Optimal Diversity-multiplexing Tradeoff Research Articles

Optimization of Fast-Decodable Full-Rate STBC with Non-Vanishing Determinants

Full-rate STBC (space-time block codes) with non-vanishing determinants achieve the optimal diversity-multiplexing tradeoff but incur high decoding complexity. To permit fast decoding, Sezginer, Sari and Biglieri proposed an STBC structure with special QR decomposition characteristics. In this paper, we adopt a simplified form of this fast-decodable code structure and present a new way to optimize the code analytically. We show that the signal constellation topology (such as QAM, APSK, or PSK) has a critical impact on the existence of non-vanishing determinants of the full-rate STBC. In particular, we show for the first time that, in order for APSK-STBC to achieve non-vanishing determinant, an APSK constellation topology with constellation points lying on square grid and ring radius √m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (m,n integers) needs to be used. For signal constellations with vanishing determinants, we present a methodology to analytically optimize the full-rate STBC at specific constellation dimension.

DMT Optimal Cooperative MAC Protocols in Wireless Mesh Networks with Minimized Signaling Overhead

In this paper, a cooperative protocol is proposed for wireless mesh networks. Two features are implemented: on-demand cooperation and selection of the best relay. First, cooperation is activated by a destination terminal when it fails in decoding the message from a source terminal. Second, a selection of the best relay is performed when cooperation is needed. The robustness of wireless links are increased while the resource consumption is minimized. The selection of the best relay is performed by a splitting algorithm, ensuring a fast selection process, the duration of which is now fully characterized. Only terminals that improve the direct link participate in the relay selection and inefficient cooperation is avoided. The proposed protocol is demonstrated to achieve an optimal diversity-multiplexing trade-off. This study focuses on Nakagami- wireless channel models to encompass a variety of fading models in the context of wireless mesh networks.

Diversity-Multiplexing Tradeoff Analysis for OFDM Systems With Subcarrier Grouping, Linear Precoding, and Linear Detection

We consider the use of linear constellation precoding and linear detection in a multicarrier OFDM system with multiple receive antennas to obtain improved performance over multipath fading channels at a low complexity. We split the full set of subcarriers into smaller groups and spread the data symbols assigned to each group via precoding matrices. We adopt the diversity-multiplexing tradeoff (DMT) framework and derive the DMT-optimal split of the subcarriers (DMT-optimal grouping) and the DMT-optimal number of symbols assigned to each group (DMT-optimal symbol loading). We determine necessary and sufficient precoder design conditions to achieve DMT optimality and give specific constructions of such precoders. Next, we consider a multiuser OFDMA system and derive an algorithm which divides the available subcarriers among the active users in order to maximize the diversity order of the system (or joint) error probability. We also extend our analysis to OFDM systems equipped with multiple transmit antennas. Finally, we obtain important insights on the role of outer codes in an OFDM system employing linear precoding and linear equalization.

Approaching the Optimal Diversity-Multiplexing Tradeoff in a Four-node Cooperative Network

A novel cooperative diversity transmission protocol is proposed for a four-node network, in which a single-antenna source communicates with an N-antenna destination with the help of two K-antenna half-duplex decode-and-forward (DF) relays. The proposed shifted successive DF relaying (SSDFR) protocol asymptotically achieves the optimal diversity-multiplexing tradeoff (DMT) the system can provide. The resulting DMT performance does not require complex coding strategies at the relays, extremely strong or weak inter-relay channel conditions, or feedback from the destination to the source, which highlights the advantages of the proposed protocol over state of the art techniques.

DMT Optimality of LR-Aided Linear Decoders for a General Class of Channels, Lattice Designs, and System Models

The work identifies the first general, explicit, and non-random MIMO encoder-decoder structures that guarantee optimality with respect to the diversity-multiplexing tradeoff (DMT), without employing a computationally expensive maximum-likelihood (ML) receiver. Specifically, the work establishes the DMT optimality of a class of regularized lattice decoders, and more importantly the DMT optimality of their lattice-reduction (LR)-aided linear counterparts. The results hold for all channel statistics, for all channel dimensions, and most interestingly, irrespective of the particular lattice-code applied. As a special case, it is established that the LLL-based LR-aided linear implementation of the MMSE-GDFE lattice decoder facilitates DMT optimal decoding of any lattice code at a worst-case complexity that grows at most linearly in the data rate. This represents a fundamental reduction in the decoding complexity when compared to ML decoding whose complexity is generally exponential in rate. The results' generality lends them applicable to a plethora of pertinent communication scenarios such as quasi-static MIMO, MIMO-OFDM, ISI, cooperative-relaying, and MIMO-ARQ channels, in all of which the DMT optimality of the LR-aided linear decoder is guaranteed. The adopted approach yields insight, and motivates further study, into joint transceiver designs with an improved SNR gap to ML decoding.

On the Limitations of the Naive Lattice Decoding

In this paper, the inherent drawbacks of the naive lattice decoding (NLD) for MIMO fading systems is investigated. We show that using the NLD for MIMO systems has considerable deficiencies in terms of the diversity-multiplexing tradeoff. Unlike the case of maximum-likelihood decoding, in this case, even the perfect lattice space-time codes which have the nonvanishing determinant property cannot achieve the optimal diversity-multiplexing tradeoff. Indeed, we show that in the case of NLD, when we fix the underlying lattice, all the codes based on full-rate lattices have the same diversity-multiplexing tradeoff as V-BLAST. Also, we derive a lower bound on the symbol error probability of the NLD for the fixed-rate MIMO systems (with equal numbers of receive and transmit antennas). This bound shows that asymptotically, the NLD has an unbounded loss in terms of the required SNR, compared to the maximum likelihood decoding.

Cooperative Transmission in a Wireless Relay Network Based on Flow Management

Cooperative transmission protocols using a flow optimization approach for a general multi-node half-duplex wireless relay network are presented. The proposed design involves solving a convex flow optimization problem on a graph that models the relay network. Two protocols are developed using the techniques of broadcasting (BC), multiple-access (MA), and time sharing (TS), and both are shown to achieve the optimal diversity-multiplexing tradeoff (DMT). Simulation results are used to quantify the performances of the proposed protocols in terms of outage probabilities, and to compare them against the max-flow-min-cut bound.

Optimal Diversity–Multiplexing Tradeoff and Code Constructions of Some Constrained Asymmetric MIMO Systems

In multiple-input-multiple-output (MIMO) communications, the notion of asymmetric channel refers to the situation when the number of transmit antennas is strictly larger than the number of receive antennas. Such channels can often be found in MIMO downlink transmissions. While existing cyclic-division-algebra (CDA)-based codes can still be employed to achieve the optimal diversity-multiplexing tradeoff (DMT) at high signal-to-noise ratio (SNR) regime, such codes cannot be directly decoded using, for example, the pure sphere decoding method. Although other means of decoding methods such as minimum mean square error generalized decision feedback equalizer (MMSE-GDFE) with lattice search and regularized lattice decoding are available, an alternative approach is to constrain the number of active transmit antennas in each channel use to be no larger than the number of receive antennas. The resulting system is coined constrained asymmetric MIMO system. Two general types of asymmetrical channels are considered in this paper, namely, 1) when there are two receive antennas and the number of transmit antennas is arbitrary, and 2) when the number of transmit antennas is one larger than the number of receive antennas. Explicit optimal transmission schemes as well as the corresponding code constructions for such constrained asymmetric MIMO channels are presented, and are shown to achieve the same DMT performance as their unconstrained counterparts.

Distributed MIMO Receiver—Achievable Rates and Upper Bounds

A multiple-input multiple-output (MIMO) system with a distributed receiver is considered. The system consists of a nomadic transmitter with several antennas, whose signal is received by multiple agents, exhibiting independent channel gains and an additive circular-symmetric Gaussian noise. In the nomadic regime, we assume that the agents do not have any decoding ability. These agents process their channel observations and forward them to the final destination through unidirectional lossless links with a fixed capacity. We propose new achievable rates based on elementary compression and on Wyner-Ziv (WZ)or chief executive officer (CEO) processing, for both fast-fading and block-fading channels, as well as for general discrete channels. The simpler two agents scheme is solved, up to an implicit equation with a single variable. Limiting the nomadic transmitter to circular-symmetric Gaussian signaling, new upper bounds are derived, based on the vector version of the entropy power inequality. Several asymptotic settings are analyzed. In addition, the upper bounds are analytically shown to be tight for several examples, while numerical calculations reveal a rather small gap in a finite 2 times 2 setting. The advantage of the WZ approach over elementary compression is shown, where only the former can achieve the optimal diversity-multiplexing tradeoff (DMT).

Efficient maximum-likelihood decoding of spherical lattice codes

A new framework for efficient exact maximum-likelihood (ML) decoding of spherical lattice codes is developed. It employs a double-tree structure: The first is that which underlies established tree-search decoders; the second plays the crucial role of guiding the primary search by specifying admissible candidates and is our present focus. Lattice codes have long been of interest due to their rich structure, leading to decoding algorithms for unbounded lattices, as well as those with axis-aligned rectangular shaping regions. Recently, spherical Lattice Space-Time (LAST) codes were proposed to realize the optimal diversity-multiplexing tradeoff of MIMO channels. We address the so-called boundary control problem arising from the spherical shaping region defining these codes. This problem is complicated because of the varying number of candidates to consider at each search stage; it is not obvious how to address it effectively within the frameworks of existing decoders. Our proposed strategy is compatible with all sequential tree-search detectors, as well as auxiliary processing such as the MMSEGDFE and lattice reduction. We demonstrate the superior performance and complexity profiles achieved when applying the proposed boundary control in conjunction with two current efficient ML detectors and show an improvement of 1dB over the state-of-the-art at a comparable complexity.

Coding and Decoding for the Dynamic Decode and Forward Relay Protocol

We study the dynamic decode-and-forward (DDF) protocol for a single half-duplex relay, single-antenna channel with quasi-static fading. The DDF protocol is well known and has been analyzed in terms of the diversity-multiplexing tradeoff (DMT) in the infinite block length limit. We characterize the finite block length DMT and give new explicit code constructions. The finite block length analysis illuminates a few key aspects that have been neglected in the previous literature: 1) we show that one dominating cause of degradation with respect to the infinite block length regime is the event of decoding error at the relay; 2) we explicitly take into account the fact that the destination does not generally know a priori the relay decision time at which the relay switches from listening to transmit mode. Both of the above problems can be tackled by a careful design of the decoding algorithm. In particular, we introduce a decision rejection criterion at the relay based on Forney's decision rule (a variant of the Neyman-Pearson rule), such that the relay triggers transmission only when its decision is reliable. Also, we show that a receiver based on the generalized likelihood ratio test (GLRT) rule that jointly decodes the relay decision time and the information message achieves the optimal DMT. Our results show that no cyclic redundancy check (CRC) for error detection or additional protocol overhead to communicate the decision time are needed for DDF. Finally, we investigate the use of minimum mean-squared error generalized decision feedback equalizer (MMSE-GDFE) lattice decoding at both the relay and the destination, and show that it provides near-optimal performance at moderate complexity.

End-to-End Joint Antenna Selection Strategy and Distributed Compress and Forward Strategy for Relay Channels

Multihop relay channels use multiple relay stages, each with multiple relay nodes, to facilitate communication between a source and destination. Previously, distributed space-time codes were proposed to maximize the achievable diversity-multiplexing tradeoff; however, they fail to achieve all the points of the optimal diversity-multiplexing tradeoff. In the presence of a low-rate feedback link from the destination to each relay stage and the source, this paper proposes an end-to-end antenna selection (EEAS) strategy as an alternative to distributed space-time codes. The EEAS strategy uses a subset of antennas of each relay stage for transmission of the source signal to the destination with amplifying and forwarding at each relay stage. The subsets are chosen such that they maximize the end-to-end mutual information at the destination. The EEAS strategy achieves the corner points of the optimal diversity-multiplexing tradeoff (corresponding to maximum diversity gain and maximum multiplexing gain) and achieves better diversity gain at intermediate values of multiplexing gain, versus the best-known distributed space-time coding strategies. A distributed compress and forward (CF) strategy is also proposed to achieve all points of the optimal diversity-multiplexing tradeoff for a two-hop relay channel with multiple relay nodes.

Quantifying the Loss of Compress–Forward Relaying Without Wyner–Ziv Coding

The compress-and-forward (CF) strategy achieves the optimal diversity-multiplexing tradeoff (DMT) of a three-node half-duplex relay network in slow fading, under the assumption that the relay has perfect knowledge of all three channel coefficients and that the relay makes use of Wyner-Ziv (WZ) source coding with side information. This paper studies the achievable DMT of the same network when the relay is constrained to make use of standard (non-WZ) source coding. Under a short-term power constraint at the relay, using source coding without side information results in a significant loss in terms of the DMT. For multiplexing gains <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> les <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> / <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> , this loss can be fully compensated for by using power control at the relay. On the contrary, for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> isin ( <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> / <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> ,1), the loss with respect to WZ coding remains significant.

Space–Time Codes From Structured Lattices

We present constructions of space-time (ST) codes based on lattice coset coding. First, we focus on ST code constructions for the short block-length case, i.e., when the block length is equal to or slightly larger than the number of transmit antennas. We present constructions based on dense lattice packings and nested lattice (Voronoi) shaping. Our codes achieve the optimal diversity-multiplexing tradeoff (DMT) of quasi-static multiple-input multiple-output (MIMO) fading channels for any fading statistics, and perform very well also at practical, moderate values of signal-to-noise ratios (SNR). Then, we extend the construction to the case of large block lengths, by using trellis coset coding. We provide constructions of trellis coded modulation (TCM) schemes that are endowed with good packing and shaping properties. Both short-block and trellis constructions allow for a reduced complexity decoding algorithm based on minimum mean-squared error generalized decision feedback equalizer (MMSE-GDFE) lattice decoding and a combination of this with a Viterbi TCM decoder for the TCM case. Beyond the interesting algebraic structure, we exhibit codes whose performance is among the state-of-the art considering codes with similar encoding/decoding complexity.

Asymptotic-Information-Lossless Designs and the Diversity&amp;#x2013;Multiplexing Tradeoff

It is known that neither the Alamouti nor the V-BLAST scheme achieves the Zheng-Tse diversity-multiplexing tradeoff (DMT) of the multiple-input multiple-output (MIMO) channel. With respect to the DMT curve, the Alamouti scheme achieves the point corresponding to maximum diversity gain only, whereas V-BLAST meets only the point corresponding to maximum multiplexing gain. It is also known that D-BLAST achieves the optimal DMT for n transmit and n receive antennas, but only under the assumption that the leading and trailing zeros are ignored. When these zeros are taken into account, D-BLAST achieves the point corresponding to zero multiplexing gain, but not the point corresponding to zero diversity gain. The first scheme to achieve the DMT is the coding scheme of Yao and Wornell for the case of two transmit and two receive antennas. In this paper, we introduce the notion of an asymptotic-information-lossless (AILL) design and obtain a necessary and sufficient condition under which a design is AILL. Analogous to the result that full-rank designs achieve the point corresponding to the zero multiplexing gain of the optimal DMT curve, we show AILL to be a necessary and sufficient condition for a design to achieve the point on the DMT curve corresponding to zero diversity gain. We also derive a lower bound on the tradeoff achieved by designs from field extensions and show that the tradeoff is very close to the optimal tradeoff in the case of a single receive antenna. A lower bound to the tradeoff achieved by designs from division algebras is presented which indicates that these designs achieve both extreme points (corresponding to zero diversity and zero multiplexing gain) of the optimal DMT curve. Finally, we present simulations results for n transmit and n receive antennas, for n=2,3,4, which suggest that designs from division algebras are likely to have the property of being DMT achieving.

Diversity–Multiplexing Tradeoff in ISI Channels

The optimal diversity-multiplexing tradeoff curve for the intersymbol interference (ISI) channel is computed and various equalizers are analyzed using this performance metric. Maximum-likelihood signal decoding (MLSD) and decision feedback equalization (DFE) equalizers achieve the optimal tradeoff without coding, but zero forcing (ZF) and minimum mean-square-error (MMSE) equalizers do not. However if each transmission block is ended with a period of silence lasting the coherence time of the channel, both ZF and MMSE equalizers become diversity-multiplexing optimal. This suggests that the bulk of the performance gain obtained by replacing linear decoders with computationally intensive ones such as orthogonal frequency-division multiplexing (OFDM) or Viterbi, can be realized in much simpler fashion-with a small modification to the transmit scheme.

Hybrid Forwarding in Cooperative Communications

A cooperative communication scheme was developed in which the relay node performs hybrid forwarding, where the relay node will not forward data for a transmission pair unless the corresponding direct transmission fails, and the relay node will not forward data for a transmission pair unless the quality of the channel between the corresponding source and the relay is good. A performance analysis of the outage probability and the diversity-multiplexing tradeoff for the scheme shows that the forwarding mechanism efficiently reduces fading in wireless channels by achieving full diversity order of magnitude improvement. A considerable signal-to-noise ratio gain is provided by this scheme compared with existing cooperation schemes with fixed forwarding. This flexible forwarding mechanism enables the scheme to provide the optimal diversity-multiplexing tradeoff performance.

A Case for Amplify–Forward Relaying in the Block-Fading Multiple-Access Channel

This correspondence demonstrates the significant gains that multiple-access users can achieve from <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sharing</i> a single amplify-forward relay in slow-fading environments. The proposed protocol, namely, multiple-access amplify-forward (MAF), allows for a low-complexity relay and achieves the optimal diversity-multiplexing tradeoff (DMT) at high multiplexing gains. Analysis of the protocol further reveals that it outperforms both the compress-forward strategy at low multiplexing gains and the dynamic decode-forward protocol at high multiplexing gains. An interesting feature of the proposed protocol is that, at high multiplexing gains, it resembles a multiple-input single-output (MISO) system, and at low multiplexing gains, it provides each user with the same DMT as if there were no contention for the relay from the other users.

On the Optimality of the ARQ-DDF Protocol

In this correspondence, the performance of the automatic repeat request-dynamic decode and forward (ARQ-DDF) cooperation protocol is analyzed in two distinct scenarios. The first scenario is the multiple access relay channel where a single relay is dedicated to simultaneously help two multiple access users. For this setup, it is shown that the ARQ-DDF protocol achieves the channel's optimal diversity multiplexing tradeoff (DMT). The second scenario is the cooperative vector multiple access channel where two users cooperate in delivering their messages to a destination equipped with two receiving antennas. For this setup, a new variant of the ARQ-DDF protocol is developed where the two users are purposefully instructed not to cooperate in the first round of transmission. Lower and upper bounds on the achievable DMT are then derived. These bounds are shown to converge to the optimal tradeoff as the number of transmission rounds increases.

Optimal Space–Time Codes for the MIMO Amplify-and-Forward Cooperative Channel

In this work, we extend the nonorthogonal amplify-and-forward (NAF) cooperative diversity scheme to the multiple-input multiple-output (MIMO) channel. A family of space-time block codes for a half-duplex MIMO NAF fading cooperative channel with N relays is constructed. The code construction is based on the nonvanishing determinant (NVD) criterion and is shown to achieve the optimal diversity-multiplexing tradeoff (DMT) of the channel. We provide a general explicit algebraic construction, followed by some examples. In particular, in the single-relay case, it is proved that the Golden code and the 4times4 Perfect code are optimal for the single-antenna and two-antenna cases, respectively. Simulation results reveal that a significant gain (up to 10 dB) can be obtained with the proposed codes, especially in the single-antenna case