Optimal Multiuser Transmit Beamforming: A Difficult Problem with a Simple Solution Structure

Transmit beamforming is a versatile technique for signal transmission from an array of $N$ antennas to one or multiple users [1]. In wireless communications, the goal is to increase the signal power at the intended user and reduce interference to non-intended users. A high signal power is achieved by transmitting the same data signal from all antennas, but with different amplitudes and phases, such that the signal components add coherently at the user. Low interference is accomplished by making the signal components add destructively at non-intended users. This corresponds mathematically to designing beamforming vectors (that describe the amplitudes and phases) to have large inner products with the vectors describing the intended channels and small inner products with non-intended user channels. While it is fairly easy to design a beamforming vector that maximizes the signal power at the intended user, it is difficult to strike a perfect balance between maximizing the signal power and minimizing the interference leakage. In fact, the optimization of multiuser transmit beamforming is generally a nondeterministic polynomial-time (NP) hard problem [2]. Nevertheless, this lecture shows that the optimal transmit beamforming has a simple structure with very intuitive properties and interpretations. This structure provides a theoretical foundation for practical low-complexity beamforming schemes. (See this lecture note for the complete abstract/introduction)Comment: Accepted for publication as lecture note in IEEE Signal Processing Magazine, 11 pages, 3 figures. The results can be reproduced using the following Matlab code: https://github.com/emilbjornson/optimal-beamformin

Constrained Bayesian Active Learning of Interference Channels in Cognitive Radio Networks

In this paper, a sequential probing method for interference constraint learning is proposed to allow a centralized Cognitive Radio Network (CRN) accessing the frequency band of a Primary User (PU) in an underlay cognitive scenario with a designed PU protection specification. The main idea is that the CRN probes the PU and subsequently eavesdrops the reverse PU link to acquire the binary ACK/NACK packet. This feedback indicates whether the probing-induced interference is harmful or not and can be used to learn the PU interference constraint. The cognitive part of this sequential probing process is the selection of the power levels of the Secondary Users (SUs) which aims to learn the PU interference constraint with a minimum number of probing attempts while setting a limit on the number of harmful probing-induced interference events or equivalently of NACK packet observations over a time window. This constrained design problem is studied within the Active Learning (AL) framework and an optimal solution is derived and implemented with a sophisticated, accurate and fast Bayesian Learning method, the Expectation Propagation (EP). The performance of this solution is also demonstrated through numerical simulations and compared with modified versions of AL techniques we developed in earlier work.Comment: 14 pages, 6 figures, submitted to IEEE JSTSP Special Issue on Machine Learning for Cognition in Radio Communications and Rada

Robust Monotonic Optimization Framework for Multicell MISO Systems

The performance of multiuser systems is both difficult to measure fairly and to optimize. Most resource allocation problems are non-convex and NP-hard, even under simplifying assumptions such as perfect channel knowledge, homogeneous channel properties among users, and simple power constraints. We establish a general optimization framework that systematically solves these problems to global optimality. The proposed branch-reduce-and-bound (BRB) algorithm handles general multicell downlink systems with single-antenna users, multiantenna transmitters, arbitrary quadratic power constraints, and robustness to channel uncertainty. A robust fairness-profile optimization (RFO) problem is solved at each iteration, which is a quasi-convex problem and a novel generalization of max-min fairness. The BRB algorithm is computationally costly, but it shows better convergence than the previously proposed outer polyblock approximation algorithm. Our framework is suitable for computing benchmarks in general multicell systems with or without channel uncertainty. We illustrate this by deriving and evaluating a zero-forcing solution to the general problem.Comment: Published in IEEE Transactions on Signal Processing, 16 pages, 9 figures, 2 table

Successive Convex Approximation Algorithms for Sparse Signal Estimation with Nonconvex Regularizations

In this paper, we propose a successive convex approximation framework for sparse optimization where the nonsmooth regularization function in the objective function is nonconvex and it can be written as the difference of two convex functions. The proposed framework is based on a nontrivial combination of the majorization-minimization framework and the successive convex approximation framework proposed in literature for a convex regularization function. The proposed framework has several attractive features, namely, i) flexibility, as different choices of the approximate function lead to different type of algorithms; ii) fast convergence, as the problem structure can be better exploited by a proper choice of the approximate function and the stepsize is calculated by the line search; iii) low complexity, as the approximate function is convex and the line search scheme is carried out over a differentiable function; iv) guaranteed convergence to a stationary point. We demonstrate these features by two example applications in subspace learning, namely, the network anomaly detection problem and the sparse subspace clustering problem. Customizing the proposed framework by adopting the best-response type approximation, we obtain soft-thresholding with exact line search algorithms for which all elements of the unknown parameter are updated in parallel according to closed-form expressions. The attractive features of the proposed algorithms are illustrated numerically.Comment: submitted to IEEE Journal of Selected Topics in Signal Processing, special issue in Robust Subspace Learnin

Optimality Properties, Distributed Strategies, and Measurement-Based Evaluation of Coordinated Multicell OFDMA Transmission

The throughput of multicell systems is inherently limited by interference and the available communication resources. Coordinated resource allocation is the key to efficient performance, but the demand on backhaul signaling and computational resources grows rapidly with number of cells, terminals, and subcarriers. To handle this, we propose a novel multicell framework with dynamic cooperation clusters where each terminal is jointly served by a small set of base stations. Each base station coordinates interference to neighboring terminals only, thus limiting backhaul signalling and making the framework scalable. This framework can describe anything from interference channels to ideal joint multicell transmission. The resource allocation (i.e., precoding and scheduling) is formulated as an optimization problem (P1) with performance described by arbitrary monotonic functions of the signal-to-interference-and-noise ratios (SINRs) and arbitrary linear power constraints. Although (P1) is non-convex and difficult to solve optimally, we are able to prove: 1) Optimality of single-stream beamforming; 2) Conditions for full power usage; and 3) A precoding parametrization based on a few parameters between zero and one. These optimality properties are used to propose low-complexity strategies: both a centralized scheme and a distributed version that only requires local channel knowledge and processing. We evaluate the performance on measured multicell channels and observe that the proposed strategies achieve close-to-optimal performance among centralized and distributed solutions, respectively. In addition, we show that multicell interference coordination can give substantial improvements in sum performance, but that joint transmission is very sensitive to synchronization errors and that some terminals can experience performance degradations.Comment: Published in IEEE Transactions on Signal Processing, 15 pages, 7 figures. This version corrects typos related to Eq. (4) and Eq. (28

Energy efficiency optimization in MIMO interference channels: A successive pseudoconvex approximation approach

In this paper, we consider the (global and sum) energy efficiency optimization problem in downlink multi-input multi-output multi-cell systems, where all users suffer from multi-user interference. This is a challenging problem due to several reasons: 1) it is a nonconvex fractional programming problem, 2) the transmission rate functions are characterized by (complex-valued) transmit covariance matrices, and 3) the processing-related power consumption may depend on the transmission rate. We tackle this problem by the successive pseudoconvex approximation approach, and we argue that pseudoconvex optimization plays a fundamental role in designing novel iterative algorithms, not only because every locally optimal point of a pseudoconvex optimization problem is also globally optimal, but also because a descent direction is easily obtained from every optimal point of a pseudoconvex optimization problem. The proposed algorithms have the following advantages: 1) fast convergence as the structure of the original optimization problem is preserved as much as possible in the approximate problem solved in each iteration, 2) easy implementation as each approximate problem is suitable for parallel computation and its solution has a closed-form expression, and 3) guaranteed convergence to a stationary point or a Karush-Kuhn-Tucker point. The advantages of the proposed algorithm are also illustrated numerically.Comment: submitted to IEEE Transactions on Signal Processin

Receive Combining vs. Multi-Stream Multiplexing in Downlink Systems with Multi-Antenna Users

In downlink multi-antenna systems with many users, the multiplexing gain is strictly limited by the number of transmit antennas $N$ and the use of these antennas. Assuming that the total number of receive antennas at the multi-antenna users is much larger than $N$, the maximal multiplexing gain can be achieved with many different transmission/reception strategies. For example, the excess number of receive antennas can be utilized to schedule users with effective channels that are near-orthogonal, for multi-stream multiplexing to users with well-conditioned channels, and/or to enable interference-aware receive combining. In this paper, we try to answer the question if the $N$ data streams should be divided among few users (many streams per user) or many users (few streams per user, enabling receive combining). Analytic results are derived to show how user selection, spatial correlation, heterogeneous user conditions, and imperfect channel acquisition (quantization or estimation errors) affect the performance when sending the maximal number of streams or one stream per scheduled user---the two extremes in data stream allocation. While contradicting observations on this topic have been reported in prior works, we show that selecting many users and allocating one stream per user (i.e., exploiting receive combining) is the best candidate under realistic conditions. This is explained by the provably stronger resilience towards spatial correlation and the larger benefit from multi-user diversity. This fundamental result has positive implications for the design of downlink systems as it reduces the hardware requirements at the user devices and simplifies the throughput optimization.Comment: Published in IEEE Transactions on Signal Processing, 16 pages, 11 figures. The results can be reproduced using the following Matlab code: https://github.com/emilbjornson/one-or-multiple-stream

Task-Based Information Compression for Multi-Agent Communication Problems with Channel Rate Constraints

A collaborative task is assigned to a multiagent system (MAS) in which agents are allowed to communicate. The MAS runs over an underlying Markov decision process and its task is to maximize the averaged sum of discounted one-stage rewards. Although knowing the global state of the environment is necessary for the optimal action selection of the MAS, agents are limited to individual observations. The inter-agent communication can tackle the issue of local observability, however, the limited rate of the inter-agent communication prevents the agent from acquiring the precise global state information. To overcome this challenge, agents need to communicate their observations in a compact way such that the MAS compromises the minimum possible sum of rewards. We show that this problem is equivalent to a form of rate-distortion problem which we call the task-based information compression. We introduce a scheme for task-based information compression titled State aggregation for information compression (SAIC), for which a state aggregation algorithm is analytically designed. The SAIC is shown to be capable of achieving near-optimal performance in terms of the achieved sum of discounted rewards. The proposed algorithm is applied to a rendezvous problem and its performance is compared with several benchmarks. Numerical experiments confirm the superiority of the proposed algorithm.Comment: 13 pages, 9 figure

Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization

In this paper, we propose an inexact block coordinate descent algorithm for large-scale nonsmooth nonconvex optimization problems. At each iteration, a particular block variable is selected and updated by inexactly solving the original optimization problem with respect to that block variable. More precisely, a local approximation of the original optimization problem is solved. The proposed algorithm has several attractive features, namely, i) high flexibility, as the approximation function only needs to be strictly convex and it does not have to be a global upper bound of the original function; ii) fast convergence, as the approximation function can be designed to exploit the problem structure at hand and the stepsize is calculated by the line search; iii) low complexity, as the approximation subproblems are much easier to solve and the line search scheme is carried out over a properly constructed differentiable function; iv) guaranteed convergence of a subsequence to a stationary point, even when the objective function does not have a Lipschitz continuous gradient. Interestingly, when the approximation subproblem is solved by a descent algorithm, convergence of a subsequence to a stationary point is still guaranteed even if the approximation subproblem is solved inexactly by terminating the descent algorithm after a finite number of iterations. These features make the proposed algorithm suitable for large-scale problems where the dimension exceeds the memory and/or the processing capability of the existing hardware. These features are also illustrated by several applications in signal processing and machine learning, for instance, network anomaly detection and phase retrieval