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## Universal Space-Time Coding (2003)

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Venue: | IEEE Trans. Inform. Theory |

Citations: | 141 - 7 self |

### Citations

2353 | On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun - Foschini, Gans - 1998 |

2074 | A simple transmit diversity technique for wireless communications
- Alamouti
- 1998
(Show Context)
Citation Context ...time codes in quasi-static fading channels [4], [5]. Subsequent works resulted in new trellis and graphical space–time codes that exhibit better coding advantages [6]–[16], new block space–time =-=codes [17]–[-=-22], and extensions to more realistic frequency-selective and time-selective channels [23]–[30]. Despite this recent progress, Manuscript received January 2, 2002; revised November 19, 2002. The mat... |

1751 | Space-time codes for high data rate wireless communication: Performance criterion and code construction
- Tarokh, Seshadri, et al.
- 1998
(Show Context)
Citation Context ...nput–multiple-output (MIMO) systems, number theory. I. INTRODUCTION TAROKH et al. coined the term “space–time codes” to describe the two-dimensional signals used in multiple transmit antennas =-=systems [4]. -=-In the coherent scenario, where the channel state information (CSI) is available a priori only at the receiver, Guey et al. and Tarokh et al. derived the design criteria for full diversity space–tim... |

1496 | Space-time block codes from orthogonal designs
- Tarokh, Jafarkhani, et al.
- 1999
(Show Context)
Citation Context ... the layered space–time architectures in [32] achieve high transmission rates with low complexity receivers at the expense of reduced diversity advantages. Finally, the orthogonal space–time codes=-= in [18] and-=- the diagonal algebraic space–time (DAST) codes in [19]–[21] achieve full diversity and allow for a low-complexity receiver, but entail a significant loss in the transmission rates, in general. On... |

1404 | Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas - Foschini - 1996 |

1393 |
Geometric Algorithms and Combinatorial Optimization
- Grötschel, Lovász, et al.
- 1988
(Show Context)
Citation Context ...rection. APPENDIX I PRELIMINARIES In this appendix, we summarize some key results and definitions from number theory which are relevant to our work. The interested reader is referred to [36] and [58]�=-=��[60]-=-. 16 As described in [47]. Definition 3: Let , then is called an algebraic number if there exists a polynomial such that .We call an algebraic integer if, in addition, one can choose to be monic (i.e.... |

1190 |
A Course in Computational Algebraic Number Theory
- Cohen
- 1993
(Show Context)
Citation Context ....If is complex, one can also define its minimal polynomial over coefficients). (i.e., with complex rational Example 7: The cyclotomic numbers, which are the th roots of unity , are algebraic integers =-=[59]-=-. The degree of over equals , the Euler -function [59], which represents the number of integers and prime with . Definition 4: The height of a polynomial is defined as the maximum modulus of its coeff... |

998 |
Linear Algebra and its Applications
- Strang
- 1988
(Show Context)
Citation Context ...ll permutations, where each term is a product of entries , with each row and column represented once; i.e., each term corresponds to a path that goes down through the matrix and uses each column once =-=[62]-=-. Second, each thread of the TAST code occupies the column (respectively, row ) only once, by construction (3). We first need to prove that the exponents of in all the cross terms in (3) are integers.... |

664 | Error Control Coding: Fundamentals and Applications - Lin, Castello - 1983 |

398 |
A Universal Lattice Code Decoder for Fading Channels
- Viterbo, Boutros
- 1999
(Show Context)
Citation Context ...al complexity in “ ” [41]. This observation suggests that by restricting the number of threads to threads, one ensures that ML decoding can be performed using the polynomial complexity sphere deco=-=der [40]-=-, [34], [41]. This choice for the number of threads also agrees with the fact that the information-theoretic capacity of the system grows with . While our TAST coding framework is general for an arbit... |

347 |
Improved methods for calculating vectors of short length in a lattice, including a complexity analysis
- Fincke, Pohst
- 1985
(Show Context)
Citation Context ...oise level [34], [41]. When the number of unknowns is less than or equal to the number of equations (i.e., ), the complexity of the sphere decoder is exponential in and polynomial in the lattice rank =-=[42]-=-. Therefore, in this scenario, the proper choice of the initial radius will allow for a decoder with a polynomial complexity in (roughly cubic) at medium-tolarge SNRs [41]. In general, when , one can ... |

278 |
Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels
- Guey, Fitz, et al.
- 1999
(Show Context)
Citation Context ... state information (CSI) is available a priori only at the receiver, Guey et al. and Tarokh et al. derived the design criteria for full diversity space–time codes in quasi-static fading channels [4]=-=, [5]. Subseq-=-uent works resulted in new trellis and graphical space–time codes that exhibit better coding advantages [6]–[16], new block space–time codes [17]–[22], and extensions to more realistic frequen... |

270 | Differential unitary space-time modulation
- Hochwald, Sweldens
- 2000
(Show Context)
Citation Context ...nd approach extends differential phase-shift keying (DPSK) modulation to the space–time paradigm, where two-dimensional unitary space–time codes are constructed to exploit the spatial diversity [5=-=2], [53]-=-. In this approach, the symbols are first drawn from a group of unitary matrices, then differentially encoded before transmission across the channel. The differential coding approach assumes that the ... |

262 | Differential space-time modulation
- Hughes
- 2000
(Show Context)
Citation Context ...e second approach extends differential phase-shift keying (DPSK) modulation to the space–time paradigm, where two-dimensional unitary space–time codes are constructed to exploit the spatial divers=-=ity [52]-=-, [53]. In this approach, the symbols are first drawn from a group of unitary matrices, then differentially encoded before transmission across the channel. The differential coding approach assumes tha... |

233 | Lattice code decoder for space–time codes
- Damen, Chkeif, et al.
- 2000
(Show Context)
Citation Context ...mples of space–time codes are known in the literature to achieve full rate and full diversity with a polynomial complexity receiver; for , with -quadrature amplitude modulation (QAM) constellation i=-=n [34], [3-=-3], and for with all QAM and pulse amplitude modulation (PAM) constellations in [22]. In this paper, we develop a universal framework for constructing multiple-input–multiple-output (MIMO) space–t... |

194 | R.L.: Systematic design of unitary space-time constellations
- Hochwald, Marzetta, et al.
- 1962
(Show Context)
Citation Context ...erent space–time signals (e.g., [48]). Two approaches for space–time signal design in this scenario have been investigated. In the first approach, noncoherent space–time codes have been proposed=-= [49]–[51]-=-. These codes assume that the channel is fixed over only one matrix symbol duration and can exploit the spatial diversity without the need of any training symbols. The second approach extends differen... |

186 | Signal space diversity: A power and bandwidth efficient diversity technique for the Rayleigh fading channel
- Boutros, Viterbo
- 1998
(Show Context)
Citation Context ...AM or PAM). The rotation matrix (complex or real) is constructed on an algebraic number field with an algebraic number of degree . In this paper, we consider the “locally” optimal rotations from [=-=36]–[38]-=- (see the Appendixes for more details). One can easily verify that the DAST codes achieve full diversity, and their coding gains are proportional to the minimum product distances associated with the r... |

181 | Orthogonal Arrays: Theory and Applications - HEDAYAT, SLOANE, et al. - 1999 |

132 |
On coding for block fading channels
- Knopp, Humblet
- 2000
(Show Context)
Citation Context ...thers (each thread will lie in a different algebraic subspace). It is also worth noting that a suitable class of SISO codes satisfying Theorem 4 contains the codes optimized for block-fading channels =-=[25]. Th-=-is is because by exploiting all the degrees of freedom in the MIMO channel, a “thread” transforms the quasi-static MIMO fading channel into a single-input multiple-output block-fading channel. Not... |

109 |
Capacity of multi-antenna Gaussian channels
- Teletar
- 1995
(Show Context)
Citation Context ...en using algebraically rotated constellations; however, there is no benefit of sending more than symbols per channel use. This is because, on one hand, the capacity of the channel increases only with =-=[1]-=-, 5 and on the other hand, one cannot increase the coding gain by sending more than symbols per channel use, since one can always set the additional symbols to zero in the coding gain expression which... |

107 |
Functional Analysis,’’ 2nd ed
- Rudin
- 1991
(Show Context)
Citation Context ..., if is defined over , and if is defined over . Hence, for , . Recall that a norm over a vector space (defined over or over ), in the usual Euclidean sense, should have the following three properties =-=[61]-=-. 1) It is multiplicative, i.e., . , for or 2) For , , and . 3) The triangle inequality: . The algebraic norm defined in Proposition 2 is multiplicative, and satisfies the relation ; however, it can t... |

107 | G.Taricco, “Limiting performance of block-fading channels with multiple antennas - Biglieri - 2001 |

103 | Gamal, “On the theory of space-time codes for PSK modulation
- Hammons, El
- 2000
(Show Context)
Citation Context ...gn criteria for full diversity space–time codes in quasi-static fading channels [4], [5]. Subsequent works resulted in new trellis and graphical space–time codes that exhibit better coding advanta=-=ges [6]–[16],-=- new block space–time codes [17]–[22], and extensions to more realistic frequency-selective and time-selective channels [23]–[30]. Despite this recent progress, Manuscript received January 2, 20... |

103 | Diagonal algebraic space-time block codes
- Damen, Meraim, et al.
- 2002
(Show Context)
Citation Context ...transmission rates with low complexity receivers at the expense of reduced diversity advantages. Finally, the orthogonal space–time codes in [18] and the diagonal algebraic space–time (DAST) codes=-= in [19]–[-=-21] achieve full diversity and allow for a low-complexity receiver, but entail a significant loss in the transmission rates, in general. Only few sporadic examples of space–time codes are known in t... |

94 |
Algebraic tools to build modulation schemes for fading channels
- Giraud, Boutillon, et al.
- 1997
(Show Context)
Citation Context ...scussed in the next section. A DAST code is obtained by rotating an -dimensional (i.e., ) information symbol vector by an rotation , which maximizes the associated minimum product distance defined as =-=[36]-=- where belong to the multidimensional constellation considered (QAM or PAM). The rotation matrix (complex or real) is constructed on an algebraic number field with an algebraic number of degree . In t... |

81 | A construction of a spacetime code based on number theory
- Damen, Tewfik, et al.
- 2002
(Show Context)
Citation Context ...codes in quasi-static fading channels [4], [5]. Subsequent works resulted in new trellis and graphical space–time codes that exhibit better coding advantages [6]–[16], new block space–time codes=-= [17]–[22], -=-and extensions to more realistic frequency-selective and time-selective channels [23]–[30]. Despite this recent progress, Manuscript received January 2, 2002; revised November 19, 2002. The material... |

81 | Space-time block codes: a capacity perspective
- Sandhu, Paulraj
- 2000
(Show Context)
Citation Context ...finition 1: The information loss of a TAST code is given by the difference between the ergodic capacity of the equivalent uncoded system in (12) and the ergodic capacity of the original channel [31], =-=[45]-=-. From (12), the ergodic capacity of symmetric TAST precoded system with transmit and receive antennas is given by [31] (27) where is the TAST precoder, is the covariance matrix of the information sym... |

80 | Cayley differential unitary space-time codes
- Hassibi, Hochwald
- 2002
(Show Context)
Citation Context ...–time coding have focused on systems with relatively small numbers of transmit antennas and low data rates [52], [53]. More recently, a novel approach based on the Cayley transform have been propose=-=d [54]-=-. The so-called Cayley differential (CD) codes allow for high transmission rates with reasonable receiver complexity. The proposed CD codes, however, fail to achieve full spatial diversity [54]. In th... |

58 | Multiple-antenna signal constellations for fading channels - Agrawal, Richardson, et al. - 2001 |

51 | A new approach to layered space-time coding and signal processing - Gamal, Hammons - 1997 |

37 | Improved codes for space-time trellis coded modulation - Baro, Bauch, et al. |

37 | Further results in space-time coding for Rayleigh fading - Grimm, Fitz, et al. - 1998 |

37 |
Linear dispersion codes for MIMO systems based on frame theory
- Paulraj
- 2002
(Show Context)
Citation Context ...HEORY, VOL. 49, NO. 5, MAY 2003 Fig. 9. The performances of the differential TAST code „ and the differential code by Hassibi and Hochwald. real rotation [38] and , 13 and the corresponding code fro=-=m [57]-=- when using a 4-QAM constellation, at the same rate and number of transmit and receive antennas. 14 Here again, one notes the gain offered by the TAST code. In addition, and as in the preceding figure... |

35 | Optimum space-time convolutional codes - Yan, Blum - 2000 |

35 | Space–frequency codes for broadband fading channels - Bölcskei, Paulraj - 2001 |

31 | Full Rate Space–Time Turbo Codes - Liu, Takeshita - 2001 |

30 | On the design of space-time and space-frequency codes for MIMO frequencyselective fading channels
- Gamal, Jr, et al.
- 2003
(Show Context)
Citation Context ... the threads (e.g., one refers to the first thread). frequency-division multiplexing (OFDM) front end(s) such that the channel seen by the TAST code can be well approximated by the block-fading model =-=[26]. -=-The previous TAST construction can then be used as a full diversity space–frequency code. V. DIFFERENTIAL AND NONCOHERENT TAST CODES: (N-CSI) Here, we suppose that neither the transmitter nor the re... |

29 | Generalized sphere decoder for asymmetrical space-time communication architecture
- Damen, Abed-Meraim, et al.
- 2000
(Show Context)
Citation Context ...6 Suboptimal detectors based on the turbo principle suitable for large array sizes will be considered in a separate paper.s1102 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 5, MAY 2003 [41], =-=[43]. Th-=-e generalized sphere decoder can be implemented via the quadratic-residue (QR) decomposition of the equivalent channel matrix and has an additional exponential complexity in “ ” [41]. This observa... |

27 | Space-Time Coded OFDM for High Data Rate Wireless Communication Over Wide-band Channels - Agrawal, Tarokh, et al. - 1998 |

26 | Turbo coded modulation for systems with transmit and receive antenna diversity over block fading channels: System model, decoding approaches and practical considerations - Stefanov, Duman - 2001 |

26 | The performance of convolutional codes on the block erasure channel using various finite interleaving techniques - Lapidoth - 1994 |

25 | Robust Space-Time Codes for Correlated Rayleigh Fading Channels - Siwamogsatham, Fitz |

21 | Signal design and convolutional coding for noncoherent space-time communication on the block-Rayleigh-fading channel
- McCloud, Brehler, et al.
- 2002
(Show Context)
Citation Context ...ve a local optima of the coding gain for a 4-QAM constellation. 14 A typographical error in the SNR scaling in the corresponding figure in [57] was corrected after consultation with the first author. =-=[55] 1-=-5 at a rate of 2 bits per channel use. Both schemes have comparable performances (according to the authors’ knowledge, the code in [55] currently represents the state of the art). In addition, the T... |

19 | Hammons Jr., “A new approach to layered space-time coding and signal processing - Gamal, R - 2001 |

18 | Turbo decoding of concatenated space-time codes - Narayanan - 1999 |

16 |
Capacity of a Mobile Multiple– Antenna Communication Link in Rayleigh Flat Fading
- Marzetta, Hochwald
- 1999
(Show Context)
Citation Context ...ent information-theoretic calculations suggest that high capacities are still possible with multiple-antenna systems in the noncoherent scenario if the channel coherence time is not very small (e.g., =-=[3]). F-=-ollow-up works have investigated the signal design criterion for noncoherent space–time signals (e.g., [48]). Two approaches for space–time signal design in this scenario have been investigated. I... |

15 |
Further results on the sphere decoder
- Damen, A-Meraim, et al.
(Show Context)
Citation Context ...ion symbols of the threads. The received signal in (12) represents an uncoded system with a new channel matrix with equations and unknowns. This motivates ML decoding using the sphere decoder 6 [34], =-=[41]-=-. The sphere decoder computes the ML metric over all constellation points enclosed in a sphere of a given radius . Finding the closest constellation point to the received signal is done by enumerating... |

14 |
Unitary space-time modulation for multiple antenna communications in rayleigh ?at fading
- Hochwald, Marzetta
- 2000
(Show Context)
Citation Context ...ystems in the noncoherent scenario if the channel coherence time is not very small (e.g., [3]). Follow-up works have investigated the signal design criterion for noncoherent space–time signals (e.g.=-=, [48]). Two-=- approaches for space–time signal design in this scenario have been investigated. In the first approach, noncoherent space–time codes have been proposed [49]–[51]. These codes assume that the ch... |

14 | Hammons Jr., “On the design of algebraic space-time codes for MIMO blockfading channels - Gamal, R - 2003 |

14 | Giannakis, “Space-time block coded multiple access through frequency-selective fading channels - Liu, B - 2001 |

12 |
Transmitter diversity code design for achieving full diversity on Rayleigh fading channels
- Grimm
- 1998
(Show Context)
Citation Context ...riteria for full diversity space–time codes in quasi-static fading channels [4], [5]. Subsequent works resulted in new trellis and graphical space–time codes that exhibit better coding advantages =-=[6]–[16], new -=-block space–time codes [17]–[22], and extensions to more realistic frequency-selective and time-selective channels [23]–[30]. Despite this recent progress, Manuscript received January 2, 2002; r... |

12 |
Joint coding/decoding in a multiple access system: Applications to mobile communications
- Damen
- 1999
(Show Context)
Citation Context ...of space–time codes are known in the literature to achieve full rate and full diversity with a polynomial complexity receiver; for , with -quadrature amplitude modulation (QAM) constellation in [34]=-=, [33], an-=-d for with all QAM and pulse amplitude modulation (PAM) constellations in [22]. In this paper, we develop a universal framework for constructing multiple-input–multiple-output (MIMO) space–time co... |

11 | Design of coded modulation schemes for Orthogonal transmit diversity,” submitted for publication - Borran, Memarzadeh, et al. - 2001 |

10 | A Randomization Technique for Non-Orthogonal Space-Time Block Codes - Hottinen, Tirkkonen - 2001 |

9 | On the design and performance of algebraic space-time codes for BPSK and QPSK modulation - Gamal, Hammons - 2002 |

8 |
Linear dispersion codes
- Hassibi, Hochwald
- 2001
(Show Context)
Citation Context ...m-likelihood (ML) decoding for full diversity space–time trellis codes grows exponentially with the transmission rate. On the other hand, the polynomial complexity linear dispersion codes proposed i=-=n [31] f-=-ail to achieve full diversity for high transmission rates. Similarly, the layered space–time architectures in [32] achieve high transmission rates with low complexity receivers at the expense of red... |

7 | Improved 8and 16-state space-time codes for 4PSK with two transmit antennas - Ionescu, Mukkavilli, et al. - 2001 |

7 |
Transmit diversity using rotated constellations with Hadamard transform
- Damen, Abed-Meraim, et al.
- 2000
(Show Context)
Citation Context ...mission rates with low complexity receivers at the expense of reduced diversity advantages. Finally, the orthogonal space–time codes in [18] and the diagonal algebraic space–time (DAST) codes in [=-=19]–[21] a-=-chieve full diversity and allow for a low-complexity receiver, but entail a significant loss in the transmission rates, in general. Only few sporadic examples of space–time codes are known in the li... |

7 |
Space-time block codes in multipath CDMA systems
- Geng, Mitra, et al.
- 2001
(Show Context)
Citation Context ... graphical space–time codes that exhibit better coding advantages [6]–[16], new block space–time codes [17]–[22], and extensions to more realistic frequency-selective and time-selective channe=-=ls [23]–[30]-=-. Despite this recent progress, Manuscript received January 2, 2002; revised November 19, 2002. The material in this paper was presented in part at the IEEE International Symposium on Information Theo... |

7 |
Transcendental Numbers
- Shidlovskii
- 1989
(Show Context)
Citation Context ... is the Golden mean given in (14). The choice of to be the Golden mean is motivated by the fact that the latter number is the irrational number which is “worst” approximated by rational numbers [2=-=2], [58]-=-; thus, this choice maximizes the determinant of the , which equals with . Second, one constructs the code matrix (40) It is clear that the code matrix defined above is Hermitian and has a full rank w... |

6 |
Algebraic space-time codes for block fading channels
- Gamal, Hammons
- 2001
(Show Context)
Citation Context ...be fixed across one fading block, but change independently from one block to the next. It is straightforward to see that full diversity codes will achieve “ ” levels of diversity in this scenario.=-= In [24]-=-, it was shown that the diversity advantage of the space-time code is given by (33) where is the th block of . Full diversity codes should then be constructed such that is full rank for every . The in... |

5 |
Optimal linear labeling for the minimization of both source and channel distortion
- Belore, Giraud, et al.
- 2000
(Show Context)
Citation Context ...l. The advantage of the TAST code in this scenario is, however, evident in the fact that the equivalent precoded channel is very close to a nonfaded additive white Gaussian noise (AWGN) channel [19], =-=[37]-=-, [38]. This transformation reduces the code design problem to the extensively studied AWGN channel paradigm. 8 2) : Using (12) one can easily show that, in the ergodic scenario, the symmetric TAST co... |

5 | On diagonal space-time codes using algebraic rotations
- Damen, Beaulieu
(Show Context)
Citation Context ...annel use (using a 16-QAM constellation). Note that the code 12 For more details on the advantages/disadvantages of the complex and real rotations when used with DAST codes, the reader is referred to =-=[46].sEL GAM-=-AL AND DAMEN: UNIVERSAL SPACE–TIME CODING 1109 Fig. 4. The performances of the codes „ (0 aIand 0 a � ), and the linear dispersion code @w aPY„aPA, over quasi-static fading channel with x aPre... |

4 | A space-time coding concept for a multi-element transmitter - Hottinen, Kuchi, et al. - 2001 |

3 | Code design metrics for space-time systems under arbitrary fading - Kose, Wesel - 2001 |

3 | Optimum space–time convolutional codes,” presented at the WCNC’00 - Yan, Blum - 2000 |

2 | Full Diversity PSK Space-Time Codes - Guidi, Grant, et al. - 2001 |

2 |
Sphere packing in the Grassmann manifold: a geometric approach to the noncoherent multiantenna channel
- Zheng, Tse
- 2000
(Show Context)
Citation Context ...oncoherent space–time signals (e.g., [48]). Two approaches for space–time signal design in this scenario have been investigated. In the first approach, noncoherent space–time codes have been pro=-=posed [49]��-=-�[51]. These codes assume that the channel is fixed over only one matrix symbol duration and can exploit the spatial diversity without the need of any training symbols. The second approach extends dif... |

2 |
On signal design for the noncoherent multi-antenna block-rayleigh-fading channel,” submitted to
- Brehler, Varanasi
- 2002
(Show Context)
Citation Context ...nally, we remark that the proposed codes belong to the class of training-based schemes. The good performance of this class of schemes was recently, and independently, observed by Brehler and Varanasi =-=[56]-=-. VI. SIMULATION RESULTS Simulation results are presented for transmit antennas and receive antennas, R-CSI, and N-SCI Rayleigh quasi-static as well as time-selective fading channels. Comparisons are ... |

1 | Transmit diversity as a combination of spatial and delay diversity - Damen, Abed-Meraim, et al. - 2000 |

1 |
On the design of adaptive space-time codes,” presented at the Information Theory Workshop
- Gamal, Bourles
- 2001
(Show Context)
Citation Context ... of time is . 3 The extension of the TAST codes to the adaptive scenario (i.e., the CSI is available partially at the transmitter and fully at the receiver, TR-CSI) is rather straightforward based on =-=[47]-=-, and is omitted here. It is, however, worth noting that significant improvement in the performances of the TAST codes can be obtained by exploiting partial transmitter CSI as shown in [47].sEL GAMAL ... |

1 | Gamal, “On the design of space-frequency codes,” presented at the Fourth Int - El - 2001 |

1 | Space-time block codes - Geng, Mitra, et al. - 2001 |

1 | Distance specta for space-time trellis coded modulations,” presented at the Int - Aktas, Fitz - 2001 |