Precoder Optimization for Interference Channels with Interference Alignment

Interference Alignment in Multi-user MIMO Systems

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 proposes a new strategy combining interference alignment (IA) and rate splitting for a three user interference channel with a cognitive transmitter. Although a half of available dimensions are sacrificed to align interference signals from other users in conventional IA, the proposed scheme recycles the sacrificed dimensions with the help of a cognitive transmitter. The achievable rate region and generalized degrees of freedom are derived. Compared with conventional IA, the proposed scheme improves achievable rate and generalized degrees of freedom.

We consider the effective degrees of freedom (DoF) achieved by interference alignment when channel state information (CSI) is acquired by training and feedback. For a flat block- fading K × K interference channel with power P per transmitter and M antennas per node, we show that interference alignment achieves higher DoF than orthogonal transmission (e.g., TDMA) only if the channel coherence time is large and the capacity of feedback link is at least as Θ(log P). Under this condition, to maximize the effective DoF, each receiver needs to feed back CSI via (M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -1)log P bits per coherence interval; smaller growth rate of feedback bits will decrease the effective DoF. We also show that in the presence of training and feedback cost, K = 3 achieves the optimal DoF for a broad range of channel characteristics; with larger number of user pairs, the DoF falls short of its optimum and beyond a certain point becomes a decreasing function of K.

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.

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.

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.

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.

Many works considered the CSI uncertainty effect on interference alignment (IA) in MIMO narrow-band interference channel (IC). However, much less attention has been paid to CSI uncertainty effect on IA in SISO frequency-selective IC. This is partially due to the impractical requirement of infinitely large number of subcarriers to implement IA in such channel. Recently, a new approach was proposed to alleviate this limitation through applying IA in time domain while utilizing the guard interval [1]. In this paper, we investigate the performance of this approach under CSI imperfection. We also propose robust transceiver design algorithm that maximizes the received SINR accounting for the CSI imperfection in SISO frequency selective IC. In addition, the average sum stream rate under CSI imperfection is derived.

This paper presents a new Interference Alignment (IA) scheme for K-User Multiple Input Multiple Output (MIMO) Interference Channel (IC) based on two metaheuristics, namely Particle Swarm Optimization (PSO) and Artificial Bee Colony (ABC) Algorithm. Tackling interference is an essential issue in wireless communications to which Interference Alignment (IA) provides a promising solution. However, IA still lacks of explicit and straightforward design procedures. In fact, most of IA procedures aim to minimize a certain Interference Leakage (IL) which measures the effect of the interference on the network, this results in complex optimization tasks involving a large amount of decision variables, together with a problem of convergence of the IA solutions. In this paper the IA optimization is performed using PSO, ABC and their cooperative counterparts, more suitable for large scale optimization. A comparison between the four algorithms is also carried out. The cooperative proposed approaches seem to be promising.

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.

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.

The achievable degrees of freedom (DOF) in an interference relay channel are examined. We highlight some differences between two relaying strategies in two different setups. Namely, we study the DOF achieved using interference alignment and interference cancellation in a cognitive relay setup and in a half-duplex relay setup. The difference between the two setups is causality, which leads to a difference in the achievable DOF. We show that 3/2 DOF are achievable almost surely in a 3-user complex Gaussian interference relay channel with quasistatic channel coefficients and with one antenna at each node. This result holds for both the cognitive as well as for the more practical half-duplex non-cognitive relay. However, while this is achieved in the cognitive relay setup using one symbol duration, it needs two symbol extensions in the half-duplex case. On the other hand, we show that in the cognitive case, interference cancellation performs better than interference alignment, since it achieves 2 DOF almost surely. Moreover, interference cancellation achieves full, i.e. K DOF for a family of K-user interference channels with a cognitive relay. Interestingly, for the half-duplex setup the outcome is swapped. Here, interference alignment outperforms interference cancellation, since the latter achieves less than 3/2 DOF almost surely in the 3 user case.

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.

The explosive demand for high data rates and the need ubiquitous wireless connectivity has led to the phenomenon of network densification, the deployment of large number of base station within a geographic area. Since the radio spectrum is limited, wireless transmissions need to share a common resource which results in interference. Interference in turn creates a bottleneck on the communication rate. Recently three breakthroughs have been made to increase the capacity of wireless networks while mitigating interference : multiple input multiple output (MIMO) beamforming, interference alignment (IA) and pattern reconfigurable antennas (PRA). While a great deal of progress has been made on understanding each individually, relatively little is known about how to use these techniques in combination to increase system performance. In this dissertation, we analyze and experimentally evaluate how these new technologies can be leveraged to enhance the rate performance of communication networks. Specifically, we address three problems: i) enhancing multi-user MIMO performance with PRA, ii) improving IA performance with PRA-based antenna selection, and iii) enabling practical blind IA with PRA. To understand the impact PRAs on MU-MIMO and IA, it is necessary to evaluate their performance through measurements. Simulation-based studies often reiterate over set of simplistic channel models and unrealistic assumptions. This dissertation focuses on a measurement-based evaluation approach. For each of these problems, a hardware testbed implementation and measurement methodology is developed for evaluating the performance of the aforementioned technologies through efficient and repeatable experiments. The first two major contributions of this dissertation focus on multi-user MIMO (MU-MIMO) and IA. While the theoretical rate gains of MU-MIMO and IA are substantial, the performance gains in practical systems and measured channels have been limited. One of the key limiting factors are spatially correlated user channels. In this work, we propose MU-MIMO and IA transmission schemes that leverage the pattern diversity provided by PRAs to mitigate the effects of spatial correlation in wireless channels. PRAs are capable of dynamically adjusting their radiation pattern, providing an additional degree of freedom that can be exploited to improve MU-MIMO and IA performance by treating the array configuration and radiation characteristics as additional components in the joint optimization of adaptive system parameters. We quantify the benefits of pattern diversity on the rate performance of MU-MIMO and IA systems through extensive measurements. Furthermore, we develop efficient antenna mode selection algorithms to realize those benefits without increasing the system complexity or overhead, making these techniques suitable for practical implementation. The third contribution of this dissertation focuses on a transmission technique known as blind IA. In contrast to MU-MIMO and conventional IA, blind IA is signaling scheme that suppresses interference in multi-user systems, without the knowledge of channel state information at the transmitter (CSIT). The key to performing IA without CSIT is the use of PRAs that are capable of dynamically switching among a fixed number of radiation patterns to introduce artificial fluctuations in the channel. In this dissertation, we develop a novel implementation of a blind IA system on a software defined radio platform where each of the receivers is equipped with a PRA. Our work provides the first experimental evaluation of PRA-based blind IA schemes and demonstrates significant gains in rate and reliability over traditional orthogonal transmission schemes such as TDMA.