Interference Alignment

Interference channels, where multiple transmit and receive user pairs communicate using the same radio resources, are a building block of wireless networks. The interference channel is a good model for communication in cellular networks, wireless local area networks, and ad-hoc networks. Conventional thinking about the interference channel is that each user pair has no information about other users in the network and therefore its optimum strategy is to be greedy and maximize its own rate. Unfortunately, the sum of the data rates achieved across all user pairs with this strategy is of the same order as the rate of a single communication link. Recent work on the interference channel by Jafar’s group and Khandani’s group, however, has shown that sum rates can scale linearly with the number of users at high SNR, using a transmission strategy known as interference alignment.

Interference alignment is a linear precoding technique that attempts to align interfering signals in time, frequency, or space. In MIMO networks, interference alignment uses the spatial dimension offered by multiple antennas for alignment. The key idea is that users coordinate their transmissions, using linear precoding, such that the interference signal lies in a reduced dimensional subspace at each receiver.

Allowing some coordination between transmit and receive user pairs enables interference alignment. In this way, it is possible to design the transmit strategies such that the interference aligns at each receiver. From a sum rate perspective, with K user pairs, an interference alignment strategy achieves a sum throughput on the order of K/2 interference free links! Basically each user can effectively get half the system capacity. Thus unlike the conventional interference channel, there is a net sum capacity increase with the number of active user pairs. This result has special importance in cellular and ad hoc networks, showing that coordination between users can help overcome the limiting effects of interference generated by simultaneous transmission.

We have been studying several aspects of interference alignment at UT, with an emphasis on its practice. Our areas of interest include algorithms for computing interference alignment solutions and more general precoding strategies for the interference channel, interference alignment performance in measured channels, clustering to reduce overhead in interference alignment, and analysis of interference alignment in the presence of channel estimation error. You can find a recent presentation on this topic here.

Select Publications

O. El Ayach, S. W. Peters, and R. W. Heath, Jr., “The Practical Challenges of Interference Alignment”, *to appear in IEEE Wireless Communications Magazine, Feb., 2013.*

Since IA’s inception, researchers have investigated its performance and proposed improvements, verifying IA’s ability to achieve the maximum degrees of freedom (an approximation of sum capacity) in a variety of settings, developing algorithms for determining alignment solutions, and generalizing transmission strategies that relax the need for perfect alignment but yield better performance. This article provides an overview of the concept of interference alignment as well as an assessment of practical issues including performance in realistic propagation environments, the role of channel state information at the transmitter, and the practicality of interference alignment in large networks.

O. El Ayach, A. Lozano, and R. W. Heath, Jr., “On the Overhead of Interference Alignment: Training, Feedback, and Cooperation,” *to appear in IEEE Transactions on Wireless Communications* , Dec, 2012. (Earlier conference version in * Proc. of the IEEE Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, November 6-9, 2011*).

This paper characterizes the effective throughput performance of interference alignment in MIMO systems where channel knowledge is acquired by training and analog feedback. We derive throughput-maximizing training and feedback resource allocations accounting for estimation error, training and feedback overhead, and channel selectivity (i.e. Doppler). We accurately characterize the effective sum rate with overhead in relation to various parameters such as signal-to-noise ratio and Doppler spread. We show that the overhead of IA can be optimized to ensure good performance in a wide range of fading scenarios.

O. El Ayach and R. W. Heath, Jr., “Grassmannian Differential Limited Feedback for Interference Alignment”, to appear in IEEE Transactions on Signal Processing, Dec. 2012, (Earlier conference version in *Proc. European Signal Processing Conference (EUSIPCO), 2011*)

This paper proposes a low overhead feedback strategy to be used with interference alignment in temporally correlated frequency selective channels. The proposed differential strategy exploits both the CSI’s Grassmannian structure as well as temporal correlation to significantly reduce the codebook sizes needed to achieve reasonable performance via alignment. Both analytical and numerical results are given to characterize the quantization distortion achieved by the proposed algorithm, as well as the resulting loss in sum rate incurred by quantized feedback.

B. Nosrat-Makouei, J. G. Andrews, and R. W. Heath, Jr., “User Arrival in MIMO Interference Alignment Networks,” *IEEE Trans. on Wireless, vol 11, no. 2, pp 842-851, 2011*. (Previous conference version in IEEE ICASSP 2011)

In this paper we analyze a constant MIMO interference channel where a set of active users cooperate via interference alignment while a set of secondary users desire access to the channel. We derive the minimum number of secondary transmit antennas required so that a secondary user can use the channel without affecting the sum rate of the active users and derive several precoder designs that approximately maximize the secondary users’ sum rate. When the secondary users do not have enough antennas, we perform numerical optimization to find secondary user precoders that cause minimum degradation to the sum rate of the active users.

J. Starr, O. El Ayach, and R. W. Heath, Jr., “Interference Alignment with Per-Antenna Power Constraints,” to appear in *Proc. of the 2011 IEEE International Symposium on Information Theory*, pp. 2892-2896, Saint-Petersburg, Russia, July 31 – Aug. 5, 2011.

This paper focuses on analyzing interference alignment under a set of practically motivated power constraints. In practice, each transmit antenna has its own power amplifier which constrains the power radiated from each antenna. This paper proposes and analyzes two algorithms for performing interference alignment with per-antenna equality and inequality power constraints. In addition to the algorithmic solutions, we derive feasibility conditions for per-antenna constrained alignment and show that incorporating such power constraints may render alignment infeasible in systems previously considered feasible.

I. Santamaria, O. Gonzalez, R. W. Heath, Jr., and S. W. Peters, “Maximum Sum-Rate Interference Alignment Algorithms for MIMO Channels,” *Proc. of **IEEE Global Telecommunications Conf.*, Miami, FL, December 6-10, 2010.

The interference aligned solution for precoding in the interference channel is not unique. Consequently, a particular interference aligned solution may not lead to the best possible alignment in terms of sum capacity. In this paper we suggest a procedure for finding the capacity maximizing interference alignment. The idea is to employ alternating minimization with an additional step on the Grassmann manifold in the direction of a higher sum capacity solution. This algorithm leads to interference aligned solutions with higher sum capacity performance.

O. El Ayach and R. W. Heath, Jr., “Interference Alignment with Analog Channel State Feedback”, IEEE Transactions on Wireless Communications, vol. 11, no.2, pp 626-636, Feb. 2012, (Earlier conference version in *Proc. of the* *Military Communications Conference*, San Jose, CA, October 2010)

This paper proposes using analog feedback to fulfill interference aligment CSI requirement. It shows that the full multiplexing gain is preserved by analog feedback and the degradation experienced is bounded under mild conditions on feedback quality. In the journal version we consider the overhead of training and feedback to numerically optimize the system’s effective throughput. We present both theoretical and numerical results to demonstrate the performance of IA with analog feedback.

B. Nosrat-Makouei, J. G. Andrews, and R. W. Heath, Jr., “A Simple SINR Characterization for Linear Interference Alignment over Uncertain MIMO Channels,” *Proc. of the **2010 IEEE International Symposium on Information Theory*, June 13-18, 2010, Austin, TX. See also the preprint of the journal version at ArXiv.

This paper derives expressions for the SINR distribution in a MIMO interference channel in the presence of channel estimation error and antenna correlation. It assumes that an interference alignment precoding strategy exists and is employed, along with a zero-forcing receiver. The SINR distributions for Rayleigh fading are used to compute quantities like the ergodic capacity and symbol-error-rate which can be used to compare IA to other transmission techniques such as spatial multiplexing and beamforming.

S. W. Peters and R. W. Heath, Jr., “Orthogonalization to Reduce Interference Alignment Overhead in the MIMO Interference Channel”, *Proc. of the* International Zurich Seminar on Communications, Zurich, Switzerland, pp. 126-129, March 3-5, 2010. See also the preprint of the journal version at ArXiv.

The degrees of freedom with interference alignment scale linearly with the number of users. Unfortunately, achieving interference alignment requires overhead to perform functions like synchronization, channel estimation, and feedback. These overheads may scale superlinearly with the number of users. In this paper we propose orthogonalization to reduce the effects of overhead. The idea is to break users into smaller interference alignment groups, on orthogonal channels, to balance sum rates gains and overhead losses.

S. W. Peters and R. W. Heath, Jr., “Cooperative Algorithms for MIMO Interference Channels”, to appear in *IEEE Trans. on Veh. Tech.*. Preprint available at ArXiv.

This paper generalizes our work on alternating minimization for other objective functions; the idea is to alternately solve for precoders and decoders until a convergence target is met. One approach is leakage-plus-noise, which is motivated by the presence of (different) non-coordinated interference at the receivers. Another approach is MMSE precoding and decoding, motivated by the ubiquity of MMSE techniques in single user and multiple user MIMO systems. The final approach is a particular SINR (signal to interference plus noise) ratio maximization. The SINR optimization gives the best performance, followed by the MMSE approach. Both result in non-semi-unitary precoders compared with conventional interference alignment.

O. El Ayach, S. W. Peters, and R. W. Heath, Jr., “Feasibility of Interference Alignment of Measured MIMO-OFDM Channels”, *IEEE Trans. on Veh. Tech.*. vol. 59. No. 9. pp. 4309-4321, Nov. 2010. Older preprint available at ArXiv.

This paper presents the first experimental study of interference alignment in measured multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) interference channels. We show that interference alignment achieves the claimed scaling factors in a wide variety of measured channel settings for a 3 user, 2 antennas per node setup. We also characterize the effect of several realistic system imperfections such as channel estimation error and channel spatial correlation, on sum rate performance. This paper includes both indoor and outdoor results, as well as more comprehensive measurements and analysis, when compared to the older conference version.

S. W. Peters and R. W. Heath, Jr., “Interference Alignment Via Alternating Minimization,” *Proc. of the **IEEE Int. Conf. on Acoustics, Speech, and Signal Proc.*, Taipei, Taiwan, April 2009, pp. 2445 – 2448.

This paper formulates an alternating minimization algorithm to find precoders for the MIMO interference channel. The idea is to alternate between finding precoders for each transmitter and reduced-dimensionality interference subspaces at each receivers. The result is a general, low complexity, algorithmic solution for interference alignment.

Sponsors

We have been fortunate to have several outstanding sponsors of our work including the DARPA IT-MANET program, Grant W911NF-07-1-0028, the Office of Naval Research under grant N000141010337, the Army Research Labs under grant Grant W911NF1010420, and Huawei Technologies Co. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the aforementioned sponsors.