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A Construction of a Space-Time Code Based on Number Theory
Abstract
We construct a full data rate space-time (ST) block code over M=2
transmit antennas and T=2 symbol periods, and we prove that it achieves
a transmit diversity of 2 over all constellations carved from Z[i]<sup>4
</sup>. Further, we optimize the coding gain of the proposed code and
then compare it to the Alamouti code. It is shown that the new code
outperforms the Alamouti (see IEEE J Select. Areas Commun., vol.16,
p.1451-58, 1998) code at low and high signal-to-noise ratio (SNR) when
the number of receive antennas N>1. The performance improvement is
further enhanced when N or the size of the constellation increases. We
relate the problem of ST diversity gain to algebraic number theory, and
the coding gain optimization to the theory of simultaneous Diophantine
approximation in the geometry of numbers. We find that the coding gain
optimization is equivalent to finding irrational numbers "the furthest,"
from any simultaneous rational approximations
... 5.4.1. Le code B 1 , appelé aussi code 1 dans les figures, est introduit par Damen et al. [32], le code B 2 appelé aussi code 2 ou «golden code» a été introduit par Belfiore et al. [33]. En fin le code B 3 , pour le cas G 6,3 , a été présenté dans [27, pag. ...
... We hope that by choosing a good coherent STB code we can generate a good non-coherent STB code. Besides, if the parameters of the coherent STB code can be controlled (see for example [27], [32]), we hope to control the parameters of the non-coherent STB code, at least partially. Another important point is the decoding algorithm: the the coherent STB has sufficient algebric structure to have a simplified decoding algorithm, we would like to exploit the knowledge. ...
... Some interesting references are [55,27,78]. Coherent STB codes considered here are carved from lattices [27,78,32]. ...
This thesis is composed of two parts, the most important one deals with space-time block coding and decoding for systems in which no channel knowledge is available at the receiver and at the transmitter. We focus on a particular family of non-coherent space-time block codes, obtained by a non-linear map called the exponential map. This part is composed of four chapters. In Chapter 2, we introduce the channel model. Basic mathematical tools, results on non-coherent space-time codes and previous propositions are also recalled. In Chapter 3, an in-depth investigation of non-coherent space-time codes obtained via the exponential map is carried out. We explain the geometrical interpretation of this coding procedure and we solve some open problems on code design. In Chapter 4, we propose a new simplified decoder for the case of one transmit antenna and many receive antennas. Discussion on the expected complexity and parameters of the decoder are detailed, as well as simulations and comparisons with other propositions. In Chapter 5, we propose two simplified decoders in the general multiple antenna case. Discussion about the expected complexity and about decoder parameters is provided. Simulations and comparisons conclude this chapter. The second part of the thesis is composed of Chapter 6, which deals with CPM signal decompositions in PAM pulses. CPM is a non-linear modulation, while signal decomposition as a linear combination of waveforms can simplify the code and receiver design both for coherent and non-coherent systems. A final conclusion is reported. A summary of the thesis in french is provided in Chapter 1.
... 5.4.1. Le code B 1 , appelé aussi code 1 dans les figures, est introduit par Damen et al. [32], le code B 2 appelé aussi code 2 ou «golden code» a été introduit par Belfiore et al. [33]. En fin le code B 3 , pour le cas G 6,3 , a été présenté dans [27, pag. ...
... We hope that by choosing a good coherent STB code we can generate a good non-coherent STB code. Besides, if the parameters of the coherent STB code can be controlled (see for example [27], [32]), we hope to control the parameters of the non-coherent STB code, at least partially. Another important point is the decoding algorithm: the the coherent STB has sufficient algebric structure to have a simplified decoding algorithm, we would like to exploit the knowledge. ...
... Some interesting references are [55,27,78]. Coherent STB codes considered here are carved from lattices [27,78,32]. ...
... At this stage, we invoke the Lindenmann's theorem [16], which states that, if a 1 , a 2 , · · · , a m are distinct algebraic numbers, and if c 1 , c 2 , · · · , c m are algebraic and not all equal to zero, then c 1 e a1 + c 2 e a2 + · · · + c m e am = 0. ...
... N ω vl are all algebraic [16]. Therefore, comparing (51) and (50), if the terms φ q = 0, real, distinct and algebraic ensures that all the eigen values of ∆ ′ ij are non-zero, making it full rank (i.e., rank P ). ...
Orthogonal time frequency space (OTFS) is a 2-dimensional (2D) modulation technique designed in the delay-Doppler domain. A key premise behind OTFS is the transformation of a time varying multipath channel into an almost non-fading 2D channel in delay-Doppler domain such that all symbols in a transmission frame experience the same channel gain. It has been suggested in the recent literature that OTFS can extract full diversity in the delay-Doppler domain, where full diversity refers to the number of multipath components separable in either the delay or Doppler dimension, but without a formal analysis. In this paper, we present a formal analysis of the diversity achieved by OTFS modulation along with supporting simulations. Specifically, we prove that the asymptotic diversity order of OTFS (as SNR $\rightarrow \infty$) is one. However, in the finite SNR regime, potential for a higher order diversity is witnessed before the diversity one regime takes over. Also, the diversity one regime starts at lower BER values for increased frame sizes. We also propose a phase rotation scheme for OTFS using transcendental numbers. We show that OTFS with this proposed scheme extracts the full diversity in the delay-Doppler domain.
... One of the important factors in deriving bounds for the optimization of constellations is considering their complexity. For example, well-known union bounds on the performance of constellations in the AWGN channel based on Q(·) function in [38] are quite difficult to optimize for medium-to-large constellations since each evaluation of Q(·) takes a relatively long time in comparison to the simplified bounds presented in (17), (20), (21), and (22). Furthermore, in many cases, optimization based on more complex bounds results in very little improvement. ...
... However, for the irregular constellations, the decision regions are very complex and designing a decoder based on these regions may be infeasible. In both cases, sphere decoders may decrease the decoding complexity [20]. However, for regular constellations, sphere decoders with lower complexity can be designed. ...
Finding the best signal constellation for different communication channels is one of the fundamental problems in digital communication. This problem has been studied widely from different angles and many methods have been proposed for designing good practical signal constellations. There has been a rejuvenated interest in designing good constellations during last decade, in part due to the advent of novel optimization techniques. Nevertheless, most of the recent work, similar to the older work in this area, aims to optimize the constellation within a presumed structure (such as points lying on concentric rings). In this paper, we develop a different approach: we aim to optimize constellations based on a Chernoff bound on the probability of error in the versatile Nakagami-m fading channel. We derive two general bounds on the symbol error rate and bit error rate performance of orthogonal transmission in Nakagami-m fading channel for single-input single-output and orthogonal space-time block codes and we show that a substantial improvement in the error probability is achieved with the novel constellations that are optimized using these bounds.
... No with STBCs from [43], [47]- [49] bpcu : bits per channel use ⌊c⌋ 2 p denotes the largest integer that is a power of two and smaller than c Q : No. of DMs in [31], [32] d : transmit diversity order Q ′ : No. of cyclic signal matrices in [32] † Depends on the STBC employed Section VI concludes the paper. 2 II. ...
... Thus, the schemes in [31] and [33] pose the issue of UDCS. Furthermore, it is worth mentioning that the UDCS issue highlighted above can be seen in other differential transmission schemes in the literature, which employ 1) orthogonal space-time block codes [45], [46] combined with QAM signal sets, 2) non-orthogonal STBCs constructed both from division algebras [43] and from number theory [47], the Golden code [48] and the perfect STBCs [49]. The performance of these STBCs in the context of differential transmission scheme can be found in [50]. ...
We show that certain signal constellations invoked for classic differential encoding result in a phenomenon we term as the {\em unbounded differential constellation size} (UDCS). Various existing differential transmission schemes that suffer from this issue are identified. Then, we propose an enhanced algebraic field extension based differential spatial modulation scheme (AFE-DSM) and its enhanced counterpart that strikes a diversity-rate trade-off (AFE-DSM-DR), both of which overcome the UDCS issue without compromising its full transmit diversity advantage. Furthermore, the proposed schemes are extended to incorporate amplitude and phase shift keying (APSK) in order to exploit all the available degrees of freedom. Additionally, we propose a pair of detection schemes specially designed for APSK aided differential transmission schemes. Explicitly, we conceive the buffered minimum mean squared error (B-MMSE) detector and buffered maximum likelihood (B-ML) detector, which exploit the knowledge of previously detected symbols in order to further improve the detection performance. Our simulation results have shown that the proposed detectors are capable of bridging the performance gap between the conventional differential detector (CDD) and the coherent detector that has full channel state information. Specifically, when employing the proposed APSK aided AFE-DSM scheme operating at a rate of 2 bits per channel use (bpcu), the B-MMSE and B-ML detectors are observed to give about 3 dB and 3.5 dB signal-to-noise ratio gain with respect to their CDD counterpart at a bit error ratio of $10^{-5}$.
... However, it is found that OSTBC has a low code rate that cannot be above 3/4 symbols per channel use for more than two transmit antennas [8]. To improve the code rate of the STBC, numerous code designs have been developed including quasi-orthogonal STBC [9]- [19] and STBC based on algebraic number theory [20]- [30]. Two typical designs of those codes are threaded algebraic ST (TAST) codes [21], [23] and cyclic division algebra based ST codes [24]- [30] which have been shown to obtain full rate and full diversity. ...
... Assume that p is the minimal index such thatX i p = 0. Then, we can write (20) as ...
Partial interference cancellation (PIC) group decoding proposed by Guo and Xia is an attractive low-complexity alternative to the optimal processing for multiple-input multiple-output (MIMO) wireless communications. It can well deal with the tradeoff among rate, diversity and complexity of space-time block codes (STBC). In this paper, a systematic design of full-diversity STBC with low-complexity PIC group decoding is proposed. The proposed code design is featured as a group-orthogonal STBC by replacing every element of an Alamouti code matrix with an elementary matrix composed of multiple diagonal layers of coded symbols. With the PIC group decoding and a particular grouping scheme, the proposed STBC can achieve full diversity, a rate of $(2M)/(M+2)$ and a low-complexity decoding for $M$ transmit antennas. Simulation results show that the proposed codes can achieve the full diversity with PIC group decoding while requiring half decoding complexity of the existing codes.
... Such a design has been presented for n t = 4 in [12]. It is known how to design information lossless codes [13] for the case where n r ≥ n t . However, it is not known how to design information lossless codes when n r < n t . ...
... It has been shown that if the generator matrix is unitary, the STBC does not reduce the ergodic capacity of the MIMO channel [13], [22]. For the generator matrix to be unitary, a prerequisite is that the number of receive antennas should be atleast equal to the number of transmit antennas, because only then will the generator matrix be square. ...
For an $n_t$ transmit, $n_r$ receive antenna system ($n_t \times n_r$ system), a {\it{full-rate}} space time block code (STBC) transmits $n_{min} = min(n_t,n_r)$ complex symbols per channel use and in general, has an ML-decoding complexity of the order of $M^{n_tn_{min}}$ (considering square designs), where $M$ is the constellation size. In this paper, a scheme to obtain a full-rate STBC for $2^a$ transmit antennas and any $n_r$, with reduced ML-decoding complexity of the order of $M^{n_t(n_{min}-3/4)}$, is presented. The weight matrices of the proposed STBC are obtained from the unitary matrix representations of a Clifford Algebra. For any value of $n_r$, the proposed design offers a reduction from the full ML-decoding complexity by a factor of $M^{3n_t/4}}$. The well known Silver code for 2 transmit antennas is a special case of the proposed scheme. Further, it is shown that the codes constructed using the scheme have higher ergodic capacity than the well known punctured Perfect codes for $n_r < n_t$. Simulation results of the symbol error rates are shown for $8 \times 2$ systems, where the comparison of the proposed code is with the punctured Perfect code for 8 transmit antennas. The proposed code matches the punctured perfect code in error performance, while having reduced ML-decoding complexity and higher ergodic capacity.
... Although the first spacetime (ST) coding was developed by Tarok et al. [6], proposing the construction criteria for ST codes. Since then many works have proposed a variety of ST transmission schemes in order to attain a good compromise between error performance and transmission rate in different system contexts [14,15,16,17,18,19,20,21,22,23,24]. In order to exploit the spatial and frequency diversities over frequency-selective MIMO channels, the space-frequency (SF) coding was proposed in [25]. ...
... 16: Influence of the number of receive antennas: BER versus SNR. ...
Since the growing success of mobile systems in the 1990s, new wireless technologies have been developed in order to support a growing demand for high-quality multimedia services with low error rates. An interesting way to improve the error performance and to achieve better transmission rates is to combine the use of various diversities and multiplexing access techniques in the MIMO system context. The incorporation of oversampling, spreading and multiplexing operations and additional diversities on wireless systems lead to multidimensional received signals which naturally satisfy tensor models. This thesis proposes a new tensorial approach based on a tensor space-time (TST) coding for MIMO wireless communication systems. The signals received by multiple antennas form a fourth-order tensor that satisfies a new tensor model, referred to as PARATUCK-(2,4) (PT-(2,4)) model. A performance analysis is carried out for the proposed TST system and a recent space-time-frequency (STF) system, which allows to derive expressions for the maximum diversity gain over a at fading channel. An uplink processing based on the TST coding with allocation resources is proposed. A new tensor decomposition is introduced, the so-called PT-(N1,N), which generalizes the standard PT-2 and our PT-(2,4) model. This thesis establishes uniqueness conditions for the PARATUCK-(N1,N) model. From these results, joint symbol and channel estimation is ensured for the TST and STF systems. Semi-blind receivers are proposed based on the well-known Alternating Least Squares algorithm and the Levenberg-Marquardt method, and also a new receiver based on the Kronecker Least Squares (KLS) for both systems.
... Comparisons with [6], [17] and [18] were not made in this work but we propose it as an interesting research opportunity. We note that the design technique presented in this work cannot increase the FER union upper bound of the initial code but can only improve it or, in the worst-case, keep it the same. ...
... Damen's code in [6], Lu's code in [17] or the Maddah-Ali code in [18]. ...
... These are codes whose codebook C is a finite subset of a lattice Λ ⊂ C n . It is common to resort to number fields and division algebras to obtain lattices for reliability and security purposes, as they provide a highly structured way of studying their properties and measuring their performance [1], [2]. We will focus on the reliability problem at the physical layer [3]. ...
... Also, it has been shown in [42] that random rotations perform as good as algebraic rotations in a high-diversity high-dimensional environment. In [43] [44] [45] among others, the authors proposed then algebraic constructions of space-time codes for uncoded multiple-antenna systems, and they outperformed orthogonal designs as they were full-rate, i.e. one symbol is sent per transmit antenna per ...
This thesis deals with the design of coded modulations for block fading channels with iterative detection and decoding. First, the design of coded modulations for multiple antenna fading channels is discussed. The design consists of choosing a space-time precoding matrix that minimizes the discrete-input outage probability for such systems, while taking into account several parameters that make it suitable for optimized iterative decoding. To conclude this part, the design of turbo codes for multiple-antenna fading systems is proposed based on channel multiplexers. Introducing no additional complexity at the receiver end, these mutliplexers allow to achieve full diversity and high coding gains. Second, coded modulations for cooperative fading channels is proposed, in which relays transmit sequentially to each others. Bounds on the achievable diversity orders are derived, and space-time transmission strategies are proposed to achieve optimal performance. Finally, the design of irregular turbo codes for block-fading channels is presented. Based on the density evolution method, degree profiles that perform very close to the fundamental limits are proposed.
... Nonvanishing determinants are of interest to preserve the spectral efficiency. Many such codes are presented in the literature [8][9][10][11][12]. But the determinant of all these codes approaches to zero when the number of bits per symbol is increased [6]. ...
The 5th Generation (5G) of wireless communication will be heterogeneous to support various traffic types and applications. Generalized Frequency Division Multiplexing (GFDM) is a 5G non-orthogonal waveform contender that offers scalable and spectrally compliant air interface which addresses the needs of 5G requirements. To aid the demand of high-capacity and reliable transmission, multiple antenna techniques are relied upon. In this work, the 5G modulation scheme GFDM is combined with a full rate Space–Time Block Code (STBC) with nonvanishing determinant called Golden Codes to exploit the diversity to the fullest. The proposed work is implemented in a real-time SDR test-bed called Wireless Open Access Research Platform (WARP). The proposed work is compared with 4G OFDM modulation in both Golden Coded and Uncoded case. The performance is assessed in terms of BER and Capacity/Bandwidth resolution. From the investigations, it is seen that the BER performance of Golden Coded GFDM outperforms the Uncoded GFDM by 3 dB SNR gain and attains a near-equal performance as OFDM. Also, it is evident from the analysis that, there is a 3.5 bps/Hz gain in capacity when compared to OFDM.
... The use of division algebras for space-time coding is usually attributed to the seminal work by Sethuraman et al. [32]. Number fields and cyclic algebras were discussed, which have been favourite tools for space-time design; see, for example, [6,11,12]. Other algebras have been explored, such as Clifford algebras [13], crossed product algebras [35] and non-crossed product algebras [15]. ...
In this paper, we present some recent instances of applying algebraic tools from different categories, in the design of digital communications systems. More specifically, we present a structure based on the elements of a group ring for constructing and encoding QC-LDPC codes. As another instance, we present the construction of space-time block codes from the rings of twisted Laurent series which include some instances of crossed product and non-crossed product division algebras.
... Furthermore, we note that the TPC matrix introduced in Sec. II.C may invoke any of the richly documented historical full-rank constructions in the space-time domain such as the lattice-based [62]- [64], field extension aided [65] and division algebras assited [66]- [68] constructions. However, it was astutely pointed out in [66] that the TPC matrices that achieve the optimal diversitymultiplexing tradeoff do not always lead to the best BER performance. ...
Index modulation (IM) is a recently proposed multi-carrier transmission scheme, which conveys information both by conventional symbols as well as by the specific subcarrier activation patterns conveying them. However, an impediment of IM is that it lacks transmit diversity gain. In this paper, we circumvent this limitation by proposing a limited-feedback assisted IM transmission scheme. Specifically, Euclidean distance based subcarrier subset selection (ED-SSS) is proposed and its attainable transmit diversity order is shown to be
$N_c-N_{IM}+1$
, where
$N_c$
is the total number of subcarriers in an IM block and
$N_{IM}$
is the number of subcarriers used for IM. Furthermore, the ED-SSS is shown to be amenable to low-complexity implementation owing to the orthogonality of its subcarriers. In order to attain the maximum transmit diversity order of
$N_c$
, ED-SSS is further extended with the aid of transmit precoding and its transmit diversity order is quantified. The proposed precoding assisted ED-SSS is shown to subsume several of the existing precoding aided IM transmission schemes. Simulation studies are conducted for validating our theoretical claims and also for quantifying the attainable performance gains of the proposed schemes. Specifically, at a BER of
$10^{-3}$
an SNR gain as high as 8dB is observed in case of precoding when compared to its counterpart operating without precoding.
... Space time transmit and receive diversities are some examples of this technology. To improve the performance of the wireless systems, several space time trellis codes (STTC) and linear dispersion codes were presented [3][4][5][6][7][8]. A prominent drawback of these approaches is that they have high complexity in the decoder. ...
In this work, we propose a transmit diversity coding scheme with partial channel state information (CSI) at the transmitter to improve the performance of wireless systems. This scheme is equipped with 4 transmit and n receive antennas. For the case of flat fading channel, one, two or three feedback bits are used. Based on these feedback bits, 3 out of 4 transmit antennas are used for each transmission period. For frequency selective fading channel, one feedback frame is used to improve the performance. The proposed approach guarantees diversity order of 3n and very simple decoding complexity at the receiver. This is due to the fact that our approach is based on space time block coding (STBC). Our scheme significantly outperforms the conventional scheme with comparable bandwidth efficiency. Simulation results show that significant improvement in the performance of the proposed scheme is achieved compared to that of the conventional scheme. At bit error probability of 6.5*10 ⁻⁶ , the performance of the proposed scheme is approximately 4 dB better than that of the conventional scheme. For the proposed scheme, as the number of feedback bits increases the performance increases.
... There have been several kinds of space-time block code designs, for example, orthogonal space-time code designs [1], [2], [3], [4], unitary space-time designs [5], [6], [7], lattice based diagonal space-time code designs using algebraic number theory [8], [9], [10] and lattice based space-time block code designs from algebraic number fields, that have attracted much attention [11], [12], [13], [14], [15], [16]. ...
The main of this paper is to prove that in terms of normalized density, a space-time block code based on an irreducible quadratic polynomial over the Eisenstein integers is an optimal space-time block code compared with any quadratic space-time block code over the ring of integers of imaginary quadratic fields. In addition we find the optimal design of space-time block codes based on an irreducible quadratic polynomial over some rings of imaginary quadratic fields.
... The detected classic modulated symbol index as well as the detected TA activation index may now be translated back to bits. The complexity order of the list-normalized-MRC-based SM detector is given by O (N T + 2N List ), where the demodulator has to be invoked N List times in (129) before comparing the N List candidates in (130). ...
A pair of salient tradeoffs have driven the MIMO systems developments. More explicitly, the early era of MIMO developments was predominantly motivated by the multiplexingdiversity tradeoff between the Bell Laboratories Layered Space- Time (BLAST) and Space-Time Block Coding (STBC). Later, the Linear Dispersion Code (LDC) concept was introduced to strike a flexible tradeoff. The more recent MIMO system designs were motivated by the performance-complexity tradeoff, where the Spatial Modulation (SM) and Space-Time Shift Keying (STSK) concepts eliminate the problem of Inter-Antenna Interference (IAI) and perform well with the aid of low-complexity linear receivers without imposing a substantial performance loss on generic ML/MAP aided MIMO detection. Against the background of the MIMO design tradeoffs in both uncoded and coded MIMO systems, in this treatise, we offer a comprehensive survey of MIMO detectors ranging from hard-decision to softdecision. The soft-decision MIMO detectors play a pivotal role in approaching to the full performance potential promised by the MIMO capacity theorem. Having said that, in the near-capacity system design, the soft-decision MIMO detection dominates the total complexity, because all the MIMO signal combinations have to be examined, when both the channel’s output signal and the a priori LLRs gleaned from the channel decoder are taken into account. Against this background, we provide reducedcomplexity design guidelines, which are conceived for a widerange of soft-decision MIMO detectors.
... Le rang de la matrice construite avec les différences entre chaque pair distincte de codes, doit être maximal. Belfiore et al. ont montré que l'on pouvait construire d'autres codes spatio-temporels aux propriétés très intéressantes [60]. En se basant sur des résultats de la théorie des nombres, les auteurs construisent un code bloc spatio-temporel à diversité maximale et débit maximal pour 2 antennes de transmission et sur 2 périodes symbole. ...
This thesis focuses on analytical performance studies of radiocommunication systems in shadowing environments. The shadowing implied by the presence of large scale obstacles between transmitter and receiver, induces a change in the average symbol error probability (SEP). The symbol error outage (SEO), defined as the probability to observe a given average SEP over a fading channel in a shadowing environment, is hence a more adequate metric. The most challenging issue in the SEO study is that an inversion of the SEP with respect to the signal to noise (SNR) is needed. We have studied the single input single output (SISO) case first. Thanks to the Laplace method, we derive some new tight SEP approximations in Nakagami-m and Ricean fading channels. These new expressions are tight for a large SNR range which improves results available in the open literature. We prove that our expressions are invertible w.r.t. the SNR, facilitating the SEO to be expressed in closed form as a function of the target SEP. We also derive the packet error outage (PEO) in closed form considering channel block and convolutional codes with hard decision decoding. We then extend this approach to multiple input multiple output (MIMO) systems. We derive closed form expressions for the SEO of MIMO space-time block encoded as well as spatial multiplexing systems. Finally, interfering environments are considered. A SEP invertible expression is derived when the signal of interest and the interfering one experience a Rayleigh fading channel. We also study the performance of an improved multi-user maximum likelihood sequence estimation algorithm in an interfering WLAN context.
... The detected classic modulated symbol index as well as the detected TA activation index may now be translated back to bits. The complexity order of the list-normalized-MRC-based SM detector is given by O (N T + 2N List ), where the demodulator has to be invoked N List times in (129) before comparing the N List candidates in (130). ...
Unlike a generic PSK/QAM detector, which may visit a constellation diagram only once, a depth-first Sphere Decoder (SD) has to re-visit the same constellation diagram multiple times. Therefore, in order to prevent the SD from repeating the detection operations, the Schnorr-Euchner search strategy of Schnorr and Euchner may be invoked for optimizing the nodes' search-order, where the ideal case is for the SD to visit the constellation nodes in a zigzag fashion. However, when the hard-decision Multiple-Symbol Differential Sphere Detection (MSDSD) of Lampe et al. is invoked for using multiple receive antennas , the Schnorr-Euchner search strategy has to visit and sort all the constellation points. A similar situation is encountered for the soft-decision MSDSD of Pauli et al., when the a priori LLRs gleaned from the channel decoder are taken into account. In order to tackle these open problems, in this paper, we propose a correlation process for the hard-decision MSDSD of Lampe et al. and a reduced-complexity design for the soft-decision MSDSD of Pauli et al., so that the Schnorr-Euchner search strategy always opts for visiting the constellation points in a zigzag fashion. Our simulation results demonstrate that a substantial complexity reduction is achieved by our reduced-complexity design without imposing any performance loss. Explicitly, up to 88.7% complexity reduction is attained for MSDSD aided D16PSK. This complexity reduction is quite substantial, especially when the MSDSD is invoked several times- during turbo detection. Furthermore, in order to offer an improved solution and a comprehensive study for the soft-decision MSDSD, we also propose to modify the output of the SD to harmonize its operation with the near-optimum Approx-Log-MAP. Then the important subject of coherent versus noncoherent is discussed in the context of coded systems, which suggests that MSDSD aided DPSK is an eminently suitable candidate for turbo detection assisted coded systems operating at high Doppler frequencies.
... One challenge faced in some designed STBCs with the above criteria (e.g. [6]) is that the values of their coding gains vanish when the data transmission rate increases. Belfiore et al. [7] and Oggier et al. [8] define the non-vanishing determinant (NVD) property as a refinement criterion that determines whether the coding gain of an STBC remains constant when the constellation size increases. ...
This paper presents a comprehensive study on the Full-Rate and Linear-Receiver (FRLR) STBC proposed as a newly coding scheme with the low decoding complexity for a 2×2 MIMO system. It is shown that the FRLR code suffers from the lack of the non-vanishing determinant (NVD) property that is a key parameter in designing a full-rate STBC with a good performance in higher data rates, across QAM constellation. To overcome this drawback, we show that the existence of the NVD feature for the FRLR code depends on the type of the modulation. In particular, it is analytically proved that the FRLR code fulfills the NVD property across the PAM constellation but not for the QAM scheme. Simulation results show that, at a BER equal to 10-4, utilising the PAM modulation for the FRLR-STBC, provides about 2 dB gain over a use of the QAM when the bandwidth efficiency is 6b/s/Hz. In addition, for the PAM constellation, the FRLR code significantly outperforms some existing full-rate STBCs. Finally, we utilise the moment generating function approach to derive an exact closed-form expression for the average error probability of the FRLR code with the BPSK modulation.
... The main draw back of this type of code is the complexity of ML decoder which rises exponentially with number of transmit antennas. Threaded Algebraic codes [5] based on Diophantine approximation theory and number field were further generalized in [6,7] for arbitrary number of transmit and receive antennas, retaining full rate and maximum diversity. Such types of high rate STBC codes have also been constructed using division algebras [8,9]. ...
... constructed a space-time block code of R sym LDC = 1 over N t = 2 transmit antennas and T = 2 symbol periods using algebraic number theory in [22]. Xin et. ...
This chapter provides readers a systematic and panoramic view of LDC formula- tion and various LDC designs under different criteria.
... A higher data rate can be achieved by transmitting symbols simultaneously from M transmit antennas. Full rate and full diversity codes for the 2 × 2 coherent multiple-input multiple-output (MIMO) systems, were first constructed by Damen et al. (2002) , using algebraic number theory. Tarokh et al. (1998) showed that the main code design criterion for the Space Time Block Codes (STBCs) is the rank criterion , i.e, the rank of the difference of two distinct codeword matrices has to be maximal. ...
In this work we present a new class of space-time block codes based on arithmetic Fuchsian groups. This new class of codes satisfies the property full-diversity, linear dispersion and full-rate. Keywords— MIMO-systems, space-time code,diversity, division ring, arithmetic fuchsian groups Resumo— Neste trabalho apresentaremos uma nova classe de códigos temporais de bloco baseado na teoria de grupos fuchsianos aritméticos. Esta nova classe de códigos satisfaz as propriedades de diversidade máxima, dispersão linear e e de taxa máxima.
Orthogonal time-frequency space (OTFS) is a two-dimensional (2D) modulation technique designed in the delay-Doppler domain. A key premise behind OTFS is the transformation of a time-varying multipath channel into an almost non-fading 2D channel in the delay-Doppler domain such that all symbols in a transmission frame experience the same channel gain. It has been suggested in the recent literature that the OTFS can extract full diversity in the delay-Doppler domain, where full diversity refers to the number of multipath components separable in either the delay or Doppler dimension, but without formal analysis. In this paper, we present a formal analysis of the diversity achieved by the OTFS modulation along with supporting simulations. Specifically, we prove that the asymptotic diversity order of the OTFS (as SNR → ∞) is one. However, in the finite SNR regime, the potential for a higher order diversity is witnessed before the diversity one regime takes over. Also, the diversity one regime is found to start at lower BER values for increased frame sizes. We also propose a phase rotation scheme for the OTFS using transcendental numbers and show that the OTFS, with this proposed scheme, extracts full diversity in the delay-Doppler domain.
We apply a polarization-time (PT) code based on number theory for PDL mitigation. The PT code has comparable PDL mitigation performance to the best PT codes reported so far, and higher tolerance to fiber nonlinearities.
This chapter analyzes various possible transceivers that tackle inter-antenna interference (IAI) under different criteria such as maximal mutual information, zero forcing (ZF), and minimum mean-square error (MMSE). The maximum data rate achievable by a multiple-input multiple-output (MIMO) system depends on the available channel state information at the transmitter and receiver. The chapter considers that channel state information (CSI) is always available at the receiver side (CSIR) but not necessarily at the transmitter side (CSIT). It investigates the channel capacity of various MIMO channels, and presents possible transceivers for its exploitation. The chapter studies four possible receiver structures for MIMO channels with only CSIR are optimal beamformer-based receiver, ZF receiver, MMSE receiver, and VBLST. It expounds the two space-time coding techniques, Alamouti's codes and general space-time block codes (STBCs). To get some idea of diversity-multiplexing tradeoff (DMT), the chapter deals with the asymptotic behavior of the MIMO channel capacity.
A multiple-input-multiple-output automatic repeat request (MIMO-ARQ) system with two transmit antennas using modified multistrata space-time codes (MSSTCs) is proposed. We propose a low-complexity detection algorithm composed of interlayer interference cancelation, linear postprocessing after which intralayer interference does not exist, and a slicing operation. The proposed detection algorithm is performed in a successive layer-by-layer manner through multiple transmissions. Phases of two layers are adaptively assigned to mitigate the effect of interlayer interference through multiple transmissions with a low amount of additional feedback system overhead. We analytically obtain the approximation for the block error rate (BLER) of the proposed MIMO-ARQ system, and based on this, we propose a transmit power-allocation scheme, which minimizes the BLER of the MIMO-ARQ system at each transmission round. Numerical analyses and simulations show that the proposed MIMO-ARQ system results in low BLER and high throughput performances with low computational complexity.
This paper investigates distributed linear constellation precoding (DLCP) for two-way relaying communication systems in conjunction with the techniques of bit-interleaved coded modulation (BICM) and BICM with iterative decoding (BICM-ID). First, a decoding strategy for the relay node that is based on quaternary code representation is developed. Then, the union bounds (for the case of BICM) and error-free feedback bounds (for the case of BICM-ID) on the quaternary digit error probability and bit error probability under network coding in the multiple-access (MA) phase are obtained. Based on the obtained bounds, the impact of DLCP on the error performance is analyzed by considering three error types in the MA phase. It is shown that type-3 errors need to be carefully taken into account in the design of a DLCP scheme. By developing a performance metric related to type-3 errors, the design parameter of DLCP is optimized when BICM is used, whereas it is shown that DLCP is not needed when BICM-ID is used. Extensive simulation results are provided to corroborate the analysis and demonstrate the performance superiority of the proposed decoding strategy over the one that directly decodes the exclusive-OR (xor) code. For the case of BICM, the performance advantage achieved by properly designing DLCP is also illustrated.
We present a reliability-based near-maximum-likelihood (near-ML) decoder of Golden code with significantly reduced average decoding complexity over the current state of the art We first pre-process the received signal via zero-forcing (ZF) filtering and compute symbol reliabilities at the filter output Reliable symbols are directly decoded and removed from the received signal. The remaining symbols are decoded by reduced-dimension ML or near-ML decoding. Computational studies included herein reveal extensive complexity savings when compared with state-of-the art ML and near-ML decoders for the Golden code. For instance, for the 16-QAM signal constellation at pre-detection signal-to-noise ratio of 28dB that corresponds to bit-error-rate of about 10-4, the presented algorithm achieves more than 96.6% average-complexity savings, while maintaining indistinguishable to ML performance.
Information theory predicts that Multiple-Input Multiple-Output (MIMO) channels are able to provide huge gains in terms of reliability and transmission rate. This chapter derives practical methodologies to achieve these gains. It addresses the method to increase link performance and data rates through coding across space and time. It presents two design methodologies motivated by an error rate optimization perspective or by information theory. It also gives an overview of schemes known as space–time codes and discusses their performance in terms of both error and transmission rates. Two groups of encoding schemes are reviewed: space–time block coding in which encoding is based on a block definition and space–time trellis coding in which codes are described by a trellis. Among space–time block codes, the chapter discusses the broad class of Spatial Multiplexing (V-BLAST and D-BLAST) schemes, orthogonal and quasi-orthogonal codes, linear dispersion codes as well as the more recently developed algebraic codes. Regarding trellis codes, the chapter considers the classical space–time trellis codes as well as the super-orthogonal space–time trellis codes.
The second edition of MIMO Wireless Networks is unique in bridging the gap between multiple-input/multiple-output (MIMO) radio propagation and signal processing techniques and presenting robust MIMO network designs for real-world wireless channels. The book emphasizes how propagation mechanisms impact MIMO system performance under practical power constraints, examining how real-world propagation affects the capacity and the error performance of MIMO transmission schemes. Combining a solid mathematical analysis with a physical and intuitive approach, the book progressively derives innovative designs, taking into consideration that MIMO channels are usually far from ideal. Reflecting developments since the first edition was published, the book has been updated and now includes several new chapters covering Multi-user MIMO, Multi-Cell MIMO, Receiver Design, MIMO in 3GPP and WiMAX and realistic system level evaluations of MIMO network performance. Reviews physical models and analytical representations of MIMO propagation channels, highlighting their strengths and weaknesses Gives overview of space-time coding techniques, covering both classical and more recent schemes Shows innovative and practical designs of robust space-time coding, precoding and antenna selection techniques for realistic propagation (including single-carrier and MIMO-OFDM transmissions).
Until recently restricted to office applications, wireless local area networks (WLAN), propelled by advances in electronics and signal processing, is ermerging as the most promising technology to bring connectivity in home environment. Obviously more exible
and less expensive than their wired counterparts, the future of WLAN is hindered by the capacity of keeing the pace with the exponential growth in data rate sustained by multimedia communications. A major breakthrough came recently with the multiple-Input Multiple-Output (MIMO) communication architecture that uses multiple-antenna arrays at both transmitter and receiver ends. This concept generalizes all previously known transmission systems and allows a far more sophisticated processing of space to
yield unprecedented spectral efficiencies.
Few existing WLAN standards can cope with high data rate multimedia services. The HIPERLAN2 specification, achieving data rates up to 54 Mbps in the 5 GHz spectrum, is among them and the question naturally arises as to determine the potential of a MIMO-based physical interface. This thesis, led in the framework of a CIFRE grant between THOMSON multimedia and the IETR (Rennes Institute for Electronics and Telecommunications), provides some clue to this problem.
We construct full-diversity, arbitrary rate STBCs for specific number of transmit antennas over an apriori specified signal set using twisted Laurent series rings. Constructing full-diversity space-time block codes from algebraic constructions like division algebras has been done by Shashidhar et al. Constructing STBCs from crossed product algebras arises this question in mind that besides these constructions, which one of the well-known division algebras are appropriate for constructing space-time block codes. This paper deals with twisted Laurent series rings and their subrings twisted function fields, to construct STBCs. First, we introduce twisted Laurent series rings over field extensions of Q. Then, we generalize this construction to the case that coefficients come from a division algebra. Finally, we use an algorithm to construct twisted function fields, which are noncrossed product division algebras, and we propose a method for constructing STBC from them.
In MIMO system multiple antennas are used at transmitter and receiver side. MIMO has many advantages in comparison to SISO in terms of capacity, bit-rate and reliability. MIMO is classified mainly in three categories: spatial multiplexing, spatial diversity and beam forming. Spatial multiplexing provide higher data rate and spatial diversity effectively reduce bit-error rate. Here, we are mainly focusing on the space time coding which is basically a spatial diversity technique. The objective of this literature survey is to provide a comprehensive overview on space time coding technique. Here, large number of papers are provided on space time coding technique. In this paper, topics such as channel coding, space time coding for frequency flat fading channels, MIMO & MIMO-OFDM channels with ISI are discussed. The cited papers will serve as a good beginning for further reading.
Interference alignment is known to achieve the maximum sum DoF of 4M/3 in the 2 × 2 X-Network (i.e., two-transmitter (Tx) two-receiver (Rx) X-Network) with M antennas at each node, as demonstrated by Jafar and Shamai. Recently, an Alamouti code based transmission scheme, which we call the Li-Jafarkhani-Jafar (LJJ) scheme, was proposed for the 2×2 X-Network with two antennas at each node. This scheme achieves a sum degrees of freedom (DoF) of 8/3 and also a diversity gain of two with fixed finite constellation inputs. This work first proposes a new STBC for a three transmit antenna single user MIMO system. Building on this STBC, we extend the LJJ scheme to the 2×2 X-Network with three antennas at each node. As in the LJJ scheme and unlike the Jafar-Shamai scheme, only local channel knowledge is assumed at each Tx. It is shown that the proposed scheme achieves the maximum possible sum DoF of 4. A diversity gain of 3 is also guaranteed when fixed finite constellation inputs are used.
This paper is concerned with the design of distributed precoding in the multiple access (MA) phase for two-way relaying communication (TWRC) systems using OFDM. The error probability analysis is conducted to establish the diversity and coding gains for three error types in the MA phase. Then, the design criteria of distributed precoding to achieve the maximum diversity and coding gains are given. The frequency-grouped linear constellation precoding (F-GLCP) is first investigated and shown not to be able to achieve the maximum diversity gain under type-3 errors. A novel frequency–time GLCP (FT-GLCP) that performs precoding in both the frequency and time domains is then proposed. It is proved that the proposed FT-GLCP is able to achieve the maximum diversity gain under type-3 errors while maintaining the maximum diversity and coding gains under type-1 and type-2 errors. To corroborate the theoretical analysis, simulation results are provided to show the advantage of the proposed FT-GLCP over other schemes in both Rayleigh and Rician fading channels.
This paper is concerned with the design of distributed precoding in the multiple access (MA) phase for two-way relaying communication (TWRC) systems using OFDM. The error probability analysis is conducted to establish the diversity and coding gains for three error types in the MA phase. Then the design criteria of distributed precoding to achieve the maximum diversity and coding gains are given. The frequency-grouped linear constellation precoding (F-GLCP) is first investigated and shown not to be able to achieve the maximum diversity gain under type-3 errors. Then a novel frequency-time GLCP (FT-GLCP) which performs precoding in both frequency and time domains is proposed. It is proved that the proposed FT-GLCP is able to achieve the maximum diversity gain under type-3 errors, while maintaining the maximum diversity and coding gains under type-1 and type-2 errors. To corroborate the theoretical analysis, simulation results are provided to show the advantage of the proposed FT-GLCP over other schemes in both Rayleigh and Rician fading channels.
The problem of constructing unitary space-time codes with high diversity product has been studied in many prior works. Recently, constructions of parametric fully diverse unitary space-time codes for prime number antennas system have been introduced. In this paper, the authors propose new construction methods based on these constructions. And fully diverse codes of any number antennas are obtained from these constructions. Unitary space-time codes from present constructions are found to have better error performance than many best known ones.
The increasing need for high data-rate transmissions over time- or
frequency-selective fading channels has drawn attention to modulation
schemes with high spectral efficiency such as QAM. With the aim of
increasing the “diversity order” of the signal set we
consider multidimensional rotated QAM constellations. Very high
diversity orders can be achieved and this results in an almost Gaussian
performance over the fading channel, This multidimensional modulation
scheme is essentially uncoded and enables one to trade diversity for
system complexity, at no power or bandwidth expense
The signal detection algorithm of the vertical BLAST (Bell
Laboratories Layered Space-Time) wireless communications architecture is
briefly described. Using this joint space-time approach, spectral
efficiencies ranging from 20-40 bit/s/Hz have been demonstrated in the
laboratory under flat fading conditions at indoor fading rates. Early
results are presented
Transmitter diversity wireless communication systems over Rayleigh
fading channels using pilot symbol assisted modulation (PSAM) are
studied. Unlike conventional transmitter diversity systems with PSAM
that estimate the superimposed fading process, we are able to estimate
each individual fading process corresponding to the multiple
transmitters by using appropriately designed pilot symbol sequences.
With such sequences, special coded modulation schemes can then be
designed to access the diversity provided by the multiple transmitters
without having to use an interleaver or expand the signal bandwidth. The
code matrix notion is introduced for the coded modulation scheme, and
its design criteria are also established. In addition to the reduction
in receiver complexity, simulation results are compared to, and shown to
be superior to, that of an intentional frequency offset system over a
wide range of system parameters
We present a range of coding schemes for OFDM transmission using
binary, quaternary, octary, and higher order modulation that give high
code rates for moderate numbers of carriers. These schemes have tightly
bounded peak-to-mean envelope power ratio (PMEPR) and simultaneously
have good error correction capability. The key theoretical result is a
previously unrecognized connection between Golay complementary sequences
and second-order Reed-Muller codes over alphabets Z<sub>2</sub>h. We
obtain additional flexibility in trading off code rate, PMEPR, and error
correction capability by partitioning the second-order Reed-Muller code
into cosets such that codewords with large values of PMEPR are isolated.
For all the proposed schemes we show that encoding is straightforward
and give an efficient decoding algorithm involving multiple fast
Hadamard transforms. Since the coding schemes are all based on the same
formal generator matrix we can deal adaptively with varying channel
constraints and evolving system requirements
We present a maximum-likelihood decoding algorithm for an
arbitrary lattice code when used over an independent fading channel with
perfect channel state information at the receiver. The decoder is based
on a bounded distance search among the lattice points falling inside a
sphere centered at the received point. By judicious choice of the
decoding radius we show that this decoder can be practically used to
decode lattice codes of dimension up to 32 in a fading environment
Abstract ● ● ● Transmit Antenna Array,, ● ● Receive
We present a generalized sphere decoding (GSD) algorithm and its application for detecting information symbols in the case where one has a system with more inputs (symbols) than outputs or the opposite situation. We study the special case of a multi-antenna scenario in a cellular system with N antennas at the mobile and M/spl ges/N antennas at the base station. GSD reaches the maximum likelihood (ML) performance for both up and down link with a moderate complexity.
We investigate the use of multiple transmitting and/or receiving antennas for single user communications over the additive Gaussian channel with and without fading. We derive formulas for the capacities and error exponents of such channels, and describe computational procedures to evaluate such formulas. We show that the potential gains of such multi-antenna systems over single-antenna systems is rather large under independenceassumptions for the fades and noises at different receiving antennas.
We study an iterative detection algorithm of an uncoded multi-transmittermulti-receiver system. The main data stream is demultiplexed into Msubstreams, and each substream is modulated independently then transmitted byits dedicated antenna. The receiver disposes of N M antennas. Over eachreceive antenna the signal is a superposition of the M substreams affected byindependent fades and disturbed by AWGN. The detection algorithm is based onthe QR decomposition of the channel transfer matrix which is then used toperform hard or soft inter-substream interference cancellation. Comparisonsare done with the V-BLAST optimal order (OPT) detection algorithm. Theproposed algorithm is 4 to 8 times less complex than that of the V-BLAST OPT,while maintaining comparable performance.
The fast development of digital communications hardware allows for
the application of very powerful algorithms at the expense of a small
increase in complexity compared to the traditionally implemented
algorithms. In this paper we give further results on the sphere decoder
(SD) algorithm, and its applications to a broad range of digital
communications problems related to the separation of m independent
sources by n sensors. First, we discuss practical implementation issues
and propose an efficient method to initialize the SD parameters based on
computing an estimate of the packing radius of the lattice. We relate
the initializing method to the expected performance of the SD, and show
that at high SNR, one obtains near optimum performance. The complexity
of the SD is then shown to be much less than the upper bound on the
complexity of the Fincke and Pohst (1985) algorithm for the problem of
finding short length vectors in an m-dimensional lattice. Simulations
show that the SD of an m-dimensional lattice needs at most O(m<sup>4.5
</sup>) arithmetic operations at low SNR, and O(m<sup>3</sup>) at high
SNR. The obtained results offer a very powerful tool to reach near the
maximum likelihood (ML) decoding performance in several cases such as
lattice codes decoding over the Gaussian and Rayleigh fading channels,
multiuser detection, uncoded multi-antenna systems detection and
space-time codes decoding, and vector quantization
Multiple-antenna systems that operate at high rates require simple yet effective space-time transmission schemes to handle the large traffic volume in real time. At rates of tens of bits/sec/Hz, V-BLAST, where every antenna transmits its own independent substream of data, has been shown to have good performance and simple encoding and decoding.
Yet V-BLAST suffers from its inability to work with fewer receive antennas than transmit antennas. Furthermore, because V-BLAST transmits independent data streams on its antennas there is no built-in spatial coding to guard against deep fades from any given transmit antenna. On
the other hand, there are many previously-proposed space-time codes that have good fading resistance and simple decoding, but these codes generally have poor performance at high data rates or with many
antennas.
We propose a high-rate coding scheme that can handle any configuration of transmit and receive antennas and that subsumes both V-BLAST and many proposed space-time block codes as special cases. The scheme transmits substreams of data in linear combinations over space
and time. The codes are designed to optimize the mutual information between the transmitted and received signals. Because of their linear structure, the codes retain the decoding simplicity of V-BLAST, and because of their information theoretic optimality, they possess many
coding advantages. We give examples of the codes and show that their performance is generally superior to earlier proposed methods over a wide range of rates and SNRs.
Theoretical and practical aspects of diagonal algebraic space-time block codes over n transmit and m receive antennae are examined. These codes are obtained by sending a rotated version of the information symbols over the principal diagonal of the n × n space-time matrix over n transmit antennae and n symbol periods. The output signal-to-noise ratios of two predecoding filters and two decoding algorithms are derived. Analysis of the information loss incurred by using the codes considered is used to clarify their structures, and the expected performances. Different algebraic real and complex rotations presented in the literature are analyzed and compared as regards the achieved coding gains, the complexities, performances, and peak-to-mean envelope power ratios.
The peak-to-average power ratio PAPR(𝒞) of a code 𝒞 is an important characteristic of that code when it is used in OFDM communications. We establish bounds on the region of achievable triples (R, d, PAPR(𝒞)) where R is the code rate and d is the minimum Euclidean distance of the code. We prove a lower bound on PAPR in terms of R and d and show that there exist asymptotically good codes whose PAPR is at most 8logn. We give explicit constructions of error-correcting codes with low PAPR by employing bounds for hybrid exponential sums over Galois fields and rings
Geometric algorithms and combinatorial optimization / Martin Grötschel ; László Lovász ; Alexander Schrijver. - Berlin u. a. : Springer, 1988. - XII, 362 S. - (Algorithms and combinatorics ; 2).
Incluye bibliografía e índice
We establish a relationship between optimization of the diversity
product for two-antenna diagonal space-time codes and the elementary
theory of continued fractions
Space-time block codes are a remarkable modulation scheme discovered recently for the multiple antenna wireless channel. They have an elegant mathematical solution for providing full diversity over the coherent, flat-fading channel. In addition, they require extremely simple encoding and decoding. Although these codes provide full diversity at low computational costs, we show that they incur a loss in capacity because they convert the matrix channel into a scalar AWGN channel whose capacity is smaller than the true channel capacity. In this letter the loss in capacity is quantified as a function of channel rank, code rate, and number of receive antennas.
We explore the lattice sphere packing representation of a multi-antenna system and the algebraic space-time (ST) codes. We apply the sphere decoding (SD) algorithm to the resulted lattice code. For the uncoded system, SD yields, with small increase in complexity, a huge improvement over the well-known V-BLAST detection algorithm. SD of algebraic ST codes exploits the full diversity of the coded multi-antenna system, and makes the proposed scheme very appealing to take advantage of the richness of the multi-antenna environment. The fact that the SD does not depend on the constellation size, gives rise to systems with very high spectral efficiency, maximum-likelihood performance, and low decoding complexity.
A generalised sphere decoding (GSD) algorithm is presented. Its
application to the detection of information symbols in a multiantenna
cellular system with N antennas at the mobile and M⩾N antennas at
the base station is explored. The GSD algorithm has the maximum
likelihood (ML) performance for both the uplink and downlink with
moderate complexity
This paper presents a simple two-branch transmit diversity scheme.
Using two transmit antennas and one receive antenna the scheme provides
the same diversity order as maximal-ratio receiver combining (MRRC) with
one transmit antenna, and two receive antennas. It is also shown that
the scheme may easily be generalized to two transmit antennas and M
receive antennas to provide a diversity order of 2M. The new scheme does
not require any bandwidth expansion or any feedback from the receiver to
the transmitter and its computation complexity is similar to MRRC
This paper proposes a bandwidth-efficient fading-resistant
transmission scheme which implements transmitter diversity using L
antennas at the base station. When the antennas are spaced sufficiently
far apart, the transmission from each antenna undergoes a different
degree of fading. These transmissions are coordinated to mitigate the
effects of Rayleigh fading, and the mobile receiver can recover the
entire L-dimensional transmitted vector signal as long as the signal
energy of at least one coordinate is large enough. L-dimensional
fading-resistant signal constellations are generated by maximizing a
figure of merit for the Rayleigh fading channel. This scheme offers a
significant performance improvement over a conventional single-antenna
binary phase-shift keying (BPSK) scheme when coding is ineffective due
to slow fading
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
The first lower bound on the peak-to-average power ratio (PAPR) of
a constant energy code of a given length n, minimum Euclidean distance
and rate is established. Conversely, using a nonconstructive
Varshamov-Gilbert style argument yields a lower bound on the achievable
rate of a code of a given length, minimum Euclidean distance and maximum
PAPR. The derivation of these bounds relies on a geometrical analysis of
the PAPR of such a code. Further analysis shows that there exist
asymptotically good codes whose PAPR is at most 8 log n. These bounds
motivate the explicit construction of error-correcting codes with low
PAPR. Bounds for exponential sums over Galois fields and rings are
applied to obtain an upper bound of order (log n)<sup>2</sup> on the
PAPRs of a constructive class of codes, the trace codes. This class
includes the binary simplex code, duals of binary, primitive
Bose-Chaudhuri-Hocquenghem (BCH) codes and a variety of their nonbinary
analogs. Some open problems are identified
We introduce space-time block coding, a new paradigm for
communication 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. Space-time
block codes are designed to achieve the maximum diversity order for a
given number of transmit and receive antennas subject to the constraint
of having a simple decoding algorithm. The classical mathematical
framework of orthogonal designs is applied to construct space-time block
codes. It is shown that space-time block codes constructed in this way
only exist for few sporadic values of n. Subsequently, a generalization
of orthogonal designs is shown to provide space-time block codes for
both real and complex constellations for any number of transmit
antennas. These codes achieve the maximum possible transmission rate for
any number of transmit antennas using any arbitrary real constellation
such as PAM. For an arbitrary complex constellation such as PSK and QAM,
space-time block codes are designed that achieve 1/2 of the maximum
possible transmission rate for any number of transmit antennas. For the
specific cases of two, three, and four transmit antennas, space-time
block codes are designed that achieve, respectively, all, 3/4, and 3/4
of maximum possible transmission rate using arbitrary complex
constellations. The best tradeoff between the decoding delay and the
number of transmit antennas is also computed and it is shown that many
of the codes presented here are optimal in this sense as well
We investigate the use of multiple transmitting and/or receiving antennas for single user communications over the additive Gaussian channel with and without fading. We derive formulas for the capacities and error exponents of such channels, and describe computational procedures to evaluate such formulas. We show that the potential gains of such multi-antenna systems over single-antenna systems is rather large under independence assumptions for the fades and noises at different receiving antennas. 1 Introduction We will consider a single user Gaussian channel with multiple transmitting and/or receiving antennas. We will denote the number of transmitting antennas by t and the number of receiving antennas by r. We will exclusively deal with a linear model in which the received vector y 2 C r depends on the transmitted vector x 2 C t via y = Hx+ n (1) where H is a r Theta t complex matrix and n is zero-mean complex Gaussian noise with independent, equal variance real and imaginary p...
Space–timeblockcodes:Acapacityperspec-tive
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Lattice codes decoder for space&ndash,time codes
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Transcendental Numbers