Universal Space–Time Codes From Demultiplexed Trellis Codes

In broadcast scenarios or in the absence of accurate channel probability distribution information, code design for consistent channel-by-channel performance, rather than average performance over a channel distribution, may be desirable. Root and Varaiya's compound channel theorem for linear Gaussian channels promises the existence of universal codes that operate reliably whenever the channel mutual information (MI) is above the transmitted rate. This paper presents 2-D trellis codes that provide such universal performance over the compound linear vector Gaussian channel when demultiplexed over two, three, and four transmit antennas. The presented trellis codes are found by an exhaustive search that guarantees consistent performance on every matrix channel that supports the information transmission rate with an MI gap that is similar to the capacity gap of a well-designed additive white Gaussian noise (AWGN)-specific code on the AWGN channel. As a result of their channel-by-channel consistency, the universal trellis codes presented here also deliver comparable, or in some cases, superior frame-error rate and bit-error rate performance under quasi-static Rayleigh fading to trellis codes of similar complexity that are designed specifically for the quasi-static Rayleigh fading scenario.     

Multistage Iterative Decoding With Complexity Reduction for Concatenated Space–Time Codes

Space–time encoders exploiting concatenated coding structures are efficient in attaining the high rates available to large-dimensional multiple-transmitter, multiple-receiver wireless systems under fading conditions, while also providing maximal diversity benefits. We present a multistage iterative decoding structure that takes full advantage of the concatenated nature of the transmission path, treating the modulator and channel stages as an additional encoder in serial concatenation. This iterative decoder architecture allows an encoder employing decoupled coding and modulation to reach the performance of coded modulation systems. It also admits reduced-complexity decoding with a computational load that is nonexponential in the number of antennas or the transmission bit rate, and makes practical decoding for large transmitter arrays possible. The performance curves for these methods follow the shape of the Fano bound, with only a modest power penalty.     

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     

Space–Time Coding for Multiuser Ultra-Wideband Communications

In this paper, we present the construction of full-rate, fully diverse, and totally real space–time (ST) codes for ultra-wideband (UWB) transmissions. In particular, we construct two families of codes adapted to real carrierless UWB communications that employ pulse position modulation, pulse amplitude modulation, or a combination of the two. The first family encodes adjacent symbols and is constructed from totally real cyclic division algebras. The second family encodes the pulses used to convey one information symbol, and permits achieving high performance levels with reduced complexity. The first family of codes achieves only a fraction of the coding gain of the second one. Moreover, these coding gains are independent from the size of the transmitted constellation. For time-hopping multiple-access channels, the amplitude spreading code associated with the second family of codes is taken to be user-specific. In this case, a simple design criterion is proposed, and spreading matrices constructed according to this criterion permit reducing the level of multiple-access interference (MAI). Performance over realistic indoor UWB channels verify the theoretical claims, and show high performance levels and better immunity against MAI.     

Information-Lossless Space–Time Block Codes From Crossed-Product Algebras

It is known that the Alamouti code is the only complex orthogonal design (COD) which achieves capacity and that too for the case of two transmit and one receive antenna only. Damen proposed a design for two transmit antennas, which achieves capacity for any number of receive antennas, calling the resulting space-time block code (STBC) when used with a signal set an information-lossless STBC. In this paper, using crossed-product central simple algebras, we construct STBCs for arbitrary number of transmit antennas over an a priori specified signal set. Alamouti code and quasi-orthogonal designs are the simplest special cases of our constructions. We obtain a condition under which these STBCs from crossed-product algebras are information-lossless. We give some classes of crossed-product algebras, from which the STBCs obtained are information-lossless and also of full rank. We present some simulation results for two, three, and four transmit antennas to show that our STBCs perform better than some of the best known STBCs and also that these STBCs are approximately 1 dB away from the capacity of the channel with quadrature amplitude modulation (QAM) symbols as input     

Space–Time Code Design for CPFSK Modulation Over Frequency-Nonselective Fading Channels

We derive a novel space–time code-design criterion for continuous-phase frequency-shift keying (CPFSK) over frequency-nonselective fading channels. Our derivation is based on a specific matrix that is related to the input symbols of the CPFSK modulators. With this code-design criterion, we propose a simple interleaved space–time encoding scheme for CPFSK modulation over frequency-nonselective correlated fading channels to exploit potential temporal and spatial diversity advantages. Such an encoding scheme consists of a ring convolutional encoder and a spatial encoder, between which a convolutional interleaver is placed. A decoding algorithm that generates symbol metrics for the Viterbi decoder of convolutional codes from the spatial modulation trellis is examined. Simulation results confirm that the advantages of combination of the interleaved convolutional encoding (for temporal diversity) and the spatial encoding (for spatial diversity) are promising for various system parameters.     

Proportional Fair Space–Time Scheduling for Wireless Communications

We extend the proportional fair (PF) scheduling algorithm to systems with multiple antennas. There are$K$client users (each with a single antenna) and one base station (with$n_R$antennas). We focus on the reverse link of the system, and assume a slow-fading channel where clients are moving with pedestrian speed. Qualcomm's original PF scheduling algorithm satisfies the PF criteria only when the communication is constrained to one user at a time with no power waterfilling. However, the original PF algorithm does not generalize easily when we have$n_R$receive antennas at the base station. In this paper, we shall formulate the PF scheduling design as a convex optimization problem. One challenge is in the optimal power allocation over the multiantenna multiaccess capacity region, which is still an open problem. For practical consideration, we consider multiuser minimum mean-square error processing at the base station. To obtain first-order insight, we propose an asymptotically optimal PF scheduling solution. Using the proposed PF solution for a multiantenna base station, the system capacity is enhanced by exploiting the multiuser selection diversity, as well as the distributed multiple-input multiple-output configuration. It is found that the PF scheduler achieves a good balance between fairness and system capacity gain.     

Spectral-Efficient Differential Space–Time Coding Using Non-Full-Diverse Constellations

In this paper, a method is proposed to construct spectral-efficient unitary space–time codes for high-rate differential communications over multiple-antenna channels. Unlike most of the known methods, which are designed to maximize the diversity product (the minimum determinant distance), our objective is to increase the spectral efficiency. The simulation results indicate that for high spectral efficiency and for more than one receive antenna, the new method significantly outperforms the existing alternatives. In the special case of two transmit antennas, which is the main focus of this paper, the relation between the proposed code and the Alamouti scheme helps us to provide an efficient maximum-likelihood decoding algorithm. Also, we demonstrate that similar ideas can be applied for designing codes for more than two transmit antennas. As an example, we present a construction for a 4$times$4 unitary constellations which has a good performance, as compared with the other known codes.     

Spectrally Efficient Differential Space–Time Coding Using Non-Full-Diverse Constellations

In this paper, a method is proposed to construct spectrally efficient unitary space–time codes for high-rate differential communications over multiple-antenna channels. Unlike most of the known methods which are designed to maximize the diversity product (the minimum determinant distance), our objective is to increase the spectral efficiency. The simulation results indicate that for high spectral efficiency and for more than one receive antenna, the new method significantly outperforms the existing alternatives. In the special case of two transmit antennas, which is the main focus of this paper, the relation between the proposed code and the Alamouti scheme helps us to provide an efficient maximum-likelihood (ML) decoding algorithm. Also, we demonstrate that similar ideas can be applied to designing codes for more than two transmit antennas. As an example, we present a construction for 4-by-4 unitary constellations which has a good performance, compared with the other known codes.     

Channel-Optimized Quantization With Soft-Decision Demodulation for Space–Time Orthogonal Block-Coded Channels

We introduce three soft-decision demodulation channel-optimized vector quantizers (COVQs) to transmit analog sources over space–time orthogonal block (STOB)-coded flat Rayleigh fading channels with binary phase-shift keying (BPSK) modulation. One main objective is to judiciously utilize the soft information of the STOB-coded channel in the design of the vector quantizers while keeping a low system complexity. To meet this objective, we introduce a simple space–time decoding structure that consists of a space–time soft detector, followed by a linear combiner and a scalar uniform quantizer with resolution$q$. The concatenation of the space–time encoder/modulator, fading channel, and space–time receiver can be described by a binary-input,$2^q$-output discrete memoryless channel (DMC). The scalar uniform quantizer is chosen so that the capacity of the equivalent DMC is maximized to fully exploit and capture the system's soft information by the DMC. We next determine the statistics of the DMC in closed form and use them to design three COVQ schemes with various degrees of knowledge of the channel noise power and fading coefficients at the transmitter and/or receiver. The performance of each quantization scheme is evaluated for memoryless Gaussian and Gauss–Markov sources and various STOB codes, and the benefits of each scheme is illustrated as a function of the antenna-diversity and soft-decision resolution$q$. Comparisons to traditional coding schemes, which perform separate source and channel coding operations, are also provided.     

Linear MMSE Space–Time Equalizer for MIMO Multicode CDMA Systems

We propose an enhanced chip-level linear space–time (ST) equalizer for multiple-input multiple-output (MIMO) multicode code-division multiple-access (CDMA) systems. In the MIMO multicode CDMA systems, the reuse of the same spreading codes in different transmit antennas significantly degrades the equalization performance if the ST equalizer uses a minimum mean-squared error (MMSE) weighting vector that minimizes the mean-squared error of the equalizer output chip sequence. As the CDMA despreader concatenated to an ST equalizer distorts interstream interference components differently from multipath interference and background noise components, the chip-level MMSE weighting vector usually steers in suboptimal directions in the signal space. Therefore, we propose a new MMSE weighting vector that takes the despreading effect into account in this paper. Simulation results show a substantial performance improvement through the new weighting vector.     

Semiorthogonal Space–Time Block Codes

In this paper, a new class of full-diversity, rate-one space-time block codes (STBCs) called semiorthogonal algebraic space-time block codes (SAST codes) is proposed. SAST codes are delay optimal when the number of transmit antennas is even. The SAST codeword matrix has a generalized Alamouti structure where the transmitted symbols are replaced by circulant matrices and the commutativity of circulant matrices simplifies the detection of transmit symbols. SAST codes with maximal coding gain are constructed by using rate-one linear threaded algebraic space-time (LTAST) codes. Compared with LTSAT codes, SAST codes not only reduce the complexity of maximum-likelihood detection, but also provide remarkable performance gain. They also outperform other STBC with rate one or less. SAST codes also perform well with suboptimal detectors such as the vertical-Bell Laboratories layered space-time (V-BLAST) nulling and cancellation receiver. Finally, SAST codes attain nearly 100% of the Shannon capacity of open-loop multiple-input-single-output (MISO) channels.     

Space–Time Precoding for Mean and Covariance Feedback: Application to Wideband OFDM

Space–Time Precoding for Mean and Covariance Feedback: Application to Wideband OFDM We consider optimization of the capacity of a multi-input single-output wideband cellular “downlink,” in which the base station has estimates of the statistics of the spatial channel. Our main focus is on orthogonal frequency-division multiplexed systems, although some of our results apply to single-carrier systems as well. Prior work has shown that estimates of the channel spatial covariance can be obtained without overhead for both frequency-division duplex (FDD) and time-division duplex (TDD) systems by suitably averaging uplink measurements. In this paper, we investigate the benefits of supplementing this “free” covariance feedback with mean feedback, where the latter refers to estimates of the spatial channel realization in each subcarrier. Mean feedback can be obtained using reciprocity for TDD systems, and requires explicit feedback for FDD systems. We first devise strategies for using both covariance and mean feedback, mainly restricting attention to beamforming, which is optimal or near-optimal for many outdoor channels with narrow spatial spread. Second, since mean feedback degrades rapidly with feedback delay for mobile channels, we develop quantitative rules of thumb regarding the accuracy required for the mean feedback to be a useful supplement to the already available, and robust, covariance feedback. Our results validate the following intuition: the accuracy requirements for mean feedback to be useful are more relaxed for channels with larger spatial spread, or for a larger number of transmit elements.     

Explicit Space–Time Codes Achieving the Diversity–Multiplexing Gain Tradeoff

A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity-multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space-time (ST) code. This tradeoff is precisely known in the case of independent and identically distributed (i.i.d.) Rayleigh fading, for Tgesn_t+n_r-1 where T is the number of time slots over which coding takes place and n_t,n_r are the number of transmit and receive antennas, respectively. For Tt+n_r-1, only upper and lower bounds on the D-MG tradeoff are available. In this paper, we present a complete solution to the problem of explicitly constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff for any number of receive antennas. We do this by showing that for the square minimum-delay case when T=n_t=n, cyclic-division-algebra (CDA)-based ST codes having the nonvanishing determinant property are D-MG optimal. While constructions of such codes were previously known for restricted values of n, we provide here a construction for such codes that is valid for all n. For the rectangular, T>n_t case, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all Tgesn_t is the same as that previously known to hold for Tgesn_t+n _r-1     

Tools for Performance Analysis and Design of Space–Time Block Codes

Space-time block codes (STBCs) have attracted recent interest due to their ability to take advantage of both space and time diversity to reliably transmit data over a wireless fading channel. In many cases, their design is based on asymptotically tight performance criteria, such as the worst-case pairwise error probability (PEP) or the union bound. However, these quantities fail to give an accurate performance picture, especially at low signal-to-noise ratio, because the classical union bound is known to be loose in this case. This paper develops tighter performance criteria for STBCs which yield considerably better bounds. First, the union bound is developed as the average of the exact PEPs. By noting that some of the terms in the bound are redundant, a second bound is obtained by expurgation. Since this still yields a loose bound, a tighter bound, denoted as the progressive union bound (PUB), is obtained. Because the PUB cannot be computed in closed form, in its most general case, and to avoid computing a high-dimensional numerical integration, its saddlepoint approximation is developed. In addition to the significant improvement of the PUB analysis over other bounding methods, it is also shown that codes designed to optimize the PUB can perform better than those obtained by the looser criteria     

On Fast-Decodable Space–Time Block Codes

We focus on full-rate, fast-decodable space–time block codes (STBCs) for $2times2$ and $4times2$ multiple-input multiple-output (MIMO) transmission. We first derive conditions and design criteria for reduced-complexity maximum-likelihood (ML) decodable $2times2$ STBCs, and we apply them to two families of codes that were recently discovered. Next, we derive a novel reduced-complexity $4times2$ STBC, and show that it outperforms all previously known codes with certain constellations.      

Space–charge conditions in a reflected flow of electrons

Some characteristic features of a reflected flow of electrons are described, in particular variations of the virtual cathode and transit time with respect to current. This has been accomplished by finding new solutions to well-known basic equations treated earlier by Fay, Samuel, Shockley, Salzberg, Haeff and others. The results are applicable to problems where the current is varied while earlier solutions were considering the potential as variable. The theoretical results are found to be in agreement with experimental results obtained on reflex klystrons and space-charge deflection tubes.     

Robust Linear Receivers for Multiaccess Space–Time Block-Coded MIMO Systems: A Probabilistically Constrained Approach

Traditional multiuser receiver algorithms developed for multiple-input–multiple-output (MIMO) wireless systems are based on the assumption that the channel state information (CSI) is precisely known at the receiver. However, in practical situations, the exact CSI may be unavailable because of channel estimation errors and/or outdated training. In this paper, we address the problem of robustness of multiuser MIMO receivers against imperfect CSI and propose a new linear technique that guarantees the robustness against CSI errors with a certain selected probability. The proposed receivers are formulated as probabilistically constrained stochastic optimization problems. Provided that the CSI mismatch is Gaussian, each of these problems is shown to be convex and to have a unique solution. The fact that the CSI mismatch is Gaussian also enables to convert the original stochastic problems to a more tractable deterministic form and to solve them using the second-order cone programming approach. Numerical simulations illustrate an improved robustness of the proposed receivers against CSI errors and validate their better flexibility as compared with the robust multiuser MIMO receivers based on the worst case designs.     

On Space–Time Trellis Codes Achieving Optimal Diversity Multiplexing Tradeoff

Multiple antennas can be used for increasing the amount of diversity (diversity gain) or increasing the data rate (the number of degrees of freedom or spatial multiplexing gain) in wireless communication. As quantified by Zheng and Tse, given a multiple-input-multiple-output (MIMO) channel, both gains can, in fact, be simultaneously obtained, but there is a fundamental tradeoff (called the Diversity-Multiplexing Gain (DM-G) tradeoff) between how much of each type of gain, any coding scheme can extract. Space-time codes (STCs) can be employed to make use of these advantages offered by multiple antennas. Space-Time Trellis Codes (STTCs) are known to have better bit error rate performance than Space-Time Block Codes (STBCs), but with a penalty in decoding complexity. Also, for STTCs, the frame length is assumed to be finite and hence zeros are forced towards the end of the frame (called the trailing zeros), inducing rate loss. In this correspondence, we derive an upper bound on the DM-G tradeoff of full-rate STTCs with nonvanishing determinant (NVD). Also, we show that the full-rate STTCs with NVD are optimal under the DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros. Next, we give an explicit generalized full-rate STTC construction for any number of states of the trellis, which achieves the optimal DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros