Channel State Information Uncertainty Research Articles

On Analytical Achievable Rate for MIMO Linear Interference Alignment with Imperfect CSI

This paper studies the impact of channel error on the achievable rate of symmetrical K-user multiple-input multiple-output linear interference alignment (IA) networks. The upper and lower bounds of the achievable sum rate are derived analytically with the assumption of orthonormal transmit precoders and receive filters designed from imperfect channel state information (CSI) over both the uncorrelated and correlated channels. For uncorrelated channels, quite tight lower and upper bounds are obtained. The impact of channel error on the degrees of freedom (DoF) and the DoF persistence conditions are also investigated. Results show that the DoF of IA networks persists only if the channel error decreases in an order higher than the signal-to-noise ratio. For correlated channel, the lower and upper bounds for one realization of IA are derived. The derived upper bound can be used to characterize the achievable rate approximately. Simulation results indicate that the achievable rate of IA network is influenced significantly by CSI uncertainty. The obtained analytical bounds provide an intuitive way to show the impact of channel error on the achievable rate and thus can help practical systems deign.

Robust Rank-Two Multicast Beamforming Under a Unified CSI Uncertainty Model

We study a beamformed Alamouti coding scheme for a multigroup multicast downlink system. In this letter, we take into account imperfect channel state information (CSI) at the base station, where the CSI uncertainty region is formed by the intersection of multiple ellipsoids. This unified CSI uncertainty model leads to a challenging robust rank-two multicast beamforming problem. We first develop a conservative approximation method to handle the intractable constraints in the formulated robust beamforming problem under the considered CSI uncertainty region. Then, we apply the semidefinite relaxation (SDR) technique to approximately solve this problem. Our rank-profile analysis shows that the SDR is tight when there are at most two users in each multicast group and the CSI uncertainty region is not too large. Finally, we carry out simulations to validate the effectiveness of the proposed conservative approximation method. The simulation results coincide with the rank-profile analysis.

Transmit Precoding in Underlay MIMO Cognitive Radio With Unavailable or Imperfect Knowledge of Primary Interference Channel

This paper addresses the problem of precoder design that maximizes the throughput of a secondary user (SU) in an underlay multiple-input multiple-output cognitive radio network, where the channel state information (CSI) from the SU to the primary user (PU) is unavailable or inaccurate. The design maintains the quality of service for the PU through an interference amount measure in terms of the interference temperature or the leakage rate. For the case of unknown CSI, we propose an iterative adaptation algorithm by exploiting the side information in the primary communication network. For the case of imperfect CSI, we model the amount of uncertainty to be within a convex set defined by the Schatten norm and apply the maximin optimization to obtain the solution. To complete this paper, we derive the conditions on the CSI uncertainty radius under which the robust design with imperfect CSI would not perform better than the one with unknown CSI, when using the interference temperature metric. The proposed techniques are supported by numerical simulations.

Joint Secondary User Transceiver Optimization and User/Antenna Selection for MIMO–OFDM Cognitive Radio Networks with CSI Uncertainty

A novel algorithm is proposed for joint minimum mean-squared error based optimal secondary user transceiver design and user/antenna selection for multiple input multiple-output orthogonal frequency division multiplexing cognitive radio networks. Closed form expressions are developed for the optimal transceiver considering the availability of both full as well as partial channel state information (CSI) of the primary user at the secondary users. Further, optimal transceiver design is also considered for a realistic scenario with CSI uncertainty. In this regard, a low complexity algorithm is presented for joint secondary user and antenna selection towards sum-MSE minimization in cognitive radio networks. Simulation results demonstrate the improved performance of the proposed transceiver design and user/antenna selection algorithm in terms of the MSE performance at the receiver.

Robust Energy-Efficient MIMO Transmission for Cognitive Vehicular Networks

This work investigates a robust energy-efficient solution for multiple-input–multiple-output (MIMO) transmissions in cognitive vehicular networks. Our goal is to design an optimal MIMO beamforming for secondary users (SUs), considering imperfect interference channel-state information (CSI). Specifically, we optimize the energy efficiency (EE) of SUs, given that the transmission power constraint, the robust interference power constraint, and the minimum transmission rate are satisfied. To solve the optimization problem, we first characterize the uncertainty of CSI by bounding it in a Frobenius-norm-based region and then equivalently convert the robust interference constraint to a linear matrix inequality (LMI). Furthermore, a feasible ascent direction approach is proposed to reduce the optimization problem into a sequential linearly constrained semidefinite program, which leads to a distributed iterative optimization algorithm for deriving the robust and optimal beamforming. The feasibility and convergence of the proposed algorithm is theoretically validated, and the final experimental results are also supplemented to show the strength of the proposed algorithm over some conventional schemes in terms of the achieved EE performance and robustness.

CSI Impaired Precoding Optimization for Energy-Efficient MIMO Communications Under Total Power Constraint

This letter considers robust precoding for improving the energy efficiency (EE) in a multiple-input multiple-output (MIMO) channel when only imperfect or partial channel state information (CSI) is available at the transmitter. From the perspective of worst case robustness, we aim to maximize the worst case EE subject to additive or multiplicative bounded CSI uncertainties and under total power constraint. Theoretically, we prove that the robust energy-efficient precoding optimization leads to a channel-diagonalizing structure. Accordingly, the complicated matrix-valued problem is equivalently transformed into a simpler scalar optimization problem, which can be solved efficiently with a water-filling interpretation for the optimal power allocation. The obtained worst case EE was finally verified and evaluated via numerical simulations.

Robust Power Allocation and Outage Analysis for Secrecy in Independent Parallel Gaussian Channels

This letter studies parallel independent Gaussian channels with uncertain eavesdropper channel state information (CSI). Firstly, we evaluate the probability of zero secrecy rate in this system for (i) given instantaneous channel conditions and (ii) a Rayleigh fading scenario. Secondly, when non-zero secrecy is achievable in the low SNR regime, we aim to solve a robust power allocation problem which minimizes the outage probability at a target secrecy rate. We bound the outage probability and obtain a linear fractional program that takes into account the uncertainty in eavesdropper CSI while allocating power on the parallel channels. Problem structure is exploited to solve this optimization problem efficiently. We find the proposed scheme effective for uncertain eavesdropper CSI in comparison with conventional power allocation schemes.

Robust Ergodic Uplink Resource Allocation in Underlay OFDMA Cognitive Radio Networks

The ergodic resource allocation (ERA) problem for uplink transmission in underlay cognitive radio networks (CRNs) is investigated. The objective is to maximize the ergodic sum-rate of secondary users (SUs) considering the unavailability of perfect channel state information (CSI), and subject to transmit power limitations of SUs, and the interference threshold constraint to guarantee the quality of service of primary users. Since with average-based formulation of ERA, the interference threshold constraint and transmit power limitations of SUs do not hold instantaneously, one can replace the average-based constraints in ERA with their outage-based counterparts. For the uncertainty on the CSI values, we utilize the robust optimization theory where the uncertain parameters are modeled as a sum of the estimated value and error which is assumed to be bounded. We then map the considered ERA problems to their robust counterparts. Generally, the robust approaches degrade the performance (e.g., sum rate of SU), as they conservatively consider the error to be in the maximum extent and try to preserve the constrains under any condition of error (worst-case scenario). We aim to moderate this effect by using appropriate models for uncertain parameters, relaxing the worst-case scenario, and stochastically preserving the constraints. Moreover, robust problems are in general non-convex and suffer from high computational complexity due to the existence of uncertain system parameters. Therefore, we use effective suboptimal approaches to solve them with a reasonable complexity. This includes methods based on chance constraint approach as well as an iterative scheme. The proposed solutions provide a trade-off between robustness, performance, and complexity. Simulation results reveal that by using the proposed schemes, stable sum-rate of SUs in the presence of CSI uncertainties can be achieved while the instantaneous power and interference constraints are met with a desired probability.

Robust Masked Beamforming for MISO Cognitive Radio Networks With Unknown Eavesdroppers

This paper studies a cognitive radio network (CRN), in which a multiple-input single-output (MISO) secondary transmitter (SU-Tx) aims to send confidential messages to its receiver (SU-Rx) in the presence of unknown eavesdroppers, while having to control its generated interference to the primary users (PUs) under a given threshold. Because the eavesdroppers are unknown, this paper considers the artificial noise (AN) approach or masked beamforming to provide the information secrecy. The objective of this paper is to maximize the power of AN in order to degrade the eavesdroppers, subject to the signal-to-interference-plus-noise ratio (SINR) constraint at the SU-Rx, as well as the interference temperature limits (ITLs) for the PUs. In the case of perfect channel state information (CSI), we reveal that the optimal strategy for the information-bearing signal is beamforming. Imperfect CSI cases of bounded and stochastic uncertainties are investigated. In the case of ellipsoid-bounded errors, we derive the equivalent forms for the SINR and ITL constraints and then transform the optimization problem into a form of semidefinite programming (SDP). For the case of probabilistic CSI uncertainties, we propose an outage-constrained robust formulation where the CSI errors are Gaussian distributed. With the aid of two kinds of Bernstein-type inequalities, we reexpress the probabilistic constraints into deterministic forms, which results in a safe approximate solution.

Optimal GLRT-Based Robust Spectrum Sensing for MIMO Cognitive Radio Networks With CSI Uncertainty

In this paper, we develop generalized likelihood ratio test (GLRT)-based detectors for robust spectrum sensing in multiple-input multiple-output (MIMO) cognitive radio networks considering uncertainty in the available channel state information (CSI). Initially, for a scenario with known CSI uncertainty statistics, we derive the novel robust estimator-correlator detector (RECD) and the robust generalized likelihood detector (RGLD), which are robust against the uncertainty in the available estimates of the channel coefficients. Subsequently, for a scenario with unknown CSI uncertainty statistics, we develop a generalized likelihood ratio test (GLRT) based composite hypothesis robust detector (CHRD) for spectrum sensing. Closed form expressions are presented for the probability of detection $(P_D)$ and the probability of false alarm $(P_{FA})$ to characterize the detection performance of the proposed robust spectrum sensing schemes. Further, a deflection coefficient based optimization framework is also developed and solved to derive closed form expressions for the optimal beacon sequences. Simulation results are presented to demonstrate the performance improvement achieved by the proposed robust spectrum sensing schemes and to verify the analytical results derived.

Ergodic Interference Alignment With Limited Feedback: Power Control and Rate Adaptation

Considering the time-varying $K$ -user single-antenna interference channel (IC), it has been shown that, when terminals have perfect global channel state information (CSI) and they can tolerate asymptotically long delay, applying an ergodic interference alignment (EIA) scheme can achieve half of the interference-free achievable rate. However, in practice, obtaining such CSI is challenging, and only a limited delay is acceptable. This paper addresses data transmission over the IC by taking these concerns into account. Specifically, we consider the case that each transmitter attains only quantized CSI via limited feedback signals. This causes imperfect interference alignment and a degraded performance. We propose adaptive schemes to compensate the impact of the CSI uncertainties. We first study a power control problem which is concerned with communicating at fixed rates using minimum transmit powers. A power control algorithm is used to reach the solution. Next, we address a throughput maximization problem when the transmit powers are fixed. Through the analysis of system outage probability, we propose a rate adaptation scheme to maximize throughput. Finally, we quantify the throughput loss in delay-limited systems. Our results show that, even with limited feedback, performing the EIA scheme with proper power control or rate adaptation strategies can still outperform conventional orthogonal transmission approaches.

Optimal Stochastic Coordinated Beamforming for Wireless Cooperative Networks With CSI Uncertainty

Transmit optimization and resource allocation for wireless cooperative networks with channel state information (CSI) uncertainty are important but challenging problems in terms of both the uncertainty modeling and performance optimization. In this paper, we establish a generic stochastic coordinated beamforming (SCB) framework that provides flexibility in the channel uncertainty modeling, while guaranteeing optimality in the transmission strategies. We adopt a general stochastic model for the CSI uncertainty, which is applicable for various practical scenarios. The SCB problem turns out to be a joint chance constrained program (JCCP) and is known to be highly intractable. In contrast to all of the previous algorithms for JCCP that can only find feasible but sub-optimal solutions, we propose a novel stochastic DC (difference-of-convex) programming algorithm with optimality guarantee, which can serve as the benchmark for evaluating heuristic and sub-optimal algorithms. The key observation is that the highly intractable probability constraint can be equivalently reformulated as a dc constraint. This further enables efficient algorithms to achieve optimality. Simulation results will illustrate the convergence, conservativeness, stability and performance gains of the proposed algorithm.

Capacity Analysis of Interference Alignment With Bounded CSI Uncertainty

Interference alignment (IA) has been demonstrated to achieve the degree-of-freedom (DoF) of an interference channel given perfect global channel state information (CSI). In this letter, we consider the case of imperfect CSI with bounded errors and derive a capacity lower bound of the channel using IA. We show that this lower bound is within 1 bps/Hz of the capacity of the perfect CSI case up to a certain signal-to-noise ratio (SNR) which we refer to it as the saturating SNR. Further, we introduce a new metric called modified DoF (mDoF) in order to characterize the multiplexing performance of IA with imperfect CSI at finite SNR. Simulation results for the 3-user case are provided to illustrate the region within which the actual capacity of IA falls.

Robust Distributed Beamforming With Interference Coordination in Downlink Cellular Networks

This paper focuses on multicell coordinated beamforming in the presence of channel state information (CSI) errors, where base stations (BSs) collaboratively mitigate their intercell interference (ICI). Assuming that the CSI errors are hyper-spherically bounded, we consider an optimization problem that minimizes the overall transmission power of BSs subject to signal-to-interference-plus-noise-ratio (SINR) constraints at each mobile station (MS). We solve this problem in a distributed manner with a limited information exchange among BSs. Using semidefinite relaxation (SDR) and the S-Lemma, we first reformulate our optimization problem into a numerically tractable one. Since the SINR constraints are coupled, we introduce an algorithm by which each BS can obtain a local version of its coupling variables via a small data exchange with other BSs. Then, we propose an iterative algorithm that employs the projected gradient method to coordinate ICI across multiple BSs. Finally, we extend the application of the proposed algorithm to solve the problem of robust and distributed per-user SINR maximization under per-BS power constraints. Simulation results confirm the effectiveness of the proposed algorithm in terms of power efficiency and convergence in the presence of CSI uncertainties.

Robust Estimator-Correlator for Spectrum Sensing in MIMO CR Networks with CSI Uncertainty

This work introduces novel detection schemes which are robust with respect to the uncertainty in the estimate of the signal covariance matrix for non-coherent spectrum sensing in multiple-input multiple-output (MIMO) cognitive radio networks. We employ an eigenvalue perturbation theory based approach to model the uncertainty in the estimated signal covariance matrix. Subsequently, we derive an optimization framework for the generalized likelihood ratio test (GLRT) based robust test statistic detector (RTSD) and robust estimator-correlator detector (RECD) towards primary user detection, which incorporate the channel state information (CSI) uncertainty inherent in such scenarios. Further, employing the Karush-Kuhn-Tucker (KKT) conditions, we derive closed form expressions for the proposed robust spectrum sensing schemes. Simulation results demonstrate the superior performance of the proposed robust detectors in comparison to the uncertainty agnostic estimator-correlator (EC) detector for spectrum sensing in MIMO cognitive radio networks with CSI uncertainty.

Power Allocation in MISO Interference Channels with Stochastic CSIT

This paper considers multiuser interference channels in which the transmitters have imperfect channel state information (CSI) where CSI perturbations are modeled stochastically. Transmitters are assumed to be equipped with multiple antennas serving single-antenna receivers. Transmitters use pre-designed discrete codebooks for beamforming directions and dynamically (based on the available CSI) select the best set of beamformers from the given codebook. The objective is to perform optimal power allocation to different users while certain quality of service (QoS) guarantees are ensured for the users. Imposed by stochastic CSI uncertainties, guarantees provided for the QoS measures have a stochastic nature too. The primary focus is placed on the interference channels for which two power allocation problems are considered. The first problem minimizes power consumption subject to serving users at certain data rates and the second problem considers max-min rate allocation subject to given power budgets for the transmitters. The core step in formalizing these problems in mathematically tractable forms relies on using Bernstein approximation, which approximates and convexifies the non-convex stochastic guarantees by conservative convex and deterministic counterparts. For solving this resulting convex and deterministic optimization problem, a specialized version of the long-step logarithmic barrier cutting plane (LLBCP) algorithm is used. Effectiveness of the proposed solutions and comparisons with other existing methods are assessed via extensive simulation results.

Robust Optimization for Amplify-and-Forward MIMO Relaying From a Worst-Case Perspective

In this paper, we consider robust optimization of amplify-and-forward (AF) multiple-input multiple-output (MIMO) relay precoders in presence of deterministic imperfect channel state information (CSI), when the CSI uncertainty lies in a norm bounded region. Two widely used performance metrics, mutual information (MI) and mean square error (MSE), are adopted as design objectives. According to the philosophy of worst-case robustness, the robust optimization problems with respect to maximizing the worst-case MI and minimizing the worst-case MSE are formulated as maximin and minimax problems, respectively. Due to the fact that these two problems do not have a concave-convex or convex-concave structure, we cannot rely on the conventional saddle point theory to find the robust solutions. Nevertheless, by exploiting majorization theory, we show that the formulated maximin and minimax problems both admit saddle points. We further analytically characterize the saddle points, and provide closed-form solutions to robust relay precoder designs. Interestingly, we find that, under both MI and MSE metrics, the robust relay optimization leads to a channel-diagonalizing structure, meaning that eigenmode transmission is optimal from the worst-case robustness perspective. The proposed robust designs can improve the spectral efficiency and reliability of AF MIMO relaying against CSI uncertainties at the similar cost of computational complexity as the existing non-robust schemes.

Robust Relay Beamforming in Cognitive Two-Way Relay Networks

We propose a novel restriction and relaxation (RAR) method for robust relay beamforming design in cognitive two-way relay networks with imperfect channel state information (CSI). Due to the uncertainty in CSI, the self-interference cannot be completely canceled. Hence, both channel uncertainties and residual self-interference should be considered. The proposed RAR method aims to minimize the transmit power of relay nodes while guaranteeing the worst-case signal-to-interference plus noise ratio (SINR) of secondary users (SUs) as well as satisfying the interference temperature (IT) constraints. Specifically, in the restriction step of RAR, infinite number of complicated constraints are reformulated into a finite number of linear matrix inequalities (LMIs). In the relaxation step of RAR, the nonconvex robust design problem is transformed into a convex semidefinite program (SDP), which can be efficiently solved by interior-point methods. Simulation results verify the effectiveness and robustness of the proposed method.

Robust Design of Widely Linear Pre-Equalization Filters for Pre-Rake UWB Systems

Pre-rake ultra-wideband (UWB) systems are appealing for UWB communications applications which include devices with different processing capabilities so that signal processing complexity need to be shifted from the receiver of one or more devices to the transmitter of another. Recently, basic pre-rake schemes have been extended to include full pre-equalization, multiple-antenna, and multi-user interference processing. All these design approaches for pre-rake UWB systems have relied on the availability of accurate channel state information (CSI) at the transmitter. However, uncertainties in the acquisition of CSI can drastically affect the overall system performance. Therefore, in this paper, we present robust design methods for pre-equalization filters (PEFs) for pre-rake UWB systems that take CSI uncertainties into account. We treat the general case of a broadcast (i.e., multiuser) pre-rake UWB system, which includes single-user communication often considered in literature as a special case. For this general setting, we derive new PEF designs that improve system performance with imperfect CSI. Similar to the literature on robust filter designs for multiple-input multiple output (MIMO) systems, we consider two uncertainty models, namely stochastic and bounded uncertainty, which correspond to different performance optimization paradigms, and we adjust these according to channel estimation of UWB channels. As most of previous work on (pre-rake) UWB, we focus on binary transmission. We argue that widely linear filter design should be applied in this case and thus extend the robust filter design methodology accordingly. Our numerical results for typically UWB test channels demonstrate the efficacy of the proposed design procedures to achieve reliable communication in multiuser pre-rake UWB systems.

Robust joint signal and interference alignment in cognitive radio networks with ellipsoidal channel state information uncertainties

The authors propose a distributed robust joint signal and interference alignment design for multiple-input-multiple-output cognitive radio (CR) networks where single primary link coexists with multiple secondary links. Considering two practical challenges of interference alignment: imperfect channel state information (CSI) and finite signal-to-noise ratio, the proposed scheme aims to minimise both the leakage of interference signals and that of the desired signals, while maintaining interference to the primary user below a permissible level. Under the assumption of the ellipsoidal CSI uncertainties, the joint worst-case optimisation problem is decomposed and reformulated as semi-definite programming form by using S -lemma, orthogonal relaxation and semi-definite relaxation. Simulation results verify the effectiveness of the joint design, and robustness of the worst-case design against channel uncertainties.