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Achievable rate regions for bi-directional relaying
Sang Joon Kim, Natasha Devroye, Patrick Mitran, Vahid Tarokh
TL;DR
The paper addresses half-duplex bidirectional relaying that does not fully exploit wireless broadcast and derives achievable regions for four new protocols. It specializes eight achievable regions and two outer bounds to Gaussian channels, finding that the preferred protocol depends on channel and SNR conditions.
Problem
Half-duplex relaying can fail to fully exploit the broadcast nature of the wireless channel when only a single node transmits at a time.
Method
The paper derives achievable rate regions for four new half-duplex bidirectional relaying protocols and specializes eight achievable regions and two outer bounds to the Gaussian case.
Results
The mixed TDBC protocol outperforms other proposed protocols under some channel conditions, while MABC generally outperforms TDBC at low SNR and TDBC reverses this relation at high SNR.
Takeaways & Limitations
The optimal relaying scheme depends on the channel and SNR regime, and mixed TDBC can be advantageous under particular channel conditions.
Abstract
from arXiv · showhide
In a bi-directional relay channel, two nodes wish to exchange independent messages over a shared wireless half-duplex channel with the help of a relay. In this paper, we derive achievable rate regions for four new half-duplex protocols and compare these to four existing half-duplex protocols and outer bounds. In time, our protocols consist of either two or three phases. In the two phase protocols, both users simultaneously transmit during the first phase and the relay alone transmits during the second phase, while in the three phase protocol the two users sequentially transmit followed by a transmission from the relay. The relay may forward information in one of four manners; we outline existing Amplify and Forward (AF), Decode and Forward (DF) and Compress and Forward (CF) relaying schemes and introduce the novel Mixed Forward scheme. The latter is a combination of CF in one direction and DF in the other. We derive achievable rate regions for the CF and Mixed relaying schemes for the two and three phase protocols. In the last part of this work we provide a comprehensive treatment of 8 possible half-duplex bi-directional relaying protocols in Gaussian noise, obtaining their respective achievable rate regions, outer bounds, and their relative performance under different SNR and relay geometries.
I. INTRODUCTION
The paper studies spectrally efficient half-duplex bi-directional relay protocols, deriving achievable rate regions and outer bounds while comparing protocol and relaying-scheme tradeoffs under varying channel conditions.
- Half-duplex operation restricts each node to transmitting or receiving at a given time, addressing practical limitations of simultaneous same-frequency transmission and reception.
- Protocol rate regions are not generally nested, so their relative benefits depend on channel conditions and relay processing capabilities.The paper reports that different protocols may be optimal under different channel conditions.
- The paper compares two temporal protocols: three-phase TDBC, which permits side information, and two-phase MABC, which exploits simultaneous multiple access but lacks that side information.TDBC uses a →r, b →r, and a ←r →b; MABC combines the first two phases into a →r ←b phase.
- Four relaying schemes are considered: AF, DF, CF, and Mixed Forward, which combines DF in one direction with CF in the other.AF forwards amplified noise, DF decodes both messages, and CF compresses the received signal without decoding the source messages.
- The work derives achievable regions for new CF and Mixed schemes in both TDBC and MABC, obtains cut-set outer bounds, and specializes eight regions and two bounds to Gaussian noise.The comparison includes existing regions and a simple AF scheme in Gaussian noise.
- Under some channel conditions, Mixed TDBC outperforms the other protocols, while CF TDBC has the best bound under other conditions.
II. PRELIMINARIES
The paper compares eight half-duplex bi-directional relaying protocols across two-phase MABC and three-phase TDBC settings. It formalizes rates, alphabets, phase durations, error events, and achievable regions for discrete-time memoryless channels.
- Protocol scope: Eight protocols combine AF, DF, CF, and Mixed forwarding across MABC and TDBC versions.The Mixed scheme uses DF in one direction and CF in the other.
- Channel model: The half-duplex constraint prevents a node from transmitting and receiving simultaneously.
- Protocol timing: Protocols use either L = 2 MABC phases or L = 3 TDBC phases, with relative durations Δ_ℓ.
- Notation and messages: Messages W_a and W_b are independent and uniformly distributed, with rates R_a and R_b.
- Compression model: Compress-and-forward schemes represent received signals using compressed variables Ŷ_i that need not equal the original outputs Y_i.
- Achievability: Achievability requires vanishing decoding error probabilities as blocklength n grows while phase durations converge to fixed values.
B. Previous results
The paper states prior outer bounds and achievable DF results, then introduces the CF and Mixed rate regions developed in the paper. It applies these schemes to both MABC and TDBC protocols.
- Existing results: Prior work provides outer bounds and DF achievable regions for both MABC and TDBC protocols.
- CF decoding: CF relaying does not decode w_a or w_b, so the schemes use joint typicality decoders instead of algebraic network coding.
- New results: The paper presents three new achievable rate regions in Theorems 8, 12, and 14, plus an improvement to the CF MABC region.
- MABC: MABC has simultaneous terminal transmissions in phase 1 followed by relay broadcasting in phase 2.
- Mixed and TDBC protocols: Theorem 8 mixes DF for w_a with CF for w_b, while the three-phase TDBC results use sequential terminal transmissions before relay transmission.
A. MABC Protocol
The MABC protocols let both terminals transmit simultaneously before the relay broadcasts, with CF or Mixed forwarding determining how the relay handles the two directions. The paper derives their achievable regions and analyzes decoding and Gaussian extensions.
- Decoding: The relay does not decode CF messages; each terminal uses joint typicality decoding and its own transmitted message as side information.
- CF region: The CF MABC achievable region is defined as the closure of rate pairs satisfying the theorem’s constraints over joint distributions.
- Mixed forwarding: The Mixed MABC strategy uses DF on the a → r → b link and CF on the b → r → a link, with Gelfand-Pinsker coding protecting w_a.
- Mixed region: Theorem 8 gives an achievable Mixed MABC region with phase-specific distributions and auxiliary variables.
- Comparison: The Mixed MABC region is contained within the DF MABC region, so it achieves no rate pair unavailable to DF, while potentially reducing relay decoding requirements.
B. TDBC Protocol
The TDBC protocol uses three phases in which the users transmit sequentially and the relay broadcasts in the final phase. The section develops CF and mixed relaying variants, including a mixed scheme that uses DF in one direction and CF in the other.
- Protocol structure: TDBC has three phases: node a transmits, node b transmits, and the relay then broadcasts to both nodes.The relay and the other node receive during each user’s transmission phase.
- CF relaying: The CF TDBC protocol divides the relay broadcast into two sub-phases using distinct broadcast strategies.One sub-phase uses Marton’s broadcast scheme without receiver side information; the other uses a compound-channel transmission with different side information.
- CF relaying: The CF TDBC achievable region is characterized by a joint distribution covering the two user transmissions, relay compression, and relay broadcasting.The distribution includes phase-specific channel laws and a cardinality bound |Q| ≤13.
- Mixed relaying: The mixed TDBC protocol uses DF on the a →r →b link and CF on the b →r →a link.This design targets channels whose two directional links have different strengths.
- Comparisons: The mixed TDBC protocol can achieve a larger rate region than DF TDBC in cases where the r ↔a channel is very strong.The section also states that mixed TDBC is not outer bounded by the DF TDBC region in contrast to the MABC comparison.
IV. GAUSSIAN CASE
The Gaussian-case analysis applies the preceding half-duplex relaying results to reciprocal complex Gaussian channels with power constraints and unit-power Gaussian noise. It evaluates AF, DF, CF, and Mixed schemes for MABC and TDBC protocols and applies protocol outer bounds.
- Gaussian extension: Gaussian input distributions and a continuous-domain Markov lemma transfer the preceding achievable-region arguments to Gaussian channels.Strong typicality does not directly apply to continuous domains, so the continuous-domain extension is used.
- Channel model: The Gaussian model assumes reciprocal channel gains, full channel-state information, unit-power circularly symmetric complex Gaussian noise, and average transmit-power constraints.The effective gain between transmitter i and receiver j is denoted h_ij.
- Schemes and bounds: The analysis considers AF, DF, CF, and Mixed relaying for both MABC and TDBC protocols, together with their outer bounds.This yields a comprehensive comparison of four relaying schemes across the two half-duplex protocol families.
A. Amplify and Forward
The AF analysis fixes equal phase durations because relaying operates on a symbol-by-symbol basis, then derives Gaussian achievable rate regions under relay power scaling.
- AF construction: AF uses equal phase durations because the relay performs symbol-by-symbol relaying.The relay scales its received symbol to satisfy its transmit-power constraint.
- AF regions: The Gaussian AF achievable rate regions are obtained for the considered half-duplex protocols.The regions are expressed using channel gains and transmit powers.
B. Decode and Forward
The Gaussian DF section applies the earlier DF results and optimizes phase durations for the given channel mutual informations to maximize the achievable rate regions.
- DF regions: The DF achievable rate regions follow by applying the preceding theorems to the Gaussian channel.The resulting regions are evaluated using Gaussian input distributions.
- Optimization: The phase durations are optimized for the given channel mutual informations to maximize the achievable rate regions.The optimization is performed over the Δ_l parameters.
C. Compress and Forward
The Gaussian analysis applies prior theorems to obtain achievable rate regions and shows that the relevant outer bounds coincide with Gaussian broadcast-channel capacity regions.
- Applying Theorems 5 and 12 yields achievable rate regions for the Gaussian case.
- Under Costa’s setup with |h_ra| > |h_rb|, the outer bound is equivalent to the Gaussian broadcast-channel capacity region.
- The corresponding bounds for |h_ra| ≤ |h_rb| are also the Gaussian broadcast-channel capacity region.
- The Gaussian regions are numerically evaluated after optimizing the phase powers P^(ℓ).
D. Mixed Forward
The Mixed Forward analysis constructs Gaussian relay-broadcasting channels for Mixed MABC and Mixed TDBC, then optimizes their parameters and phase powers to obtain rate regions and outer bounds.
- Costa’s setup is used to model the relay-broadcasting phase of the Mixed MABC protocol.
- The Mixed MABC channel uses auxiliary variables V_r and U_b with parameter β constrained to 0 ≤ β ≤ 1.
- An analogous relay-broadcasting channel is constructed for the Mixed TDBC protocol.
- The achievable rate regions are numerically optimized over α, β, and the phase powers P^(ℓ) to maximize their boundary.
- Gaussian outer bounds are obtained from Theorems 1 and 2 by optimizing the phase durations Δℓ for given channel mutual informations.
V. ACHIEVABLE RATE REGIONS IN THE GAUSSIAN CHANNEL
The Gaussian evaluation compares ten half-duplex protocols and outer bounds across symmetric and asymmetric channel configurations, SNRs, relay positions, and rate constraints. The results show that protocol performance depends on channel conditions, with distinct advantages for MABC, TDBC, DF, CF, AF, and Mixed schemes in different regimes.
- Evaluation setup: The comparison includes AF, DF, CF, Mixed, and outer-bound variants of both MABC and TDBC protocols.
- Evaluation setup: Ten schemes are compared using achievable regions, outer bounds, maximal sum-rates, constrained sum-rates, SNR, relay position, and symmetric or asymmetric channel gains.
- Main conclusion: Different schemes are optimal under different channel conditions.
- Symmetric case: In the low-SNR symmetric case, DF MABC dominates the other protocols, while MABC generally outperforms TDBC.
- Symmetric case: CF does not outperform DF in TDBC because DF uses two parallel channels while CF uses one channel with two receivers.
- Symmetric case: In high SNR, CF MABC outperforms DF MABC because simultaneous transmissions create interference for DF but not for CF.
- Asymmetric cases: The Mixed TDBC region lies between the CF TDBC and DF TDBC regions, while DF TDBC dominates at high SNR in an asymmetric case with a strong direct link.
- Power assumption: Under full-power transmission, increasing interference with SNR widens the intercept-point gap between Mixed and DF schemes; optimizing transmission power could enlarge the Mixed MABC region.
2) Asymmetric Cases:
Across asymmetric channels and relay positions, protocol performance depends on SNR, link strengths, forwarding scheme, and rate constraints. MABC tends to favor low SNR, while TDBC tends to favor high SNR, with mixed schemes showing direction-dependent behavior.
- Asymmetric channel comparisons: When har > hbr, mixed TDBC achieves a larger region because the first link is more critical to DF performance; the difference decreases as SNR increases.The opposite asymmetric ordering produces a smaller mixed-forwarding region under the stated input assumptions.
- SNR dependence: In the symmetric case, DF MABC is better below about 12 dB, CF MABC is better above it, and AF MABC is always worse than CF MABC.The mixed TDBC sum-rate lies between the DF and CF schemes in the symmetric comparison.
- Relay position: With σ = 2, DF MABC has a strongly asymmetric sum-rate that peaks and nearly reaches the outer bound at ζ = 0.6.The relay-position optimizations are intended to help determine favorable placement under particular rate constraints.
- Overall comparison: Overall, the MABC protocol generally outperforms TDBC at low SNR, whereas TDBC generally outperforms MABC at high SNR.The paper evaluates eight Gaussian half-duplex protocols and two outer bounds under varied channel conditions.
APPENDIX I
The appendix presents the random coding, encoding, decoding, and error-analysis steps for an achievable-rate construction. It uses codebooks, compression, binning, and typicality arguments to establish reliability under the stated conditions.
- Code construction: The construction generates independent codebooks for the users, relay compression, and relay transmission, with message indices assigned to randomly generated bins.The relay codebooks include u_r^(3)(w_r) and compressed descriptions indexed by w_r′.
- Encoding: During the phases, the users transmit codewords, the relay estimates or decodes one message, compresses a received signal, and forwards a relay codeword.The relay selects a bin using the decoded message and searches for a jointly typical relay codeword before transmission.
- Decoding: The destination nodes decode relay indices and user messages using side information, bin indices, and joint typicality.Knowing its own message reduces the relay-index search space at a destination.
- Error analysis: The error analysis partitions failures by phase and decoding ambiguity, then bounds the resulting error probability through typicality and covering arguments.The listed events include missing compression indices, missing relay codewords, and competing message pairs.
- Reliability: As n →∞, the relevant error-probability terms vanish when the theorem conditions and rate constraints hold.The appendix also restricts the time-sharing alphabet to |Q| ≤ 6 using Carathéodory's theorem.