with only one device being power limited, we show that the power control for the power-limited device has an interesting channel-inversion water-filling structure, while those for other devices (with sufficiently large power budgets) reduce to channel-inversion power control. Numerical results show that the derived power control significantly reduces the computation error as compared with the conventional designs.

Coordinated Beamforming for Secure Transmission in the Downlink Multicell MIMO Systems

In this paper, we consider secure transmission in the downlink of a multicell multiple-input multiple-output (MIMO) system with M cells; each cell consists of an N-antenna base station (BS) and K single-antenna users. Coordinated beamforming (CBf) is employed for secrecy enhancement. For CBf, the BSs can jointly design their respective beamformers to control intercell interference and information leakage. Based on regularized channel inversion (RCI) precoding structures, we derive the concise formulas for the signal-to-interference-plus-noise ratio (SINR) for the legitimate users and the colluding malicious users and the achievable secrecy sum rate for the CBf. Furthermore, we obtain a novel closed-form expression of the secrecy sum in the large-system regime where N and K tend to infinity with a fixed ratio β=K/N. Based on the large-system regime result, the regularization parameter is optimized to maximize the secrecy sum rate. It shows that the optimal regularization parameter decreases with the signal-to-noise ratio (SNR) and stays constant at high SNR if β&gt;1/M. Compared with single-cell processing (SCP) without BS cooperation, simulation results show that the secrecy performance can be significantly enhanced by exploiting CBf. The analytical results are validated with the finite-size system by simulation.

Coordinated Beamforming and Artificial Noise in the Downlink Secure Multi-cell MIMO Systems under Imperfect CSI

In this letter, we consider the problem of secure communication in the downlink of a multi-cell multiple-input multiple-output (MIMO) system under imperfect channel state information (CSI) at the base stations (BSs). The passive eavesdroppers are considered, whose CSI are unavailable at the BSs. Coordinated Beamforming (CBf) and artificial noise (AN) are employed for secrecy enhancement. Based on regularized channel inversion (RCI) precoding, we derive a novel closed-form expression of secrecy sum-rate in the large-system regime. Based on the result, the regularization parameter and the power allocation ratio between the information signal and AN are optimized. Compared to the perfect CSI case, it shows that the channel estimation error can drastically reduce the secrecy sum-rate, and more power needs to be allocated for AN when the channel estimation error grows larger. The analytical results are validated with the finite-size system by simulation.

Low-complexity Kaczmarz precoding in DL massive MIMO with partial CSI and correlation

The capacity achieved by three low-complexity precoders is compared considering a single-cell massive MIMO (M-MIMO) broadcast channel; these linear precoders include the conventional zero-forcing (ZF) beamforming, regularized channel inversion (RCI) precoding, and a precoding version based on the iterative randomized Kaczmarz algorithm (rKA). The capacity vs complexity tradeoff is analyzed in the downlink (DL) with a base-station (BS) deploying massive uniform linear array (ULA) antennas, while mobile terminals (MT) are equipped with single-antenna. Based on the precoding matrix associated with each scheme, we first derive the interference matrix and, consequently, the capacity achieved by each analyzed precoder. Numerical results demonstrate that the rKA’s performance-complexity tradeoff is superior compared to conventional ZF and RCI precoding when operating in M-MIMO scenario, since it holds a relative robustness against system loading increasing, appears to be equally efficient under a simple equal power assignment (EPA) policy and implies in a considerably reduced number of complex operations.

Large-Scale Analysis of Regularized Block Diagonalization Precoding for Physical Layer Security of Multi-User MIMO Wireless Networks

In this paper, we propose regularized block diagonalization (RBD) precoding using artificial noise (AN) for physical layer security in downlink multi-user MIMO wireless networks. We derive secrecy sum rate and asymptotic secrecy sum rate for the proposed scheme. The optimum power allocation for legitimate users and the AN signal are derived in closed form for optimal asymptotic secrecy sum rate. Our analysis shows that to achieve best performance, it is more efficient to degrade the performance of the eavesdropper than improving legitimate users' rates. We also study the impact of error in channel estimation on the system and derive closed-form SINR and secrecy sum-rate expressions for this case. Simulations show that the secrecy sum rate of RBD precoding outperforms the secrecy sum rate of regularized channel inversion by 0.5 bits/s/Hz. We also show that in the presence of channel estimation error, our approximate closed forms for secrecy sum rate are very closed to the actual secrecy sum rate over a wide range of SNR and channel estimation error values.

An inter user interference cancellation scheme for two users in full-duplex MIMO system

The inter-user interference is a critical performance degradation factor in a full-duplex multi-user system. However, the conventional pre-processing filters, such as null space projection filter and regularized channel inversion filter, are unsuitable for user pre-processing filter when each user has the same number of transmit and receive antennas. Thus, we propose an inter-user-interference cancellation scheme applicable to the users with the same number of transmit and receive antennas. By summing the received signals for two symbol periods, the proposed scheme can cancel inter-user interference and obtain a twofold diversity order. The simulation results show that the proposed scheme obtains higher spectral efficiency than the half-duplex system.

Improving Secrecy via Extension to Regularized Channel Inversion Precoding

This letter extends the regularized channel inversion (RCI) precoding to include more regularization parameters. Compared with the RCI precoding, more practical user channels are considered and the information obtained from the eigen-decomposition of the channel covariance matrix is exploited. The weighted sum-rate problem with different confidentiality requirements of the users is discussed, and the simple gradient descent method is used to solve the problem and determine the regularization parameters. The numerical results show that the proposed precoder outperform the RCI precoder in terms of the weighted sum-rate.

Pairing and Scheduling for Large Array MIMO Using Regularized Channel Inversion Receivers Over Nakagami-m Fading

Multiple-antenna (MIMO) technology is becoming mature for wireless communications and has been incorporated into wireless broadband standards like LTE and WiFi. Basically, the more the number of antennas the transmitter/receiver is equipped with, the more the possible independent signal paths and the better the performance in terms of data rate and link reliability. Massive MIMO (also known as large-scale antenna systems, very large MIMO, hyper MIMO and full-dimension MIMO) makes a clean break with current practice through the use of a very large number of service antennas (e.g., hundreds or thousands) that are operated fully coherently and adaptively. Extra antennas help by focusing the transmission and reception of signal energy into ever-smaller regions of space. This brings huge improvements in throughput and energy efficiency, in particularly when combined with simultaneous scheduling for a large number of user terminals (e.g., tens or hundreds). This Paper focuses on the efficiency of the throughput and fairness of large MIMO. In multi-user MIMO (MU-MIMO), fewer antennas were used, so the only concern was to mitigate the interference of the users. The interference was mitigated by using zero forcing (ZF) receivers while the fairness of the system was maintained by using Rayleigh Fading channel. MU-MIMO was sufficient for only smaller number of users as the number of antennas were fixed and all the users had to be entertained within the same bandwidth. But, in Massive MIMO, along with the interference of the users, the interference of the antennas themselves is also a big concern, as, the number of antennas were increased contrary to the fixed fewer number of antennas in MU-MIMO. Rayleigh Fading channel and ZF receivers were not sufficient for increased number of users and antennas respectively. The problem of interference amongst the antennas was addressed by using regularized channel inversion receivers that increase the throughput of the system while, the fairness of the system is increased by using Nakagami-m channel.

Regularized Channel Inversion for Simultaneous Confidential Broadcasting and Power Transfer: A Large System Analysis

We propose for the first time new transmission schemes based on linear precoding to enable simultaneous confidential broadcasting and power transfer in a multiuser multi-input single-output network, where a base station with $N$ antennas simultaneously transmits power and confidential messages to $K$ single-antenna users. We first design two transmission schemes based on the rules of regularized channel inversion (RCI) for both power splitting (PS) and time switching (TS) receiver architectures, namely, RCI-PS and RCI-TS schemes. For each scheme, we derive channel-independent expressions to approximate the secrecy sum rate and the harvested power in the large-system regime where $K,N\rightarrow \infty$ with a fixed ratio $\beta =K/N$ . Based on the large-system results, we jointly optimize the regularization parameter of the RCI and the PS ratio or the TS ratio such that the secrecy sum rate is maximized subject to an energy-harvesting constraint. We then present the tradeoff between the secrecy sum rate and the harvested power achieved by each scheme, and find that neither scheme always outperforms the other one. Motivated by this fact, we design an RCI-hybrid scheme based on the RCI and a newly proposed hybrid receiver architecture. The hybrid receiver architecture takes advantages of both the PS and TS receiver architectures. We show that the RCI-hybrid scheme outperforms both the RCI-PS and RCI-TS schemes.

Clustered linear precoding for downlink network MIMO systems with partial CSI

AbstractWe propose two novel clustered linear precoding schemes applicable to network multi‐input multi‐output systems using only partial channel state information to enhance the sum‐rate of the system. Using a channel model that decomposes a multi‐input multi‐output channel matrix into transmit and receive steering vectors and assuming that only transmit steering vectors are available at the base transmit stations, we, first, propose a regularized channel inversion precoding scheme to enhance the sum‐rate assuming only single‐antenna users are available in the system. Next, because of the limitation of regularized channel inversion to handle users with multiple receive antennas, a novel block diagonalization method is proposed. We construct the precoding matrices that jointly eliminate inter‐cell interference and maximize the sum‐rate for a given input covariance matrix. Assuming total power constraint and per‐base‐station power constraints, optimal power allocation schemes are further developed to optimize the sum‐rate. We analytically show that the sum‐rate increases linearly with the number of users when only single‐antenna users are present in the system. Numerical results show that at low signal‐to‐noise ratios, the block diagonalization precoding outperforms the regularized channel inversion in terms of the bit error rate; while at high signal‐to‐noise ratios, the regularized channel inversion provides a better performance. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Channel inversion and regularization revisited

In multiuser multiple-input-single-output (MISO) downlink, linear precoders like channel inversion (CI) and regularized CI (RCI) are more desirable than their nonlinear counterparts due to their reduced complexity. To achieve the full benefits of linear precoding, the availability of perfect channel state information (CSI) at base stations (BSs) is necessary. Since in practice, having access to perfect CSI is not pragmatic, in this paper, we evaluate the performance of CI and RCI under a generalized, imperfect CSI model where the variance of the channel estimation error depends on the signal-to-noise ratio (SNR) and thus covers digital and analog feedbacks as two special cases. Then, based on this imperfect CSI model, we quantify the asymptotic mean loss in sum rate and the achievable degrees of freedom (DoFs) by deriving the received signal-to-interference-plus-noise ratio (SINR) of each user. For example, it is shown that the achievable DoF is directly related to the SNR exponent of the channel estimation error variance. Also, two asymptotic gaps to capacity for the analog feedback are derived: mean loss in sum rate and power loss. In addition, we propose an adaptive RCI technique by deriving an appropriate regularization parameter as a function of the error variance and without imposing any restrictions on the number of users or antennas. It is shown that in the presence of CSI mismatch, while the comparative improvement of the standard RCI to CI becomes negligible, the adaptive RCI compensates this degraded performance of the standard RCI without introducing any extra computational complexity.

Base Station Cooperation for Confidential Broadcasting in Multi-Cell Networks

We design linear precoders that perform confidential broadcasting in multi-cell networks for two different forms of base station (BS) cooperation, namely, multi-cell processing (MCP) and coordinated beamforming (CBf). We consider a two-cell network where each cell consists of an $N$ -antenna BS and $K$ single-antenna users. For such a network, we design a linear precoder based on the regularized channel inversion (RCI) for the MCP and a linear precoder based on the generalized RCI for the CBf. For each form of BS cooperation, we derive new channel-independent expressions to approximate the secrecy sum rate achieved by the precoder in the large system regime where $K,N\rightarrow\infty$ with a fixed ratio $\beta=K/N$ . Using these results, we determine the optimal regularization parameters of the RCI and the generalized RCI precoders that maximize the secrecy sum rate for the MCP and the CBf, respectively. We further propose power-reduction strategies that significantly increase the secrecy sum rate at high transmit signal-to-noise ratios when the network load is high. Our numerical results substantiate the derived expressions, verify the optimality of the determined optimal regularization parameters, and demonstrate the performance improvement offered by the proposed power-reduction strategies.

Adaptively Regularized Phase Alignment Precoding for Multiuser Multiantenna Downlink

In multiantenna downlink, linear precoders like channel inversion (CI), regularized CI (RCI), and phase alignment (PA) are more desirable than their nonlinear counterparts due to their reduced complexity. However, to exploit the full benefits of multiuser downlink, the availability of perfect channel state information (CSI) at the base station (BS) is mandatory. Since such an assumption is not realistic in practice, subject to CSI mismatch, the downlink precoders may undergo a severe degradation in performance. Therefore, an adaptive linear precoder is required to achieve better performance under the availability of imperfect CSI at the BS. In this paper, we propose such a linear precoder by judiciously picking up an optimized regularization parameter based on the knowledge of the channel estimation error variance in advance. To do so, we first introduce a generalized CSI mismatch model where the variance of the channel estimation error is a function of the signal-to-noise ratio (SNR) and is thus able to accommodate a variety of distinct scenarios such as reciprocal channels and CSI feedback. It is shown that the proposed precoding scheme, dubbed adaptively regularized PA (RPA), is capable of achieving lower bit error rates (BERs) compared to standard linear precoders under both perfect and imperfect CSI.

Linear Precoding of Data and Artificial Noise in Secure Massive MIMO Systems

In this paper, we consider secure downlink transmission in a multi-cell massive multiple-input multiple-output (MIMO) system where the numbers of base station (BS) antennas, mobile terminals, and eavesdropper antennas are asymptotically large. The channel state information of the eavesdropper is assumed to be unavailable at the BS and hence, linear precoding of data and artificial noise (AN) are employed for secrecy enhancement. Four different data precoders (i.e., selfish zero-forcing (ZF)/regularized channel inversion (RCI) and collaborative ZF/RCI precoders) and three different AN precoders (i.e., random, selfish/collaborative null-space based precoders) are investigated and the corresponding achievable ergodic secrecy rates are analyzed. Our analysis includes the effects of uplink channel estimation, pilot contamination, multi-cell interference, and path-loss. Furthermore, to strike a balance between complexity and performance, linear precoders that are based on matrix polynomials are proposed for both data and AN precoding. The polynomial coefficients of the data and AN precoders are optimized respectively for minimization of the sum mean squared error of and the AN leakage to the mobile terminals in the cell of interest using tools from free probability and random matrix theory. Our analytical and simulation results provide interesting insights for the design of secure multi-cell massive MIMO systems and reveal that the proposed polynomial data and AN precoders closely approach the performance of selfish RCI data and null-space based AN precoders, respectively.

Multi-User OFDM MIMO in IEEE 802.11ac: A Simulation Framework to Analysis and Synthesis

In this paper a simulation framework to analyze, design and optimize the physical layer of IEEE 802.11ac wireless local networks focusing on multi-user operation scenario is described and validated. The performance of channel inversion, regularized channel inversion and block diagonalization precoding schemes with zero forcing and minimum mean squared error multiple input multiple output (MIMO) detectors is evaluated over different configurations of 802.11 Task Group n (TGn) and TGac channel models. The expressive power loss due to multi-user operation has shown that more advanced precoding and MIMO detection schemes must be designed to allow satisfactory operation into feasible signal-to-noise-ratio ranges.

Transmit-Power Efficient Linear Precoding Utilizing Known Interference for the Multiantenna Downlink

It has been shown that the knowledge of both channel and data information at the base station prior to downlink transmission can help increase the received signal-to-noise ratio (SNR) of each user without the need to increase the transmitted power. Achievability is based on the idea of phase alignment (PA) precoding, where instead of nulling out the destructive interference, it judiciously rotates the phases of the transmitted symbols. In this way, they add up coherently at the intended user and yield higher received SNRs. In addition, it is well known that regularized channel inversion (RCI) precoding improves the performance of channel inversion (CI) in multiantenna downlink communications. In line with this and similar to the RCI precoding, in this paper, we propose the idea of regularized PA (RPA), which is shown to improve the performance of original PA precoding. To do this, we first rectify the original PA precoding, deriving a closed-form expression to evaluate the amount of transmit-power reduction achieved for the same average output SNR compared with CI precoding. We then use this new analysis to select the appropriate regularization factor for our proposed RPA scheme. It is shown by means of theoretical analysis and simulations that the proposed RPA precoding outperforms CI, RCI, and PA precoders from both symbol error rate (SER) and throughput perspectives and provides a more power-efficient alternative. This is particularly pronounced as the number of transmit antennas becomes larger, where up to a 50-times reduction in the transmit power is achieved by RPA (PA) compared with RCI (CI) precoding for a given performance.

Confidential Broadcasting via Linear Precoding in Non-Homogeneous MIMO Multiuser Networks

We propose linear precoding with power control to achieve confidential broadcasting in multi-input-multi-output multiuser networks such that the base station (BS) with N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> antennas securely broadcasts messages to K users with N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> antennas each. We focus on the practical non-homogeneous scenario where the distances between the BS and the users are not equal. We first design a linear precoder based on regularized channel inversion, and derive new channel-independent expressions for the achievable secrecy sum-rate in the large system regime. With the aid of these expressions, we examine the impact of user dispersion, N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> , and K on the secrecy sum-rate. We then propose a power reduction strategy and power allocation algorithms to increase the secrecy sum-rate. We demonstrate that our power reduction strategy increases the secrecy sum-rate at high signal-to-noise ratios. We also show the secrecy sum-rate advantage of optimal power allocation over equal power allocation. Furthermore, we consider channel correlation and derive an easy-to-compute expression for the secrecy sum-rate to examine its impact on the secrecy performance.