Linear Zero-forcing Research Articles

Linear and Nonlinear Precoding Schemes for Centralized Multicell MIMO-OFDM Systems

The aim of this article is to propose and compare linear and nonlinear precoding schemes for multicell multiuser MIMO-OFDM based systems. The considered linear precoder is designed in two phases: first the intercell interference is removed by applying a linear zero-forcing algorithm. Then the system is further optimized by proposing three power allocation algorithms with per base station power constraint and different complexity tradeoffs: one optimal to minimize the average bit-error-rate and two suboptimal. The proposed nonlinear precoding is designed to minimize average bit-error-rate, over the users, conditioned to a channel realization and the transmitted data. In the high SNR regime, this problem simplifies from a constrained quadratic nonlinear optimization to a single quadratic problem with a scaling, allowing to reduce the complexity. The performance of the proposed schemes is evaluated, considering typical pedestrian scenarios based on LTE specifications. Numerical results show that the performance of the nonlinear scheme outperforms the linear ones. The nonlinear algorithm selects and inverts part of the correlation matrix unlike the linear zero-forcing where full inversion is required. This leads to a better performance as the selection allows to get a better conditioned matrix. Also, it is shown that the complexity of the nonlinear scheme is similar to the linear suboptimal closed-form one.

Space–time coding scheme for time–frequency asynchronous two‐way relay networks

In this study, the authors develop a distributed space–time coding scheme for a two-way relay network that contains multiple distributed relay nodes. Both the time and frequency asynchronous nature of the distributed system are considered in the design. Distributed convolutional coding is employed to handle multiple timing errors in the networks. The authors prove that under perfect frequency synchronisation, the proposed scheme can achieve both spatial and multipath diversity by linear receivers, such as linear zero-forcing or minimum mean-square-error receiver, thus providing a low decoding complexity. The authors further find that frequency asynchronism has little effect on the designed scheme and show that the diversity can still be achieved almost surely (in the measured theoretic sense) under both time and frequency asynchronous scenarios. The authors also provide numerical results to corroborate the proposed studies.

Network MIMO With Linear Zero-Forcing Beamforming: Large System Analysis, Impact of Channel Estimation, and Reduced-Complexity Scheduling

We consider the downlink of a multicell system with multiantenna base stations and single-antenna user terminals, arbitrary base station cooperation clusters, distance-dependent propagation pathloss, and general “fairness” requirements. Base stations in the same cooperation cluster employ joint transmission with linear zero-forcing beamforming, subject to sum or per-base station power constraints. Intercluster interference is treated as noise at the user terminals. Analytic expressions for the system spectral efficiency are found in the large-system limit where both the numbers of users and antennas per base station tend to infinity with a given ratio. In particular, for the per-base station power constraint, we find new results in random matrix theory, yielding the squared Frobenius norm of submatrices of the Moore-Penrose pseudo-inverse for the structured non-i.i.d. channel matrix resulting from the cooperation cluster, user distribution, and path-loss coefficients. The analysis is extended to the case of nonideal Channel State Information at the Transmitters obtained through explicit downlink channel training and uplink feedback. Specifically, our results illuminate the trade-off between the benefit of a larger number of cooperating antennas and the cost of estimating higher-dimensional channel vectors. Furthermore, our analysis leads to a new simplified downlink scheduling scheme that preselects the users according to probabilities obtained from the large-system results, depending on the desired fairness criterion. The proposed scheme performs close to the optimal (finite-dimensional) opportunistic user selection while requiring significantly less channel state feedback, since only a small fraction of preselected users must feed back their channel state information.

Low-Complexity Groupwise OSIC-ZF Detection for $N \times N$ Spatial Multiplexing Systems

In this paper, we present a low-complexity groupwise ordered successive interference canceler (OSIC) with the zero-forcing (ZF) criterion for N × N spatial multiplexing systems. The proposed detection is composed of a number of processing stages, and at each stage, a fraction of the transmitted data streams are nulled to zero. In this way, the original matrix-inverse operations are replaced by a series of inversions with smaller sizes, and the computational complexity can then be largely reduced. We show that this groupwise OSIC-ZF approach can provide a good tradeoff between the computational complexity and the error-rate performance and, therefore, is very attractive as a new detection method standing between the linear ZF detection and the optimal OSIC-ZF detection.

Space–Time Block Coding for Doubly-Selective Channels

In this paper, we present a new space-time block code for time- and frequency-selective (doubly-selective) channels. It can be interpreted as the extension of the Alamouti code to doubly-selective channels, and relies on a joint time-frequency reversal of the transmitted sequences. Under certain channel conditions, the proposed space-time block code belongs to the class that achieves full spatial, delay, and Doppler diversity using a maximum-likelihood (ML) receiver, as well as a linear zero-forcing (LZF) or linear minimum mean-squared error (LMMSE) receiver. For realistic doubly-selective channels, a real-valued linear data model is presented, for which different receiver structures can be developed.

On the performance of space-time coded and spatially multiplexed MIMO systems with linear receivers

We consider multi-antenna systems utilizing groupwise layered space-time coding. Simple, yet accurate approximations for the probability density function (PDF) of the signalto- interference-plus-noise ratio (SINR) at the output of a linear zero-forcing (ZF) and minimum mean square error (MMSE) data estimators are presented. It is shown that the derived PDFs are exact in the limit of large number of receive antennas. We also demonstrate via numerical examples that the approximations are accurate already for relatively small numbers of receive antennas. Closed form expressions in terms of some well-known special functions for the uncoded bit error probability as well as frame error probability (FEP) of turbo-like coded system with iterative sum-product based decoder are derived. For comparison, we also give analytical lower bounds for the FEP of a coded multiantenna system utilizing conventional spatial multiplexing and linear ZF or MMSE based estimator at the receiver.

SER bound for ordered ZF-SIC receiver in M-QAM MIMO system

With the advances of the 4th generation (4G) mobile communications targeting multi-user QoS-guaranteed multimedia services, multiple input multiple output (MIMO) combined with multi-level quadrature amplitude modulation (M-QAM) technique gains great popularity for its capability of supporting high data rate. In this article, the statistical distribution of signal to noise plus interference ratio (SINR) in each recursive step of optimal ordered zero-forcing perfect successive interference cancellation (ZF-OOPSIC) detection is deduced based on order statistics theory. The symbol error rate (SER) of M-QAM MIMO system with ZF-OOPSIC detector is analyzed under flat Rayleigh fading channel. This precise analytical SER result could be taken as the lower bound of various zero-forcing ordered successive interference cancellation (ZF-OSIC) receivers. Combining the analytical SER expression for linear zero-forcing (ZF) receiver as upper bound, the rough range of SER for various ZF-OSIC detectors could be obtained. Monte Carlo (MC) simulations validate the analytical results and prove the conclusions.

Low-Complexity Linear Zero-Forcing for the MIMO Broadcast Channel

Maximizing the sum capacity in the multiple-input multiple-output (MIMO) broadcast channel requires the use of dirty paper coding (DPC). However, practical implementations of DPC which are nearly optimum exhibit high computational complexity. As an alternative to DPC linear zero-forcing can be used where the multiuser interference is completely canceled by linear beamforming. Determining the optimum user allocation, transmit and receive filters thereby constitutes a combinatorial and nonconvex optimization problem. To circumvent its direct solution and therefore reduce complexity, we propose an algorithm that successively allocates data streams to users and, in contrast to state-of-the-art approaches, includes the receive filters into the optimization. We then show several steps that reduce the complexity of the algorithm at marginal performance losses. Thus, performance of state-of-the-art approaches can be maintained while the computational complexity is reduced considerably, as it is shown by a detailed complexity analysis and simulation results.

Advanced Receiver Design for Quadrature OFDMA Systems

Quadrature orthogonal frequency division multiple access (Q-OFDMA) systems have been recently proposed to reduce the peak-to-average power ratio (PAPR) and complexity, and improve carrier frequency offset (CFO) robustness and frequency diversity for the conventional OFDMA systems. However, Q-OFDMA receiver obtains frequency diversity at the cost of noise enhancement, which results in Q-OFDMA systems achieving better performance than OFDMA only in the higher signal-to-noise ratio (SNR) range. In this paper, we investigate various detection techniques such as linear zero forcing (ZF) equalization, minimum mean square error (MMSE) equalization, decision feedback equalization (DFE), and turbo joint channel estimation and detection, for Q-OFDMA systems to mitigate the noise enhancement effect and improve the bit error ratio (BER) performance. It is shown that advanced detections, for example, DFE and turbo receiver, can significantly improve the performance of Q-OFDMA.

Asymptotic Performance of Linear Receivers in MIMO Fading Channels

Linear receivers are an attractive low-complexity alternative to optimal processing for multi-antenna MIMO communications. In this paper we characterize the information-theoretic performance of MIMO linear receivers in two different asymptotic regimes. For fixed number of antennas, we investigate the limit of error probability in the high-SNR regime in terms of the Diversity-Multiplexing Tradeoff (DMT). Following this, we characterize the error probability for fixed SNR in the regime of large (but finite) number of antennas. As far as the DMT is concerned, we report a negative result: we show that both linear Zero-Forcing (ZF) and linear Minimum Mean-Square Error (MMSE) receivers achieve the same DMT, which is largely suboptimal even in the case where outer coding and decoding is performed across the antennas. We also provide an approximate quantitative analysis of the markedly different behavior of the MMSE and ZF receivers at finite rate and non-asymptotic SNR, and show that while the ZF receiver achieves poor diversity at any finite rate, the MMSE receiver error curve slope flattens out progressively, as the coding rate increases. When SNR is fixed and the number of antennas becomes large, we show that the mutual information at the output of a MMSE or ZF linear receiver has fluctuations that converge in distribution to a Gaussian random variable, whose mean and variance can be characterized in closed form. This analysis extends to the linear receiver case a well-known result previously obtained for the optimal receiver. Simulations reveal that the asymptotic analysis captures accurately the outage behavior of systems even with a moderate number of antennas.

Linear beamformer schemes with simple relay selection in MIMO relay networks

This article considers a wireless multi-hop/mesh network where single multi-antenna source-destination pair communicates through a selected relay subset using simple relay selection under the constraint of fixed relay's number. Compared with random selection, the simple relay selection can yield certain capacity advantages while linear zero-forcing (ZF) receiver and linear beamformer are considered at the relay. For match-filter (MF) beamformer and amplified-and-forward (AF) beamformer with a fixed number of relays, the capacities are given. Furthermore, we extend the simple selection methods to the relaying scheme with orthogonal-triangular (QR) beamformer and investigate these linear beamformer schemes over spatially correlated multi-input multi-output (MIMO) links for both the backward and forward channel over the two-hop MIMO relay networks.

Analysis of Tomlinson–Harashima Precoding in Multiuser MIMO Systems With Imperfect Channel State Information

Tomlinson-Harashima precoding (THP) is a successive pre-equalization technique that arranges all the receivers into a group of layers and then suppresses the interlayer interference in sequence at the transmitter. There are two basic THP structures according to the positions of the diagonal weighted filter-decentralized at the receivers or centralized at the transmitter-which are denoted dTHP or cTHP, respectively. In this paper, the investigation of THP in downlink multiuser multiple-input-multiple-output (MIMO) systems with imperfect channel state information (CSI) is performed. The interference power caused by imperfect CSI is first derived, and then, the comparison between the two basic THP structures and the linear zero-forcing (ZF) precoding is carried out in terms of the interference power and the system capacity. Furthermore, the cases with different CSI errors among users are considered, through which the independence of the interference power among layers or subchannels is presented, including the cases with single-antenna and multiantenna receivers. Analytical and simulation results indicate that with imperfect CSI, THP leads to less interference power than linear-ZF, and dTHP leads to larger system capacity than cTHP and linear-ZF.

Optimizing Zero-Forcing Precoders for MIMO Broadcast Systems

In this paper, we develop optimization techniques for linear zero-forcing (ZF) multiple-input, multiple-output (MIMO) broadcast (BC) precoders. A computationally efficient technique based on QR-update algorithms for implementing this optimization is then proposed. The application of the proposed ZF precoders and decoders to systems with both large and small numbers of users is considered. The proposed optimization technique is also shown to perform very well in conjunction with nonlinear Tomlinson--Harashima precoding. Both coded and uncoded performances are considered.

Optimizing ZF Precoders for MIMO Broadcast Systems

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we develop optimization techniques for linear zero-forcing (ZF) multiple-input multiple-output (MIMO) broadcast (BC) precoders. A computationally efficient technique based on QR-update algorithms for implementing this optimization is then proposed. The application of the proposed ZF precoders and decoders to both large and small (in terms of the number of users) systems is considered. It is also shown that the proposed optimization technique is robust, and performs very well with nonlinear Tomlinson--Harashima precoding. Both coded and uncoded performances are considered. </para>

Successive narrowband interference cancellation for OFDM systems

We consider the problem of mitigating narrowband interference (NBI) in conventional orthogonal frequency-division multiplexing (OFDM) systems, wherein the presence of the NBI dramatically degrades the performances of both the linear zero-forcing and minimum mean-square error equalizers. A new receiver based on successive NBI cancellation is synthesized, which is capable of assuring good NBI rejection, without requiring insertion of additional redundancy in the OFDM signal

Error performance analysis of linear zero forcing and MMSE precoders for MIMO broadcast channels

Linear precoding (LP) techniques for a multiuser multiple-input-multiple-output broadcast channel is investigated and analytical results of the achievable sum capacity and error performance for zero forcing (ZF) linear precoders is presented. It is shown that the detection signal-to-noise ratio of ZF-LP can be accurately approximated by a shrinking Chi-square distribution. The symbol error rate and its achievable diversity gain of ZF-LP are given. Then, an improved linear precoder based on the minimum mean-square error (MMSE) criterion is derived. Its error performance and sum capacity are analysed and compared with that of the ZF-LP. It is shown that the MMSE-LP can achieve much better error performance and a high sum capacity than the existing ZF-LP.

A Near-Optimal Linear Crosstalk Canceler for Upstream VDSL

Crosstalk is the major source of performance degradation in VDSL. Several crosstalk cancelers have been proposed to address this. Unfortunately, they suffer from error propagation, high complexity, and long latency. This paper presents a simple, linear zero-forcing (ZF) crosstalk canceler. This design has a low complexity and no latency and does not suffer from error propagation. Furthermore, due to the well-conditioned structure of the VDSL channel matrix, the ZF design causes negligible noise enhancement. A lower bound on the performance of the linear ZF canceler is derived. This allows performance to be predicted without explicit knowledge of the crosstalk channels, which simplifies service provisioning considerably. This bound shows that the linear ZF canceler operates close to the single-user bound. Therefore, the linear ZF canceler is a low-complexity, low-latency design with predictable near-optimal performance. The combination of spectral optimization and crosstalk cancellation is also considered. Spectra optimization in a multiaccess channel generally involves a complex optimization problem. Since the linear ZF canceler decouples transmission on each line, the spectrum on each modem can be optimized independently, leading to a significant reduction in complexity

Transmit antenna shuffling for quasi-orthogonal space-time block codes with linear receivers

In this letter, we propose a transmit antenna shuffling scheme for quasi-orthogonal space-time block codes (QO-STBCs). We show that by adaptively mapping the space-time sequences of the QO-STBC to the appropriate transmit antennas depending on the channel condition, the proposed scheme can improve its transmit diversity with limited feedback information. The performance of the scheme with various numbers of shuffling patterns is analyzed. The bit error probability of the schemes is evaluated by simulations. It is demonstrated that with the linear zero-forcing (ZF) and the minimum mean squared error (MMSE) receivers, the closed-loop QO-STBC using two feedback bits can achieve almost the same performance as the ideal 4-path diversity and it is about 4-5 dB better than the open loop schemes

Linear and decision-feedback per tone equalization for DMT-based transmission over IIR channels

The per-tone equalizer (PTEQ) has been presented as an attractive alternative for the classical time-domain equalizer (TEQ) in discrete multitone (DMT) based systems, such as ADSL systems. The PTEQ is based on a linear minimum mean-square-error (L-MMSE) equalizer design for each separate tone. In this paper, we reconsider DMT modulation and equalization in the ADSL context under the realistic assumption of an infinite impulse response (IIR) model for the wireline channel. First, optimum linear zero-forcing (L-ZF) block equalizers for arbitrary IIR model orders and cyclic prefix (CP) lengths are developed. It is shown that these L-ZF block equalizers can be decoupled per tone, hence they lead to an L-ZF PTEQ. Then, based on the L-ZF PTEQ, low-complexity L-MMSE PTEQ extensions are developed: the linear PTEQ extension exploits frequency-domain transmit redundancy from pilot and unused tones; alternatively, a closely related decision-feedback PTEQ extension can be applied. The PTEQ extensions then add flexibility to a DMT-based system design: the CP overhead can be reduced by exploiting frequency-domain transmit redundancy instead, so that a similar bitrate as with the original PTEQ is achieved at a lower memory and computational cost or, alternatively, a higher bitrate is achieved without a considerable cost increase. Both PTEQ extensions are also shown to improve the receiver's robustness to narrow-band interference.

On the existence of FIR zero-forcing equalizers for nonredundantly precoded transmissions through FIR channels

We consider in this letter the problem of perfectly equalizing finite-impulse response (FIR) channels in digital communication systems employing nonredundant precoding at the transmitter. In this case, as it is well known, there is no linear FIR zero-forcing (ZF) block equalizer that is capable of perfectly recovering information symbols in the absence of noise. We show that, when the symbol constellation is improper, it is possible to design widely linear FIR-ZF block equalizers that, unlike their linear counterparts, process both the received signal and its complex-conjugate version. In addition, we propose a simple procedure to synthesize universal modulation-induced cyclostationary (MIC) precoders that assures perfect symbol recovery, regardless of the channel-zero locations.