Perfect Space–Time Block Codes

In this paper, we introduce the notion of perfect space–time block codes (STBCs). These codes have full-rate, full-diversity, nonvanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We present algebraic constructions of perfect STBCs for 2, 3, 4, and 6 antennas

Cyclic division algebras: a tool for space-time coding

Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space–Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion requires that the difference of any two distinct codewords has full rank. Extensive work has been done on Space–Time coding, aiming at finding fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space–Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to improve the design of good codes. The aim of this work is to provide a tutorial introduction to the algebraic tools involved in the design of codes based on cyclic division algebras. The different design criteria involved will be illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property, and the diversity multiplexing trade-off. The final target is to give the complete mathematical background underlying the construction of the Golden code and the other Perfect Space–Time block codes

Golden Space-Time Trellis Coded Modulation

In this paper, we present a concatenated coding scheme for a high rate $2\times 2$ multiple-input multiple-output (MIMO) system over slow fading channels. The inner code is the Golden code \cite{Golden05} and the outer code is a trellis code. Set partitioning of the Golden code is designed specifically to increase the minimum determinant. The branches of the outer trellis code are labeled with these partitions. Viterbi algorithm is applied for trellis decoding. In order to compute the branch metrics a lattice sphere decoder is used. The general framework for code optimization is given. The performance of the proposed concatenated scheme is evaluated by simulation. It is shown that the proposed scheme achieves significant performance gains over uncoded Golden code.Comment: 33 pages, 13 figure

High-rate codes that are linear in space and time

Multiple-antenna systems that operate at high rates require simple yet effective space-time transmission schemes to handle the large traffic volume in real time. At rates of tens of bits per second per hertz, Vertical Bell Labs Layered Space-Time (V-BLAST), where every antenna transmits its own independent substream of data, has been shown to have good performance and simple encoding and decoding. Yet V-BLAST suffers from its inability to work with fewer receive antennas than transmit antennas-this deficiency is especially important for modern cellular systems, where a base station typically has more antennas than the mobile handsets. Furthermore, because V-BLAST transmits independent data streams on its antennas there is no built-in spatial coding to guard against deep fades from any given transmit antenna. On the other hand, there are many previously proposed space-time codes that have good fading resistance and simple decoding, but these codes generally have poor performance at high data rates or with many antennas. We propose a high-rate coding scheme that can handle any configuration of transmit and receive antennas and that subsumes both V-BLAST and many proposed space-time block codes as special cases. The scheme transmits substreams of data in linear combinations over space and time. The codes are designed to optimize the mutual information between the transmitted and received signals. Because of their linear structure, the codes retain the decoding simplicity of V-BLAST, and because of their information-theoretic optimality, they possess many coding advantages. We give examples of the codes and show that their performance is generally superior to earlier proposed methods over a wide range of rates and signal-to-noise ratios (SNRs)

Full-Rate, Full-Diversity, Finite Feedback Space-Time Schemes with Minimum Feedback and Transmission Duration

In this paper a MIMO quasi static block fading channel with finite N-ary delay-free, noise-free feedback is considered. The transmitter uses a set of N Space-Time Block Codes (STBCs), one corresponding to each of the N possible feedback values, to encode and transmit information. The feedback function used at the receiver and the N component STBCs used at the transmitter together constitute a Finite Feedback Scheme (FFS). Although a number of FFSs are available in the literature that provably achieve full-diversity, there is no known universal criterion to determine whether a given arbitrary FFS achieves full-diversity or not. Further, all known full-diversity FFSs for T<N_t where N_t is the number of transmit antennas, have rate at the most 1. In this paper a universal necessary condition for any FFS to achieve full-diversity is given, using which the notion of Feedback-Transmission duration optimal (FT-Optimal) FFSs - schemes that use minimum amount of feedback N given the transmission duration T, and minimum transmission duration given the amount of feedback to achieve full-diversity - is introduced. When there is no feedback (N=1) an FT-optimal scheme consists of a single STBC with T=N_t, and the universal necessary condition reduces to the well known necessary and sufficient condition for an STBC to achieve full-diversity: every non-zero codeword difference matrix of the STBC must be of rank N_t. Also, a sufficient condition for full-diversity is given for the FFSs in which the component STBC with the largest minimum Euclidean distance is chosen. Using this sufficient condition full-rate (rate N_t) full-diversity FT-Optimal schemes are constructed for all (N_t,T,N) with NT=N_t. These are the first full-rate full-diversity FFSs reported in the literature for T<N_t. Simulation results show that the new schemes have the best error performance among all known FFSs.Comment: 12 pages, 5 figures, 1 tabl

Full Diversity Unitary Precoded Integer-Forcing

We consider a point-to-point flat-fading MIMO channel with channel state information known both at transmitter and receiver. At the transmitter side, a lattice coding scheme is employed at each antenna to map information symbols to independent lattice codewords drawn from the same codebook. Each lattice codeword is then multiplied by a unitary precoding matrix ${\bf P}$ and sent through the channel. At the receiver side, an integer-forcing (IF) linear receiver is employed. We denote this scheme as unitary precoded integer-forcing (UPIF). We show that UPIF can achieve full-diversity under a constraint based on the shortest vector of a lattice generated by the precoding matrix ${\bf P}$. This constraint and a simpler version of that provide design criteria for two types of full-diversity UPIF. Type I uses a unitary precoder that adapts at each channel realization. Type II uses a unitary precoder, which remains fixed for all channel realizations. We then verify our results by computer simulations in $2\times2$, and $4\times 4$ MIMO using different QAM constellations. We finally show that the proposed Type II UPIF outperform the MIMO precoding X-codes at high data rates.Comment: 12 pages, 8 figures, to appear in IEEE-TW

Euclidean Distance Decomposition to Generate New 16-QAM and 64-QAM Space-Time Trellis Codes

International audienceIn this paper, the generation of 16-QAM and 64-QAM space-time trellis codes (STTCs) for several transmit antennas is considered. The main problem with an exhaustive search is the important time to find the best 2^{2n}-QAM STTCs, especially for great values of n and for great numbers of transmit antennas. In order to reduce this search time, an efficient method must be used to generate optimal 2^{2n}-QAM STTCs. Thus, a new method based on the decomposition of the Euclidean distance between 2 codewords is proposed to design the optimal STTCs. Thanks to this new method, the first 16-QAM STTCs with more than 2 transmit antennas and the first 64-QAM STTCs with 2 and 3 transmit antennas are proposed. The performance of these new codes is evaluated by simulation

Embedded Rank Distance Codes for ISI channels

Designs for transmit alphabet constrained space-time codes naturally lead to questions about the design of rank distance codes. Recently, diversity embedded multi-level space-time codes for flat fading channels have been designed from sets of binary matrices with rank distance guarantees over the binary field by mapping them onto QAM and PSK constellations. In this paper we demonstrate that diversity embedded space-time codes for fading Inter-Symbol Interference (ISI) channels can be designed with provable rank distance guarantees. As a corollary we obtain an asymptotic characterization of the fixed transmit alphabet rate-diversity trade-off for multiple antenna fading ISI channels. The key idea is to construct and analyze properties of binary matrices with a particular structure induced by ISI channels.Comment: Submitted to IEEE Transactions on Information Theor