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Approaching the Capacity of Wireless Networks through Distributed Interference Alignment
Krishna Gomadam, Viveck R. Cadambe, Syed A. Jafar
TL;DR
The paper addresses unresolved finite-dimensional feasibility and global-channel-knowledge requirements for interference alignment. It develops iterative algorithms based on local channel information, reciprocity, and minimizing interference to unintended receivers, and reports numerical feasibility insights with performance close to theoretical predictions.
Problem
Finite-dimensional interference-alignment feasibility is unknown, and existing schemes may require impractical global channel knowledge.
Method
The paper develops iterative algorithms that use local channel information, network reciprocity, and a cognitive principle of minimizing interference to unintended receivers.
Results
The algorithms provide numerical insights into alignment feasibility and show significant benefits with performance close to theoretical predictions.
Takeaways & Limitations
Distributed interference alignment is numerically achievable through reciprocity and a “do no harm” interference-management approach.
Abstract
from arXiv · showhide
Recent results establish the optimality of interference alignment to approach the Shannon capacity of interference networks at high SNR. However, the extent to which interference can be aligned over a finite number of signalling dimensions remains unknown. Another important concern for interference alignment schemes is the requirement of global channel knowledge. In this work we provide examples of iterative algorithms that utilize the reciprocity of wireless networks to achieve interference alignment with only local channel knowledge at each node. These algorithms also provide numerical insights into the feasibility of interference alignment that are not yet available in theory.
I. INTRODUCTION
Interference alignment can approach high-SNR network capacity, but practical schemes face unresolved finite-dimension feasibility and global-channel-knowledge challenges. The paper proposes distributed iterative algorithms using local information, reciprocity, and a cognitive “minimize interference to others” principle.
- At high SNR, every user can achieve nearly one half of the interference-free capacity simultaneously and almost surely.
- Interference alignment overlaps interfering signals in half of each receiver’s signal space, leaving the other half available for desired transmission.
- Closed-form alignment solutions require global channel knowledge, while finite-dimensional feasibility and general analytical solutions remain open problems.
- The proposed algorithms require only local channel knowledge, including each receiver’s desired-transmitter channel and effective-noise covariance.
- Their cognitive principle has each transmitter primarily minimize interference to unintended receivers, an approach reported to lead to interference alignment.
- The algorithms use reciprocity to obtain interference information naturally available in time-division duplex networks.
A. Interference Avoidance and Iterative Waterfilling
Prior iterative waterfilling and interference-avoidance methods optimize each transmitter’s own receiver, whereas this work develops reciprocal, distributed alignment algorithms for broader interference networks.
- Interference avoidance and iterative waterfilling are selfish approaches in which each transmitter chooses power or signaling directions for its desired receiver.
- Interference alignment differs by coordinating interference across receivers rather than seeking a unilateral best response.
- For the three-user, two-antenna case, aligned interfering signals are co-linear at each receiver, while desired signals need not be orthogonal to interference.
- The proposed iterative algorithms combine reciprocity with prior iterative and duality-based approaches to achieve distributed interference alignment.
- The reciprocal channel switches transmitter and receiver roles and uses reciprocal channel matrices H̅[kl](n) = H[lk]†(n).
IV. INTERFERENCE ALIGNMENT OVER LIMITED DIMENSIONS - AN OPEN PROBLEM
Interference alignment over limited signalling dimensions remains an open problem, despite long symbol extensions approaching theoretical degrees-of-freedom bounds. The formulation uses transmit precoders and receive suppression filters to separate desired signals from interference.
- Long symbol extensions make achieved degrees of freedom approach the theoretical outer bound for time-varying interference and X networks.
- The maximum degrees of freedom achievable by aligning interference signal vectors over a limited number of dimensions is not known in general.
- Each transmitter uses an M[k]×d[k] precoding matrix V[k] whose columns span its transmitted signal space.
- Each receiver uses an N[k]×d[k] matrix U[k] to select an interference-free desired-signal subspace.
- When interference lies in the null space of U[k], the desired signal passes through a full-rank d[k]×d[k] channel while interference is eliminated.
- Under these conditions, user k achieves d[k] degrees of freedom.
A. Feasibility of Alignment
The feasibility problem asks whether transmit and receive filters can satisfy the interference-alignment conditions for a specified degrees-of-freedom allocation. Its general solution is unknown, motivating numerical distributed algorithms.
- A degrees-of-freedom allocation is feasible when transmit precoders V[k] and receive suppression matrices U[k] satisfy the alignment feasibility conditions.
- For random channel matrices, satisfying the interference-nullification condition implies the desired effective channel is full rank with probability 1.
- The general feasibility of finding filters for a random channel and specified allocation is unknown, so the distributed algorithm provides a numerical approach to this open problem.
B. Reciprocity of Alignment
Reciprocity lets interference-alignment feasibility transfer between an interference network and its reciprocal, enabling an iterative distributed algorithm. The algorithm alternates receiver updates across the two directions while reducing leakage interference.
- B. Reciprocity of Alignment: Switching transmit and receive filters transfers any feasible degrees-of-freedom allocation between the original and reciprocal networks.
- B. Reciprocity of Alignment: Reciprocity is the key property used to construct distributed interference-alignment algorithms.
- B. Reciprocity of Alignment: For multiple-antenna nodes without symbol extensions, alignment suppresses all interference at each receiver while preserving the allocated interference-free dimensions.
- B. Reciprocity of Alignment: The algorithm measures alignment quality by total leakage power remaining after receive suppression filters are applied.
- B. Reciprocity of Alignment: The pictorial algorithm represents MIMO channels as links and uses transmit power P per node in both communication directions.
- B. Reciprocity of Alignment: Each iteration alternates between the original and reciprocal networks, where receivers update suppression filters to minimize leakage interference.
- B. Reciprocity of Alignment: The receive filters from the reciprocal network become transmit precoders in the original network before the next iteration.
A. Proof of Convergence
The iterative algorithm is proven to converge because each receiver-side update cannot increase weighted leakage interference, whose nonnegative value decreases monotonically. However, convergence to a global minimum is not guaranteed.
- The weighted leakage interference metric quantifies residual interference after receive suppression and is the quantity minimized by the algorithm.
- Because weighted leakage interference is bounded below by zero, monotonic reduction guarantees convergence.
- The algorithm initializes arbitrary precoding matrices and alternates interference-covariance and suppression-filter computations in the original and reciprocal networks.
- With transmit filters fixed, Step 4 minimizes weighted leakage interference over all receive suppression filters and therefore cannot increase it.
- The reciprocal-network update also reduces weighted leakage interference, so each complete iteration monotonically decreases the metric.
- Convergence to a global minimum is not guaranteed because the interference-optimization problem is non-convex.
B. Max-SINR Algorithm
The Max-SINR extension modifies iterative interference alignment to optimize receiver SINR rather than only minimizing leakage interference. It alternates forward and reciprocal-network updates until convergence.
- Max-SINR algorithm: The Max-SINR algorithm chooses receive filters to maximize each stream's SINR instead of only minimizing leakage interference.The receive combining vectors need not be orthogonal across streams.
- Max-SINR algorithm: The algorithm is presented as an iterative implementation of the modified interference-alignment scheme described by the paper.Algorithm 2 specifies the Max-SINR iteration.
- Iteration steps: The procedure starts from linearly independent unit-norm precoding vectors and repeats the updates until convergence.The initial precoding matrix V[k] has d[k] linearly independent columns.
- Iteration steps: Each iteration computes interference-plus-noise covariance matrices and receive combining vectors in the forward and reciprocal networks.The reciprocal direction reuses receive combining vectors as precoding vectors.
VI. PERFORMANCE RESULTS AND APPLICATIONS
In the three-user, two-antenna interference channel, decentralized iterative alignment performs close to theoretical alignment based on global channel knowledge. The SINR-maximizing modification improves performance at low and intermediate transmit powers while matching alignment at high power.
- Three-user two-antenna channel: The decentralized iterative algorithm performs very close to theoretical interference alignment in the three-user, two-antenna channel.The comparison uses independent zero-mean unit-variance circularly symmetric complex Gaussian channel coefficients.
- Three-user two-antenna channel: The modified interference-alignment scheme provides a considerable performance gain at low and intermediate P.The modification chooses receive-combining vectors to maximize SINR.
- Three-user two-antenna channel: At high P, the modified scheme achieves the same performance as interference alignment.The comparison also includes interference avoidance and isotropic transmission.
A. Feasibility of Interference Alignment
The iterative algorithm evaluates finite-dimensional alignment feasibility through the interference power remaining in each receiver's desired signal space. The results suggest feasible stream configurations for four-user networks without channel extension, while additional streams increase residual interference.
- Feasibility criterion: The algorithm measures the fraction of interference power in the desired signal space as total network stream count varies.Perfect alignment corresponds to zero interference in that space, up to numerical errors.
- Four-user feasibility: Four users with five antennas each can support two streams per transmitter with interference alignment, according to the plotted results.This configuration corresponds to 8 degrees of freedom without channel extension.
- Four-user feasibility: Increasing the number of streams increases interference in the desired signal space, indicating that alignment becomes infeasible for those configurations.The paper notes that approaching the 10-degree-of-freedom upper bound may require channel extensions.
- Four-user feasibility: For the four-antenna case, the plot indicates feasibility up to a total of 6 streams in the four-user network.
B. Networks with single antenna nodes
For single-antenna networks, relays can create a virtual MIMO structure that reduces the signalling dimensions needed for interference alignment. In the three-source, three-destination relay channel, two time slots achieve 3/2 degrees of freedom, whereas the no-relay approach requires infinitely long extensions asymptotically.
- Relay-assisted alignment: Relays create a virtual MIMO system that can avoid long symbol extensions for interference alignment in single-antenna networks.The relay channel uses three sources, three destinations, and a half-duplex relay.
- Relay-assisted alignment: Two time slots achieve the 3/2 degrees-of-freedom outer bound in the three-source, three-destination interference relay channel.Without relays, the same network approaches 3/2 degrees of freedom only with infinitely long symbol extensions.
- Relay-assisted alignment: Over two time slots, the relay network becomes a three-user MIMO interference channel with a non-diagonal channel structure.For random independent channels, the resulting multiplexing gain is 3/2 with probability 1.
- Relay-assisted alignment: Using fewer signalling dimensions makes iterative algorithms work better through faster convergence.This is identified as an advantage of the relay-assisted construction.
- Time-extension performance: With time extensions, the no-relay scheme achieves multiplexing gain 4 in 3 time slots, while adding a relay achieves multiplexing gain 3/2 in two slots and improves low-P performance.The relay example is intended to show benefits when nodes lack multiple antennas.
VII. CONCLUSION
Distributed interference alignment is achievable through iterative, reciprocity-based algorithms, with benefits that are significant and close to theoretical predictions.
- Iterative algorithms based on network reciprocity achieve distributed interference alignment.
- Numerical comparisons show significant benefits over orthogonal, simultaneous-transmission, and selfish-interference-avoidance schemes.
- The distributed algorithms perform close to theoretical predictions.
- The algorithms use a “do no harm” approach with applications to wireless networks, where mitigating interference is fundamental.