## High-Rate Codes that are Linear in Space and Time (2000)

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Venue: | IEEE Trans. Inform. Theory |

Citations: | 331 - 11 self |

### BibTeX

@ARTICLE{Hassibi00high-ratecodes,

author = {Babak Hassibi and Bertrand Hochwald},

title = {High-Rate Codes that are Linear in Space and Time},

journal = {IEEE Trans. Inform. Theory},

year = {2000},

volume = {48},

pages = {1804--1824}

}

### Years of Citing Articles

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### Abstract

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---this deficiency is especially important for modern cellular systems where a basestation typically has more antennas than the mobile handsets. 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...

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Citation Context ...h at high rates will be an important part of future telecommunications systems. One way to get high rates on a scattering-rich wireless channel is to use multiple transmit and/or receive antennas. In =-=[1, 2]-=-, theoretical and experimental evidence demonstrates that channel capacity grows linearly as the number of transmit and receive antennas grow simultaneously. Early uses of multiple transmit antennas i... |

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Citation Context ...uence of nulling and cancelling steps. Since then there has been considerable work on a variety of space-time transmission schemes and performance measures [9] such as the space-time trellis codes of =-=[10]-=- and the space-time block codes of [11, 12] for the known channel and [13, 14, 15] for the unknown channel. At very high rates and with a large number of antennas, many of these space-time codes suffe... |

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Citation Context ...Since then there has been considerable work on a variety of space-time transmission schemes and performance measures [9] such as the space-time trellis codes of [10] and the space-time block codes of =-=[11, 12]-=- for the known channel and [13, 14, 15] for the unknown channel. At very high rates and with a large number of antennas, many of these space-time codes suffer from complexity or performance difficulti... |

1046 |
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321 | How much training is needed in multiple-antenna wireless links
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Citation Context ...ption is reasonable if training or pilot signals are sent to learn the channel, which is then constant for some coherence interval. The coherence interval of the channel should be large compared to M =-=[18]-=-. When the channel is known at the receiver, the resulting channel capacity (often referred to as the perfectknowledgescapacity) is [2, 1] C(ae; M;N) = max Rs0;tr Rs=M E log det i I N + ae M HR s H j ... |

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Citation Context ...lexity low, linear processing is often preferred [3]. To achieve the high data rates promised in [2], however, new approaches for space-time transmission are needed. One such approach is presented in =-=[7, 8]-=- where a practical scheme, called V-BLAST (Vertical Bell Labs Layered Space-Time), encodes and decodes rates of tens of bits/sec/Hz with eight transmit and twelve receive antennas. The V-BLAST archite... |

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Citation Context ... receiver decodes the substreams using a sequence of nulling and cancelling steps. Since then there has been considerable work on a variety of space-time transmission schemes and performance measures =-=[9]-=- such as the space-time trellis codes of [10] and the space-time block codes of [11, 12] for the known channel and [13, 14, 15] for the unknown channel. At very high rates and with a large number of a... |

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Citation Context ...o encode 5. Can be decoded in a variety of ways including simple linear-algebraic techniques such as: (a) Successive nulling and cancelling (V-BLAST [7], square-root V-BLAST [16]) (b) Sphere decoding =-=[17]-=- 6. Are designed with the numbers of both the transmit and receive antennas in mind 7. Satisfy an information-theoretic optimality criterion - the codes are designed to maximize the mutual information... |

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Citation Context ... no longer "full-rate" (see [12]) and therefore generally perform poorly in the maximum mutual information they can achieve, even when N = 1. We give an example of these non-square orthogona=-=l designs [12, 20, 21]-=-. For M = 3, we have, for example, the rate 3 4 orthogonal design S = r 4 3 2 6 6 6 6 6 6 6 6 6 6 4 s 1 s 2 s 3 \Gammas 2 s 1 0 \Gammas 3 0 s 1 0 \Gammas 3 s 2 3 7 7 7 7 7 7 7 7 7 7 5 : (14) The facto... |

124 |
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Citation Context ...ble work on a variety of space-time transmission schemes and performance measures [9] such as the space-time trellis codes of [10] and the space-time block codes of [11, 12] for the known channel and =-=[13, 14, 15]-=- for the unknown channel. At very high rates and with a large number of antennas, many of these space-time codes suffer from complexity or performance difficulties. The number of states in the trellis... |

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76 | An efficient square-root algorithm for BLAST
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Citation Context ...ennas 4. Are very simple to encode 5. Can be decoded in a variety of ways including simple linear-algebraic techniques such as: (a) Successive nulling and cancelling (V-BLAST [7], square-root V-BLAST =-=[16]-=-) (b) Sphere decoding [17] 6. Are designed with the numbers of both the transmit and receive antennas in mind 7. Satisfy an information-theoretic optimality criterion - the codes are designed to maxim... |

71 |
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Citation Context ...ironment achieve reliability through "diversity, " where redundant information is sent or received on two or more antennas in the hope that at least one path from the transmitter reaches the=-= receiver [3, 4, 5, 6]-=-. To keep the transmitter and receiver complexity low, linear processing is often preferred [3]. To achieve the high data rates promised in [2], however, new approaches for space-time transmission are... |

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Citation Context |

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Citation Context ...o encode 5. Can be decoded in a variety of ways including simple linear-algebraic techniques such as: (a) Successive nulling and cancelling (V-BLAST [7], square-root V-BLAST [18]) (b) Sphere decoding =-=[19, 20]-=- 6. Are designed with the numbers of both the transmit and receive antennas in mind 7. Satisfy an information-theoretic optimality criterion - the codes are designed to maximize the mutual information... |

41 |
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Citation Context ... no longer "full-rate" (see [12]) and therefore generally perform poorly in the maximum mutual information they can achieve, even when N = 1. We give an example of these non-square orthogona=-=l designs [12, 20, 21]-=-. For M = 3, we have, for example, the rate 3 4 orthogonal design S = r 4 3 2 6 6 6 6 6 6 6 6 6 6 4 s 1 s 2 s 3 \Gammas 2 s 1 0 \Gammas 3 0 s 1 0 \Gammas 3 s 2 3 7 7 7 7 7 7 7 7 7 7 5 : (14) The facto... |

32 |
Differential unitary space time modulation,” tech
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Citation Context ...ble work on a variety of space-time transmission schemes and performance measures [9] such as the space-time trellis codes of [10] and the space-time block codes of [11, 12] for the known channel and =-=[13, 14, 15]-=- for the unknown channel. At very high rates and with a large number of antennas, many of these space-time codes suffer from complexity or performance difficulties. The number of states in the trellis... |

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A construction of a spacetime code based on the theory of numbers
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Citation Context ...l efficiencies. The LD code (31) may be modified as in Remark 4 in Section 3.2, without changing its mutual information, by premultiplying the transmitted signal vector by an orthogonal matrix t . In =-=[24]-=- it is shown that full diversity is achieved for this two-antenna code transmitting QPSK if we choose to be the block diagonal matrix = diag 02 @4 1 0 5 ; 0 1 3 2 4 R I , I R 3 5 ; 2 4 R I , I R 3 5 ;... |

8 |
Differential space-time modulation," Submitted to
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Citation Context ...ble work on a variety of space-time transmission schemes and performance measures [9] such as the space-time trellis codes of [10] and the space-time block codes of [11, 12] for the known channel and =-=[13, 14, 15]-=- for the unknown channel. At very high rates and with a large number of antennas, many of these space-time codes suffer from complexity or performance difficulties. The number of states in the trellis... |

4 |
Space-time autocoding," submitted to
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- 1999
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Citation Context ...; 1) entries), the optimizing covariance is R s;opt = I M , and (3) becomes C(ae; M;N) = E log det i I N + ae M HH j : (4) This expectation can sometimes be computed in closed form (see, for example, =-=[19]-=-). 1 Equation (3) actually slightly generalizes [2, 1], which assume that H has independent CN (0; 1) entries. 6 When the channel is constant for at least T channel uses we may write xs= r ae M Hss+ v... |

3 |
Space-time autocoding,” submitted to
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Citation Context ...endentCN(0; 1) entries), the optimizing covariance is Rs;opt = IM, and (3) becomes C( ; M; N ) = E log det IN + M HH : (4) This expectation can sometimes be computed in closed form (see, for example, =-=[22]-=-). When the channel is constant for at least T channel uses we may write x = so that defining X = x 1 x 2 ::: xT (where the superscript t denotes “transpose”), we obtain r Hs + v ; =1;:::;T; M t , S =... |

1 | A wireless communication technique of extraordinary spectral efficiency - Foschini, Golden, et al. - 1999 |