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Non-orthogonal Multiple Access in Large-Scale Heterogeneous Networks

Yuanwei Liu, Zhijin Qin, Maged Elkashlan, Arumugam Nallanathan, Julie A. McCann

arXiv:1705.03325v1cs.IT

TL;DR

The paper addresses the limited systematic evaluation of NOMA-enhanced hybrid HetNets by combining NOMA small cells with massive-MIMO macro cells. It develops biased user association and stochastic-geometry-based analytical evaluations of coverage, spectrum efficiency, and energy efficiency. The reported results show that coverage depends strongly on rates and power sharing, macro-cell spectrum efficiency benefits from more antennas, and dense NOMA small-cell deployment improves network energy efficiency relative to OMA-based HetNets.

  • Problem

    The impact of NOMA-enhanced hybrid HetNets lacks systematic evaluation across coverage probability and energy efficiency, while introducing interference, SIC ordering, and power-sharing association challenges.

  • Method

    The paper combines NOMA-enhanced small cells with massive-MIMO macro cells, develops biased user association, and uses stochastic geometry for analytical performance evaluation.

  • Results

    The analysis covers small-cell coverage probability, small-cell spectrum efficiency, a macro-cell spectrum-efficiency lower bound, and whole-network energy efficiency.

  • Takeaways & Limitations

    NOMA-enhanced small cells achieve higher energy efficiency than macro cells, while increasing macro-cell antenna numbers enhances spectrum efficiency but can reduce energy efficiency.

Abstract

from arXiv · show

In this paper, the potential benefits of applying non-orthogonal multiple access (NOMA) technique in $K$-tier hybrid heterogeneous networks (HetNets) is explored. A promising new transmission framework is proposed, in which NOMA is adopted in small cells and massive multiple-input multiple-output (MIMO) is employed in macro cells. For maximizing the biased average received power for mobile users, a NOMA and massive MIMO based user association scheme is developed. To evaluate the performance of the proposed framework, we first derive the analytical expressions for the coverage probability of NOMA enhanced small cells. We then examine the spectrum efficiency of the whole network, by deriving exact analytical expressions for NOMA enhanced small cells and a tractable lower bound for massive MIMO enabled macro cells. Lastly, we investigate the energy efficiency of the hybrid HetNets. Our results demonstrate that: 1) The coverage probability of NOMA enhanced small cells is affected to a large extent by the targeted transmit rates and power sharing coefficients of two NOMA users; 2) Massive MIMO enabled macro cells are capable of significantly enhancing the spectrum efficiency by increasing the number of antennas; 3) The energy efficiency of the whole network can be greatly improved by densely deploying NOMA enhanced small cell base stations (BSs); and 4) The proposed NOMA enhanced HetNets transmission scheme has superior performance compared to the orthogonal multiple access~(OMA) based HetNets.

I. INTRODUCTION

The paper proposes a hybrid HetNets framework combining NOMA-enhanced small cells with massive-MIMO macro cells, addressing the limited systematic evaluation of this design. It develops biased user association and analytical performance evaluation for coverage, spectrum efficiency, and energy efficiency.

  • I. INTRODUCTION: The framework evaluates coverage probability, spectrum efficiency, and energy efficiency using stochastic geometry to model large-scale network randomness.The paper presents numerical results after deriving the corresponding analytical performance measures.
  • A. Motivation and Related Works: The paper identifies a gap in systematic evaluation of NOMA-enhanced hybrid HetNets, including coverage probability and energy efficiency.It also highlights added interference, SIC channel-ordering, and power-sharing challenges.
  • B. Contributions and Organization: The proposed architecture applies NOMA in small cells and massive MIMO with linear ZFBF in macro cells for multi-tier downlink HetNets.The framework considers single-antenna small-cell BSs and large antenna arrays at macro BSs.
  • B. Contributions and Organization: NOMA-enhanced small cells target high spectrum efficiency, low complexity, and a fairness/throughput tradeoff in dense HetNets.Single-antenna small cells avoid the complex cluster-based precoding and detection designs used in MIMO-NOMA systems.
  • B. Contributions and Organization: A flexible biased association policy incorporates NOMA power sharing and massive-MIMO effects into maximum biased received-power selection.The paper derives analytical coverage expressions for NOMA small cells, exact small-cell spectrum-efficiency expressions, and a tractable macro-cell lower bound.

1) Average received power in NOMA enhanced small cells:

The small-cell association model selects users using biased average received power, while NOMA transmission distinguishes near and far users and accounts for power sharing, path loss, and interference.

  • 1) Average received power in NOMA enhanced small cells:: Small-cell association uses the averaged received power from candidate BSs, with near users selected as the typical users for NOMA pairing.The received-power model includes BS transmit power, near-user power sharing, path loss, distance, and bias.
  • 1) NOMA enhanced small cell transmission:: The model assumes each small-cell BS already serves one user from the previous association round before pairing a typical user through NOMA.Associated users and their BSs are modeled with equal distance r_k for analytical simplicity.
  • 1) NOMA enhanced small cell transmission:: When the typical user is closer to the BS than the connected user, it is treated as the near user and first decodes the connected far user's message.This case is defined by x ≤ r_k, with the connected user's distance set to r_k.
  • 1) NOMA enhanced small cell transmission:: The SINR expressions include power-sharing coefficients, AWGN, macro-cell interference, other small-cell interference, fading, and path loss.Small-scale fading is modeled as exponential for small-cell links and Gamma(N,1) for macro-cell links.
  • 1) NOMA enhanced small cell transmission:: When the typical user is farther from the BS than the connected user, the connected near user first decodes the typical far user's message and applies SIC.This case is defined by x > r_k, after which the near user's SINR excludes the canceled far-user signal.

2) Massive MIMO aided macro cell transmission:

The framework combines massive MIMO macro cells with NOMA-enhanced small cells and associates users by maximizing biased average received power. Increasing macro-cell antennas favors macro-cell association through array gains, while increasing the NOMA power-sharing coefficient favors small-cell association.

  • Massive MIMO aided macro cell transmission: Macro BSs use massive MIMO, while small-cell analysis accounts for interference from both macro and small cells.The macro-user SINR includes desired massive-MIMO reception and interference terms from other macro BSs and small-cell BSs.
  • User association: User association is based on maximizing each tier’s biased average received power.The framework permits users to access any tier providing the best coverage under flexible association.
  • User association: The analysis derives association probabilities and distance distributions for users connected to both NOMA small cells and macro cells.Separate distance PDFs are obtained for connected small-cell and macro-cell BSs.
  • User association: Increasing the number of macro-cell antennas increases macro-cell association probability and decreases small-cell association probability.The paper attributes this trend to the large array gains provided by macro cells.
  • User association: Increasing the NOMA power-sharing coefficient a_n increases small-cell association probability.As a_n approaches 1, the association rule becomes the same as the conventional OMA-based approach.

B. The Laplace Transform of Interference

The interference analysis decomposes total interference into small-cell and macro-cell components using Laplace transforms. Coverage is then evaluated separately for near and far NOMA users, whose decoding conditions depend on target rates and power allocation.

  • Interference Laplace transforms: Total interference for a small-cell user is decomposed into small-cell and macro-cell interference, with a factorized Laplace transform.The paper writes the total-interference transform as the product of the transforms for the two interference sources.
  • Interference Laplace transforms: The interference transforms use special functions including the Gauss hypergeometric function and incomplete Beta function.The expressions also impose nearest-distance constraints for the connected small-cell or macro-cell BS.
  • Coverage probability: Near-user coverage requires decoding the connected user’s message and then decoding the near user’s own message.The conditional event is represented by simultaneous SINR thresholds for both decoding steps.
  • Coverage probability: Far-user coverage treats the near user’s message as noise when decoding the far user’s own message.A closed-form conditional coverage expression is given when a_m,k − τ_t a_n,k is nonnegative; otherwise the coverage probability is zero.
  • Coverage probability: The overall small-cell coverage probability combines the near- and far-user cases over the user-distance distribution.An exact analytical expression is provided, with a closed-form special case under equal path-loss exponents and negligible thermal noise.
  • Coverage probability: Coverage depends on both users’ target rates and can be invalidated by inappropriate power allocation.The target rate of the connected user and the typical user both enter the coverage conditions.

IV. SPECTRUM EFFICIENCY

The paper evaluates spectrum efficiency tier by tier for the proposed NOMA-enhanced hybrid HetNets framework.

  • Spectrum efficiency: Spectrum efficiency is evaluated separately for each tier of the NOMA-enhanced hybrid HetNets.

A. Ergodic Rate of NOMA enhanced Small Cells

For NOMA-enhanced small cells, achievable ergodic rates are determined opportunistically from users’ channel conditions rather than fixed target rates. The paper derives expressions for near-user, far-user, and aggregate small-cell ergodic rates, while noting that the aggregate result generally remains a double integral.

  • Ergodic rate: NOMA small-cell ergodic rates are determined opportunistically by users’ channel conditions instead of fixed targeted rates.If the far user decodes its own message, the near user can decode the far-user message because it has better channel conditions.
  • Ergodic rate: The paper derives achievable ergodic-rate expressions separately for near and far users in each small-cell tier.
  • Ergodic rate: The achievable ergodic rate of the small cells is then expressed conditionally on the HPPPs using the near- and far-user rate results.
  • Ergodic rate: The aggregate ergodic-rate expression is generally a double integral because closed-form solutions remain challenging even in some special cases.The derived expression is described as more efficient and more accurate than Monte Carlo simulation.

B. Ergodic Rate of Macro Cells

The paper derives a tractable lower bound for macro-cell ergodic rate because exact evaluation becomes intractable with many antennas. The bound shows that increasing macro-cell antennas enhances achievable ergodic rate through larger array gains.

  • Massive MIMO can improve macro-cell ergodic rate through multiple-antenna array gains, but increases power consumption and complexity.Exact analytical results require high-order Laplace-transform derivatives, which become mathematically intractable as antenna count grows.
  • Theorem 3 provides a lower bound for the achievable ergodic rate of macro cells.
  • Under equal path loss exponents, the macro-cell ergodic-rate lower bound has a closed-form expression.
  • Increasing the number of macro-cell antennas enhances achievable ergodic rate because users experience larger array gains.

C. Spectrum Efficiency of the Proposed Hybrid Hetnets

The paper combines macro-cell ergodic-rate lower bounds with exact small-cell spectrum efficiencies to obtain a tractable spectrum-efficiency expression for the proposed hybrid HetNets.

  • Proposition 1 gives a tractable lower-bound expression for the spectrum efficiency of the proposed hybrid HetNets.
  • The network expression uses the macro-cell lower-bound spectrum efficiency and the exact spectrum efficiency of each small-cell tier.

V. ENERGY EFFICIENCY

The paper evaluates energy efficiency by modeling small-cell and macro-cell power consumption and combining these models with their respective spectrum efficiencies. The resulting hybrid-network energy efficiency is summarized in Proposition 2.

  • Energy efficiency is investigated because it is an important performance metric for 5G systems.
  • The analysis first models power consumption for both small-cell and macro-cell base stations.
  • Small-cell power consumption includes static hardware consumption and the power-amplifier efficiency factor for each tier.
  • Macro-cell power consumption includes static hardware use, amplifier efficiency, and practical transceiver, precoding, and coding/decoding parameters.
  • The energy-efficiency formulation uses total energy consumption together with the spectrum efficiencies of NOMA small cells and macro cells.
  • Proposition 2 gives the energy efficiency of the proposed hybrid HetNets.

VI. NUMERICAL RESULTS

Numerical results show that association, coverage, spectrum efficiency, and energy efficiency depend strongly on antenna counts, bias factors, power allocation, and targeted rates. The proposed NOMA-enhanced HetNets outperform OMA-based small cells while revealing important energy-efficiency trade-offs.

  • User Association Probability and Coverage Probability: Increasing macro BS antennas shifts user association toward macro cells by increasing their array gain, while larger bias factors encourage small-cell association.The association trends are attributed respectively to massive-MIMO array gains and bias-based small-cell coverage or load balancing.
  • User Association Probability and Coverage Probability: Coverage probability depends strongly on targeted rates and NOMA power sharing, with inappropriate selections producing zero coverage.For fixed coefficients, optimal targeted rates exist; for given targeted rates, the plotted surfaces indicate an optimal power allocation.
  • Spectrum Efficiency: NOMA-enhanced small cells outperform OMA-based small cells in spectrum efficiency, although increasing the bias factor decreases small-cell spectrum efficiency.A larger bias factor associates more low-SINR macro users with small cells, degrading their spectrum efficiency.
  • Spectrum Efficiency: Macro cells achieve higher spectrum efficiency than small cells, and their spectrum efficiency improves as the bias factor increases.The paper attributes the macro-cell advantage to simultaneous multiuser service and array gains; shifting low-SINR users away from macro cells also improves macro spectrum efficiency.
  • Energy Efficiency: Macro-cell energy efficiency decreases as antenna count increases, whereas NOMA-enhanced small cells achieve higher energy efficiency than massive-MIMO macro cells.Additional antenna-driven array gains increase spectrum efficiency but also add substantial baseband processing power consumption.
  • Numerical Evaluation: The proposed framework combines NOMA small cells and massive-MIMO macro cells, deriving analytical coverage, spectrum-efficiency, and energy-efficiency results.The numerical evaluation uses Monte Carlo simulations to verify the analytical results.

APPENDIX A: PROOF OF LEMMA 2

Appendix A derives the near-user ergodic rate and the relevant SINR distribution by evaluating interference Laplace transforms for small-cell and macro-cell BSs.

  • The small-cell interference Laplace transform is derived using Campbell’s theorem, the PPP generating functional, and exponential fading.
  • The macro-cell interference transform uses the PPP generating functional and Gamma-distributed channel gain with parameter (N, 1).
  • The proof first formulates the near-user ergodic rate in the k-th small-cell tier.
  • Combining the interference transforms with prior expressions yields the complete CCDF of the near-user SINR γ_kn.
  • For z above the near-user threshold, the far-user SINR CCDF is zero; below the threshold, a similar derivation gives the existing user’s ergodic rate.

APPENDIX D: PROOF OF THEOREM 3

Appendix D derives a lower bound for macro-cell ergodic rate by applying Jensen’s inequality, large-system channel approximations, and stochastic-geometry expectations.

  • Jensen’s inequality provides the lower bound for the achievable ergodic rate of macro cells.
  • The law of large numbers approximates the channel quantity ρ_o,1 by GM, enabling an approximation of τ_1,L.
  • The proof evaluates expected macro- and small-cell interference conditioned on d_o,1 = x using Campbell’s theorem.
  • Substituting the evaluated expectation terms into the preceding expression produces the result in (35).
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