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37
The capacity region of the Gaussian multipleinput multipleoutput broadcast channel
 IEEE TRANS. INF. THEORY
, 2006
"... The Gaussian multipleinput multipleoutput (MIMO) broadcast channel (BC) is considered. The dirtypaper coding (DPC) rate region is shown to coincide with the capacity region. To that end, a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequa ..."
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Cited by 170 (3 self)
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The Gaussian multipleinput multipleoutput (MIMO) broadcast channel (BC) is considered. The dirtypaper coding (DPC) rate region is shown to coincide with the capacity region. To that end, a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality, to show that a superposition of Gaussian codes is optimal for the degraded vector broadcast channel and that DPC is optimal for the nondegraded case. Furthermore, the capacity region is characterized under a wide range of input constraints, accounting, as special cases, for the total power and the perantenna power constraints.
Transmitter Optimization for the MultiAntenna Downlink with PerAntenna Power Constraints
 IEEE Transactions on Signal Processing
, 2007
"... Abstract—This paper considers the transmitter optimization problem for a multiuser downlink channel with multiple transmit antennas at the basestation. In contrast to the conventional sumpower constraint on the transmit antennas, this paper adopts a more realistic perantenna power constraint, bec ..."
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Cited by 57 (5 self)
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Abstract—This paper considers the transmitter optimization problem for a multiuser downlink channel with multiple transmit antennas at the basestation. In contrast to the conventional sumpower constraint on the transmit antennas, this paper adopts a more realistic perantenna power constraint, because in practical implementations each antenna is equipped with its own power amplifier and is limited individually by the linearity of the amplifier. Assuming perfect channel knowledge at the transmitter, this paper investigates two different transmission schemes under the perantenna power constraint: a minimumpower beamforming design for downlink channels with a single antenna at each remote user and a capacityachieving transmitter design for downlink channels with multiple antennas at each remote user. It is shown that in both cases, the perantenna downlink transmitter optimization problem may be transformed into a dual uplink problem with an uncertain noise. This generalizes previous uplink–downlink duality results and transforms the perantenna transmitter optimization problem into an equivalent minimax optimization problem. Further, it is shown that various notions of uplink–downlink duality may be unified under a Lagrangian duality framework. This new interpretation of duality gives rise to efficient numerical optimization techniques for solving the downlink perantenna transmitter optimization problem. Index Terms—Beamforming, broadcast channel, capacity region, dirtypaper coding, Lagrangian duality. I.
Wornell, “Secure transmission with multiple antennas II: The MIMOME wiretap channel
 IEEE Trans. Inf. Theory
"... Abstract—The role of multiple antennas for secure communication is investigated within the framework of Wyner’s wiretap channel. We characterize the secrecy capacity in terms of generalized eigenvalues when the sender and eavesdropper have multiple antennas, the intended receiver has a single antenn ..."
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Cited by 57 (7 self)
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Abstract—The role of multiple antennas for secure communication is investigated within the framework of Wyner’s wiretap channel. We characterize the secrecy capacity in terms of generalized eigenvalues when the sender and eavesdropper have multiple antennas, the intended receiver has a single antenna, and the channel matrices are fixed and known to all the terminals, and show that a beamforming strategy is capacityachieving. In addition, we study a masked beamforming scheme that radiates power isotropically in all directions and show that it attains nearoptimal performance in the high SNR regime. Insights into the scaling behavior of the capacity in the large antenna regime as well as extensions to ergodic fading channels are also provided. Index Terms—Artificial noise, broadcast channel, cryptography, generalized eigenvalues, masked beamforming, MIMO systems, multiple antennas, secrecy capacity, secure spacetime codes, wiretap channel. I.
The MIMOME channel
 in Proceedings of the 45th Annual Allerton Conference on Communication, Control and Computing
, 2007
"... Abstract — The MIMOME channel is a Gaussian wiretap channel in which the sender, receiver, and eavesdropper all have multiple antennas. We characterize the secrecy capacity as the saddlevalue of a minimax problem. Among other implications, our result establishes that a Gaussian distribution maximiz ..."
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Cited by 16 (1 self)
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Abstract — The MIMOME channel is a Gaussian wiretap channel in which the sender, receiver, and eavesdropper all have multiple antennas. We characterize the secrecy capacity as the saddlevalue of a minimax problem. Among other implications, our result establishes that a Gaussian distribution maximizes the secrecy capacity characterization of Csiszár and Körner when applied to the MIMOME channel. We also determine a necessary and sufficient condition for the secrecy capacity to be zero. Large antenna array analysis of this condition reveals several useful insights into the conditions under which secure communication is possible. I.
Degrees of freedom in wireless multiuser spatial multiplex systems with multiple antennas
 IEEE Transactions on Communications
, 2006
"... Abstract—This letter investigates the structure of the optimal spatial multiplex scheme in a multiuser multiantenna wireless fading environment. Based on a sumcapacity criterion, this letter shows that the optimal transmission strategy in an uplink or downlink channel with n antennas at the basest ..."
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Cited by 12 (0 self)
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Abstract—This letter investigates the structure of the optimal spatial multiplex scheme in a multiuser multiantenna wireless fading environment. Based on a sumcapacity criterion, this letter shows that the optimal transmission strategy in an uplink or downlink channel with n antennas at the basestation involves more than n users at the same time. In particular, when remote users are equipped with m antennas each, the maximum number of data streams is shown to be upper bounded by n 2, with each user transmitting or receiving up to m 2 data streams. This gives a dimensioncounting interpretation for multiuser diversity. Multiple antennas at the basestation increases the total number of dimensions, thus allowing more users to transmit and receive at the same time. By contrast, multiple antennas at the remote terminal allow a single user to occupy multiple dimensions, which increases its transmission rate, but also has the potential effect of precluding simultaneous transmission by other users. Index Terms—Broadcast channel, multipleaccess channel, multipleinput multipleoutput (MIMO) system, multiuser diversity, spatial diversity, spatial multiplex. I.
Input optimization for multiantenna broadcast channels and perantenna power constraints
 in Proc. IEEE GLOBECOM
, 2004
"... Abstract — This paper considers a Gaussian multiantenna broadcast channel with individual power constraints on each antenna, rather than the usual sum power constraint over all antennas. Perantenna power constraints are more realistic because in practical implementations each antenna has its own p ..."
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Cited by 9 (0 self)
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Abstract — This paper considers a Gaussian multiantenna broadcast channel with individual power constraints on each antenna, rather than the usual sum power constraint over all antennas. Perantenna power constraints are more realistic because in practical implementations each antenna has its own power amplifier. The main contribution of this paper is a new derivation of the duality result for this class of broadcast channels that allows the input optimization problem to be solved efficiently. Specifically, we show that uplinkdownlink duality is equivalent to Lagrangian duality in minimax optimization, and the dual multipleaccess problem has a much lower computational complexity than the original problem. This duality applies to the entire capacity region. Further, we derive a novel application of Newton’s method for the dual minimax problem that finds an optimal search direction for both the minimization and the maximization problems at the same time. This new computational method is much more efficient than the previous iterative waterfillingbased algorithms and it is applicable to the entire capacity region. Finally, we show that the previous QRbased precoding method can be easily modified to accommodate the perantenna constraint. I.
Conjugate Gradient Projection Approach for MultiAntenna Gaussian Broadcast Channels
"... It has been shown recently that the dirtypaper coding is the optimal strategy for maximizing the sum rate of multipleinput multipleoutput Gaussian broadcast channels (MIMO BC). Moreover, by the channel duality, the nonconvex MIMO BC sum rate problem can be transformed to the convex dual MIMO mult ..."
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Cited by 9 (4 self)
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It has been shown recently that the dirtypaper coding is the optimal strategy for maximizing the sum rate of multipleinput multipleoutput Gaussian broadcast channels (MIMO BC). Moreover, by the channel duality, the nonconvex MIMO BC sum rate problem can be transformed to the convex dual MIMO multipleaccess channel (MIMO MAC) problem with a sum power constraint. In this paper, we design an efficient algorithm based on conjugate gradient projection (CGP) to solve the MIMO BC maximum sum rate problem. Our proposed CGP algorithm solves the dual sum power MAC problem by utilizing the powerful concept of Hessian conjugacy. We also develop a rigorous algorithm to solve the projection problem. We show that CGP enjoys provable convergence, nice scalability, and great efficiency for large MIMO BC systems. 1
Precoding for the multiantenna downlink: Multiuser SNR gap and optimal user ordering
 IEEE Trans. Commun
, 2007
"... Abstract—This paper develops a practical design method for implementing Tomlinson–Harashima precoding (THP) in a downlink channel with multiple antennas at the transmitter and a single antenna at each receiver. A twostep design process is proposed for minimizing the total transmit power while satis ..."
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Cited by 8 (1 self)
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Abstract—This paper develops a practical design method for implementing Tomlinson–Harashima precoding (THP) in a downlink channel with multiple antennas at the transmitter and a single antenna at each receiver. A twostep design process is proposed for minimizing the total transmit power while satisfying every user’s minimum data rate and maximum biterror rate (BER) requirements. First, the BER and rate requirements are converted to “virtual rate ” requirements, which account for the gaptocapacity introduced by practical quadrature amplitude modulation (QAM) and THP. The second step is to determine the transmit covariance matrices (which specify the entire THP system) that will provide these virtual rates at the minimum total transmit power. As one of the main features in the proposed scheme, an algorithm for finding the optimal user encoding (or presubtraction) order in polynomial time is proposed. In addition, we also propose an algorithm that finds a nearoptimal order, but which is much less complex. The proposed method outperforms existing zeroforcingbased THP systems in term of power efficiency. Index Terms—Broadcast channel (BC), multipleaccess channel (MAC), multiple antennas, signaltonoise ratio (SNR) gap approximation, spatial multiplex, Tomlinson–Harashima precoding (THP), uplink–downlink duality. I.
Spatial Multiplex in Downlink Multiuser MultipleAntenna Wireless Environments, in
 Proc. of IEEE Global Communications Conf. (GLOBECOM
, 2003
"... Abstract — This paper studies the optimal spatial multiplex scheme in a downlink multipleantenna environment with perfect transmitter and receiver channel knowledge. Using recent results on the sum capacity of the Gaussian vector broadcast channel, the optimal number of precoded data streams in a d ..."
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Cited by 5 (0 self)
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Abstract — This paper studies the optimal spatial multiplex scheme in a downlink multipleantenna environment with perfect transmitter and receiver channel knowledge. Using recent results on the sum capacity of the Gaussian vector broadcast channel, the optimal number of precoded data streams in a downlink channel is characterized. The main result is the following: the sumcapacity achieving transmission strategy in a random downlink channel with n transmit antennas at the basestation and K receivers each equipped with m antennas involves between n to 1 n(n +1) 2 data streams in total, with each user receiving between m to 1 m(m +1) data streams. This gives a dimension count2 ing interpretation for multiuser diversity. In particular, it shows that the throughput maximizing transmission strategy in a downlink channel with n transmit antennas should involve between n to 1 n(n +1) active users at any time. 2 I.
Improved Capacity Scaling in Wireless Networks With Infrastructure
, 2008
"... This paper analyzes the impact and benefits of infrastructure support in improving the throughput scaling in networks of n randomly located wireless nodes. The infrastructure uses multiantenna base stations (BSs), in which the number of BSs and the number of antennas at each BS can scale at arbitra ..."
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Cited by 5 (4 self)
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This paper analyzes the impact and benefits of infrastructure support in improving the throughput scaling in networks of n randomly located wireless nodes. The infrastructure uses multiantenna base stations (BSs), in which the number of BSs and the number of antennas at each BS can scale at arbitrary rates relative to n. Two schemes are introduced in this study: a BSbased singlehop routing protocol with multipleaccess uplink and broadcast downlink and a BSbased multihop routing protocol. Then, the throughput scaling laws of each are analyzed here. These schemes are compared against two conventional schemes without BSs: the multihop (MH) transmission and hierarchical cooperation (HC) schemes. It is shown that the BSbased routing schemes do not improve the throughput scaling in dense networks. In contrast, the proposed BSbased routing schemes can, under realistic