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On Design of Collaborative Beamforming for Two-Way Relay Networks

Meng Zeng, Rui Zhang, Shuguang Cui

arXiv:1010.4980v1cs.ITmath.IT

TL;DR

The paper studies how to characterize achievable rate regions for two-way relay networks using collaborative beamforming under different channel-reciprocity assumptions. It derives closed-form reciprocal-channel beamformers and numerical algorithms for non-reciprocal channels; the reported comparisons favor reciprocal channels and TDD-based relaying.

  • Problem

    The paper asks how to efficiently characterize the achievable rate region for collaborative two-way relay networks under reciprocal and non-reciprocal channels.

    Source: §II · p. 5

  • Method

    It characterizes reciprocal-channel regions with WSISMin beamformers and closed-form solutions, and non-reciprocal regions with numerical algorithms.

    Source: §I · p. 3, §VII · p. 20

  • Results

    The reciprocal-channel rate region is larger than the non-reciprocal region under the same power and noise settings, and the conclusion favors TDD-based relaying.

    Source: §VI · p. 18, §VII · p. 21

  • Takeaways & Limitations

    TDD-based relaying is more favorable when channel coherence time exceeds an operation period and transmit-receive chain calibration is properly done.

    Source: §VI · p. 19, §VII · p. 21

    For non-reciprocal channels, closed-form solutions are unavailable; individual-power constraints yield approximate rather than optimal beamformer solutions.

    Source: §I · p. 3

Abstract

from arXiv · show

We consider a two-way relay network, where two source nodes, S1 and S2, exchange information through a cluster of relay nodes. The relay nodes receive the sum signal from S1 and S2 in the first time slot. In the second time slot, each relay node multiplies its received signal by a complex coefficient and retransmits the signal to the two source nodes, which leads to a collaborative two-way beamforming system. By applying the principle of analog network coding, each receiver at S1 and S2 cancels the "self-interference" in the received signal from the relay cluster and decodes the message. This paper studies the 2-dimensional achievable rate region for such a two-way relay network with collaborative beamforming. With different assumptions of channel reciprocity between the source-relay and relay-source channels, the achievable rate region is characterized under two setups. First, with reciprocal channels, we investigate the achievable rate regions when the relay cluster is subject to a sum-power constraint or individual-power constraints. We show that the optimal beamforming vectors obtained from solving the weighted sum inverse-SNR minimization (WSISMin) problems are sufficient to characterize the corresponding achievable rate region. Furthermore, we derive the closed form solutions for those optimal beamforming vectors and consequently propose the partially distributed algorithms to implement the optimal beamforming, where each relay node only needs the local channel information and one global parameter. Second, with the non-reciprocal channels, the achievable rate regions are also characterized for both the sum-power constraint case and the individual-power constraint case. Although no closed-form solutions are available under this setup, we present efficient numerical algorithms.

I. INTRODUCTION

This paper addresses the lack of closed-form characterization and efficient distributed algorithms for collaborative two-way relay beamforming. It models two sources exchanging information through a relay cluster and studies achievable rate regions under reciprocal and non-reciprocal channels.

I. INTRODUCTION

  • Prior collaborative two-way relay rate-region work offered only numerical solutions, motivating closed-form beamforming characterizations and distributed algorithms.The paper considers single-antenna relays under both sum-power and individual-power constraints, unlike the multiple-antenna single-relay setup in.Source: §I · p. 2
  • For reciprocal channels, the paper seeks closed-form optimal beamformers; for non-reciprocal channels, it characterizes the regions with numerical SDP algorithms, optimal under sum power and approximate under individual limits.The non-reciprocal algorithms use semi-definite relaxation.Source: §I · pp. 2–3

II. SYSTEM MODEL

  • Two single-antenna sources communicate through K single-antenna relays in two time slots, with no direct source-to-source link.The sources transmit simultaneously to the relays; each relay scales its received mixture by a complex coefficient and forwards it.Source: §II · pp. 3–4
  • Each source knows its own transmitted signal and cancels its self-interference; the rate region is the convex hull of achievable rate pairs under sum-power or individual relay-power limits.The paper characterizes this region for reciprocal and non-reciprocal channel assumptions.Source: §II · pp. 4–5

III. RECIPROCAL CHANNEL CASE

With reciprocal source-relay and corresponding relay-source channels, beamforming phases can be aligned to maximize both link rates for fixed amplitudes. The paper uses weighted sum inverse-SNR minimization to characterize the rate-region boundary with closed-form optimal beamformers.

III. RECIPROCAL CHANNEL CASE

  • Under reciprocity, setting each beamforming phase opposite the sum of its two channel phases maximizes both rates for fixed beamformer amplitudes.The reciprocity assumption is associated with TDD relaying.Source: §III · p. 5
  • The usual weighted sum-rate maximization route has no derived closed-form solution, whereas the inverse-SNR formulation admits closed-form solutions.The received SNR numerators differ only by a scalar constant.Source: §III · p. 6
  • The achievable rate region can be constructed from the inverse-SNR region using closed-form solutions to a sequence of WSISMin problems.The inverse-SNR and rate pairs are bijectively mapped, enabling the inverse-SNR Pareto boundary to characterize the rate region.Source: §III · p. 6

A. Characterizing the Achievable Rate Region

The achievable rate region can be constructed from points obtained by weighted-sum inverse-SNR minimization. This follows by mapping the inverse-SNR Pareto boundary to the rate boundary and excluding mapped points unnecessary for the convex hull.

A. Characterizing the Achievable Rate Region

  • Inverse-SNR boundary points not obtained by WSISMin map to rate points unnecessary for constructing the achievable region by convex hulling.The proof identifies these mapped points as unnecessary, leaving the WSISMin-derived points sufficient for construction.Source: A. Characterizing the Achievable Rate… · p. 7
  • The inverse-SNR Pareto boundary maps to the rate-region Pareto boundary in both directions.The paper uses this correspondence to construct the rate region from the inverse-SNR region.Source: A. Characterizing the Achievable Rate… · p. 7
  • The points obtained by WSISMin are sufficient to construct the achievable rate region by convex hulling their mapped rate points.The paper therefore studies WSISMin solutions instead of directly studying the rate-region problems.Source: A. Characterizing the Achievable Rate… · pp. 7–8

B. Collaborative Beamforming under Sum-power Constraint

Under a relay-cluster sum-power constraint, the section solves weighted-sum inverse-SNR minimization to obtain beamforming vectors and rate pairs. It also describes a partially distributed implementation using a lookup table and relay-local channel information.

B. Collaborative Beamforming under Sum-power Constraint

  • For 0 < µ < 1, the optimal vector uses the full sum-power budget because scaling it up would decrease the objective.At µ = 0 or 1, the problem reduces to finding an optimal collaborative one-way relay beamforming vector.Source: B. Collaborative Beamforming under… · p. 8
  • The weighted inverse-SNR problem is rewritten as a Rayleigh-Ritz ratio to obtain the optimal solution.The formulation is developed from the SNR expressions under the sum-power constraint.Source: B. Collaborative Beamforming under… · p. 8
  • Searching over the weight µ yields optimal beamforming vectors whose rate pairs, after convex hulling, form the achievable rate region.The sum-power constraint bounds the relay cluster's total transmit power.Source: B. Collaborative Beamforming under… · pp. 8–9
  • A control center can look up the desired weight µ and broadcast it with one global constant, allowing each relay to calculate its coefficient from local channel information.The lookup table is indexed by µ, and the relay uses its local channels h1,i and h2,i.Source: B. Collaborative Beamforming under… · p. 9

C. Collaborative Beamforming under Individual-Power Constraints

Under individual relay-power limits, the paper derives a closed-form solution for the reciprocal-channel beamforming design. Its implementation is partially distributed: relays use local information and a broadcast global parameter, and some may transmit below maximum power.

C. Collaborative Beamforming under Individual-Power Constraints

  • The optimal solution to (44) has the threshold form α(k*) in (45), where k* is the smallest k satisfying the stated condition.For each given μ, (44) can be solved analytically.Source: C. Collaborative Beamforming under… · p. 10
  • The partially distributed implementation requires the control center to broadcast λ_k* alongside μ, while each relay determines φ_i using local information.Each relay then follows (45) to set its coefficient and transmit power.Source: C. Collaborative Beamforming under… · p. 11
  • Some relays may transmit below their maximum allowable power under the optimal solution.The individual-power design uses p_R,i ≤ p_i, with p_i the maximum allowable power for relay i.Source: C. Collaborative Beamforming under… · pp. 9, 11

IV. NON-RECIPROCAL CHANNEL CASE

For non-reciprocal channels, the reciprocal-channel approach does not apply, and the usual weighted-rate boundary problem is non-convex. The paper instead uses a rate-profile formulation and converts the beamforming design to an SDP after relaxation.

IV. NON-RECIPROCAL CHANNEL CASE

  • Without channel reciprocity, the approach used in the preceding section does not apply to characterizing the region boundary.The passage identifies FDD as a possible source of non-reciprocal channels.Source: §IV · p. 11
  • The rate-profile method is used to characterize the achievable-region boundary because the weighted-rate objective is non-concave.The profile specifies the rate proportions κ and 1 − κ.Source: §IV · p. 11
  • The rate-profile problem is converted to an SDP by relaxing the rank-one constraint on X.The constraint rank(X) = 1 follows from X = ww^H; the feasible set also retains the power-constraint assumptions.Source: §IV · p. 12

A. Sum-power Constrained Case

For non-reciprocal channels under a sum-power limit, the paper characterizes achievable rate profiles through relay-power minimization and feasibility checks. A bisection search over the target sum rate uses convex power-feasibility problems after removing the rank-one constraint.

A. Sum-power Constrained Case

  • The sum-power problem removes the rank-one constraint and uses bisection over the target sum rate, solving convex power-feasibility problems at each step.The search maintains lower and upper rate bounds until their gap is below ε.Source: A. Sum-power Constrained Case · pp. 12–13
  • If the minimum relay sum-power for target SNRs γ1 and γ2 is no greater than P_R, the SNR pair is achievable; otherwise it is not.The targets correspond to the rate profile for a given sum rate.Source: A. Sum-power Constrained Case · p. 13
  • The rate upper bound is set to twice the larger rate obtained by decoupling the two-way relay channel into two one-way relay channels.The one-way collaborative relay rate is derived from prior sum-power results.Source: A. Sum-power Constrained Case · p. 13

1) Rank-one solution:

For non-reciprocal channels, the sum-power relaxation admits exact rank-one reconstruction, whereas the individual-power case is handled through a sequence of convex feasibility problems. The latter uses bisection over the target rate under K individual relay-power limits.

1) Rank-one solution:

  • With only two linear constraints, a non-rank-one relaxed solution can be transformed into an exact rank-one solution for the sum-power case.The resulting non-reciprocal beamforming solution is fully centralized, not partially distributed.Source: 1) Rank-one solution · p. 13

B. Individual-Power Constrained Case

  • Under individual relay-power limits, the design solves a sequence of convex feasibility problems and bisects over the target rate.Feasibility at a given rate determines whether that rate is achievable, with the search narrowing an interval containing the optimum.Source: B. Individual-Power Constrained Case · pp. 14–15
  • The individual-power formulation has K separate relay-power constraints rather than one cluster-wide sum-power constraint.The per-relay transmit power is D_i,i X_i,i and is limited by P_i^R.Source: B. Individual-Power Constrained Case · p. 14
  • At the end of Algorithm 2, X_opt may not be rank-one, and the cited rank-one decomposition technique cannot be applied with K + 2 linear constraints.The passage does not state a rank-one reconstruction result for this individual-power case.Source: B. Individual-Power Constrained Case · p. 15

1) Rank-one solution based on randomization:

The paper presents lower-complexity suboptimal beamforming schemes alongside a randomization method for obtaining approximate rank-one solutions under individual-power constraints. Reciprocal and non-reciprocal channels use different phase and power choices.

1) Rank-one solution based on randomization:

  • For individual-power constraints, randomization generates candidate beamforming vectors from Xopt and selects one by minimizing constraint violations.The randB routine generates candidates while preserving the individual power levels; the resulting rate region is approximate.Source: 1) Rank-one solution based on… · pp. 15–16
  • Randomization provides only approximate solutions, so the corresponding rate region is not exact.Source: 1) Rank-one solution based on… · p. 15

V. SUB-OPTIMAL SCHEMES

  • The proposed suboptimal schemes reduce implementation complexity relative to the optimal schemes established earlier.In the reciprocal case, relays need only local channel information to set their transmit phase and coefficient.Source: §V · p. 16, A. Reciprocal Channel Case · p. 17

A. Reciprocal Channel Case

  • For reciprocal channels, equal-power beamforming uses equal relay powers under a sum-power constraint, while max-power beamforming uses each relay’s maximum allowable power under individual constraints.The relay phases are matched to the channels.Source: A. Reciprocal Channel Case · p. 16

B. Non-reciprocal Channel Case

  • For non-reciprocal channels, each relay greedily selects a phase to maximize its own contribution to one directional SNR, then uses equal or maximum power as constrained.Equal power is used for the sum-power constraint; maximum power is used for individual-power constraints.Source: B. Non-reciprocal Channel Case · p. 17

VI. NUMERICAL RESULTS

The numerical results compare achievable rate regions across channel reciprocity assumptions, power constraints, and suboptimal schemes. They report larger regions for reciprocal channels, while the non-reciprocal suboptimal schemes are effective only at small relay power.

Rate-region comparisons

  • The reciprocal-channel rate region is larger than the non-reciprocal region under the same power and noise settings.The authors attribute the difference to phase matching being possible in the reciprocal case but not the non-reciprocal case.Source: Figure 2 · p. 18, §VI · pp. 18–19

Suboptimal-scheme performance

  • Equal-power beamforming is strictly suboptimal for reciprocal channels with a sum-power constraint, while max-power beamforming approaches the boundary as the power budget falls.Source: Figure 3 · p. 18, §VI · p. 19, Figure 4 · p. 19

Rate-region comparisons

  • The individual-power rate region is smaller than the sum-power region, and the plotted region is symmetric under the assumed symmetric transmit powers and channel statistics.When S2 transmits no power, the rate pairs collapse to the horizontal-axis segment for the one-way link from S1.Source: §VI · p. 18, Figure 2 · p. 18

Suboptimal-scheme performance

Rate-region comparisons

  • Reciprocal channels permit phase matching to the overall channel, and the control center broadcasts one scalar rather than a K-dimensional complex beamforming vector.The paper says TDD is more favorable when channel coherence exceeds an operation period and transmit-receive chains are properly calibrated.Source: §VI · p. 19

VII. CONCLUSION

The conclusion summarizes how the achievable rate region is characterized under reciprocal and non-reciprocal channels and the two relay power constraints. It also reports a larger region for reciprocal channels and favors TDD-based relaying within the stated conditions.

VII. CONCLUSION

  • For reciprocal channels, Pareto-optimal beamforming vectors from WSISMin problems characterize the rate region and have closed-form solutions enabling partially distributed implementation.For non-reciprocal channels, the paper uses rate profiles with relaxed SDP methods and rank-one reconstruction or randomization, depending on the power constraint.Source: §VII · p. 20
  • The achievable rate region is larger for reciprocal than non-reciprocal channels, and the paper concludes that TDD-based relaying is more favorable for this collaborative-beamforming network.Source: §VII · pp. 20–21

A. Proof of Proposition 1

B. Proof of Proposition 2

The proof establishes that the inverse-SNR-to-rate mapping sends relevant line segments to convex curves. It uses this geometry to show that certain inverse-SNR boundary points are unnecessary for constructing the achievable region.

B. Proof of Proposition 2

  • Lemma 1 shows that the mapping U sends a positive, decreasing linear relation to a nonnegative, decreasing convex curve.The proof defines the mapping through y = log2(1 + 1/x) and p(y) = log2(1 + 1/q(x)).Source: B. Proof of Proposition 2 · p. 21
  • For a fixed weight, any boundary point outside S(µ, I) between two minimizers lies above the segment joining them.Thus, these points are not attainable by solving WSISMin.Source: B. Proof of Proposition 2 · p. 22
  • The omitted boundary portions are unnecessary for constructing O because convex hulling the mapped segments suffices.The proof applies this argument to the inverse-SNR boundary portions outside S(I).Source: B. Proof of Proposition 2 · p. 23
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