Linear Dispersion Codes Research Articles

High-rate diversity across time and frequency using linear dispersion

To improve performance of orthogonal frequency division multiplexing (OFDM) for fading channels, this paper proposes increasing frequency and time diversity using linear dispersion codes (LDC-OFDM). Methods of LDC-OFDM processing are proposed for both zero-padding (ZP) and cyclic-prefix (CP) type guard intervals. A two-step-estimation (TSE) decoding strategy is proposed that decouples symbol estimation from LDC decoding. This paper analyzes the upper bound diversity order of LDC-CP-OFDM, which is equal to the full diversity order available in the channels. A criterion for full frequency-time diversity design is derived, a rate-one code is provided and performance is examined through simulations. This paper also investigates LDC-CP-OFDM and LDC-ZP-OFDM performance under imperfect channel estimation and low complexity receiver structures, respectively. In addition, TSE is shown to have performance close to that of full complexity one-step estimation (OSE).

Lattice-Based Linear Precoding for MIMO Channels With Transmitter CSI

Herein, the design of linear dispersion codes for multiple-input multiple-output communication systems is investigated. The receiver as well as the transmitter are assumed to have perfect knowledge of the channel, and the receiver is assumed to employ maximum likelihood detection. We propose to use linear precoding and lattice invariant operations to transform the channel matrix into a lattice generator matrix with large minimum distance separation. With appropriate approximations, it is shown that this corresponds to selecting lattices with good sphere-packing properties. Lattice invariant transformations are then used to minimize the power consumption. An algorithm for this power minimization is presented along with a lower bound on the optimization. Numerical results indicate significant gains by using the proposed method compared to channel diagonalization with adaptive bit loading.

Linear-Dispersion Division Multiple-Access for MIMO systems

Comprehensive study on novel Linear-Dispersion Division Multiple-Access (LDDMA) for multi-user uplink Multiple-Input Multiple-Output (MIMO) systems is proposed. In the new multiplexing scheme, each user’s information symbol is dispersed by a User-Specific Matrix (USM) both in space and time domain and linearly combined at base-station side. And a simple random search algorithm, based on capacity maximization criteria, is developed to generate a bank of USMs. Simulation results are presented to demonstrate the advantages of LDDMA. When the Bit Error Rate (BER) reaches 10−3, the performance gains are 3dB and 5dB, compared with Time-Division Linear Dispersion Codes (TD-LDC) and BLAST, respectively.

An insight into space–time block codes using Hurwitz–Radon families of matrices

It is shown that for four-transmitter systems, a family of four-by-four unit-rate complex quasi-orthogonal space–time block codes, where each entry equals a symbol variable up to a change of sign and/or complex conjugation, can be generated from any two independent codes via elementary operations. The two independent groups of codes in the family generally have different properties of diversity, but the codes in each group have the same diversity provided that the differential symbol constellation is symmetric. It is also shown that for four-transmitter systems, an eight-by-four unit-rate complex linear dispersion space–time block code can be constructed by using Hurwitz–Radon families of matrices of size eight such that diversity three is guaranteed even when all symbols are independently selected from any given constellation. This code is so far the only known unit-rate linear dispersion code that has diversity no less than three for four transmitters under any given constellation.

A Common Function of Different Union Bounds for Optimization and BER Evaluation of Space–Time codes

Optimizing Space-Time (S---T) codes, in terms of minimum error rates, based on Union bounds often produces results which vary with signal-to-noise ratio (SNR) and so are not easy to use. Error-rate evaluation of S---T codes using Union bounds sometimes requires very heavy computational loads and so is impractical. In this paper, a Common function shared by different Union bounds is derived and proposed for optimization of S---T codes. The Common function produces the optimization results which are SNR independent and so requires a substantially lower computation load. Results of Monte-Carlo simulation on two 2 × 2 rotation-based diagonal S---T codes (D codes) and two Linear-Dispersion codes (LDCs) show that the bit-error-rate (BER) performances using the optimization results based on the Common function, the exact Union bound (EUB) and Chernoff Union bound (CUB) are all nearly identical. The Common function is subsequently used to develop a modified Union bound (MUB) and modified Chernoff Union bound (MCUB). Results of numerical calculations show that the MUB can provide a good compromise between the accuracy and computational load for BER evaluation.

Optimal linear dispersion codes for correlated MIMO channels

The design of space-time codes for frequency flat, spatially correlated MIMO fading channels is considered. The focus of the paper is on the class of space-time block codes known as linear dispersion (LD) codes, introduced by Hassibi and Hochwald. The LD codes are optimized with respect to the mutual information between the inputs to the space-time encoder and the output of the channel. The use of the mutual information as both a design criterion and a performance measure is justified by allowing soft decisions at the output of the space-time decoder. A spatial Fourier (virtual) representation of the channel is exploited to allow for the analysis of MIMO channels with quite general fading statistics. Conditions, known as generalized orthogonal conditions (GOC's), are derived for an LD code to achieve an upper bound on the mutual information, with the understanding that LD codes that achieve the upper bound, if they exist, are optimal. Explicit code constructions and properties of the optimal power allocation schemes are also derived. In particular, it is shown that optimal LD codes correspond to beamforming to a single virtual transmit angle at low SNR, and a necessary and sufficient condition for beamforming to be optimal is provided. Due to the nature of the code construction, it is further observed that the optimal LD codes can be designed to adapt to the statistics of different scattering environments. Finally, numerical results are provided to illustrate the optimal code design for three examples of sparse scattering environments. The performance of the optimal LD codes for these scattering environments is compared with that of LD codes designed assuming the i.i.d. Rayleigh fading (rich scattering) model, and it is shown that the optimal LD codes perform significantly better. The optimal LD codes are also compared to beamforming LD codes and it is shown that beamforming is nearly optimal over a range of SNR's of interest.

Multilayered linear dispersion codes for flat fading multiple-input multiple-output channels

The authors investigate the design of linear dispersion (LD) codes, aiming at flexible encoding schemes that allow various rate-performance tradeoffs under a common coding structure. First, the capacity of LD codes is studied. It is shown that the maximum attainable multiplexing gain of a linear dispersion code is the number of symbols per channel use of the code (i.e. coding rate in symbols). In addition, conditions on the construction of linear dispersion matrices for various multiplexing gains are established. A general multilayered linear dispersion coding scheme that allows various multiplexing gains is then proposed. In the proposed scheme, coding rate can be adapted by employing different numbers of dispersion matrices. Furthermore, phase shifting among input symbols is applied to optimise the error performance without loss of multiplexing gain. The construction of dispersion matrices and the optimisation of the phase shifts together constitute a structured approach for the design of linear dispersion codes. Simulation results demonstrate that the new codes outperform conventional LD codes at various data rates.

LDC in MIMO Ricean Channels: Optimal Transmit Strategy With MMSE Detection

In this paper, we study a MIMO system with a transmitter using a linear dispersion code (LDC) and a linear minimum mean square-error (MMSE) detector at the receiver in a Ricean flat-fading environment. We assume that the receiver has perfect channel state information and the transmitter knows only the mean channel matrix either by feedback or channel estimation. The focus of our work is the analysis of the optimal transmit strategy using different types of LDC. On the one hand, we consider spatial multiplexing schemes that achieve high data rates, but sacrifice diversity. On the other hand, we have schemes that achieve full diversity like quasi-orthogonal space-time block codes or orthogonal space-time block code. Depending on the LDC in use, the optimization problem is either convex or nonconvex. For both of these classes of LDC, we first derive the properties of the average normalized MSE and then analyze the impact of the mean component on the MSE, the optimal transmit strategy and the optimal power allocation. Finally, we derive some bounds on the error rate performance for different scenarios with the MMSE receiver.

Diversity–multiplexing trade-off performance of linear dispersive codes

The diversity-multiplexing trade-off function provides a complete view to evaluate both the diversity-based and the multiplexing-based space-time schemes. The trade-off performance of the linear dispersive codes (LDCs) over both the block fading and the fast fading channels is analysed. Results show that LDCs can achieve better trade-off performance than some of the existing space-time schemes over block fading channels. For fast fading channels, the time factor is clearly expressed in the trade-off function, and the analysis shows that LDCs can achieve both the space diversity and the time diversity simultaneously.

Closed-loop linear dispersion coded eigen-beam transmission and its capacity

A flexible closed-loop multiple-input-multiple-output (MIMO) system amalgamating linear dispersion codes and eigen-beam transmission is proposed that achieves a higher capacity than the equivalent open‐loop scheme, at the cost of feeding back a modest number of feedback bits from the mobile to the base station using Grassmannian beamforming-vector quantisation.

Statistical-mechanics-based analysis of multiuser MIMO channels with linear dispersion codes

We provide a mean field description of a linear dispersion (LD) coded multiuser multiple-input multiple-output (MU-MIMO) channel, using the replica method. Our main statement is that the LD coded MU-MIMO channel is asymptotically decoupled into a bank of single-user space-time coded MIMO channels, in which their space-time coding schemes are determined by the statistics of the original LD codes. We apply our mean field theory to evaluating the performance of the LD coded MU-MIMO channel on the basis of the outage formulation.

A novel linear Differential Space-Time Modulation

Based on the Complex Orthogonal Linear Dispersion (COLD) code, a novel linear Differential Space-Time Modulation (DSTM) design is proposed in this paper. Compared with the existing nonlinear DSTM schemes based on group codes, the proposed linear DSTM scheme is easier to design, enjoys full diversity and allows for a simplified differential receiver, which can detect the transmitted symbols separately. Furthermore, compared with the existing linear DSTM based on orthogonal design, our new construction can be applied to any number of transmit antennas. Similar to other algorithms, the proposed scheme also can be demodulated with or without channel estimates at the receiver, but the performance degrades approximately by 3dB when estimates are not available.

Linear Dispersion Codes with Limited Feedback

LDCs system with finite-rate error-free feedback is proposed in this letter. The optimal transmission codeword is selected at the receiver and the codeword index is sent to the transmitter. A simple random search algorithm is introduced for codebook generation. Moreover, the max-min singular value criterion is adopted for codeword selection. Simulation results showed that, with only 3-4 feedback bits, the low-complexity Zero-Forcing receiver can approach the Maximum-Likelihood (ML) performance.

Minimum Error-Rate Linear Dispersion Codes for Cooperative Relays

Cooperative diversity systems have recently been proposed as a solution to provide spatial diversity for terminals where multiple antennas are not feasible to be implemented. As in multiple-input-multiple-output systems, space-time codes can be used to efficiently exploit the increase in capacity provided in cooperative diversity systems. In this paper, we propose a two-layer linear dispersion (LD) code for cooperative diversity systems and derive a simulation-based optimization algorithm to optimize the LD code and power allocation in terms of block error rate. The proposed code design paradigm can obtain optimal codes under arbitrary fading statistics. Performance comparisons are made to other cooperative diversity schemes. The effect that distances between source, relays, and destination terminals have on the energy allocation between the broadcast and cooperative intervals is also studied. Cooperative diversity, gradient estimation, linear dispersion (LD) codes, multiple-input-multiple-output (MIMO), stochastic approximation.

Design of Linear Dispersion Codes for Practical MIMO-OFDM Systems

In this work, we consider the problem of designing space-time-frequency linear dispersion (LD) codes in wideband multiple-input multiple-output (MIMO) antenna systems employing orthogonal-frequency-division-multiplexing (OFDM). Three criteria are presented and discussed in detail, which involve: 1) minimizing the average uncoded block error rate, 2) maximizing the ergodic mutual information, and 3) a two-step procedure considering the optimization of the ergodic mutual information as well as the average uncoded block error rate, respectively. For any set of subcarriers, any number of OFDM symbol intervals, any number of transmit/receive antennas and any statistical fading channel model, the corresponding optimized LD code matrices are numerically computed via a stochastic gradient descent algorithm assuming either maximum likelihood or linear zero-forcing decoding. As an application of the proposed criteria, code design examples are presented for practical next-generation communication systems which operate over a realistic 3GPP spatial channel model. The robustness of the resulting codes over several operating conditions is comprehensively demonstrated via numerical simulations

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

Design and Performance of Space–Time Codes for Spatially Correlated MIMO Channels

Space-time code (STC) designs classically rely on the assumption of independent and identically distributed (i.i.d.) Rayleigh channels. However, poor scattering conditions may have detrimental effects on the performance of STCs. In this letter, we derive code-design criteria leading to robust STCs in a large variety of slow-fading propagation conditions. No channel knowledge is assumed at the transmitter. Codes satisfying these criteria are shown to perform much better on real-world channels than codes designed only for i.i.d. channels. As examples, the robustness of various spatial multiplexing schemes, linear dispersion codes, and space-time trellis codes is discussed based on those criteria

Dynamic Nulling-and-Canceling for Efficient Near-ML Decoding of MIMO Systems

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> It is known that conventional nulling-and-canceling (NC) detection for multiple-input/multiple-output (MIMO) systems cannot exploit all of the available diversity, and, thus, its performance is significantly inferior to that of maximum likelihood (ML) detection. Conventional NC employs the layerwise postequalization signal-to-noise ratios (SNRs) as reliability measures for layer sorting. These SNRs are &lt;emphasis emphasistype="boldital"&gt;average&lt;/emphasis&gt; quantities that do not depend on the received vector. In this paper, we propose the novel &lt;emphasis emphasistype="boldital"&gt;dynamic nulling-and-canceling&lt;/emphasis&gt; (DNC) technique that uses approximate &lt;emphasis emphasistype="boldital"&gt;a posteriori&lt;/emphasis&gt; probabilities as measures of layer reliability. The DNC technique is a minimum mean-square error (MMSE) nulling scheme combined with an improved “dynamic” layer sorting rule that exploits the information contained in the current received vector. We calculate the error probability of DNC for a simple special case and show that it is upper bounded by the error probability of conventional NC. Simulation results are presented for spatial multiplexing systems and for systems using linear dispersion codes. It is demonstrated that the DNC technique can yield near-ML performance for a wide range of system sizes and channel SNRs at a fraction of the computational complexity of the sphere-decoding algorithm for ML detection. </para>

Distributed Space-Time Coding in Wireless Relay Networks

We apply the idea of space-time coding devised for multiple-antenna systems to the problem of communications over a wireless relay network with Rayleigh fading channels. We use a two-stage protocol, where in one stage the transmitter sends information and in the other, the relays encode their received signals into a linear dispersion (LD) code, and then transmit the coded signals to the receive node. We show that for high SNR, the pairwise error probability (PEP) behaves as (logP/P)min{TH}, with T the coherence interval, that is, the number of symbol periods during which the channels keep constant, R the number of relay nodes, and P the total transmit power. Thus, apart from the log P factor, the system has the same diversity as a multiple-antenna system with R transmit antennas, which is the same as assuming that the R relays can fully cooperate and have full knowledge of the transmitted signal. We further show that for a network with a large number of relays and a fixed total transmit power across the entire network, the optimal power allocation is for the transmitter to expend half the power and for the relays to collectively expend the other half. We also show that at low and high SNR, the coding gain is the same as that of a multiple-antenna system with R antennas. However, at intermediate SNR, it can be quite different, which has implications for the design of distributed space-time codes

Automatic design of orthogonal or near-orthogonal linear dispersive space-time block codes

This paper considers a wireless multiple-input multiple-output (MIMO) communication system in a frequency-nonselective scenario with spatially uncorrelated Rayleigh fading channel coefficients and investigates the design of linear dispersive (LD) space-time block codes. Efficient LD codes are obtained by optimizing the constituent weight matrices so that an upper bound on the union bound of the codeword error probability is minimized. Interestingly, the proposed design procedure automatically generates LD codes that either correspond to, or are close to, the well-known class of orthogonal space-time block (OSTB) codes. A theoretical analysis confirms this by proving that OSTB codes are indeed optimal, when the setup under study permits their existence. Simulation results demonstrate the excellent performance of the designed codes. In particular, the importance of the codes' near-orthogonal property is illustrated by showing that low-complexity linear equalizer techniques can be used for decoding purposes while incurring a relatively moderate performance loss compared with optimal maximum-likelihood (ML) decoding