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Max-min Fairness of K-user Cooperative Rate-Splitting in MISO Broadcast Channel with User Relaying

Yijie Mao, Bruno Clerckx, Jian Zhang, Victor O. K. Li, Mohammed Arafah

arXiv:1910.07843v2eess.SPcs.ITmath.IT

TL;DR

The paper extends cooperative rate-splitting from two users to K users, jointly optimizing transmission and relay scheduling for max-min fairness. It proposes channel-strength-based relay selection and an SCA optimizer; CRS gains over NRS and the SCA approach reduces computational complexity relative to prior optimization.

  • Problem

    The K-user CRS design must address joint time-allocation optimization, relay scheduling, and the expanded search space as users share the common rate.

    Source: A. Motivations and Contributions · p. 2, §I · p. 2

  • Method

    The approach combines channel-strength-based relay selection with an SCA algorithm that jointly optimizes precoders, message split, and time allocation.

    Source: 2) Decentralized relaying protocol · p. 7, 2) Complexity · p. 9

  • Results

    143.4% relative rate gain for CRS: Proposed SCA over NRS: WMMSE with four transmit antennas, rising to 192.9% with two.

    Source: §V · p. 11

  • Takeaways & Limitations

    A single strongest-channel relay can achieve performance close to optimal scheduling, while the proposed SCA method has the same or higher rate performance than WMMSE at lower computational complexity.

    Source: §V · p. 11

Abstract

from arXiv · show

Cooperative Rate-Splitting (CRS) strategy, relying on linearly precoded rate-splitting at the transmitter and opportunistic transmission of the common message by the relaying user, has recently been shown to outperform typical Non-cooperative Rate-Splitting (NRS), Cooperative Non-Orthogonal Multiple Access (C-NOMA) and Space Division Multiple Access (SDMA) in a two-user Multiple Input Single Output (MISO) Broadcast Channel (BC) with user relaying. In this work, the existing two-user CRS transmission strategy is generalized to the K-user case. We study the problem of jointly optimizing the precoders, message split, time slot allocation, and relaying user scheduling with the objective of maximizing the minimum rate among users. An efficient self-organizing relaying protocol is first proposed followed by a Successive Convex Approximation (SCA)-based algorithm to jointly optimize time slot, precoders and message split. Numerical results show that the worst-case achievable rate achieved by CRS is significantly increased over that of NRS and SDMA in a wide range of network loads and user deployments. Importantly, the proposed SCA-based algorithm dramatically reduces the computational complexity without any rate loss compared with the conventional algorithm in the literature of CRS. Therefore, we conclude that the proposed K-user CRS is more powerful than the existing transmission schemes.

I. INTRODUCTION

This work generalizes two-user cooperative rate-splitting to K users, addressing joint fairness optimization and relay selection challenges. It proposes channel-strength-based relaying protocols and an SCA-based joint optimizer.

A. Motivations and Contributions

  • The K-user max-min fairness problem jointly optimizes precoders, message split, time allocation, and relay scheduling, extending the two-user framework in.The earlier approach used exhaustive search for time allocation, while K-user relay selection and common-rate sharing expand the design challenge.Source: A. Motivations and Contributions · p. 2
  • The optimal relay group’s worst common rate is always higher than that of other users in direct transmission, motivating channel-strength-based centralized and decentralized protocols.Source: A. Motivations and Contributions · pp. 2–3
  • The SCA algorithm jointly optimizes precoders, message split, and time allocation through one iterative layer, reducing complexity relative to the two-layer procedure in.Source: A. Motivations and Contributions · p. 3
  • A single relaying user achieves performance very close to optimal scheduling across varied network loads, user deployments, and relay power constraints.Source: A. Motivations and Contributions · p. 3
  • CRS’s worst-case rate gains over NRS become more significant with fewer transmit antennas, more users, or greater channel-strength disparity.The strongest-channel user forwards the common stream to users with weaker channels.Source: A. Motivations and Contributions · p. 3

II. SYSTEM MODEL

The system divides K single-antenna users into relaying and receiving groups and transmits in direct and cooperative phases. Rate-splitting sends a common stream and user-specific private streams, with relaying users forwarding the common stream to the other group.

II. SYSTEM MODEL

  • Transmission uses a direct phase of duration fraction θ, followed by a cooperative phase in which group-1 users forward signals to group-2.The two phases need not receive equal time.Source: §II · p. 3

A. Direct transmission phase

  • In the direct phase, each user’s message is split into common and private parts; the common stream is decoded by all users and each private stream by its intended user.Users decode the common stream before their private stream using successive interference cancellation.Source: A. Direct transmission phase · pp. 3–4

B. Cooperative transmission phase

  • The model assumes perfect transmitter and receiver channel information and perfect successive interference cancellation; imperfect CSIT and imperfect SIC are outside this work’s scope.Source: A. Direct transmission phase · p. 4, B. Cooperative transmission phase · p. 4
  • In the cooperative phase, group-1 users use non-regenerative decode-and-forward to re-encode and forward the common stream to group-2.Group-2 users combine common-stream decoding across both time slots; the common rate must be decodable by all users and is allocated among their messages.Source: B. Cooperative transmission phase · p. 4

III. PROBLEM FORMULATION

The section formulates K-user CRS as a max-min fairness problem and outlines a two-stage approach: select relaying users, then jointly optimize transmission resources. It also gives an optimal-grouping insight that motivates selection using users’ channel strengths.

III. PROBLEM FORMULATION

  • The max-min formulation jointly optimizes precoders, common-rate allocation, time-slot allocation, and relay selection to maximize the minimum user rate under a transmit-power constraint.The common-stream rate constraint reflects the NDF relaying model.Source: §III · p. 4
  • The paper distinguishes NDF from RDF by independent codebooks in NDF versus a shared BS-and-relay codebook in RDF, and focuses on NDF CRS.The formulation includes time-slot allocation and relay selection as variables, unlike conventional max-min rate problems without user relaying.Source: §III · p. 4
  • The joint selection and resource-allocation problem is mixed-integer and non-convex, with relaying-group search requiring consideration of P_K−1 possible groups.The passage identifies the scheduling search as computationally complex, particularly as the user count grows.Source: §III · p. 5
  • The proposed low-complexity solution first selects relaying users and then jointly optimizes precoders, message split, and time allocation.Both centralized and decentralized selection are considered in Stage 1.Source: §III · p. 5

A. Stage 1: Relaying user selection protocol

  • For 0 < θ⋆ < 1, Proposition 1 states that the common rates achieved by the two user groups are equal at the global optimum and gives an optimal grouping rule.The supplied passages do not provide the full mathematical expression of that rule.Source: A. Stage 1: Relaying user selection… · p. 5
  • The grouping guidance places users with large R[1]c,k in K1 and users with low R[1]c,k in K2.The paper notes that these common rates are strongly influenced by users’ channel strengths.Source: A. Stage 1: Relaying user selection… · p. 6
  • The proposed channel-strength-based selection protocols use RTS/CTS handshaking and consider both centralized and decentralized relay selection.The decentralized and centralized protocols are illustrated for K1 = 1.Source: A. Stage 1: Relaying user selection… · p. 6

1) Centralized relaying protocol:

The centralized protocol has the BS select the K1 users with the highest channel gains after the handshake, then broadcast their selection in a flag packet.

1) Centralized relaying protocol:

  • After obtaining all users’ CSI, the BS selects the K1 highest-channel-gain users and broadcasts the selection to all users in a flag packet.Each user decodes the packet to learn its selection result.Source: 1) Centralized relaying protocol · p. 6

2) Decentralized relaying protocol:

The decentralized protocol selects relays using user channel strengths and timers, while avoiding dependence on accurate CSIT at the BS. The protocol supports multiple relays, although selecting one best relay is reported to achieve nearly the optimal rate.

2) Decentralized relaying protocol:

  • When all common-rate constraints are active, users in group 1 achieve the same optimal common rate, which exceeds the rate achieved by users in group 2.Searching relay groups is computationally difficult; the passage states that the scheduling problem is NP-hard and that P_K−1 groups may need to be searched.Source: 2) Decentralized relaying protocol · p. 6
  • A decentralized selection approach is preferred because it is not influenced by CSIT inaccuracy, whereas centralized selection performance drops as CSIT worsens.The distinction arises because the BS selects centrally using CSIT, while users select locally using CSIR.Source: 2) Decentralized relaying protocol · p. 6
  • Each user sets a countdown timer inversely proportional to its channel strength; the first to reach zero announces itself as the best relay.The process repeats until K1 relaying users have been selected.Source: 2) Decentralized relaying protocol · pp. 6–7
  • The protocol is generalized to select multiple relaying users, and selecting a single best relay achieves a rate almost the same as optimal scheduling.The number of relays K1 remains a variable in the proposed algorithm.Source: 2) Decentralized relaying protocol · p. 7
  • Compared with the decentralized protocol without RS in, the CRS-assisted protocol requires each user to know only its BS channel, rather than channels to every other user.The cited comparison also notes that selects only one relay, whereas this protocol supports multiple relays.Source: 2) Decentralized relaying protocol · p. 7

3) Comparison:

The comparison identifies different overhead and synchronization trade-offs across centralized selection, decentralized selection, and strategies without user relaying.

3) Comparison:

  • Centralized selection has relatively higher signaling overhead because its flag packet carries the selection information for all users.The centralized packet is larger than a decentralized flag carrying the result for one selected user.Source: 3) Comparison · p. 7
  • Strategies without user relaying have the lowest overhead; decentralized selection takes more time because each relay requires an additional timer.With spatial multiplexing of K BS flag packets, each centralized flag packet can be the same size as a decentralized one.Source: 3) Comparison · p. 7
  • User-relaying strategies require more accurate synchronization than strategies without relaying, though the information-transmission phases assume perfect synchronization.The relay-selection phase relies on CTS or flag timing, and a larger λ reduces collision probability.Source: 3) Comparison · p. 7
  • Decentralized selection is insensitive to CSIT inaccuracy, while centralized selection performance degrades as CSIT worsens.The decentralized protocol selects relays using CSIR, whereas centralized selection uses BS-obtained CSIT.Source: 3) Comparison · p. 7

B. Stage 2: Joint precoder design and resource allocation

Stage 2 jointly optimizes precoders, message split, and time allocation using SCA. It reformulates the max-min problem and approximates its nonconvex constraints to produce convex subproblems.

B. Stage 2: Joint precoder design and resource allocation

C. Convergence and Complexity

The proposed SCA algorithm repeatedly solves an approximated problem until convergence, using a specified tolerance.

C. Convergence and Complexity

  • The SCA algorithm iteratively solves the approximated problem (22) until convergence within tolerance ε.Source: 1) Convergence · p. 8

1) Convergence:

The SCA algorithm converges to a stationary point from any feasible initial point. Its proof uses monotonic objective improvement, bounded iterates, and asymptotically tight approximations.

1) Convergence:

  • For any feasible initial point, the proposed SCA algorithm is guaranteed to converge to a stationary point of problem (11).Source: 1) Convergence · p. 8
  • The objective sequence improves monotonically and is bounded above by the transmit-power constraint, ensuring convergence.Source: 1) Convergence · pp. 8–9
  • Asymptotically tight approximations and equivalent KKT conditions establish convergence to stationary points of problem (11).Source: 1) Convergence · p. 9
  • Approximating the exponential-cone constraints with SOC constraints gives worst-case complexity O(log(ε^-1)[KNt]^3.5).Source: 2) Complexity · p. 9

2) Complexity:

The proposed SCA algorithm jointly optimizes time allocation, precoders, and message split rather than using a one-dimensional exhaustive search. Its worst-case complexity is δ^-1-fold lower than the earlier algorithm.

2) Complexity:

  • The SCA algorithm jointly optimizes θ, P, and c without one-dimensional exhaustive search, yielding δ^-1-fold lower worst-case complexity than.Source: 2) Complexity · p. 9

V. NUMERICAL RESULTS

Numerical evaluations show that the proposed CRS/SCA method improves max-min rates over non-cooperative baselines while retaining performance comparable to WMMSE at lower complexity. A single best-relay choice performs well across tested user loads and relay-power settings.

V. NUMERICAL RESULTS

  • In Fig. 4(a), CRS: Proposed SCA gains 143.4% over NRS: WMMSE, 41.4% over ERS: WMMSE, and 166.1% over SDMA: WMMSE, while matching or exceeding CRS: WMMSE at lower complexity.The comparisons average rates across SNRs; the SCA algorithm has much lower computational complexity than CRS: WMMSE.Source: Figure 4 · p. 10, §V · p. 11
  • With two rather than four transmit antennas, CRS's average-SNR gain over NRS rises from 143.4% to 192.9%, and its gain over SDMA rises from 166.1% to 266.1%.The reported CRS–NRS average rate gap is 23.9% in the lower-disparity case and 192.9% in the higher-disparity case; CRS is reported as more capable under stronger multi-user interference.Source: §V · p. 11
  • In Fig. 5, one best relay improves average rate by 57.83% over K/2 best relays and 13.5% over one random relay; exhaustive optimal scheduling has the highest selection complexity.For K=3, one best relay performs almost as well as optimal scheduling, whereas K/2 best relays degrades max-min rate.Source: §V · pp. 9–11, Figure 5 · p. 11
  • As K increases, NRS loses rate more noticeably while ERS improves relative to CRS; CRS also improves over NRS and ERS, and RS-based schemes outperform SDMA in both underloaded and overloaded settings.The text attributes the trend to increased multi-user interference and a larger share of messages transmitted through the common stream.Source: §V · pp. 11–12
  • In Fig. 6, SCA and WMMSE converge within a few iterations, with similar convergence speed; different initializations affect speed but yield close convergence rates.SCA jointly optimizes θ, P, and c, while the comparison WMMSE algorithm jointly optimizes P and c.Source: Figure 6 · p. 12, §V · p. 12
  • In Fig. 7, one best relay reaches 95.6% of optimal scheduling's average rate and a 105.6% relative rate improvement over SDMA as relay power increases from 0 to 20 dBW.The text presents this protocol as a preferred trade-off between total power consumption, Pt + K1Pr, and max-min rate.Source: Figure 7 · p. 12, §V · p. 13

VI. CONCLUSIONS

The conclusion presents a two-stage approach to K-user CRS max-min fairness and reports gains over non-cooperative RS and SDMA under several challenging conditions. Its proposed relaying protocols and SCA-based optimization also reduce scheduling or computational burden without substantial rate loss.

VI. CONCLUSIONS

  • Single-relayer selection achieves nearly the same rate performance as optimal relaying scheduling.The proposed protocols also allow decentralized scheduling using only BS-to-user CSI at each user.Source: §VI · p. 13
  • The SCA-based algorithm solves the max-min fairness problem more efficiently than the literature algorithm, without rate loss.Source: §VI · p. 13
  • CRS rate improvement over non-cooperative RS and SDMA becomes more significant at low SNR, with fewer antennas, greater channel disparity, or more users.Source: §VI · p. 13
  • K-user CRS enhances common-stream transmission and is reported to better manage strong multi-user interference and channel-strength disparities.Source: §VI · p. 13
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