## Sum power iterative water-filling for multi-antenna Gaussian broadcast channels (2005)

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Venue: | IEEE Trans. Inform. Theory |

Citations: | 89 - 17 self |

### BibTeX

@ARTICLE{Jindal05sumpower,

author = {Nihar Jindal and Wonjong Rhee and Syed Jafar and Goldsmith Fellow Ieee},

title = {Sum power iterative water-filling for multi-antenna Gaussian broadcast channels},

journal = {IEEE Trans. Inform. Theory},

year = {2005},

volume = {51},

pages = {1570--1580}

}

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### Abstract

In this paper we consider the problem of maximizing sum rate of a multiple-antenna Gaussian broadcast channel. It was recently found that dirty paper coding is capacity achieving for this channel. In order to achieve capacity, the optimal transmission policy (i.e. the optimal transmit covariance structure) given the channel conditions and power constraint must be found. However, obtaining the optimal trans-mission policy when employing dirty paper coding is a computationally complex non-convex problem. We use duality to transform this problem into a well-structured convex multiple-access channel problem. We exploit the structure of this problem and derive simple and fast iterative algorithms that provide the optimum transmission policies for the multiple-access channel, which can easily be mapped to the optimal broadcast channel policies.

### Citations

8867 |
Elements of Information Theory
- Cover, Thomas
- 1991
(Show Context)
Citation Context ... L and j > L. Since |Sij| ≤ � SiiSjj for any S ≥ 0, this implies Sii > 0 and Sjj > 0, i.e. at least one diagonal entry of S is strictly positive below the L-th row/column. Using Hadamard’s inequality =-=[5]-=- and the fact that Dii = 0 for i > L, we have |I + SD| ≤ �M i=1 (1+SiiDii) = �L i=1 (1+SiiDii). We now construct another matrix S ′ that achieves a strictly larger objective than S. We define S ′ to b... |

1907 | Capacity of multi-antenna Gaussian channels
- Telatar
- 1995
(Show Context)
Citation Context ...t the capacity of a point-to-point MIMO channel is achieved by choosing the input covariance along the eigenvectors of the channel matrix and by water-filling on the eigenvalues of the channel matrix =-=[9]-=-. Thus, Q (n+1) i should be chosen as a water-fill of the channel Gi, i.e. the eigenvectors of Q (n+1) i should equal the left eigenvectors of Gi, with the eigenvalues chosen by the water-filling proc... |

696 |
Writing on dirty paper
- Costa
- 1983
(Show Context)
Citation Context ... [1]K = 1, [K]K = K, and so forth. III. SUM RATE CAPACITY In [3], [10], [12], [16], the sum rate capacity of the MIMO BC (denoted as CBC(H1,...,HK,P)) was shown to be achievable by dirty-paper coding =-=[4]-=-. From these results, the sum rate capacity can be written in terms of the following maximization: CBC(H1,... ,HK,P) = max {Σi} K i=1 : Σi≥0, P K i=1 Tr(Σi)≤P 2 � � log �I + H1Σ1H † � � � + (4) � � �I... |

338 |
Shitz), “On the achievable throughput of a multi-antenna gaussian broadcast channel
- Caire, Shamai
- 2003
(Show Context)
Citation Context ...rest in characterizing and computing the capacity region of multiple-antenna broadcast (downlink) channels in recent years. An achievable region for the multiple-antenna downlink channel was found in =-=[3]-=-, and this achievable region was shown to achieve the sum rate capacity in [3], [10], [12], [16], and was more recently shown to achieve the full capacity region in [14]. Though these results show tha... |

199 | Sum capacity of Gaussian vector broadcast channels
- Yu, Cioffi
- 2004
(Show Context)
Citation Context ...k) channels in recent years. An achievable region for the multiple-antenna downlink channel was found in [3], and this achievable region was shown to achieve the sum rate capacity in [3], [10], [12], =-=[16]-=-, and was more recently shown to achieve the full capacity region in [14]. Though these results show that the general dirty paper coding strategy is optimal, one must still optimize over the transmit ... |

195 | Iterative water-filling for Gaussian vector multiple-access channels
- Yu, Rhee, et al.
- 2004
(Show Context)
Citation Context ... for computing (1). This algorithm is inspired by and is very similar to the iterative waterfilling algorithm for the conventional individual power constraint MAC problem by Yu, Rhee, Boyd and Cioffi =-=[17]-=-. This paper is structured as follows. In the next section, the system model is presented. In Section III, expressions for the sum capacity of the downlink and dual uplink channels are stated. In Sect... |

187 | Sum capacity of the vector gaussian broadcast channel and uplink-downlink duality
- Viswanath, Tse
- 2003
(Show Context)
Citation Context ...ownlink) channels in recent years. An achievable region for the multiple-antenna downlink channel was found in [3], and this achievable region was shown to achieve the sum rate capacity in [3], [10], =-=[12]-=-, [16], and was more recently shown to achieve the full capacity region in [14]. Though these results show that the general dirty paper coding strategy is optimal, one must still optimize over the tra... |

162 |
Shitz), “The capacity region of the gaussian mimo broadcast channel
- Weingarten, Steinberg, et al.
(Show Context)
Citation Context ...a downlink channel was found in [3], and this achievable region was shown to achieve the sum rate capacity in [3], [10], [12], [16], and was more recently shown to achieve the full capacity region in =-=[14]-=-. Though these results show that the general dirty paper coding strategy is optimal, one must still optimize over the transmit covariance structure (i.e. how transmissions over different antennas shou... |

97 | Nonlinear Programming: A Unified Ap- proach - Zangwill - 1969 |

73 | On the duality of gaussian multiple-access and broadcast channels
- Vishwanath, Jindal, et al.
- 2004
(Show Context)
Citation Context ...ct optimization for sum rate capacity is a computationally complex non-convex problem. Therefore, obtaining the optimal rates and transmission policy is difficult 1 . A duality technique presented in =-=[7]-=-, [10] transforms the non-convex downlink problem into a convex sum power uplink (multiple-access channel, or MAC) problem, which is much easier to solve, from which the optimal downlink covariance ma... |

61 |
Introduction to Convex Optimization with Engineering Applications
- Boyd, Vandenberghe
- 2003
(Show Context)
Citation Context ... have a joint power constraint instead of individual constraints as in the conventional MAC. As in the case of the conventional MAC, there exist standard interior point convex optimization algorithms =-=[2]-=- that solve (1). An interior point algorithm, however, is considerably more complex than our algorithms and does not scale well when there are large numbers of users. Recent work by Lan and Yu based o... |

59 |
Downlink capacity evaluation of cellular networks with known interference cancellation
- Viswanathan, Venkatesan, et al.
- 2003
(Show Context)
Citation Context ...and Yu based on minimax optimization techniques appears to be promising but suffers from much higher complexity than our algorithms [8]. A steepest-descent method was proposed by Viswanathan, et. al. =-=[13]-=-, and an alternative, dual decomposition based algorithm was proposed by Yu in [15]. The complexity of these two algorithms is on the same order as the complexity of the algorithms proposed here. Howe... |

40 |
Duality, achievable rates, and sum-rate capacity of MIMO broadcast channels
- Vishwanath, Jindal, et al.
- 2003
(Show Context)
Citation Context ...ast (downlink) channels in recent years. An achievable region for the multiple-antenna downlink channel was found in [3], and this achievable region was shown to achieve the sum rate capacity in [3], =-=[10]-=-, [12], [16], and was more recently shown to achieve the full capacity region in [14]. Though these results show that the general dirty paper coding strategy is optimal, one must still optimize over t... |

29 | A dual decomposition approach to the sum power Gaussian vector multiple access channel sum capacity
- Yu
- 2003
(Show Context)
Citation Context ... from much higher complexity than our algorithms [8]. A steepest-descent method was proposed by Viswanathan, et. al. [13], and an alternative, dual decomposition based algorithm was proposed by Yu in =-=[15]-=-. The complexity of these two algorithms is on the same order as the complexity of the algorithms proposed here. However, we find our algorithm to converge more rapidly, and our algorithm is also cons... |

11 |
Sum power iterative waterfilling for Gaussian vector broadcast channels
- Vishwanath, Wonjong, et al.
- 2003
(Show Context)
Citation Context ...≤P 3) Compute the updated covariance matrices Q (n) i as: Q (n) i K� � � � log � �I + i=1 G (n) i � † SiG (n) i 1 = K S(n) K − 1 i + K Q(n−1) i i = 1,... ,K. (29) Algorithm 2 (which first appeared in =-=[11]-=-) differs from the original algorithm only in the addition of the third step. 6 As discussed in Section IX, the original algorithm can be used to generate an excellent starting point for Algorithm 2. ... |

7 | Input optimization for multi-antenna broadcast channels and per-antenna power constraints
- Lan, Yu
- 2004
(Show Context)
Citation Context ...cale well when there are large numbers of users. Recent work by Lan and Yu based on minimax optimization techniques appears to be promising but suffers from much higher complexity than our algorithms =-=[8]-=-. A steepest-descent method was proposed by Viswanathan, et. al. [13], and an alternative, dual decomposition based algorithm was proposed by Yu in [15]. The complexity of these two algorithms is on t... |