Gaussian MIMO Broadcast Channel Research Articles

On the Sum Capacity Achieving Precoders and Their CSI Requirements for MIMO-BC

Dirty paper coding (DPC) precoders achieve the sum capacity of Gaussian MIMO broadcast channels. However, most systems employ zero-forcing (ZF) and MMSE precoders that are near-optimal at high SNR if the base station (BS) has a large number of antennas, but achievable sum rates with ZF and MMSE precoders are significantly below sum capacity otherwise. We show that capacity achieving DPC precoding requires CSI that scales only as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(M^{2})$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -antenna user (not full CSI as is generally assumed) irrespective of the number of BS antennas. Moreover, easily computable generalized decision feedback equalizers significantly outperform MMSE and ZF precoders.

Secret Key Generation From Vector Gaussian Sources With Public and Private Communications

In this paper, we consider the problem of secret key generation with one-way communication through both a rate-limited public channel and a rate-limited secure channels where the public channel is from Alice to Bob and Eve and the secure channel is from Alice to Bob. In this model, we do not pose any constraints on the sources, i.e. Bob is not degraded to or less noisy than Eve. We obtain the optimal secret key rate in this problem, both for the discrete memoryless sources and vector Gaussian sources. The vector Gaussian characterization is derived by suitably applying the enhancement argument, and proving a new extremal inequality. The extremal inequality can be seen as coupling of two extremal inequalities, which are related to the degraded compound MIMO Gaussian broadcast channel, and the vector generalization of Costa's entropy power inequality, accordingly.

MIMO Gaussian Broadcast Channels With Common, Private, and Confidential Messages

The two-user multiple-input multiple-output Gaussian broadcast channel with common, private, and confidential messages is considered. The transmitter sends a common message to both users, a confidential message to the User 1 and a private (non-confidential) message to the User 2. The secrecy-capacity region is characterized by showing that certain inner and outer bounds coincide and that the boundary points are achieved by Gaussian inputs, which enables the development of a tight converse. The proof relies on the factorization of upper concave envelopes and a variant of dirty-paper coding (DPC). It is shown that the entire region is exhausted by using DPC to cancel out the signal of the non-confidential message at Receiver 1, thus making DPC against the signal of the confidential message unnecessary. A numerical example illustrates the secrecy-capacity results.

Converse Results for the Downlink Multicell Processing With Finite Backhaul Capacity

In this paper, we study outer bounds on the capacity region of the downlink multicell processing model with finite backhaul capacity for the simple case of two base stations and two mobile users. It is modeled as a two-user multiple access diamond channel. It consists of a first hop from the central processor to the base stations via orthogonal links of finite capacity and the second hop from the base stations to the mobile users via a Gaussian interference channel. The outer bound is derived using the converse tools of the multiple access diamond channel and that of the Gaussian MIMO broadcast channel. Through numerical results, it is shown that our outer bound improves upon the existing outer bounds greatly in the medium backhaul capacity range, and as a result, the gap between the outer bounds and the rate of the time-sharing of the known achievable schemes is significantly reduced.

On the Capacity Region of the Parallel Degraded Broadcast Channel With Three Receivers and Three-Degraded Message Sets

We consider a broadcast channel with three receivers and three-degraded message sets, i.e., the transmitter has a common message intended for all three receivers, a message intended for receivers 2 and 3, and a private message intended only for receiver 3. The messages are transmitted over a family of parallel degraded broadcast channels. In the most general case, the broadcast channel consists of the product of six parallel degraded broadcast channels, each with a different order of degradedness. We first consider an achievable rate region of Nair and El Gamal, by appropriately choosing independent input random variables and auxiliary random variables for each subchannel. We then show that the achievable rate region attains the capacity region for two different classes of such broadcast channels, one consisting of the product of five parallel degraded broadcast channels and another consisting of the product of three parallel degraded broadcast channels. To accomplish this, we make use of an information-theoretic inequality that may be proven using the Csiszar-sum identity. Next, we extend the result to the Gaussian case. We consider the aligned Gaussian MIMO broadcast channel consisting of the product of six parallel degraded Gaussian broadcast channels, where the Gaussian noise vectors for each of the users in each of the subchannels follow a degradedness order, i.e., the noise covariance matrices may be ordered in a positive semi-definite sense. We show that the Nair–El Gamal achievable rate region considered in this paper is maximized by Gaussian inputs. To prove that Gaussian inputs are optimal, we prove an extremal entropy inequality employing a new method recently introduced by Geng and Nair to prove the capacity region of the two-user Gaussian MIMO broadcast channel with common and private messages.

Iterative Waterfilling With User Selection in Gaussian MIMO Broadcast Channels

We consider a sum rate maximization problem with user scheduling in Gaussian MIMO broadcast channels, which is a combinatorial optimization problem. We transform it into a cardinality problem taking into account that selecting inactive users is equivalent to allocating zero power to unselected users. Then, we relax the cardinality constraint by introducing a penalty function to promote sparse power allocation among users. The proposed iterative waterfilling with user selection algorithm is a generalization of the well-known iterative waterfilling that maximizes the sum rate under a sum power constraint in MIMO broadcast channels. Numerical results show that it achieves a very high sum rate with a moderate complexity only proportional to the number of users.

The DoF Region of the Three-Receiver Gaussian MIMO Broadcast Channel With Receiver Message Side Information

We consider the three-receiver Gaussian multiple-input multiple-output broadcast channel with an arbitrary number of antennas at the transmitter and the receivers. We investigate the degrees-of-freedom (DoF) region of the channel when each receiver requests a private message, and may know some of the messages requested by the other receivers as receiver message side information (RMSI). We establish the DoF region of the channel for all 16 possible non-isomorphic RMSI configurations by deriving tight inner and outer bounds on the region. To derive the inner bounds, we first propose a scheme for each RMSI configuration, which exploits both the null space and the side information of the receivers. We then use these schemes in conjunction with time sharing for 15 RMSI configurations, and with time sharing and two-symbol extension for the remaining one. To derive the outer bounds, we construct enhanced versions of the channel for each RMSI configuration, and upper bound their DoF region. After establishing the DoF region, in the case where all the nodes have the same number of antennas, we introduce some common properties of the DoF region, and the capacity region of the index coding problem.

On the Role of Transmit Correlation Diversity in Multiuser MIMO Systems

Correlation across transmit antennas, in multiple antenna systems (MIMO), has been studied in various scenarios and has been shown to be detrimental or provide benefits depending on the particular system and underlying assumptions. In this paper, we investigate the effect of transmit correlation on the capacity of the Gaussian MIMO broadcast channel (BC), with a particular interest in the large-scale array (or massive MIMO) regime. To this end, we introduce a new type of diversity, referred to as transmit correlation diversity, which captures the fact that the channel vectors of different users may have different, and often nearly mutually orthogonal, large-scale channel eigen-directions. In particular, when taking the cost of downlink training properly into account, transmit correlation diversity can yield significant capacity gains in all regimes of interest. Our analysis shows that the system multiplexing gain can be increased by a factor up to $\lfloor{M}/{r}\rfloor$, where $M$ is the number of antennas and $r\le M$ is the common rank of the users transmit correlation matrices, with respect to standard schemes that are agnostic of the transmit correlation and treat the channels as if they were isotropically distributed. Thus, this new form of diversity reveals itself as a valuable "new resource" in multiuser communications.

An Extremal Inequality and the Capacity Region of the Degraded Compound Gaussian MIMO Broadcast Channel With Multiple Users

The two-user compound Gaussian MIMO broadcast channel models the situation where each user has a finite set of possible realizations. The transmitter sends two messages, one for each user, such that each user must be able to decode its message regardless of the actual realization. This channel also models a broadcast channel (BC) with two groups of users and two messages, with one message intended for each group of users. Weingarten et al. established the capacity region for the degraded case where the realizations/users exhibit a degradedness order. The degradedness order is defined through an additional realization/user where the realizations/users from one set are degraded with respect to him and where he is degraded with respect to the realizations/users from the other set. To show that Gaussian inputs attain the capacity region, they proved a new extremal inequality and employed the use of the channel enhancement technique as well. In this paper, we extend the result to the N-user degraded compound Gaussian MIMO BC, where the N users exhibit a degradedness order similar to the two-user case. We first prove a generalization of the extremal inequality considered by Weingarten et al.; instead of considering the difference between the weighted sum of two sets of conditional differential entropies, we consider the summation of N - 1 sets of such differences, where the conditioning random variables of the N - 1 sets form a Markov chain. Our proof relies on the Gaussian perturbation approach, the necessary KKT conditions as well as a data processing inequality. Finally, we specialize the generalized extremal inequality to characterize the capacity region of the N-user degraded compound Gaussian MIMO BC. By making appropriate use of the necessary KKT conditions, we are able to do away with the use of the channel enhancement technique that was employed in the proof of the capacity region of the two-user case.

Corrections to “Vector Precoding for Gaussian MIMO Broadcast Channels: Impact of Replica Symmetry Breaking” [Mar 12 1413-1440

There are a number of corrections for the above titled paper (ibid., vol. 58, no. 3, pp. 1413-1440, Mar. 2012). They are presented here.

Vector Precoding for Gaussian MIMO Broadcast Channels: Impact of Replica Symmetry Breaking

The “replica method” of statistical physics is employed for the large-system analysis of vector precoding for the Gaussian multiple-input multiple-output broadcast channel. The transmitter comprises a linear front-end combined with nonlinear precoding, minimizing transmit energy by means of input alphabet relaxation. For the common discrete lattice-based relaxation, the problem violates replica symmetry and a replica symmetry breaking (RSB) ansatz is taken. The limiting empirical distribution of the precoder's output and the limiting transmit energy are derived for one-step RSB. Particularizing to a “zero-forcing” (ZF) linear front-end, a decoupling result is derived. For discrete lattice-based relaxations, the impact of RSB is demonstrated for the transmit energy. The spectral efficiencies of the aforementioned precoding methods are compared to linear ZF and Tomlinson-Harashima precoding (THP). Focusing on quaternary phase shift-keying (QPSK), significant performance gains of both lattice and convex relaxations are revealed for medium to high signal-to-noise ratios (SNRs) when compared to linear ZF precoding. THP is shown to be outperformed as well. Comparing certain lattice-based relaxations for QPSK against a convex counterpart, the latter is found to be superior for low and high SNRs but slightly inferior for medium SNRs in terms of spectral efficiency.

Cognitive Radio MIMO Gaussian Broadcast Channels with the Power Constraint

The cognitive radio multiple-input multiple-output Gaussian broadcast channels are studied where multiple antennas are available for both primary users and secondary users in a spectrum sharing environment, and the sum-rate capacity is also obtained under both the SUs’ transmit power constraint and interference power constraint at the primary receivers. The paper principally consists of two steps. First, a duality technique and dirty paper coding are adopted to simplify the channels. Second, we propose an iterative power allocation algorithm to obtain the maximum sum-rate capacity and examine the effects of the constraint parameters on the concerned quantities. Finally, numerical simulation results are presented to validate the proposed theoretical analysis.

Correction of Convergence Proof for Iterative Water-Filling in Gaussian MIMO Broadcast Channels

This paper first identifies a flaw in the proof of the convergence theorem of Algorithm 2 of the paper by Jindal [“Sum power iterative water-filling for multi-antenna Gaussian broadcast channels,” IEEE Trans. Inf. Theory, vol. 54, no. 4, pp. 1570-1580, Apr. 2005] and then presents a corrected convergence proof for that algorithm.

Closed-Form Sum-MSE Minimization for the Two-User Gaussian MIMO Broadcast Channel

We study the Gaussian multiple-input, multiple-output broadcast channel, where a base station with N_T antennas transmits K independent messages to K users, each having a single receive antenna. The messages consist of independent, identically distributed Gaussian random variables and we study linear transmission with an end-to-end distortion criterion. By using an already established uplink/downlink duality and a recently discovered special relation between beamforming vectors and channel vectors, we present a closed-form expression for the optimal power allocation in the two-user case. We also outline an iterative algorithm that finds the optimal power allocation for an arbitrary number of users.

Secret Key Agreement From Vector Gaussian Sources by Rate Limited Public Communication

We investigate the secret key agreement from correlated vector Gaussian sources in which the legitimate parties can use the public communication with limited rate. For the class of protocols with the one-way public communication, we show that the optimal trade-off between the rate of key generation and the rate of the public communication is characterized as an optimization problem of a Gaussian random variable. The characterization is derived by using the enhancement technique introduced by Weingarten et.al. for MIMO Gaussian broadcast channel.

Reduced Feedback Schemes Using Random Beamforming in MIMO Broadcast Channels

A random beamforming scheme for the Gaussian MIMO broadcast channel with channel quality feedback is investigated and extended. Considering the case where the n receivers each have N receive antennas, the effects of feeding back various amounts of signal-to-interference-plus-noise ratio (SINR) information are analyzed. Using the results from order statistics of the ratio of a linear combination of exponential random variables, the distribution function of the maximum order statistic of the SINR observed at the receiver is found. The analysis from viewing each antenna as an individual user is extended to allow combining at the receivers, where it is known that the linear MMSE combiner is the optimal linear receiver and the CDF for the SINR after optimal combining is derived. Analytically, using the Delta Method, the asymptotic distribution of the maximum order statistic of the SINR with and without combining is shown to be, in the nomenclature of extreme order statistics, of type 3. The throughput of the feedback schemes are shown to exhibit optimal scaling asymptotically in the number of users. Finally, to further reduce the amount of feedback, a hard threshold is applied to the SINR feedback. The amount of feedback saved by implementing a hard threshold is determined and the effect on the system throughput is analyzed and bounded.

The Gaussian MIMO Broadcast Channel under Receive Power Protection Constraints

A Gaussian MIMO broadcast channel (GMBC) models the MIMO transmission of Gaussian signals from a transmitter to one or more receivers. Its capacity region and different precoding schemes for it have been well investigated, especially for the case wherein there are only transmit power constraints. In this paper, a special case of GMBC is investigated, wherein receive power constraints are also included. By imposing receive power constraints, the model, called protected GMBC (PGMBC), can be applied to certain scenarios in spatial spectrum sharing, secretive communications, mesh networks and base station cooperation. The sum capacity, capacity region, and application examples for the PGMBC are discussed in this paper. Sub-optimum precoding algorithms are also proposed for the PGMBC, where standard user precoding techniques are performed over a BC with a modified channel, which we refer to as the “protection-implied BC.” In the protection-implied BC, the receiver protection constraints have been implied in the channel, which means that by satisfying the transmit power constraints on the protection implied channel, receiver protection constraints are guaranteed to be met. Any standard single-user or multi-user MIMO precoding scheme may then be performed on the protection-implied channel. When SINR-matching duality-based precoding is applied on the protection-implied channel, sum-capacity under full protection constraints (zero receive power), and near-sum-capacity under partial protection constraints (limited non-zero receive power) are achieved, and were verified by simulations.

On the optimal transmit strategy for the MIMO bidirectional broadcast channel

In this work the transmit covariance matrix optimization problem for the discrete memoryless MIMO Gaussian bidirectional broadcast channel is studied. A half-duplex relay node establishes bidirectional communication between two nodes using a decode-and-forward protocol. In the initial multiple access phase both nodes transmit their messages to the relay node. In the succeeding phase the relay broadcasts an optimal re-encoded message so that both nodes can decode the other's message using their own message as side information. The capacity region of the bidirectional broadcast channel is completely characterized by a weighted rate sum maximization problem, which can be solved by a simple iterative fixed point algorithm. If an efficient transmit covariance matrix is invariant with respect to the joint subspace spanned by the channels, then different combinations of the part transmitted on the orthogonal subspaces result in equivalent transmit strategies with different ranks. A closed-form procedure to obtain the optimal transmit covariance is derived for the case where the rank of the channels is equal to the number of antennas at the relay node and a full-rank transmission is optimal. It shows the complicated structure of the optimal eigenspace, which depends on the weights and the mean transmit power constraint. For parallel channels the optimal solution is completely characterized and discussed, which also solves the optimal power allocation problem for a single-antenna OFDM system.

Linear beamforming and superposition coding with common information for the gaussian MIMO broadcast channel

A Gaussian multiple-input multiple-output broadcast channel (MIMO GBC) is considered. Throughout the paper it is assumed that 1) input signals are Gaussian and 2) perfect channel state information is available at the transmitter and at the receivers. By considering each data stream as a single user, the uplink-downlink signal-to-interference-plus-noise (SINR) duality is generalized to the MIMO case with general cross-talk matrix. The duality is subsequently applied to finding the solution for the SINR-balancing problem. The result serves as a tool for characterizing achievable rate regions of different coding strategies. Next, we investigate a superposition coding scheme proposed by Cover-van der Meulen-Hajek and Pursley (nicknamed CMHP), where there is a common message to both users. We consider a MIMO broadcast channel with two users, each user has two antennas and the transmitter has four antennas. Assuming one common stream is sent by CMHP coding and successive decoding, a lower bound to the CMHP rate region is found. Behaviors of the CMHP rate region and sumrate are analyzed. We find the sumrate gaps between DPC, CMHP, and MMSE at high SNR for general 2-user multiple-input singleoutput (MISO) Gaussian broadcast channel. The result suggests when CMHP is beneficial for sumrate.

How much does transmit correlation affect the sum-rate scaling of MIMO gaussian broadcast channels?

This paper considers the effect of spatial correlation between transmit antennas on the sum-rate capacity of the MIMO Gaussian broadcast channel (i.e., downlink of a cellular system). Specifically, for a system with a large number of users n, we analyze the scaling laws of the sum-rate for the dirty paper coding and for different types of beamforming transmission schemes. When the channel is i.i.d., it has been shown that for large n, the sum rate is equal to M log log n + M log P/M + o(1) where M is the number of transmit antennas, P is the average signal to noise ratio, and o(1) refers to terms that go to zero as n rarr infin. When the channel exhibits some spatial correlation with a covariance matrix R (non-singular with tr(R) = M), we prove that the sum rate of dirty paper coding is M log log n + M log P/M + log det(R) + o(1). We further show that the sum-rate of various beamforming schemes achieves M log log n + M log P/M + M log c + o(1) where c les 1 depends on the type of beamforming. We can in fact compute c for random beamforming proposed in and more generally, for random beamforming with preceding in which beams are pre-multiplied by a fixed matrix. Simulation results are presented at the end of the paper.