Large system analysis of the secrecy sum-rates with Regularized Channel Inversion precoding

In this paper, we consider an optimal power allocation problem that maximizes the sum-rate of a single-cell MISO broadcast channel with regularized channel inversion (RCI) beamforming at the base station (BS). Unlike the channel inversion or zero forcing beamforming, the optimal power allocation with RCI precoding at the base station is a nonlinear non-convex optimization problem with many local optima. Here, we investigate this problem in the large system limit, i.e., when the number of users K and antennas at the base station N tend to infinity with their ratio β = K/N being held constant. We assume each user has data symbol with power p k , and slow-varying path-loss a k . In this setting, we first derive the expression for the signal to interference plus noise ratio (SINR). Then, we divide all K users into L groups where all users in each group are assumed to be co-located or to have approximately the same distance from the BS. In other words, the users in each group have the same path-loss which is distance dependent. Moreover, in each group, we assume that the power allocated to each user is the same. Based on this system model, we investigate optimal power allocation schemes that maximize the sum-rate per antenna, firstly under average power constraint, and then under both average and per-user power constraints. We show that both problems are convex and the power allocation mainly follows the well-known water-filling strategy.

The algorithm of getting the regularization parameter of conventional regularized channel inversion (RCI) assumes that all the users have equal path loss, also referred to as homogeneous user channels. In this paper, we modify the RCI to improve the performance in the scenario of unequal path loss. The minimum mean-square-error (MMSE) criterion is used to interpret RCI and a new precoder is proposed. The proposed algorithm overcomes the problem that the optimal RCI precoder was derived under the assumption of equal path loss. The proposed precoder exploits the path loss information. Due to the MMSE interpretation, the regularization parameters of the proposed precoder can be obtained directly from the path losses of the users, thus the computational complexity is low. The secrecy sum-rate for confidential broadcasting is also discussed. Simulation results show better performance of the modified RCI when the practical scenario with unequal path loss is considered.

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.

In this paper, we consider the downlink of a multiuser wireless communication system with multiple antennas at the base station and users each with a single receive antenna. It is known that when channel state information (CSI) is available at the transmitter a large performance gain can be achieved. In a system employing time-division duplexing (TDD), CSI can be obtained at the base station if there is reciprocity between the forward and reverse channels. CSI can also be conveyed from the users to the base station via a limited-rate feedback channel in a frequency-division duplexing (FDD) system. In any case, channel estimation errors are inevitable due to the presence of background noise in the estimated signal and due to the finite number of feedback bits used in a limited-rate feedback system model. In this paper, we first consider the general case when partial CSI is available at the transmitter. We derive an MMSE based precoding technique that considers channel estimation errors as an integral part of the system design. Using rate-distortion theory and the generalized Lloyd vector quantization algorithm, we then specialize our results for the more practical limited-rate feedback system model. Compared to previously proposed precoding techniques such as channel inversion and regularized channel inversion, it is shown that the proposed precoding technique significantly improves the average bit error rate (BER) in the system. Furthermore, the performance of the proposed technique is investigated in the high signal-tonoise ratio (SNR) regime. It is shown that the proposed technique suffers from a ceiling effect that asymptotically limits the system performance.

We consider regularized channel inversion (RI) precoding in the multiuser multiple input multiple output (Mu-MIMO) broadcast channels. To make the RI operate in quantized channel feedback condition, we analyze the quantization error variance and its effect on the signal to interference plus noise power ratio (SINR) of the user signals. The regularization parameter ? is optimized to balance the additive noise, the multiuser interference and the interference due to the quantization error and to maximize SINR. Analysis and simulations show that regularization enhances the performance of zero forcing beam forming (ZFBF) over the whole signal to noise power ratio (SNR) range. This is contrasting to the perfect channel status information (CSI) case, where the enhancement is observed in the low SNR region.

In this paper, we study the problem of secure transmission in the downlink of a multi-cell massive multiple-input multiple-output (MIMO) system, where pilot contamination is considered due to non-orthogonal uplink training sequence. The channel state information (CSI) of the eavesdropper is assumed to be unavailable at the base station (BS) and hence, the BS cooperation and an artificial noise(AN) are investigated for secrecy enhancement. Based on regularized channel inversion (RCI), we derive a closed-form expression for ergodic secrecy sum rate in the large-system regime, in which the number of users per cell K, the number of antenna at BS N and the number of antenna at eavesdropper N E tend to infinity. Our analysis shows that the secrecy sum-rate saturates at high signal-to-noise ratio (SNR) with BSs cooperation and AN under pilot contamination.

In this work, we consider the downlink of a single-cell multi-user MIMO system in which the base station (BS) makes use of N antennas to communicate with K single-antenna user equipments (UEs). The UEs move around in the cell according to a random walk mobility model. We aim at determining the energy consumption distribution when different linear precoding techniques are used at the BS to guarantee target rates within a finite time interval T. The analysis is conducted in the asymptotic regime where N and K grow large with fixed ratio under the assumption of perfect channel state information (CSI). Both recent and standard results from large system analysis are used to provide concise formulae for the asymptotic transmit powers and beamforming vectors for all considered schemes. These results are eventually used to provide a deterministic approximation of the energy consumption and to study its fluctuations around this value in the form of a central limit theorem. Closed-form expressions for the asymptotic means and variances are given. Numerical results are used to validate the accuracy of the theoretical analysis and to make comparisons. We show how the results can be used to approximate the probability that a battery-powered BS runs out of energy and also to design the cell radius for minimizing the energy consumption per unit area. The imperfect CSI case is also briefly considered.

Based on directional modulation (DM), robust beamforming matrix design for sum secrecy rate maximization is investigated in this article in multiuser systems. First, in case of random estimation error of direction angles toward eavesdroppers, a Von Mises distribution-sum secrecy rate maximization (VMD-SSRM) method is proposed to maximize the sum secrecy rate. Semidefinite relaxation and first-order approximation based on Taylor expansion are employed to solve the optimization problem in VMD-SSRM. Then in order to optimize the sum secrecy rate in the case of the worst estimation errors of direction angles, a maximum angle estimation error-SSRM (MAEE-SSRM) method is proposed. The optimization problem is constructed based on the upper and lower bounds of the estimated eavesdropping channel related coefficient and then solved by the change of the variable. Simulation results show that our two proposed methods have better sum secrecy rates than the state-of-the-art methods. Furthermore, when the estimation error of direction angle toward each eavesdropper is random but within a certain range, the secrecy performance of our VMD-SSRM method is better than that of our MAEE-SSRM method. On contrary, when the direction angle estimation error reaches its maximum, our MAEE-SSRM method performs better.

Interference exploitation has recently been shown to provide significant security benefits in multiuser communication systems. In this technique, the known interference is designed to be constructive to the legitimate users and disruptive to the malicious receivers. Accordingly, this paper analyzes the secrecy performance of constructive interference (CI) precoding technique in multi-user multiple-input single-output (MU-MISO) systems with phase-shift-keying (PSK) signals and in the presence of multiple passive eavesdroppers. The secrecy performance of CI technique is comprehensively investigated in terms of symbol error probability (SEP), and secrecy sum-rate. Firstly, new and exact analytical expressions for the average SEP of the legitimate users and the eavesdroppers are derived. Departing from classical Gaussian rate analysis, we employ finite constellation rate expressions to investigate the secrecy sum-rate. In this regard, closed-form analytical expression of the ergodic secrecy sum-rate is obtained. Then, based on the new secrecy sum-rate expression we revisit adaptive modulation (AM) scheme with the aim to enhance the secrecy performance. The numerical results in this work demonstrate that, the interference exploitation technique achieves a significant performance gain over the interference suppression schemes. Furthermore, the proposed AM scheme provides significant improvement in terms of the secrecy sum-rate.

This paper studies the over-the-air computation (AirComp) in an orthogonal frequency division multiplexing (OFDM) system with imperfect channel state information (CSI), in which multiple single-antenna wireless devices (WDs) simultaneously send uncoded signals to a multi-antenna access point (AP) for distributed functional computation over multiple subcarriers. In particular, we consider two scenarios with best-effort and error-constrained computation tasks, with the objectives of minimizing the average computation mean squared error (MSE) and the computation outage probability over the multiple subcarriers, respectively. Towards this end, we jointly optimize the transmit coefficients at the WDs and the receive beamforming vectors at the AP over subcarriers, subject to the maximum transmit power constraints at individual WDs. First, for the special case with a single receive antenna at the AP, we propose the semi-closed-form globally optimal solutions to the two problems using the Lagrange-duality method. It is shown that at each subcarrier, the WDs’ optimized power control policy for average MSE minimization follows a regularized channel inversion structure, while that for computation outage probability minimization follows an on-off regularized channel inversion, with the regularization dependent on the transmit power budget and channel estimation error. Next, for the general case with multiple receive antennas at the AP, we present efficient algorithms based on alternating optimization and convex optimization to find converged solutions to both problems. It is shown that with finite receive antennas at the AP, a non-zero computation MSE for AirComp is inevitable due to the channel estimation errors even when the transmit powers at WDs tend to infinity, while with massive receive antennas, the average MSE and outage probability vanish when the channel vectors are independent and identically distributed. Finally, numerical results are provided to demonstrate the effectiveness of the proposed designs.

The performance of a downlink multi-user MIMO system seriously degrades in time-varying environments because MIMO channels change during the time interval between transmission parameter determination and actual MIMO transmission. Channel prediction techniques have been proposed to improve the performance in dynamic environments. In this paper, based on indoor measurement data, we show the effect of the linear channel extrapolation, and evaluate the bit error rate performance of the multi-user MIMO system with the channel prediction technique.

For the poor transmission reliability of data, this paper focuses on the precoding techniques of the multi-user MIMO system in the broadcast phase in, and the single-user MIMO and multi-user MIMO precoding techniques have been studied. In the multi-user MIMO system, BD-MMSE-VP coding algorithm is proposed to improve the BER performance of the system by using the MMSE criterion to optimize the perturbation vector, and the system uses QR technique to decompose BD channel matrix. Simulation results verify the effective of the proposed precoding algorithm.

By combining adaptive modulation (AM) and automatic repeat request (ARQ) protocol, cross-layer design (CLD) scheme of multiuser MIMO system with antenna selection (AS) is presented. By means of exact expression of packet error rate (PER), the switching thresholds (STs) of fading gain subject to a target PER constraint are derived, which is more accurate than the conventional STs based on approximate PER expression. According to these results, and using mathematical manipulation, we obtain the tightly closed-form expressions of the average spectral efficiency (SE) and average PER. Simulation results show that the multiuser MIMO system with CLD-AS scheme using the derived STs provides higher SE than that using existing approximate STs while meeting the PER requirement. Besides, simulations for average SE and PER are in good agreement with the derived theoretical expressions, which demonstrates the validity of the theoretical analysis.

We introduce the Two-Way MIMO Wire-tap channel and convey its potential to provide information-theoretic secure communications. We mainly address two challenges, namely the channel estimation and the single-user decodability issues. For the former, as the channel estimation becomes trickier owing to self-interference, we propose that users project their training signals on orthogonal subspaces to ensure their separability upon reception. For the latter, we suggest that each user uses, among the available antennas only the antenna subset that ensures both peers are not single-user decodable while maximizing their achievable secrecy sum-rate. Thus, unlike the result reported in literature about the two-way single-antenna wire-tap channel, we demonstrate that it is possible to achieve a positive secrecy rate even when a user is single-user decodable, provided that a sufficient number of transmit antennas be available. Finally, we observe that the proposed scheme, unlike secrecy schemes based on Wyner's degraded wire-tap channel, provides positive secrecy sum-rate even when the eavesdropper's channels incur as little noise as the legitimate users', and maintains the secrecy rate's growth in the high SNR regime, a fact we demonstrate by computer simulations. This makes the proposed scheme a better candidate for high-throughput secrecy applications.

In this paper, the system-level performance is evaluated for the feedback scheme on the Precoding Matrix Index (PMI) and Channel Quality Indication (CQI), which are required for user selection in the multi-user MIMO system. Our analysis demonstrates that the number of users, the number of selected users, and codebook size are the key factors that govern the performance of the Best Companion Grouping (BCG)-based user scheduling. Accordingly, we have confirmed that the probability of forming the co-scheduled user group is determined by these factors, implying that the number of PMI’s and codebook size can be dynamically determined to maximize the average system throughput subject to the given feedback overhead, as the number of users varies in the cell.