Interference Alignment as a Rank Constrained Rank Minimization

This paper investigates the linear precoder design for multiple-input multiple-output (MIMO) K-user interference channels with finite alphabet inputs. We first obtain the general explicit expressions of the achievable rate of each user in MIMO interference channel systems. We study optimal transmission strategies in both high signal-to-noise ratio (SNR) and low SNR regions. We show that given finite alphabet inputs, a simple power allocation design can achieve optimal performance. In contrast, the well-known interference alignment technique for Gaussian input scenarios, only utilizes a partial interference-free signal space for transmission and leads to a constant performance loss when it is applied to finite-alphabet input scenarios. We determine this constant rate loss at high SNR. Moreover, we establish necessary conditions for the linear precoder design of the weighted sum-rate maximization. We also develop an efficient iterative algorithm for determining precoding matrices of all the users. Our numerical results show that for the practical digital modulated signals from discrete constellations, the proposed iterative algorithm achieves considerably higher sum-rate than the existing methods.

Interference alignment (IA) with symbol extensions in the quasi-static flat-fading K-user multipleinput multiple-output (MIMO) interference channel (IC) is considered in this paper. In general, long symbol extensions are required to achieve the optimal fractional degrees of freedom (DOF). However, long symbol extensions over orthogonal dimensions produce structured (diagonal or block diagonal) channel matrices from transmitters to receivers. Most of existing approaches are limited in cases where the channels have some special structures, because they align the interference without preserving the dimensionality of the desired signal explicitly. To overcome this common drawback of most existing IA algorithms, two novel iterative algorithms for IA with symbol extensions are proposed. The first algorithm designs transceivers for IA based on the mean square error (MSE) criterion which minimizes the total MSE of the system while preserving the dimensionality of the desired signal. The novel IA algorithm is a constrained optimization problem which can be solved by Lagrangian method. Its convergence is proven as well. Utilizing the reciprocity of alignment, the second algorithm is proposed based on the maximization of the multidimensional case of the generalized Rayleigh Quotient. It maximizes each receiver’s signal to interference plus noise ratio (SINR) while preserving the dimensionality of the desired signal. In simulation results, we show the superiority of the proposed algorithms in terms of four aspects, i.e., average sum rate, the fraction of the interfering signal power in the desired signal subspace, bit error rate (BER) and the relative power of the weakest desired data stream.

Interference alignment(IA) with symbol extensions in the K-user multiple-input multiple-output (MIMO) interference channel(IC) is considered in this paper. Symbol extensions produce the channels of special structure. Most of existing approaches are limited in cases where the channels have some special structure, because they align the interference without preserving the dimensionality of the desired signal. For that reason, two novel iterative algorithms for IA with symbol extensions are proposed. The first algorithm designs transceivers for IA based on minimizing the maximum per-user mean square error(MSE) while preserving the dimensionality of the desired signal. Utilizing channel reciprocity, the second algorithm is proposed which is the constrained optimization problem. It maximizes each receiver's SINR while preserving the dimensionality of the desired signal. The simulation results show that the proposed algorithms not only achieve good performance in terms of BER performance but also achieve good performance on maximizing sum rate of the system.

In this work we investigate the degrees of freedom (DoF) performance of the multiple-input multiple-output (MIMO) interference channel (IC) with limited channel state information (CSI) feedback. We generalise two space-time interference alignment (STIA) algorithms to the K user MIMO IC with a transmit power constraint and show how these schemes exploit both current and outdated CSI to achieve an improved DoF performance. We show that by combining the STIA schemes with traditional methods major DoF gain can be achieved for a variety of limited CSI feedback scenarios in comparison to what would be obtained via the use of the traditional techniques on their own. In particular we prove that optimal DoF can still be achieved in situations where current CSI is unavailable for a derived fraction of the time. Finally we validate the promised DoF results via system simulation and show how these DoF gains reflect in terms of sum rate achievable.

We show that the three-user M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> × M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> Multiple-Input Multiple-Output (MIMO) interference channel where each transmitter is equipped with M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> antennas and each receiver is equipped with M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> antennas has (M) d(M, N) = <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sup> min (M/2-1/κ, N/2+1/κ) degrees of freedom (DoF) normalized by time, frequency, and space dimensions, where = <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sup> min(M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> , M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> ), N = <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sup> max(M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> , M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> ), κ = <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sup> ⌈M/N-M⌉. While the DoF outer bound of d(M, N) is established for every M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> , M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> value, the achievability of d(M, N) DoF is established in general subject to a normalization with respect to spatial extensions, i.e., the scaling of the number of antennas at all nodes. In particular, we show that qd(M, N) DoF are achievable for the three-user qMT × qMR MIMO interference channel, for some positive integer q, which may be seen as a spatial extension factor. q is the scaling factor needed to make the value qd(M, N) an integer. Given spatial extensions, the achievability relies only on linear beamforming based interference alignment schemes and requires neither channel extensions nor channel variations in time or frequency. In the absence of spatial extensions, it is shown through examples how essentially the same interference alignment scheme may be applied over time-extensions over either constant or time-varying channels. The central new insight to emerge from this paper is the notion of subspace alignment chains as the DoF bottlenecks. The subspace alignment chains are instrumental both in identifying the extra dimensions to be provided by a genie to a receiver for the DoF outer bound, as well as in the construction of the optimal interference alignment schemes. The DoF value d(M, N) is a piecewise linear function of M, N, with either M or N being the bottleneck within each linear segment, whereas the other value contains some redundancy, i.e., it can be reduced without reducing the DoF. The corner points of these piecewise linear segments correspond to two sets, A={1/2, 2/3, 3/4, ...}and B={1/3, 3/5, 5/7, ...}. The set A contains all those values of M/N and only those values of M/N for which there is redundancy in both M and N, i.e., either can be reduced without reducing the DoF. The set B contains all those values of M/N and only those values of M/N for which there is no redundancy in either M or N, i.e., neither can be reduced without reducing the DoF. Because A and B represent settings with maximum and minimum redundancy, essentially they are the basis for the DoF outer bounds and inner bounds, respectively. Our results settle the question of feasibility of linear interference alignment, introduced previously by Cenk et al., for the three-user M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> × M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> MIMO interference channel, completely for all values of M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> , M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> . In particular, we show that the linear interference alignment problem (M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> × M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> , d) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> (as defined in previous paper by Cenk et al.) is feasible if and only if d ≤ d(M, N). With the exception of the values M/N ∈ B, and only with that exception, we show that for every M/N value there are proper systems (as defined by Cenk et al.) that are not feasible. Evidently the redundancy contained in all other values of M/N manifests itself as superfluous variables that are not discounted in the definition of proper systems, thus creating a discrepancy between proper and feasible systems. Our results show that M/N ∈ A are the only values for which there is no DoF benefit of joint processing among co-located antennas at the transmitters or receivers. This may also be seen as a consequence of the maximum redundancy in the M/N ∈ A settings.

The multiple-input-multiple-output (MIMO) interference channel (IC) is considered with constant channel coefficients, N transmit and receive antennas at each user, and coordinated multipoint (CoMP) transmission, in which each message is jointly transmitted by Mt successive transmitters. It is shown that the degrees of freedom with CoMP transmission using a two-stage scheme involving both zero forcing and interference alignment can be significantly greater than without CoMP transmission. In particular, all K users can achieve d degrees of freedom almost surely if d ≤ ⌊(2N)/(K - M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> + 1)⌋ and M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> ≤ K - 2. Additionally, the first M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> - 1 users can achieve d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> degrees of freedom, and the last K - M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> + 1 users can achieve d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> degrees of freedom almost surely if d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ≤ ⌊(2N)/(K -M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> +1)⌋, M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> ≤ K - 2, and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> + (K - M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> + 1) d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ≤ 2N. Next, the analysis of the constant MIMO IC with CoMP transmission is applied to the development of iterative algorithms to design transmit and receive vectors analogous to those without CoMP transmission. The degrees of freedom analysis is shown through simulations to provide a good guideline for choosing how many beams each user should send for the iterative algorithms. Furthermore, it is seen in the numerical results that CoMP transmission can result in greatly improved convergence speed for the iterative interference alignment algorithms.

We consider a two-user multiple-input multiple-output (MIMO) interference channel (IC), where a single data stream is transmitted and each receiver applies the minimum mean square error (MMSE) filter. In this paper, we study an open problem on the computation of the Pareto boundary of the achievable rate region by optimal joint transmit/receive beamforming design. The Pareto boundary can be divided by two turning points into the weak Pareto boundary and the strict Pareto boundary. The weak Pareto boundary and turning points can be computed exactly. For the strict Pareto boundary, we propose a computationally efficient method called iterative alternating algorithm (IAA) to maximize the rate of one user while the rate of the other user is fixed. Numerical simulations show that the IAA provides a better lower bound on the strict Pareto boundary compared with the existing methods.

This paper proposes an algorithm to compute the transmit beamformers for linear interference alignment for the MIMO interference channel and the MIMO interfering multiple-access/broadcast channel without symbol extensions. We first formulate the interference alignment problem as a rank minimization problem with linear constraints, then approximate the matrix rank by the nuclear norm. We further propose the use of an iterative reweighted nuclear norm approach and show that adaptive reweighting can significantly improve the algorithm's ability to find aligned beamformers. Simulation results show that the proposed algorithm is able to provide more interference-free dimensions and also converges faster than a previously proposed rank-constrained rank-minimization approach for interference alignment.

In this paper, we present uplink interference alignment (IA) schemes for two-cell multiple-input–multiple-output (MIMO) channels. Assuming that each cell has $K$ interfering users, we propose an IA scheme for the full interference channel (FIC) and derive the achievable degrees of freedom (DOFs). The proposed scheme is extended to a two-cell Z-interference channel (ZIC), where only one of the two cells is subject to interference. The achievable DOF for the two-cell ZIC channel is also derived, and it is compared with that of the FIC with the IA and that of the conventional orthogonal scheme based on numerical results.

In this paper, we consider a new design problem of optimizing a linear transceiver for correlated multiple-input multiple-output (MIMO) interference channels in the presence of channel state information (CSI) errors, which is a more realistic and practical scenario than those considered in the previous studies on uncorrelated MIMO interference channels. By taking CSI errors into account, the optimization problem is initially formulated to minimize the average mean square error (MSE) under the general power constraints. Since the objective function is not jointly convex in precoders and receive filters, we split the original problem into two convex subproblems, and then linear precoders and receive filters are obtained by solving two subproblems iteratively. It is shown that the proposed algorithm is guaranteed to converge to a local minimum. The numerical results show that the proposed algorithm can significantly reduce the sensitivity to CSI errors compared with the existing robust schemes in the correlated MIMO interference channel.

In this paper, the weighted sum outage-rate maximization for time-invariant multiple-input multiple-output (MIMO) interference channels is considered. The cumulative distribution function (CDF) of outage events in MIMO interference channels is characterized under the assumption of Gaussian distribution for channel uncertainty. Based on the derived expression for the outage probability an iterative beam design algorithm for maximizing the weighted sum outage-rate under outage constraints is newly proposed. Numerical results show that the proposed algorithm shows good sum outage-rate performance.

Interference alignment (IA) is a suggested technique for multiple input multiple output (MIMO) based multipath interference channels to achieve large multiplexed gain. CSI (Channel state information) is required by IA schemes to design precoding vectors at the transmitter side. There will be significant capacity overhead in feedback link. This paper concentrates on the MIMO channels which are having overtime-correlated differential CSI feedback employed by IA design. Utmost multiplexing gain is obtained by the lowest amount of differential feedback. Because of restricted differential feedback, the interference in the upper bound is avoided by using IA system through differential CSI feedback. The analytical performance of the proposed algorithm is done using simulation results and achieved better Sum rate by using normal differential feedback and by using differential feedback through the Lloyd's algorithm.

When Multi-Input Multi-Output (MIMO) radar and communication transceivers are co-existing and operating simultaneously in the same frequency band, interference can be managed by designing signal spaces that facilitate spectrum sharing. In this paper, a spatial precoder-decoder design based on Interference Alignment (IA) is proposed assuming that an interference channel is shared by K c communication users and K r radar users. In order to find an IA based precoder-decoder solution guaranteeing desired multiplexing gain or diversity order, the design problem is formulated using a rank minimization criterion with rank constraint for the communication subsystem and rank minimization criterion with l 0 norm constraint for the radar, respectively. To deal with the non-convex nature of the optimization problem, we relax the problem by using nuclear norm. A generalized likelihood ratio test for target detection and a maximum likelihood estimator for target direction parameter are derived. The performance of the proposed IA based precoder-decoder design is analyzed using the GLRT detector and the ML estimator for MIMO radar. Analytical and simulation results show that radar detection performance is similar to the interference-free case with desired diversity order if the decoder matrix is semi-unitary and IA is perfect.

It is already well-known that interference alignment (IA) achieves the sum capacity of the K-user interference channel at the high interference regime. On the other hand, it is intuitively clear that when the interference levels are very low, a sum-rate scaling of K (as opposed to K/2 for IA) should be accessed at high signal-to-noise ratio values by simple ("myopic") singlelink multiple-input multiple-output (MIMO) techniques such as waterfilling. Recent results have indicated that in certain low-to-moderate interference cases, treating interference as noise may in fact be preferable. In this paper, we present a distributed iterative algorithm for K-user MIMO interference networks which attempts to adjust itself to the interference regime at hand, in the above sense, as well as to the channel conditions. The proposed algorithm combines the system-wide mean squared error minimization with the waterfilling solution to adjust to the interference levels and channel conditions and maximize accordingly each user's transmission rate. Sum-rate computer simulations for the proposed algorithm over Ricean fading channels show that, in the interference-limited regime, the proposed algorithm reconfigures itself in order to achieve the IA scaling whereas, in the low-to-moderate interference regime, it leads itself towards interference-myopic MIMO transmissions.

This article studies the problem of Sum-secure Degrees of Freedom (SDoF) of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(M,M,N,N)$ </tex-math></inline-formula> multiple-input–multiple-output (MIMO) interference channel with local output feedback, so as to build an information-theoretic foundation and provide practical transmission schemes for 6G-enabled Vehicles-to-Vehicles (V2V). For this problem, we propose two novel transmission schemes, i.e., the interference decoding scheme and the interference alignment scheme, and thus establish a sum-SDoF lower bound. In particular, to optimize the phase duration, we analyze the security and decoding constraints and formulate a linear-fractional optimization problem. Furthermore, we show that the derived sum-SDoF lower bound is the sum-SDoF for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M \le N/2$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N=M$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2N \le M$ </tex-math></inline-formula> antenna configurations, and reveal that for a fixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> , the optimal <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> to maximize the sum-SDoF is not less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2N$ </tex-math></inline-formula> . Through simulations, we examine the secure sum-rate performance of proposed transmission schemes and reveal that using local output feedback can lead to a higher secure sum-rate than that by using delayed channel state information at the transmitter (CSIT).