Finite-SNR Diversity–Multiplexing Tradeoff for Correlated Rayleigh and Rician MIMO Channels

A nonasymptotic framework is presented to analyze the diversity-multiplexing tradeoff of a multiple-input-multiple-output (MIMO) wireless system at finite signal-to-noise ratios (SNRs). The target data rate at each SNR is proportional to the capacity of an additive white Gaussian noise (AWGN) channel with an array gain. The proportionality constant, which can be interpreted as a finite-SNR spatial multiplexing gain, dictates the sensitivity of the rate adaptation policy to SNR. The diversity gain as a function of SNR for a fixed multiplexing gain is defined by the negative slope of the outage probability versus SNR curve on a log-log scale. The finite-SNR diversity gain provides an estimate of the additional power required to decrease the outage probability by a target amount. For general MIMO systems, lower bounds on the outage probabilities in correlated Rayleigh fading and Rician fading are used to estimate the diversity gain as a function of multiplexing gain and SNR. In addition, exact diversity gain expressions are determined for orthogonal space-time block codes (OSTBC). Spatial correlation significantly lowers the achievable diversity gain at finite SNR when compared to high-SNR asymptotic values. The presence of line-of-sight (LOS) components in Rician fading yields diversity gains higher than high-SNR asymptotic values at some SNRs and multiplexing gains while resulting in diversity gains near zero for multiplexing gains larger than unity. Furthermore, as the multiplexing gain approaches zero, the normalized limiting diversity gain, which can be interpreted in terms of the wideband slope and the high-SNR slope of spectral efficiency, exhibits slow convergence with SNR to the high-SNR asymptotic value. This finite-SNR framework for the diversity-multiplexing tradeoff is useful in MIMO system design for realistic SNRs and propagation environments     

Joint Source and Channel Coding for MIMO Systems: Is it Better to be Robust or Quick?

A framework is developed for optimizing the tradeoff between diversity, multiplexing, and delay in multiple-input multiple-output (MIMO) systems to minimize end-to-end distortion. The goal is to find the optimal balance between the increased data rate provided by antenna multiplexing, the reduction in transmission errors provided by antenna diversity and automatic repeat request (ARQ), and the delay introduced by ARQ. First, closed-form analytical results are developed to minimize end-to-end distortion of a vector quantizer concatenated with a space-time MIMO channel code in the high SNR regime. The minimization determines the optimal point on the diversity-multiplexing tradeoff curve. For large but finite SNR this optimal point is found via convex optimization, which is illustrated with an example of a practical joint source-channel code design. It is then shown that for MIMO systems with ARQ retransmission, sources without a delay constraint have distortion minimized by maximizing the ARQ window size. This results in a new multiplexing-diversity tradeoff region enhanced by ARQ. However, under a source delay constraint the problem formulation changes to account for delay distortion associated with random message arrival and random ARQ completion times. In this case, the simplifications associated with a high SNR assumption break down, and a dynamic programming formulation is required to capture the channel diversity-multiplexing tradeoff as well as the random arrival and retransmission dynamics. Results based on this formulation show that a delay-sensitive system obtains significant performance gains by adapting its operating point on the diversity-multiplexing-delay region to system dynamics.     

Analysis on the Diversity-Multiplexing Tradeoff for Ordered MIMO SIC Receivers

The diversity-multiplexing tradeoff for multiple-input multiple-output (MIMO) point-to-point channels and multiple access channels were first proposed and studied by Zheng and Tse recently. While the optimal tradeoff curves for MIMO channels have been explicitly explored, those corresponding to some suboptimal and practical MIMO schemes are still open. One such important problem is the diversity-multiplexing tradeoff for a V-BLAST type system employing ordered successive interference cancellation (SIC) receivers with zero forcing (ZF) or minimum mean square error (MMSE) processing at each stage. In this paper, we take a novel geometrical approach and rigorously verify that under general settings, the optimal ordering rule for a V-BLAST SIC receiver will not improve its performance regarding diversity-multiplexing tradeoff in point-to- point channels. The same geometrical tool is then applied to MIMO spatial-division multiple access channels, leading to some first results in this area. Particularly, we reveal that when the rates of data streams are fixed (i.e., zero spatial multiplexing gain), the diversity order is not improved by user ordering.     

Explicit Space–Time Codes Achieving the Diversity–Multiplexing Gain Tradeoff

A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity-multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space-time (ST) code. This tradeoff is precisely known in the case of independent and identically distributed (i.i.d.) Rayleigh fading, for Tgesn_t+n_r-1 where T is the number of time slots over which coding takes place and n_t,n_r are the number of transmit and receive antennas, respectively. For Tt+n_r-1, only upper and lower bounds on the D-MG tradeoff are available. In this paper, we present a complete solution to the problem of explicitly constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff for any number of receive antennas. We do this by showing that for the square minimum-delay case when T=n_t=n, cyclic-division-algebra (CDA)-based ST codes having the nonvanishing determinant property are D-MG optimal. While constructions of such codes were previously known for restricted values of n, we provide here a construction for such codes that is valid for all n. For the rectangular, T>n_t case, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all Tgesn_t is the same as that previously known to hold for Tgesn_t+n _r-1     

Lattice coding and decoding achieve the optimal diversity-multiplexing tradeoff of MIMO channels

This paper considers communication over coherent multiple-input multiple-output (MIMO) flat-fading channels where the channel is only known at the receiver. For this setting, we introduce the class of LAttice Space-Time (LAST) codes. We show that these codes achieve the optimal diversity-multiplexing tradeoff defined by Zheng and Tse under generalized minimum Euclidean distance lattice decoding. Our scheme is based on a generalization of Erez and Zamir mod-/spl Lambda/ scheme to the MIMO case. In our construction the scalar "scaling" of Erez-Zamir and Costa Gaussian "dirty-paper" schemes is replaced by the minimum mean-square error generalized decision-feedback equalizer (MMSE-GDFE). This result settles the open problem posed by Zheng and Tse on the construction of explicit coding and decoding schemes that achieve the optimal diversity-multiplexing tradeoff. Moreover, our results shed more light on the structure of optimal coding/decoding techniques in delay-limited MIMO channels, and hence, open the door for novel approaches for space-time code constructions. In particular, 1) we show that MMSE-GDFE plays a fundamental role in approaching the limits of delay-limited MIMO channels in the high signal-to-noise ratio (SNR) regime, unlike the additive white Gaussian noise (AWGN) channel case and 2) our random coding arguments represent a major departure from traditional space-time code designs based on the rank and/or mutual information design criteria.     

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.     

The Throughput–Reliability Tradeoff in Block-Fading MIMO Channels

We build on Zheng and Tse's elegant formulation of diversity-multiplexing tradeoff (DMT) to provide a better understanding of the asymptotic interplay between transmission rate, error probability, and signal-to-noise ratio (SNR) in block-fading multiple-input multiple-output (MIMO) channels. In particular, we identify the limitation imposed by the notion of multiplexing gain and develop a new formulation called the throughput-reliability tradeoff (TRT), that avoids this limitation. The new characterization is then used to elucidate the asymptotic trends exhibited by the outage probability curves of block-fading MIMO channels     

Performance of Hybrid ARQ Schemes with ML Code Combining over a Block-Fading Rayleigh Channel

This paper considers two well-known selective-repeat retransmission schemes, namely, hybrid type-I ARQ and hybrid type-II ARQ, using convolutional coding, in conjunction with maximum-likelihood code combining. Our theoretical analysis, based upon the concept of generalized weight distribution, shows that the use of code combining yields a significant throughput at very high channel error rates not only in constant AWGN channels but also in fading channels. To demonstrate this, we consider a widely-used block-fading Rayleigh channel model, in which the channel is assumed to be constant during each block of data and the fading is assumed to be independent from block to block. A key parameter in designing retransmission protocols for delay-limited applications in such channels is the minimum number of retransmissions, needed to achieve error-free decoding at almost all channel conditions (low outage probability). This number can be reduced significantly when code combining is employed.     

On Space–Time Trellis Codes Achieving Optimal Diversity Multiplexing Tradeoff

Multiple antennas can be used for increasing the amount of diversity (diversity gain) or increasing the data rate (the number of degrees of freedom or spatial multiplexing gain) in wireless communication. As quantified by Zheng and Tse, given a multiple-input-multiple-output (MIMO) channel, both gains can, in fact, be simultaneously obtained, but there is a fundamental tradeoff (called the Diversity-Multiplexing Gain (DM-G) tradeoff) between how much of each type of gain, any coding scheme can extract. Space-time codes (STCs) can be employed to make use of these advantages offered by multiple antennas. Space-Time Trellis Codes (STTCs) are known to have better bit error rate performance than Space-Time Block Codes (STBCs), but with a penalty in decoding complexity. Also, for STTCs, the frame length is assumed to be finite and hence zeros are forced towards the end of the frame (called the trailing zeros), inducing rate loss. In this correspondence, we derive an upper bound on the DM-G tradeoff of full-rate STTCs with nonvanishing determinant (NVD). Also, we show that the full-rate STTCs with NVD are optimal under the DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros. Next, we give an explicit generalized full-rate STTC construction for any number of states of the trellis, which achieves the optimal DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros     

Finite-SNR diversity-multiplexing tradeoffs in fading relay channels

We analyze the diversity-multiplexing tradeoff in a fading relay channel at finite signal-to-noise ratios (SNRs). In this framework, the rate adaptation policy is such that the target system data rate is a multiple of the capacity of an additive white Gaussian noise (AWGN) channel. The proportionality constant determines how aggressively the system scales the data rate and can be interpreted as a finite-SNR multiplexing gain. The diversity gain is given by the negative slope of the outage probability with respect to the SNR. Finite-SNR diversity performance is estimated using a constrained max-flow min-cut upper bound on the relay channel capacity. Moreover, the finite-SNR diversity-multiplexing tradeoff is characterized for three practical decode and forward half-duplex cooperative protocols with different amounts of broadcasting and simultaneous reception. For each configuration, system performance is computed as a function of SNR under a system-wide power constraint on the source and relay transmissions. Our analysis yields the following findings; (i) improved multiplexing performance can be achieved at any SNR by allowing the source to transmit constantly, (ii) both broadcasting and simultaneous reception are desirable in half-duplex relay cooperation for superior diversity-multiplexing performance, and (iii) the diversity-multiplexing tradeoff at finite-SNR is impacted by the power partitioning between the source and the relay terminals. Finally, we verify our analytical results by numerical simulations     

On the Diversity-Multiplexing Tradeoff for Wireless Cooperative Multiple Access Systems

A spectrally efficient strategy is proposed for cooperative multiple access (CMA) channels in a centralized communication environment with N users. By applying superposition coding, each user will transmit a mixture containing its own information as well as the other users', which means that each user shares parts of its power with the others. The use of superposition coding in cooperative networks was first proposed in [E. G. Larsson and B. R. Vojcic, Cooperative transmit diversity based on superposition modulation, IEEE Commun. Lett., vol. 9, pp. 778 780, Sep. 2005. ] , which will be generalized to a multiple-user scenario in this paper. Since the proposed CMA system can be seen as a precoded point-to-point multiple-antenna system, its performance can be best evaluated using the diversity-multiplexing tradeoff. By carefully categorizing the outage events, the diversity-multiplexing tradeoff can be obtained, which shows that the proposed cooperative strategy can achieve larger diversity/multiplexing gain than the compared transmission schemes at any diversity/multiplexing gain. Furthermore, it is demonstrated that the proposed strategy can achieve optimal tradeoff for multiplexing gains 0 les tau les1 whereas the compared cooperative scheme is only optimal for 0 les tau les (1/N) . As discussed in the paper, such superiority of the proposed CMA system is due to the fact that the relaying transmission does not consume extra channel use and, hence, the deteriorating effect of cooperative communication on the data rate is effectively limited.     

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.     

Diversity‐multiplexing tradeoff over correlated Rayleigh fading channels: a non‐asymptotic analysis

In this paper, we present a finite‐signal‐to‐noise ratio (finite‐SNR) framework to establish tight bounds on the diversity‐multiplexing tradeoff of a multiple input multiple output (MIMO) system. We focus on a more realistic propagation environment where MIMO channel fading coefficients are correlated and where SNR values are finite. The impact of spatial correlation on the fundamental diversity‐multiplexing tradeoff is investigated. We present tight lower bounds on the outage probability of both spatially uncorrelated and correlated MIMO channels. Using these lower bounds, accurate finite‐SNR estimates of the diversity‐multiplexing tradeoff are derived. These estimates allow to gain insight on the impact of spatial correlation on the diversity‐multiplexing tradeoff at finite‐SNR. As expected, the diversity‐multiplexing tradeoff is severely degraded as the spatial correlation increases. For example, a MIMO system operating at a spectral efficiency of R bps/Hz and at an SNR of 5 dB in a moderately correlated channel, achieves a better diversity gain than a system operating at the same spectral efficiency and at an SNR of 10 dB in a highly correlated channel, when the multiplexing gain r is greater than 0.8. Another interesting point is that provided that the spatial correlation channel matrix is of full rank, the maximum diversity gain is not affected by the spatial correlation. Copyright © 2009 John Wiley & Sons, Ltd.     

一种基于分布式空时码的协作MIMO分集复用折衷新方案

协作MIMO通过多个单天线节点的相互协作构造多发射天线,以此形成一种虚拟MIMO多天线阵列获得空间分集增益。考虑到协作MIMO特点,天线间采用分布式空时编码进行编码协作。文章研究了协作MIMO中基于分布式空时码(DSTC)的分集复用折衷(DMT)新方案,该方案通过推导两种DSTC的中断概率与分集增益表达式,结合两类DSTC的DMT策略,根据改变复用增益阈值自适应获得最佳DMT与中断性能。数值仿真表明,所提的DMT策略可以逼近协作MIMO的DMT上限,协作节点采用该策略的中断性能仅次于上限的中断性能。在多节点构成协作MIMO网络分布式空时编码协作中,提出的DMT新方案可使系统高效地获得协作分集增益与中断性能。     

A quasi-orthogonal group space-time architecture to achieve a better diversity-multiplexing tradeoff

Most existing MIMO (multiput-input multiput-output) schemes optimize only either the diversity gain or the multiplexing gain. To obtain a good tradeoff between these two, the quasi-orthogonal group space-time (QoGST) architecture is proposed, wherein the transmit stream is subgrouped but encoded via an inter-group space-time block encoder, with group interference suppression at the receiver. This paper also considers another combined space-time coding and layered space-time architecture, which we refer to as group layered space-time (GLST), where space-time block coding is employed within each group. Under the assumption of Rayleigh fading and a prior perfect channel state information at the receiver, a performance analysis will demonstrate that both QoGST and GLST can achieve a good diversity-multiplexing tradeoff. QoGST is even superior to GLST. Simulation results will validate our analysis and further show that compared to the existent layered space-time block code (LSTBC) scheme, both QoGST and GLST can achieve a significant performance gain     

Co-ordinate Interleaved Spatial Multiplexing with Channel State Information

Performance of spatial multiplexing multiple-input multiple-output (MIMO) wireless systems can be improved with channel state information (CSI) at both ends of the link. This paper proposes a new linear diagonal MIMO transceiver, referred to as co-ordinate interleaved spatial multiplexing (CISM). With CSI at transmitter and receiver, CISM diagonalizes the MIMO channel and interleaves the co-ordinates of the input symbols (from rotated QAM constellations) transmitted over different eigenmodes. The analytical and simulation results show that with co-ordinate interleaving across two eigenmodes, the diversity gain of the data stream transmitted over the weaker eigenmode becomes equal to that of the data transmitted on the stronger eigenmode, resulting in a significant improvement in the overall diversity. The diversity-multiplexing tradeoff (DMT) is analyzed for CISM and is shown that it achieves higher diversity gain at all positive multiplexing gains compared to existing diagonal transceivers. Over rank n MIMO channels, with input symbols from rotated n-dimensional constellations, the DMT of CISM is a straight line connecting the endpoints (0,N_tN_r) and (min{N_t,N_r}, 0), where N_t, and N_r} are the number of transmit and receive antennas, respectively.     

On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels

We propose novel cooperative transmission protocols for delay-limited coherent fading channels consisting of N (half-duplex and single-antenna) partners and one cell site. In our work, we differentiate between the relay, cooperative broadcast (down-link), and cooperative multiple-access (CMA) (up-link) channels. The proposed protocols are evaluated using Zheng-Tse diversity-multiplexing tradeoff. For the relay channel, we investigate two classes of cooperation schemes; namely, amplify and forward (AF) protocols and decode and forward (DF) protocols. For the first class, we establish an upper bound on the achievable diversity-multiplexing tradeoff with a single relay. We then construct a new AF protocol that achieves this upper bound. The proposed algorithm is then extended to the general case with (N-1) relays where it is shown to outperform the space-time coded protocol of Laneman and Wornell without requiring decoding/encoding at the relays. For the class of DF protocols, we develop a dynamic decode and forward (DDF) protocol that achieves the optimal tradeoff for multiplexing gains 0/spl les/r/spl les/1/N. Furthermore, with a single relay, the DDF protocol is shown to dominate the class of AF protocols for all multiplexing gains. The superiority of the DDF protocol is shown to be more significant in the cooperative broadcast channel. The situation is reversed in the CMA channel where we propose a new AF protocol that achieves the optimal tradeoff for all multiplexing gains. A distinguishing feature of the proposed protocols in the three scenarios is that they do not rely on orthogonal subspaces, allowing for a more efficient use of resources. In fact, using our results one can argue that the suboptimality of previously proposed protocols stems from their use of orthogonal subspaces rather than the half-duplex constraint.     

Approximately universal codes over slow-fading channels

Performance of reliable communication over a coherent slow-fading multiple-input multiple-output (MIMO) channel at high signal-to-noise ratio (SNR) is succinctly captured as a fundamental tradeoff between diversity and multiplexing gains. This paper studies the problem of designing codes that optimally tradeoff the diversity and multiplexing gains. The main contribution is a precise characterization of codes that are universally tradeoff-optimal, i.e., they optimally tradeoff the diversity and multiplexing gains for every statistical characterization of the fading channel. This characterization is referred to as approximate universality; the approximation is in the connection between error probability and outage capacity with diversity and multiplexing gains, respectively. The characterization of approximate universality is then used to construct new coding schemes as well as to show optimality of several schemes proposed in the space-time coding literature.     

几种接收机在MIMO信道下的性能比较

多入多出（MIMO）无线信道具有空间复用增益和分集增益特性,因此MIMO系统和单入单出（SISO）无线系统相比能够获得更高的频谱效率。本文在不同天线组合下分析了几种MIMO空时信号处理算法的性能,仿真结果和理论分析表明：空间复用增益和分集增益不能同时获得最大,因此在设计MIMO通信系统时可根据实际情况选择天线数,即不仅考虑系统抵抗信道衰落的分集增益,还要考虑能够提供更高的数据传输速率,通过折衷考虑空间复杂增益和分集增益,从更全面的观点评估系统的性能。     

On the Diversity Order of Spatial Multiplexing Systems With Transmit Antenna Selection: A Geometrical Approach

In recent years, the remarkable ability of multiple-input-multiple-output (MIMO) wireless communication systems to provide spatial diversity or multiplexing gains has been clearly demonstrated. For MIMO diversity schemes, it is well known that antenna selection methods that optimize the postprocessing signal-to-noise ratio (SNR) can preserve the diversity order of the original full-size MIMO system. On the other hand, the diversity order achieved by antenna selection in spatial multiplexing systems, especially those exploiting practical coding and decoding schemes, has not thus far been rigorously analyzed. In this paper, a geometrical framework is proposed to theoretically analyze the diversity order achieved by transmit antenna selection for separately encoded spatial multiplexing systems with linear and decision-feedback receivers. When two antennas are selected from the transmitter, the exact achievable diversity order is rigorously derived, which previously only appears as conjectures based on numerical results in the literature. If more than two antennas are selected, we give lower and upper bounds on the achievable diversity order. Furthermore, the same geometrical approach is used to evaluate the diversity-multiplexing tradeoff in spatial multiplexing systems with transmit antenna selection