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Recovering Multiplexing Loss Through Successive Relaying Using Repetition Coding
Yijia Fan, Chao Wang, John Thompson, H. Vincent Poor
TL;DR
Half-duplex relay networks can lose multiplexing performance, while practical repetition-coded relaying must also address interference between relays. This paper analyzes a two-relay successive-relaying protocol, derives achievable rates, and develops V-BLAST detection; it reports multiplexing-loss recovery with retained diversity or combining gain and advantages over the classic protocol under the stated conditions.
Problem
Half-duplex relay protocols can incur multiplexing loss, while practical repetition-coded relaying must handle co-channel interference between relays and at the destination.
Method
The paper analyzes successive relaying with repetition coding, derives achievable rates under channel and interference constraints, and develops low-complexity V-BLAST detection.
Results
The protocol recovers multiplexing loss while retaining diversity or combining gain, and offers significant performance advantages over the classic protocol in the analyzed scenarios.
Takeaways & Limitations
Successive repetition-coded relaying can provide a practical rate-diversity tradeoff, with especially significant advantages when relay interference is sufficiently strong.
Abstract
from arXiv · showhide
In this paper, a transmission protocol is studied for a two relay wireless network in which simple repetition coding is applied at the relays. Information-theoretic achievable rates for this transmission scheme are given, and a space-time V-BLAST signalling and detection method that can approach them is developed. It is shown through the diversity multiplexing tradeoff analysis that this transmission scheme can recover the multiplexing loss of the half-duplex relay network, while retaining some diversity gain. This scheme is also compared with conventional transmission protocols that exploit only the diversity of the network at the cost of a multiplexing loss. It is shown that the new transmission protocol offers significant performance advantages over conventional protocols, especially when the interference between the two relays is sufficiently strong.
I. INTRODUCTION
The paper develops a practical two-relay repetition-coded protocol to recover multiplexing lost by half-duplex relaying while retaining diversity or combining gain. It analyzes achievable rates, interference conditions, adaptive choices, and V-BLAST detection.
- Protocol and motivation: Conventional repetition-coded relaying can achieve full diversity but requires orthogonal relay transmissions, causing significant multiplexing loss.Prior simultaneous-relay approaches recover the multiplexing factor to 1/2, but remain spectrally inefficient at high SNR.
- Protocol and motivation: The protocol uses repetition-coded digital relaying with two relays transmitting successively in turn while the source transmits continuously.L codewords are transmitted in L + 1 time slots and jointly decoded at the destination.
- Implementation and adaptation: The proposed protocol offers significant advantages over the classic protocol, especially when the two relays are close to each other.The conclusions report advantages in both low- and high-SNR regions for the stated scenario, while source-relay strengthening increases capacity for both protocols.
- Implementation and adaptation: A low-rate-feedback V-BLAST decoder is proposed to approach theoretical achievable rates in slow-fading environments.The analysis also derives channel constraints and achievable rates for different source-relay and interference conditions.
- Achievable performance: Each codeword is received through both direct and delayed relay links, allowing diversity or combining gain while recovering multiplexing loss.The paper specifically reports a maximal diversity gain of two with almost no multiplexing loss in one analyzed scenario.
C. Relations to Previous and Concurrent Work
The paper positions successive digital relaying as a practical alternative to more theoretical schemes, incorporating the direct link and relay interference into its analysis. Its protocol continuously transmits source codewords while two relays forward them successively, and the destination jointly decodes the resulting observations.
- Relation to prior work: Earlier successive-relaying studies considered amplify-and-forward or omitted the direct link, limiting their treatment of cooperative diversity and achievable rates.The paper states that including the direct link yields a diversity gain of 2 and changes the rate and signalling analysis.
- Relation to prior work: This work analyzes capacity under channel and interference constraints not given in the cited prior study and uses a V-BLAST decoder.The paper describes its capacity analysis as more general and says it contains the earlier scenario as a special case.
- Protocol design: The protocol uses a four-node network in which two relays decode and forward successive source codewords while the destination receives direct and relay signals.The schedule alternates relay forwarding and source transmission across L + 1 slots before joint decoding all L codewords.
- Protocol design: In each intermediate slot, one relay forwards the previous codeword while the other decodes the next source codeword under interference.The destination simultaneously receives the forwarded and direct codewords, and the process repeats through slot L.
- Protocol design: The protocol has multiplexing ratio L/(L + 1), approaching 1 for large L, while the destination receives two copies of each codeword.This structure preserves the possibility of diversity gain but creates co-channel interference at the relays and destination.
III. ACHIEVABLE RATES
The achievable-rate analysis assumes fading and power conditions, constrains source-relay rates for correct relay decoding, and treats relay interference according to channel-dependent cancellation or noise-limited decoding.
- Assumptions and notation: The analysis considers adaptive transmission with receiver CSI feedback, capacity C(x) = log2(1 + x), and equal transmit power.It uses a slow, flat, block-fading model in which channels remain static over each L + 1-slot frame, while noting applicability to faster flat fading.
- Source-relay constraints: Relay decoding requires the source transmission rate to remain below the Shannon capacity of the source-relay channels.The achievable rates are indexed by relay choices and codewords through the relay index vector and rates R_i.
- Relay interference: Interference among relays is a major protocol defect because a transmitting relay interferes with the other relay’s reception of the source signal.The intermediate-slot network resembles a two-user Gaussian interference channel, whose optimal solution remains open.
- Relay interference: When relay interference is stronger than the desired signal, the receiving relay can decode and subtract the interference before decoding its desired source signal.Otherwise, it may decode the desired signal while treating interference as Gaussian noise, producing an additional rate constraint.
- Relay interference: The achievable rate depends on channel conditions linking the source, relays, and destination, with distinct constraints for cancellation and direct interference-as-noise decoding.The relevant inequalities are adapted to consecutive rates R_i−1 and R_i across the transmission slots.
C. Space-Time Processing at the Destination
The relay network is represented as a multiple-access MIMO channel in the time domain, enabling capacity analysis through rate constraints. The resulting bound is described as tight and achievable with space-time V-BLAST detection in slow fading.
- Channel representation: The relay network’s input-output relation is equivalent to a multiple-access MIMO channel with time-expanded signal, received, and noise vectors.The received vector has dimension (L + 1) × 1, while the transmitted signal vector has dimension L × 1.
- Achievability: The sum-capacity bound is extremely tight and achievable with space-time V-BLAST decoding at the destination in a slow fading scenario.The V-BLAST algorithm decodes the signals under the stated channel condition.
- Capacity bound: The combined constraints provide a way to calculate the proposed protocol’s network capacity upper bound.The analysis combines individual rate constraints with the sum-capacity constraint.
- Rate calculation: Rate constraints are imposed for each transmitted codeword and updated across time slots according to relay decoding criteria.A logical if statement selects the relay decoding order and determines which constraint applies.
I + HHHSNR
The capacity analysis identifies relay interference as a major constraint but shows that sufficiently strong interference can be decoded and canceled. Under additional channel conditions, the proposed protocol attains a MIMO-like rate with multiplexing scaling.
- Interference conditions: Relay interference can significantly degrade network capacity, but sufficiently strong interference can allow interference decoding and subtraction without affecting overall capacity.This behavior follows the Gaussian interference-network result discussed for the model.
- Capacity comparison: The proposed rate bounds are significantly larger than those of the comparison bounds.The passage refers specifically to the bounds provided by equation (15).
- Capacity result: Under the stated conditions, the rate equals the MIMO channel capacity with a multiplexing scaling.The result is conditioned on inequalities (14) and (16), which ensure correct relay decoding without affecting network capacity.
- Relay selection: Choosing nearby fixed relays can satisfy both required conditions, and relay selection may provide higher capacity than the classic multicast relay protocol at high SNR.The paper suggests combining the proposed protocol with relay-selection techniques.
F. The V-BLAST Algorithm
The paper applies low-rate-feedback V-BLAST MMSE detection to signals multiplexed across time rather than space. Under sufficiently strong interference and source-to-relay channels, this detector can achieve the proposed protocol’s target rate.
- Detection method: The destination uses low-rate-feedback V-BLAST MMSE detection, iteratively decoding the strongest signal and subtracting it from the received vector.The detector performs one iteration per signal stream while treating undecoded signals as interference.
- Space-time processing: Unlike traditional MIMO, the protocol independently encodes signal streams along the time dimension.The time-expanded representation preserves the multiple-access MIMO capacity structure.
- Rate adaptation: The SINR for each signal is used to set a transmission rate below log2(1 + SINR_i).Adaptive modulation and coding is proposed for this rate selection.
- Achievable rate: V-BLAST can achieve rate (19) when the relay-interference and source-to-relay channels are sufficiently strong.The relevant conditions are stated to be more likely than the earlier capacity conditions.
- Implementation conditions: The V-BLAST application requires signalling overhead and is limited to slow fading over at least L + 1 transmission slots.SINR information must be fed back before transmission begins.
G. Comparison with Classic Protocols
Classic relay protocols use additional time slots and repetition or distributed Alamouti coding to exploit diversity, incurring multiplexing loss. The proposed successive-relaying approach is positioned as a higher-capacity alternative, especially with suitable relay selection.
- Classic Protocol I: Classic Protocol I divides each message transmission into three time slots, with each relay decoding, re-encoding, and retransmitting the message.The destination combines signals received across all three slots.
- Classic Protocol I: Classic Protocol I incurs a multiplexing loss compared with direct transmission.The passage identifies the loss through the protocol’s capacity expression.
- Protocol comparison: Classic Protocol II has the same diversity gain as Classic Protocol I but reduced multiplexing loss.The comparison is made directly between the two classic protocols.
- Proposed protocol: The proposed protocol’s multiplexing ratio relative to direct transmission is L/(L+ 1).The paper also considers combining relaying with direct transmission and relay selection.
3) Performance Comparison:
The proposed successive relaying protocol is compared with classic protocols through achievable-capacity and diversity–multiplexing analyses. Larger relay-block lengths reduce multiplexing loss, while the protocol can retain diversity gain under relay-decoding conditions.
- Capacity comparison: The proposed protocol can achieve its best capacity with higher probability than classic protocols under the stated relay conditions.This follows from the relative likelihood of conditions (16) and (31), which determine whether the respective capacity bounds apply.
- Capacity comparison: Larger values of L reduce multiplexing loss and provide higher capacity gains as SNR increases.The capacity gain is plotted against SNR for different L values.
- Diversity–multiplexing tradeoff: The diversity–multiplexing analysis assumes that the relays correctly decode the transmitted signals, characterizing the scheme’s possible performance.The analysis is for a slow-fading scenario and is conditioned on relay decoding.
- Diversity–multiplexing tradeoff: A maximal diversity gain of 2 is achievable, while the multiplexing gain approaches 1 for large L.The result implies a spectral-efficiency advantage for the successive relaying scheme.
- Scope and adaptation: The theorem provides a high-SNR upper bound because the probability that condition (14) holds decreases as SNR increases.The same theorem extends to faster fading when signals transmitted in each slot are independently encoded, and adaptive protocols can preserve the tradeoff performance.
2. In case III, the relays are located
The simulations compare the proposed protocol with direct transmission and Classic Protocol II across network geometries. Performance depends on source–relay strength and relay interference, with V-BLAST approaching the capacity bounds.
- Simulation setup: V-BLAST approaches the capacity bounds in all three simulated geometries.This supports the use of low-complexity V-BLAST detection to implement the proposed protocol.
- Geometric effects: Relaying is generally unhelpful when source–relay and direct links have similar quality because the relaying link gain is small.Even in these cases, the proposed protocol retains a performance gain over direct transmission at both low and high SNR.
- Geometric effects: When source–relay links are stronger and relay interference is sufficiently strong, the relays can support interference-free transmission.In this geometry, the proposed protocol gains from combining while incurring negligible multiplexing loss.
- Geometric effects: The classic protocol can perform worse than direct transmission at high SNR because of its significant multiplexing loss.Its low-SNR performance advantage over direct transmission is improved, but the high-SNR penalty remains.
- Conclusions: The analysis concludes that successive relaying retains combining or diversity gain while recovering the multiplexing loss associated with classic relaying.Across geometries, capacity increases with stronger source–relay links, while the proposed protocol offers advantages at both low and high SNR.
APPENDIX
The appendix derives the diversity–multiplexing tradeoff by modeling successive relaying as a multiple-access MIMO channel and analyzing outage constraints and determinant expressions.
- Model and rate: Conditioned on correct relay decoding, successive relaying can be represented as a multiple-access MIMO channel with L inputs and L + 1 outputs.The protocol transmits L symbols over L + 1 time slots, yielding an equivalent rate of Lr log ρ.
- Outage analysis: The error analysis bounds conditional pairwise error probability and identifies outage probability as the dominant error event.Codeword length can be selected so non-outage error events become sufficiently small.
- Outage analysis: The diversity–multiplexing tradeoff is obtained by minimizing the exponential outage order over the relevant outage set.For m = 1, the outage exponent is expressed as ρ^-d_o(r), with d_o(r) defined through the channel exponents.
- Determinant analysis: For m > 1, determinant analysis of the equivalent MIMO matrices shows that the resulting diversity gain exceeds that for m = 1.The proof uses recursive determinant calculations and comparisons among submatrix constraints.
- Determinant analysis: Under slow fading, alternating channel exponents simplify the analysis because v1 = v3 = ... and v2 = v4 = ....The resulting outage constraints correspond to the network’s rate constraints.