Optimal Space–Time Codes for the MIMO Amplify-and-Forward Cooperative Channel

In this work, we extend the nonorthogonal amplify-and-forward (NAF) cooperative diversity scheme to the multiple-input multiple-output (MIMO) channel. A family of space-time block codes for a half-duplex MIMO NAF fading cooperative channel with N relays is constructed. The code construction is based on the nonvanishing determinant (NVD) criterion and is shown to achieve the optimal diversity-multiplexing tradeoff (DMT) of the channel. We provide a general explicit algebraic construction, followed by some examples. In particular, in the single-relay case, it is proved that the Golden code and the 4times4 Perfect code are optimal for the single-antenna and two-antenna cases, respectively. Simulation results reveal that a significant gain (up to 10 dB) can be obtained with the proposed codes, especially in the single-antenna case     

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     

Space–Time Turbo Equalization With Successive Interference Cancellation for Frequency-Selective MIMO Channels

This paper addresses the issue of iterative space–time equalization for multiple-input–multiple-output (MIMO) frequency-selective fading channels. A new soft equalization concept based on successive interference cancellation (SIC) is introduced for a space–time bit-interleaved coded modulation (STBICM) transmission. The proposed equalizer allows us to separate intersymbol interference (ISI) and multiantenna interference (MAI) functions. Soft ISI is successively suppressed using a low-complexity suboptimum minimum mean square error (MMSE) criterion. The decoupling of ISI and MAI offers more flexibility in the design of the whole space–time equalizer. Different multiantenna detection criteria can be considered, ranging from simple detectors to the optimal maximum a posteriori (MAP) criterion. In particular, we introduce two soft equalizers, which are called SIC/SIC and SIC/MAP, and we show that they can provide a good performance-to-complexity tradeoff for many system configurations, as compared with other turbo equalization schemes. This paper also introduces an MMSE-based iterative channel state information (CSI) estimation algorithm and shows that attractive performance can be achieved when the proposed soft SIC space–time equalizer iterates with the MMSE-based CSI estimator.      

Analysis of Differential Orthogonal Space–Time Block Codes Over Semi-Identical MIMO Fading Channels

We study the performance of differential orthogonal space-time block codes (OSTBC) over independent and semi-identically distributed block Rayleigh fading channels. In this semiidentical fading model, the channel gains from different transmit antennas to a common receive antenna are identically distributed, but the gains associated with different receive antennas are nonidentically distributed. Arbitrary fluctuation rates of the fading processes from one transmission block to another are considered. We first derive the optimal symbol-by-symbol differential detector, and show that the conventional differential detector is suboptimal. We then derive expressions of exact bit-error probabilities (BEPs) for both the optimal and suboptimal detectors. The results are applicable for any number of receive antennas, and any number of transmit antennas for which OSTBCs exist. For two transmit antennas, explicit and closed-form BEP expressions are obtained. For an arbitrary number of transmit antennas, a Chernoff bound on the BEP for the optimal detector is also derived. Our results show that the semi-identical channel statistics degrade the error performance of differential OSTBC, compared with the identical case. Also, the proposed optimal detector substantially outperforms the conventional detector when the channel fluctuates rapidly. But in near-static fading channels, the two detectors have similar performances     

Nonredundant Precoding-Assisted Blind Channel Estimation for Single-Carrier Space–Time Block-Coded Transmission With Frequency-Domain Equalization

Relying on nonredundant diagonal precoding and independent and identically distributed (i.i.d.) source assumption, this paper proposes a blind channel estimation scheme for single-carrier frequency-domain equalization-based space-time block coded systems. The proposed method exploits the precoding-induced linear signal structure in the conjugate cross correlation between the two temporal block received signals as well as the circulant channel matrix property and can yield exact solutions whenever the channel noise is circularly Gaussian and the receive data statistic is perfectly obtained. The channel estimation formulation builds on rearranging the set of linear equations relating the entries of conjugate cross-correlation matrix and products of channel impulse responses into one with a distinctive block-circulant with circulant-block (BCCB) structure. This allows a simple identifiability condition depending on precoder parameters alone and also provides a natural yet effective optimal precoder design framework for improving solution accuracy when imperfect data estimation occurs. We consider two models of data mismatch, from both deterministic and statistical points of view, and propose the associated design criteria. The optimization problems are formulated to take advantage of the BCCB system matrix property and are solved analytically. The proposed optimal precoder aims to optimize solution robustness against deterministic error perturbation and also minimize the mean-square error when the data mismatch is modeled as a white noise. Pairwise error probability analysis is conducted for investigating the equalization performance. Numerical examples are used to illustrate the performance of the proposed method     

High-Rate Full-Diversity Space–Time–Frequency Codes for Broadband MIMO Block-Fading Channels

A systematic design of high-rate full-diversity space-time-frequency (STF) codes is proposed for multiple-input multiple-output frequency-selective block-fading channels. It is shown that the proposed STF codes can achieve rate M_t and full-diversity M_tM_rM_bL, i.e., the product of the number of transmit antennas M_t, receive antennas M_r, fading blocks M_b, and channel taps L. The proposed STF codes are constructed from a layered algebraic design, where each layer of algebraic coded symbols are parsed into different transmit antennas, orthogonal frequency-division multiplexing tones, and fading blocks without rate loss. Simulation results show that the proposed STF codes achieve higher diversity gain in block-fading channels than some typical space-frequency codes     

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     

Diversity–Multiplexing Tradeoff and Outage Performance for Rician MIMO Channels

In this paper, we analyze the diversity–multiplexing tradeoff (DMT), originally introduced by Zheng and Tse, and outage performance for Rician multiple-input–multiple-output (MIMO) channels. The DMT characteristics of Rayleigh and Rician channels are shown to be identical. In a high signal-to-noise ratio (SNR) regime, the log–log plot of outage probability versus SNR curve for a Rician channel is a shifted version of that for the corresponding Rayleigh channel. The SNR gap between the outage curves of the Rayleigh and Rician channels is derived. The DMT and outage performance are also analyzed for Rician multiple-input–single-output (MISO)/single-input–multiple-output (SIMO) channels over a finite SNR regime. A closed-form expression for the outage probability is derived and the finite SNR DMT characteristic is analyzed. It is observed that the maximum diversity gain can be achieved at some finite SNR–the maximum gain tends to increase linearly with the Rician factor. The finite SNR diversity gain is shown to be a linear function of the finite SNR multiplexing gain. The consistency between the DMTs for finite and infinite SNRs is also shown.      

On the Performance of Space–Time Block-Coded MIMO Video Communications

The problem of efficient video communications over multiple-input-multiple-output (MIMO) wireless systems is of great significance due to the high capacity of the multiple antenna system. The high data rates provided by the MIMO system can be traded off with diversity gain by using different channel-coding schemes. Also, by using different video source-coding methods, high compression gain can be traded off with the error resilience gain. One should jointly consider source coding and channel coding when designing a MIMO wireless video system. However, little is known so far about what combinations of channel-coding and source-coding methods have the best overall performance in a MIMO system. In this paper, by comparing the performances of several different typical combinations through both theoretical and simulation studies, we show that no single combination is the best for the entire range of channel conditions, but rather, different combinations may be best for a subrange     

On the Design of Space–Time Trellis Codes for Transmit-Correlated Fading Channels

The analysis and design of space-time codes for correlated fading channels when the diversity gain is large enough is considered. We derive a simple form for a distance metric that characterizes the code performance in the presence of transmit correlation, and propose some design criteria to build good space-time trellis codes (STTCs) for correlated channels. For the case of two transmit antennas, we show that in strongly correlated channels, performance is governed by the constellation that results from the sum of the constellations associated with the transmit antennas. This suggests the use of new constellations to design better codes for correlated channels. The design criteria are then extended to any number of transmit antennas. Based on these criteria, we derive new STTCs for two and three transmit antennas that perform much better in correlated channels than the STTC optimized for the independent and identically distributed case. We also consider set partitioning applied to the sum constellation as a simple technique to design good codes for correlated channels. The codes derived show performance close to the codes found by an exhaustive search. Finally, we consider antenna selection as an alternative to build good codes for more than two antennas in fading-correlated scenarios     

Finite-SNR Diversity–Multiplexing Tradeoff for Correlated Rayleigh and Rician MIMO Channels

A nonasymptotic framework is presented to analyze the diversity-multiplexing tradeoff of a multiple-input-multiple-output (MIMO) wireless system at finite signal-to-noise ratios (SNRs). The target data rate at each SNR is proportional to the capacity of an additive white Gaussian noise (AWGN) channel with an array gain. The proportionality constant, which can be interpreted as a finite-SNR spatial multiplexing gain, dictates the sensitivity of the rate adaptation policy to SNR. The diversity gain as a function of SNR for a fixed multiplexing gain is defined by the negative slope of the outage probability versus SNR curve on a log-log scale. The finite-SNR diversity gain provides an estimate of the additional power required to decrease the outage probability by a target amount. For general MIMO systems, lower bounds on the outage probabilities in correlated Rayleigh fading and Rician fading are used to estimate the diversity gain as a function of multiplexing gain and SNR. In addition, exact diversity gain expressions are determined for orthogonal space-time block codes (OSTBC). Spatial correlation significantly lowers the achievable diversity gain at finite SNR when compared to high-SNR asymptotic values. The presence of line-of-sight (LOS) components in Rician fading yields diversity gains higher than high-SNR asymptotic values at some SNRs and multiplexing gains while resulting in diversity gains near zero for multiplexing gains larger than unity. Furthermore, as the multiplexing gain approaches zero, the normalized limiting diversity gain, which can be interpreted in terms of the wideband slope and the high-SNR slope of spectral efficiency, exhibits slow convergence with SNR to the high-SNR asymptotic value. This finite-SNR framework for the diversity-multiplexing tradeoff is useful in MIMO system design for realistic SNRs and propagation environments     

Space–Time Coding Techniques With Bit-Interleaved Coded Modulations for MIMO Block-Fading Channels

The space-time bit-interleaved coded modulation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a block-fading multiple-input multiple-output (MIMO) channel. Its maximum-likelihood (ML) performance is computed under ideal interleaving conditions, which enables a global optimization taking into account channel coding. Thanks to a diversity upper bound derived from the Singleton bound, an appropriate choice of the time dimension of the space-time coding is possible, which maximizes diversity while minimizing complexity. Based on the analysis, an optimized interleaver and a set of linear precoders, called dispersive nucleo algebraic (DNA) precoders are proposed. The proposed precoders have good performance with respect to the state of the art and exist for any number of transmit antennas and any time dimension. With turbo codes, they exhibit a frame error rate which does not increase with frame length.     

Mapping Optimization for Space–Time Bit-Interleaved Coded Modulation With Iterative Decoding

For space-time bit-interleaved coded modulation (ST-BICM) systems with iterative decoding, the overall performance is affected by the chosen mapping. In bit-error rate (BER) curves, one mapping reaches an error floor (EF) at a low signal-to-noise ratio (SNR), while other mappings result in a lower EF at a higher SNR. The constellation mappings are divided into groups where each group exhibits a distinctive BER curve. We show that the convergence abscissa of the system depends on the average total bit errors and the harmonic mean of the minimum squared Euclidean distance. In this letter, we characterize all mapping groups for ST-BICM with 8-phase-shift keying and present the optimal selection for each mapping group over independent fading channels     

On Space–Time Coding With Pulse Position and Amplitude Modulations for Time-Hopping Ultra-Wideband Systems

In this work, we propose novel families of space-time (ST) block codes that can be associated with impulse radio ultra-wideband (IR-UWB) communication systems. The carrier-less nature of this nonconventional totally real transmission technique necessitates the construction of new suitable coding schemes. In fact, the last generation of complex-valued ST codes (namely, the perfect codes) cannot be associated with IR-UWB systems where the phase reconstitution at the receiver side is practically infeasible. On the other hand, while the perfect codes were considered mainly with quadrature amplitude modulation (QAM) and hexagonal (HEX) constellations, IR-UWB systems are often associated with pulse-position modulation (PPM) and hybrid PPM-PAM (pulse-amplitude modulation) constellations. In this paper, instead of adopting the classical approach of constructing ST codes over infinite fields or for the perfect codes), we study the possibility of constructing modulation-specific codes that are exclusive to PPM and PPM-PAM. The proposed full-rate codes are totally real, information lossless, and have a uniform average energy per transmit antenna. They permit to achieve a full diversity order with any number of transmit antennas. In some situations, the proposed schemes have an optimal nonvanishing coding gain and satisfy all the construction constraints of the perfect codes in addition to the constraint of being totally real. Simulations performed over realistic indoor UWB channels showed that the proposed schemes outperform the best known codes constructed from cyclic division algebras.     

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     

Coded Unitary Space–Time Modulation With Iterative Decoding: Error Performance and Mapping Design

This paper studies the bit error probability of coded unitary space-time modulation with iterative decoding where neither the transmitter nor the receiver knows the channel fading coefficients. The tight error bound with respect to the asymptotic performance is first analytically derived for any given unitary constellation and mapping rule. Design criteria regarding the choice of unitary constellation and mapping are then established. Furthermore, using the unitary constellation obtained from orthogonal design with quadrature phase-shift keying (QPSK or 4-PSK) and 8-PSK, two different mapping rules are proposed. The first mapping rule gives the most suitable mapping for systems that do not implement iterative processing, which is similar to a Gray mapping in coherent channels. The second mapping rule yields the best mapping for systems with iterative decoding. In particular, analytical and simulation results show that with the proposed mappings of the unitary constellations obtained from orthogonal designs, the asymptotic error performance of the iterative systems can closely approach a lower bound which is applicable to any unitary constellation and mapping     

Low-Complexity Equalization for SC-FDMA Uplink Systems Over Time and Frequency Selective Channels

Wireless Personal Communications - In this paper, we propose novel equalization schemes for compensating the adverse effects caused by carrier frequency offsets (CFOs), timing errors and doubly...     

Space–Time Code Design for Correlated Ricean MIMO Channels at Finite SNR

Space–time code (STC) designs commonly rely on the assumptions of independent and identically distributed (i.i.d.) Rayleigh channels (being either slow or fast fading) and high signal-to-noise ratio (SNR). However, it has been shown that poor scattering conditions can have detrimental effects on the performance of STCs and that the behavior of codes at high SNR is radically different from the finite SNR behavior. This calls for new design criteria that correctly predict the behavior of codes in correlated channels at finite SNR. In this paper, we investigate how spatially and temporally correlated Ricean fading affects the performance of STCs at finite SNR. We derive a code design criterion leading to robust STCs in a wide variety of propagation conditions and do not require any channel knowledge at the transmitter. Codes satisfying this criterion are shown to perform sensibly better in correlated channels than codes designed only for i.i.d. slow or fast Rayleigh-fading channels. Examples of space–time trellis codes and algebraic codes are proposed in order to illustrate the developed criterion.      

Achieving Single-User Performance in an FEC-Coded DS-CDMA System for Frequency Selective and Flat Fading Channels

This paper presents a reduced-complexity soft-input soft-output trellis/tree multiuser equalizer for an iterative DS-CDMA system undergoing Rayleigh frequency selective fading. The algorithm first expands the equalizer-trellis to an equivalent trellis/tree structure. Then it applies the M-algorithm to the equivalent structure twice, once to reduce the number of states in the trellis and the other to reduce the number of branches emanating from each state. To compute soft-information, the algorithm utilizes not only those fully-extended paths reaching the end of the trellis but also paths that are traversed and discarded in the pruned trellis. Through a simple update-and-discard procedure, reliable soft-information is extracted from discarded paths which enables an extremely large trellis to be successfully decoded with modest complexity. BER performance is presented for a convolutional-coded DS-CDMA system employing random spreading sequences. Our results demonstrate that the proposed algorithm is capable of achieving single-user performance with a much reduced complexity. The proposed algorithm can also be applied to reduce the complexity of multiuser detection where the transmission channel is frequency flat. Single-user performance can also be achieved with the proposed technique.