Channel Covariance Research Articles

Robust Coordinated Beamforming for Secure MISO Interference Channels with Bounded Ellipsoidal Uncertainties

This letter addresses the robust optimization problem of downlink coordinated beamforming over the multiple-input single-output (MISO) interference channel with confidential messages (IFC-CM) exploiting imperfect channel state information (CSI) at the transmitters. Our aim is to minimize the total transmitted power subject to the worst-case individual signal-to-interference-plus-noise-ratio (SINR) and equivocation rates. To model the imperfect CSI, bounded ellipsoidal uncertainties for channel vector and covariance are both considered. By deriving the equivalent forms for the SINR and secrecy constraints, the worst-case robust problem can be reformulated as a semi-definite programming (SDP) with the help of rank relaxation. Simulation results show the effectiveness of the proposed algorithm.

Minimum mean‐square‐error expression of LMMSE channel estimation in SISO OFDM systems

An expression of the minimum mean square error (MMSE) of the linear MMSE channel estimation is given in the case of a non-invertible channel covariance matrix, as in single-input single-output (SISO) OFDM system. A matrix expression, already proposed for a multi-input multi-output OFDM system in a previous article, is not valid in SISO. A new proof is then proposed, by deriving a scalar expression of the MMSE, which leads to solve an optimisation problem. Furthermore, we show that the proposed solution is the global minimum. Simulations validate the proposed development.

Application of a joint and iterative MMSE-based estimation of SNR and frequency-selective channel for OFDM systems

This article presents an iterative minimum mean square error- (MMSE-) based method for the joint estimation of signal-to-noise ratio (SNR) and frequency-selective channel in an orthogonal frequency division multiplexing (OFDM) context. We estimate the SNR thanks to the MMSE criterion and the channel frequency response by means of the linear MMSE (LMMSE). As each estimation requires the other one to be performed, the proposed algorithm is iterative. In this article, a realistic case is considered; i.e., the channel covariance matrix used in LMMSE is supposed to be totally unknown at the receiver and must be estimated. We will theoretically prove that the algorithm converges for a relevantly chosen initialization value. Furthermore simulations show that the algorithm quickly converges to a solution that is close to the one in which the covariance matrix is perfectly known. Compared to existing SNR estimation methods, the algorithm improves the trade-off between the number of required pilots and the SNR estimation quality.

Optimal Channel Estimation in Beamformed Systems for Common-Randomness-Based Secret Key Establishment

Establishing secret keys from the commonly-observed randomness of reciprocal wireless propagation channels has recently received considerable attention. We consider such key establishment between two multiantenna nodes that use beamforming for communication, showing that the upper bound on the number of key bits that can be generated from the channel observations can be maximized by properly probing the channel. Specifically, we demonstrate that the eigenvectors of the channel spatial covariance matrix should be used as beamformer weights during channel estimation and we optimize the energy allocated to channel estimation for each beamformer weight under a total energy constraint. Finally, by assuming that the channel covariance is separable, we illustrate implementation of the technique for practical beamforming systems and more advanced signal models that incorporate antenna mutual coupling.

Robust Downlink Beamforming With Partial Channel State Information for Conventional and Cognitive Radio Networks

We address the problem of robust multiuser downlink beamforming under the assumption that the transmitter has partial covariance-based channel state information (CSI). In our approach, the uncertainty on the channel covariance matrices is assumed to be confined in an ellipsoid of given size and shape, where prior knowledge about the statistical distribution of the CSI mismatch is taken into account. The goal is to minimize the transmitted power under the worst-case quality-of-service (QoS) constraints. We extend the developed robust problem to downlink beamforming in cognitive radio (CR) networks where QoS constraints apply to the users of the secondary network (SN) and interference leaked to the primary users (PUs) is required to be below a given interference threshold. We avoid the coarse approximations used by previous solutions and obtain exact reformulations for both worst-case problems based on Lagrange duality. The resulting problems can then be approximated using semidefinite relaxation (SDR). Further, we consider a popular alternative robust approach that is based on probabilistic QoS and interference constraints and show that both approaches are generally equivalent. Computer simulations show that the proposed techniques provide substantial performance improvements over earlier robust downlink beamforming techniques for both the conventional and the CR scenarios.

Capacity Pre-Log of Noncoherent SIMO Channels Via Hironaka's Theorem

We find the capacity pre-log of a temporally correlated Rayleigh block-fading single-input multiple-output (SIMO) channel in the noncoherent setting. It is well known that for block-length L and rank of the channel covariance matrix equal to Q, the capacity pre-log in the single-input single-output (SISO) case is given by 1-Q/L. Here, Q/L can be interpreted as the pre-log penalty incurred by channel uncertainty. Our main result reveals that, by adding only one receive antenna, this penalty can be reduced to 1/L and can, hence, be made to vanish for the block-length L→∞, even if Q/L remains constant as L→∞. Intuitively, even though the SISO channels between the transmit antenna and the two receive antennas are statistically independent, the transmit signal induces enough statistical dependence between the corresponding receive signals for the second receive antenna to be able to resolve the uncertainty associated with the first receive antenna's channel and thereby make the overall system appear coherent. The proof of our main theorem is based on a deep result from algebraic geometry known as Hironaka's Theorem on the Resolution of Singularities.

Artificial channel aided LMMSE estimation for time–frequency selective channels in OFDM context

This paper proposes a linear minimum mean square error-based (LMMSE) channel estimation method, which allows avoiding the necessary knowledge of the channel covariance matrix or its estimation. To do so, a perfectly tunable filter acting like an artificial channel is added at the receiver side. We show that an LMMSE estimation of the sum of this artificial channel and the physical channel only needs the covariance matrix of the artificial channel, and the channel estimation is finally obtained by subtracting the frequency coefficients of the added filter. We call this method artificial channel aided-LMMSE (ACA-LMMSE). Theoretical developments and simulations prove that its performance is close to theoretical LMMSE, and we show that this method reduces the computational complexity, compared to usual LMMSE, due to the covariance matrix used for ACA-LMMSE is computed only once throughout the transmission duration. We put the conditions on the artificial channel parameters to get the expected mask effect. Simulations display the performance of the proposed method, in terms of MMSE and bit error rate (BER). Indeed, the difference of BER between our method and the theoretical LMMSE is less than 2dB.

Robust Training Sequence Design for Correlated MIMO Channel Estimation

We study how to design a worst-case robust training sequence for multiple-input multiple-output (MIMO) channel estimation. We consider mean-squared error of channel estimates as the figure of merit which is a function of second-order statistics of the MIMO channel, i.e., channel covariance matrix, in order to optimize training sequences under a total power constraint. In practical applications, the channel covariance matrix is not known perfectly. Thus the main aspect of our design is to improve robustness of the training sequences against possible uncertainties in the available channel covariance matrix. Using a deterministic uncertainty model, we formulate a robust training sequence design as a minimax optimization problem where we take such imperfections into account. We investigate the robust design problem assuming the general case of an arbitrarily correlated MIMO channel and a non-empty compact convex uncertainty set. We prove that such a problem admits a globally optimal solution by exploiting the convex-concave structure of the objective function, and propose numerical algorithms to address the robust training design problem. We proceed the analysis by considering multiple-input single-output (MISO) channels and Kronecker structured MIMO channels along with unitarily-invariant uncertainty sets. For these scenarios, we show that the problem is diagonalized by the eigenvectors of the nominal covariance matrices so that the robust design is significantly simplified from a complex matrix-variable problem to a real vector-variable power allocation problem. For the MISO channel, we provide closed-form solutions for the robust training sequences with the uncertainty sets defined by the spectral norm and nuclear norm.

Capacity Analysis of MIMO Spatial Channel Model using Novel Adaptive Semi Blind Estimation Scheme

Multiple Input Multiple Output (MIMO) has been remained in much importance in recent past because of high capacity gain over a single antenna system. In this article, analysis over the capacity of the MIMO channel systems with spatial channel has been considered when the channel state information (CSI) is considered imperfect or partial. The dynamic CSI model has also been tried consisting of channel mean and covariance which leads to extracting of the channel estimates and error covariance. Both parameters indicated the CSI quality since these are the functions of temporal correlation factor, and based on this, the model covers data from perfect to statistical CSI, either partial or full blind. It is found that in case of partial and imperfect CSI, the capacity depends on the statistical properties of the error in the CSI. Based on the knowledge of statistical distribution of the deviations in CSI knowledge, a new estimation approach which maximizes the capacity of spatial channel model has been tried. The interference interactively reduced by employing the iterative channel estimation and data detection approach.

Comparative analysis of reduced Kalman filters with guaranteed estimation quality

The paper gives a comparison of two methods for synthesis of reduced Kalman filters with guaranteed estimation quality. Their distinguishing feature is the fact that the matrix calculated in the covariance channel of a synthesized suboptimal filter is an upper bound for the real covariance matrix. In the first method, guaranteed estimation is provided due to the increase in disturbing noises, and in the second, due to two factors: the increase in disturbing noises and changes in the dynamics matrix, which improves the actual accuracy of filtering in some cases.

Techniques for MIMO Channel Covariance Matrix Quantization

In this correspondence, we develop techniques to efficiently quantize channel covariance matrices in multiple-input multiple-output (MIMO) Rayleigh fading environments. While these covariance matrices change less frequently than the channel matrices themselves, this information needs to be updated when it does change. Furthermore, these covariance matrices have significantly more parameters to quantize than their channel matrix counterparts. Since many applications focus on utilizing the strongest eigenmodes of the channel covariance matrix and since these matrices tend to be low-rank, we focus on efficiently quantizing the dominant eigenvectors of these matrices. We develop Lloyd-type algorithms based on training data from the environment to develop our codebooks. We also develop an algorithm based on the reduced-parameter Kronecker and Weichselberger models to generate codebooks with reduced real-time codeword search.

Precoder design in cognitive radio networks with channel covariance information

In cognitive radio communications, the coexistence of a secondary network is allowed as long as the interference power from the secondary network measured at the primary user (PU) is kept below a given threshold. In this paper, we consider a communication scenario for cognitive radio networks where the channel state information (CSI) between the secondary base station (SBS) and the PU is perfectly known at the SBS transmitter whereas only the covariance information is known for the channel of the SBS-secondary user (SU) link at the SBS transmitter. A closed-form precoder design method is proposed for the purpose of maximizing the upper bound of the expected achievable rate of the SU. Even though the proposed scheme uses only partial CSI of the SBS-SU link, the simulation results show that the expected achievable rate of the proposed scheme approaches to that of a conventional scheme with full CSI as the channel correlation increases.

Distributed superdense coding over noisy channels

We study multipartite superdense coding in the presence of a covariant noisy channel. We investigate the case of many senders and one receiver, considering both unitary and nonunitary encoding. We study the scenarios where the senders apply local encoding or global encoding. We show that, up to some preprocessing on the original state, the senders cannot do better encoding than local, unitary encoding. We then introduce general Pauli channels as a significant example of covariant maps. Considering Pauli channels, we provide examples for which the superdense coding capacity is explicitly determined.

On the Beamforming Capacity of MISO Channels

This letter addresses the calculation of the ergodic beamforming capacity of a multiple-input single-output (MISO) communication channel in the presence of additive Gaussian noise and known channel distribution information at the transmitter (CDIT). Necessary conditions to achieve the ergodic beamforming capacity are presented in the general case and derived explicitly when the CDIT is Gaussian. This case is commonly addressed in the literature as the joint channel mean information (CMI) and channel covariance information (CCI) case. An iterative algorithm based on the projected gradient approach is derived in order to find the beamforming capacity and the corresponding capacity-achieving beamforming vector.

A Coordinated Approach to Channel Estimation in Large-Scale Multiple-Antenna Systems

This paper addresses the problem of channel estimation in multi-cell interference-limited cellular networks. We consider systems employing multiple antennas and are interested in both the finite and large-scale antenna number regimes (so-called "massive MIMO"). Such systems deal with the multi-cell interference by way of per-cell beamforming applied at each base station. Channel estimation in such networks, which is known to be hampered by the pilot contamination effect, constitute a major bottleneck for overall performance. We present a novel approach which tackles this problem by enabling a low-rate coordination between cells during the channel estimation phase itself. The coordination makes use of the additional second-order statistical information about the user channels, which are shown to offer a powerful way of discriminating across interfering users with even strongly correlated pilot sequences. Importantly, we demonstrate analytically that in the large-number-of-antennas regime, the pilot contamination effect is made to vanish completely under certain conditions on the channel covariance. Gains over the conventional channel estimation framework are confirmed by our simulations for even small antenna array sizes.

Robust Transmitter Design for Amplify-and-Forward MIMO Relay Systems Exploiting Only Channel Statistics

In this paper, we address statistically robust transmit design problems that maximize the average mutual information of amplify-and-forward (AF) multiple-input multiple-output (MIMO) two-hop relay channels having perfect channel state information at the receiver (CSIR) and statistical channel state information at the transmitter (CSIT). In the selected scenario, the source, relay, and destination terminals are equipped with correlated antennas where a direct link between the source and the destination terminals can be found. Moreover, the statistical CSIT consists of the channel means and covariance matrices of various links. The design problem is taken into account for regimes in large systems because there are no closed-form expressions of the average mutual information in the MIMO relay channel. The contribution of this study includes the derivation of asymptotic mutual information expressions for the MIMO relay channel in the large system limit. Additionally, an efficient optimization algorithm based on the asymptotic mutual information is proposed in order to find the asymptotically optimal source covariance matrix and the relay amplifying matrix. Numerical simulation results show that the new approach achieves indistinguishable performance compared to those of the maximization approaches in finite-dimensional systems, even for a small number of antennas at each link. The impact of the CSI of various links on the throughput of the MIMO relay channel is studied using the new approach.

Interference mitigation techniques for clustered multicell joint decoding systems

Multicell joint processing has originated from information-theoretic principles as a means of reaching the fundamental capacity limits of cellular networks. However, global multicell joint decoding is highly complex and in practice clusters of cooperating Base Stations constitute a more realistic scenario. In this direction, the mitigation of intercluster interference rises as a critical factor towards achieving the promised throughput gains. In this paper, two intercluster interference mitigation techniques are investigated and compared, namely interference alignment and resource division multiple access. The cases of global multicell joint processing and cochannel interference allowance are also considered as an upper and lower bound to the interference alignment scheme, respectively. Each case is modelled and analyzed using the per-cell ergodic sum-rate throughput as a figure of merit. In this process, the asymptotic eigenvalue distribution of the channel covariance matrices is analytically derived based on free-probabilistic arguments in order to quantify the sum-rate throughput. Using numerical results, it is established that resource division multiple access is preferable for dense cellular systems, while cochannel interference allowance is advantageous for highly sparse cellular systems. Interference alignment provides superior performance for average to sparse cellular systems on the expense of higher complexity.

Beamforming Capacity Optimization for MISO Systems with Both Mean and Covariance Feedback

The beamforming capacity optimization problem in MISO systems, when the transmitter has both mean and covariance feedback of the channel, has been tackled only with the SNR maximization approach, which is known to give a sub-optimal solution. Numerical solutions of the full rank input covariance matrix, presented in the literature, are capable of tracking the beamforming vector only if it is the optimal capacity achieving solution. In this paper, we solve the beamforming capacity optimization problem by following an analytical approach that projects the beamforming vector on an orthonormal basis defined by the eigenvectors of the channel covariance matrix. The proposed formulation reduces the complexity of calculating the solution and provides intuition into the problem itself. In particular, we express the necessary conditions for beamforming capacity maximization as a system of two equations, which can be solved numerically very efficiently using the secant method. Surprisingly, our indicative numerical results for the 2 x 1 and 10 x 1 MISO systems, showed that for some cases the performance gain through beamforming capacity optimization compared to the SNR maximization approach can reach 0.4 bps/Hz. This means that the SNR maximization solution deviates considerably from the optimal beamforming vector. Finally, the optimality of the SNR maximization solution is also examined.

Hybrid multiple-input multiple-output scheme combining precoded and non-precoded transmission

A hybrid multiple-input multiple-output (MIMO) scheme capable of switching between non-precoded transmission and covariance-based precoded transmission is proposed. Such a hybrid scheme is desired as the covariance-based precoding gain is insignificant when transmission takes place in lowly correlated channels. Such a minor performance gain is not sufficient to justify the high-complexity precoding employed at the transmitter. The proposed scheme does not require a feedback channel or pilot symbols as knowledge of the downlink channel covariance and correlation is made available to the downlink transmitter through direct estimation from the uplink received signal energy. Performance studies carried out in both time division duplex and frequency division duplex systems reveal that the performance of the proposed scheme is on par with that of an existing MIMO precoding scheme with the added advantage of lower complexity and improved system efficiency.

Optimum Linear Design of Two-Hop MIMO Relay Networks With QoS Requirements

This paper investigates the optimal design issue of general two-hop amplify-and-forward (AF) MIMO relay networks with one source, one destination and multiple relays, each having multiple antennas. Beginning with the study of the mean-square error (MSE) matrix of the system and the use of Wiener filter for the destination processing, a constrained optimization problem with respect to the unknown relay precoder matrix is formulated with a goal of minimizing the total relay transmit power subject to the MSE-based quality of service (QoS) requirement on each data stream. As the original optimization problem is very difficult to solve, a lower bound of the cost function is pursued to achieve an approximate optimization problem. The modified problem is then solved under the perfect channel state information (CSI) condition using the diagonalization method along with the majorization theory to obtain optimal relay precoder with either nonoptimal or optimal source precoding. It is shown that a convex relay precoding solution is available in the case of nonoptimal source precoding whereas a joint optimal source and relay precoder can be obtained through a multilevel waterfilling algorithm. The optimal design of the relay network under imperfect CSI condition is also considered by modifying the optimization problem with the channel mean and covariance matrix, leading to a robust optimization design of the relay precoder. The proposed technique is validated by Monto Carlo simulations and is also compared with one existing method.