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Aligned Interference Neutralization and the Degrees of Freedom of the 2x2x2 Interference Channel
Tiangao Gou, Syed A. Jafar, Sang-Woon Jeon, Sae-Young Chung
TL;DR
The paper asks whether the 2 × 2 × 2 interference channel can attain its 2-DoF min-cut despite having only two relays. It introduces aligned interference neutralization, combining alignment across hops with over-the-air cancellation, and shows that 2 DoF is achieved almost surely for time-varying and constant channels.
Problem
The challenge is achieving interference-free transmission at the 2-DoF min-cut when direct interference neutralization appears to require three relays but the network has only two.
Method
The paper combines interference alignment and interference neutralization so interference terms align across hops and can be cancelled at the final hop.
Results
2 DoF is achieved almost surely for the 2 × 2 × 2 interference channel with either time-varying or constant channel coefficients.
Takeaways & Limitations
For two sources and two destinations, the min-cut outer bound of 2 DoF remains achievable almost surely regardless of the number of hops.
Takeaways & Limitations
The constant-channel scheme requires the rational dimensions framework because the time-varying scheme cannot be applied directly to constant channels.
Abstract
from arXiv · showhide
We show that the 2x2x2 interference channel, i.e., the multihop interference channel formed by concatenation of two 2-user interference channels achieves the min-cut outer bound value of 2 DoF, for almost all values of channel coefficients, for both time-varying or fixed channel coefficients. The key to this result is a new idea, called aligned interference neutralization, that provides a way to align interference terms over each hop in a manner that allows them to be cancelled over the air at the last hop.
1 Introduction
The 2 × 2 × 2 interference channel is a layered multihop network whose DoF were previously unresolved for two single-antenna distributed relays. Existing approaches either achieve at most 1 DoF by treating hops as interference channels or achieve 4/3 DoF by treating them as X channels.
- 1 Introduction: Relatively little is known about multihop interference networks with few distributed relays, and even a coarse DoF characterization was unavailable for the 2 × 2 × 2 IC.The paper identifies this network as a fundamental setting of interest.
- 1 Introduction: The 2 × 2 × 2 IC concatenates two single-hop interference channels with distributed, single-antenna relays and two nodes per layer.It is presented as the simplest layered multihop interference network with few distributed relays.
- Interference Channel Approach: 1 DoF is the maximum achievable when either hop is treated as a 2-user interference channel, because that channel creates a bottleneck.This approach also precludes interference alignment at high SNR.
- Interference Alignment Approach: 4/3 DoF is achievable by treating both hops as 2 × 2 X channels, where each source splits its message into parts intended for the two relays.The relays decode these messages and create an X-channel configuration on the second hop.
- Interference Alignment Approach: 4/3 DoF applies for almost all channel coefficients under time-varying or constant channels, including fixed real channels, through interference alignment.The passage identifies 4/3 DoF as the highest previously known result applicable broadly across channel coefficient settings.
- Interference Neutralization Approach: At least 3 relays are necessary for interference neutralization in the relevant K × R × K network, whereas the 2 × 2 × 2 IC has only 2 relays.This makes direct interference neutralization apparently infeasible in the network under study.
2 sources
The paper studies whether the 2 × 2 × 2 interference channel can achieve its 2-DoF min-cut with fully connected generic channels. It introduces aligned interference neutralization and proves 2 DoF for almost all time-varying or constant channel coefficients.
- Main Question: Fully connected generic channels create a challenge because conventional interference neutralization requires at least 3 relays, while this network has only 2.Direct amplify-and-forward neutralization equations cannot generally be satisfied simultaneously.
- Main Result: Aligned interference neutralization combines interference alignment across relay dimensions with interference neutralization across the final hop.With M dimensions, the construction gives W1 access to M interference-free dimensions and W2 access to M − 1.
- Alignment at Relays: For M = 2, source beamforming aligns selected symbols at each relay, allowing the relays to isolate desired combinations through channel inversion.One beamforming vector can be chosen randomly, while the others are solved from the alignment equations.
- Aligned Interference Neutralization: The relays forward aligned combinations using beamforming vectors chosen so that interference cancels at the destinations.Destination D1 decodes two symbols successively, while D2 receives its desired symbol interference-free after discarding an interference dimension.
2 Channel Model
The network is a two-hop channel with two sources, two single-antenna relays, and two destinations. Each hop is modeled by a 2 × 2 channel matrix with Gaussian noise, power constraints, and either time-varying or constant coefficients.
- Network Structure: The 2 × 2 × 2 interference channel contains two sources, two relays, and two destinations, with each source carrying a message for its corresponding destination.The first hop connects sources to relays, and the second connects relays to destinations.
- First Hop: At relay Rk, the first-hop signal is determined by channel coefficients Fkj(t), source inputs Xj(t), and independent Gaussian noise Zk(t).The channel coefficient Fkj(t) describes the link from source Sj to relay Rk.
- Second Hop: At destination Dk, the second-hop signal is determined by channel coefficients Gkj(t), relay inputs XRj(t), and independent Gaussian noise Nk(t).The coefficient Gkj(t) describes the link from relay Rj to destination Dk.
- Assumptions: Every node has average power constraint P, the relays operate full-duplex, and channel knowledge is distributed across sources, relays, and destinations.Sources know first-hop channels, relays know both hops, and destinations know channel information as specified by the model.
- Channel Variations: The model includes both time-varying channels, whose coefficients are independently redrawn each use, and constant channels, whose coefficients remain fixed during transmission.In both cases, coefficients are drawn from a continuous distribution.
- Performance Definition: Achievable rates require both message error probabilities to vanish as blocklength grows, and sum-capacity is defined as the maximum achievable sum rate.The paper defines degrees of freedom from the high-SNR behavior of the sum-capacity.
3.1 Time-varying channel coefficients - linear scheme
The time-varying scheme uses symbol extensions and beamforming to align interference at relays, then neutralizes it across the second hop so destinations decode desired signals.
- 3.1 Time-varying channel coefficients - linear scheme: 2M−1 DoF are achieved over M symbol extensions, yielding normalized DoF (2M−1)/M and approaching 2 as M grows.Source 1 carries M sub-messages, while source 2 carries M−1.
- 3.1 Time-varying channel coefficients - linear scheme: Beamforming vectors align F11v1,i+1 with F12v2,i at R1 and F21v1,i with F22v2,i at R2.These alignments consolidate signals into shared dimensions at the relays.
- 3.1 Time-varying channel coefficients - linear scheme: Distinct time-varying channel coefficients make the induced Vandermonde structure nonsingular almost surely, proving the required beamforming vectors are linearly independent.The vectors at both sources are therefore available as independent signaling dimensions.
- 3.1 Time-varying channel coefficients - linear scheme: Relays isolate each dimension by inverting their effective M×M channel matrices, then amplify and forward the resulting signals using new beamforming vectors.This two-stage relay operation separates aligned dimensions before second-hop transmission.
- 3.1 Time-varying channel coefficients - linear scheme: At D1, aligned relay signals produce x1,1 followed by differences x1,i+1−x1,i, allowing sequential decoding; D2 similarly decodes differences x2,i−x2,i−1.Complementary-sign alignment cancels the undesired symbols over the air at the destinations.
3.2 Constant channel coefficients - rational dimension framework
For constant channels, the time-varying linear construction loses independence after symbol extension, so the scheme uses rational dimensions and hard decisions to retain aligned interference neutralization.
- 3.2 Constant channel coefficients - rational dimension framework: Constant channel coefficients make the extended channel matrices scaled identities, causing the time-varying beamforming vectors to become linearly dependent.The constant-channel section therefore restricts the construction to real channels and adopts rational dimensions.
- 3.2 Constant channel coefficients - rational dimension framework: Relays make hard decisions on received rational dimensions before forwarding, preventing relay noise from accumulating as in amplify-and-forward.The relay estimates become reliable in the high-power limit.
- 3.2 Constant channel coefficients - rational dimension framework: Rationally independent monomial beamforming coefficients provide a one-to-one mapping between relay constellations and transmitted sub-symbols almost surely.The minimum constellation distance grows with power, enabling reliable relay estimation as P approaches infinity.
- 3.2 Constant channel coefficients - rational dimension framework: After aligned interference cancellation, D1 and D2 decode their desired symbol chains with vanishing estimation error as P→∞.D1 and D2 recover source messages through sequential estimation and decoding.
- 3.2 Constant channel coefficients - rational dimension framework: The constant-channel construction achieves arbitrarily close to 2 DoF by taking M arbitrarily large and ε arbitrarily small.Each source-1 sub-message and source-2 sub-message contributes the stated rational-dimension DoF under the construction.
3.3 Constant channel coefficients with linear scheme
For constant complex channels, asymmetric complex signaling and linear beamforming achieve 2 complex DoF for almost all channel coefficients when phase-based independence conditions hold.
- 3.3 Constant channel coefficients with linear scheme: The constant-channel scheme uses asymmetric complex signaling and represents the complex SISO channel as a real 2 × 2 MIMO channel with scaled rotation matrices.This differs from the diagonal channel matrices used in the SISO time-varying case.
- 3.3 Constant channel coefficients with linear scheme: Linear independence in the first hop requires the phase combination φ12 + φ21 −φ11 −φ22 to avoid producing the identity rotation.The beam directions are related through a scaled rotation determined by this phase combination.
- 3.3 Constant channel coefficients with linear scheme: The second hop requires an analogous linear-independence condition for vR1,1 and vR1,2.Together, the first- and second-hop conditions yield the stated DoF result.
- 3.3 Constant channel coefficients with linear scheme: 2 complex DoF are achieved for the original complex SISO channel when the required linear-independence conditions hold.The corresponding real 2 × 2 MIMO representation achieves 3 real DoF.
4 Extensions
The results extend to longer layered networks and MIMO nodes: two-source, two-destination networks retain 2 DoF, while the MIMO version achieves 2M −1 DoF under eigenvalue conditions.
- 4 Extensions: 2 DoF remain achievable almost surely for two-source, two-destination layered multihop interference networks regardless of the number of hops.For more than two hops, relays after the second can amplify and forward with generic amplification factors, reducing the network to an effective 2-hop setting.
- 4 Extensions: For networks with more than two hops, simple amplify-and-forward relaying may achieve exactly 2 DoF without aligned interference neutralization.The two-hop case differs because amplify-and-forward leaves two neutralization equations but only one freely adjustable amplification factor after normalization.
- 4 Extensions: 2M −1 DoF are achievable in the 2 × 2 × 2 MIMO interference network when each node has M antennas and the stated eigenvalue conditions hold.The relevant matrix must have M distinct eigenvalues for the beamforming directions to be linearly independent.
5 Conclusion
The paper shows that aligned interference neutralization achieves the 2-DoF min-cut for almost all channel coefficients in both time-varying and constant channels, with extensions to arbitrary hop counts.
- 5 Conclusion: 2 DoF, equal to the min-cut outer bound, are achievable for almost all channel coefficients in the 2 × 2 × 2 interference channel.The result holds whether channel coefficients are time-varying or constant.
- 5 Conclusion: Aligned interference neutralization combines interference alignment and interference neutralization to align interference across hops for cancellation at the last hop.Time-varying channels use linear beamforming over symbol extensions, while constant channels use rational dimensions.
- 5 Conclusion: 2 DoF remain achievable almost surely for two-source, two-destination multihop interference channels regardless of the number of hops.Extensions to more than two sources and two destinations remain challenging and are identified as future work.