Frequency-selective Interference Channel Research Articles

Game Theoretic Dynamic Channel Allocation for Frequency-Selective Interference Channels

We consider the problem of distributed channel allocation in large networks under the frequency-selective interference channel. Performance is measured by the weighted sum of achievable rates. Our proposed algorithm is a modified Fictitious Play algorithm that can be implemented distributedly, and its stable points are the pure Nash equilibria of a given game. Our goal is to design a utility function for a non-cooperative game, such that all of its pure Nash equilibria have close to optimal global performance. This will make the algorithm close to optimal while requiring no communication between users. We propose a novel technique to analyze the Nash equilibria of a random interference game, determined by the random channel gains. Our analysis is asymptotic in the number of users. First, we present a natural non-cooperative game where the utility of each user is his achievable rate. It is shown that, asymptotically in the number of users and for strong enough interference, this game exhibits many bad equilibria. Then, we propose a novel non-cooperative M frequency-selective interference channel game as a slight modification of the former, where the utility of each user is artificially limited. We prove that even its worst equilibrium has asymptotically optimal weighted sum rate for any interference regime and even for correlated channels. This is based on an order statistics analysis of the fading channels that is valid for a broad class of fading distributions (including Rayleigh, Rician, m-Nakagami, and more). We carry out simulations that show fast convergence of our algorithm to the proven asymptotically optimal pure Nash equilibria.

Degrees of Freedom in Vector Interference Channels

This paper continues the Wu-Shamai-Verdu program [3] on characterizing the degrees of freedom (DoF) of interference channels (ICs) through Renyi information dimension. Specifically, we find a single-letter formula for the DoF of vector ICs, encompassing multiple-input multiple-output (MIMO) ICs, time- and/or frequency-selective ICs, and combinations thereof, as well as scalar ICs as considered in [3]. The DoF-formula we obtain lower-bounds the DoF of all channels--with respect to the choice of the channel matrix--and upper-bounds the DoF of almost all channels. It applies to a large class of noise distributions, and its proof is based on an extension of a result by Guionnet and Shlyakthenko [3] to the vector case in combination with the Ruzsa triangle inequality for differential entropy introduced by Kontoyiannis and Madiman [4]. As in scalar ICs, achieving full DoF requires the use of singular input distributions. Strikingly, in the vector case it suffices to enforce singularity on the joint distribution of each individual transmit vector. This can be realized through signaling in subspaces of the ambient signal space, which is in accordance with the idea of interference alignment, and, most importantly, allows the scalar entries of the transmit vectors to have non-singular distributions. The DoF-formula for vector ICs we obtain enables a unified treatment of "classical" interference alignment a la Cadambe and Jafar [5], and Maddah-Ali et al. [6], and the number-theoretic schemes proposed in [7], [8]. Moreover, it allows to calculate the DoF achieved by new signaling schemes for vector ICs. We furthermore recover the result by Cadambe and Jafar on the non-separability of parallel ICs [9] and we show that almost all parallel ICs are separable in terms of DoF. Finally, our results apply to complex vector ICs, thereby extending the main findings of [2] to the complex case.

A Parallel Decomposition Method for Nonconvex Stochastic Multi-Agent Optimization Problems

Consider the problem of minimizing the expected value of a (possibly nonconvex) cost function parameterized by a random (vector) variable, when the expectation cannot be computed accurately (e.g., because the statistics of the random variables are unknown and/or the computational complexity is prohibitive). Classical sample stochastic gradient methods for solving this problem may empirically suffer from slow convergence. In this paper, we propose for the first time a stochastic parallel Successive Convex Approximation-based (best-response) algorithmic framework for general nonconvex stochastic sum-utility optimization problems, which arise naturally in the design of multi-agent systems. The proposed novel decomposition enables all users to update their optimization variables in parallel by solving a sequence of strongly convex subproblems, one for each user. Almost surely convergence to stationary points is proved. We then customize our algorithmic framework to solve the stochastic sum rate maximization problem over Single-Input-Single-Output (SISO) frequency-selective interference channels, multiple-input-multiple-output (MIMO) interference channels, and MIMO multiple-access channels. Numerical results show that our algorithms are much faster than state-of-the-art stochastic gradient schemes while achieving the same (or better) sum-rates.

Optimal Spectrum Management in Multiuser Interference Channels

In this paper, we study the problem of continuous frequency optimal spectrum management in multiuser frequency selective interference channels. We assume that interference is treated as noise by the decoders, and separate encoding is applied. First, a simple pair-wise channel condition for frequency division multiple access schemes to achieve all Pareto optimal points of the rate region is derived. It enables fully distributed global optimal decision making on whether any two users should use orthogonal channels. Next, we present an analytical solution to finding the maximum sum-rate in two-user symmetric frequency flat channels. Generalizing this solution to frequency selective channels, a convex optimization is established that yields the global optimum. Finally, we show that our method generalizes to <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K$</tex></formula> -user <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(K \geq 2)$</tex></formula> weighted sum-rate maximization in asymmetric frequency selective channels, and we transform this classic nonconvex optimization to an equivalent convex optimization in the primal domain.

Joint Sensing and Power Allocation in Nonconvex Cognitive Radio Games: Nash Equilibria and Distributed Algorithms

In this paper, we propose a novel class of Nash problems for cognitive radio (CR) networks, modeled as Gaussian frequency-selective interference channels, wherein each secondary user (SU) competes against the others to maximize his own opportunistic throughput by choosing jointly the sensing duration, the detection thresholds, and the vector power allocation. The proposed general formulation allows us to accommodate several (transmit) power and (deterministic/probabilistic) interference constraints, such as constraints on the maximum individual and/or aggregate (probabilistic) interference tolerable at the primary receivers. To keep the optimization as decentralized as possible, global (coupling) interference constraints are imposed by penalizing each SU with a set of time-varying prices based upon his contribution to the total interference; the prices are thus additional variable to optimize. The resulting players' optimization problems are nonconvex; moreover, there are possibly price clearing conditions associated with the global constraints to be satisfied by the solution. All this makes the analysis of the proposed games a challenging task; none of classical results in the game theory literature can be successfully applied. The main contribution of this paper is to develop a novel optimization-based theory for studying the proposed nonconvex games; we provide a comprehensive analysis of the existence and uniqueness of a standard Nash equilibrium, devise alternative best-response based algorithms, and establish their convergence. Some of the proposed algorithms are totally distributed and asynchronous, whereas some others require limited signaling among the SUs (in the form of consensus algorithms) in favor of better performance; overall, they are thus applicable to a variety of CR scenarios, either cooperative or noncooperative, which allows the SUs to explore the existing tradeoff between signaling and performance.

Conjectural Equilibrium in Multiuser Power Control Games

This paper considers a noncooperative game in which competing users sharing a frequency-selective interference channel selfishly optimize their power allocation in order to improve their achievable rates. Previously, it was shown that a user having the knowledge of its opponents' channel state information can make foresighted decisions and substantially improve its performance compared with the case in which it deploys the conventional iterative water-filling algorithm, which does not exploit such knowledge. This paper discusses how a foresighted user can acquire this knowledge by modeling its experienced interference as a function of its own power allocation. To characterize the outcome of the multiuser interaction, the conjectural equilibrium is introduced, and the existence of this equilibrium for the investigated water-filling game is proven. Importantly, we show that both the Nash equilibrium and the Stackelberg equilibrium are special cases of the conjectural equilibrium. We also develop practical algorithms to form accurate beliefs and select desirable power allocation strategies. Numerical simulations indicate that a foresighted user without any a priori knowledge of its competitors' private information can effectively learn how the other users will respond to its actions, and induce the entire system to an operating point that improves both its own achievable rate as well as the rates of the other participants in the water-filling game.

A New Perspective on Multi-User Power Control Games in Interference Channels

This paper considers the problem of how to allocate power among competing users sharing a frequency-selective interference channel. We model the interaction between selfish users as a non-cooperative game. As opposed to the existing iterative water-filling algorithm that studies the myopic users, this paper studies how a foresighted user, who knows the channel state information and response strategies of its competing users, should optimize its transmission strategy. To characterize this multi-user interaction, the Stackelberg equilibrium is introduced, and the existence of this equilibrium for the investigated noncooperative game is shown. We analyze this interaction in more detail using a simple two-user example, where the foresighted user determines its transmission strategy by solving as a bi-level program which allows him to account for the myopic user's response. It is analytically shown that a foresighted user can improve its performance, if it has the necessary information about its competitors. Since the optimal solution of Stackelberg equilibrium is computationally prohibitive, we propose a practical low-complexity approach based on Lagrangian duality theory. Numerical simulations verify the performance improvements. Possible ways to acquire the required information and to extend the formulation to more than two users are also discussed.

Game theory and the frequency selective interference channel

This paper provides a tutorial overview of game theoretic techniques used for communication over frequency selective interference channels. We discuss both competitive and cooperative techniques. Keywords: Game theory, competitive games, cooperative games, Nash Equilibrium, Nash bargaining solution, Generalized Nash games, Spectrum optimization, distributed coordination, interference channel, multiple access channel, iterative water-filling.

Optimal Power Allocation for Spectrum Sharing in Frequency-Selective Unlicensed Bands

Power allocation is an important issue for spectrum sharing of unlicensed bands, in which multiple unlicensed systems may coexist and operate. Recently some works have been reported on spectrum sharing in frequency-flat unlicensed bands. However, there has not been much work on power allocation for spectrum sharing in frequency-selective unlicensed bands. In this letter, we propose an optimal power allocation strategy on frequency-selective interference channels (IC). This strategy, also referred as general power allocation (GPA), allows the transmission power density to vary within one subcarrier. By showing the duality of frequency-selective and parallel frequency- flat channels, we can therefore compute the achievable rate region of GPA. Our method can be regarded as an extension of previous results for frequency-flat scenarios. It will also be shown that over frequency-selective Gaussian IC, the proposed GPA can achieve larger rate region than conventional power allocation strategies, where the transmission power density on each subcarrier is set equal.

Asynchronous Iterative Water-Filling for Gaussian Frequency-Selective Interference Channels

This paper considers the maximization of information rates for the Gaussian frequency-selective interference channel, subject to power and spectral mask constraints on each link. To derive decentralized solutions that do not require any cooperation among the users, the optimization problem is formulated as a static noncooperative game of complete information. To achieve the so-called Nash equilibria of the game, we propose a new distributed algorithm called asynchronous iterative water-filling algorithm. In this algorithm, the users update their power spectral density (PSD) in a completely distributed and asynchronous way: some users may update their power allocation more frequently than others and they may even use outdated measurements of the received interference. The proposed algorithm represents a unified framework that encompasses and generalizes all known iterative water-filling algorithms, e.g., sequential and simultaneous versions. The main result of the paper consists of a unified set of conditions that guarantee the global converge of the proposed algorithm to the (unique) Nash equilibrium of the game.

The Capacity Region of Frequency-Selective Gaussian Interference Channels Under Strong Interference

This paper presents the capacity region of frequency-selective Gaussian interference channels under the condition of strong interference, assuming an average power constraint per user. First, a frequency-selective Gaussian interference channel is modeled as a set of independent parallel memoryless Gaussian interference channels. Using nonfrequency selective results, the capacity region of frequency-selective Gaussian interference channels under strong interference is expressed mathematically. Exploiting structures inherent in the problem, a dual problem is constructed for each independent memoryless channel, in which both mathematical and numerical analysis are performed. Furthermore, three suboptimal methods are compared to the capacity-achieving coding and power allocation scheme. Iterative waterfilling, a suboptimal scheme, provides close-to-optimum performance and has a distributed coding and power allocation scheme, which are attractive in practice.

A multi-user power control algorithm for digital subscriber lines

The paper investigates the power control problem in a frequency-selective interference channel. A centralized power control algorithm is developed with the objective of maximizing the achievable rate region given an average power constraint for each user. The proposed algorithm is based on the multi-user discrete bit-loading algorithm that considers the power allocation over frequency and users simultaneously. Simulation results for very high-speed digital subscriber line (VDSL) systems show that the proposed algorithm achieves a larger rate region than that achieved by a power control algorithm based on iterative waterfilling.

A multi-user power control algorithm for digital subscriber lines

The paper investigates the power control problem in a frequency-selective interference channel. A centralized power control algorithm is developed with the objective of maximizing the achievable rate region given an average power constraint for each user. The proposed algorithm is based on the multi-user discrete bit-loading algorithm that considers the power allocation over frequency and users simultaneously. Simulation results for very high-speed digital subscriber line (VDSL) systems show that the proposed algorithm achieves a larger rate region than that achieved by a power control algorithm based on iterative waterfilling.

Distributed multiuser power control for digital subscriber lines

This paper considers the multiuser power control problem in a frequency-selective interference channel. The interference channel is modeled as a noncooperative game, and the existence and uniqueness of a Nash equilibrium are established for a two-player version of the game. An iterative water-filling algorithm is proposed to efficiently reach the Nash equilibrium. The iterative water-filling algorithm can be implemented distributively without the need for centralized control. It implicitly takes into account the loop transfer functions and cross couplings, and it reaches a competitively optimal power allocation by offering an opportunity for loops to negotiate the best use of power and frequency with each other. When applied to the upstream power backoff problem in very-high bit-rate digital subscriber lines and the downstream spectral compatibility problem in asymmetric digital subscriber lines, the new power control algorithm is found to give a significant performance improvement when compared with existing methods.