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The Multi-way Relay Channel

Deniz Gunduz, Aylin Yener, Andrea Goldsmith, H. Vincent Poor

arXiv:1004.2434v2cs.IT

TL;DR

The paper studies coding schemes for a general communication setup and compares their performance. CF achieves exchange rates within a constant, power-independent gap of exchange capacity, while nested lattice codes achieve a finite-gap result in the two-user-per-cluster case.

  • Problem

    The paper considers a general communication setup and seeks performance comparisons among coding schemes.

  • Method

    The paper characterizes achievable rate regions using AF, DF, and CF schemes, and also characterizes nested-lattice-coding rates when each cluster has two users.

  • Results

    CF achieves exchange rates within a constant bit offset of exchange capacity independent of cluster count and node power constraints; nested lattice codes achieve a finite bit gap in the two-user-per-cluster case.

  • Takeaways & Limitations

    The results compare fundamental coding schemes and show near-capacity exchange rates for CF and nested lattice coding in their stated settings.

Abstract

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The multiuser communication channel, in which multiple users exchange information with the help of a relay terminal, termed the multi-way relay channel (mRC), is introduced. In this model, multiple interfering clusters of users communicate simultaneously, where the users within the same cluster wish to exchange messages among themselves. It is assumed that the users cannot receive each other's signals directly, and hence the relay terminal in this model is the enabler of communication. In particular, restricted encoders, which ignore the received channel output and use only the corresponding messages for generating the channel input, are considered. Achievable rate regions and an outer bound are characterized for the Gaussian mRC, and their comparison is presented in terms of exchange rates in a symmetric Gaussian network scenario. It is shown that the compress-and-forward (CF) protocol achieves exchange rates within a constant bit offset of the exchange capacity independent of the power constraints of the terminals in the network. A finite bit gap between the exchange rates achieved by the CF and the amplify-and-forward (AF) protocols is also shown. The two special cases of the mRC, the full data exchange model, in which every user wants to receive messages of all other users, and the pairwise data exchange model which consists of multiple two-way relay channels, are investigated in detail. In particular for the pairwise data exchange model, in addition to the proposed random coding based achievable schemes, a nested lattice coding based scheme is also presented and is shown to achieve exchange rates within a constant bit gap of the exchange capacity.

I. INTRODUCTION

The paper introduces the multi-way relay channel, where multiple user clusters exchange information through a relay without direct user-to-user reception. For the Gaussian model, it characterizes achievable schemes and exchange-capacity bounds, showing finite-gap performance for CF and nested lattice coding.

  • Model: The multi-way relay channel has N = KL users grouped into L clusters of K users, with intra-cluster message exchange enabled by a relay.Users do not receive each other’s signals directly; communication uses a Gaussian MAC to the relay and a Gaussian broadcast channel from the relay.
  • Motivation: The model covers clustered information exchange in peer-to-peer, social, sensor, and satellite-assisted networks.Examples include users sharing files, friend groups exchanging information, sensors sharing measurements, and geographically distributed ad-hoc networks.
  • Protocols: The paper derives achievable rate regions for multi-way extensions of decode-and-forward, amplify-and-forward, and compress-and-forward protocols.The schemes exploit each user’s knowledge of its own message during decoding; DF decodes all messages at the relay, whereas CF forwards a quantized relay observation.
  • Exchange-rate analysis: The analysis characterizes an exchange-capacity upper bound and total exchange rates achievable by AF, DF, and CF in a symmetric Gaussian network.Exchange rate is the common user rate, while exchange capacity is the supremum of achievable total exchange rates.
  • Special cases: For pairwise data exchange, nested lattice coding achieves rates within a finite-bit gap of exchange capacity and allows the relay to decode a function rather than every individual message.The paper also investigates full data exchange and pairwise data exchange as special cases of the multi-way relay channel.
  • Main results: CF achieves total exchange rates within a finite-bit gap of exchange capacity for any number of clusters and users, with the gap independent of node power constraints.The CF rate loss is attributed to forwarding relay noise, whose relative effect becomes less important as the number of users per cluster increases.

II. SYSTEM MODEL

The mRC has multiple user clusters that exchange messages through a full-duplex relay because users cannot directly overhear one another. Restricted encoders generate inputs only from messages, and the model specifies Gaussian channel noises, power constraints, coding, and reliable-achievability definitions.

  • The relay and users operate full-duplex, with independent Gaussian noises and average power constraints on all transmitted signals.
  • Users in each cluster decode the messages of all other users in that same cluster, while the relay enables communication between users.
  • Restricted encoders ignore received outputs, so each user’s channel input depends only on its messages.
  • The outer bound and finite-bit-gap arguments apply only under the restricted-encoder assumption, although the achievable schemes also apply without it.
  • The capacity region is defined through rate tuples supported by a sequence of codes whose average error probability tends to zero as blocklength grows.

III. THE ACHIEVABLE RATE REGION

The paper develops inner and outer bounds for the Gaussian mRC using classical relaying strategies and bounds adapted to multi-way exchange. Multi-way users exploit knowledge of their own transmitted signals to improve achievable rate regions.

  • The Gaussian mRC capacity region is bounded using a combination of cut-set and genie-aided outer bounds.
  • AF, DF, and CF relaying provide the proposed achievable rate regions, extending schemes from the classical one-way relay channel.
  • Knowledge of users’ own transmit signals can improve the achievable rate region in the multi-way relay setting.

A. Outer bound

The outer-bound analysis combines cut-set constraints with a genie-aided broadcast-channel bound under restricted encoders. The section also characterizes AF-based achievable rates using cluster time-sharing and Gaussian multiple-access decoding.

  • A. Outer bound: The cut-set bound evaluates information flow across cuts separating selected users from the relay and remaining users.
  • A. Outer bound: A genie gives all network messages to the relay and remaining users, reducing the problem to transmitting each cluster’s messages to one chosen user.
  • A. Outer bound: The resulting broadcast-channel constraints, intersected with the cut-set constraints, form an outer bound on the restricted-encoder capacity region.
  • A. Outer bound: The genie-aided bound ignores feedback to encoders and therefore is valid only for restricted encoders.
  • B. Amplify-and-forward (AF) Relaying: In AF relaying, the relay amplifies its received signal within its power constraint, while users decode the other users’ messages in their cluster.
  • B. Amplify-and-forward (AF) Relaying: AF time-shares among clusters because transmissions from different clusters interfere, applying the strategy separately within each cluster’s timeslot.
  • B. Amplify-and-forward (AF) Relaying: The AF rate region is achievable through the union of rate tuples satisfying the stated inequalities and time-sharing constraints.

C. Decode-and-forward (DF) Relaying

The paper presents DF and CF schemes for multi-way exchange, exploiting users’ own messages as side information. DF decodes all messages at the relay, whereas CF forwards a quantized relay observation without requiring users to decode that quantization first.

  • C. Decode-and-forward (DF) Relaying: DF uses a multiple-access phase in which the relay decodes all users’ messages, followed by a broadcast phase to the users.
  • C. Decode-and-forward (DF) Relaying: The relay broadcasts all cluster messages simultaneously, and each user uses its own message as correlated side information to decode the remaining messages.
  • C. Decode-and-forward (DF) Relaying: The DF rate region is achievable under the proposition’s inequalities, relay-power, and cluster time-sharing constraints.
  • D. Compress-and-forward (CF) Relaying: CF quantizes the relay’s received signal and broadcasts the quantized output while exploiting users’ correlated side information.
  • D. Compress-and-forward (CF) Relaying: Users directly decode other users’ message indices without first decoding the quantized relay codeword.
  • D. Compress-and-forward (CF) Relaying: For the discrete memoryless mRC, the CF theorem gives an achievable rate region under a specified factorized distribution and all user subsets.
  • D. Compress-and-forward (CF) Relaying: The Gaussian CF region uses cluster time-sharing, relay power allocation, Gaussian codebooks, and Gaussian quantization noise.

E. Lattice Coding

The pairwise data-exchange scheme uses nested lattice codes so the relay decodes and broadcasts modulo sums rather than individual messages. This structured approach achieves rates close to exchange capacity but is specialized to two-user pairs.

  • Scope: The structured scheme does not directly scale beyond K = 2 because a modulo sum of more than two messages does not reveal all remaining messages to a user.Accordingly, the paper concentrates lattice coding on pairwise exchange.
  • Lattice-coded exchange: Nested lattice codes let paired users transmit lattice points whose modulo sum is decoded by the relay and broadcast back to both users.Each user subtracts its own message point to recover the partner’s message.
  • Lattice construction: The scheme uses common coarse and fine lattices, with the coarse lattice shaping codewords to satisfy the power constraint.The coarse lattice satisfies σ^2(Λc) = P, while the fine lattice is chosen for channel coding.
  • Transmission procedure: Users transmit in time division among pairs, and each pair uses one portion of the timeslot with the same nested lattice structure.For L pairs, each pair transmits over 1/L of the timeslot.
  • Decoding: The relay decodes modulo sums with vanishing error probability, after which the broadcast phase is constrained by the rate deliverable to each user.The broadcast rate is bounded by the weakest user in each pair.

IV. EXCHANGE RATE FOR A SYMMETRIC NETWORK

The paper analyzes exchange capacity and achievable rates in a symmetric Gaussian mRC using AF, DF, and CF relaying. Although the exact capacity bounds generally do not match, CF remains within a constant bit gap independent of power constraints.

  • Capacity and achievable rates: For a symmetric Gaussian mRC with L clusters of K users, the exchange capacity is bounded above and achievable rates are derived for AF, DF, and CF.The analysis assumes equal user powers and unit noise variances.
  • Capacity bounds: The lower and upper bounds generally do not match, so the exact exchange capacity remains open.The paper nevertheless bounds their separation by a finite number of bits independent of user power constraints.
  • Decode-and-forward: DF achieves exchange capacity when relay broadcasting is the bottleneck, and its optimality range expands with clusters, users per cluster, or user power.This corresponds to the regime where relay-to-user transmission limits the exchange capacity.
  • Amplify-and-forward: AF achieves a lower total exchange rate than CF, but their gap is bounded independently of power constraints and the number of clusters.AF’s simplicity may make it attractive despite the rate difference.
  • Compress-and-forward: CF achieves rates within 2(K−1) bits of exchange capacity for arbitrary cluster and user counts, independently of available user and relay power.The constant gap depends only on K.
  • Compress-and-forward: The CF gap is bounded by one bit independently of K and decays to half a bit as K increases.At high power, the finite gap becomes negligible, making CF nearly optimal.

A. The Multi-way Relay Channel with Full Data Exchange

In full data exchange, every user in a single cluster seeks all other users’ messages. Exchange rates depend nonmonotonically on the number of users and on how relay power scales with user power.

  • Model: The full-data-exchange model has one cluster in which each user must decode all other users’ messages.It is one extreme of the multi-way relay channel.
  • User scaling: The total exchange rate initially decreases as users are added because the new users introduce interference.When relay power scales as Pr = KP, the rate can fall sharply at first.
  • Relaying schemes: CF maintains a finite gap from the upper bound at all power values, while its gap from AF is also finite and AF approaches CF as users increase.For small user counts, CF can dominate AF and DF over broad power ranges.
  • Relaying schemes: DF exceeds CF in the low-power regime, and the power range where DF dominates expands with the number of users.The paper attributes this behavior to CF forwarding more noise under increased interference.
  • Relay-power scaling: When relay power equals user power, Pr = P, both DF and CF approach the upper bound as the number of users increases.For the plotted power constraint, DF reaches the upper bound with fewer users.
  • User scaling: After interference saturates, the total exchange rate begins increasing, and exchange capacity diverges as the number of users grows when relay power scales with users.With non-scaling relay power, the capacity instead saturates.

B. The multi-way Relay Channel with Pairwise Data Exchange

The pairwise data-exchange model consists of multiple two-way relay channels served by one relay. Nested lattice coding approaches exchange capacity, while CF and AF remain within a uniform finite gap.

  • Model: Pairwise data exchange pairs users so each user is interested only in its partner’s data, yielding multiple two-way relay channels sharing one relay.This is the other extreme of the multi-way relay channel alongside full data exchange.
  • Lattice coding: The lattice scheme time-shares among clusters during both lattice-coded multiple-access and broadcast phases.Each pair transmits over 1/L of the timeslot and uses a common nested lattice code.
  • Lattice coding: The relay broadcasts each pair’s modulo sum to both users, with the broadcast rate bounded by the rate deliverable to each user.This produces the stated nested-lattice achievable exchange rate.
  • Performance: Lattice coding achieves exchange capacity when 0 ≤ LP − 1/2, and otherwise remains within log 3/2 bits under the stated condition LP ≥ 1/2.The gap decays to zero as LP tends to infinity.
  • Numerical comparison: As power increases, lattice coding quickly outperforms the other schemes and approaches the exchange-capacity upper bound.The comparison uses L = 8 pairs with relay power Pr = 2LP.
  • Numerical comparison: As the number of pairs increases, lattice coding improves and approaches the upper bound, while CF and AF remain within a finite bit gap.This comparison uses P = −5 dB and Pr = 2LP.

VI. CONCLUSION

The paper characterizes achievable regions for Gaussian multi-way relay channels and shows that CF remains within a constant bit offset of exchange capacity, independent of cluster count and node powers. For pairwise exchange, nested lattice coding is also within a finite gap and outperforms the other schemes in the multiple-cluster setting.

  • The Gaussian mRC contains multiple user clusters communicating through one relay, with no cross-reception between users in different clusters.
  • The paper characterizes achievable rate regions for AF, DF, and CF, and additionally for nested lattice coding when each cluster has two users.
  • CF achieves exchange rates within a constant bit offset of exchange capacity, independent of the number of clusters and node power constraints.
  • The gap between CF and AF total exchange rates is bounded by a finite number of bits.
  • Nested lattice codes achieve rates within a finite bit gap of exchange capacity for multiple clusters with two users each, and outperform all other schemes in this setup.
  • The results suggest that relay decoding requirements may limit DF exchange rates, while relaxing those requirements can approach capacity in some scenarios.

APPENDIX A

The appendix presents the CF coding construction using block Markov encoding, relay quantization, and sequential decoding. Its error analysis yields the quantization and message-rate constraints required for reliable communication.

  • The CF scheme uses block Markov encoding across B messages and B+1 channel blocks, with the relay forwarding each block’s information in the next block.
  • Transmitters send only new messages in each block, enabling sequential decoding because there is no coherent combining.
  • The relay generates quantization and relay codebooks, then selects a quantization index based on its received signal and forwards the corresponding codeword.
  • The error analysis treats channel blocks separately and bounds the total error probability by their sum.
  • The relay quantization rate must satisfy RQ = I(Yr; ˆYr) + ǫ for the associated error probability to vanish as n →∞.
  • The achievable message rate obeys R(S) < min{I(X(S); ˆYr|X(Sc)), I(XK; ˆYr) + I(Xr; Yi) −I(Yr; ˆYr)} −¯ǫ for all user subsets S.
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