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Optimal Beamforming for Two-Way Multi-Antenna Relay Channel with Analogue Network Coding
Rui Zhang, Ying-Chang Liang, Chin Choy Chai, Shuguang Cui
TL;DR
The paper studies how to characterize achievable bidirectional rates in an ANC-based two-way relay channel with a multi-antenna relay under transmit-power constraints. It derives a reduced optimal beamforming structure and an efficient rate-profile-based optimization method, then compares low-complexity matched-filter and zero-forcing schemes with the optimal design.
Problem
The paper seeks the capacity region and suitable relay beamforming designs for ANC/AF-based two-way relay channels with a multi-antenna relay.
Method
It derives an optimal relay beamforming structure, reduces the design to four complex variables when M > 2, and uses convex relay-power minimization within a rate-profile algorithm.
Results
Matched-filter relay beamforming achieves sum-rates and rate regions close to the optimal ones under various SNR and channel conditions.
Takeaways & Limitations
Matched-filter beamforming can be a good implementation-oriented solution, while zero-forcing may perform poorly under strong channel correlation.
Abstract
from arXiv · showhide
This paper studies the wireless two-way relay channel (TWRC), where two source nodes, S1 and S2, exchange information through an assisting relay node, R. It is assumed that R receives the sum signal from S1 and S2 in one time-slot, and then amplifies and forwards the received signal to both S1 and S2 in the next time-slot. By applying the principle of analogue network (ANC), each of S1 and S2 cancels the so-called "self-interference" in the received signal from R and then decodes the desired message. Assuming that S1 and S2 are each equipped with a single antenna and R with multi-antennas, this paper analyzes the capacity region of an ANC-based TWRC with linear processing (beamforming) at R. The capacity region contains all the achievable bidirectional rate-pairs of S1 and S2 under the given transmit power constraints at S1, S2, and R. We present the optimal relay beamforming structure as well as an efficient algorithm to compute the optimal beamforming matrix based on convex optimization techniques. Low-complexity suboptimal relay beamforming schemes are also presented, and their achievable rates are compared against the capacity with the optimal scheme.
I. INTRODUCTION
The paper studies an ANC-based two-way relay channel in which two single-antenna sources exchange information through a multi-antenna relay using two time-slots. It characterizes bidirectional achievable rates under source and relay power constraints and develops optimal and low-complexity beamforming designs.
- Motivation: ANC reduces one information-exchange round to two time-slots by allowing S1 and S2 to transmit simultaneously to R.Each source later removes its own self-interference from the relay’s broadcast signal.
- Contributions: The paper derives an optimal relay beamforming structure and an efficient convex-optimization-based algorithm for computing boundary rate-pairs.The structure reduces the number of complex-valued design variables from M^2 to 4 when M > 2.
- Contributions: Two low-complexity schemes based on matched filtering and zero forcing are evaluated against the optimal sum-capacity.The introduction reports that matched-filter beamforming achieves sum-rates close to sum-capacity under various SNR and channel conditions.
- System model: The model uses single-antenna sources, an M-antenna relay with M ≥2, flat-fading channels, and equal-duration uplink and downlink slots.The relay linearly processes the concurrent source transmissions before broadcasting.
- Capacity objective: The capacity region contains all bidirectional rate-pairs achievable under transmit-power constraints P1, P2, and PR at S1, S2, and R.For fixed source powers, the relay beamforming matrix realizes different boundary rate tradeoffs.
III. CAPACITY REGION CHARACTERIZATION
The capacity-region analysis shows that optimal relay beamforming can be confined to the channel subspace associated with the two sources. This reduces the general design to a 2 × 2 matrix problem and yields further simplifications for orthogonal or parallel channels.
- Optimal structure: The optimal relay beamforming matrix A lies in the space spanned by the two source-to-relay channel vectors.This differs from separate unidirectional OWRC designs, whose optimal matrices generally differ unless the channel vectors are parallel.
- Special cases: For orthogonal source channels, the optimal structure contains two nonnegative diagonal parameters c and d.For parallel source channels, it instead contains a single nonnegative parameter a.
- General case: Outside the orthogonal and parallel cases, the paper optimizes the general 2 × 2 matrix B for each boundary rate-pair.The corresponding relay matrix A is reconstructed from B using the reduced-dimensional structure.
B. Optimization Problems
The paper avoids direct weighted-sum-rate maximization because the resulting problem is non-convex, instead using rate profiles and relay-power minimization to trace the capacity-region boundary.
- Optimization formulation: Direct weighted-sum-rate maximization is difficult because its objective is not concave in B, despite convex constraints.The paper therefore uses an alternative rate-profile method.
- Rate-profile method: A rate profile fixes the ratio between each user’s rate and the sum-rate, so maximizing the sum-rate finds a boundary point along a specified direction.In the two-user case, α = [α21, α12]^T specifies the direction.
- Relay power minimization: The alternative problem minimizes relay power subject to rate constraints for a target rate-pair.If the required minimum relay power is at most PR, the target is feasible; otherwise it lies outside the rate region.
- Algorithm: A bisection algorithm updates lower and upper sum-rate bounds according to whether the minimum required relay power satisfies PR.The procedure stops when the interval width is at most δr, and the converged lower bound is optimal for the chosen rate-profile direction.
C. Power Minimization under SNR Constraints
The paper converts rate constraints into receiver-SNR constraints and reformulates relay power minimization using vectorization and semidefinite relaxation. Despite the relaxation's potential higher rank, the special problem structure permits recovery of an exact rank-one optimum.
- Problem reformulation: Receiver rate constraints are expressed as equivalent SNR targets γ1 ≥ γ̄1 and γ2 ≥ γ̄2 at S1 and S2.The targets γ̄1 and γ̄2 are defined from the desired rates and rate-profile parameter.
- Problem reformulation: Vectorization rewrites relay power as pR = ∥Φb∥2 and received-signal terms as ∥G_i b∥2.Here b = Vec(B), enabling the matrix problem to be represented through vector variables.
- SDP relaxation: The resulting optimization remains non-convex because of a rank-one constraint, but removing that constraint yields a convex SDP relaxation.The real matrix variable X is formed from the real and imaginary parts of b.
- Exact recovery: The SDP can be solved efficiently, and the paper shows that an optimal rank-one solution can be reconstructed without sacrificing optimality.This avoids the usual loss associated with randomized rank-one recovery from higher-rank SDP solutions.
IV. LOW-COMPLEXITY RELAY BEAMFORMING SCHEMES
The paper develops MRR-MRT and ZFR-ZFT relay beamforming schemes as lower-complexity alternatives to the optimal design. Their power-balancing structures remain within the optimal beamforming space, while their performance and applicability depend on channel geometry.
- Proposed schemes: Two lower-complexity schemes are proposed: MRR-MRT and ZFR-ZFT.They are designed to reduce implementation complexity relative to the optimal relay beamforming scheme.
- MRR-MRT: MRR-MRT uses matched-filter reception and transmission to maximize forwarded signal power without suppressing inter-source interference at R.The coefficients a_x and b_x balance relay power allocated to the two directions.
- ZFR-ZFT: ZFR-ZFT completely removes self-interference at S1 and S2, simplifying their receivers compared with ANC implementations requiring receiver-side cancellation.Its advantage is receiver simplification rather than universal rate superiority.
- Relation to the optimum: Both suboptimal schemes satisfy the optimal beamforming structure, but their associated coefficient matrices are generally suboptimal.The schemes differ from the optimum through their choices of B rather than through the structural subspace.
- Special channel cases: MRR-MRT is optimal when h1 and h2 are orthogonal or parallel, whereas ZFR-ZFT is optimal in the orthogonal case only.When h1 ∥ h2, ZFR-ZFT does not exist because its matrix construction is singular.
V. PERFORMANCE ANALYSIS
The performance analysis derives an upper bound on optimal-scheme sum-capacity and lower bounds for MRR-MRT and ZFR-ZFT. It uses these bounds to compare achievable sum-rates and study their high-SNR behavior.
- Sum-capacity bounds: The optimal scheme's sum-capacity is characterized through a weighted-sum-rate problem, with an upper bound used because the original problem is non-convex.The bound is tightened by optimizing the directional time and relay-power allocations.
- Bound construction: The directional subproblems optimize relay beamforming for each corresponding one-way relay channel before combining the two directions under a common relay-power constraint.This decomposition supports tractable evaluation of the sum-rate upper bound.
- Sum-capacity bounds: CUB requires numerical search in general because its optimization over κ21, κ12, P21, and P12 has no closed-form solution.A simpler, looser bound fixes κ21 = κ12 = 1/2 and P21 = P12 = PR.
- Sum-capacity bounds: The upper bound C(0)UB is valid regardless of the rate-profile vector α because it imposes no rate-allocation constraint between r21 and r12.It can also serve as the sum-rate target for Algorithm 3.1.
- Suboptimal-scheme bounds: The analysis derives lower bounds RMRLB and RZFLB for the MRR-MRT and ZFR-ZFT schemes, respectively.These bounds assume equal balancing coefficients within each scheme.
B. Asymptotic Results
The paper examines high-SNR behavior because TWRC is intended to recover spectral-efficiency losses caused by half-duplex transmission. It presents asymptotic limits for several upper and lower rate bounds.
- Motivation: High-SNR analysis is motivated by TWRC's role in recovering half-duplex spectral-efficiency loss.The analysis lets source and relay powers grow while studying limiting rate bounds.
- Asymptotic bounds: The asymptotic theorem gives limiting values for the upper and lower rate bounds in equations (34), (35), and (36).The supplied passage identifies convergence of the lower bounds to values stated in equations (37), (38), and (39).
UB, RMR
At high SNR, MRR-MRT and ZFR-ZFT share the sum-capacity upper bound’s asymptotic pre-log factor, but their constant rate gaps depend on channel correlation. MRR-MRT loses at most 0.1699 bits/complex dimension, while ZFR-ZFT becomes less favorable as channel correlation increases.
- High-SNR behavior: At high SNR, MRR-MRT and ZFR-ZFT asymptotically achieve the same sum-rate pre-log factor as the sum-capacity upper bound.Their rate gaps from the upper bound remain constants independent of PR.
- High-SNR behavior: 0.1699 bits/complex dimension is the maximum asymptotic sum-rate loss of MRR-MRT from sum-capacity.This bound follows from the maximum value 9/8 attained at ρ = 1/3.
- Correlation effects: ZFR-ZFT’s asymptotic sum-rate loss increases as channel correlation ρ increases.Zero-forcing incurs more SNR loss when separating increasingly correlated signals at the relay.
- Correlation effects: At ρ = 0 or 1, MRR-MRT has zero asymptotic sum-rate loss, while ZFR-ZFT has zero loss at ρ = 0.These cases agree with the stated optimality lemmas for the two schemes.
- Capacity-region construction: The capacity region is obtained by taking the union of achievable regions over feasible source powers, with relay beamforming controlling boundary rate tradeoffs.For M = 4, P1 = P2 = PR = 10, and ρ = 0.5, the capacity region is symmetric over r12 and r21.
B. Achievable Rates of Suboptimal Beamforming Schemes
Numerical comparisons show that MRR-MRT remains close to the optimal scheme across channel correlations, whereas ZFR-ZFT degrades as correlation grows. At ρ = 1/3, their high-SNR gaps are 0.1699 and 0.5850 bits/complex dimension, respectively, and DF generally has a larger region than AF except in a strongly correlated regime where AF can be preferable in practice.
- Suboptimal beamforming schemes: MRR-MRT’s achievable rate region is close to the optimal region for small, moderate, and large channel correlation.At ρ = 0.5, its observed rate loss is negligible.
- Suboptimal beamforming schemes: ZFR-ZFT is close to optimal at small ρ but degrades significantly as ρ increases.Low correlation allows zero-forcing to separate signals with small SNR losses; higher correlation makes separation more costly.
- Sum-rate comparison: At ρ = 1/3, MRR-MRT converges to the sum-capacity upper bound with a 0.1699 bits/complex dimension gap, versus 0.5850 for ZFR-ZFT.These observations agree with Corollary 5.1.
- Heuristic baselines: Direct relaying and one-way alternative relaying have significant high-SNR gaps from MRR-MRT.The paper attributes these gaps to absent relay beamforming gain and approximately half the spectral efficiency from alternative relaying, respectively.
- AF versus DF: DF-based TWRC generally has a larger capacity region than AF-based TWRC, with the gain increasing as ρ decreases.When channel correlation is strong, AF can outperform fixed-time DF in parts of the rate region and may be more suitable because it has lower relay encoding/decoding complexity.
VII. CONCLUSION AND FUTURE WORK
The paper characterizes ANC/AF-based multi-antenna TWRC capacity, develops optimal and low-complexity relay beamforming, and compares ANC/AF with DF under channel-correlation conditions.
- The capacity-region characterization uses a rate-profile method because WSRMax is not directly applicable to the non-convex ANC/AF optimization problem.
- MF-based MRR-MRT achieves sum-rates and rate regions close to the optimal scheme under various conditions, whereas ZF-based ZFR-ZFT may perform poorly with strong channel correlation.
- Future directions include joint source-relay beamforming with multi-antenna sources, multiple source-pair transmission, hybrid AF/DF, and estimate-and-forward operations.
APPENDIX II PROOF OF LEMMA 3.1
The appendix establishes structural simplifications for optimal relay beamforming and proves that Algorithm 3.1 converges to the optimal solution of its rate-profile problem.
- For orthogonal channel vectors, maximizing both rates while minimizing relay power sets the off-diagonal parameters a and b to zero.
- For parallel channel vectors, the corresponding optimal structure sets b, c, and d to zero, leaving only the parameter a active.
- Algorithm 3.1 converges because its feasible lower rate is within δr of the optimum, and δr can be made arbitrarily small.
- The rank-one recovery routine decomposes the SDP solution, solves basic linear programs, and compares subproblem objectives to obtain a global minimum.
APPENDIX VI PROOF OF LEMMA 5.1
The appendix derives lower bounds for the proposed MF- and ZF-based relay beamforming schemes by evaluating their rates at maximum relay power.
- Setting the MF beamforming parameters equal to ν and using maximum relay power yields the lower bound on sum-rate in (35).
- For ZF beamforming, Jensen’s inequality and positive-semidefinite trace bounds produce a lower bound that becomes (36) after substituting θ1θ2(1−ρ).