The Minimum Decoding Delay of Maximum Rate Complex Orthogonal Space–Time Block Codes

The growing demand for efficient wireless transmissions over fading channels motivated the development of space-time block codes. Space-time block codes built from generalized complex orthogonal designs are particularly attractive because the orthogonality permits a simple decoupled maximum-likelihood decoding algorithm while achieving full transmit diversity. The two main research problems for these complex orthogonal space-time block codes (COSTBCs) have been to determine for any number of antennas the maximum rate and the minimum decoding delay for a maximum rate code. The maximum rate for COSTBCs was determined by Liang in 2003. This paper addresses the second fundamental problem by providing a tight lower bound on the decoding delay for maximum rate codes. It is shown that for a maximum rate COSTBC for 2m - 1 or 2m antennas, a tight lower bound on decoding delay is r = (_m-1 ^2m) . This lower bound on decoding delay is achievable when the number of antennas is congruent to 0, 1, or 3 modulo 4. This paper also derives a tight lower bound on the number of variables required to construct a maximum rate COSTBC for any given number of antennas. Furthermore, it is shown that if a maximum rate COSTBC has a decoding delay of r where r < r les 2r, then r=2r. This is used to provide evidence that when the number of antennas is congruent to 2 modulo 4, the best achievable decoding delay is 2(_m-1 ^2m_).     

Orthogonal designs with maximal rates

Orthogonal designs have been used as space-time block codes for wireless communications with multiple transmit antennas, which can achieve full transmit diversity and have a very simple decoupled maximum-likelihood decoding algorithm. The rate of an orthogonal design is defined as the ratio of the number of transmitted information symbols in a block of channel uses to the length of the given block, which reflects the bandwidth efficiency of the employed space-time block code constructed from the orthogonal design. This paper focuses on the analysis and synthesis of orthogonal designs with the maximum possible rates, which may be real or complex and square or rectangular matrices. We first provide several representations of orthogonal designs and their characterizations in terms of Hurwitz-Radon matrix equations. Next, we observe that the real orthogonal designs, square or rectangular, and the complex square orthogonal designs with maximal rates have been well understood from the existing results in the mathematics literature which can be dated back to 1890s. However, unfortunately, it is not the case for the complex rectangular orthogonal designs with rates as high as possible. We then construct a class of complex orthogonal designs for any number of transmit antennas. The proposed complex orthogonal designs for the number of transmit antennas n=2m-1 and 2m have the same rate m+1 and 2m, where m is any natural number. Finally, we demonstrate that, for n=2m-1 and 2m with any given natural number m, the value m+1 and 2m is the maximum possible rate that the complex orthogonal designs, square or rectangular, with n transmit antennas can achieve.     

Gaussian integers and interleaved rank codes for space–time block codes

This paper presents the design of space–time block codes (STBCs) over maximum rank distance (MRD) codes, energy‐efficient STBCs, STBCs using interleaved‐MRD codes, the use of Gaussian integers for STBCs modulation, and Gabidulin's decoding algorithm for decoding STBCs. The design fundamentals of STBCs using MRD codes are firstly put forward for different number of transmit antennas. Extension finite fields (Galois fields) are used to design these linear block codes. Afterward, a comparative study of MRD‐based STBCs with corresponding orthogonal and quasi‐orthogonal codes is also included in the paper. The simulation results show that rank codes, for any number of transmit antennas, exhibit diversity gain at full rate contrary to orthogonal codes, which give diversity gain at full rate only for two transmit antennas case. Secondly, an energy‐efficient MRD‐STBC is proposed, which outperforms orthogonal STBC at least for 2 × 1 antenna system. Thirdly, interleaved‐MRD codes are used to construct higher‐order transmit antenna systems. Using interleaved‐MRD codes further reduces the complexity (compared with normal MRD codes) of the decoding algorithm. Fourthly, the use of Gaussian integers is utilized in mapping MRD‐based STBCs to complex constellations. Furthermore, it is described how an efficient and computationally less complex Gabidulin's decoding algorithm can be exploited for decoding complex MRD‐STBCs. The decoding results have been compared against hard‐decision maximum likelihood decoding. Under this decoding scheme, MRD‐STBCs have been shown to be potential candidate for higher transmit antenna systems as the decoding complexity of Gabidulin's algorithm is far less, and its performance for decoding MRD‐STBCs is somewhat reasonable. Copyright © 2015 John Wiley & Sons, Ltd.     

On the nonexistence of rate-one generalized complex orthogonal designs

Orthogonal space-time block coding proposed recently by Alamouti (1998) and Tarokh et al. (1999) is a promising scheme for information transmission over Rayleigh-fading channels using multiple transmit antennas due to its favorable characteristics of having full transmit diversity and a decoupled maximum-likelihood (ML) decoding algorithm. Tarokh et al. extended the theory of classical orthogonal designs to the theory of generalized, real, or complex, linear processing orthogonal designs and then applied the theory of generalized orthogonal designs to construct space-time block codes (STBC) with the maximum possible diversity order while having a simple decoding algorithm for any given number of transmit and receive antennas. It has been known that the STBC constructed in this way can achieve the maximum possible rate of one for every number of transmit antennas using any arbitrary real constellation and for two transmit antennas using any arbitrary complex constellation. Contrary to this, in this correspondence we prove that there does not exist rate-one STBC from generalized complex linear processing orthogonal designs for more than two transmit antennas using any arbitrary complex constellation.     

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.     

Linear Decoupled and Quasi-Decoupled Space–Time Codes

Space-time transmit diversity results in coupling of transmitted symbols across different antennas, which increases the complexity of maximum-likelihood decoding. Symbol coupling can be completely or partially avoided if the space-time code (STC) satisfies specific decoupling conditions; examples of such codes are orthogonal space-time block codes and quasi-orthogonal codes. In this letter, we study decoupling conditions for a linear full-diversity STC. Quasi-decoupled codes are proposed as a partially decoupled full-diversity STC of any rate for any number of transmit antennas with minimum decoding delay. By optimizing the coding gain of quasi-decoupled codes, it is shown that quasi-orthogonal codes have competitive performance with respect to the Alamouti code, and the more-recent threaded algebraic space-time (TAST) codes and ABBA codes. A general full-diversity decoupling condition is considered, and the general solution to this case, which also encompasses previously known orthogonal STCs, is derived     

Spectral efficiency of MIMO multiaccess systems with single-user decoding

The use of multiple antennas at the transmitter and the receiver is considered for the uplink of cellular communication systems. The achievable spectral efficiency in bits/s/Hz is used as the criterion for comparing various design choices. The focus is on wideband code-division multiple-access (CDMA) systems when the receiver uses the matched-filter or the minimum mean-squared error detector, followed by single-user decoders. The spreading sequences of the CDMA system are assumed to be random across the users, but could be dependent across the transmit antennas of each user. Using analytical results in the large system asymptote, guidelines are provided for the sequence design across the transmit antennas and for choosing the number of antennas. In addition, comparisons are made between (random) CDMA and orthogonal multiaccess with multiple antennas. It is shown that CDMA, even with single-user decoding, can outperform orthogonal multiaccess when the number of receive antennas is sufficiently large.     

Space-time block coding for wireless communications: performanceresults

We document the performance of space-time block codes, which provide a new paradigm for transmission over Rayleigh fading channels using multiple transmit antennas. Data is encoded using a space-time block code, and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas. The received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise. Maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection. This uses the orthogonal structure of the space-time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver. We review the encoding and decoding algorithms for various codes and provide simulation results demonstrating their performance. It is shown that using multiple transmit antennas and space-time block coding provides remarkable performance at the expense of almost no extra processing     

Fast differential unitary space-time demodulation via square orthogonal designs

This paper presents a natural concatenation between the real or complex square orthogonal designs and the differential unitary space-time modulation scheme for multiple transmit antennas. This concatenation renders the differential unitary space-time demodulator to detect in a decoupled way the information symbols and hence to have a very low decoding complexity. This fast differential demodulation method for multiple transmit antennas can be regarded as a generalization of the existing differential detection method proposed by Tarokh and Jafarkhani for two and four transmit antennas based on orthogonal designs. By using a systematic matrix-vector representation, a unified and simplified form of differential unitary demodulation via square orthogonal designs for any number of transmit antennas and any transmission rate in bits per channel use is provided.     

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     

A space-time coding modem for high-data-rate wirelesscommunications

This paper presents the theory and practice of a new advanced modem technology suitable for high-data-rate wireless communications and presents its performance over a frequency-flat Rayleigh fading channel. The new technology is based on space-time coded modulation (STCM) with multiple transmit and/or multiple receive antennas and orthogonal pilot sequence insertion (O-PSI). In this approach, data is encoded by a space-time (ST) channel encoder and the output of the encoder is split into N streams to be simultaneously transmitted using N transmit antennas. The transmitter inserts periodic orthogonal pilot sequences in each of the simultaneously transmitted bursts. The receiver uses those pilot sequences to estimate the fading channel. When combined with an appropriately designed interpolation filter, accurate channel state information (CSI) can be estimated for the decoding process. Simulation results of the proposed modem, as applied to the IS-136 cellular standard, are presented. We present the frame error rate (FER) performance results as a function of the signal-to-noise ratio (SNR) and the maximum Doppler frequency, in the presence of timing and frequency offset errors. Simulation results show that for a 10% FER, a 32-state eight-phase-shift keyed (8-PSK) ST code with two transmit and two receive antennas can support data rates up to 55.8 kb/s on a 30-kHz channel, at an SNR of 11.7 dB and a maximum Doppler frequency of 180 Hz. Simulation results for other codes and other channel conditions are also provided. We also compare the performance of the proposed STCM scheme with delay diversity schemes and conclude that STCM can provide significant SNR improvement over simple delay diversity     

随机布局多天线信号联合时差估计Cramer-Rao下界

该文针对随机布局多天线信号联合时差估计Cramer-Rao下界（CRLB）开展研究,在深入研究多路联合参数估计和经典时差估计算法的基础上,首先建立信号模型,进而得到频域的联合概率密度函数,然后推导出Fisher信息矩阵和Cramer-Rao下界的解析表达式。最后,对结果进行了讨论分析,并同两路时差估计Cramer-Rao下界进行了对比。结果表明,多天线联合时差估计能够利用各信号的相同信息,有效提升时差估计性能,而且在低信噪比条件下估计性能改善更为明显。此外,可以看到增加天线数目不可能无限降低时差估计Cramer-Rao下界,其受待估时差的两路信号信噪比限制。      

A zero-forcing approximate log-likelihood receiver for MIMO bit-interleaved coded modulation

This letter considers multiple-input multiple-output (MIMO) systems with bit-interleaved coded modulation. An approximate log-likelihood decoding approach is presented based on a zero-forcing receiver. The implementation complexity is low compared to the maximum likelihood (ML) receiver. We show that the performance gap, compared to ML, reduces when either the number of receive antennas or the modulation order is increased. Results are presented for both narrowband fast-fading, and orthogonal frequency division multiplexing (OFDM) channels. In the OFDM case, we demonstrate performance gaps as low as 0.5 dB for MIMO extensions to the IEEE 802.11a wireless local area network physical layer standard.     

A squaring method to simplify the decoding of orthogonal space-timeblock codes

We present a squaring method to simplify the decoding of orthogonal space-time block codes in a wireless communication system with an arbitrary number of transmit and receive antennas. Using this squaring method, a closed-form expression of signal-to-noise ratio after space-time decoding is also derived. It gives the same decoding performance as the maximum-likelihood ratio decoding while it shows much lower complexity     

Square-matrix embeddable space-time block codes for complex signalconstellations

Space-time block codes for providing transmit diversity in wireless communication systems are considered. Based on the principles of linearity and unitarity, a complete classification of linear codes is given in the case when the symbol constellations are complex, and the code is based on a square matrix or restriction of such by deleting columns (antennas). Maximal rate delay optimal codes are constructed within this category. The maximal rates allowed by linearity and unitarity fall off exponentially with the number of transmit antennas     

Coding for radio channels

The conventional remedy to time and/or frequency variability of radio channels is diversity. Redundant coding is a kind of diversity, as each coded symbol can be recovered from other symbols. Only linear binary block codes are considered. Any binary random variable can be represented by its algebraic value,a real number whose sign indicates its most likely value and whose absolute value measures the probability of this value. The algebraic value of a received binary symbol is itself a random variable, whose distribution obeys a particular constraint. The algebraic value associated with the maximum likelihood decision on a binary symbol, given a set of independent received replicas of it, and that associated with the sum modulo 2 of binary random variables are also considered. The symbol-by-symbol decoding is then analysed in the case of threshold decoding, then in the general case. An approximate bound on the decoding error probability for additive Gaussian noise and coherent demodulation is used to assess the advantage of coding when unequalenergy symbols are received, according to a deterministic or a Rayleigh distribution. Simulation results are given for the Hamming (15,11) code. Coding affords a significant advantage provided the channel is good enough, while conventional diversity always provides gain.     

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 complex orthogonal space-time block code for 8 transmit antennas

We present a new complex orthogonal space-time block code (STBC) for 8 transmit antennas, which is generated simply by padding a transmission vector for the 8th transmit antenna to the transmission matrix of the existing complex orthogonal STBC for 7 transmit antennas. The presented complex orthogonal STBC for 8 transmit antennas achieves the same maximal rate 5/8 as well as the same minimal decoding delay 56 as those of the previously constructed complex orthogonal STBC for 7 transmit antennas.     

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.     

An efficient generalized sphere decoder for rank-deficient MIMO systems

We derive a generalized sphere decoder (GSD) for rank-deficient multiple input multiple output (MIMO) systems using N transmit antennas and M receive antennas. This problem arises when N>M or when the channel gains are strongly correlated. The upper triangular factorization of the Grammian yields an under-determined system and the standard sphere decoding (SD) fails. For constant modulus constellations, we modify the maximum likelihood (ML) cost metric so that the equivalent Grammian is rank N. The resulting GSD algorithm has significantly lower complexity than previous algorithms. A method to handle nonconstant modulus constellations is also developed.