Interference Alignment under Training and Feedback Constraints

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algorithms to the K user MIMO IC with a transmit power constraint and show how these schemes exploit both current and outdated CSI to achieve an improved DoF performance.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.

In wireless interference networks, orthogonal medium access techniques like time division multiple access (TDMA) and frequency division multiple access (FDMA) achieve merely one degree of freedom (DoF). However, by using interference alignment (IA) the achievable DoF can be linearly scaled up with the number of users. Nevertheless, this outperformance of IA heavily depends on the availability of perfect channel state information (CSI), which is not a realistic assumption in practice. Since under imperfect CSI, the performance of IA may become severely degraded, design of enhanced IA algorithms by relying on the availability of merely imperfect CSI is of particular interest. In this paper, we propose a least squares (LS)-based IA algorithm which adaptively designs the beamformers based on the knowledge of the channel estimation error variance, which is possible to be known in advance. Then, we compare the performance of the proposed scheme with interference leakage minimization algorithms. It is shown that under both perfect and imperfect CSI, the proposed LS-based IA outperforms interference leakage minimization algorithms.

This paper studies the impact of channel error on the achievable rate of symmetrical K-user multiple-input multiple-output linear interference alignment (IA) networks. The upper and lower bounds of the achievable sum rate are derived analytically with the assumption of orthonormal transmit precoders and receive filters designed from imperfect channel state information (CSI) over both the uncorrelated and correlated channels. For uncorrelated channels, quite tight lower and upper bounds are obtained. The impact of channel error on the degrees of freedom (DoF) and the DoF persistence conditions are also investigated. Results show that the DoF of IA networks persists only if the channel error decreases in an order higher than the signal-to-noise ratio. For correlated channel, the lower and upper bounds for one realization of IA are derived. The derived upper bound can be used to characterize the achievable rate approximately. Simulation results indicate that the achievable rate of IA network is influenced significantly by CSI uncertainty. The obtained analytical bounds provide an intuitive way to show the impact of channel error on the achievable rate and thus can help practical systems deign.

Interference alignment (IA) is a linear precoding strategy that can achieve optimal capacity scaling at high SNR in interference networks. Most of the existing IA designs require full channel state information (CSI) at the transmitters, which induces a huge CSI signaling cost. Hence it is desirable to improve the feedback efficiency for IA and in this paper, we propose a novel IA scheme with a significantly reduced CSI feedback. To quantify the CSI feedback cost, we introduce a novel metric, namely the feedback dimension. This metric serves as a first-order measurement of CSI feedback overhead. Due to the partial CSI feedback constraint, conventional IA schemes can not be applied and hence, we develop a novel IA precoder/decorrelator design and establish new IA feasibility conditions. Via dynamic feedback profile design, the proposed IA scheme can also achieve a flexible tradeoff between the degree of freedom (DoF) requirements for data streams, the antenna resources and the CSI feedback cost. We show by analysis and simulations that the proposed scheme achieves substantial reductions of CSI feedback overhead under the same DoF requirement in MIMO interference networks.

In the presence of perfect channel state information (CSI), the achievable degrees of freedom (DoF) in wireless interference networks can be linearly scaled up with the number of users. Achievability is based on the idea of interference alignment (IA). However, in the presence of imperfect CSI, the sum rate becomes degraded, and full DoF may no longer be achievable. In this paper, the authors propose novel least squares (LS)- and minimum mean square error (MMSE)-based IA schemes that adaptively design beamformers by relying on the availability of imperfect CSI and knowledge of the channel estimation error variance in advance. Interestingly and unlike the other robust algorithms, the proposed adaptive schemes do not impose extra computational complexity compared to their nonadaptive counterparts. It is shown that the LS-based IA is able to outperform interference leakage minimization algorithms under both perfect and imperfect CSI. Furthermore, the authors compare the performance of the proposed MMSE-based IA with maximum signal-to-interference-plus-noise ratio (Max-SINR) algorithm. The authors show that while under perfect CSI, the MMSE-based IA achieves the same performance as that of Max-SINR, the former outperforms the latter under CSI mismatch. Meanwhile, it is shown that the proposed MMSE-based IA needs less CSI to be available and has less computational complexity compared to Max-SINR.

It is very important that the radar sensor network has autonomous capabilities such as self-managing, etc. Quite often, MIMO interference channels are applied to radar sensor networks, and for self-managing purpose, interference management in MIMO interference channels is critical. Interference alignment (IA) has the potential to dramatically improve system throughput by effectively mitigating interference in multi-user networks at high signal-to-noise (SNR). However, the implementation of IA predominantly relays on perfect and global channel state information (CSI) at all transceivers. A large amount of CSI has to be fed back to all transmitters, resulting in a proliferation of feedback bits. Thus, IA with limited feedback has been introduced to reduce the sum feedback overhead. In this paper, by exploiting the advantage of heterogeneous path loss, we first investigate the throughput of IA with limited feedback in interference channels while each user transmits multi-streams simultaneously, then we get the upper bound of sum rate in terms of the transmit power and feedback bits. Moreover, we propose a dynamic feedback scheme via bit allocation to reduce the throughput loss due to limited feedback. Simulation results demonstrate that the dynamic feedback scheme achieves better performance in terms of sum rate.

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.

This paper fully determines the degree-of-freedom (DoF) region of two-user interference channels with arbitrary number of transmit and receive antennas in the case of isotropic and independent (or block-wise independent) fading, where the channel state information is available to the receivers but not to the transmitters. Thus, the DoF gap between previous upper and lower bounds due to the authors, Vaze and Varanasi, and Huang, Jafar, Shamai, and Vishwanath is closed. The DoF result characterizes the capacity region to the first order of the logarithm of the signal-to-noise ratio (SNR) in the high-SNR regime. The DoF region is achievable by using random Gaussian codebooks independent of the channel states, which implies that it is impossible to increase the DoF using beamforming and interference alignment in the absence of channel state information at the transmitters.

Interference Alignment in Multi-user MIMO Systems

Interference alignment (IA) is a promising technique for interference mitigation in multicell networks due to its ability to completely cancel the intercell interference through linear precoding and receive filtering. In small networks, the amount of required channel state information (CSI) is modest and IA is, therefore, typically applied jointly over all base stations. In large networks, where the channel coherence time is short in comparison to the time needed to obtain the required CSI, base station clustering must be applied however. We model such clustered multicell networks as a set of coalitions, where CSI acquisition and IA precoding are performed independently within each coalition. We develop a long-term throughput model, which includes both CSI acquisition overhead and the level of interference mitigation ability as a function of the coalition structure. Given the throughput model, we formulate a coalitional game where the involved base stations are the rational players. Allowing for individual deviations by the players, we formulate a distributed coalition formation algorithm with low complexity and low communication overhead that leads to an individually stable coalition structure. The dynamic clustering is performed using only long-term CSI, but we also provide a robust short-term precoding algorithm, which accounts for the intercoalition interference when spectrum sharing is applied between coalitions. Numerical simulations show that the distributed coalition formation is generally able to reach long-term sum throughputs within 10% of the global optimum.

It was shown recently that the 2-user interference channel with a cognitive relay (IC-CR) has full degrees of freedom (DoF) almost surely, that is, 2 DoF. The purpose of this work is to check whether the DoF of the K-user IC-CR, consisting of K user pairs and a cognitive relay, follow as a straight forward extension of the 2-user case. As it turns out, this is not the case. The k-user IC-CR is shown to have 2K/3 DoF if K > 2 for the when the channel is time varying, achievable using interference alignment. Thus, while the basic k-user IC with time varying channel coefficients has 1/2 DoF per user for all K, the k-user IC-CR with varying channels has 1 DoF per user if K = 2 and 2/3 DoF per user if K > 2. Furthermore, the DoF region of the 3-user IC-CR with constant channels is characterized using interference neutralization, and a new upper bound on the sum-capacity of the 2-user IC-CR is given.

Interference alignment (IA) is a multi-user wireless communication strategy that enables several transmitters simultaneous communicate with their paired receivers without interfering each other. IA is a multiplexing gain optimal strategy at the expense of requiring the global channel state information (CSI) at the transmitters. Most previous works assume the availability of perfect CSI which is not possible in practice. The performance of IA algorithms based on perfect CSI drop dramatically in circumstance where CSI errors exist. In this paper we propose a robust minimum mean square error (MMSE) IA algorithm with imperfect CSI. Based on a Gaussian statistic channel error model, we give the mean square error objective function with per transmitter power constraint. The objective function is not joint convex in precoders and decoders, but convex in precoders with fixed decoders and vice versa. We convert the problem to two convex sub problems and iteratively find the solutions. Numerical results are presented to validate the effectiveness of proposed algorithm when CSI errors exist.

MIMO researches penetrate the network community due to recent advances in MIMO degree-of-freedom (DoF) frameworks. Independent from MIMO, Successive Interference Cancellation (SIC) is a powerful multi-user detection technology in physical layer, and Interference Alignment (IA) is a powerful technique for handling interference. Based on the understanding of the strengths and weaknesses of SIC and DoF, we propose mutual assistance by joint IA, SIC and DoF-based Interference Cancellation (DoF IC) to (i) conserve precious DoF resources through IA and SIC; (ii) meet strict SINR threshold criteria through IC and IA. In this paper, we develop the mathematical model to achieve the two ideas in a multi-hop MIMO network. Combined with scheduling and routing constraints, we developed a cross-layer optimization framework to the throughput maximization problem with joint IA and SIC for MIMO networks. Simulation results show that the join of IA and SIC can significantly conserve DoFs for IC and thereby improve the throughput.

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.

A novel interference alignment (IA) scheme based on the errors-in-variables (EIV) mathematic model has been proposed to overcome the channel state information (CSI) estimation error for the MIMO interference channels. By solving an equivalently unconstrained optimization problem, the proposed IA scheme employing a weighted total least squares (WTLS) algorithm can obtain the solution to a constrained optimization problem for transmit precoding (TPC) matrices and minimizes the distortion caused by imperfect CSI according to the EIV model. It is shown that the design of TPC matrices can be realized through an efficient iterative algorithm. The convergence of the proposed scheme is presented as well. Simulation results show that the proposed IA scheme is robust over MIMO interference channels with imperfect CSI, which yields significantly better sum rate performance than the existing IA schemes such as distributed iterative IA, maximum signal-to-interference-plus-noise ratio (Max SINR), and minimum mean square error (MMSE) schemes.