Construction Of Space-time Codes Research Articles

Selective-Fading Multiple-Access MIMO Channels: Diversity-MultiplexingTradeoff and Dominant Outage Event Regions

We establish the optimal diversity-multiplexing (DM) tradeoff for coherent selective-fading multiple-access MIMO channels and provide corresponding code design criteria. As a byproduct, on the conceptual level, we find an interesting relation between the DM tradeoff framework and the notion of dominant error event regions, first introduced in the AWGN case by Gallager, IEEE Trans. IT, 1985. This relation allows us to accurately characterize the error mechanisms in MIMO fading multiple-access channels. In particular, we find that, for a given rate tuple, the maximum achievable diversity order is determined by a single outage event that dominates the total error probability exponentially in SNR. Finally, we examine the distributed space-time code construction proposed by Badr and Belfiore, Int. Zurich Seminar on Commun., 2008, using the code design criteria derived in the present paper.

Single-Carrier Rotation-Interleaved Space-Time Code for Frequency-Selective Fading Channels

A novel single-carrier-based space-time code construction scheme to exploit the advantages of a frequency-selective fading channel is investigated in this paper. The proposed construction scheme is based on multiplexing independent streams of phase-rotated space-time codes in a time-interleaved fashion. The advantage of such design is that it guarantees full space-time-multipath diversity by using traditional space-time codes or MIMO signaling schemes originally designed for flat fading channels as the constituent codes. Another advantage is that this approach incurs no loss in bandwidth efficiency and it alleviates the problem of high PAPR in OFDM-based space-time codes. By employing random or algebraic rotations, the design is potentially suitable for any number of transmit antennas or multipaths. The simulation results indicate that full space-time-multipath diversity is attained using this new approach, and comparisons with some existing space-time codes designed for frequency-selective channels are made to show its performance advantage.

Mathematics of Digital Communications: From Finite Fields to Group Rings and Noncommutative Algebra

In this paper, we present some recent instances of applying algebraic tools from different categories, in the design of digital communications systems. More specifically, we present a structure based on the elements of a group ring for constructing and encoding QC-LDPC codes. As another instance, we present the construction of space-time block codes from the rings of twisted Laurent series which include some instances of crossed product and non-crossed product division algebras.

Quadratic Extension Field Codes for Free Space Optical Intensity Communications

In this paper, we propose a space time code construction with a symbol rate of two for free space optical multiple input multiple output (FSO-MIMO) communication systems. Unlike the radio-frequency space time coding design that deals with complex symbols holding phase and magnitude information, the phase detection in wireless optical communications remains very challenging. This restricts optical MIMO codes to carry positive magnitude information belonging to a finite range. To meet these requirements, we propose to design the FSO-MIMO codes considering first the whole real space $\mathbb {R}$ , followed by a bipolar to unipolar conversion to fit in the intensity margin. We consider the MIMO configurations with two or four transmitters (lasers). We base our shape-preserving construction on quadratic extension fields that linearly combine the real symbols to form a full diversity space time code with a symbol rate equal to two. For each configuration, we show how to optimize the choice of the algebraic field numbers to jointly optimize the non-vanishing code determinant and the bipolar to unipolar attenuation factor.

Iterated Space-Time Code Constructions From Cyclic Algebras

We propose a full-rate iterative space-time code construction, which uses an algebraic space-time code in order to design a new space-time code of double the code rank. We give a condition for determining whether the resulting codes satisfy the full diversity property, and study their maximum likelihood decoding complexity with respect to sphere decoding. Particular emphasis is given to the asymmetric multiple input double output codes. In the process, we derive an interesting way of obtaining division algebras, and study their centers and maximal subfields.

Constructions of unitary space-time codes with full diversity

The problem of constructing unitary space-time codes with high diversity product has been studied in many prior works. Recently, constructions of parametric fully diverse unitary space-time codes for prime number antennas system have been introduced. In this paper, the authors propose new construction methods based on these constructions. And fully diverse codes of any number antennas are obtained from these constructions. Unitary space-time codes from present constructions are found to have better error performance than many best known ones.

Cyclic codes over [formula omitted

The ring in the title is perhaps the first noncommutative ring to have been used as alphabet for block codes. The original motivation was the construction of some quaternionic modular lattices from codes. The new application is the construction of space time codes obtained by concatenation from the Golden code. In this paper, we derive structure theorems for cyclic codes over that ring, and use them to characterize the lengths where self dual cyclic codes exist. These codes in turn give rise to formally self dual F4-codes.

Pragmatic Space-Time Trellis Codes: GTF-Based Design for Block Fading Channels

The pragmatic approach for the construction of space-time codes over block fading channels leads to low complexity encoders/decoders, making them suitable for various applications including cooperative communications. We propose a design methodology for pragmatic space-time codes, which is based on an extension of the concept of generalized transfer function. Our search algorithm provides code generators for arbitrary number of states, antennas, and fading rates. Numerical results show that the performance of the pragmatic space-time codes obtained by our design methodology is generally better than that of previously known space-time trellis codes in block fading channels.

Full-Diversity Space-Time Error Correcting Codes with Low-Complexity Receivers

We propose an explicit construction of full-diversity space-time block codes, under the constraint of an error correction capability. Furthermore, these codes are constructed in order to be suitable for a serial concatenation with an outer linear forward error correcting (FEC) code. We apply the binary rank criterion, and we use the threaded layering technique and an inner linear FEC code to define a space-time error-correcting code. When serially concatenated with an outer linear FEC code, a product code can be built at the receiver, and adapted iterative receiver structures can be applied. An optimized hybrid structure mixing MMSE turbo equalization and turbo product code decoding is proposed. It yields reduced complexity and enhanced performance compared to previous existing structures.

D-MG Tradeoff and Optimal Codes for a Class of AF and DF Cooperative Communication Protocols

Cooperative relay communication in a fading channel environment under the orthogonal amplify-and-forward (OAF), nonorthogonal and orthogonal selection decode-and-forward (NSDF and OSDF) protocols is considered here. The diversity-multiplexing gain tradeoff (DMT) of the three protocols is determined and DMT-optimal distributed space-time (ST) code constructions are provided. The codes constructed are sphere decodable and in some instances incur minimum possible delay.

Space–Time Codes Achieving the DMD Tradeoff of the MIMO-ARQ Channel

For the quasi-static, Rayleigh-fading multiple-input multiple-output (MIMO) channel with n t transmit and n r receive antennas, Zheng and Tse showed that there exists a fundamental tradeoff between diversity and spatial-multiplexing gains, referred to as the diversity-multiplexing gain (D-MG) tradeoff. Subsequently, El Gamal, Caire, and Damen considered signaling across the same channel using an L -round automatic retransmission request (ARQ) protocol that assumes the presence of a noiseless feedback channel capable of conveying one bit of information per use of the feedback channel. They showed that given a fixed number L of ARQ rounds and no power control, there is a tradeoff between diversity and multiplexing gains, termed the diversity-multiplexing-delay (DMD) tradeoff. This tradeoff indicates that the diversity gain under the ARQ scheme for a particular information rate is considerably larger than that obtainable in the absence of feedback. In this paper, a set of sufficient conditions under which a space-time (ST) code will achieve the DMD tradeoff is presented. This is followed by two classes of explicit constructions of ST codes which meet these conditions. Constructions belonging to the first class achieve minimum delay and apply to a broad class of fading channels whenever nr ges nt and either L\nt or nt\L. The second class of constructions do not achieve minimum delay, but do achieve the DMD tradeoff of the fading channel for all statistical descriptions of the channel and for all values of the parameters nr, nt, L.

Construction Methods for Asymmetric and Multiblock Space–Time Codes

In this paper, the need for the construction of asymmetric and multiblock space-time codes is discussed. Above the trivial puncturing method, i.e., switching off the extra layers in the symmetric multiple-input multiple-output (MIMO) setting, two more sophisticated asymmetric construction methods are proposed. The first method, called the block diagonal method (BDM), can be converted to produce multiblock space-time codes that achieve the diversity-multiplexing tradeoff (DMT). It is also shown that maximizing the density of the newly proposed block diagonal asymmetric space-time (AST) codes is equivalent to minimizing the discriminant of a certain order, a result that also holds as such for the multiblock codes. An implicit lower bound for the density is provided and made explicit for an important special case that contains e.g., the systems equipped with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4Tx</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+2Rx</i> antennas. Further, an explicit scheme achieving the bound is given. Another method proposed here is the smart puncturing method (SPM) that generalizes the subfield construction method proposed in earlier work by Hollanti and Ranto and applies to any number of transmitting and lesser receiving antennas. The use of the general methods is demonstrated by building explicit, sphere decodable codes using different cyclic division algebras (CDAs). Computer simulations verify that the newly proposed methods can compete with the trivial puncturing method, and in some cases clearly outperform it. The conquering construction exploiting maximal orders improves upon the punctured perfect code and the DjABBA code as well as the Icosian code. Also extensive DMT analysis is provided.

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.

Pragmatic Space-Time Codes for Cooperative Relaying in Block Fading Channels

We address the problem of construction of space-time codes for cooperative communications in block fading channels. More precisely, we consider a pragmatic approach based on the concatenation of convolutional codes and BPSK/QPSK modulation to obtain cooperative codes for relay networks, for which we derive the pairwise error probability, an asymptotic bound for frame error probability, and a design criterion to optimize both diversity and coding gain. Based on this framework, we set up a code search procedure to obtain a set of good pragmatic space-time codes (P-STCs) with overlay construction, suitable for cooperative communication with a variable number of relays in quasistatic channel, which outperform in terms of coding gain other space-time codes (STCs) proposed in the literature. We also find that, despite the fact that the implementation of pragmatic space-time codes requires standard convolutional encoders and Viterbi decoders with suitable generators and branch metric, thus having low complexity, they perform quite well in block fading channels, including quasistatic channel, even with a low number of states and relays.

On constructions of algebraic space-time codes with AM-PSK constellations satisfying rate-diversity tradeoff

Constructions of space-time codes having amplitude-modulated phase-shift keying (AM-PSK) constellations are presented in this paper. The first construction, termed p-radii construction, is obtained by extending Hammons' dyadic dual-radii construction to the cases when the size of the constellation is a power of a prime p, p /spl ges/ 2. The resultant code is optimal with respect to the rate-diversity tradeoff and has an AM-PSK constellation with signal points distributed over p- concentric circles in the complex plane, i.e., there are p radii. Also contained in this paper is the identification of rich classes of nontrivial subset-subcodes of the newly constructed space-time codes and it is shown that these subset-subcodes are again, all optimal. Finally, a new generalization of the super-unified construction by Hammons is presented. It is shown that codes obtained from several previously known constructions are subset-subcodes of the one derived from this generalized construction.

Nonlinear hierarchical space-time block codes: construction and regular MTCM design

The performance criteria of multiple-input multiple-output (MIMO) fading channels, diversity gain and coding gain, demand quite different coding techniques than those for Gaussian channels. This paper proposes a new approach to space-time block code (STBC) construction for coherent systems different from algebraic constructions (unitary, orthogonal) by explicitly manipulating the relationship between codewords rather than properties of the individual codeword. Algebraic constructions may incur performance losses of several decibels, relative to code sets determined by exhaustive search. By characterizing the structure of exhaustively searched code sets, a set of isometries between code words is determined. These isometries, in turn, suggest a greedy code construction method: nonlinear hierarchical codes (NHCs). NHCs of arbitrary block sizes/constellations can be constructed bottom-up by optimizing the coding gain layer-by-layer and reusing the optimized structure. The strong symmetry and layered structure make NHCs ideal constituent codes for regular multiple trellis-coded modulation (RMTCM). New RMTCM design procedures are proposed for various rates/block sizes/constellation sizes. The match between the distance spectrum of NHCs to the regular trellis structure optimizes coding gain directly, and full diversity is naturally maintained given the full rank of the constituent NHCs. Several factors which affect performance are analyzed and exploited to improve the overall performance. Set expansion is used to improve high-rate MTCM designs. In each design, the performance/rate/complexity tradeoffs are gracefully balanced and optimized. With limited growth in complexity, the proposed designs can achieve more than 2 dB gain over current MTCM designs.

Systematic space-time trellis code construction for correlated Rayleigh-fading channels

The potential for capacity increase in multiple-antenna wireless communication systems has drawn considerable attention to space-time codes. However, most of the existing space-time code construction methods have assumed ideal channel models: either quasi-static fading or fast fading. In this work, we propose a systematic space-time trellis code construction method for correlated fading channels. We derive the performance criteria that take into account both spatial and temporal channel correlation, assuming that the space-time correlation matrix is of full rank. Moreover, we introduce a new design criterion and analyze the properties of the space-time trellis codes satisfying the proposed criterion. Using the design criterion, we develop a code construction procedure that jointly considers diversity advantage and coding advantage for an arbitrary number of transmit antennas and any memoryless modulation. The flexibility of the proposed code construction method is demonstrated by designing space-time trellis codes for two, three, and four transmit antennas with quaternary phase-shift keying (QPSK), 8PSK (phase-shift keying), and 4ASK (amplitude-shift keying) modulations. The simulation results demonstrate that not only can our method generate new codes for an arbitrary number of antennas, but in some special cases, where previously proposed codes exist, our codes also perform very well.

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-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.

Performance evaluation of super-orthogonal space-time trellis codes using a moment generating function-based approach

A new class of space-time codes called super-orthogonal trellis codes was introduced that combine set-partitioning with a super set of orthogonal space-time block codes in such a way as to provide full diversity with increased rate and improved coding gain over previous space-time trellis code (STTC) constructions. Here, we extend the moment generating function-based method, which was previously applied to analyzing the performance of space-time block orthogonal and trellis codes, to the above-mentioned super-orthogonal codes. It is shown that the maximum-likelihood metric and expressions for the pairwise error probability previously developed for the Alamouti (1998) space-time block code combined with multidimensional trellis-coded modulation can be readily extended to the super-orthogonal case. As such, the evaluation of the pairwise error probability for the latter can be performed in a similar manner to that previously described with the specific results depending on the particular trellis code design.

On the design of algebraic space-time codes for MIMO block-fading channels

The availability of multiple transmit antennas allows for two-dimensional channel codes that exploit the spatial transmit diversity. These codes were referred to as space-time codes by Tarokh et al. (see ibid., vol.44, p.744-765, Mar. 1998) Most prior works on space-time code design have considered quasi-static fading channels. We extend our earlier work on algebraic space-time coding to block-fading channels. First, we present baseband design criteria for space-time codes in multi-input multi-output (MIMO) block-fading channels that encompass as special cases the quasi-static and fast fading design rules. The diversity advantage baseband criterion is then translated into binary rank criteria for phase shift keying (PSK) modulated codes. Based on these binary criteria, we construct algebraic space-time codes that exploit the spatial and temporal diversity available in MIMO block-fading channels. We also introduce the notion of universal space-time codes as a generalization of the smart-greedy design rule. As a part of this work, we establish another result that is important in its own right: we generalize the full diversity space-time code constructions for quasi-static channels to allow for higher rate codes at the expense of minimal reductions in the diversity advantage. Finally, we present simulation results that demonstrate the excellent performance of the proposed codes.