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## How Much Training is Needed in Multiple-Antenna Wireless Links? (2000)

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Venue: | IEEE Trans. Inform. Theory |

Citations: | 508 - 9 self |

### Citations

2922 | Capacity of multi-antenna gaussian channels
- Telatar
- 1999
(Show Context)
Citation Context ...ications 1 Introduction Multiple-antenna wireless communication links promise very high data rates with low error probabilities, especially when the wireless channel response is known at the receiver =-=[1, 2]-=-. To learn the channel, the receiver often requires the transmitter to send known training signals during some portion of the transmission interval. An early study of the effect of training on channel... |

1682 |
Microwave Mobile Communications,
- Jakes
- 1974
(Show Context)
Citation Context ...s for another interval T , and so on. This is an appropriate model for TDMA- or frequency-hopping-based systems, and is a tractable approximation of a continuously fading channel model such as Jakes' =-=[4]-=-. We further assume that channel estimation (via training) and data transmission is to be done within the interval T , after which new training allows us to estimate the channel for the next T symbols... |

1435 |
Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas.
- Foschini
- 1996
(Show Context)
Citation Context ...ications 1 Introduction Multiple-antenna wireless communication links promise very high data rates with low error probabilities, especially when the wireless channel response is known at the receiver =-=[1, 2]-=-. To learn the channel, the receiver often requires the transmitter to send known training signals during some portion of the transmission interval. An early study of the effect of training on channel... |

439 |
System Identification,
- Soderstrom, Stoica
- 1989
(Show Context)
Citation Context ...ors with the same covariance matrix, the one with a Gaussian distribution has the largest entropy. The following lemma gives a crucial property of cov(SjX; H). Its proof can be found in, for example, =-=[12]-=-. 26 Lemma 1 (Minimum Covariance Property of E jX;H S). Let S = f(X; H) be any estimate of S given X and H . Then we have cov(SjX; H) = E (S \Gamma E jX;H S) (S \Gamma E jX;H S) 6 E (S \Gamma S) (S \G... |

426 | Fading channels: informationtheoretic and communications aspects
- Biglieri, Proakis, et al.
- 1998
(Show Context)
Citation Context ... infer that, at low power, the capacity with training is given by (40) and decays as ae 2 . However, 17 the true channel capacity (which does not necessarily require training to achieve) decays as ae =-=[9]-=-, [10]. We therefore must conclude that training is highly suboptimal when ae is small. 3.4 Equal training and data power A communication system often does not have the luxury of varying the power dur... |

379 | Design Algorithm and Initial Laboratory Results using
- Golden
- 1999
(Show Context)
Citation Context ...sOur solution is based on a lower bound on the information-theoretic capacity achievable with trainingbased schemes. An example of a training-based scheme that has attracted recent attention is BLAST =-=[4, 5]-=- where an experimental prototype has achieved 20 bits/sec/Hz data rates with 8 transmit and 12 receive antennas. The lower bound allows us to compute the optimal amount of training as a function of , ... |

281 |
Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels.
- Guey, Fitz, et al.
- 1999
(Show Context)
Citation Context ...+ m=1 M m S S = T IM; (18) as the optimal solution; i.e., the training signal must be a multiple of a matrix with orthonormal columns. A similar conclusion is drawn in [3] when training for BLAST and =-=[13]-=- when training with so-called “shiftinvariant” sequences to minimize total estimation error. With this choice of training signal, we obtain In fact, we have the stronger result R ~ H = which implies t... |

271 | The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,”
- Medard
- 2000
(Show Context)
Citation Context ...r EV 0 d V 0sd =NT d sup p S d (\Delta);tr ES d S 0sd =MT d I(X d ; S d j H): A similar argument for lower-bounding the mutual information in a scalar and multiple-access wireless channel is given in =-=[5]-=-. The worst-case noise is the content of the next theorem, which is proven in Appendix A. Theorem 1 (Worst-Case Uncorrelated Additive Noise). Consider the matrix-valued additive noise known channel X ... |

269 | Differential space-time modulation,”
- Hughes
- 2000
(Show Context)
Citation Context ... for training when T > 2M, more power for data transmission when T < 2M, and the same power when T = 2M. We note that there have been some proposals for multiple-antenna differential modulation [14], =-=[15]-=- that use M transmit antennas and an effective block size of T =2M. These proposals can be thought of as a natural extension of standard single-antenna DPSK where the first half of the transmission (c... |

256 | Simplified Processing for High Spectral Efficiency Wireless Communication Employing Multi-Element Arrays,
- Foschini, Golden, et al.
- 1999
(Show Context)
Citation Context ...sOur solution is based on a lower bound on the information-theoretic capacity achievable with trainingbased schemes. An example of a training-based scheme that has attracted recent attention is BLAST =-=[4, 5]-=- where an experimental prototype has achieved 20 bits/sec/Hz data rates with 8 transmit and 12 receive antennas. The lower bound allows us to compute the optimal amount of training as a function of , ... |

215 | A differential detection scheme for transmit diversity
- Tarokh, Jafarkhani
- 1999
(Show Context)
Citation Context ...smission (comprising M time samples across M transmit antennas) acts as a reference for the second half (also comprising M time samples). A differential scheme using orthogonal designs is proposed in =-=[8]-=-. In these proposals, both halves of the transmission are given equal power. But because T = 2M , Corollary 2 says that giving each half equal power is optimal in the sense of maximizing the capacity ... |

123 |
The capacity of discrete-time memoryless Rayleigh-fading channels
- Abou-Faycal, Shamai, et al.
- 2001
(Show Context)
Citation Context ...ve SNR (31) decays as ; the quality of the channel estimate is very poor. The true channel capacity, however, (which does not necessarily require training to achieve) decays as , rather than as [17], =-=[18]-=-. These simple power considerations therefore suggest that training and using the channel estimate as if it were correct is highly suboptimal when is small. D. Equal Training and Data Power A communic... |

113 | BLAST training: Estimating channel characteristics for high capacity space-time wireless.
- Marzetta
- 1999
(Show Context)
Citation Context ...e channel, the receiver often requires the transmitter to send known training signals during some portion of the transmission interval. An early study of the effect of training on channel capacity is =-=[3]-=- where it is shown that, under certain conditions, by choosing the number of transmit antennas to maximize the throughput in a wireless channel, one generally spends half the coherence interval traini... |

105 | On information rates for mismatched decoders.
- Merhav, Kaplan, et al.
- 1994
(Show Context)
Citation Context ...least as good as our bound. Receivers that assume the channel estimate after training to be perfect are generally suboptimal (sometimes also called “mismatched”) and their analyses can be complicated =-=[7, 8]-=-. We do not pursue any such analyses here. We only remark that the bound we compute does not directly apply to the best mismatched receiver—instead, our bound applies to the best overall transmitter/r... |

68 |
Nearest neighbor decoding for additive non-gaussian noise channels
- Lapidoth
- 1996
(Show Context)
Citation Context ...least as good as our bound. Receivers that assume the channel estimate after training to be perfect are generally suboptimal (sometimes also called “mismatched”) and their analyses can be complicated =-=[7, 8]-=-. We do not pursue any such analyses here. We only remark that the bound we compute does not directly apply to the best mismatched receiver—instead, our bound applies to the best overall transmitter/r... |

56 |
Least sum of squared errors (LSSE) channel estimation,”
- Crozier, Falconer, et al.
- 1991
(Show Context)
Citation Context ...atrix of the MMSE estimate (which in this case is also the LMMSE estimate) 1Maximizing SNRs has been studied in many other contexts as well; for a study in intersymbol interference (ISI) channels see =-=[21]-=-. HASSIBI AND HOCHWALD: HOW MUCH TRAINING IS NEEDED IN MULTIPLE-ANTENNA WIRELESS LINKS? 955 where we have used the equation to compute , , and . It follows that we need to choose to solve In terms of ... |

39 |
Communication on the Grassman manifold: A geometric approach to the noncoherent multiple-antenna channel
- Zheng, Tse
- 2002
(Show Context)
Citation Context ...ed at versus the predicted high SNR value of , and in Fig. 6 at versus the predicted . We now ask whether the high-SNR bound (46) is tight? The answer to this question can be found in the recent work =-=[19]-=- of Zheng and Tse, where it is shown that at high SNR, the leading term of the actual channel capacity (without imposing any constraints such as training) is . Thus, in the leading SNR term (as ), tra... |

37 |
On the capacity of channels with additive non-Gaussian noise
- Ihara
- 1978
(Show Context)
Citation Context ...ribution on H is right rotationally invariant, i.e. when p(H ) = p(H) for all such that = =IN, then RV;opt = IN: The notion that Gaussian additive noise is the worst for mutual information is not new =-=[10, 11, 12]-=-. Theorem 1 is, however, tailored for our purposes since the noise is uncorrelated with the signal (rather than independent as is usually assumed in these references), and we are also able to compute ... |

34 |
unitary space time modulation
- Hochwald, Sweldens, et al.
- 2000
(Show Context)
Citation Context ... for training when T ? 2M , more power for data transmission when T ! 2M , and the same power when T = 2M . We note that there have been some proposals for multiple-antenna 16 differential modulation =-=[6]-=-, [7] that use M transmit antennas and an effective block size of T = 2M . These proposals can be thought of as a natural extension of standard single-antenna DPSK where the first half of the transmis... |

19 |
Shitz). The capacity of discrete-time Rayleigh fading channels
- Abou-Faycal, Trott, et al.
- 1997
(Show Context)
Citation Context ...r that, at low power, the capacity with training is given by (40) and decays as ae 2 . However, 17 the true channel capacity (which does not necessarily require training to achieve) decays as ae [9], =-=[10]-=-. We therefore must conclude that training is highly suboptimal when ae is small. 3.4 Equal training and data power A communication system often does not have the luxury of varying the power during th... |

9 |
An information theoretic study of communication in the presence of jamming,” in
- McEliece, Stark
- 1981
(Show Context)
Citation Context ...ribution on H is right rotationally invariant, i.e. when p(H ) = p(H) for all such that = =IN, then RV;opt = IN: The notion that Gaussian additive noise is the worst for mutual information is not new =-=[10, 11, 12]-=-. Theorem 1 is, however, tailored for our purposes since the noise is uncorrelated with the signal (rather than independent as is usually assumed in these references), and we are also able to compute ... |

8 |
Differential space-time modulation," Submitted to
- Hughes
- 1999
(Show Context)
Citation Context ...training when T ? 2M , more power for data transmission when T ! 2M , and the same power when T = 2M . We note that there have been some proposals for multiple-antenna 16 differential modulation [6], =-=[7]-=- that use M transmit antennas and an effective block size of T = 2M . These proposals can be thought of as a natural extension of standard single-antenna DPSK where the first half of the transmission ... |

8 |
Packing spheres in the Grassman manifold: a geometric approach to the non-coherent multi-antenna channel
- Zheng, Tse
- 2002
(Show Context)
Citation Context ...ole process of training is highly suboptimal at low SNR. We now ask whether the same is true at high SNR, and whether our bounds are tight? The answer to this question can be found in the recent work =-=[11]-=- of Zheng and Tse where it is shown that at high SNR the leading term of the actual channel capacity (without imposing any constraints such as training) is \Gamma 1 \Gamma K T \Delta K log ae. Thus, i... |

3 |
Calculation of the rate of information production by means of stationary random processes and the capacity of a stationary channel,” Dokl
- Pinsker
- 1956
(Show Context)
Citation Context ...ribution on H is right rotationally invariant, i.e. when p(H ) = p(H) for all such that = =IN, then RV;opt = IN: The notion that Gaussian additive noise is the worst for mutual information is not new =-=[10, 11, 12]-=-. Theorem 1 is, however, tailored for our purposes since the noise is uncorrelated with the signal (rather than independent as is usually assumed in these references), and we are also able to compute ... |