Effect of keyholes on the symbol error rate of space-time block codes

In multiple-input multiple-output (MIMO) fading environments, degenerate channel phenomena, so-called keyholes or pinholes, may exist under realistic assumptions where a MIMO channel has uncorrelated spatial fading between antenna arrays but a rank-deficient transfer matrix. In this letter, we analyze the average symbol error rate (SER) of orthogonal space-time block codes (STBCs) with M-PSK and M-QAM constellations over keyhole channels.     

On the error probability of quasi‐orthogonal space–time block codes

In this paper, we derive the exact pairwise error probabilities (PEPs) of various quasi‐orthogonal space–time block codes (QO‐STBCs) using the moment generating function. By classifying the exact PEPs of QO‐STBCs into three types, we derive the closed‐form expression for each type of PEP. Based on these closed‐form expressions, we obtain the union bounds on the symbol error probability and bit error probability for QPSK modulation. Through simulation, it is shown that these union bounds are quite tight. Copyright © 2008 John Wiley & Sons, Ltd.     

Performance of Alamouti Scheme with Transmit Antenna Selection for M-ray Signals

In this letter, we present a comprehensive performance analysis of orthogonal space-time block codes (STBCs) with transmit antenna selection under uncorrelated Rayleigh fading channels. Two best transmit antennas that maximize the instantaneous received signal-to-noise (SNR) are selected. Using the well-known moment generating function-based analysis approach, we derive the exact average symbol error rate (SER) for M-ary signals. Furthermore, we provide tight upper bounds on the SER for any number of transmit antennas and receive antennas. The tightness is verified by simulation results. It is shown that the diversity order, with antenna selection, is maintained as that of the full complexity systems     

Performance analysis of space-time block codes over keyhole Nakagami-m fading channels

In multiple-input-multiple-output (MIMO) fading environments, degenerate channel phenomena, called keyholes or pinholes, may exist under the realistic assumption that the spatial fading is uncorrelated at the transmitter and the receiver, but the channel has a rank-deficient transfer matrix. In this paper, we analyze the exact average symbol error rate (SER) of orthogonal space-time block codes (STBCs) with M-PSK and M-QAM constellations over Nakagami-m fading channels in the presence of the keyhole. We derive the moment generating function (MGF) of instantaneous signal-to-noise ratio (SNR) after space-time block decoding (signal combining) in such channels. Using a well-known MGF-based analysis approach, we express the average SER of the STBC in the form of single finite-range integrals whose integrand contains only the derived MGF. Numerical results show that the keyhole significantly degrades the SER performance of the STBC from idealistic behaviors in independent identically distributed MIMO channels.     

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     

Symbol Error Rate Performance Analysis for Generalized Coordinate Interleaved Orthogonal Designs over Nakagami-m Fading Channels

Space-time block codes (STBC) using generalized coordinate interleaved orthogonal designs (GCIOD) proposed by Khan and Rajan (IEEE Trans Inf Theory 52(5):2062–2091, 2006) allow single-complex symbol decoding and offer higher data rates than orthogonal STBC. In this paper, we present an exact closed-form formula (i.e., not in integral form) for the average symbol pair-wise error probability of GCIODs derived in quasi-static frequency-nonselective independent identically distributed Nakagami- \(m\) fading channels. By using the formula, a tight union bound (UB) on symbol error rate (SER) is presented. Extensive simulation results show that the curves for the UB are asymptotically tight with the simulated SER curves for various antenna configurations and different modulations. The UB thus can be used to accurately predict and optimize the performance of GCIODs without the need of time-consuming simulations.     

On channel orthogonalization using space-time block coding with partial feedback

Orthogonal space-time block codes (OSTBCs) yield full diversity gain even while requiring only a linear receiver. Such full-rate (rate-one) orthogonal designs are available for complex symbol constellations only for N=2 transmit antennas. In this paper, we propose a new family of full-rate space-time block codes (STBCs) using a single parameter feedback for communication over Rayleigh fading channels for N=3,4 transmit antennas and M receive antennas. The proposed rate-one codes achieve full diversity, and the performance is similar to maximum receiver ratio combining. The decoding complexity of these codes are only linear even while performing maximum-likelihood decoding. The partial channel information is a real phase parameter that is a function of all the channel gains, and has a simple closed-form expression for N=3,4. This feedback information enables us to derive (channel) orthogonal designs starting from quasi-orthogonal STBCs. The feedback complexity is significantly lower than conventional closed-loop transmit beamforming. We compare the proposed codes with the open-loop OSTBCs and also with the closed-loop equal gain transmission (EGT) scheme which uses equal power loading on all antennas. Simulated error-rate performances indicate that the proposed channel orthogonalized STBCs significantly outperform the open-loop orthogonal designs, for the same spectral efficiency. Moreover, even with significantly lower feedback and computational complexity, the proposed scheme outperforms the EGT technique for M>N.     

The Performance of Space-Time Block Codes from Coordinate Interleaved Orthogonal Designs over Nakagami-m Fading Channels

Space-time block codes (STBCs) from coordinate interleaved orthogonal designs (CIODs) offer several advantages including full-diversity and single-symbol decodability. In an effort to assess their performance in quasi-static frequencynonselective i.i.d. Nakagami-m fading channels, we analyze the error rate, outage capacity, and information outage probability. First, based on an accurate closed-form formula for the average symbol pairwise error rate (SPER), we derive tight union upper and lower bounds on the symbol-error rate (SER). Second, we apply Gaussian and Gamma approximations to provide closedform expressions for the outage capacity. Third, using high signal-to-noise ratio (SNR) and moment-matching approximation techniques, we also derive accurate closed-form approximations for the information outage probability (IOP). Finally, we show that STBCs from CIODs provide full-diversity by deriving SERbased and IOP-based asymptotic and instantaneous diversity orders. Monte-Carlo simulations show that the analytical results agree with simulation experiments.     

Space-time block codes based on coordinate symmetric orthogonal designs

Space-time block codes based on coordinate symmetric orthogonal designs are proposed. Compared with space-time block codes from complex orthogonal design when the code rate is the same and the transmission rate is fixed, space-time block codes from coordinate symmetric orthogonal design with more transmit antennas can reduce the bit error rate and symbol error rate. Also these new codes have the same low decoding complexity as space-time block codes from complex orthogonal designs.     

Novel Constructions of Improved Square Complex Orthogonal Designs for Eight Transmit Antennas

Constructions of square, maximum rate complex orthogonal space-time block codes (CO STBCs) are well known, however codes constructed via the known methods include numerous zeros, which impede their practical implementation. By modifying the Williamson and Wallis-Whiteman arrays to apply to complex matrices, we propose two methods of construction of square, order-4n CO STBCs from square, order-n codes which satisfy certain properties. Applying the proposed methods, we construct square, maximum rate, order-8 CO STBCs with no zeros, such that the transmitted symbols are equally dispersed through transmit antennas. Those codes, referred to as the improved square CO STBCs, have the advantages that the power is equally transmitted via each transmit antenna during every symbol time slot and that a lower peak-to-average power ratio (PAPR) is required to achieve the same bit error rates as the conventional CO STBCs with zeros.     

Quasi-orthogonal STBC with minimum decoding complexity: performance analysis, optimal signal transformations, and antenna selection diversity

This letter presents a new method to directly analyze and optimize symbol error rate (SER) performance of minimum decoding complexity (MDC) ABBA space-time block codes based on a tight union bound on SER. Additionally, a new signal transformation for rectangular quadrature amplitude modulation is proposed to provide better performance than the existing ones with lower encoding/decoding complexities. It is also shown that MDC-ABBA codes achieve full-diversity with antenna selection and limited feedback.     

Performance Analysis of Space-Time Block Codes with Transmit Antenna Selection in Nakagami-m Fading Channels

In this paper, multiple-input multiple-output systems employing space-time block codes (STBCs) with transmit antenna selection (TAS) are examined for flat Nakagami-m fading channels. Exact symbol error rate (SER) expressions for M-ary modulation techniques are derived by using the moment generating function based analysis method. In the SER analysis, the receiver is assumed to use maximal ratio combining whereas a subset of transmit antennas that maximizes the instantaneous received signal-to-noise ratio (SNR) is selected for STBC transmission. The analytical SER results are validated by Monte Carlo simulations. By deriving upper and lower bounds for SER expressions, it is shown that TAS/STBC schemes achieve full diversity orders at high SNRs.     

Decoding, Performance Analysis, and Optimal Signal Designs for Coordinate Interleaved Orthogonal Designs

Space-time block codes (STBC) using coordinate interleaved orthogonal designs (CIOD) proposed recently by Khan and Rajan allow single-complex symbol decoding and offer higher data rates than orthogonal STBC. In this paper, we present the channel decoupling property of CIOD codes. A new general maximum likelihood method is derived, enabling the calculation of the symbol pair-wise error probability and union bound (UB) on symbol error rate (SER). Extensive simulation results show that the UB is within 0.1 dB from the simulated SER when SER < 10^-2. The UB thus can be used to accurately predict and optimize the performance of CIOD codes. Furthermore, a new signal design combining signal rotation and power allocation is presented for constellations with uneven powers of real and imaginary parts such as rectangular quadrature amplitude modulation.     

Differential spatial modulation scheme based on orthogonal space-time block coded

Focusing on the problem that differential spatial modulation (DSM) couldn’t obtain transmit diversity and has high decoding complexity,a new differential spatial modulation scheme based on the orthogonal space-time block code was proposed and the proposed scheme is called OSTBC-DSM.There were two matrices in this scheme:the spatial modulation matrix and the symbol matrix.The former was aimed to activate different transmit antennas by setting the position of nonzero elements,and the latter structured symbolic matrix by using orthogonal space-time block codes (OSTBC) as the basic code block.The proposed scheme could obtain full transmit diversity and higher spectral efficiency compared with the conventional DSM schemes.Moreover,the OSTBC-DSM supported linear maximum likelihood (ML) decoding.The simulation results show that under different spectral efficiencies,the proposed OSTBC-DSM scheme has better bit error rate (BER) performance than other schemes.     

High-throughput error correcting space-time block codes

This letter investigates using an error correction code (ECC) to construct the space-time block code (STBC). Block ECCs over several Galois fields are considered. The resulting STBCs have significantly higher throughput and better performance than orthogonal STBCs, at the cost of increased decoding complexity.     

Performance Analysis of Alamouti Space-Time Block Code Over Time-Selective Fading Channels

Alamouti orthogonal space-time block code (Alamouti in IEEE J Sel Areas Commun 16(8):1451–1458, 1998) has been applied widely in wireless communication, e.g., IEEE 802.16e-2005 standard. In this paper, theoretical analysis of symbol error rate performance for Alamouti orthogonal space-time block code (AOSTBC) over time-selective fading channels with a zero-forcing linear receiver is derived. Firstly, a closed-form expression (i.e., not in integral form) is derived for the average symbol pair-wise error probability (SPEP) in time-selective frequency-nonselective independent identically distributed Rayleigh fading channels. Then, the SPEP is used to derive a tight upper bound (UB) for the symbol-error rate (SER) of AOSTBC via establishing algorithmic Bonferroni-type upper bound. Extensive simulation results show that the curves for the UB coincide with the simulated SER curves for various antenna configurations even at very low signal-to-noise ratio regimes. The UB thus can be used to accurately predict the performance of AOSTBC code over time-selective fading channels when a zero-forcing receiver is used.     

BER analysis of QAM on fading channels with transmit diversity

In this letter, we derive analytical expressions for the bit error rate (BER) of space-time block codes (STBC) from complex orthogonal designs (COD) using quadrature amplitude modulation (QAM) on Rayleigh fading channels. We take a bit log-likelihood ratio (LLR) based approach to derive the BER expressions. The approach presented here can be used in the BER analysis of any STBC from COD with linear processing for any value of M in an M-QAM system. Here, we present the BER analysis and results for a 16-QAM system with i) (2-Tx, L-Rx) antennas using Alamouti code (rate-1 STBC), ii) (3-Tx, L-Rx) antennas using a rate-1/2 STBC, and iii) (5-Tx, L-Rx) antennas using a rate-7/11 STBC. In addition to being used in the BER analysis, the LLRs derived can also be used as soft inputs to decoders for various coded QAM schemes, including turbo coded QAM with space-time coding as in high speed downlink packet access (HSDPA) in 3G.     

Single-Symbol Maximum Likelihood Decodable STBCs with Fewer Zero-Entries

Two generalized constructions of single-symbol maximum likelihood (ML) decodable space-time block codes (SSDCs), the coordinate interleave orthogonal design (CIOD) and the generalized CIOD (GCIOD), have been proposed by Khan et al. Their main disadvantage is that there are too many zero entries in the codeword matrix. The transmission of these zero-valued symbols causes switching off some of the transmit antennas resulting in a high peak-to-average power ratio and also imposing a severe constraint on hardware implementation of the code. Based on the orthogonal construction of orthogonal space-time block codes and coordinate interleaving, a new generalized construction of SSDCs is proposed. Our proposed codes not only maintain the desirable properties of high rate, full transmit diversity and single-symbol ML decoding similar to CIODs and GCIODs, but also have fewer zero entries in its codeword matrix than CIODs and GCIODs.     

多天线对角空频编码传输

将平坦衰落信道的对角代数空时码(DAST)推广到频率选择性衰落信道,提出了对角空频分组码(DSF).基于多输入多输出天线和正交频分复用(OFDM),DSF码将满秩的旋转信号星座和子载波分组结合起来,以对角发送方式(每时刻只有一个天线发射)发射旋转信息符号向量的每个分量.成对错误概率分析表明:在频率选择性信道中,通过选择最佳的旋转矩阵,这种DSF-OFDM系统能实现满分集增益和最大的编码增益.系统采用了球型解码器对DSF码实施最大似然解码,它的解码复杂性是中等的,并且,解码算法的复杂性与信号星座的维数无关.此外,和先前所提出的一些方法相比,提出的空频码还具有频谱效率高(1symbol/s/Hz)的性能特点.     

Peroformance analysis of decode-and-forward MIMO relay channels with OSTBCs

Performance analysis is presented for multiple-input multiple-output(MIMO) relay channels employing transmit antenna diversity with orthogonal space-time block codes(OSTBCs),where the source and the destination are equipped with Ns and Nd antennas,and communicate with each other with the help of a multiple-antenna relay operating in decode-and-forward(DF) mode.Over independent,not necessarily identical Rayleigh fading channels,exact closed-form symbol error rate(SER) expressions are derived for various digi...