#### DMCA

## Distributed Cooperative Beamforming in Multi-Source Multi-Destination Clustered Systems

### Citations

3235 | Capacity of wireless networks
- Gupta, Kumar
- 1999
(Show Context)
Citation Context ...uke.edu. Yupeng Liu is with Alcatel-Lucent, New Providence, NJ, 07974, USA, yupeng.liu@alcatel-lucent.com. Athina Petropulu is with the Dept. of Electrical and Computer Engineering, Rutgers, the State University of New Jersey, Piscataway, NJ, 08854, USA, athinap@rutgers.edu. communications of multiple, distinct, single-antenna, sourcedestination pairs that overlap both in time and frequency. This scenario is also referred to as multiuser peer-to-peer relay networks [5]–[16]. In general, the per-node throughput capacity of a wireless ad-hoc network reduces rapidly as the network size increases [22]. Therefore, it is often preferable to divide the network nodes into multiple clusters, with each cluster containing nodes which have distinct sub-goals, or are geographically close to each other, e.g., applications involving networks of mobile wireless robots [23, 24]. In this paper we consider a multi-cluster network, in which multiuser peer-to-peer relay communications take place inside each cluster, while the intra-cluster communications cause inter-cluster interference. In this context, the relay weights are computed based on channel second-order statistics, so that the total relay transm... |

1098 | Semidefinite programming
- Vandenberghe, Boyd
- 1996
(Show Context)
Citation Context ...r network, in which multiuser peer-to-peer relay communications take place inside each cluster, while the intra-cluster communications cause inter-cluster interference. In this context, the relay weights are computed based on channel second-order statistics, so that the total relay transmit power is minimized, while meeting certain signal-to-interference-plus-noise-ratio (SINR) constraints at the destinations. First, we show that a computationally efficient approximate solution is attainable by relaxing the original NP-hard non-convex problem employing semidefinite relaxation (SDR) techniques [25]–[28]. Second, we propose a distributed approach to solve the relaxed problem, which allows for each cluster to compute its optimal beamforming weights based on information exchanges with neighboring clusters only. Such a distributed approach obviates the need for a a central processing unit that has access to the channel statistics of all clusters and obtains the relay weights; centralized approaches do not scale well with the number of network nodes, resulting in high complexity and long delays. Our proposed distributed approach is based on Accelerated Distributed Augmented Lagrangians (ADAL... |

996 |
Parallel and distributed computation: numerical methods. Englewood Cliffs: Prentice-Hall. Republished by Athena Scientific
- Bertsekas, Tsitsiklis
- 1989
(Show Context)
Citation Context ...ccess to the channel statistics of all clusters and obtains the relay weights; centralized approaches do not scale well with the number of network nodes, resulting in high complexity and long delays. Our proposed distributed approach is based on Accelerated Distributed Augmented Lagrangians (ADAL) [29]. ADAL is a distributed optimization method that relies on augmented Lagrangians (AL), a regularization technique that is obtained by adding a quadratic penalty term to the ordinary Lagrangian of a problem [30]. Compared to standard distributed optimization techniques, such as dual decomposition [31], AL methods converge very fast and do not require strict convexity of the objective function [30, 31]. The latter is a necessary feature for our multi-cluster relay beamforming problem, since the objective function under consideration is affine. At the same time, it was shown in [29] that, for a number of different applications, ADAL exhibits a significant improvement in convergence speed compared to existing AL techniques, such as the ADMM [32] and the DQA [33]. We propose two different ways to apply ADAL to the multicluster beamforming problem, termed Direct and Indirect, 2 that allow us to... |

993 | Distributed optimization and statistical learning via the alternating direction method of multipliers
- Boyd, Parikh, et al.
- 2011
(Show Context)
Citation Context ...ratic penalty term to the ordinary Lagrangian of a problem [30]. Compared to standard distributed optimization techniques, such as dual decomposition [31], AL methods converge very fast and do not require strict convexity of the objective function [30, 31]. The latter is a necessary feature for our multi-cluster relay beamforming problem, since the objective function under consideration is affine. At the same time, it was shown in [29] that, for a number of different applications, ADAL exhibits a significant improvement in convergence speed compared to existing AL techniques, such as the ADMM [32] and the DQA [33]. We propose two different ways to apply ADAL to the multicluster beamforming problem, termed Direct and Indirect, 2 that allow us to model different message exchange patterns (necessary for the iterative execution of ADAL) between the individual clusters. Specifically, the message exchange pattern in the Direct method is determined by the coupling SINR constraints due to inter-cluster interference. On the other hand, the message exchanges in the Indirect method can be defined arbitrarily by the user. Both approaches rely on transforming the SINR coupling constraints to a line... |

666 | User cooperation diversity. part i. system description
- Sendonaris, Erkip, et al.
- 1927
(Show Context)
Citation Context ...exchanges. Two different approaches are presented, differing in the message exchange patterns between clusters. The performance of the decentralized scheme is demonstrated via simulations. Index Terms—Cooperative beamforming, multi-cluster systems, multi-source multi-destination systems, multiuser peer-topeer relay networks, distributed optimization, augmented Lagrangian. I. INTRODUCTION Cooperative (or relaying) approaches for wireless communications have the potential for significant performance improvement, such as extended coverage of the network, throughput enhancement and energy savings [1]–[21]. In cooperative beamforming, a set of relays form a “virtual antenna array” and retransmit weighted versions of the source signals (decode-and-forward (DF) relaying), or weighted versions of the received signals (amplify-and-forward (AF) relaying). By exploiting constructive interference effects, the relays focus the transmitted power on the destinations’ locations, thus increasing the directional channel gain. By achieving spatial multiplexing, cooperative beamforming can support the Copyright (c) 2014 IEEE. Personal use of this material is permitted. However, permission to use this mat... |

544 | Information flow and cooperative control of vehicle formations
- Fax, Murray
- 2004
(Show Context)
Citation Context ...tgers.edu. communications of multiple, distinct, single-antenna, sourcedestination pairs that overlap both in time and frequency. This scenario is also referred to as multiuser peer-to-peer relay networks [5]–[16]. In general, the per-node throughput capacity of a wireless ad-hoc network reduces rapidly as the network size increases [22]. Therefore, it is often preferable to divide the network nodes into multiple clusters, with each cluster containing nodes which have distinct sub-goals, or are geographically close to each other, e.g., applications involving networks of mobile wireless robots [23, 24]. In this paper we consider a multi-cluster network, in which multiuser peer-to-peer relay communications take place inside each cluster, while the intra-cluster communications cause inter-cluster interference. In this context, the relay weights are computed based on channel second-order statistics, so that the total relay transmit power is minimized, while meeting certain signal-to-interference-plus-noise-ratio (SINR) constraints at the destinations. First, we show that a computationally efficient approximate solution is attainable by relaxing the original NP-hard non-convex problem employing... |

114 |
Nonlinear Optimization
- Ruszczyński
- 2006
(Show Context)
Citation Context ...y. Such a distributed approach obviates the need for a a central processing unit that has access to the channel statistics of all clusters and obtains the relay weights; centralized approaches do not scale well with the number of network nodes, resulting in high complexity and long delays. Our proposed distributed approach is based on Accelerated Distributed Augmented Lagrangians (ADAL) [29]. ADAL is a distributed optimization method that relies on augmented Lagrangians (AL), a regularization technique that is obtained by adding a quadratic penalty term to the ordinary Lagrangian of a problem [30]. Compared to standard distributed optimization techniques, such as dual decomposition [31], AL methods converge very fast and do not require strict convexity of the objective function [30, 31]. The latter is a necessary feature for our multi-cluster relay beamforming problem, since the objective function under consideration is affine. At the same time, it was shown in [29] that, for a number of different applications, ADAL exhibits a significant improvement in convergence speed compared to existing AL techniques, such as the ADMM [32] and the DQA [33]. We propose two different ways to apply A... |

105 | Transmit beamforming for physical-layer multicasting
- Sidiropoulos, Davidson, et al.
- 2006
(Show Context)
Citation Context ...ng SDR of problem (7) becomes min {Xn,∀n∈N} ∑ n∈N Tr(XnRnT ) (8) s.t. Tr(XnQnnm) + ∑j 6=n j∈N Tr(XjQjnm) ≥ 1, Xn ∈ SL+, ∀n ∈ N , m ∈Mn. Note that, by dropping the rank constraints, we essentially enlarge the feasible set. Hence, in general, the relaxation (8) will only yield an approximate solution to (7), with an optimal value that provides a lower bound for the original problem. Therefore, the optimizers X∗j , ∀j ∈ N of (8) will not be rank-one in general, due to the relaxation. If they are, then they will be the optimal solution to the original problem (7). If not, randomization techniques [37] can be employed to obtain a rank one matrix. Remark 1 Observe that, similar to [4, 14, 15, 20], the above formulation assumes knowledge of the second order statistics of channel state information (CSI). In a practical setting, this can be obtained based on past observations. Remark 2 The inequality constraints in (8) must be active at the optimal solution (satisfied as equalities), because if they were not, we would be able to decrease the magnitudes of Xn further, thus invalidating the optimality assumption. III. DISTRIBUTED RELAY BEAMFORMING Since the beamforming decisions in (8) are couple... |

86 | Network beamforming using relays with perfect channel information,” Information Theory - Jing, Jafarkhani - 2009 |

67 | Distributed beamforming for relay networks based on second-order statistics of the channel state information
- Havary-Nassab, Shahbazpanahi, et al.
- 2008
(Show Context)
Citation Context ...j∈M |gTmWf jsj |2 represents interference at user Um caused by signals intended for other users, the term ||gTmWv||2 denotes noise at the relays that was propagated to the user, and |zm|2 denotes noise at the user level. The expectation in the above equation refers to everything that is random, i.e., signals, channels, noise. Observe that the average SINR is defined as the ratio of the expected values, which is different than the expected value of the ratio. This definition is frequently used in communications textbooks, e.g. [36], and in published works related to the problem considered here [4, 6, 7, 14, 20]. Similar as before, we can manipulate the SINR expression to write it in a more compact matrix form SINRm = Desired︷ ︸︸ ︷ P0w HRmS w P0w HRmI w︸ ︷︷ ︸ Interference + wHRmv w + 1︸ ︷︷ ︸ Noise . The desired signal matrix for user Um is Hermitian RmS = E{(fTm gTm)H(fTm gTm)}, with denoting the Hadamard (entry-wise) product. The corresponding interference matrix is also Hermitian RmI = j 6=m∑ j∈M E{(fTj gTm)H(fTj gTm)}, and the respective noise matrix is diagonal Rmv = diag { E{|g1m|2}, . . . ,E{|gLm|2} } . Utilizing the above notation, the single-cluster optimization problem (2) can be compactly w... |

58 | Rank-constrained separable semidefinite programming with applications to optimal beamforming,” - Huang, Palomar - 2010 |

57 | Optimal downlink beamforming using semidefinite optimization
- Bengtsson, Ottersten
- 1999
(Show Context)
Citation Context ...work, in which multiuser peer-to-peer relay communications take place inside each cluster, while the intra-cluster communications cause inter-cluster interference. In this context, the relay weights are computed based on channel second-order statistics, so that the total relay transmit power is minimized, while meeting certain signal-to-interference-plus-noise-ratio (SINR) constraints at the destinations. First, we show that a computationally efficient approximate solution is attainable by relaxing the original NP-hard non-convex problem employing semidefinite relaxation (SDR) techniques [25]–[28]. Second, we propose a distributed approach to solve the relaxed problem, which allows for each cluster to compute its optimal beamforming weights based on information exchanges with neighboring clusters only. Such a distributed approach obviates the need for a a central processing unit that has access to the channel statistics of all clusters and obtains the relay weights; centralized approaches do not scale well with the number of network nodes, resulting in high complexity and long delays. Our proposed distributed approach is based on Accelerated Distributed Augmented Lagrangians (ADAL) [29... |

40 |
A Diagonal Quadratic Approximation Method for Large Scale Linear Programs. O.R Letts
- MULVEY, RUSZCZvNSKI
- 1992
(Show Context)
Citation Context ...m to the ordinary Lagrangian of a problem [30]. Compared to standard distributed optimization techniques, such as dual decomposition [31], AL methods converge very fast and do not require strict convexity of the objective function [30, 31]. The latter is a necessary feature for our multi-cluster relay beamforming problem, since the objective function under consideration is affine. At the same time, it was shown in [29] that, for a number of different applications, ADAL exhibits a significant improvement in convergence speed compared to existing AL techniques, such as the ADMM [32] and the DQA [33]. We propose two different ways to apply ADAL to the multicluster beamforming problem, termed Direct and Indirect, 2 that allow us to model different message exchange patterns (necessary for the iterative execution of ADAL) between the individual clusters. Specifically, the message exchange pattern in the Direct method is determined by the coupling SINR constraints due to inter-cluster interference. On the other hand, the message exchanges in the Indirect method can be defined arbitrarily by the user. Both approaches rely on transforming the SINR coupling constraints to a linear form by introd... |

34 |
Principles of Digital Communication.
- Gallager
- 2008
(Show Context)
Citation Context ...rence + ||gTmWv||2 + |zm|2︸ ︷︷ ︸ Noise ) , where the term P0 ∑j 6=m j∈M |gTmWf jsj |2 represents interference at user Um caused by signals intended for other users, the term ||gTmWv||2 denotes noise at the relays that was propagated to the user, and |zm|2 denotes noise at the user level. The expectation in the above equation refers to everything that is random, i.e., signals, channels, noise. Observe that the average SINR is defined as the ratio of the expected values, which is different than the expected value of the ratio. This definition is frequently used in communications textbooks, e.g. [36], and in published works related to the problem considered here [4, 6, 7, 14, 20]. Similar as before, we can manipulate the SINR expression to write it in a more compact matrix form SINRm = Desired︷ ︸︸ ︷ P0w HRmS w P0w HRmI w︸ ︷︷ ︸ Interference + wHRmv w + 1︸ ︷︷ ︸ Noise . The desired signal matrix for user Um is Hermitian RmS = E{(fTm gTm)H(fTm gTm)}, with denoting the Hadamard (entry-wise) product. The corresponding interference matrix is also Hermitian RmI = j 6=m∑ j∈M E{(fTj gTm)H(fTj gTm)}, and the respective noise matrix is diagonal Rmv = diag { E{|g1m|2}, . . . ,E{|gLm|2} } . Utilizing t... |

25 |
Multiple peerto-peer communications using a network of relays
- Fazeli-Dehkordy, Shahbazpanahi, et al.
- 2009
(Show Context)
Citation Context ...discuss the single cluster relay beamforming scenario. Then, we formulate the multi-cluster problem and propose to pose it as a convex optimization problem using SDR. In Section III, we present two different ways to obtain a decentralized solution to the convex multi-cluster problem by applying ADAL. Finally, in Section IV, we present simulation results to verify the validity of our approach. II. RELAY BEAMFORMING To facilitate understanding of the multi-cluster scenario, we first formulate the cooperative beamforming problem for a single cluster. The solution for this problem can be found in [14]. Then, in Section II-B we formulate and solve the multicluster problem. A. Single Cluster case In the single cluster scenario, the goal is to allow communication of multiple single-antenna source-destination pairs, which transmit simultaneously using the same channel. The transmission takes place in two stages, i.e., two consecutive time-slots. In the first stage, all sources transmit, while in the second stage the relays retransmit the signals that they received in an AF fashion. A simple case with two source-destination pairs and three relays is depicted in Fig. 1. Consider a network compos... |

21 | Convex optimization-based beamforming: From receive to transmit and network designs,” - Gershman, Sidiropoulos, et al. - 2010 |

18 | A cross-layer approach to collaborative beamforming for wireless ad hoc networks,” - Dong, Petropulu, et al. - 2008 |

16 |
Distributed robust multicell coordinated beamforming with imperfect csi: An admm approach. Signal Processing,
- Shen, Chang, et al.
- 2012
(Show Context)
Citation Context ...pling constraints to a linear form by introducing appropriate auxiliary variables. We show, via numerical experiments, that the Direct method is generally more efficient than the Indirect. However, the flexibility of the Indirect method in selecting the message exchange pattern between clusters might make it more appropriate for certain applications. To the best of our knowledge, there is no prior work showing that the multi-cluster relay beamforming problem is amenable to a decentralized solution. The closest scenarios considered in the literature are those of multi-cell downlink beamfroming [34, 35], which do not involve relays and thus the formulation is considerably simpler; the two AF communication stages of the relay problem that we consider in this paper give rise to several additional interference terms that have to be taken into account. The beamforming weight design in [34] and [35] employs respectively the dual decomposition method and the ADMM. Although one could use similar methods as in [34, 35] to solve our problem, the ADAL method converges faster according to the simulation results presented in Section IV. The rest of the paper is organized as follows: In Section II, we fi... |

15 | On downlink beamforming with indefinite shaping constraints - Hammarwall, Bengtsson, et al. - 2006 |

13 | Distributed beamforming and power allocation for cooperative networks,”
- Ding, Chin, et al.
- 2008
(Show Context)
Citation Context ...anges. Two different approaches are presented, differing in the message exchange patterns between clusters. The performance of the decentralized scheme is demonstrated via simulations. Index Terms—Cooperative beamforming, multi-cluster systems, multi-source multi-destination systems, multiuser peer-topeer relay networks, distributed optimization, augmented Lagrangian. I. INTRODUCTION Cooperative (or relaying) approaches for wireless communications have the potential for significant performance improvement, such as extended coverage of the network, throughput enhancement and energy savings [1]–[21]. In cooperative beamforming, a set of relays form a “virtual antenna array” and retransmit weighted versions of the source signals (decode-and-forward (DF) relaying), or weighted versions of the received signals (amplify-and-forward (AF) relaying). By exploiting constructive interference effects, the relays focus the transmitted power on the destinations’ locations, thus increasing the directional channel gain. By achieving spatial multiplexing, cooperative beamforming can support the Copyright (c) 2014 IEEE. Personal use of this material is permitted. However, permission to use this material... |

11 |
Cooperative transmission for relay networks based on second-order statistics of channel state information
- Li, Petropulu, et al.
- 2011
(Show Context)
Citation Context ...j∈M |gTmWf jsj |2 represents interference at user Um caused by signals intended for other users, the term ||gTmWv||2 denotes noise at the relays that was propagated to the user, and |zm|2 denotes noise at the user level. The expectation in the above equation refers to everything that is random, i.e., signals, channels, noise. Observe that the average SINR is defined as the ratio of the expected values, which is different than the expected value of the ratio. This definition is frequently used in communications textbooks, e.g. [36], and in published works related to the problem considered here [4, 6, 7, 14, 20]. Similar as before, we can manipulate the SINR expression to write it in a more compact matrix form SINRm = Desired︷ ︸︸ ︷ P0w HRmS w P0w HRmI w︸ ︷︷ ︸ Interference + wHRmv w + 1︸ ︷︷ ︸ Noise . The desired signal matrix for user Um is Hermitian RmS = E{(fTm gTm)H(fTm gTm)}, with denoting the Hadamard (entry-wise) product. The corresponding interference matrix is also Hermitian RmI = j 6=m∑ j∈M E{(fTj gTm)H(fTj gTm)}, and the respective noise matrix is diagonal Rmv = diag { E{|g1m|2}, . . . ,E{|gLm|2} } . Utilizing the above notation, the single-cluster optimization problem (2) can be compactly w... |

10 |
Weighted cross-layer cooperative beamforming for wireless networks,”
- Dong, Petropulu, et al.
- 2009
(Show Context)
Citation Context ...the Dept. of Mechanical Engineering and Materials Science, Duke University, Durham, NC, 27708, USA, {n.chatzip,michael.zavlanos}@duke.edu. Yupeng Liu is with Alcatel-Lucent, New Providence, NJ, 07974, USA, yupeng.liu@alcatel-lucent.com. Athina Petropulu is with the Dept. of Electrical and Computer Engineering, Rutgers, the State University of New Jersey, Piscataway, NJ, 08854, USA, athinap@rutgers.edu. communications of multiple, distinct, single-antenna, sourcedestination pairs that overlap both in time and frequency. This scenario is also referred to as multiuser peer-to-peer relay networks [5]–[16]. In general, the per-node throughput capacity of a wireless ad-hoc network reduces rapidly as the network size increases [22]. Therefore, it is often preferable to divide the network nodes into multiple clusters, with each cluster containing nodes which have distinct sub-goals, or are geographically close to each other, e.g., applications involving networks of mobile wireless robots [23, 24]. In this paper we consider a multi-cluster network, in which multiuser peer-to-peer relay communications take place inside each cluster, while the intra-cluster communications cause inter-cluster int... |

9 | Network integrity in mobile robotic networks,” Automatic Control,
- Zavlanos, Ribeiro, et al.
- 2013
(Show Context)
Citation Context ...tgers.edu. communications of multiple, distinct, single-antenna, sourcedestination pairs that overlap both in time and frequency. This scenario is also referred to as multiuser peer-to-peer relay networks [5]–[16]. In general, the per-node throughput capacity of a wireless ad-hoc network reduces rapidly as the network size increases [22]. Therefore, it is often preferable to divide the network nodes into multiple clusters, with each cluster containing nodes which have distinct sub-goals, or are geographically close to each other, e.g., applications involving networks of mobile wireless robots [23, 24]. In this paper we consider a multi-cluster network, in which multiuser peer-to-peer relay communications take place inside each cluster, while the intra-cluster communications cause inter-cluster interference. In this context, the relay weights are computed based on channel second-order statistics, so that the total relay transmit power is minimized, while meeting certain signal-to-interference-plus-noise-ratio (SINR) constraints at the destinations. First, we show that a computationally efficient approximate solution is attainable by relaxing the original NP-hard non-convex problem employing... |

8 |
Distributed peer-to-peer beamforming for multiuser relay networks,”
- Chen, Gershman, et al.
- 2009
(Show Context)
Citation Context ...3 · 108)/(2.4 · 109) = 0.125m which is a reasonable choice for wireless transmissions utilizing ultra high frequency carrier waves (2.4GHz). In all the cases presented below, we have set the initial values of the primal variables to 0, and randomly sampled the dual variables from a uniform distribution in [0, 1]. Note that, different initialization values did not appear to affect the convergence speed significantly. Moreover, the penalty parameter ρ is in general user defined in augmented Lagrangian methods. In our simulations, we have found that fastest convergence is obtained for values ρ ∈ [1, 10], while at the same time preventing ill-conditioning. In Fig. 4 we compare the two proposed methods, Direct and Indirect, for the 2 different setups of Fig. 3. Fig. 3(a) depicts a case with 5 clusters positioned in parallel, while Fig. 3(b) presents a case with 5 clusters but denser spatial positioning of the users. In both scenarios, we consider clusters containing 2 source-destination pairs and 5 relays, i.e., |Mn |= 2 and |Ln |= 5, ∀ n ∈ N , respectively. The same SINR requirement is set for all users at γ = 10dB. For the direct method, we assume that there exists at least one user that suf... |

6 | Distributed beamforming for multi-group multicasting relay networks,” Signal Processing, - Bornhorst, Pesavento, et al. - 2012 |

6 |
Distributed coordinated multi-cell transmission based on dual decomposition,”
- Tolli, Pennanen, et al.
- 2009
(Show Context)
Citation Context ...pling constraints to a linear form by introducing appropriate auxiliary variables. We show, via numerical experiments, that the Direct method is generally more efficient than the Indirect. However, the flexibility of the Indirect method in selecting the message exchange pattern between clusters might make it more appropriate for certain applications. To the best of our knowledge, there is no prior work showing that the multi-cluster relay beamforming problem is amenable to a decentralized solution. The closest scenarios considered in the literature are those of multi-cell downlink beamfroming [34, 35], which do not involve relays and thus the formulation is considerably simpler; the two AF communication stages of the relay problem that we consider in this paper give rise to several additional interference terms that have to be taken into account. The beamforming weight design in [34] and [35] employs respectively the dual decomposition method and the ADMM. Although one could use similar methods as in [34, 35] to solve our problem, the ADAL method converges faster according to the simulation results presented in Section IV. The rest of the paper is organized as follows: In Section II, we fi... |

5 |
Multi-user twoway relay networks with distributed beamforming,”
- Wang, Chen, et al.
- 2011
(Show Context)
Citation Context ...Dept. of Mechanical Engineering and Materials Science, Duke University, Durham, NC, 27708, USA, {n.chatzip,michael.zavlanos}@duke.edu. Yupeng Liu is with Alcatel-Lucent, New Providence, NJ, 07974, USA, yupeng.liu@alcatel-lucent.com. Athina Petropulu is with the Dept. of Electrical and Computer Engineering, Rutgers, the State University of New Jersey, Piscataway, NJ, 08854, USA, athinap@rutgers.edu. communications of multiple, distinct, single-antenna, sourcedestination pairs that overlap both in time and frequency. This scenario is also referred to as multiuser peer-to-peer relay networks [5]–[16]. In general, the per-node throughput capacity of a wireless ad-hoc network reduces rapidly as the network size increases [22]. Therefore, it is often preferable to divide the network nodes into multiple clusters, with each cluster containing nodes which have distinct sub-goals, or are geographically close to each other, e.g., applications involving networks of mobile wireless robots [23, 24]. In this paper we consider a multi-cluster network, in which multiuser peer-to-peer relay communications take place inside each cluster, while the intra-cluster communications cause inter-cluster interfer... |

4 |
Distributed cooperative beamforming in multi-source multi-destination clustered systems,”
- Chatzipanagiotis, Liu, et al.
- 2014
(Show Context)
Citation Context ...j∈M |gTmWf jsj |2 represents interference at user Um caused by signals intended for other users, the term ||gTmWv||2 denotes noise at the relays that was propagated to the user, and |zm|2 denotes noise at the user level. The expectation in the above equation refers to everything that is random, i.e., signals, channels, noise. Observe that the average SINR is defined as the ratio of the expected values, which is different than the expected value of the ratio. This definition is frequently used in communications textbooks, e.g. [36], and in published works related to the problem considered here [4, 6, 7, 14, 20]. Similar as before, we can manipulate the SINR expression to write it in a more compact matrix form SINRm = Desired︷ ︸︸ ︷ P0w HRmS w P0w HRmI w︸ ︷︷ ︸ Interference + wHRmv w + 1︸ ︷︷ ︸ Noise . The desired signal matrix for user Um is Hermitian RmS = E{(fTm gTm)H(fTm gTm)}, with denoting the Hadamard (entry-wise) product. The corresponding interference matrix is also Hermitian RmI = j 6=m∑ j∈M E{(fTj gTm)H(fTj gTm)}, and the respective noise matrix is diagonal Rmv = diag { E{|g1m|2}, . . . ,E{|gLm|2} } . Utilizing the above notation, the single-cluster optimization problem (2) can be compactly w... |

4 | QoS-constrained multiuser peerto-peer amplify-and-forward relay beamforming,” Signal Processing, - Fadel, El-Keyi, et al. - 2012 |

3 |
On the sumrate of amplify-and-forward relay networks with multiple source-destination pairs,”
- Liu, Petropulu
- 2011
(Show Context)
Citation Context |

3 | Large-scale multipair two-way relay networks with distributed af beamforming,” - Ngo, Larsson - 2013 |

3 |
Joint optimization of source power allocation and distributed relay beamforming in multiuser peer-to-peer relay networks,” Signal Processing,
- Cheng, Pesavento
- 2012
(Show Context)
Citation Context ...j 6=n j∈N Tr(XjQjnm) ≥ 1, Xn ∈ SL+, ∀n ∈ N , m ∈Mn. Note that, by dropping the rank constraints, we essentially enlarge the feasible set. Hence, in general, the relaxation (8) will only yield an approximate solution to (7), with an optimal value that provides a lower bound for the original problem. Therefore, the optimizers X∗j , ∀j ∈ N of (8) will not be rank-one in general, due to the relaxation. If they are, then they will be the optimal solution to the original problem (7). If not, randomization techniques [37] can be employed to obtain a rank one matrix. Remark 1 Observe that, similar to [4, 14, 15, 20], the above formulation assumes knowledge of the second order statistics of channel state information (CSI). In a practical setting, this can be obtained based on past observations. Remark 2 The inequality constraints in (8) must be active at the optimal solution (satisfied as equalities), because if they were not, we would be able to decrease the magnitudes of Xn further, thus invalidating the optimality assumption. III. DISTRIBUTED RELAY BEAMFORMING Since the beamforming decisions in (8) are coupled in the constraint set, a central processing unit would have to be employed to gather the data... |

3 | MMSE-based distributed beamforming in cooperative relay networks,” - Choi - 2011 |

2 | QoS guarantees in AF relay networks with multiple source-destination pairs in the presence of imperfect - Liu, Petropulu - 2013 |

2 |
An augmented Lagrangian method for distributed optimization,”
- Chatzipanagiotis, Dentcheva, et al.
- 2014
(Show Context)
Citation Context ...28]. Second, we propose a distributed approach to solve the relaxed problem, which allows for each cluster to compute its optimal beamforming weights based on information exchanges with neighboring clusters only. Such a distributed approach obviates the need for a a central processing unit that has access to the channel statistics of all clusters and obtains the relay weights; centralized approaches do not scale well with the number of network nodes, resulting in high complexity and long delays. Our proposed distributed approach is based on Accelerated Distributed Augmented Lagrangians (ADAL) [29]. ADAL is a distributed optimization method that relies on augmented Lagrangians (AL), a regularization technique that is obtained by adding a quadratic penalty term to the ordinary Lagrangian of a problem [30]. Compared to standard distributed optimization techniques, such as dual decomposition [31], AL methods converge very fast and do not require strict convexity of the objective function [30, 31]. The latter is a necessary feature for our multi-cluster relay beamforming problem, since the objective function under consideration is affine. At the same time, it was shown in [29] that, for a n... |

1 | Joint optimization of source power allocation and relay beamforming in multiuser cooperative wireless networks,” - Li, Zhang, et al. - 2011 |

1 | Joint optimization of source power allocation and cooperative beamforming for SC-FDMA multi-user multirelay networks,” - Kha, Tuan, et al. - 2013 |