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On the Capacity and Diversity-Multiplexing Tradeoff of the Two-Way Relay Channel
Rahul Vaze, Robert W. Heath
TL;DR
The paper addresses capacity and diversity-multiplexing tradeoffs for MIMO two-way relay channels with multiple relays. It develops optimal and locally implementable AF strategies, bounds their achievable regions, and studies compress-and-forward strategies for diversity-multiplexing tradeoff. The results establish constant-gap capacity scaling with many relays and optimal diversity-multiplexing performance in the stated full-duplex and selected half-duplex settings.
Problem
The paper addresses capacity and diversity-multiplexing tradeoffs for MIMO two-way relay channels with multiple relays.
Method
It uses an iterative power-minimization algorithm for optimal AF beamformers, proposes local-CSI dual channel matching, and studies compress-and-forward strategies.
Results
The dual channel matching region is close to optimal AF, differs from an upper bound by a constant term asymptotically, and compress-and-forward achieves optimal diversity-multiplexing tradeoff for full duplex and some half-duplex cases.
Takeaways & Limitations
Dual channel matching simplifies CSI requirements while retaining near-optimal achievable rates, and compress-and-forward attains the stated diversity-multiplexing tradeoffs.
Takeaways & Limitations
The optimal AF strategy is restricted to single-antenna terminals and cannot be extended easily to the multi-antenna case.
Abstract
from arXiv · showhide
This paper considers a multiple input multiple output (MIMO) two-way relay channel, where two nodes want to exchange data with each other using multiple relays. An iterative algorithm is proposed to achieve the optimal achievable rate region, when each relay employs an amplify and forward (AF) strategy. The iterative algorithm solves a power minimization problem at every step, subject to minimum signal-to-interference-and-noise ratio constraints, which is non-convex, however, for which the Karush Kuhn Tuker conditions are sufficient for optimality. The optimal AF strategy assumes global channel state information (CSI) at each relay. To simplify the CSI requirements, a simple amplify and forward strategy, called dual channel matching, is also proposed, that requires only local channel state information, and whose achievable rate region is close to that of the optimal AF strategy. In the asymptotic regime of large number of relays, we show that the achievable rate region of the dual channel matching and an upper bound differ by only a constant term and establish the capacity scaling law of the two-way relay channel. Relay strategies achieving optimal diversity-multiplexing tradeoff are also considered with a single relay node. A compress and forward strategy is shown to be optimal for achieving diversity multiplexing tradeoff for the full-duplex case, in general, and for the half-duplex case in some cases.
1 University Station C0803 Austin, TX 78712-0240
The work was funded by DARPA through the IT-MANET program. Figure 1 presents the two-way relay channel communication protocol.
- DARPA funded this work through IT-MANET grant no. W911NF-07-1-0028.
- Figure 1 depicts the two-way relay channel communication protocol.
I. INTRODUCTION
The paper studies capacity and diversity-multiplexing tradeoffs in multiple-antenna two-way relay channels with multiple relays. It develops AF beamforming methods, a local-CSI dual channel matching strategy, capacity bounds, and relay strategies for optimal diversity-multiplexing tradeoff.
- The capacity region of the two-way relay channel remains open, especially with multiple relay nodes.
- The paper seeks optimal AF relay beamformers for multiple-relay two-way channels, where AF is the simplest strategy suited to multiple relays.
- For single-antenna terminals, an iterative power-minimization algorithm finds optimal beamformers under minimum-SINR constraints, with KKT conditions sufficient for optimality.
- The optimal AF solution requires global CSI at every relay and lacks a closed-form achievable rate region.
- Dual channel matching uses local CSI, supports any number of terminal antennas, and has an achievable rate region close to optimal AF when both terminals have one antenna.
- As the relay count K approaches infinity, the gap between dual channel matching and an upper bound remains constant, yielding a capacity scaling law with M/2 log K bits transmitted simultaneously in both directions.
- A modified compress-and-forward strategy achieves the optimal diversity-multiplexing tradeoff for full-duplex channels and in some half-duplex cases, while general half-duplex optimality remains conjectured.
II. SYSTEM AND CHANNEL MODEL
This section introduces the two-way relay channel system and signal models. The model uses two-phase communication between terminals through relays.
- The system model describes a two-way relay channel with terminals exchanging information through relays.
- Figure 2 depicts the two-way relay channel system model with two-phase communication.
A. System Model
The paper models two terminals exchanging information through multiple relays, primarily without a direct terminal-to-terminal path. It specifies half-duplex operation, antenna configurations, relay power constraints, and a separate single-relay model with a direct path.
- Multi-relay setting: T1 and T2 exchange information through K relays that have no data of their own.The relay-only path models coverage improvement and communication between terminals outside each other’s transmission range.
- Multi-relay setting: For Sections III–V, both terminals have M antennas, each of K relays has N antennas, and all nodes operate in half-duplex mode.Nodes cannot transmit and receive simultaneously.
- Communication protocol: During the transmit phase, both terminals send and all relays receive; during the receive phase, all relays transmit and both terminals receive.The phases occupy α and 1 − α fractions of each time slot, respectively.
- Power constraints: Terminal powers are constrained by P, while relays obey either a sum-power constraint or an individual power constraint PR.Under the sum constraint, total relay power is at most PR; under the individual constraint, each relay has power at most PR.
- Single-relay extension: Section VI considers one relay with a direct T1–T2 path, using m1, m2, and mr antennas at T1, T2, and the relay.This model differs from the multi-relay setting by including direct terminal communication.
B. Channel and Signal Model
The channel model uses slow, frequency-flat block fading and amplify-and-forward relay processing. It specifies relay and terminal signals, power constraints, noise, and distinct CSI assumptions for optimal and simplified relay strategies.
- Channel assumptions: All channels are frequency-flat slow-fading block-fading channels whose coefficients remain constant within a coherence block and change independently between blocks.The coherence time is denoted Tc.
- Signal model: Relay k receives the superposition of terminal signals and noise, then transmits tk = Wkrk using an amplify-and-forward transformation.For single-antenna relays, the corresponding processing coefficient is wk.
- CSI assumptions: Both terminals know the relevant relay channel matrices during reception, while no transmit CSI is available at either terminal.The terminals lack information about the channel realizations when transmitting to the relays.
- CSI assumptions: The optimal AF strategy gives each relay global CSI, whereas the simplified AF strategy lets relay k use only its own channel information.Section VI separately assumes CSI for the single relay and the direct channel H12.
III. OPTIMAL AF STRATEGY FOR TWO-WAY RELAY CHANNEL
The optimal AF strategy seeks relay beamformers on the achievable-rate-region boundary by iteratively solving power minimization problems with SINR constraints. KKT conditions make the non-convex subproblem efficiently solvable, while the strategy’s global-CSI and analytical limitations motivate dual channel matching.
- Objective and rate region: The strategy optimizes AF relay beamformers to maximize the achievable rate region under sum-power and individual-power constraints.The boundary is parameterized by β, with R12 = βRsum and R21 = (1 − β)Rsum.
- Iterative optimization: The boundary search uses an iterative power minimization formulation that varies the target sum rate and checks relay-power feasibility.The target is increased when feasible and decreased otherwise, with step size affecting convergence speed.
- Iterative optimization: The reformulated problem is generally non-convex, but forwarded relay noise can be treated as interference in standard SINR-constrained power minimization.This recasting provides the optimization structure used by the algorithm.
- Optimality: KKT conditions are sufficient for optimality for the strictly feasible SINR-constrained problem, enabling an efficient solution for the optimal relay beamformers.The method extends to multiple relay antennas by replacing scalar relay coefficients with beamforming matrices.
- Limitations and motivation: The optimal AF algorithm assumes every relay knows all relay-channel CSI and initially treats single-antenna terminals, while its rate-region expression lacks closed-form analytical tractability.The paper therefore introduces dual channel matching, which uses local CSI and has a closed-form rate-region expression for comparison with bounds.
- Limitations and motivation: Dual channel matching is a generally suboptimal AF strategy whose achievable rate region lower-bounds that of the optimal AF strategy and supports comparison with an upper bound.Its closed-form expression makes the loss relative to the upper bound analytically tractable.
IV. DUAL CHANNEL MATCHING STRATEGY
The paper proposes dual channel matching, a simple AF strategy using local CSI, and derives its achievable rate region and asymptotic behavior. As the number of relays grows, both directional rates scale as M/2 log K, while the strategy approaches the capacity upper bound within O(1).
- Strategy: Dual channel matching is a simple AF strategy that uses local CSI and provides a lower bound on the achievable rate region.It applies conjugate forward and backward channel coefficients directly rather than the SVD unitary matrices.
- Signal model: The strategy restricts terminal signals to circularly symmetric complex Gaussian inputs and uses equal time for transmission and reception.The achievable rates R12 and R21 are then obtained from the resulting received-signal expressions.
- Achievable region: The resulting rate-region expression is analytically tractable and supports comparison with the optimal AF strategy and capacity upper bound.The paper also studies how this region changes with the number of relays K.
V. UPPER BOUND ON THE TWO-WAY RELAY CHANNEL CAPACITY
The paper derives a cut-set upper bound on the two-way relay channel capacity by separating each terminal from the network and analyzing broadcast and multiple-access cuts. Comparing this bound with dual channel matching shows an O(1) asymptotic gap and yields the capacity scaling law.
- Cut-set bound: The cut-set bound separates T1 and T2 from the network to upper-bound R12 and R21 through broadcast and multiple-access cuts.The broadcast-cut bounds use mutual information from each terminal to the collaborating relays, while the multiple-access cuts bound relay-to-terminal flow.
- Broadcast cut: Allowing relay cooperation produces upper bounds based on maximum information flow across the broadcast and multiple-access cuts.For the broadcast cut, each directional rate is bounded by the corresponding terminal-to-relays information flow without opposite-terminal interference.
- Asymptotic comparison: The upper and lower capacity-region bounds do not match for arbitrary K, but differ by only O(1) as K →∞.The lower bound is achieved by dual channel matching.
- Finite-relay comparison: For M = 1 and N = 1, dual channel matching achieves rates quite close to optimal AF despite using only local CSI.The paper illustrates the comparison for K = 2 and K = 4.
- Half-duplex implication: With half-duplex terminals and relays, the two-way relay channel can achieve unidirectional full-duplex performance under the stated scaling result.The receive phase is identified as optimal for achieving the right capacity scaling.
VI. DIVERSITY-MULTIPLEXING TRADEOFF
The paper characterizes the diversity-multiplexing tradeoff for a single-relay MIMO two-way relay channel in both full-duplex and half-duplex settings. It derives upper bounds and proposes a modified compress-and-forward strategy to achieve them.
- Scope: The section studies the DM-tradeoff of a single-relay two-way relay channel with full-duplex and half-duplex terminals and relays.The terminals have m1 and m2 antennas, while the relay has mr antennas.
- Model: A direct link between T1 and T2 is included in the single-relay model.This distinguishes the setting from the preceding sections.
- Analysis: The analysis first derives a DM-tradeoff upper bound for both duplexing modes.The full-duplex case is discussed before the half-duplex case.
- Strategy: A modified compress-and-forward strategy is proposed to achieve the DM-tradeoff upper bound.The strategy is used after obtaining the upper bound.
A. DM-tradeoff of Full-Duplex Two-Way Relay Channel
For the full-duplex two-way relay channel, the paper upper-bounds the DM-tradeoff using cooperative MIMO cut arguments and adapts compress-and-forward to the two-way setting. The modified strategy achieves the upper bound for both communication directions.
- Definitions: The DM-tradeoff measures diversity gain as the negative SNR exponent of error probability while rates scale with multiplexing gains.Because transmission is simultaneous, each direction's error probability depends on both r12 and r21.
- Upper bound: The upper bound allows cooperation between T1 and the relay, and between T2 and the relay, converting each case into a point-to-point MIMO channel.For T1 to T2, the two resulting bounds are (m1 − r12)(mr + m2 − r12) and (m1 + mr − r12)(m2 − r12).
- Upper bound: The resulting directional bounds are d12(r12, r21) ≤ min{(m1 − r12)(mr + m2 − r12), (m1 + mr − r12)(m2 − r12)} and its symmetric counterpart.The bounds hold for all r12 and r21.
- Compress-and-forward: A modified two-way compress-and-forward strategy generates codebooks at both terminals and uses relay compression decoded at both terminals.Each terminal knows its own transmitted codeword and uses that knowledge during decoding.
- Result: The modified compress-and-forward strategy achieves the DM-tradeoff upper bound.The paper computes outage exponents for the achievable rates and shows that they match the upper-bound exponents.
B. Half-Duplex Two-Way Relay Channel
The half-duplex analysis derives upper and lower diversity-multiplexing tradeoff bounds for a three-phase protocol and identifies when compress-and-forward is optimal. The bounds match under specific bottleneck conditions, but optimality remains unresolved in the general case.
- Protocol and bounds: The three-phase protocol allocates time for T1 transmission, T2 transmission, and relay transmission to both terminals, using all direct links.The rates R12 and R21 are upper bounded for this protocol.
- Protocol and bounds: The upper-bound derivation allows collaboration between the relay and the receiving terminal, while alternative terms add the maximum mutual information available at that terminal.The resulting expressions define an upper bound on the half-duplex channel’s DMT.
- Compress-and-forward strategy: The proposed compress-and-forward strategy jointly compresses relay observations from the first two phases and must satisfy a compression-rate constraint.The compression signal is jointly typical with the relay’s received signals from both phases.
- Optimality conditions: The lower and upper DMT bounds do not match generally, but compress-and-forward is optimal when the broadcast cut is the bottleneck.The theorem gives corresponding conditions for communication in both directions.
- Discussion: For the full-duplex channel, compress-and-forward achieves optimal DMT generally, whereas for half-duplex it is optimal only in some cases.The half-duplex achievable rate region and DMT depend on the communication protocol.
- Discussion: General half-duplex optimality is not established because different mutual-information quantities and optimization over phase durations make comparison difficult.The authors state that they believe the proposed strategy should be optimal generally, but do not prove it.
VII. CONCLUSION
The paper develops AF relay strategies for MIMO two-way relay channels, addressing optimal beamforming, CSI requirements, achievable-rate gaps, capacity scaling, and diversity-multiplexing tradeoffs.
- An iterative algorithm computes optimal relay beamformers for single-antenna terminals with arbitrary relay antennas by solving power minimization problems under SINR constraints.Although each problem is non-convex, satisfying the KKT conditions is sufficient for optimality.
- The optimal AF strategy maximizes the AF rate region but is limited to single-antenna terminals, requires global CSI at each relay, and lacks a closed-form achievable-rate expression.
- Dual channel matching relaxes the single-antenna and global-CSI restrictions by requiring local CSI and supporting any number of terminal antennas.Its achievable-rate region has a closed-form expression and is close to optimal AF for single-antenna terminals.
- For finite relay counts, simulations show a small gap between dual channel matching and an upper bound, while for K →∞ the gap is only a constant term.This asymptotic result establishes the capacity scaling law for the two-way relay channel.
- Two-way communication yields a two-fold capacity increase over unidirectional communication under the established scaling law.
- With a single relay and a direct terminal path, compress-and-forward achieves the optimal diversity-multiplexing tradeoff in full-duplex operation and in some half-duplex cases.The half-duplex result uses a modified compress-and-forward strategy with a three-phase transmission protocol.