Precoding Matrices Research Articles

Precoding method interference management for quasi-EVD channel.

The Cholesky decomposition-block diagonalization (CD-BD) interference alignment (IA) for a multiuser multiple input multiple output (MU-MIMO) relay system is proposed, which designs precoders for the multiple access channel (MAC) by employing the singular value decomposition (SVD) as well as the mean square error (MSE) detector for the broadcast Hermitian channel (BHC) taken advantage of in our design. Also, in our proposed CD-BD IA algorithm, the relaying function is made use to restructure the quasieigenvalue decomposition (quasi-EVD) equivalent channel. This approach used for the design of BD precoding matrix can significantly reduce the computational complexity and proposed algorithm can address several optimization criteria, which is achieved by designing the precoding matrices in two steps. In the first step, we use Cholesky decomposition to maximize the sum-of-rate (SR) with the minimum mean square error (MMSE) detection. In the next step, we optimize the system BER performance with the overlap of the row spaces spanned by the effective channel matrices of different users. By iterating the closed form of the solution, we are able not only to maximize the achievable sum-of-rate (ASR), but also to minimize the BER performance at a high signal-to-noise ratio (SNR) region.

Energy Efficiency Optimization in Relay-Assisted MIMO Systems With Perfect and Statistical CSI

A framework for energy-efficient resource allocation in a single-user, amplify-and-forward (AF), relay-assisted, multiple-input-multiple-output (MIMO) system is devised in this paper. Previous results in this area have focused on rate maximization or sum power minimization problems, whereas fewer results are available when bits/Joule energy efficiency (EE) optimization is the goal. Here, the performance metric to optimize is the ratio between the system's achievable rate and the total consumed power. The optimization is carried out with respect to the source and relay precoding matrices, subject to quality-of-service (QoS) and power constraints. Such a challenging non-convex optimization problem is tackled by means of fractional programming and alternating maximization algorithms, for various channel state information (CSI) assumptions at the source and relay. In particular the scenarios of perfect CSI and those of statistical CSI for either the source-relay or the relay-destination channel are addressed. Moreover, sufficient conditions for beamforming optimality are derived, which is useful in simplifying the system design. Numerical results are provided to corroborate the validity of the theoretical findings.

Design of Robust Transceiver for Precoded Uplink MU-MIMO Transmission in Limited Feedback System

This paper proposes a robust transceiver design against the effect of channel state information (CSI) estimation error to optimize precoded uplink (UL) multi-user multiple-input multiple-output (MU-MIMO) transmission in limited feedback system under the consideration of the least-square technique on CSI estimation. To improve this limited feedback precoding, the constrained minimum variance (MV) approach with quadratic form to realize the computationally-efficient optimization problem, advantageously invoking the characteristics of the CSI estimation error, is proposed to suppress the effect of CSI estimation error, multiple user interference and noise. According to the Lagrange multiplier method on this MV approach, the deterministic function to resist uncertain CSI can be obtained to optimize design of the precoder and adaptive matrices jointly. With these optimum adaptive and precoder matrices, an optimum robust weighting matrix can be obtained to facilitate the user-wise detection in precoded UL MU-MIMO system. Performance analysis shows that the proposed robust weighting matrix is an unbiased design and it also can regularize the diagonal loading factor technique, and the detection performance of the proposed robust transceiver design can be predicted simplistically by applying our derived signal-to-interference-plus-noise ratio formulation. Computer simulations are conducted to confirm the efficacy of the proposed design in both perfect and imperfect CSI estimation.

Spatial Hopping in MIMO Systems for Impeded Signal Reception by Multi-Element Eavesdroppers

In multiple-input multiple-output (MIMO) systems employing precoding, knowledge of the precoding matrices used is essential for successful reception. Herein, a method is proposed to reduce the ability of a third-party eavesdropper to successfully receive precoded MIMO signals. Rather than precoding with the exact precoding matrices chosen by the intended receiver, transmission occurs by sequentially precoding using a set of precoding matrices clustered around the chosen precoding matrix. This is shown to significantly degrade eavesdropper reception while maintaining performance for the desired user.

Precoding for Outage Probability Minimization on Block Fading Channels

The outage probability limit is a fundamental and achievable lower bound on the word error rate of coded communication systems affected by fading. This limit is mainly determined by two parameters: the diversity order and the coding gain. With linear precoding, full diversity on a block fading channel can be achieved without error-correcting code. However, the effect of precoding on the coding gain is not well known, mainly due to the complicated expression of the outage probability. Using a geometric approach, this paper establishes simple upper bounds on the outage probability, the minimization of which yields to precoding matrices that achieve very good performance. For discrete alphabets, it is shown that the combination of constellation expansion and precoding is sufficient to closely approach the minimum possible outage achieved by an i.i.d. Gaussian input distribution, thus essentially maximizing the coding gain.

Linear Precoding with Resource Allocation for MIMO Relay Channels

This paper considers a half-duplex relay channel with a single source, relay, and destination, where each node has multiple antennas and the relay operates in decode-and-forward (DF) mode. The additional degrees of freedoms introduced by the MIMO channels entail increased complexity in comparison with the single antenna case. We propose a new transmission strategy for the relay channel that is able to take advantage of the MIMO gains while employing practical techniques that help reduce complexity. In the proposed scheme, the source splits its message in two parts and sends them to the destination, one part directly and the other with help from the relay. For such a transmission strategy, we formulate the problem of obtaining the rate-maximizing linear precoding matrices with time, power, and spatial stream allocation and transform the problem into a convex form. We also propose a suboptimal algorithm that provides simplifying expressions to solve the resulting problem with far less computational complexity. The numerical results show that the achievable rate of our scheme is greater than the DF rate in certain scenarios, as well as that of other practical existing strategies. In addition, the rate obtained by the suboptimal method approximates the optimal rate extremely well in all considered scenarios.

Information Leakage Neutralization for the Multi-Antenna Non-Regenerative Relay-Assisted Multi-Carrier Interference Channel

In heterogeneous dense networks where spectrum is shared, users' privacy remains one of the major challenges. On a multi-antenna relay-assisted multi-carrier interference channel, each user shares the spectral and spatial resources with all other users. When the receivers are not only interested in their own signals but also in eavesdropping other users' signals, the cross talk on the spectral and spatial channels becomes information leakage. In this paper, we propose a novel secrecy rate enhancing relay strategy that utilizes both spectral and spatial resources, termed as information leakage neutralization. To this end, the relay matrix is chosen such that the effective channel from the transmitter to the colluding eavesdropper is equal to the negative of the effective channel over the relay to the colluding eavesdropper and thus the information leakage to zero. Interestingly, the optimal relay matrix in general is not block-diagonal which encourages users' encoding over the frequency channels. We proposed two information leakage neutralization strategies, namely efficient information leakage neutralization (EFFIN) and local-optimized information leakage neutralization (LOPTIN). EFFIN provides a simple and efficient design of relay processing matrix and precoding matrices at the transmitters in the scenario of limited power and computational resources. LOPTIN, despite its higher complexity, provides a better sum secrecy rate performance by optimizing the relay processing matrix and the precoding matrices jointly. The proposed methods are shown to improve the sum secrecy rates over several state-of-the-art baseline methods.

Adaptive Pilot-Symbol Patterns for MIMO OFDM Systems

Recent standards for cellular transmission systems offer a lot of flexibility, such as the choice of transmission modes, modulation alphabets, coding rates, and precoding matrices. Despite this trend, pilot-symbol patterns in today's standards remain fixed, although such an approach is suboptimal. In this paper, we show how to design optimal pilot-symbol patterns by maximizing an upper bound of a constrained capacity that takes channel estimation errors and Inter Carrier Interference into account. Furthermore, we propose adaptive pilot-symbol patterns that follow changing channel statistics. As a proof of concept, we present throughput simulation results of two competitive systems, a transmission system compliant with the Long Term Evolution (LTE)-standard and an improved system utilizing the proposed adaptive pilot patterns. The transmission system utilizing adaptive pilot patterns outperforms an LTE-standard compliant system in all considered scenarios. The throughput gain for a single input single output system ranges between 3% and 80%. For a 4 × 4 transmission system, the performance gain is significantly higher and can reach up to 850% compared to a conventional LTE system.

Transceiver Design for SC-FDE Based MIMO Relay Systems

In this paper, we propose a joint transceiver design for single-carrier frequency-domain equalization (SC-FDE) based multiple-input multiple-output (MIMO) relay systems. To this end, we first derive the optimal minimum mean-squared error linear and decision-feedback frequency-domain equalization filters at the destination along with the corresponding error covariance matrices at the output of the equalizer. Subsequently, we formulate the source and relay precoding matrix design problem as the minimization of a family of Schur-convex and Schur-concave functions of the mean-squared errors at the output of the equalizer under separate power constraints for the source and the relay. By exploiting properties of the error covariance matrix and results from majorization theory, we derive the optimal structures of the source and relay precoding matrices, which allows us to transform the matrix optimization problem into a scalar power optimization problem. Adopting a high signal-to-noise ratio approximation for the objective function, we obtain the global optimal solution for the power allocation variables. We illustrate the excellent performance of the proposed system and compare it to that of conventional orthogonal frequency-division multiplexing MIMO relay systems based on computer simulations.

Linear Precoder Design for MIMO Interference Channels with Finite-Alphabet Signaling

This paper investigates the linear precoder design for $K$-user interference channels of multiple-input multiple-output (MIMO) transceivers under finite alphabet inputs. We first obtain general explicit expressions of the achievable rate for users in the MIMO interference channel systems. We study optimal transmission strategies in both low and high signal-to-noise ratio (SNR) regions. Given finite alphabet inputs, we show that a simple power allocation design achieves optimal performance at high SNR whereas the well-known interference alignment technique for Gaussian inputs only utilizes a partial interference-free signal space for transmission and leads to a constant rate loss when applied naively to finite-alphabet inputs. Moreover, we establish necessary conditions for the linear precoder design to achieve weighted sum-rate maximization. We also present an efficient iterative algorithm for determining precoding matrices of all the users. Our numerical results demonstrate that the proposed iterative algorithm achieves considerably higher sum-rate under practical QAM inputs than other known methods.

Pair-Aware Interference Alignment in Multi-User Two-Way Relay Networks

In this paper, K bidirectionally communicating node pairs with each node having N antennas and one amplify and forward relay having R antennas are considered. Each node wants to transmit d data streams to its communication partner. Taking into account that each node can perform self interference cancellation, a new scheme called Pair-Aware Interference Alignment is proposed. In this scheme, the transmit precoding matrices and the relay processing matrix are chosen in such a way that at any given receiver all the interfering signals except the self interference are within the interference subspace and the useful signal is in a subspace linearly independent of the interference subspace. If the number of variables is larger than or equal to the number of constraints in the system, the system is classified as proper, else as improper. Through simulations it is shown that for a proper system (2Kd≤2N+R-d), interferences can be perfectly aligned and the useful signals can be decoded interference-free at the receivers. An iterative algorithm to achieve the interference alignment solution is proposed. Also for the proper system fulfilling a certain additional condition, which will be derived in this paper, a closed form solution is proposed.

Optimal precoder design for non-regenerative multiple-input multiple-output cognitive relay systems with perfect and imperfect channel state information

This paper studies optimal precoder design for non-regenerative multiple-input multiple-output MIMO cognitive relay systems, where the secondary user SU and relay station RS share the same spectrum with the primary user PU. We aim to maximize the system capacity subject to the transmit power constraints at the SU transmitter SU-Tx and RS, and the interference power constraint at the PU. We jointly optimize precoders for the SU-Tx and RS with perfect and imperfect channel state information CSI between the SU-Tx/RS and PU, where our design approach is based on the alternate optimization method. With perfect CSI, we derive the optimal structures of the RS and SU-Tx precoding matrices and develop the gradient projection algorithm to find numerical solution of the RS precoder. Under imperfect CSI, we derive equivalent conditions for the interference power constraints and convert the robust SU-Tx precoder optimization into the form of semi-definite programming. For the robust RS precoder optimization, we relax the interference power constraint related with the RS precoder to be convex by using an upper bound and apply the gradient projection algorithm to deal with it. Simulation results demonstrate the effectiveness of the proposed schemes. Copyright © 2013 John Wiley & Sons, Ltd.

Application of Analog Network Coding to MIMO Two-Way Relay Channel in Cellular Systems

An efficient analog network coding transmission protocol is proposed in this letter for a MIMO two way cellular network. Block signal alignment is first proposed to null the inter-user interference for multi-antenna users, which makes the dimensions of aligned space larger compared with the existing signal alignment. Two algorithms are developed to jointly design the precoding matrices at the relay and BS for outage optimization. Especially, the last algorithm is designed to maximize the effective channel gain to the effective noise gain ratio. The performance of this transmission protocol is also verified by simulations.

Precoding Design of MIMO AF Two-Way Multiple-Relay Systems

The precoding joint optimization is proposed for multiple-input multiple-output (MIMO) two-way multiple-relay systems with amplify-and-forward (AF) strategy in correlated fading channels. The terminal transceivers and relay precoding matrices are designed based on the minimum mean square error (MMSE) criterion. This combined design of precoding matrices and equalizers under power constraints belongs to neither concave nor convex so that the nonlinear matrix-form conjugate gradient (MCG) algorithm is developed to acquire the local optimal solutions. Simulation results demonstrate that the proposed precoding joint architecture possesses a superior bit-error-rate (BER) performance and a faster convergence rate.

Practical linear precoder design for finite alphabet multiple‐input multiple‐output orthogonal frequency division multiplexing with experiment validation

A low complexity precoding method is proposed for practical multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems. Based on the two-step optimal precoder design algorithm that maximises the lower bound of the mutual information with finite-alphabet inputs, the proposed method simplifies the precoder design by fixing the right singular vectors of the precoder matrix, eliminating the iterative optimisation between the two steps, and discretising the search space of the power allocation vector. For a 4 × 4 channel, the computational complexity of the proposed precoder design is reduced to 3 and 6% of that required by the original two-step algorithm with quadrature phase shift keying (QPSK) and 8 phase-shift keying (8PSK), respectively. The proposed method achieves nearly the same mutual information as the two-step iterative algorithm for a large range of signal-to-noise ratio (SNR) region, especially for large MIMO size and/or high constellation systems. The proposed precoding design method is applied to a 2 × 2 MIMO-OFDM system with 2048 subcarriers by designing 1024 precoders for extended channel matrices of size 4 × 4. A transceiver test bed implements these precoding matrices in comparison with other existing precoding schemes. Indoor experiments are conducted for fixed-platform non-line-of-sight channels, and the data processing results show that the proposed precoding method achieves the lowest bit error rate compared with maximum diversity, classic water-filling and channel diagonalisation methods.

Joint Precoder Design for Distributed Transmission of Correlated Sources in Sensor Networks

We consider the problem of transmitting multiple spatially distributed correlated sources to a common destination (e.g. a fusion center or an access point) in wireless sensor networks (WSNs). The correlated data from multiple sensors are jointly transmitted to the destination via orthogonal channels. We assume that the channel between each sensor and the receiver is multiple-input multiple-output (MIMO), with each sensor and the receiver equipped with multiple transmit/receive antennas. In this framework, we study the problem of joint linear precoder design for all sensors by assuming the knowledge of the instantaneous channel state information (CSI), aiming at maximizing the mutual information between the sources and the received signals at the destination. We propose a Gauss-Seidel iterative approach which successively optimizes the precoding matrix associated with each sensor, while fixing the other precoding matrices. Numerical results show that the proposed algorithm that takes into account the spatial correlation across sensors can achieve higher capacity than conventional methods that neglect the spatial correlation.

8-PSK 성운을 이용하는 이중계층 차분 선부호화 기법의 성능 분석

LTE 및 LTE-A 시스템에서 8-PSK 성운을 알파벳으로 가지는 이중계층 차분 코드북을 제안한다. 인접하는 무선 채널의 시간 상관으로 선부호화 행렬은 천천히 변화하므로, 무선 채널 공간의 전체를 양자화 하지 않고, 시간 상관에 따른 채널 공간의 일부분의 차분 성분만을 양자화하여 피드백을 한다면, 기존과 동일한 크기의 코드북으로도 가상적으로는 코드북이 매우 커지는 효과를 얻을 수 있어 채널 용량이 증가한다. 특히 제안하는 코드북은 LTE release-8의 코드북 설계조건인 8-PSK 성운을 사용하는 동 이득 특성 및 이중 계층 코드북이 단일 계층 코드북을 포함하는 특성을 가지는 새로운 차분 코드북이다. 코드북 내 코드워드들의 인자들이 8-PSK 성운만을 사용하므로, 선부호화 및 복호 시에 계산량이 낮아지는 LTE 코드북의 장점을 그대로 유지할 수 있다. 또한 동 이득 성질은 상대적으로 저렴한 비선형 증폭기를 사용할 수 있는 장점이 있어 가격의 제한을 받는 단말기 설계에는 필수적인 요소이다. 컴퓨터 시뮬레이션을 통한 이중계층 선부호화 기법의 성능 분석에서, 동일한 피드백 비트 수를 갖는 같은 크기의 코드북에서 제안하는 차분 코드북은 정상상태에서 기존 LTE 코드북보다 최소 1.2dB 성능 향상을 보인다. Dual-layer differential codebook using 8-PSK (phase shift keying) constellation as its codeword elements, is proposed for Long term evolution (LTE) and/or LTE-Advanced systems. Due to the temporal correlation of the adjacent channel matrices, the consecutive precoding matrices are likely to be similar. This approach quantize only the differential information of the channel instead of the whole channel subspace, which virtually increase the codebook size to realize more accurate quantization of the channel. Especially, the proposed codebook has the same properties of LTE release-8 codebook that is, constant modulus, complexity reduction, and nested property. The mobile station can be designed by using less expensive non-linear amplifier utilizing constant modulus property. Computer simulations show that the capacity of the proposed dual-layer codebook performs almost 1.2dB better than those of any other non-differential codebooks with the same amount of feedback information.

Power Minimization in MIMO Cognitive Networks using Beamforming Games

We consider a multi-channel multi-user cognitive radio MIMO network in which each node controls its antenna radiation directions and allocates power for each data stream by adjusting its precoding matrices. Under a noncooperative game, we optimize the set of precoding matrices (one per channel) at each node so as to minimize the total transmit power in the network. Using recession analysis and the theory of variational inequalities, we obtain sufficient conditions that guarantee the existence and uniqueness of the game's Nash Equilibrium (NE). Low-complexity distributed algorithms are also developed by exploiting the strong duality of the convex per-user optimization problem. To improve the efficiency of the NE, we introduce pricing policies that employ a novel network interference function. Existence and uniqueness of the new NE under pricing are studied. Simulations confirm the effectiveness of our joint optimization approach.

On the Impact of Low-Rank Interference on the Post-Equalizer SINR in LTE

The standardization of the fourth generation of mobile communication systems was mainly driven by the demands for higher data-rates and improved Quality of Service. To reach these goals interference coordination has been identified as a promising research field for better exploitation of the time and frequency resources. This paradigm shift from interference avoidance to interference coordination is also reflected in the ongoing enhancement of the 4th generation of mobile communication systems such as 3GPP Long Term Evolution. In this context, numerous investigations have focused on the allocation of precoding matrices that are part of the link adaptation process by some form of base station (eNB) coordination. Within this work we develop a non-centralized interference coordination scheme by noticing that the re-allocation of a precoding matrix can lead to an uncontrolled change of the interference level at users located in neighboring cells, especially at the edge. To this end, we provide a fully closed form mathematical framework describing these changes. Based on this, we derive a simple metric that improves the precoding matrix selection process in the User Equipment with the result that interference changes can be reduced without having any standard impact. This novel scheme can also be seen as an extension to previous inter-cell interference coordination schemes without the need of base-station cooperation.

Iterative D.C. Optimization of Precoding in Wireless MIMO Relaying

Optimizations of precoding matrices in precode-and-forward (PF) MIMO relaying are nonconvex programs in precoding matrix variables. The semidefinite relaxation (SDR) technique, which relaxes the concerned nonconvex quadratic constraints by (convex) semi-definite ones, can locate the optimal solutions, provided that the numbers of relaying antennas and users are very small. The computational complexity of the SDR grows explosively even with a very moderate increase in the numbers of relaying antennas and/or users, making the existing semidefinite programming (SDP) solvers incapable. In this paper, much more efficient problem formulations of precoding matrix design that exploit the spectral matrix optimization are developed. Such formulations have a low dimensionality and are computationally-tractable nonconvex matrix programs. Furthermore, by exploiting their partial convex structures in the d.c. (difference of two convex functions) framework, new effective iterative solutions are obtained. Extensive simulation results are presented to support the computational advantage of the proposed approach and show that the proposed approach can effectively handle all three considered optimization problems of precoding matrices in MIMO PF relaying, while the SDR approach either is computationally impractical or fails.