Vector Gaussian Broadcast Channel Research Articles

The Capacity Region of the Two-Receiver Gaussian Vector Broadcast Channel With Private and Common Messages

A novel method for establishing the optimality of Gaussian auxiliary random variables in multiterminal information theory problems is developed. This method is then employed to show that Marton's inner bound achieves the capacity region of the two-receiver Gaussian vector broadcast channel with private and common messages.

Weighted Sum Rate Maximization for the Two-User Vector Gaussian Broadcast Channel

In this letter, the problem of determining the maximum weighted sum rate (WSR) for given arbitrary weights is investigated for the vector Gaussian broadcast channel (GBC). Although the solution to the problem in general involves an iterative algorithm, we focus on the important special case of the two-user channel to provide a closed-form expression for the maximum WSR. The achievable rate region is known to be identical for both the vector GBC and its dual uplink Gaussian multiple access channel (GMAC) due to the uplink-downlink duality. We first derive the optimal power allocation in the dual uplink channel, which is then applied to the capacity formula to obtain the desired result. The presented result is a generalization of the (unweighted) sum rate maximization of the two-user GBC provided by Caire and Shamai .

A Complete Description of the QoS Feasibility Region in the Vector Broadcast Channel

We characterize the complete quality-of-service (QoS) feasibility region and present a simple feasibility test for given QoS requirements in the Gaussian vector broadcast channel. While most contributions in the literature recast the QoS constraints into requirements for the signal-to-interference-and-noise ratios (SINRs), we convert them into upper bounds for the minimum mean square errors (MMSEs) instead and test feasibility in the MMSE domain. Our main contribution is a complete description of the feasible MMSE region. Its closure is shown to be a polytope and we find the complete set of its bounding half-spaces noniteratively after a finite number of steps. The polytope can easily be converted into any other QoS domain like SINR or rate. However, the simple geometry of the MMSE domain is lost in other domains. Once the bounding half-spaces are determined, any target MMSE tuple can quickly be checked for feasibility by verifying its membership to the interior of the polytope. For nondegenerate channels, the only relevant bounding half-space is essentially the lower bound on the sum mean square error. No further computations are necessary contrary to existing feasibility checks which iteratively solve eigenproblems in an alternating optimization framework for every single QoS requirement to test. For two particular user/antenna configurations, we find a noniterative closed form solution for the optimum power allocation of the signal-to-interference ratio (SIR) balancing.

A Vector Generalization of Costa's Entropy-Power Inequality With Applications

This paper considers an entropy-power inequality (EPI) of Costa and presents a natural vector generalization with a real positive semidefinite matrix parameter. The new inequality is proved using a perturbation approach via a fundamental relationship between the derivative of mutual information and the minimum mean-square error (MMSE) estimate in linear vector Gaussian channels. As an application, a new extremal entropy inequality is derived from the generalized Costa EPI and then used to establish the secrecy capacity regions of the degraded vector Gaussian broadcast channel with layered confidential messages.

Code design for MIMO downlink with imperfect CSIT

In this letter, we implement a simplified version of the Cover van der Meulen Hajek Pursley (CMHP) coding originally characterized by Wajcer, Wiesel, and Shamai. The vector Gaussian broadcast channel with imperfect channel state information at the transmitter (CSIT) is considered where the transmitter only knows the channel mean and variance. Our focus is on the implementation and performance analysis of CMHP under the imperfect CSIT model using practical codes. Turbo codes described in IEEE 802.20 draft specification and quadrature amplitude modulation are used to implement CMHP. In order to find the optimal power allocation and beamforming vectors which maximize the sum rate with practical codes, we introduce the SINR penalty factor. The SNRs that achieve various target spectral efficiency are presented and analyzed.

An Extremal Inequality Motivated by Multiterminal Information-Theoretic Problems

We prove a new extremal inequality, motivated by the vector Gaussian broadcast channel and the distributed source coding with a single quadratic distortion constraint problems. As a corollary, this inequality yields a generalization of the classical entropy-power inequality (EPI). As another corollary, this inequality sheds insight into maximizing the differential entropy of the sum of two dependent random variables

Uplink-downlink duality via minimax duality

The sum capacity of a Gaussian vector broadcast channel is the saddle point of a minimax Gaussian mutual information expression where the maximization is over the set of transmit covariance matrices subject to a power constraint and the minimization is over the set of noise covariance matrices subject to a diagonal constraint. This sum capacity result has been proved using two different methods, one based on decision-feedback equalization and the other based on a duality between uplink and downlink channels. This paper illustrates the connection between the two approaches by establishing that uplink-downlink duality is equivalent to Lagrangian duality in minimax optimization. This minimax Lagrangian duality relation allows the optimal transmit covariance and the least-favorable-noise covariance matrices in a Gaussian vector broadcast channel to be characterized in terms of the dual variables. In particular, it reveals that the least favorable noise is not unique. Further, the new Lagrangian interpretation of uplink-downlink duality allows the duality relation to be generalized to Gaussian vector broadcast channels with arbitrary linear constraints. However, duality depends critically on the linearity of input constraints. Duality breaks down when the input constraint is an arbitrary convex constraint. This shows that the minimax representation of the broadcast channel sum capacity is more general than the uplink-downlink duality representation.

Sum-capacity computation for the Gaussian vector broadcast channel via dual decomposition

A numerical algorithm for the computation of sum capacity for the Gaussian vector broadcast channel is proposed. The sum capacity computation relies on a duality between the Gaussian vector broadcast channel and the sum-power constrained Gaussian multiple-access channel. The numerical algorithm is based on a Lagrangian dual decomposition technique and it uses a modified iterative water-filling approach for the Gaussian multiple-access channel. The algorithm converges to the sum capacity globally and efficiently.

Isotropic Fading Vector Broadcast Channels: The Scalar Upper Bound and Loss in Degrees of Freedom

We propose a scalar upper bound on the capacity region of the isotropic fading vector broadcast channel in terms of the capacity region of a scalar fading broadcast channel. The scalar upper bound is applicable to the broad class of isotropic fading broadcast channels regardless of the distribution of the users' channel magnitudes, the distribution of the additive noise experienced by each user, or the amount of channel knowledge available at the receiver. Using this upper bound, we prove the optimality of the Alamouti scheme in a broadcast setting, extend the recent results on the capacity of nondegraded, fading scalar broadcast channels to nondegraded fading vector broadcast channels, and determine the capacity region of a fading vector Gaussian broadcast channel with channel magnitude feedback. We also provide an example of a Rayleigh-fading broadcast channel with no channel state information available to the receiver (CSIR), where the bound on the capacity region obtained by a naive application of the scalar upper bound is provably loose, because it fails to account for the additional loss in degrees of freedom due to lack of channel knowledge at the receiver. A tighter upper bound is obtained by separately accounting for the loss in degrees of freedom due to lack of CSIR before applying the scalar upper bound.

Sum Capacity of Gaussian Vector Broadcast Channels

This paper characterizes the sum capacity of a class of potentially nondegraded Gaussian vector broadcast channels where a single transmitter with multiple transmit terminals sends independent information to multiple receivers. Coordination is allowed among the transmit terminals, but not among the receive terminals. The sum capacity is shown to be a saddle-point of a Gaussian mutual information game, where a signal player chooses a transmit covariance matrix to maximize the mutual information and a fictitious noise player chooses a noise correlation to minimize the mutual information. The sum capacity is achieved using a precoding strategy for Gaussian channels with additive side information noncausally known at the transmitter. The optimal precoding structure is shown to correspond to a decision-feedback equalizer that decomposes the broadcast channel into a series of single-user channels with interference pre-subtracted at the transmitter.

Multiuser diversity for a dirty paper approach

Multiuser diversity has attracted significant attention recently. In this letter, we propose a greedy algorithm based on QR decomposition of the channel and dirty paper precoding to exploit the multiuser diversity of the Gaussian vector broadcast channel. Simulations show the approach provides performance which is extremely close to a well-known upper bound on the sum rate. Further, exploiting multiuser diversity can provide large gain over approaches ignoring this resource.

Sum capacity of the vector Gaussian broadcast channel and uplink–downlink duality

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 (MAC) (where the role of transmitters and receivers are reversed), and the corresponding point-to-point channel (where the transmitters can cooperate).