Optimal Transmit Covariance Matrix Research Articles

A Unified MIMO Optimization Framework Relying on the KKT Conditions

A popular technique of designing multiple-input multiple-output (MIMO) communication systems relies on optimizing the positive semidefinite covariance matrix at the source. In this paper, a unified MIMO optimization framework based on the Karush-Kuhn-Tucker (KKT) conditions is proposed. In this framework, with the aid of matrix optimization theory, <xref ref-type="theorem" rid="theorem1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorem 1</xref> presents a generic optimal transmit covariance matrix for MIMO systems with diverse objective functions subject to various power constraints and different levels of channel state information (CSI). Specifically, <xref ref-type="theorem" rid="theorem1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorem 1</xref> fundamentally reveals that for a diverse family of MIMO systems, the optimal transmit covariance matrices associated with different objective functions under various power constraints can be derived in a unified generic water-filling-like form. When applying <xref ref-type="theorem" rid="theorem1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorem 1</xref> to the case of multiple general power constraints, we firstly equivalently transform multiple power constraints into a single counterpart by introducing multiple weighting factors based on Pareto optimization theory. The optimal weighting factors can be found by the proposed modified subgradient method. On the other hand, for the imperfect MIMO system with statistical CSI errors, we firstly address the non-convexity of the robust optimization problem by following the idea of alternating optimization. Finally, our numerical results verify the optimal solution structure in <xref ref-type="theorem" rid="theorem1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorem 1</xref> and the global optimality of the proposed modified subgradient method, as well as demonstrate the performance advantages of the proposed alternating optimization algorithm.

Large System Achievable Rate Analysis of RIS-Assisted MIMO Wireless Communication With Statistical CSIT

Reconfigurable intelligent surface (RIS) is an emerging technology to enhance wireless communication in terms of energy cost and system performance by equipping a considerable quantity of nearly passive reflecting elements. This study focuses on a downlink RIS-assisted multiple-input multiple-output (MIMO) wireless communication system that comprises three communication links of Rician channel, including base station (BS) to RIS, RIS to user, and BS to user. The objective is to design an optimal transmit covariance matrix at BS and diagonal phase-shifting matrix at RIS to maximize the achievable ergodic rate by exploiting the statistical channel state information at BS. Therefore, a large-system approximation of the achievable ergodic rate is derived using the replica method in large dimension random matrix theory. This large-system approximation enables the identification of asymptotic-optimal transmit covariance and diagonal phase-shifting matrices using an alternating optimization algorithm. Simulation results show that the large-system results are consistent with the achievable ergodic rate calculated by Monte-Carlo averaging. The results verify that the proposed algorithm can significantly enhance the RIS-assisted MIMO system performance.

Energy Efficiency Optimization in Massive MIMO Secure Multicast Transmission.

Herein, we focus on energy efficiency optimization for massive multiple-input multiple-output (MIMO) downlink secure multicast transmission exploiting statistical channel state information (CSI). Privacy engineering in the field of communication is a hot issue under study. The common signal transmitted by the base station is multicast transmitted to multiple legitimate user terminals in our system, but an eavesdropper might eavesdrop this signal. To achieve the energy efficiency utility–privacy trade-off of multicast transmission, we set up the problem of maximizing the energy efficiency which is defined as the ratio of the secure transmit rate to the power consumption. To simplify the formulated nonconvex problem, we use a lower bound of the secure multicast rate as the molecule of the design objective. We then obtain the eigenvector of the optimal transmit covariance matrix into a closed-form, simplifying the matrix-valued multicast transmission strategy problem into a power allocation problem in the beam domain. By utilizing the Minorize-Maximize method, an iterative algorithm is proposed to decompose the secure energy efficiency optimization problem into a sequence of iterative fractional programming subproblems. By using Dinkelbach’s transform, each subproblem becomes an iterative problem with the concave objective function, and it can be solved by classical convex optimization. We guarantee the convergence of the two-level iterative algorithm that we propose. Besides, we reduce the computational complexity of the algorithm by substituting the design objective with its deterministic equivalent. The numerical results show that the approach we propose performs well compared with the conventional methods.

Energy Efficiency Optimization for Downlink Massive MIMO With Statistical CSIT

We investigate energy efficiency (EE) optimization for single-cell massive multiple-input multiple-output (MIMO) downlink transmission with only statistical channel state information (CSI) available at the base station. We first show that beam domain transmission is favorable for energy efficiency in the massive MIMO downlink, by deriving a closed-form solution for the eigenvectors of the optimal transmit covariance matrix. With this conclusion, the EE optimization problem is reduced to a real-valued power allocation problem, which is much easier to tackle than the original large-dimensional complex matrix-valued precoding design problem. We further propose an iterative water-filling-structured beam domain power allocation algorithm with low complexity and guaranteed convergence, exploiting the techniques from sequential optimization, fractional optimization, and random matrix theory. Numerical results demonstrate the near-optimal performance of our proposed statistical CSI aided EE optimization approach.

Algorithms for Globally-Optimal Secure Signaling Over Gaussian MIMO Wiretap Channels Under Interference Constraints

Multi-user Gaussian MIMO wiretap channel is considered under interference power constraints (IPC), in addition to the total transmit power constraint (TPC). Algorithms for global maximization of its secrecy rate are proposed. Their convergence to the secrecy capacity is rigorously proved and a number of properties are established analytically. Unlike known algorithms, the proposed ones are not limited to the MISO case and are proved to converge to a global rather than local optimum in the general MIMO case, even when the channel is not degraded. In practice, the convergence is fast as only a small to moderate number of Newton steps is required to achieve a high precision level. The interplay of TPC and IPC is shown to result in an unusual property when an optimal point of the max-min problem does not provide an optimal transmit covariance matrix in some (singular) cases. To address this issue, an algorithm is developed to compute an optimal transmit covariance matrix in those singular cases. It is shown that this algorithm also solves the dual (nonconvex) problems of globally minimizing the total transmit power subject to the secrecy and interference constraints; it provides the minimum transmit power and respective signaling strategy needed to achieve the secrecy capacity, hence allowing power savings.

Intelligent Reflecting Surface Assisted Secure Wireless Communications With Multiple- Transmit and Multiple-Receive Antennas

In this paper, we propose intelligent reflecting surfaces (IRS) assisted secure wireless communications with multi-input and multi-output antennas (IRS-MIMOME). The considered scenario is an access point (AP) equipped with multiple antennas communicates with a multi-antenna enabled legitimate user in the downlink at the present of an eavesdropper configured with multiple antennas. Particularly, the joint optimization of the transmit covariance matrix at the AP and the reflecting coefficients at the IRS to maximize the secrecy rate for the IRS-MIMOME system is investigated, with two different assumptions on the phase shifting capabilities at the IRS, i.e., the IRS has the continuous reflecting coefficients and the IRS has the discrete reflecting coefficients. For the former case, due to the non-convexity of the formulated problem, an alternating optimization (AO)-based algorithm is proposed, i.e., for given the reflecting coefficients at the IRS, the successive convex approximation (SCA)-based algorithm is used to solve the transmit covariance matrix optimization, while given the transmit covariance matrix at the AP, alternative optimization is used again in individually optimizing of each reflecting coefficient at the IRS with other fixed reflecting coefficients. For the individual reflecting coefficient optimization, the closed-form or an interval of the optimal solution is provided. Then, the proposed algorithm is extended to the discrete reflecting coefficient model at the IRS. Finally, some numerical simulations have been done to demonstrate that the proposed algorithm outperforms other benchmark schemes.

Optimal Full-Rank Signaling Over MIMO Wiretap Channels Under Interference Constraint

A closed-form full-rank solution for an optimal transmit covariance matrix over a strictly degraded MIMO wiretap channel and its secrecy capacity are obtained under an interference constraint, in addition to the transmit power constraint. A number of its properties are pointed out. The regimes when one of the constraints is inactive are studied. The interference constraint affects significantly the optimal covariance and induces its new properties, which cannot be found under the total transmit power constraint only.

The Secrecy Capacity of Gaussian MIMO Wiretap Channels Under Interference Constraints

Secure signaling over multiple-input multiple-output (MIMO) wiretap channel (WTC) is studied under interference and transmit power constraints. The classical MIMO WTC model is extended to interference-limited scenarios, so that interference to other users does not exceed a given threshold while ensuring simultaneously no information leakage to an eavesdropper. The operational secrecy capacity of the Gaussian MIMO WTC under interference and transmit power constraints is rigorously established in two forms (a non-convex max problem and a convex–concave max–min problem), to which per-antenna power constraints can be added as well. Optimal signaling directions are characterized in the general case, from which (tight) upper bounds to the rank of optimal transmit covariance matrix are derived. A sufficient condition for the optimality of beamforming and a necessary condition for optimal full-rank signaling are given. Closed-form rank-1 and high-rank solutions are obtained in the case of zero interference constraints. Sufficient and necessary conditions for non-zero secrecy capacity are established. The results are extended to multi-user scenarios. A sufficient and necessary condition for the unbounded growth of the secrecy capacity with transmit power is obtained. The interplay between transmit and interference power constraints is studied, and its significant impact on optimal signaling is demonstrated (so that neither constraint can be absorbed into the other one in general, as was sometimes suggested in the literature). Overall, these results provide insights into fundamental information-theoretic limits and optimal signaling strategies for secure communications under interference constraints.

SINR Enhancement in Colocated MIMO Radar Using Transmit Covariance Matrix Optimization

In this letter, we focus on the signal-to-interference-plus-noise ratio (SINR) enhancement using transmitting waveform covariance matrix optimization in colocated MIMO radars. In our proposed algorithm, transmitted waveform covariance matrix has been optimized to focus the transmit beampattern into the target direction and in the receiver, we try to reject maximum number of interfering objects. In the proposed algorithm, transmitted waveform covariance matrix is able to be synthesized with binary phase shift keying waveforms in closed form. Simulation results show that our proposed method has better SINR performance in comparison with methods in which transmitted covariance matrix is designed independently from the interferences locations.

Robust Transceiver Design in Multiuser MIMO System with Energy Harvesting Constraints

In this paper, considering the simultaneous wireless information and power transfer scheme, we investigate the robust transceiver design in the multiuser multiple-input–multiple-output system with the energy harvesting constraints where the channel uncertainties are modeled by the worst-case model. We formulate the robust transceiver design problem as a worst-case mean-square-error (MSE) minimization problem where the transmit covariance matrix at the transmitter and the preprocessing matrices at the information-decoding (ID) receivers are alternatively optimized. With the known preprocessing matrices at the ID receivers, the transmit covariance matrix optimization problem is transformed into a semidefinite programming (SDP) by employing the \({\mathcal {S}}\)-procedure. With the known transmit covariance matrix at the transmitter, the optimization problem is transformed into another SDP by converting the MSE minimization constraint into a linear matrix inequality. Simulation results have shown that our proposed robust transceiver design outperforms the non-robust one.

Optimal Signaling for Secure Communications Over Gaussian MIMO Wiretap Channels

Optimal signaling over the Gaussian multiple-input multiple-output wire-tap channel is studied under the total transmit power constraint. A closed-form solution for an optimal transmit covariance matrix is obtained when the channel is strictly degraded. In combination with the rank-1 solution, this provides the complete characterization of the optimal covariance for the case of two transmit antennas. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the high-SNR regime. Necessary optimality conditions and a tight upper bound on the rank of an optimal covariance matrix are established for the general case, along with a lower bound to the secrecy capacity, which is tight in a number of scenarios.

Simultaneous Wireless Information Power Transfer for MISO Secrecy Channel

This paper investigates simultaneous wireless information and power transfer (SWIPT) for a multiple-input–single-output (MISO) secrecy channel. First, transmit beamforming without artificial noise (AN) is designed to achieve the secrecy rate maximization problem subject to the transmit power and the energy harvesting (EH) constraints. This problem is not convex, but this can be solved by applying a bisection search to a sequence of the associated power minimization problems, each of which can be solved by using a novel relaxation approach. Moreover, we extend our proposed algorithms to the robust case by incorporating channel uncertainties. Then, transmit beamforming with AN is investigated for the secrecy rate maximization problem, where two relaxation approaches are presented, i.e., a two-level optimization algorithm and a successive convex approximation (SCA), to solve the secrecy rate maximization problem. In addition, tightness analyses of rank relaxation are provided to show that the optimal transmit covariance matrix exactly returns rank one. Simulation results are provided to validate the performance of the proposed algorithms.

Secrecy Sum Rates of MIMO Multi-Receiver Wiretap Channels

We consider MIMO multi-receiver wiretap channels where the transmitter sends confidential messages to legitimate receivers in the presence of an external eavesdropper. Since the secrecy sum rate maximization problem is nonconvex, we derive an equivalent alternating optimization problem. We propose an iterative algorithm to find the optimal transmit covariance matrix based on the MIMO BC-MAC duality. We prove that the limit point satisfies the KKT conditions of the original secrecy sum rate maximization problem.

Rank-Deficient Solutions for Optimal Signaling Over Wiretap MIMO Channels

Capacity-achieving signaling strategies for the Gaussian wiretap MIMO channel are investigated without the degradedness assumption. In addition to known solutions, a number of new rank-deficient solutions for the optimal transmit covariance matrix are obtained. The case of a weak eavesdropper is considered in detail and the optimal covariance is established in an explicit, closed form with no extra assumptions. This provides lower and upper bounds to the secrecy capacity in the general case with a bounded gap, which are tight for a weak eavesdropper or/and low SNR. Closed form solutions are also obtained for isotropic and omnidirectional eavesdroppers, based on which lower and upper bounds to the secrecy capacity are established in the general case. Sufficient and necessary conditions for optimality of 3 popular transmission techniques, namely the zero-forcing (ZF), the standard water-filling (WF) over the channel eigenmodes and the isotropic signaling (IS), are established for the MIMO wiretap channel. These solutions are appealing due to their lower complexity. In particular, no wiretap codes are needed for the ZF transmission, and no precoding or feedback is needed for the isotropic signaling.

Large System Secrecy Rate Analysis for SWIPT MIMO Wiretap Channels

© 2015 IEEE. In this paper, we study the multiple-input multiple-output wiretap channel for simultaneous wireless information and power transfer, in which there is a base station (BS), an information-decoding (ID) user, and an energy-harvesting (EH) user. The messages intended to the ID user is required to be kept confidential to the EH user. Our objective is to design the optimal transmit covariance matrix at the BS for maximizing the ergodic secrecy rate subject to the harvested energy requirement for the EH user exploiting only statistical channel state information at the BS. To this end, we begin by deriving an approximation for the ergodic secrecy rate using large-dimensional random matrix theory and the method of Taylor series expansion. This approximation enables us to derive the asymptotic-optimal transmit covariance matrix that achieves the tradeoff for ergodic secrecy rate and harvested energy. The simulation results are provided to verify the accuracy of the approximation and show that a bigger rate-energy region can be achieved when the Rician factor increases or the path loss exponent decreases. We also show that when the transmit correlation increases or the distance between the eavesdropper and the BS decreases, the harvested energy will be increased, while the achieved ergodic secrecy rate decreases.

Secrecy Rate Optimizations for a MISO Secrecy Channel with Multiple Multiantenna Eavesdroppers

This paper investigates secrecy rate optimization problems for a multiple-input-single-output (MISO) secrecy channel in the presence of multiple multiantenna eavesdroppers. Specifically, we consider power minimization and secrecy rate maximization problems for this secrecy network. First, we formulate the power minimization problem based on the assumption that the legitimate transmitter has perfect channel state information (CSI) of the legitimate user and the eavesdroppers, where this problem can be reformulated into a second-order cone program (SOCP). In addition, we provide a closed-form solution of transmit beamforming for the scenario of an eavesdropper. Next, we consider robust secrecy rate optimization problems by incorporating two probabilistic channel uncertainties with CSI feedback. By exploiting the Bernstein-type inequality and S-Procedure to convert the probabilistic secrecy rate constraint into the determined constraint, we formulate this secrecy rate optimization problem into a convex optimization framework. Furthermore, we provide analyses to show the optimal transmit covariance matrix is rank-one for the proposed schemes. Numerical results are provided to validate the performance of these two conservative approximation methods, where it is shown that the Bernstein-type inequality-based approach outperforms the S-Procedure approach in terms of the achievable secrecy rates.

An Algorithm for Global Maximization of Secrecy Rates in Gaussian MIMO Wiretap Channels

Optimal signaling for secrecy rate maximization in Gaussian MIMO wiretap channels is considered. While this channel has attracted a significant attention recently and a number of results have been obtained, including the proof of the optimality of Gaussian signalling, an optimal transmit covariance matrix is known for some special cases only and the general case remains an open problem. An iterative custom-made algorithm to find a globally-optimal transmit covariance matrix in the general case is developed in this paper, with guaranteed convergence to a \textit{global} optimum. While the original optimization problem is not convex and hence difficult to solve, its minimax reformulation can be solved via the convex optimization tools, which is exploited here. The proposed algorithm is based on the barrier method extended to deal with a minimax problem at hand. Its convergence to a global optimum is proved for the general case (degraded or not) and a bound for the optimality gap is given for each step of the barrier method. The performance of the algorithm is demonstrated via numerical examples. In particular, 20 to 40 Newton steps are already sufficient to solve the sufficient optimality conditions with very high precision (up to the machine precision level), even for large systems. Even fewer steps are required if the secrecy capacity is the only quantity of interest. The algorithm can be significantly simplified for the degraded channel case and can also be adopted to include the per-antenna power constraints (instead or in addition to the total power constraint). It also solves the dual problem of minimizing the total power subject to the secrecy rate constraint.

Capacity of Compound MIMO Gaussian Channels With Additive Uncertainty

This paper considers reliable communications over a multiple-input multiple-output (MIMO) Gaussian channel, where the channel matrix is within a bounded channel uncertainty region around a nominal channel matrix, i.e., an instance of the compound MIMO Gaussian channel. We study the optimal transmit covariance matrix design to achieve the capacity of compound MIMO Gaussian channels, where the channel uncertainty region is characterized by the spectral norm. This design problem is a challenging nonconvex optimization problem. However, in this paper, we reveal that this problem has a hidden convexity property, which can be exploited to map the problem into a convex optimization problem. We first prove that the optimal transmit design is to diagonalize the nominal channel, and then show that the duality gap between the capacity of the compound MIMO Gaussian channel and the min-max channel capacity is zero, which proves and generalizes a conjecture of Loyka and Charalambous. The key tools for showing these results are a new matrix determinant inequality and some unitarily invariant properties.

Low Complexity User Scheduling, Ordering and Transmit Covariance Matrix Optimization Algorithms for Successive Zero-Forcing Precoding

In this paper we consider user scheduling, ordering and transmit covariance matrix optimization problems under successive zero-forcing (SZF) precoding for multiuser multiple-input multiple-output downlink. We propose a heuristic user scheduling metric and an intermediate user grouping technique to develop a low complexity greedy scheduling algorithm. A suboptimal user ordering technique is also proposed for transmit covariance matrix optimization under SZF. Proposed algorithm is of low complexity, but performs closely to the highly complex exhaustive search algorithm. For transmit covariance optimization under SZF, a dirty paper coding based algorithm has been previously proposed, which is computationally very complex. In this paper, we propose a suboptimal but much simplified algorithm, which employs an iterative procedure similar to a known multiple access channel (MAC) covariance optimization algorithm, but does not involve multiple levels of covariance matrix transformations. With the proposed suboptimal user ordering the exhaustive search through all possible user orders is avoided during transmit covariance matrix optimization resulting in a significant complexity reduction, and without a significant performance penalty. Simulation results show that the proposed algorithm performs very close to the known algorithm in the low SNR region.

Optimal Transmission for the MIMO Bidirectional Broadcast Channel in the Wideband Regime

This paper considers a transmit strategy for an AWGN MIMO bidirectional broadcast channel in the wideband regime. In order to characterize the boundaries of the wideband capacity and energy per bit regions, the transmit strategy at the relay is designed to maximize the weighted wideband rate sum. A closed form of the optimal transmit covariance matrix is derived, which shows that a single beam transmit strategy is optimal. The transmit strategies for some special cases are also analyzed. The fairness versus energy efficiency tradeoff is then discussed. In addition, an extension to multipair MIMO bidirectional broadcast channel is studied in which we show that serving a certain pair with full power is optimal in the sense of maximizing the achievable weighted wideband rate sum. Finally, a discussion on the conjecture of the minimum energy per bit for multi-pair systems is provided.