A New Unitary Space-Time Code with High Diversity Product

In this paper, we propose new full diversity unitary space-time codes based on Hamiltonian constellation designs. Our proposed constellations can be used for any number of antennas and for any data rate. For two transmitter antennas, the constellations are constructed from cyclic group codes. For a larger number of transmitter antennas, the design employs the direct sum of 2 times 2 Hamiltonian matrices and roots of unity. We give some examples of proposed constellations, and also show that they outperform known design techniques in the literature     

Real-Domain Decoder for Full-Rate Full-Diversity STBC with Multidimensional Constellations

In this letter, we present a new maximum likelihood (ML) decoding algorithm for space time block codes (STBCs) that employ multidimensional constellations. We start with a lattice representation for STBCs which transforms complex channel models into real matrix equations. Based on the lattice representation, we propose a new decoding algorithm for quasiorthogonal STBCs (QO-STBC) which allows simpleML decoding with performance identical to the conventional ML decoder. Multidimensional rotated constellations are constructed for the QO-STBCs to achieve full diversity. As a consequence, for quasi-orthogonal designs with an arbitrary number of transmit antennas N (N ? 4), the proposed decoding scheme achieves full rate and full diversity while reducing the decoding complexity from ∂(M_c^N/2) to ∂(M_c^N/4) in a M_c-QAM constellation.     

Cayley differential unitary space-time codes

One method for communicating with multiple antennas is to encode the transmitted data differentially using unitary matrices at the transmitter, and to decode differentially without knowing the channel coefficients at the receiver. Since channel knowledge is not required at the receiver, differential schemes are ideal for use on wireless links where channel tracking is undesirable or infeasible, either because of rapid changes in the channel characteristics or because of limited system resources. Although this basic principle is well understood, it is not known how to generate good-performing constellations of unitary matrices, for any number of transmit and receive antennas and for any rate. This is especially true at high rates where the constellations must be rapidly encoded and decoded. We propose a class of Cayley codes that works with any number of antennas, and has efficient encoding and decoding at any rate. The codes are named for their use of the Cayley transform, which maps the highly nonlinear Stiefel manifold of unitary matrices to the linear space of skew-Hermitian matrices. This transformation leads to a simple linear constellation structure in the Cayley transform domain and to an information-theoretic design criterion based on emulating a Cauchy random matrix. Moreover, the resulting Cayley codes allow polynomial-time near-maximum-likelihood (ML) decoding based on either successive nulling/canceling or sphere decoding. Simulations show that the Cayley codes allow efficient and effective high-rate data transmission in multiantenna communication systems without knowing the channel     

Unitary space-time modulation via Cayley transform

A prevoiusly proposed method for communicating with multiple antennas over block fading channels is unitary space-time modulation (USTM). In this method, the signals transmitted from the antennas, viewed as a matrix with spatial and temporal dimensions, form a unitary matrix, i.e., one with orthonormal columns. Since channel knowledge is not required at the receiver, USTM schemes are suitable for use on wireless links where channel tracking is undesirable or infeasible, either because of rapid changes in the channel characteristics or because of limited system resources. Previous results have shown that if suitably designed, USTM schemes can achieve full channel capacity at high SNR and, moreover, that all this can be done over a single coherence interval, provided the coherence interval and number of transmit antennas are sufficiently large, which is a phenomenon referred to as autocoding. While all this is well recognized, what is not clear is how to generate good performing constellations of (nonsquare) unitary matrices that lend themselves to efficient encoding/decoding. The schemes proposed so far either exhibit poor performance, especially at high rates, or have no efficient decoding algorithms. We propose to use the Cayley transform to design USTM constellations. This work can be viewed as a generalization, to the nonsquare case, of the Cayley codes that have been proposed for differential USTM. The codes are designed based on an information-theoretic criterion and lend themselves to polynomial-time (often cubic) near-maximum-likelihood decoding using a sphere decoding algorithm. Simulations suggest that the resulting codes allow for effective high-rate data transmission in multiantenna communication systems without knowing the channel. However, our preliminary results do not show a substantial advantage over training-based schemes.     

A differential detection scheme for transmit diversity

We present a transmission scheme for exploiting diversity given by two transmit antennas when neither the transmitter nor the receiver has access to channel state information. The new detection scheme can use equal energy constellations and encoding is simple. At the receiver, decoding is achieved with low decoding complexity. The transmission provides full spatial diversity and requires no channel state side information at the receiver. The scheme can be considered as the extension of differential detection schemes to two transmit antennas     

Full Rate Space Time Codes for Large Number of Transmitting Antennas with Linear Complexity Decoding

Among the specification of the 5G networks two crucial aspects are the support of fast mobility and high data rates. With fast mobility, the fading channels phenomenon become crucial, resulting in the need for multiple input/output channel to create spatial diversity. Space time codes (STC) have been shown to be well used with the Multiple Input Multiple Output channel. The Orthogonal STC (OSTC) family of codes is known to achieve full diversity as well as very simple implementation of the Maximum Likelihood (ML) decoder. However, it was also proven that with a complex symbol constellation one cannot achieve a full rate code when the number of transmitting antennas is larger than two. Quasi-OSTC (QSTC) can have full rate even for more than two transmitting antennas but with the penalty of decoding complexity which becomes severe if the constellation size is large. In order to tackle this inherent drawback of the OSTC/QSTC and to be able to support the 5G high data rate demand, we have come up with a different STC code that, when used with a new transmission and decoding methods, achieves full rate while maintaining linear complexity decoding for any number of transmit antennas. It can also be shown that when the transmitter knows the strongest channel (through minimal feedback) the code also achieves full diversity along with better error rate than the OSTC and the QSTC.     

On design criteria and construction of noncoherent space-time constellations

We consider the problem of digital communication in a Rayleigh flat-fading environment using a multiple-antenna system, when the channel state information is available neither at the transmitter nor at the receiver. It is known that at high signal-to-noise ratio (SNR), or when the coherence interval is much larger than the number of transmit antennas, a constellation of unitary matrices can achieve the capacity of the noncoherent system. However, at low SNR, high spectral efficiencies, or for small values of coherence interval, the unitary constellations lose their optimality and fail to provide an acceptable performance. In this work, inspired by the Stein's lemma, we propose to use the Kullback-Leibler (KL) distance between conditional distributions to design space-time constellations for noncoherent communication. In fast fading, i.e., when the coherence interval is equal to one symbol period and the unitary construction provides only one signal point, the new design criterion results in pulse amplitude modulation (PAM)-type constellations with unequal spacing between constellation points. We also show that in this case, the new design criterion is equivalent to design criteria based on the exact pairwise error probability and the Chernoff information. When the coherence interval is larger than the number of transmit antennas, the resulting constellations overlap with the unitary constellations at high SNR, but at low SNR they have a multilevel structure and show significant performance improvement over unitary constellations of the same size. The performance improvement becomes especially more significant when an appropriately designed outer code or multiple receive antennas are used. This property, together with the facts that the proposed constellations eliminate the need for training sequences and are most suitable for low SNR, makes them a good candidate for uplink communication in wireless systems.     

Transmitter diversity for OFDM systems and its impact on high-ratedata wireless networks

Transmitter diversity and down-link beamforming can be used in high-rate data wireless networks with orthogonal frequency division multiplexing (OFDM) for capacity improvement. We compare the performance of delay, permutation and space-time coding transmitter diversity for high-rate packet data wireless networks using OFDM modulation. For these systems, relatively high block error rates, such as 10%, are acceptable assuming the use of effective automatic retransmission request (ARQ). As an alternative, we also consider using the same number of transmitter antennas for down-link beamforming as we consider for transmitter diversity. The investigation indicates that delay transmitter diversity with quaternary phase-shift keying (QPSK) modulation and adaptive antenna arrays provides a good quality of service (QoS) with low retransmission probability, while space-time coding transmitter diversity provides high peak data rates. Down-link beamforming together with adaptive antenna arrays, however, provides a higher capacity than transmitter diversity for typical mobile environments     

Signal constellations for quasi-orthogonal space-time block codes with full diversity

Space-time block codes (STBCs) from orthogonal designs proposed by Alamouti, and Tarokh-Jafarkhani-Calderbank have attracted considerable attention lately due to their fast maximum-likelihood (ML) decoding and full diversity. However, the maximum symbol transmission rate of an STBC from complex orthogonal designs for complex signals is only 3/4 for three and four transmit antennas, and it is difficult to construct complex orthogonal designs with rate higher than 1/2 for more than four transmit antennas. Recently, Jafarkhani, Tirkkonen-Boariu-Hottinen, and Papadias-Foschini proposed STBCs from quasi-orthogonal designs, where the orthogonality is relaxed to provide higher symbol transmission rates. With the quasi-orthogonal structure, the quasi-orthogonal STBCs still have a fast ML decoding, but do not have the full diversity. The performance of these codes is better than that of the codes from orthogonal designs at low signal-to-noise ratio (SNR), but worse at high SNR. This is due to the fact that the slope of the performance curve depends on the diversity. It is desired to have the quasi-orthogonal STBCs with full diversity to ensure good performance at high SNR. In this paper, we achieve this goal by properly choosing the signal constellations. Specifically, we propose that half of the symbols in a quasi-orthogonal design are chosen from a signal constellation set A and the other half of them are chosen from a rotated constellation e/sup j/spl phi// A. The resulting STBCs can guarantee both full diversity and fast ML decoding. Moreover, we obtain the optimum selections of the rotation angles /spl phi/ for some commonly used signal constellations. Simulation results show that the proposed codes outperform the codes from orthogonal designs at both low and high SNRs.     

Unitary space-time modulation for multiple-antenna communicationsin Rayleigh flat fading

Motivated by information-theoretic considerations, we propose a signaling scheme, unitary space-time modulation, for multiple-antenna communication links. This modulation is ideally suited for Rayleigh fast-fading environments, since it does not require the receiver to know or learn the propagation coefficients. Unitary space-time modulation uses constellations of T×M space-time signals (Φ_i, l=1, ..., L), where T represents the coherence interval during which the fading is approximately constant, and Mi are orthonormal. When the receiver does not know the propagation coefficients, which between pairs of transmitter and receiver antennas are modeled as statistically independent, this modulation performs very well either when the signal-to-noise ratio (SNR) is high or when T≫M. We design some multiple-antenna signal constellations and simulate their effectiveness as measured by bit-error probability with maximum-likelihood decoding. We demonstrate that two antennas have a 6-dB diversity gain over one antenna at 15-dB SNR     

Mixed-Q linear space-time codes

A new modulation method for linear space-time codes is proposed based on using constellations of different sizes for different symbols. It is shown that the proposed method significantly reduces the complexity of the sphere decoding algorithm. The complexity reduction is more pronounced in high-rate codes, where each code matrix carries a large number of symbols. We also show that the choice of constellation size provides a tradeoff between performance and complexity. Using this, some guidelines for choosing constellation size are presented. As one introduces more constellation disparity in the code, the complexity is further reduced, while the performance loss grows. Typically, a complexity reduction of one to two orders of magnitude can be achieved at the expense of about 3 dB coding gain. We suggest a simple modification in our design to reduce this loss to about 2 dB.     

Quasi-orthogonal STBC with minimum decoding complexity

In this paper, we consider a quasi-orthogonal (QO) space-time block code (STBC) with minimum decoding complexity (MDC-QO-STBC). We formulate its algebraic structure and propose a systematic method for its construction. We show that a maximum-likelihood (ML) decoder for this MDC-QO-STBC, for any number of transmit antennas, only requires the joint detection of two real symbols. Assuming the use of a square or rectangular quadratic-amplitude modulation (QAM) or multiple phase-shift keying (MPSK) modulation for this MDC-QO-STBC, we also obtain the optimum constellation rotation angle, in order to achieve full diversity and optimum coding gain. We show that the maximum achievable code rate of these MDC-QO-STBC is 1 for three and four antennas and 3/4 for five to eight antennas. We also show that the proposed MDC-QO-STBC has several desirable properties, such as a more even power distribution among antennas and better scalability in adjusting the number of transmit antennas, compared with the coordinate interleaved orthogonal design (CIOD) and asymmetric CIOD (ACIOD) codes. For the case of an odd number of transmit antennas, MDC-QO-STBC also has better decoding performance than CIOD.     

基于预编码的全码率准正交空时分组码

针对传统八天线全码率准正交空时分组码解码复杂度高的问题,该文提出了两种基于预编码的传输方案。利用四天线正交码扩展得到新的八天线准正交码,结合两个反馈相位信息构成的预编码矩阵,使信道矩阵正交化,消除码间串扰,实现码元独立最大似然解码。发送端采用交织技术,进一步提高了性能,实现了双码元联合最大似然解码。和获得满分集增益的星座图旋转方案不同,预编码方案最大程度上减少了码间串扰,提高准正交码的性能。仿真结果表明,基于预编码的两种准正交码性能好于星座旋转准正交码,而且降低了解码复杂度。     

Unitary signal constellations for differential space-time modulation with two transmit antennas: parametric codes, optimal designs, and bounds

We focus on the design of unitary signal constellations for differential space-time modulation with double transmit antennas. By using the parametric form of a two-by-two unitary matrix, we present a class of unitary space-time codes called parametric codes and show that this class of unitary space-time codes leads to a five-signal constellation with the largest possible diversity product and a 16-signal constellation with the largest known diversity product. Although the parametric code of size 16 is not a group by itself, we show that it is a subset of a group of order 32. Furthermore, the unitary signal constellations of sizes 32, 64, 128, and 256 obtained by taking the subsets of the parametric codes of sizes 37, 75, 135, and 273, respectively, have the largest known diversity products. We also use large diversity sum of unitary space-time signal constellations as another significant property for the signal constellations to have good performance in low-SNR scenarios. The newly introduced unitary space-time codes can lead to signal constellations with sizes of 5 and 9 through 16 that have the largest possible diversity sums. Subsequently, we construct a few sporadic unitary signal constellations with the largest possible diversity product or diversity sum. A four-signal constellation which has both the largest possible diversity product and the largest possible diversity sum and three unitary signal constellations with the largest possible diversity sums for sizes of 6, 7, and 8 are constructed, respectively. Furthermore, by making use of the existing results in sphere packing and spherical codes, we provide several upper and lower bounds on the largest possible diversity product and the largest possible diversity sum that unitary signal constellations of any size can achieve.     

Rate-one space-time block codes with full diversity

Orthogonal space-time block codes provide full diversity, and maximum-likelihood (ML) decoding for orthogonal codes can be realized on a symbol-by-symbol basis. It has been shown that rate-one complex orthogonal codes do not exist for systems with more than two transmit antennas. For a general system with N transmit and M receive antennas, it is very desirable to design rate-one complex codes with full diversity. In this letter, we provide a systematic method of designing rate-one codes (real or complex) for a general multiple-input multiple-output system. Full diversity of these codes is then achieved by constellation rotation. A generalized, reduced-complexity decoding method for rate-one codes is also provided.     

Receive antenna selection for unitary space-time modulation over semi-correlated Ricean channels

Receive antenna selection for unitary space-time modulation (USTM) over semi-correlated Ricean fading channels is analyzed (this work generalizes that of Ma and Tepedelenlio-glu for the independent and identically distributed (i.i.d.) Rayleigh fading case). The antenna selection rule is that the receive antennas with the largest signal powers are chosen. For single antenna selection, we derive the maximum likelihood decoding for the correlated Ricean case. We also derive the Chernoff bound on the pairwise error probability for the high signal to- noise ratio (SNR) region and obtain the coding gain and diversity order. Our results show that even when there are transmitter side correlations and a line of sight component, receive antenna selection with USTM preserves the full diversity order if the USTM constellation is of full rank. We also give an approximation to the distribution function of a quadratic form of non-zero mean complex Gaussian variates (from Nabar et al.) at the high SNR region. Based on this approximation, a closed-form expression for the coding gain is also obtained and compared with that of the i.i.d. Rayleigh case. We also analyze the case of multiple receive antenna selection and derive the coding gain and diversity order. We show that USTM constellations, which have been proposed for the i.i.d. Rayleigh channel, can be used with the correlated Ricean channel as well.     

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     

Diagonal algebraic space-time block codes

We construct a new family of linear space-time (ST) block codes by the combination of rotated constellations and the Hadamard transform, and we prove them to achieve the full transmit diversity over a quasi-static or fast fading channels. The proposed codes transmit at a normalized rate of 1 symbol/s. When the number of transmit antennas n=1, 2, or n is a multiple of four, we spread a rotated version of the information symbol vector by the Hadamard transform and send it over n transmit antennas and n time periods; for other values of n, we construct the codes by sending the components of a rotated version of the information symbol vector over the diagonal of an n × n ST code matrix. The codes maintain their rate, diversity, and coding gains for all real and complex constellations carved from the complex integers ring Z [i], and they outperform the codes from orthogonal design when using complex constellations for n > 2. The maximum-likelihood (ML) decoding of the proposed codes can be implemented by the sphere decoder at a moderate complexity. It is shown that using the proposed codes in a multiantenna system yields good performances with high spectral efficiency and moderate decoding complexity     

Geometrical and Numerical Design of Structured Unitary Space–Time Constellations

There exist two important design criteria for unitary space time codes. In the situation where the signal-to-noise ratio (SNR) is large the diversity product (DP) of a constellation should be as large as possible. It is less known that the diversity sum (DS) is a very important design criterion for codes working in a low SNR environment. So far, no general method to design good-performing constellations with large diversity for any number of transmit antennas and any transmission rate exists. In this correspondence, we propose constellations with suitable structures, which allow one to construct codes with excellent diversity using geometrical symmetry and numerical methods. The presented design methods work for any dimensional constellation and for any transmission rate     

Differential space-time modulation over frequency-selective channels

We present herein a new differential space-time-frequency (DSTF) modulation scheme for systems that are equipped with an arbitrary number of transmit antennas and operate in frequency-selective channels. The proposed DSTF modulator consists of a concatenating spectral encoder and differential encoder that offer full spatio-spectral diversity and significant coding gain. A unitary structure is imposed on the differential encoder to admit linear, decoupled maximum likelihood (ML) detection in space and time. Optimum criteria based on pairwise error probability analysis are developed for spectral encoder design. We introduce a class of spectral codes, namely, linear constellation decimation (LCD) codes, which are nonbinary block codes obtained by decimating a phase-shift-keying (PSK) constellation with a group of decimation factors that are co-prime with the constellation size. Since LCD codes encode across a minimally necessary set of subchannels for full diversity, they incur modest decoding complexity among all full-diversity codes. Numerical results are presented to illustrate the performance of the proposed DSTF modulation and coding scheme, which compares favorably with several existing differential space-time schemes in frequency-selective channels.