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Exploiting Full/Half-Duplex User Relaying in NOMA Systems

Xinwei Yue, Yuanwei Liu, Shaoli Kang, Arumugam Nallanathan, Zhiguo Ding

arXiv:1812.07389v1cs.IT

TL;DR

The paper addresses limited understanding of direct-link effects and systematic performance evaluation for FD/HD user relaying in cooperative NOMA. It analyzes two relaying scenarios using outage, ergodic-rate, and energy-efficiency metrics, finding that direct links restore far-user diversity while FD NOMA is mainly advantageous at low SNR.

  • Problem

    Existing work had not investigated direct-link effects on FD user relaying or systematically evaluated ergodic rate and energy efficiency in FD/HD NOMA systems.

  • Method

    The paper analyzes near-user decode-and-forward relaying in FD and HD modes across scenarios with and without a direct BS-to-far-user link, deriving outage and ergodic-rate expressions.

  • Results

    Using the direct link gives the far user one diversity order, while FD NOMA outperforms HD NOMA in outage probability at low SNR but not at high SNR.

  • Takeaways & Limitations

    Direct links improve far-user reliability and overcome the zero diversity order caused by residual loop interference in FD relaying.

Abstract

from arXiv · show

In this paper, a novel cooperative non-orthogonal multiple access (NOMA) system is proposed, where one near user is employed as decode-and-forward (DF) relaying switching between full-duplex (FD) and half-duplex (HD) mode to help a far user. Two representative cooperative relaying scenarios are investigated insightfully. The \emph{first scenario} is that no direct link exists between the base station (BS) and far user. The \emph{second scenario} is that the direct link exists between the BS and far user. To characterize the performance of potential gains brought by FD NOMA in two considered scenarios, three performance metrics outage probability, ergodic rate and energy efficiency are discussed. More particularly, we derive new closed-form expressions for both exact and asymptotic outage probabilities as well as delay-limited throughput for two NOMA users. Based on the derived results, the diversity orders achieved by users are obtained. We confirm that the use of direct link overcomes zero diversity order of far NOMA user inherent to FD relaying. Additionally, we derive new closed-form expressions for asymptotic ergodic rates. Based on these, the high signal-to-noise radio (SNR) slopes of two users for FD NOMA are obtained. Simulation results demonstrate that: 1) FD NOMA is superior to HD NOMA in terms of outage probability and ergodic sum rate in the low SNR region; and 2) In delay-limited transmission mode, FD NOMA has higher energy efficiency than HD NOMA in the low SNR region; However, in delay-tolerant transmission mode, the system energy efficiency of HD NOMA exceeds FD NOMA in the high SNR region.

I. INTRODUCTION

The paper investigates cooperative NOMA user relaying that switches between FD and HD modes, addressing whether FD gains depend on operating conditions and direct-link availability. It develops a comprehensive performance study spanning outage probability, ergodic rate, throughput, diversity, and energy efficiency.

  • Motivation: Cooperative NOMA uses the near user as a DF relay to assist the far user, extending coverage and potentially improving diversity.The paper combines this cooperative structure with FD operation, which can transmit and receive simultaneously in the same frequency band but suffers residual loop self-interference.
  • Research questions: The study compares FD and HD NOMA relaying in scenarios with and without a direct BS–far-user link.It also considers comparisons with conventional OMA relaying and examines delay-limited and delay-tolerant energy-efficiency regimes.
  • Approach: The paper derives closed-form and asymptotic outage expressions, diversity orders, throughput results, asymptotic ergodic rates, high-SNR slopes, and energy-efficiency expressions.The proposed system lets the near user switch between FD and HD according to channel conditions, with SIC used to detect the far user's higher-power signal.
  • Main findings: FD NOMA is superior to HD NOMA in outage probability and ergodic sum rate in the low-SNR region.Without a direct link, FD NOMA converges to an outage error floor and has zero diversity order; increasing loop-interference strength makes its superiority less apparent.
  • Main findings: A direct BS–far-user link overcomes the far user's zero diversity order inherent to FD relaying, yielding one diversity order.For delay-limited transmission, FD NOMA has higher energy efficiency than HD NOMA at low SNR, whereas HD NOMA exceeds FD NOMA in delay-tolerant transmission at high SNR.

III. OUTAGE PROBABILITY

This section formulates outage events for the near and far users and derives closed-form outage expressions for FD and HD NOMA. The analysis distinguishes decoding failures at the relay from failures at the destination.

  • Outage formulation: The outage probability is evaluated in two cooperative-relaying scenarios because users' target rates are determined by quality-of-service requirements.The first scenario considered here is outage analysis without a direct link.
  • Outage formulation: D1 is in outage when it cannot detect D2's message or subsequently cannot detect its own message.The complementary event requires successful detection of x2 followed by successful detection of x1.
  • FD NOMA: Theorem 1 gives a closed-form outage probability for D1 under FD NOMA.Its derivation uses the complementary event and integrates over the fading variables associated with the useful channel and loop self-interference.
  • HD NOMA: Corollary 1 specializes D1's outage probability to HD NOMA by setting the switching factor to zero.The target SNRs depend on the users' target rates and the HD operation mode.
  • Far-user outage: D2's outage consists of relay decoding failure or destination decoding failure after successful relay decoding.The analysis separately identifies these two events before combining them into the far-user outage expression.

2) Outage Probability of D2:

The far-user outage analysis derives FD and HD expressions without a direct link and examines their high-SNR diversity behavior. Residual loop interference produces error floors for FD NOMA, while a direct link restores one diversity order for the far user.

  • FD NOMA: Theorem 2 gives a closed-form outage probability for D2 without a direct link under FD NOMA.The derivation combines the relay-decoding and destination-decoding outage events through intermediate integrals over the useful and loop-interference channel gains.
  • HD NOMA: Corollary 2 specializes the no-direct-link D2 outage expression to HD NOMA by setting the switching factor to zero.This provides the corresponding far-user benchmark for comparing FD and HD relaying.
  • Diversity analysis: D1 has diversity order 1 for HD NOMA, unlike the zero-order FD-NOMA result reported for the users in the no-direct-link case.The HD result follows from the asymptotic outage analysis based on the corresponding HD expression.
  • Diversity analysis: Zero diversity order is achieved by D2 in FD NOMA without a direct link, matching D1's FD-NOMA diversity order.The paper states that D2's diversity order is zero, as is D1's under FD NOMA.
  • Diversity analysis: Both users exhibit outage-probability error floors in the no-direct-link FD-relaying analysis.The floor is attributed to terms that remain constant and independent of ρ at high SNR.

4) Throughput Analysis:

The paper analyzes delay-limited FD/HD NOMA throughput with and without a BS–far-user direct link, deriving throughput expressions from outage probabilities. With a direct link, the far user's outage probability is characterized through a closed-form result for FD NOMA and numerical evaluation of the exact expression.

  • Without direct link: Delay-limited transmission evaluates FD/HD NOMA system throughput at a constant transmission rate subject to outage probability.The no-direct-link FD and HD throughput expressions use outage probabilities specified by earlier results.
  • With direct link: The direct link from the BS to D2 conveys information and can improve system reliability.The direct link does not affect D1's outage probability, so the analysis focuses on D2.
  • With direct link: D2 outage under FD NOMA includes cases where D1 detects x2 but D2's received SINR remains below its target, or neither D1 nor D2 detects x2.These events motivate the direct-link outage expression for the far user.
  • With direct link: The exact FD-NOMA outage expression for D2 with a direct link lacks a successful closed-form derivation and is evaluated numerically.An upper-bound-based expression is then used to obtain a theoretical result.
  • With direct link: Theorem 3 provides a closed-form expression for D2's FD-NOMA outage probability with a direct link.The expression involves the exponential integral function Ei(·).

2) Diversity Analysis:

The diversity analysis derives high-SNR outage approximations for FD/HD NOMA and examines how a BS–D2 direct link changes far-user reliability. The direct link overcomes the far user's zero diversity order inherent to FD relaying.

  • D2 for FD NOMA case: For FD NOMA with a direct link, Gaussian–Chebyshev quadrature approximates D2's outage probability at high SNR.The parameter N controls the complexity–accuracy tradeoff.
  • D2 for FD NOMA case: The BS–D2 direct link is an effective way to overcome D2's zero diversity order.This conclusion follows from the high-SNR diversity analysis for FD NOMA.
  • D2 for HD NOMA case: HD-NOMA outage performance for D2 is taken from prior analysis, with the corresponding diversity order obtained from that result.The cited discussion references.
  • Throughput analysis: With a direct link, FD/HD delay-limited throughput is obtained by substituting the corresponding direct-link outage probabilities into the throughput expressions.For FD NOMA, D1 and D2 probabilities come from (10) and (27); for HD NOMA, D2 uses [25, Eq. (11)].

IV. ERGODIC RATE

The paper derives ergodic-rate expressions for near and far users under FD/HD NOMA, with and without a direct link, and develops high-SNR approximations. Without a direct link, the far user's ergodic rate reaches a high-SNR throughput ceiling for both duplex modes.

  • Rate formulation: Ergodic sum rate is used to evaluate FD/HD user relaying when user rates depend on channel conditions.For D1, the achievable rate is log(1 + γD1) when D1 detects x2.
  • D1 ergodic rate: Theorem 4 and Corollary 3 provide closed-form ergodic-rate expressions for D1 without a direct link under FD and HD NOMA.The FD result is stated in Theorem 4, followed by the HD counterpart.
  • D2 ergodic rate: For D2 without a direct link, the FD achievable rate depends on the minimum of the decoding SINRs at D1 and D2.The paper derives an ergodic-rate expression and a high-SNR approximation for FD NOMA, with an analogous HD approximation.
  • High-SNR analysis: The high-SNR slope is used as the key parameter determining ergodic rate in the high-SNR region.The paper introduces high-SNR approximations because the relevant CDF is difficult to obtain directly.
  • High-SNR behavior: The D2 ergodic rate converges to a throughput ceiling in the high-SNR region for both FD and HD NOMA without a direct link.The result is summarized in Remark 5 after deriving the asymptotic expressions.

4) Throughput Analysis:

The paper extends throughput analysis to delay-tolerant FD/HD NOMA with and without a direct link. It derives throughput from ergodic-rate expressions and uses high-SNR approximations for the direct-link far-user rate.

  • Without direct link: In delay-tolerant transmission, FD/HD NOMA throughput is determined by evaluating ergodic rates.Without a direct link, the system throughput uses the D1 and D2 ergodic-rate expressions.
  • With direct link: For D2 with a direct link, the relaying and direct-link signals are assumed detectable at D2 and D1 for SIC, with residual interference considered.The achievable rate is then simplified before deriving the FD ergodic rate.
  • With direct link: The FD-NOMA ergodic rate of D2 with a direct link has no closed-form expression and is approximated asymptotically at high SNR.The difficulty includes obtaining the CDF of X2 and evaluating the associated integral.
  • With direct link: HD-NOMA D2 throughput with a direct link is based on the corresponding high-SNR ergodic-rate approximation.Corollary 5 states the asymptotic expression for this rate.
  • With direct link: The ergodic sum-rate expressions for FD and HD NOMA with a direct link combine the respective D1 and D2 rate results.These combinations are stated separately for FD and HD operation.

2) Slope Analysis:

The analysis characterizes high-SNR slopes and throughput-based energy efficiency for FD/HD NOMA with a direct link. It finds that D2 reaches a throughput ceiling and that direct-link assistance does not provide an additional high-SNR slope.

  • D2 with a direct link converges to a throughput ceiling in the high-SNR region for both FD and HD NOMA.
  • The direct link does not provide D2 with an additional high-SNR slope.
  • The analysis derives FD and HD system-throughput expressions for delay-tolerant transmission with a direct link.
  • Diversity orders and high-SNR slopes are summarized in Table I for comparing FD and HD NOMA systems.The table uses D for diversity order and S for high-SNR slope.
  • Energy efficiency is defined as total data rate divided by total energy consumption, with total power including BS transmit power Ps and relay transmit power Pr.The total data rate includes throughput from the BS to D1 and D2 and from D1 to D2.

VI. NUMERICAL RESULTS

Numerical results compare FD/HD NOMA user relaying with and without a direct BS–far-user link across outage probability, throughput, ergodic rate, and energy efficiency. FD NOMA generally performs better at low SNR, while residual loop interference produces high-SNR limitations and favors HD NOMA for delay-tolerant energy efficiency.

  • Without Direct Link: Exact outage curves match Monte Carlo simulations, validating the analytical expressions for FD/HD NOMA without a direct link.The comparison uses LI with E{|hLI|2} = −15 dB.
  • Without Direct Link: FD NOMA exceeds HD NOMA and OMA in outage performance at low SNR without a direct link.Residual loop interference is not dominant in the low-SNR region.
  • Without Direct Link: FD NOMA achieves higher delay-limited throughput than HD NOMA and OMA without a direct link.Increasing LI from −20 dB to −10 dB reduces throughput in the high-SNR region because FD NOMA approaches an error floor.
  • Without Direct Link: FD NOMA provides higher achievable rates than HD NOMA for both users in the low-SNR region without a direct link.At high SNR, LI creates a rate ceiling for D1, while throughput ceilings also occur for D2 in FD/HD NOMA.
  • Without Direct Link: FD NOMA achieves the maximal ergodic sum rate relative to HD NOMA and OMA in the low-SNR region.The reported comparison attributes this result to improved system spectrum efficiency.
  • With Direct Link: With a direct link, D2 obtains one diversity order, and FD NOMA remains superior to HD NOMA at low SNR but inferior at high SNR.The direct link improves far-user reliability and removes the zero-diversity behavior inherent to FD cooperative relaying.
  • With Direct Link: Increasing LI from −20 dB to −10 dB strongly degrades FD NOMA outage performance and removes its clear superiority.The result highlights LI as an important consideration in practical FD NOMA design.
  • With Direct Link: FD NOMA outperforms HD NOMA in delay-limited throughput at low SNR with a direct link.In this region, the outage probability is small and throughput depends on the fixed BS transmission rates.

APPENDIX A: PROOF OF THEOREM 3

The proof derives the outage probability of D2 and the ergodic rate of D1 through successive substitutions, expansions, and algebraic simplifications.

  • Outage probability: The outage probability of D2 is first expressed from (26), then reduced to (27) by combining intermediate results.The derivation uses quantities J11, J12, and J13 together with auxiliary expressions such as Θ1 and χ.
  • Outage probability: The derivation applies the Binomial theorem and a referenced integral identity to simplify intermediate outage expressions.Equation (A.3) uses the Binomial theorem, while (A.4) is obtained from the cited reference identity.
  • Conclusion: The proof concludes after combining the derived intermediate expressions to obtain the target analytical results.The appendix explicitly closes the derivation after equations (27) and (31).
  • Ergodic rate: The ergodic rate of D1 is formulated as an integral and converted into a closed expression after substituting auxiliary results.The proof introduces (B.1), substitutes (B.2), and evaluates J1 and J2 using polynomial expansions before obtaining (31).

APPENDIX C: PROOF OF THEOREM 5

The proof develops a high-SNR approximation for the ergodic rate of D2 by deriving the CDF of an auxiliary variable and evaluating resulting terms.

  • High-SNR ergodic rate: The proof begins by expressing the ergodic rate of D2 and focusing on the high-SNR approximation of J1.The approximation is built from the CDF of Y and an associated integral representation.
  • CDF derivation: The CDF of Y is derived from conditions involving |h1|^2a1 and |hLI|^2, with the unit step function specifying the relevant cases.Intermediate substitutions produce the CDF used in the ergodic-rate approximation.
  • Expression simplification: Polynomial expansions and a referenced integral identity are used to calculate J2 and J3 before completing the target ergodic-rate expression.The proof obtains the final result by substituting the evaluated terms into the preceding equation.

APPENDIX D: PROOF OF COROLLARY 4

The proof derives a high-SNR ergodic-rate approximation for D2 in the HD-NOMA setting by rewriting the expression and evaluating its component terms.

  • High-SNR approximation: The derivation rewrites the ergodic-rate expression and approximates J1 in the high-SNR region.The proof introduces the rewritten form in (35) and then applies the high-SNR approximation.
  • HD-NOMA rate: The resulting approximation gives the ergodic rate of D2 for HD NOMA in equation (36).The appendix states that this expression is obtained after the high-SNR approximation of J1.
  • Derivation: The proof derives the CDF of Y, forms the high-SNR ergodic-rate approximation, and evaluates J2 and J3 through algebraic manipulation.The final expression is obtained by substituting the evaluated terms into the preceding equation.
  • Conclusion: Substitution of the intermediate results yields equation (45), completing the proof.The appendix explicitly identifies this substitution as the final step.
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