A reconfigurable iterative algorithm for the K-user MIMO interference channel

Highlights

• The algorithm adjusts itself to the interference levels and channel conditions.

• The algorithm performs interference alignment in the interference-limited regime.

• When interference is weak, the algorithm performs interference-myopic transmissions.

• Sum-rate results are presented over various asymmetric Ricean fading channels.

• The algorithm outperforms many well-known techniques for all tested cases.

Abstract

Interference alignment (IA) is a recently proposed transmission technique for the K-user interference channel (IFC), which is proven to achieve a sum-rate multiplexing gain of $K/2$ at the high interference regime. Motivated by our recent work [1] that showed how the sum-rate scaling can range between $K/2$ and K for moderate-to-low interference conditions, in this paper we present a novel iterative algorithm for the K-user multiple-input multiple-output (MIMO) IFC with arbitrary number of transceiver antennas. The proposed algorithm automatically adjusts itself to the interference regime at hand, in the above sense, as well as to the wireless channel in order to achieve the appropriate sum-rate scaling. Our reconfigurable algorithm combines the system-wide mean squared error minimization criterion with the single-user waterfilling solution to maximize each user's transmission rate according to the interference levels and channel conditions. Extensive computer simulation results for the sum-rate performance of the proposed reconfigurable algorithm over various Ricean fading channels are presented. It is shown that, in the interference-limited regime, the proposed algorithm reconfigures itself so as to achieve the IA scaling whereas, in the moderate-to-low interference regime, it chooses interference-myopic MIMO transmissions for all K communication pairs.

Introduction

Recent results on the characterization of the capacity region of the K-user interference channel (IFC) [2] have shown that the capacity of wireless systems can be substantially higher than previously believed [3], [4], [5]. Specifically, it has been shown in [5] that for K pairs of interfering users operating under symmetric Rayleigh fading channels with equal average powers, a sum-rate multiplexing gain of $K/2$ is feasible at the high interference regime. The feasibility of this astonishing result has been accomplished with interference alignment (IA) [5]. IA is a transmission technique that is based on appropriate linear precoding at the transmitters, aiming at complete post-receiver processing interference elimination, and requires only global channel state information (CSI) to be available at all participating transceivers.

Since the introduction of the principle of IA, several research works exploited the space dimension offered by multiple-input multiple-output (MIMO) systems to perform IA and investigated the feasibility of IA solutions for the K-user MIMO IFC (see e.g. [6], [7], [8], [9], [10] and references therein). For the special case of K=3 a closed-form solution for the IA-achieving precoding matrices was presented in [5], [9] that was further processed in [11], [12], [13] for increasing the sum-rate performance. The authors in [7] proposed a method for achieving IA in the K-user N×N constant MIMO IFC for the special case where $K=N+1$. However, for $K>3$ MIMO communication pairs with arbitrary number of transceiver antennas closed-form solutions for IA are in general unknown and several centralized as well as distributed iterative algorithms have been recently proposed, e.g. in [12], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. The vast majority of those algorithms targets at implicitly achieving IA through the optimization of one or more constrained objective functions. For example, Gomadam et al. [14] presented two distributed iterative IA algorithms that exploit the reciprocity of wireless channels and require only local CSI at each transceiver. The one algorithm tries to minimize the overall interference leakage in the IFC, whereas the second one attempts to maximize the signal-to-interference-plus-noise ratio (SINR) for every transmitted data stream. The minimization of the overall interference in the IFC has been also considered in the centralized approaches proposed in [17], [15], [16], where alternating minimization procedures [27] were adopted for obtaining all transceiver filters. Furthermore, Peters and Heath Jr. [16] presented two additional IA-achieving iterative algorithms; one that was based on the system-wide minimum mean squared error (MMSE) criterion [28] and another one aiming at the maximization of a global SINR-based objective function that accounts for the SINRs of all transmitted data streams in the IFC. An iterative algorithm that jointly minimizes the total interference leakage and maximizes the sum rate was presented in [18]. In [20] the original IA problem [5] was reformulated as a rank constrained rank minimization problem. As it was shown, the rank minimization formulation guarantees that the interference spaces collapse to the smallest dimensional subspaces possible. A one-sided IA approach that does not require channel reciprocity and runs at the transmitters side only was presented in [22] which eliminates the need for synchronization between each communication pair. The authors in [23] considered the interfering MIMO broadcast channel and presented an iterative algorithm for maximizing the weighted sum-rate (WSR). That algorithm was based on the iterative minimization of a matrix-weighted system-wide mean squared error (MSE). A linear transceiver design that maximizes WSR and utilizes deterministic annealing to track the WSR at any desired SINR was presented in [24]. Recently, a semidistributed iterative algorithm that maximizes the weighted sum of a utility of SINR values for each data stream in the K-user MIMO IFC was proposed in [26]. This algorithm is based on linear MMSE receive filters and utilizes semidefinite programming to compute the optimum transmit covariance matrices.

Although IA attains the optimum sum-rate scaling at the high interference regime, there are certain combinations of interference levels and channel conditions where it does not [1], [29], [30], [31]. For example, it was shown in [29] for the 2-user single-input single-output Gaussian IFC that the optimum sum-rate scaling in a regime with very weak interference is achievable by treating interference as noise. The authors in [30] characterized the sum capacity of the $(N+1)$-user $1\times N$ single-input multiple-output Gaussian IFC for the symmetric case where all intended links have the same signal-to-noise ratio (SNR) and all interference links have the same interference-to-noise ratio. It was shown that there are again certain regimes where it is more preferable in terms of sum-rate performance to treat interference as noise. In [31] the low interference regime for the 2-user MIMO Gaussian IFC was studied and the authors analyzed conditions on the intended and interference links under which using Gaussian inputs and treating interference as noise at the receivers is sum-capacity achieving. Recently, the sum-rate performance results of [1] for the downlink of a K-user MIMO cellular network with asymmetric average powers and line-of-sight (LOS) conditions among the intended and the interference links demonstrated certain regimes where interference-myopic MIMO transmissions yield superior sum-rate performance than IA. Under those interference-myopic MIMO transmissions each transmitter treats interference as noise and utilizes the transmit covariance matrix obtained from the waterfilling (WF) solution [32] for its intended single-user MIMO channel.

From all the above it is obvious that to achieve the optimum sum-rate scaling for the K-user MIMO IFC one must devise a transmission scheme that is reconfigurable to the interference levels as well as to the wireless channel conditions. To the best of our knowledge, the majority of the available transmission techniques for the K-user IFC need to know a priori the interference levels so as to choose between the two extremes: treating interference as noise or performing IA. In addition, there are certain low-to-moderate interference levels where none of the latter techniques achieves the appropriate sum-rate scaling and/or chooses the appropriate number of data streams that can be reliably transmitted in the IFC. In this paper we present a centralized¹ iterative algorithm for the K-user MIMO IFC with arbitrary number of transceiver antennas. Inspired by the iterative algorithms of [14] and the findings in [1], we first investigate certain regimes where IA is suboptimal and interference-myopic MIMO transmissions yield superior performance. We then present a transceiver design that combines the system-wide MMSE criterion with the single-user WF solution to maximize each user's transmission rate according to the interference levels and channel conditions. Extensive computer simulation results are presented for the sum-rate performance of the proposed algorithm as well as its convergence characteristics.

The remainder of this paper is organized as follows. Section 2 outlines the system and channel model. In Section 3 the IA conditions and representative algorithms for achieving them are described whereas, Section 4 presents the proposed reconfigurable iterative algorithm. In Section 5 computer simulated sum-rate performance results for all presented algorithms are demonstrated along with a relevant discussion. Finally, Section 6 concludes the paper.

Notations: Throughout this paper vectors and matrices are denoted by boldface lowercase letters and boldface capital letters, respectively. The transpose conjugate and the determinant of matrix $\mathbf{A}$ are denoted by ${\mathbf{A}}^{\mathrm{H}}$ and $\det (\mathbf{A})$, respectively. Moreover, ${[\mathbf{A}]}_{i,j}$ represents the (i,j)-element of $\mathbf{A}$, $\mathrm{span}(\mathbf{A})$ its column span, $\mathrm{Tr}\{\mathbf{A}\}$ its trace and ${\mathbf{A}}^{(n)}$ denotes the nth column of $\mathbf{A}$. In addition, ${\mathbf{I}}_n$ denotes the n×n identity matrix, ${\mathbf{0}}_{m\times n}$ is the m×n zeros matrix and $\mathrm{diag}\{\mathbf{a}\}$ represents a diagonal matrix with vector $\mathbf{a}$ in its main diagonal. Notation $\parallel \mathbf{a}\parallel$ stands for the Euclidean norm of $\mathbf{a}$ and ${\parallel \mathbf{A}\parallel }_{\mathrm{F}}$ is the Frobenius norm of $\mathbf{A}$. The expectation operator is denoted as $\mathbb{E}\{·\}$ whereas, notations $X\sim \mathcal{N}(\mu ,{\sigma }^2)$ and $X\sim \mathcal{CN}(\mu ,{\sigma }^2)$ represent a random variable X following the normal and complex normal distribution, respectively, with mean $\mu$ and variance ${\sigma }^2$.