## Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality (1912)

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Venue: | IEEE Trans. on Inform. Theory |

Citations: | 199 - 2 self |

### BibTeX

@ARTICLE{Viswanath12sumcapacity,

author = {Pramod Viswanath and David Tse},

title = {Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality},

journal = {IEEE Trans. on Inform. Theory},

year = {1912}

}

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### Abstract

We characterize the sum capacity of the vector Gaussian broadcast channel by showing that the existing inner bound of Marton and the existing upper bound of Sato are tight for this channel. We exploit an intimate four-way connection between the vector broadcast channel, the corresponding point-to-point channel (where the receivers can cooperate), the multiple access channel (where the role of transmitters and receivers are reversed), and the corresponding point-to-point channel (where the transmitters can cooperate). 1

### Citations

3760 | Convex Analysis - Rockafellar - 1970 |

735 |
Writing on dirty paper
- Costa
- 1983
(Show Context)
Citation Context ...s with . Recent work by Caire and Shamai [2] obtains the sum capacity for the special case of users. They propose a transmission scheme which uses Costa’s “Writing on Dirty Paper” precoding technique =-=[3]-=-. This scheme can also be interpreted as Marton’s broadcast coding technique [9] applied to the vector Gaussian channel. For the case of two users, Caire and Shamai showed that this scheme is optimal ... |

204 | Sum capacity of Gaussian vector broadcast channels
- Yu, Cioffi
- 2001
(Show Context)
Citation Context ...lement. Our techniques can be naturally generalized to the case when the users have multiple antennas; this is discussed in Section IV. Independent proofs of the same result were reported in [20] and =-=[22]-=-. Notations: we use lower case letters to denote scalars, upper case letters to denote matrices, and boldface to denote vectors. denotes a complex circular symmetric Gaussian distribution with mean an... |

137 |
A coding theorem for the discrete memoryless broadcast channel
- Marton
- 1979
(Show Context)
Citation Context ...ecial case of users. They propose a transmission scheme which uses Costa’s “Writing on Dirty Paper” precoding technique [3]. This scheme can also be interpreted as Marton’s broadcast coding technique =-=[9]-=- applied to the vector Gaussian channel. For the case of two users, Caire and Shamai showed that this scheme is optimal in achieving the sum capacity, by demonstrating that the achievable rate meets t... |

132 |
Coding for a channel with random parameters,” Probl
- Gel’fand, Pinsker
- 1980
(Show Context)
Citation Context ...andom binning encoding technique. In fact, the connection between the broadcast channel problem and the problem of channel coding with side information at the transmitter has been known for some time =-=[6]-=-.sVISWANTH AND TSE: SUM CAPACITY OF THE VECTOR GAUSSIAN BROADCAST CHANNEL 1917 III. CONVERSE A. Sato Upper Bound To show that the sum rate is the best that can be achieved by any strategy, we begin wi... |

82 |
Optimum decision feedback multiuser equalization and successive decoding achieves the total capacity of the Gaussian multipleaccess channel
- Varanasi, Guess
- 1997
(Show Context)
Citation Context ...g signals from users to be nonexistent and treating signals from users as noise, for every , i.e., the unnormalized vector has the expression (13) we know that the sum capacity of the MAC is achieved =-=[18]-=- and SIR (14) In the broadcast channel, we retain the bank of linear filters but use a transmission strategy that codes for the users based on known interference at the transmitter. This strategy was ... |

67 | Asymptotically optimal waterfilling in vector multiple access channels - Viswanath, Tse, et al. - 2001 |

61 | Power control and capacity of spread spectrum wireless networks - Hanly, Tse - 1999 |

60 |
An outer bound on the capacity region of broadcast channels
- Sato
- 1978
(Show Context)
Citation Context ... Gaussian channel. For the case of two users, Caire and Shamai showed that this scheme is optimal in achieving the sum capacity, by demonstrating that the achievable rate meets the Sato’s upper bound =-=[12]-=-, which is the capacity of a point-to-point channel where the receivers in the downlink can cooperate. The proof involves a direct calculation and seems difficult to be generalized to . In this paper,... |

38 | Adaptive power control and MMSE interference suppression
- Ulukus, Yates
- 1998
(Show Context)
Citation Context ...rror (MMSE) filter should be used, since it maximizes the SIR for user . The optimal allocation of powers can be obtained by a simple iterative algorithm that exploits the monotonicity of the problem =-=[17]-=-. A direct solution to the downlink is not as obvious. However, the equivalence derived above shows that the downlink can be solved by converting it to an uplink problem. The optimal transmit filters ... |

36 | On the capacity of multiple input multiple output broadcast channels
- Viswanath, Jindal, et al.
- 2002
(Show Context)
Citation Context ...antenna element. Our techniques can be naturally generalized to the case when the users have multiple antennas; this is discussed in Section IV. Independent proofs of the same result were reported in =-=[20]-=- and [22]. Notations: we use lower case letters to denote scalars, upper case letters to denote matrices, and boldface to denote vectors. denotes a complex circular symmetric Gaussian distribution wit... |

33 |
On the capacity of the multiple antenna broadcast channel
- Tse, Viswanath
- 2003
(Show Context)
Citation Context ...e reciprocal MAC (in (33)) (namely, independent vector Gaussian inputs and a total power constraint). Both the forward part and the converse for multiple-receive antennas are carefully carried out in =-=[16]-=-, which also derives other results that shed insight into the entire capacity region of the vector Gaussian broadcast channel. V. CONCLUSION In this paper, we computed the sum capacity of the vector G... |

31 | Non-negative Matrices and Markov - Seneta - 1980 |

26 |
Capacity of the Multiple antenna Gaussian channel
- Telatar
- 1999
(Show Context)
Citation Context ...annels. This equivalence has been observed in seemingly different contexts in the literature. 1) In the context of the capacity of a point-to-point multipletransmit, multiple-receive antenna channel, =-=[15]-=- shows that the capacity is unchanged when the role of the transmitters and receivers is interchanged. The author calls this a reciprocity result. 2) In the context of a downlink of a multiple antenna... |

10 |
A unifying theory for uplink and downlink multiuser beamforming
- Schubert, Boche
- 2002
(Show Context)
Citation Context ...aracterization of the maximum achievable sum rate of the Costa precoding strategy. An independent and similar derivation of the duality in the context of linear beamforming strategies is presented in =-=[13]-=-. A. Point-to-Point Reciprocity Revisited Let us start with a point-to-point vector Gaussian channel with being a fixed matrix of dimension by . The additive noise is . We consider a linear transmissi... |

7 | To code, or not to code: On the optimality of symbol-by-symbol communication
- Gastpar, Rimoldi, et al.
- 2001
(Show Context)
Citation Context ...channel is the desired one (the optimal noncooperating input for the MAC.) Interestingly, a similar line of thinking is useful in the seemingly unrelated problem of optimality of uncoded transmission =-=[5]-=-. if ifsVISWANTH AND TSE: SUM CAPACITY OF THE VECTOR GAUSSIAN BROADCAST CHANNEL 1919 D. Convex Duality Interpretation Should one be surprised by the existence of such a which leads to the desirable st... |

6 |
On the duality between multiple access and broadcast channels
- Jindal, Vishwanath, et al.
(Show Context)
Citation Context ...s, [19] and [10] show that the optimal choice of transmit and receive beamforming vectors is closely related to a virtual uplink problem. 3) In the context of the degraded Gaussian broadcast channel, =-=[8]-=- shows that the capacity region is the same as the capacity region of the corresponding MAC with the transmit power constraint of the broadcast channel translated to the sum of powers in the MAC. The ... |

4 |
Optimal beamforming using tranmit antenna arrays
- Visotsky, Madhow
- 1999
(Show Context)
Citation Context ...The author calls this a reciprocity result. 2) In the context of a downlink of a multiple antenna system employing simple linear beamforming strategies followed by single-user receivers by the users, =-=[19]-=- and [10] show that the optimal choice of transmit and receive beamforming vectors is closely related to a virtual uplink problem. 3) In the context of the degraded Gaussian broadcast channel, [8] sho... |

2 | Is maximum entropy noise the worst - Diggavi, Cover - 1997 |

2 |
Transmit beamforming and power control in wireless networks with fading channels
- R-Farrokhi, Liu, et al.
- 1998
(Show Context)
Citation Context ...r calls this a reciprocity result. 2) In the context of a downlink of a multiple antenna system employing simple linear beamforming strategies followed by single-user receivers by the users, [19] and =-=[10]-=- show that the optimal choice of transmit and receive beamforming vectors is closely related to a virtual uplink problem. 3) In the context of the degraded Gaussian broadcast channel, [8] shows that t... |

1 |
On the achievable throughput in multantenna Gaussian broadcast channel
- Caire, Shamai
- 2003
(Show Context)
Citation Context ...83 (1) 0018-9448/03$17.00 © 2003 IEEE Theorem: The sum capacity of the vector Gaussian broadcast channel is where is the set of by nonnegative diagonal matrices with . Recent work by Caire and Shamai =-=[2]-=- obtains the sum capacity for the special case of users. They propose a transmission scheme which uses Costa’s “Writing on Dirty Paper” precoding technique [3]. This scheme can also be interpreted as ... |