## On the duality of Gaussian multiple-access and broadcast channels (2004)

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Venue: | IEEE Trans. Inf. Theory |

Citations: | 82 - 13 self |

### BibTeX

@ARTICLE{Jindal04onthe,

author = {Nihar Jindal and Student Member and Sriram Vishwanath and Andrea Goldsmith and Senior Member},

title = {On the duality of Gaussian multiple-access and broadcast channels},

journal = {IEEE Trans. Inf. Theory},

year = {2004},

pages = {768--783}

}

### Years of Citing Articles

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### Abstract

Abstract—We define a duality between Gaussian multiple-access channels (MACs) and Gaussian broadcast channels (BCs). The dual channels we consider have the same channel gains and the same noise power at all receivers. We show that the capacity region of the BC (both constant and fading) can be written in terms of the capacity region of the dual MAC, and vice versa. We can use this result to find the capacity region of the MAC if the capacity region of only the BC is known, and vice versa. For fading channels we show duality under ergodic capacity, but duality also holds for different capacity definitions for fading channels such as outage capacity and minimum-rate capacity. Using duality, many results known for only one of the two channels can be extended to the dual channel as well. Index Terms—Broadcast channel (BC), channel capacity, duality, fading channels, multiple-input multiple-output (MIMO) systems, multiple-access channel (MAC). I.

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Citation Context ... codeword is decoded and then subtracted from the received signal, then the next user is decoded and subtracted out, and so on. Since the Gaussian BC is a degraded BC, superposition coding is optimal =-=[1]-=-. In the Gaussian BC, superposition coding simplifies to transmitting the sum of independent Gaussian codewords (one codeword per user). The receivers also perform successive decoding with interferenc... |

2149 | Capacity of multi antenna Gaussian channels
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Citation Context ...ingle users (i.e., a single transmitter with three antennas communicating to a receiver with two antennas, or vice versa), then duality holds due to the reciprocity of multiple-antenna Gaussian links =-=[19]-=-. By the reciprocity result we know that the capacity of a channel with gain matrix is equal to the capacity of the channel with gain matrix equal to the transpose (or Hermitian transpose since conjug... |

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Citation Context ... By minimizing the dual function over all nonnegative Lagrange multipliers, we get an upper bound on the optimal value . Due to the convexity and feasibility of the problem, this bound is tight [23], =-=[22]-=- (36) where are the optimum Lagrange multipliers that lead to . In what follows, we show that is finite and strictly positive if and is zero if . First consider such that .If , then with we get . Thus... |

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Citation Context ...d MAC are duals [8]. The MIMO BC is a nondegraded BC, for which a general expression for the capacity region remains unknown. However, an achievable region for the MIMO BC based on dirty-paper coding =-=[17]-=-, [18] is known. In [8], it is shown that the capacity region of the MIMO MAC and the dirty-paper achievable region of the BC are duals, or that the MIMO BC achievable region is equal to the union of ... |

209 | Sum Capacity of the Vector Gaussian Broadcast Channel and Uplink-Downlink Duality
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Citation Context ...e MAC and BC [7]. Duality has also been very useful in proving new results, most prominently for the multiple-antenna BC. In [8], duality is extended to multiple-antenna Gaussian channels and in [8], =-=[9]-=- this duality has been used to find the sum capacity of the multiple-antenna BC. Duality also greatly simplifies numerical computation of the multiple-antenna BC sum rate capacity and achievable regio... |

150 | Capacity and optimal resource allocation for fading broadcast channels: Part II, outage capacity
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Citation Context ... of fading channels as well. We also show that the relationship in (2) holds for fading channels. Duality also holds for outage capacity and minimum-rate capacity. Though the ergodic capacity regions =-=[2]-=-, [3] and outage capacity regions [4], [5] of both the MAC and BC have previously been found, duality ties these results together. Minimum-rate capacity has only been found for the BC [6], but using d... |

108 | Sum power iterative waterfilling for multi-antenna gaussian broadcast channels
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(Show Context)
Citation Context ...ality has been used to find the sum capacity of the multiple-antenna BC. Duality also greatly simplifies numerical computation of the multiple-antenna BC sum rate capacity and achievable region [10], =-=[11]-=-. The remainder of this paper is organized as follows. In Section II, we describe the dual Gaussian BC and MAC. In Section III, we show that the constant Gaussian BC and MAC are duals. In Section IV, ... |

107 |
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Citation Context ...ading channels as well. We also show that the relationship in (2) holds for fading channels. Duality also holds for outage capacity and minimum-rate capacity. Though the ergodic capacity regions [2], =-=[3]-=- and outage capacity regions [4], [5] of both the MAC and BC have previously been found, duality ties these results together. Minimum-rate capacity has only been found for the BC [6], but using dualit... |

89 | Gaussian multiaccess channels with ISI: Capacity region and multiuser water-filling
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Citation Context ...sily extends to frequency-selective (intersymbol interference (ISI)) channels as well. BCs and MACs with time-invariant, finite-length impulse responses and additive Gaussian noise were considered in =-=[14]-=-, [15]. The dual channels have the same impulse response on the uplink and downlink, and the same noise power at each receiver. Similar to flat-fading channels, frequency-selective channels can be dec... |

81 |
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Citation Context ...hip is not only conceptually powerful, but has also been of great use in establishing new results for Gaussian channels. A number of other information-theoretic dualities (e.g., source/channel coding =-=[20], -=-channel coding/rate distortion [21], MAC/Slepian–Wolf [1, Sec. 14.5]) have been established over the decades. It remains to be seen if the multiple-access/broadcast duality is a result ofsJINDAL et ... |

78 |
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Citation Context ...eword of the next weakest user is selected, but since the weakest user’s codeword is already known, it can be pre-subtracted (using dirty paper coding [10] or coding for known interference techniques =-=[11]-=-) such that the second receiver does not experience any interference from the signal intended for the weakest user. Using this procedure for all users, it is clear that the strongest user is the last ... |

74 |
Downlink capacity evaluation of cellular networks with known-interference cancellation
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(Show Context)
Citation Context ...his duality has been used to find the sum capacity of the multiple-antenna BC. Duality also greatly simplifies numerical computation of the multiple-antenna BC sum rate capacity and achievable region =-=[10]-=-, [11]. The remainder of this paper is organized as follows. In Section II, we describe the dual Gaussian BC and MAC. In Section III, we show that the constant Gaussian BC and MAC are duals. In Sectio... |

68 | Duality between channel capacity and rate distortion with two-sided state information
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- 2002
(Show Context)
Citation Context ..., but has also been of great use in establishing new results for Gaussian channels. A number of other information-theoretic dualities (e.g., source/channel coding [20], channel coding/rate distortion =-=[21], -=-MAC/Slepian–Wolf [1, Sec. 14.5]) have been established over the decades. It remains to be seen if the multiple-access/broadcast duality is a result ofsJINDAL et al.: ON THE DUALITY OF GAUSSIAN MULTI... |

68 |
Convex optimization with engineering applications
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Citation Context ...(35) where the set is defined in Condition 2. Consider the above maximization for some fixed . Since the objective function is linear and the set is convex, this is a convex optimization problem (see =-=[23]-=- for a general reference on convex optimization and Lagrangian duality). Furthermore, the maximization takes on some optimal value by the feasibility of the constraint set. The optimal value is finite... |

62 |
Multiaccess fading channels-Part II: Delay-Limited Capacities
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- 1998
(Show Context)
Citation Context ...ng states [4], [5]. Outage capacity is concerned with situations in which each user (in either the BC or MAC) desires a constant rate a certain percentage of the time. The zero-outage capacity 5 [4], =-=[16]-=- is a special case of outage capacity where a constant rate must be maintained in all fading states, or where . By definition, a rate vector is in if and only if there exists a power policy satisfying... |

51 | Trellis precoding for the broadcast channel
- Yu, Cioffi
- 2001
(Show Context)
Citation Context ..., we will find it convenient to consider rate vectors (some of which may lie in the interior of the capacity region) achieved by use of “dirty-paper coding” to simplify the MAC-BC duality. As seen in =-=[9]-=-, the capacity region of the broadcast channel is also achievable via “dirty-paper coding”, in which the transmitter essentially “pre-subtracts” (similar to pre-coding) certain users’ codewords (inste... |

46 |
achievable rates, and sumrate capacity of MIMO broadcast channels
- Vishwanath, Jindal, et al.
- 2003
(Show Context)
Citation Context ...ibility that a more general information-theoretic duality exists between the MAC and BC [7]. Duality has also been very useful in proving new results, most prominently for the multiple-antenna BC. In =-=[8]-=-, duality is extended to multiple-antenna Gaussian channels and in [8], [9] this duality has been used to find the sum capacity of the multiple-antenna BC. Duality also greatly simplifies numerical co... |

38 | On the capacity of multiple input multiple output broadcast channels
- Vishwanath, Jindal, et al.
- 2002
(Show Context)
Citation Context ...s y j = h j X + n j and the received signal in the MAC is y = P M j=1 h j X j + n, where n j N (0; 2 ) for all j and n N (0; 2 ) and all quantities are scalars (the vector case is analyzed in [2]). Notice that the channel gains on the uplink and downlink are the same and that the noise power at every receiver is 2 . We call this BC the dual of the MAC, and vice versa. We use CBC (P ; h) and... |

38 |
Multiple-access channels with memory with and without frame synchronism
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- 1989
(Show Context)
Citation Context ...l, its capacity region is not known. However, an achievable region for the MIMO BC based on dirty-paper coding [10] [16] is known. Additionally, the capacity region of the MIMO MAC is known [17] [18] =-=[19]-=-. In [1] it is shown that the capacity region of the MIMO MAC and the dirty-paper achievable region of the BC are duals, or that the MIMO BC achievable region is equal to the union of the dual MIMO MA... |

31 | Capacity and optimal power allocation for fading broadcast channels with minimum rates
- Jindal, Goldsmith
- 2003
(Show Context)
Citation Context ...city regions [2], [3] and outage capacity regions [4], [5] of both the MAC and BC have previously been found, duality ties these results together. Minimum-rate capacity has only been found for the BC =-=[6]-=-, but using duality we can find the minimum-rate capacity of the MAC as well. Duality is an exciting new concept that gives great insight into the similarities between the Gaussian MAC and BC, as well... |

25 | The capacity region of broadcast channels with intersymbol interference and colored gaussian noise
- Goldsmith, Effros
- 2001
(Show Context)
Citation Context ...xtends to frequency-selective (intersymbol interference (ISI)) channels as well. BCs and MACs with time-invariant, finite-length impulse responses and additive Gaussian noise were considered in [14], =-=[15]-=-. The dual channels have the same impulse response on the uplink and downlink, and the same noise power at each receiver. Similar to flat-fading channels, frequency-selective channels can be decompose... |

23 |
On achievable rates in a multi-antenna broadcast downlink
- Caire, Shamai
- 2000
(Show Context)
Citation Context ...are duals [8]. The MIMO BC is a nondegraded BC, for which a general expression for the capacity region remains unknown. However, an achievable region for the MIMO BC based on dirty-paper coding [17], =-=[18]-=- is known. In [8], it is shown that the capacity region of the MIMO MAC and the dirty-paper achievable region of the BC are duals, or that the MIMO BC achievable region is equal to the union of the du... |

22 | Optimum power and rate allocation strategies for multiple access fading channels
- Vishwanath, Jafar, et al.
- 2001
(Show Context)
Citation Context ... the boundaries touch in the sense that the weighted rate sum at the intersection point is equal to the maximum of in and in for the same . From results on the ergodic capacity region of the MAC [3], =-=[13]-=-, it is optimal to decode users in order of increasing priority in all fading states. Therefore, a fixed decoding order in all fading states is optimal for the MAC. Suppose we consider a point on the ... |

20 |
Iterative water-filling for vector multiple access channels
- Yu, Rhee, et al.
- 2001
(Show Context)
Citation Context ...ast channel, its capacity region is not known. However, an achievable region for the MIMO BC based on dirty-paper coding [10] [16] is known. Additionally, the capacity region of the MIMO MAC is known =-=[17]-=- [18] [19]. In [1] it is shown that the capacity region of the MIMO MAC and the dirty-paper achievable region of the BC are duals, or that the MIMO BC achievable region is equal to the union of the du... |

17 | Outage capacities and optimal power allocation for fading multiple-access channels
- Li, Jindal, et al.
- 2005
(Show Context)
Citation Context ...that the relationship in (2) holds for fading channels. Duality also holds for outage capacity and minimum-rate capacity. Though the ergodic capacity regions [2], [3] and outage capacity regions [4], =-=[5]-=- of both the MAC and BC have previously been found, duality ties these results together. Minimum-rate capacity has only been found for the BC [6], but using duality we can find the minimum-rate capaci... |

9 | On the duality of multiple-access and broadcast channels
- Jindal, Vishwanath, et al.
- 2001
(Show Context)
Citation Context ... P is equal to the union of capacity regions of the dual MAC with power (P1 ; : : : ; PM ) such that P M j=1 P j = P : CBC (P ; h) = [ fP : 1P=Pg CMAC (P ; h): (1) Proofs of all results are given in [=-=1]-=-. Theorem 1 indicates that the dual BC and MAC are equivalent when the MAC is given a sum power constraint instead of individual power constraints. Additionally, wesnd that points on the boundary of t... |

5 |
Multiaccess fading channels–Part I:Polymatroid structure, optimal resource allocation and throughput capacities
- Tse, Hanly
- 1998
(Show Context)
Citation Context ...ding channels. In addition we show that duality holds for 3 DRAFTsa number of different capacity definitions, namely outage capacity and minimum rate capacity. Though the ergodic capacity regions [3] =-=[4]-=- and outage capacity regions [5] [6] of both the MAC and BC have previously been found, the duality ties these previously independent results together. Minimum rate capacity has only been found for th... |

3 |
On the duality between general multiple-access/broadcast channels
- Jindal, Vishwanath, et al.
(Show Context)
Citation Context ... Gaussian MAC and BC, as well as their capacities and optimal transmission strategies. It also opens up the possibility that a more general information-theoretic duality exists between the MAC and BC =-=[7]-=-. Duality has also been very useful in proving new results, most prominently for the multiple-antenna BC. In [8], duality is extended to multiple-antenna Gaussian channels and in [8], [9] this duality... |

1 |
Trellis preceding for the broadcast channel
- Yu, Cioffi
- 2001
(Show Context)
Citation Context ...users’ signals, the second strongest user can decode all users’ signals except for the strongest user’s signal, etc. The “strongest” user refers to the user with the largest channel gain . A=-=s seen in [12], the capa-=-city region of the BC is also achievable via “dirty-paper coding,” in which the transmitter “presubtracts” (similar to precoding for equalization) certain users’ codewords instead of receive... |

1 |
On degraded two-user gaussian channels
- Sato
- 1978
(Show Context)
Citation Context ...completely symmetric channels in which the channels from left-to-right and right-to-left are indistinguishable (An example of such a channel is the two-transmitter, two-receiver channel considered in =-=[20]-=- in which the channel gains � � from � � to and � � from � � to are � gains ��� from � � to and from � to ��� are equal to a constant � ). � 36 and the If the nodes on the left and right were consider... |