Full-diversity Space-time Codes Research Articles

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.

Space-Time Encoded MISO Broadcast Channel With Outdated CSIT: An Error Rate and Diversity Performance Analysis

Studies of the MISO Broadcast Channel (BC) with delayed Channel State Information at the Transmitter (CSIT) have so far focused on the sum-rate and Degrees-of-Freedom (DoF) region analysis. In this paper, we investigate for the first time, the error rate performance at finite SNR and the diversity-multiplexing tradeoff (DMT) at infinite SNR of a space-time encoded transmission over a two-user MISO BC with delayed CSIT. We consider the so-called MAT protocol obtained by Maddah-Ali and Tse, which was shown to provide 33% DoF enhancement over TDMA. While the asymptotic DMT analysis shows that MAT is always preferable to TDMA, the Pairwise Error Probability analysis at finite SNR shows that MAT is in fact not always a better alternative to TDMA. Benefits can be obtained over TDMA only at a very high rate or once concatenated with a full-rate full-diversity space-time code. The analysis is also extended to spatially correlated channels, and the influence of transmit correlation matrices and user pairing strategies on the performance are discussed. Relying on statistical CSIT, signal constellations are further optimized to improve the error rate performance of MAT and make it insensitive to user orthogonality. Finally, other transmission strategies relying on delayed CSIT are discussed.

Low-Complexity Detection of Full-Rate SFBC in BICM-OFDM Systems

Last-generation and future wireless communication standards, such as DVB-T2 or DVB-NGH, are including multi-antenna transmission and reception in order to increase bandwidth efficiency and receiver robustness. The main goal is to combine diversity and spatial multiplexing in order to fully exploit the multiple-input multiple-output (MIMO) channel capacity. Full-rate full-diversity (FRFD) space-time codes (STC) such as the Golden code are studied for that purpose. However, despite their larger achievable capacity, most of them present high complexity for soft detection, which hinders their combination with soft-input decoders in bit-interleaved coded modulation (BICM) schemes. This article presents a novel low-complexity soft detection algorithm for the reception of FRFD space-frequency block codes in BICM orthogonal frequency-division multiplexing (OFDM) systems. The proposed detector maintains a reduced and fixed complexity, avoiding the variable nature of the list sphere decoder (LSD) due to its dependence on the noise and channel conditions. Complexity and simulation-based performance results are provided which show that the proposed detector performs close to the optimal log-maximum a posteriori (MAP) detection in a variety of DVB-T2 broadcasting scenarios.

Fast Essentially Maximum Likelihood Decoding of the Golden Code

The Golden code is a full-rate full-diversity space-time code which has been incorporated in the IEEE 802.16 (WiMAX) standard. The worst case complexity of a tree-based sphere decoder for a square QAM constellation is O(N3), where N is the size of the underlying QAM constellation; the worst case will dominate average decoding complexity on any channel with a significant line of sight component. In this paper, we present a simple algorithm with quadratic complexity for decoding the Golden code that can be employed by mobile terminals with either one or two receive antennas, that is resilient to near singularity of the channel matrix, and that gives essentially maximum likelihood (ML) performance. Dual use is an advantage, since there will likely be some IEEE 802.16 mobile terminals with one receive antenna and some with two antennas. The key to the quadratic algorithm is a maximization of the likelihood function with respect to one of the pair of signal points conditioned on the other. This choice is made by comparing the determinants of two covariance matrices, and the underlying geometry of the Golden code guarantees that one of these choices is good with high probability.

Power Allocation Strategies for Distributed Space-Time Codes in Amplify-and-Forward Mode

We consider a wireless relay network with Rayleigh fading channels and apply distributed space-time coding (DSTC) in amplify-and-forward (AF) mode. It is assumed that the relays have statistical channel state information (CSI) of the local source-relay channels, while the destination has full instantaneous CSI of the channels. It turns out that, combined with the minimum SNR based power allocation in the relays, AF DSTC results in a new opportunistic relaying scheme, in which the best relay is selected to retransmit the source's signal. Furthermore, we have derived the optimum power allocation between two cooperative transmission phases by maximizing the average received SNR at the destination. Next, assuming M-PSK and M-QAM modulations, we analyze the performance of cooperative diversity wireless networks using AF opportunistic relaying. We also derive an approximate formula for the symbol error rate (SER) of AF DSTC. Assuming the use of full-diversity space-time codes, we derive two power allocation strategies minimizing the approximate SER expressions, for constrained transmit power. Our analytical results have been confirmed by simulation results, using full-rate, full-diversity distributed space-time codes.

Serially concatenated space time convolutional codes and continuous phase modulation

This paper addresses space time convolutional code design using continuous phase modulation (CPM). The possibility of constructing full diversity space time codes is investigated. A linear modulation approximation to CPM is done. Using the Gram-Schmidt orthogonalization transform the CPM signal is generated as a vector with finite energy in a different Euclidean space. A serially concatenated CPM construction is considered in searching channel codes which are able to exploit maximum diversity. Design criteria based on the encoding scheme are derived for an arbitrary number of transmit antennas. The investigations are done for a quasi-static Rayleigh fading channel.

The Icosian Code and the $E_8$ Lattice: A New $4\,\times\,4$ Space–Time Code With Nonvanishing Determinant

This paper introduces a new rate-2, full-diversity space-time code for four transmit antennas and one receive antenna. The 4times4 codeword matrix consists of four 2times2 Alamouti blocks with entries from <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> ,radic5) , and these blocks can be viewed as quaternions which in turn represent rotations in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> . The Alamouti blocks that appear in a codeword are drawn from the icosian ring consisting of all linear combinations of 120 basic rotations corresponding to symmetries of the icosahedron. This algebraic structure is different from the Golden code, but the complex entries are taken from a common underlying field. The minimum determinant is bounded below by a constant that is independent of the signal constellation, and the new code admits a simple decoding scheme that makes use of a geometric correspondence between the icosian ring and the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sub> lattice.

Serially Concatenated Space-Time Coded Continuous Phase Modulated Signals

We investigate the detection of space time codes (STC) modulated using continuous phase modulation (CPM), in quasi-static fading channels. A symbol-by-symbol iterative detector with an optimum front-end is derived, and its reduced-complexity implementation is considered. Also, we have introduced several full-rank STC-CPM systems, constructed by combining full-diversity STCs with simple, widely-used CPM schemes, and introducing a small frequency offset. According to the simulation results, the bit error rate (BER) performance of the detector for quasi-static fading is around 1 dB from the case without fading. Furthermore, a comparison of the performance of the proposed detector with results from previous work on STC-CPM showed an improvement in the range of 2-3 dB. Moreover, the detector is robust to the errors in estimating the channel state information (CSI), which is a desirable feature for practical implementation.

Super‐orthogonal space–time trellis codes for two transmit antennas in fast fading channels

AbstractSuper‐orthogonal space–time trellis codes (SOSTTC) are full rate, full diversity space–time codes with high coding gains for quasi‐static fading channels. In this paper, this design approach is extended to fast fading channels and new SOSTTC are proposed for BPSK and QPSK modulations based on the design criteria valid for this type of channels. The new full rate codes have 4‐ and 16‐state trellises for BPSK and QPSK cases, respectively, to avoid parallel transitions which restrict the error performance. The frame error performances are evaluated through computer simulations and it is shown that the proposed codes have superior performance in the fast fading case compared to their counterparts previously given in the literature for fast and quasi‐static fading channels. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Trace-Orthonormal Full-Diversity Cyclotomic Space–Time Codes

In this paper, we consider the design of full-diversity space-time codes for a coherent multiple-input multiple-output (MIMO) communication system. Starting from both the information theoretic and detection error viewpoints, we first establish that a desirable property for general linear dispersion (LD) codes is to have an interunitary as well as an intraunitary structure-a structure we call trace-orthonormality. By imposing the trace-orthonormal structure on an LD code and applying cyclotomic number theory, we establish, for an arbitrary number of transmitter and receiver antennas, a systematic and simple method to jointly design a unitary cyclotomic matrix, the Diophantine number, and the corresponding constellation for an LD code. As a result, this enables us to construct full-diversity rectangular cyclotomic LD codes with any symbol transmission rate less than or equal to the number of transmitter antennas. In addition, for the case when the number of transmitter antennas is greater than the number of receiver antennas, by taking advantage of the delay, we also arrive at the design of a special trace-orthonormal full-diversity cyclotomic space-time block code which, for the number of transmitter antenna being equal to 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , can be proved to minimize the worst case pairwise error probability of a maximum-likelihood (ML) detector for a q-ary quadrature amplitude modulation (QAM) signal constellation and, therefore, achieves optimal coding gain. Computer simulations show that these codes have bit-error performance advantages over currently available codes

Linear Decoupled and Quasi-Decoupled Space–Time Codes

Space-time transmit diversity results in coupling of transmitted symbols across different antennas, which increases the complexity of maximum-likelihood decoding. Symbol coupling can be completely or partially avoided if the space-time code (STC) satisfies specific decoupling conditions; examples of such codes are orthogonal space-time block codes and quasi-orthogonal codes. In this letter, we study decoupling conditions for a linear full-diversity STC. Quasi-decoupled codes are proposed as a partially decoupled full-diversity STC of any rate for any number of transmit antennas with minimum decoding delay. By optimizing the coding gain of quasi-decoupled codes, it is shown that quasi-orthogonal codes have competitive performance with respect to the Alamouti code, and the more-recent threaded algebraic space-time (TAST) codes and ABBA codes. A general full-diversity decoupling condition is considered, and the general solution to this case, which also encompasses previously known orthogonal STCs, is derived

Space-Time Cooperation Diversity Using High-Rate Codes

In a multi-user communication system, cooperative diversity allows single-antenna mobile sets to achieve transmit diversity. Cooperative diversity improves the communication capacity and enhances the robustness of a wireless link when a single channel alone is not reliable. In this paper, a novel cooperative diversity scheme is introduced that enables simultaneous transmission of non-redundant data from all cooperative terminals. By taking advantages of both high-rate full-diversity space-time codes and the cooperative diversity, the proposed method provides high diversity gain beyond the number of physical transmit antennas without compromising the data rates. In essence, the proposed system aims at higher data rates over the non-cooperative counterpart, while maintaining the full diversity gain.

Novel full diversity space time codes

A novel full diversity space-time trellis code, referred to as an assembled space-time trellis code (ASTTC), is presented in this letter. For this scheme, space-time trellis coded signals are first linearly transformed, and then the transformed signals are coded by using the Alamouti space-time block code. A new design criterion is proposed. It is shown that the ASTTCs can achieve not only the full diversity order but also a significant coding gain determined by the minimum weighted code distance (MWCD). Based on this design criterion, a new 4-state code is derived by a systematic code search. Simulation results show that the new ASTTC is superior by about 1.7 dB to the TSC (TarokhSeshadri-Calderbank) code at the frame error rate (FER) of 1.0e-2 over quasi-static fading channels.

Full-rate full-diversity space-time code for four-transmit-antenna systems

This paper introduces a low-complexity full-rate full-diversity space-time (ST) code for wireless communication systems with four transmit antennas. This novel code is designed by combining delay-diversity transmission with Alamouti's orthogonal ST block code. Analytical and computer simulation results show that this code yields significant frame-error-rate (FER) performance gains over some previously reported full-rate full-diversity codes for four transmit antennas.

Intercarrier Interference Self-Cancellation Space-Frequency Codes for MIMO-OFDM

Space-frequency (SF) codes that exploit both spatial and frequency diversity can be designed using orthogonal frequency division multiplexing (OFDM). However, OFDM is sensitive to frequency offset (FO), which generates intercarrier interference (ICI) among subcarriers. We investigate the pair-wise error probability (PEP) performance of SF codes over quasistatic, frequency selective Rayleigh fading channels with FO. We prove that the conventional SF code design criteria remain valid. The negligible performance loss for small FOs (less than 1%), however, increases with FO and with signal to noise ratio (SNR). While diversity can be used to mitigate ICI, as FO increases, the PEP does not rapidly decay with SNR. Therefore, we propose a new class of SF codes called ICI self-cancellation SF (ISC-SF) codes to combat ICI effectively even with high FO (10%). ISC-SF codes are constructed from existing full diversity space-time codes. Importantly, our code design provide a satisfactory tradeoff among error correction ability, ICI reduction and spectral efficiency. Furthermore, we demonstrate that ISC-SF codes can also mitigate the ICI caused by phase noise and time varying channels. Simulation results affirm the theoretical analysis.

A new full-rate full-diversity orthogonal space-time block coding scheme

In this work, we present a new space-time orthogonal coding scheme with full-rate and full-diversity. The proposed space-time coding scheme can be used on quaternary phase-shift keyed (QPSK) transceiver systems with four transmit antennas and any number of receivers. An additional feature is that the coded signals transmitted through all four transmit antennas do not experience any constellation expansion. The performance of the proposed coding scheme is studied in comparison with that of 1/2-rate full-diversity orthogonal space-time code, quasi-orthogonal code, as well as constellation-rotated quasi-orthogonal code. Our study shows that the proposed coding scheme offers full rate and outperforms the 1/2-rate orthogonal codes as well as full-rate quasi-orthogonal codes when the signal-to-noise increases. Compared to the constellation-rotated quasi-orthogonal codes (the improved QO scheme), the newly proposed code has the advantage of not expanding the signal constellation at each transmit antenna. The performance of the newly proposed code is comparable to that of the improved QO scheme.

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.

A rank criterion for QAM space-time codes

Space-time coding has been studied extensively as a powerful error correction coding for systems with multiple transmit antennas. An important design goal is to maximize the level of space diversity that a code can achieve. Toward this goal, the only systematic algebraic coding theory so far is binary rank theory by Hammons and El Gamal (see ibid. vol. 46, p.524-42, 2000) for binary phase-shift keying (BPSK) modulated codes defined over binary field and quaternary phase-shift keying (QPSK) modulated codes defined over modulo four finite ring. To design codes with higher bandwidth efficiency, we develop an algebraic rank theory to ensure full space diversity for 2/sup 2k/ quadrature and amplitude modulated (QAM) codes for any positive integer k. The theory provides the most general sufficient condition of full space diversity so far. It includes the BPSK binary rank theory as a special case. Since the condition is over the same domain that a code is defined, the full space diversity code design is greatly simplified. The usefulness of the theory is illustrated in examples, such as analyses of existing codes, constructions of new space-time codes with better performance, including the full diversity space-time turbo codes.

On the robustness of space-time coding

Space-time (ST) coding has emerged as one of the most promising technologies for meeting the challenges imposed by the wireless channel. This technology is primarily concerned with two-dimensional (2-D) signal design for multitransmit antenna wireless systems. Despite the progress in ST coding, several important questions remain unanswered. In a practical multiuser setting, one would expect different users to experience different channel conditions. This motivates the design of robust ST codes that exhibit satisfactory performance in various environments. In this paper, we investigate the robustness of ST codes in line-of-sight and correlated Rayleigh fading channels. We develop the design criteria that govern the performance of ST codes in these environments. Our analysis demonstrates that full-diversity ST codes are essential to achieving satisfactory performance in line-of-sight channels. We further show that a simple phase randomization approach achieves significant performance gains in the line-of-sight case without affecting the performance in Rayleigh fading channels. In the correlated fading environments, we characterize the achievable diversity order based on the number of diversity degrees of freedom in the channel. This characterization supports experimental observations that suggest that the quasistatic model is not a worst-case scenario and establishes the necessary tradeoff between the transmission rate and performance robustness. Finally, we consider the design of ST codes using some prior knowledge about the channel spatio-temporal correlation function.

Space-time turbo codes with full antenna diversity

In attempting to find a spectrally and power efficient channel code which is able to exploit maximum diversity from a wireless channel whenever available, we investigate the possibility of constructing a full antenna diversity space-time turbo code. As a result, both three-antenna and two-antenna (punctured) constructions are shown to be possible and very easy to find. To check the decodability and performance of the proposed codes, we derive non-binary soft-decoding algorithms. The performance of these codes are then simulated and compared with two existing space-time convolutional codes (one has minimum worst-case symbol-error probability; the other has maximal minimum free distance) having similar decoding complexity. As the simulation results show, the proposed space-time turbo codes give similar or slightly better performance than the convolutional codes under extremely slow fading. When fading is fast, the better distance spectra of the turbo codes help seize the temporal diversity. Thus, the performance advantage of the turbo codes becomes evident. In particular, 10/sup -5/ bit-error rate and 10/sup -3/ frame-error rate can be achieved at less than 6-dB E/sub b//N/sub 0/ with 1 b/s/Hz and binary phase-shift keying modulation. The practical issue of obtaining the critical channel state information (CSI) is also considered by applying an iteratively filtered pilot symbol-assisted modulation technique. The penalty when the CSI is not given a priori is about 2-3 dB.