Linear Precoder Design Research Articles

An optimized non-unitary linear precoding design for OSTBCs

This paper tackles the optimization of non-unitary linear precoding design for orthogonal space-time block codes (OSTBCs). We dig out the transmission potentials by the analysis from eigen-space point of view according to the unique structure of OSTBCs. The proposed precoding form is proven to be theoretically optimized. Compared with the classical unitary Grassmannian codebook design, the non-unitary codebook further improves the overall performance of practical systems. The constraint on codebook size to guarantee full diversity order is given and proven. Based on this, we investigate the codebook construction method combined with a water-filling (WF) technique to reduce the impact of antenna correlation. The advantages of the novel design are verified via numerical simulations.

Linear precoder designs for K-user interference channels

This paper studies linear precoding and decoding schemes for K-user interference channel systems. It was shown by Cadambe and Jafar that the interference alignment (IA) algorithm achieves a theoretical bound on degrees of freedom (DOF) for interference channel systems. Based on this, we first introduce a non-iterative solution for the precoding and decoding scheme. To this end, we determine the orthonormal basis vectors of each user's precoding matrix to achieve the maximum DOF, then we optimize precoding matrices in the IA method according to two different decoding schemes with respect to individual rate. Second, an iterative processing algorithm is proposed which maximizes the weighted sum rate. Deriving the gradient of the weighted sum rate and applying the gradient descent method, the proposed scheme identifies a local-optimal solution iteratively. Simulation results show that the proposed iterative algorithm outperforms other existing methods in terms of sum rate. Also, we exhibit that the proposed non-iterative method approaches a local optimal solution at high signal-to-noise ratio with reduced complexity.

Adaptive antenna selection at mobile stations for SDMA in WiMAX networks

AbstractThe IEEE 802.16/WiMAX standard has fully embraced multi‐antenna technology and can, thus, deliver robust performance and high transmission rates. Nevertheless, due to its inherent cost concerns, a WiMAX mobile station (MS) should preferably contain fewer radio frequency (RF) chains than antenna elements. This is because the former is often substantially more expensive than the latter. Thus, antenna selection, wherein a subset of antennas is dynamically selected to connect to the limited number of RF chains for transceiving, is a highly appealing performance enhancement technique for multi‐antenna WiMAX terminals. In this paper, a novel protocol for antenna selection in space division multiple access (SDMA) transmission is developed for the next‐generation IEEE 802.16 mobile stations. Both locally and globally optimal selection rules are considered together with various linear precoding designs at the base station (BS). The proposed protocol can readily accommodate various channel situations (e.g., reciprocal and non‐reciprocal channels). As demonstrated by analysis and simulation, the proposed protocol delivers considerable performance improvement over conventional IEEE 802.16 terminals that lack antenna selection capability. Moreover, the proposed protocol leverages the existing signaling method defined in IEEE 802.16, thereby incurring a negligible signaling overhead and requiring only minimal modifications of the standard. To the best of our knowledge, this paper represents the first effort to support antenna selection capability for SDMA transmissions in IEEE 802.16 mobile stations. Copyright © 2009 John Wiley & Sons, Ltd.

Design of Multiuser Downlink Linear MIMO Precoding Systems With Quantized Feedback

We treat the problem of sum-rate maximization via scheduling and linear precoding in multiuser (MU) multiple-input-multiple-output (MIMO) systems with quantized feedback. We formulate the optimal quantized linear precoder design problem and provide the numerical procedure for finding the solution. We also provide a simple scheduling scheme that exploits the MU diversity gain. With the aim of sum-rate maximization, we further address the quantization codebook design based on the capacity measure by introducing a new distance metric. It is demonstrated that, with or without scheduling, the proposed optimal linear precoding scheme achieves significant gain over the conventional linear precoder designs with similar feedback overhead, such as the zero-forcing precoder. Moreover, although the proposed quantization codebook design improves the performance of other existing MU-MIMO precoding schemes, the performance gain offered by it is typically much higher when it is coupled with the proposed linear precoding design.

Exploiting Quantized Channel Norm Feedback Through Conditional Statistics in Arbitrarily Correlated MIMO Systems

In the design of narrowband multi-antenna systems, a limiting factor is the amount of channel state information (CSI) available at the transmitter. This is especially evident in multi-user systems, where the spatial user separability determines the multiplexing gain, but it is also important for transmission-rate adaptation in single-user systems. To limit the feedback load, the unknown and multi-dimensional channel needs to be represented by a limited number of bits. When combined with long-term channel statistics, the norm of the channel matrix has been shown to provide substantial CSI that permits efficient user selection, linear precoder design, and rate adaptation. Herein, we consider quantized feedback of the squared Frobenius norm in a Rayleigh fading environment with arbitrary spatial correlation. The conditional channel statistics are characterized and their moments are derived for both identical, distinct, and sets of repeated eigenvalues. These results are applied for minimum mean square error (MMSE) estimation of signal and interference powers in single- and multi-user systems, for the purpose of reliable rate adaptation and resource allocation. The problem of efficient feedback quantization is discussed and an entropy-maximizing framework is developed where the post-user-selection distribution can be taken into account in the design of the quantization levels. The analytic results of this paper are directly applicable in many widely used communication techniques, such as space-time block codes, linear precoding, space division multiple access (SDMA), and scheduling.

Optimized non-unitary linear precoding for orthogonal space-time block codes

This letter proposes a novel optimized non-unitary linear precoding design for orthogonal space-time block codes (OSTBCs). We dig out the transmission potentials by the analysis from eigen-space point of view according to the unique structure of OSTBCs. The proposed precoding form is proved to be theoretically optimized. Compared with the classical unitary Grassmannian codebook design, in the sense of restriction on the time average power of transmit signal, the proposed non-unitary codebook further improves the overall performance of practical systems. The new constraint on codebook size to guarantee full diversity order is given and proved. The advantages of our proposed design are verified in the numerical simulations.

Linear precoding and finite rate feedback design for V-BLAST architecture

Linear precoding is a simple method to improve the performance of V-BLAST architecture. However, it requires perfect channel state information at the transmitter, which necessitates a great amount of feedback. In this paper, we propose a linear precoding scheme for the V-BLAST architecture with an MMSE-SIC receiver. The whole scheme includes the design of an unquantized precoder and its finite rate feedback. The unquantized precoder, consisting of a beamforming matrix and a power allocation matrix, is designed by maximizing the minimum post-detection SINR of the MMSE-SIC receiver. In addition to the optimal precoder, we give a suboptimal solution which is shown to be asymptotically optimal at low SNR. The advantage of this suboptimal solution is that its extension to finite rate feedback is straightforward. Specifically, beamforming vectors are selected sequentially from a predetermined codebook and the indices are fed back. Besides, elements of the power allocation matrix are scalar quantized and fed back. Theoretical analysis reveals that the unquantized beamforming vectors are uniformly distributed on the unit sphere, leading to a very natural codebook design criterion - the Grassmannian criterion. Moreover, the proposed scheme is flexible enough to allow for a change in the number of data streams without redesigning the codebook. Simulation results are presented to demonstrate the effectiveness of the proposed scheme.

Linear Precoding for MIMO Channels with Outdated Channel State Information in Multiuser Space-Time Block Coded Systems with Multi-Packet Reception

We propose a joint set of linear precoder designs for single cell uplink multiuser space-time block coded multiple- input multiple-output systems with multi-packet reception by exploiting outdated channel state information. By deriving the pairwise error probability with respect to both minimum and average codeword distance design metrics, we formulate the design as an optimization problem subject to transmit power constraint for each user. Due to the non-convex nature of the optimization problem, we devise an iterative algorithm to solve for linear precoding structure for general space-time block code. For orthogonal space-time block code, we also propose a simplified distributed algorithm to solve for a closed-form solution. Asymptotic analysis on the effect of quality of the outdated channel state information (CSI) on the precoder structure is also presented. Simulation results are provided to demonstrate the effectiveness of the proposed designs for different space-time block codes at various CSI qualities.

Progressive Linear Precoder Optimization for MIMO Packet Retransmissions Exploiting Channel Covariance Information

This work investigates the design of linear precoders for ARQ packet retransmissions in multi-input multi-output (MIMO) systems. We consider transmitter precoder design based on partial MIMO channel information in the form of their covariance feedback. Our objective is to maximize the ergodic mutual information provided by multiple (re)transmissions of a packet subject to transmission power constraint. We propose a set of near-optimal successive linear ARQ precoders for flat fading MIMO channels. This progressive linear ARQ precoder combines the appropriate power loading and the reverse-order pairing of singular values in the current retransmission with previous transmissions. This reverse-order pairing is a special feature unique to our sequential ARQ preceding approach with demonstrated performance gains.

Symbol Error-Rate Analysis of OSTB Codes and Linear Precoder Design for MIMO Correlated Keyhole Channels

Analytical expressions for the symbol error rate (SER) of MIMO systems are important from both design and analysis point of views. This paper derives exact and easy to evaluate analytical expression for the SER of an orthogonal space-time block coded (OSTBC) MIMO system with spatially correlated antennas, in which the channel signal propagation suffers from a degenerative effect known as a keyhole. These expressions are valid for complex valued correlations between antenna elements. Numerical results obtained by simulations are presented to confirm the validity of the analytical SER expressions for several multilevel modulation schemes. Subsequently, a procedure for designing a linear precoder which exploits the knowledge of the channel correlation matrix at the transmitter to enhance the performance of the above systemis given. To this end, the exact SER expression is minimized by using a gradient descent algorithm. To demonstrate the performance improvements achievable with the proposed precoder, numerical results obtained in several design examples are presented and compared.

Convex Conic Formulations of Robust Downlink Precoder Designs With Quality of Service Constraints

We consider the design of linear precoders (beamformers) for broadcast channels with Quality of Service (QoS) constraints for each user, in scenarios with uncertain channel state information (CSI) at the transmitter. We consider a deterministically-bounded model for the channel uncertainty of each user, and our goal is to design a robust precoder that minimizes the total transmission power required to satisfy the users' QoS constraints for all channels within a specified uncertainty region around the transmitter's estimate of each user's channel. Since this problem is not known to be computationally tractable, we will derive three conservative design approaches that yield convex and computationally-efficient restrictions of the original design problem. The three approaches yield semidefinite program (SDP) formulations that offer different trade-offs between the degree of conservatism and the size of the SDP. We will also show how these conservative approaches can be used to derive efficiently-solvable quasi-convex restrictions of some related design problems, including the robust counterpart to the problem of maximizing the minimum signal-to-interference-plus-noise-ratio (SINR) subject to a given power constraint. Our simulation results indicate that in the presence of uncertain CSI the proposed approaches can satisfy the users' QoS requirements for a significantly larger set of uncertainties than existing methods, and require less transmission power to do so.

Eigenstructures of MIMO Fading Channel Correlation Matrices and Optimum Linear Precoding Designs for Maximum Ergodic Capacity

The ergodic capacity of MIMO frequency-flat and -selective channels depends greatly on the eigenvalue distribution of spatial correlation matrices. Knowing the eigenstructure of correlation matrices at the transmitter is very important to enhance the capacity of the system. This fact becomes of great importance in MIMO wireless systems where because of the fast changing nature of the underlying channel, full channel knowledge is difficult to obtain at the transmitter. In this paper, we first investigate the effect of eigenvalues distribution of spatial correlation matrices on the capacity of frequency-flat and -selective channels. Next, we introduce a practical scheme known as linear precoding that can enhance the ergodic capacity of the channel by changing the eigenstructure of the channel by applying a linear transformation. We derive the structures of precoders using eigenvalue decomposition and linear algebra techniques in both cases and show their similarities from an algebraic point of view. Simulations show the ability of this technique to change the eigenstructure of the channel, and hence enhance the ergodic capacity considerably.

Optimal linear precoders for MIMO wireless correlated channels with nonzero mean in space-time coded systems

This paper proposes linear precoder designs exploiting statistical channel knowledge at the transmitter in a multiple-input multiple-output (MIMO) wireless system. The paper focuses on channel statistics, since obtaining real-time channel state information at the transmitter can be difficult due to channel dynamics. The considered channel statistics consist of the channel mean and transmit antenna correlation. The receiver is assumed to know the instantaneous channel precisely. The precoder operates along with a space-time block code (STBC) and aims to minimize the Chernoff bound on the pairwise error probability (PEP) between a pair of block codewords, averaged over channel fading statistics. Two PEP design criteria are studied-minimum distance and average distance. The optimal precoder with an orthogonal STBC is established, using a convex optimization framework. Different relaxations then extend the solution to systems with nonorthogonal STBCs. In both cases, the precoder is a function of both the channel mean and the transmit correlation. A linear precoder acts as a combination of a multimode beamformer and an input shaping matrix, matching each side to the channel and to the input signal structure, respectively. Both the optimal beam direction and the power of each mode, obtained via a dynamic water-filling process, depend on the signal-to-noise ratio (SNR). Asymptotic analyses of the results reveal that, for all STBCs, the precoder approaches a single-mode beamformer on the dominant right singular vector of the channel mean as the channel K factor increases. On the other hand, as the SNR increases, it approaches an equipower multiple-mode beamformer, matched to the eigenvectors of the transmit correlation. Design examples and numerical simulation results for both orthogonal and nonorthogonal STBC precoding solutions are provided, illustrating the precoding array gain.

Progressive linear precoder optimization for MIMO packet retransmissions

This paper investigates the optimal linear precoder design for packet retransmissions in multi-input-multi-output (MIMO) systems. To fully utilize the time diversity provided by automatic repeat request (ARQ), we derive a sequence of successive optimal linear ARQ precoders for flat fading MIMO channels, which minimize the mean-square error between the transmitted data and the joint receiver output. The optimization is subject to an overall transmit power constraint. This progressive linear ARQ precoder combines the appropriate power loading and the optimal pairing of channel matrix singular values in the current retransmission with previous transmissions. This optimal pairing is a special feature unique to our sequential ARQ precoding approach. Simulation results demonstrate the effectiveness of this optimized ARQ precoding in reducing symbol MSE and detection bit-error rate.

Joint linear optimization with receiver constraint for MIMO systems

In this article, we propose a new linear multiple-input multiple-output (MIMO) system with the receiver matched to the transmit filter plus channel, leading to a linear transmission system which diagonalizes the MIMO channel into its eigenspace similar to joint transmit and receive filter optimization (joint TX/RX optimization). The proposed transmission scheme is shown to achieve comparable bit-error performance to joint TX/RX optimization in the case of adaptive modulation. Furthermore, the proposed scheme outperforms joint TX/RX optimization in terms of complexity: if neither the possibility of feedback, nor sufficient computational resources at both sides of the link are available, the joint TX/RX optimization approach is not applicable contrary to the proposed approach. We investigate different optimization criteria well known from linear precoder design and joint TX/RX optimization: maximizing the signal-to-noise ratio, removing the interference, and minimizing the mean square error. Closed-form solutions are derived for these cases.

Linear transmitter precoding design for downlink of multiuser MIMO systems

Presented is the design of a set of linear transmitter precoding vectors for the downlink of a multiuser MIMO system, using the maximum signal-to-jamming and noise ratio for each user criterion, subject to a transmit power constraint. The proposed scheme does not impose any restrictions on the number of transmit antennas as do many conventional methods. Simulation results have shown the superiority of the proposed scheme compared to the conventional transmitter precoding schemes.

Design and analysis of linear precoders under a mean square error criterion, Part I: foundations and worst case designs

Much debate surrounds the pros and cons of linear precoders in wireless communication systems. This two-part paper contributes to the debate by formulating the precoder design problem as an optimisation problem and studying the optimal solutions, thereby gaining a better understanding of how precoders work and what they can do. Part I builds a mathematical foundation for the study of linear precoders. It is shown that under a mean square error criterion, a natural convex geometry arises. This geometry facilitates the derivation of a necessary and sufficient condition for a linear precoder to be maximally robust, meaning the precoder's worst case performance is no worse than that of any other linear precoder. Part II studies precoders having the lowest average mean square error, deriving closed form solutions in special cases and developing a stochastic optimisation algorithm for computing optimal precoders in general.

Design and analysis of linear precoders under a mean square error criterion, Part II: MMSE designs and conclusions

Part I of this two-part paper formulated the precoder design problem as an optimisation problem and solved it with respect to a worst case criterion. Part II studies a similar optimisation problem but with respect to a criterion measuring average performance. A stochastic optimisation algorithm is proposed for solving this problem. For special cases, closed form solutions are also given. These results indicate linear precoders reduce the effects of frequency distortion caused by multipath channels but are powerless to counteract additive white Gaussian noise. The conclusion is linear precoders should introduce only a small amount of redundancy and be used in conjunction with an error correcting code capable of combatting additive noise.

A frequency domain deterministic approach to channel identification

This letter studies the problem of identifying a single-input single-output channel by linearly precoding the input. Although it is known that filterbanks induce cyclostationarity if the input is stationary, this work demonstrates that filterbanks induce a special form of spectral redundancy even if the input is deterministic. This frequency domain interpretation of filterbanks nicely characterizes all linear precoders. It not only helps to identify the strengths and aid in the design of linear precoders, it leads to frequency domain subspace methods for the identification of AR, MA and ARMA channels.