PaperlayerView exact revision on arXiv

Source-linked AI summary

On the Performance Gain of NOMA over OMA in Uplink Communication Systems

Zhiqiang Wei, Lei Yang, Derrick Wing Kwan Ng, Jinhong Yuan, Lajos Hanzo

arXiv:1903.01683v1cs.IT

TL;DR

The paper asks how much ergodic sum-rate gain NOMA provides over OMA in practical uplink cellular systems and how that gain varies across antenna configurations and deployments. It develops a unified analysis spanning single-antenna, multi-antenna, massive-MIMO, single-cell, and multi-cell systems. The analysis identifies distinct gain sources and reports scaling with cell size, antennas, and users, including a constant small-scale fading gain of γ = 0.57721 nat/s/Hz in Rayleigh fading.

  • Problem

    The literature lacks comprehensive ESG analysis of NOMA over OMA across practical detection techniques, antenna configurations, and single-cell or multi-cell deployments.

  • Method

    The paper develops a unified analytical treatment of uplink NOMA-versus-OMA ESG for single-antenna, multi-antenna, and massive-MIMO systems in single-cell and multi-cell deployments.

  • Results

    NOMA’s ESG includes a cell-size-dependent large-scale near-far gain, a Rayleigh-fading small-scale gain of γ = 0.57721 nat/s/Hz, and gains that scale with antennas and users.

  • Takeaways & Limitations

    The paper shows that NOMA’s ESG can grow with cell size, the number of base-station antennas, and the number of users across the analyzed uplink scenarios.

Abstract

from arXiv · show

In this paper, we investigate and reveal the ergodic sum-rate gain (ESG) of non-orthogonal multiple access (NOMA) over orthogonal multiple access (OMA) in uplink cellular communication systems. A base station equipped with a single-antenna, with multiple antennas, and with massive antenna arrays is considered both in single-cell and multi-cell deployments. In particular, in single-antenna systems, we identify two types of gains brought about by NOMA: 1) a large-scale near-far gain arising from the distance discrepancy between the base station and users; 2) a small-scale fading gain originating from the multipath channel fading. Furthermore, we reveal that the large-scale near-far gain increases with the normalized cell size, while the small-scale fading gain is a constant, given by $γ$ = 0.57721 nat/s/Hz, in Rayleigh fading channels. When extending single-antenna NOMA to $M$-antenna NOMA, we prove that both the large-scale near-far gain and small-scale fading gain achieved by single-antenna NOMA can be increased by a factor of $M$ for a large number of users. Moreover, given a massive antenna array at the base station and considering a fixed ratio between the number of antennas, $M$, and the number of users, $K$, the ESG of NOMA over OMA increases linearly with both $M$ and $K$. We then further extend the analysis to a multi-cell scenario. Compared to the single-cell case, the ESG in multi-cell systems degrades as NOMA faces more severe inter-cell interference due to the non-orthogonal transmissions. Besides, we unveil that a large cell size is always beneficial to the ergodic sum-rate performance of NOMA in both single-cell and multi-cell systems. Numerical results verify the accuracy of the analytical results derived and confirm the insights revealed about the ESG of NOMA over OMA in different scenarios.

I. INTRODUCTION

The paper addresses limited unified evidence on NOMA’s ergodic sum-rate gain over OMA across practical antenna configurations and cellular deployments. It analyzes the gain’s sources and behavior from single-antenna systems through massive-MIMO and multi-cell settings.

  • Research gap: Existing literature lacks a comprehensive ESG analysis of NOMA over OMA using practical detection techniques across single-antenna, multi-antenna, massive-array, single-cell, and multi-cell systems.The paper identifies missing comparisons across these practical scenarios.
  • Approach: The study provides a unified analysis of NOMA’s ESG over OMA in single-antenna, multi-antenna, and massive-MIMO uplink systems across single-cell and multi-cell deployments.Its stated aim is to clarify ESG behavior across these system configurations.
  • SNR behavior: In the high-SNR regime, NOMA achieves high ESG, whereas the ESG vanishes at low SNR.This regime-dependent behavior is stated for the cases considered.
  • Single-antenna findings: In single-antenna systems, NOMA’s ESG comprises a large-scale near-far gain that increases with cell size and a small-scale fading gain that equals γ = 0.57721 nat/s/Hz in Rayleigh fading.The two gains are attributed to distance discrepancy and small-scale fading, respectively.
  • Multi-antenna findings: For sufficiently many users, using M base-station antennas increases the SISO-NOMA-over-SISO-OMA ESG by M-fold.The paper analytically quantifies the gain from adding antennas.
  • Multi-antenna findings: Compared with MIMO-OMA using MRC, MIMO-NOMA achieves an (M − 1)-fold degrees-of-freedom gain, with high-SNR ESG slope (M − 1) versus SNR in dB.This result concerns the high-SNR regime.

K , the ESG of mMIMO-NOMA over mMIMO-OMA

With a massive antenna array and a fixed antenna-to-user ratio, the ESG of mMIMO-NOMA over mMIMO-OMA increases linearly with both K and M. In multi-cell systems, inter-cell interference degrades ESG, while larger cells remain beneficial.

  • The ESG of mMIMO-NOMA over mMIMO-OMA increases linearly with both K and M using MRC detection.
  • Without joint multi-cell signal processing, inter-cell interference degrades NOMA's ESG, especially for small cells with dense deployment.No degrees-of-freedom gain is achieved in multi-cell systems under this setting.
  • The ESG of NOMA over OMA saturates in the high-SNR regime across single-antenna, multi-antenna, and massive-MIMO multi-cell systems.
  • A larger cell size benefits NOMA in both single-cell and multi-cell systems.In multi-cell systems, larger cells reduce inter-cell interference and prevent severe ESG degradation; in single-cell systems, they enhance the near-far gain.

B. Signal and Channel Model

The paper models uplink NOMA and OMA with distance-dependent Rayleigh channels, a total transmit-power constraint, and perfect CSI at the base station. It compares SIC-based NOMA with FDMA-based OMA using specified multi-antenna detection and grouping strategies.

  • Signal Model: A total transmit-power constraint Pmax is imposed, with perfect uplink CSI assumed at the base station for coherent detection.
  • Channel Model: The channel model combines Rayleigh fading with distance-based path loss, using user distance d_k and path-loss exponent α.Shadowing is ignored to simplify analysis while retaining the distance-based characterization of near-far gain.
  • Signal Detection: SISO-NOMA uses SIC, while MIMO-NOMA uses MMSE-SIC and massive-MIMO NOMA uses MRC-SIC.OMA uses FDMA with ZF or MRC detection depending on the antenna regime.
  • Resource Allocation: The analysis adopts equal resource allocation, random OMA user grouping, and channel-gain-based SIC ordering for NOMA.The selected SIC order is not generally optimal for maximizing MIMO-NOMA or massive-MIMO-NOMA sum rate.
  • Limitations: Optimal user grouping is generally intractable, and optimal SIC ordering in multi-antenna and massive-MIMO systems remains open.

III. ESG OF SISO-NOMA OVER SISO-OMA

The paper derives ergodic sum rates for SISO-NOMA and SISO-OMA and compares their asymptotic ESG under equal resource allocation. NOMA's gain separates into a distance-driven near-far component and a Rayleigh-fading component.

  • ESG Decomposition: The asymptotic ESG of SISO-NOMA over SISO-OMA is composed of the large-scale component ϑ(D,D0) and the small-scale component γ.
  • Comparison with OMA: SISO-NOMA provides a higher asymptotic ergodic sum rate than SISO-OMA because the near-far component is nonnegative.
  • Large-Scale Near-Far Gain: The near-far component depends on normalized cell size η = D/D0, not the absolute values of D and D0.It represents the gain from exploiting distance discrepancies among NOMA users.
  • Small-Scale Fading Gain: When all users are at the same distance, the near-far gain vanishes and the ESG equals the Rayleigh small-scale gain γ = 0.57721 nat/s/Hz.This is the minimum asymptotic ESG in the stated K →∞ and Pmax →∞ regime.
  • Large-Scale Near-Far Gain: Different absolute cell radii yield the same ESG when they have the same normalized cell size η.
  • Large-Scale Near-Far Gain: The large-scale near-far gain increases with normalized cell size because greater distance heterogeneity strengthens large-scale fading differences among users.

IV. ESG OF MIMO-NOMA OVER MIMO-OMA

This section derives ergodic sum-rates for MIMO-NOMA and MIMO-OMA and compares their asymptotic ESG under many-user conditions. With MMSE-SIC, MIMO-NOMA becomes asymptotically equivalent to SISO-NOMA with M-fold spatial degrees of freedom.

  • MIMO-NOMA analysis: MMSE-SIC is capacity-achieving for the instantaneous MIMO-NOMA channel, although its determinant-based capacity expression is difficult to simplify.The section therefore develops an asymptotically tight upper bound for large K.
  • Ergodic-rate derivation: The ergodic sum-rate is obtained by averaging the instantaneous sum-rate over Gamma-distributed channel gains conditioned on user distance.The analysis uses the conditional PDF and CDF of the channel gain and Gaussian-Chebyshev quadrature for the resulting distributions.
  • Ergodic-rate derivation: Equal power allocation and the independence of system sum-rate from MMSE-SIC decoding order simplify the large-user ergodic sum-rate analysis.The users are treated as identically distributed within the cell under the stated assumption.
  • Asymptotic behavior: For K→∞ with M≪K, MIMO-NOMA behaves asymptotically like SISO-NOMA with M-fold spatial degrees of freedom.Diverse user channel directions span the M-dimensional signal space, allowing MMSE-SIC to exploit the available spatial DoF.
  • Asymptotic behavior: MIMO-NOMA fully exploits the system’s spatial DoF and is approximated by SISO-NOMA with M-fold DoF when K is much larger than M.This follows from the received signals spanning the M-dimensional signal space.

B. Ergodic Sum-rate of MIMO-OMA with FDMA-ZF

This section formulates MIMO-OMA ergodic sum-rates under FDMA-ZF and FDMA-MRC, then compares their asymptotic ESG against MIMO-NOMA. The comparison shows distinct spatial-DoF and power-gain effects across detection schemes and SNR regimes.

  • FDMA-ZF: FDMA-ZF groups M users per frequency subband and uses normalized zero-forcing vectors derived from each group’s composite channel matrix.Random user grouping and equal resource allocation are assumed.
  • FDMA-ZF: For large K, MIMO-NOMA’s ESG over FDMA-ZF is M times the corresponding single-antenna ESG.Both the large-scale near-far gain and small-scale fading gain are multiplied by M.
  • FDMA-ZF: MIMO-NOMA and FDMA-ZF both exploit the maximal spatial DoF M, while ZF incurs an additional power gain of ln(M) in the asymptotic comparison.The ZF projection loses average power within each user group to suppress inter-user interference.
  • FDMA-MRC: FDMA-MRC has no MIMO-NOMA performance gain in the low-SNR regime.The section states that the asymptotic ESG vanishes as Pmax→0.
  • FDMA-MRC: A closed-form ESG is unavailable for FDMA-MRC, although its high-SNR ESG is expected to be dominated by a term increasing linearly with SNR in dB.The remaining term is constant for fixed outer and inner cell radii.
  • FDMA-MRC: At high SNR, MIMO-NOMA achieves an (M−1)-fold DoF gain over FDMA-MRC, whereas FDMA-MRC provides only an ln(M) power gain.The resulting ESG therefore has a power reduction by a factor of ln(M) in its second term.

V. ESG OF mMIMO-NOMA OVER mMIMO-OMA

This section analyzes massive-MIMO NOMA and OMA with a large antenna array and fixed user-to-antenna scaling. It derives asymptotic ergodic sum-rates under MRC-based reception and considers both D>D0 and D=D0.

  • Massive-MIMO-NOMA: Massive-MIMO-NOMA uses MRC-SIC at the base station, with all single-antenna users transmitting simultaneously as M grows.The asymptotic analysis assumes equal resource allocation.
  • Massive-MIMO-OMA: Massive-MIMO-OMA groups W=ςM users per frequency subband and uses orthogonal subbands with low-complexity MRC detection.Favorable propagation makes inter-user interference negligible under the stated grouping model.
  • Massive-MIMO-OMA: The massive-MIMO-OMA instantaneous rate expression is an upper bound because the analysis assumes an interference-free grouped subband.The ergodic sum-rate is derived under equal resource allocation and the condition D>D0.
  • Asymptotic comparison: For D=D0, Theorem 3 gives asymptotic ergodic sum-rates for both massive-MIMO-NOMA and massive-MIMO-OMA when K and M grow with fixed scaling.The result uses equal resource allocation and K=δς.

C. ESG in Massive-antenna Systems

In massive-antenna systems, the asymptotic ESG grows linearly with both the number of antennas and users under fixed ratios, but this scaling does not constitute an additional DoF gain. The multi-cell extension is motivated by stronger inter-cell interference for NOMA than OMA.

  • Scope and regime: The massive-MIMO expressions for D>D0 are limited to that cell-size regime, while the D=D0 case is used to expose clearer insights.Simulations are stated to show that the D=D0 insights also apply when D>D0.
  • Massive-antenna scaling: With fixed average received sum SNR and fixed ratios δ and ς, the asymptotic ESG scales linearly with both M and K.The asymptotic ESG per user and per antenna are constant under this scaling.
  • Massive-antenna scaling: The linear ESG scaling arises because the spatial DoF of both massive-MIMO-NOMA and massive-MIMO-OMA increase linearly with M and K.NOMA uses an M×K system, while OMA’s DoF is limited by its grouped M×W structure.
  • Massive-antenna scaling: Unlike the finite-antenna high-SNR comparison, massive-MIMO-NOMA gains no additional DoF over massive-MIMO-OMA despite linear ESG scaling.The extra benefit ζ does not increase linearly with SNR in dB.
  • Multi-cell extension: Multi-cell analysis is needed because NOMA experiences more severe inter-cell interference than OMA, whose orthogonal allocation limits interference-producing users.Single-cell NOMA gain serves as an upper bound for non-cooperative multi-cell systems approached through conservative frequency reuse.

A. Inter-cell Interference in NOMA and OMA Systems

The multi-cell analysis models inter-cell interference (ICI) for NOMA and OMA under shared-frequency operation, then derives their SINR and interference-power expressions. It assumes perfect serving-cell CSI, unknown ICI channels, and no cooperative multi-cell processing.

  • The analysis uses unity frequency reuse, so adjacent-cell users transmit over the same frequency band as the serving cell.
  • Each cell contains one M-antenna base station serving K single-antenna uplink users, with L adjacent cells imposing interference.
  • The serving base station knows all serving-cell channels but not the ICI channels, and uses receive beamformers based on serving-cell channels and interference structure.
  • After beamforming, each interfering link is equivalent to a single-antenna Rayleigh fading channel, regardless of the number of serving-base-station antennas.
  • The detector treats ICI as additive white Gaussian noise, making multi-cell performance depend on SINR rather than the single-cell SNR.
  • NOMA interference includes all adjacent-cell users, whereas OMA partitions users into G frequency groups so only 1/G of adjacent-cell users transmit on each subband.

B. ESG in Multi-cell Systems

In multi-cell systems, ICI reduces NOMA’s ergodic sum-rate gain over OMA, while the analytical expressions extend the single-cell results by replacing noise with interference-plus-noise. Simulations closely match the analyses across user, antenna, and SNR settings.

  • Without joint multi-cell processing, the effective ICI channel remains single-antenna Rayleigh fading and its interference power is independent of the serving base station’s antenna count.
  • NOMA rates are obtained from single-cell analyses by replacing N0 with INOMA_inter + N0, while OMA faces reduced interference because only 1/G users transmit per subband.
  • ICI degrades NOMA’s ESG relative to single-cell systems because non-orthogonal transmissions create stronger inter-cell interference than OMA.
  • The ESG increases with user count, approaches its K →∞ asymptote, and becomes nearly constant per user in massive-MIMO systems with a fixed antenna-to-user ratio.
  • Additional antennas substantially increase multi-antenna ESG, while larger normalized cell sizes increase ESG through stronger large-scale near-far gain.

B. ESG versus the SNR in Single-cell Systems

Single-cell simulations examine how NOMA’s ESG varies with SNR, antennas, users, and cell size. The results show distinct SNR trends across antenna regimes, with fading, near-far, spatial, and degrees-of-freedom gains shaping the comparison with OMA.

  • SNR dependence: MIMO-NOMA with FDMA-MRC has an (M −1)-fold DoF gain over MIMO-OMA, producing ESG growth linear in SNR in the high-SNR regime.
  • SNR dependence: mMIMO-NOMA ESG first increases and then decreases with SNR because MRC-SIC becomes interference-limited while mMIMO-OMA remains interference-free.
  • SNR dependence: 0.575 nat/s/Hz is the SISO-NOMA ESG over SISO-OMA at SNRsum = 40 dB when η = 1, verifying the small-scale fading gain γ.
  • SNR dependence: The ESG increases with normalized cell size in single-antenna, multi-antenna, and massive-MIMO systems because larger cells strengthen the large-scale near-far gain.
  • Antenna dependence: Increasing the number of antennas raises MIMO-NOMA ESG linearly through the DoF gain, while an additional ln(M) power-gain factor appears with FDMA-ZF.
  • Antenna dependence: In massive-MIMO systems, ESG per user remains almost constant as M increases under a fixed M/K ratio, while larger cells provide higher ESG per user.

VIII. CONCLUSIONS AND FUTURE WORK

The paper quantifies NOMA’s ergodic sum-rate gain over OMA across antenna configurations and cell deployments, identifying near-far, fading, spatial, and interference effects. It also states idealized assumptions and future work addressing imperfect CSI and SIC errors.

  • Scope and contributions: The study analyzes NOMA-over-OMA ESG in single-antenna, multi-antenna, massive-MIMO, single-cell, and multi-cell uplink systems.The analytical results are extended from single-cell systems to multi-cell systems by characterizing effective inter-cell interference.
  • Single-antenna systems: Two single-antenna gains are identified: a large-scale near-far gain and a small-scale fading gain.The large-scale gain increases with cell size, while the small-scale fading gain is constant at γ = 0.57721 nat/s/Hz in Rayleigh fading channels.
  • Multi-antenna systems: M antennas increase the SISO-NOMA-over-SISO-OMA ESG by M times through the extra spatial degrees of freedom.This result concerns the considered multi-antenna extension of the single-antenna system.
  • Massive-MIMO systems: In massive-MIMO single-cell systems, NOMA’s ESG over OMA increases linearly with both the number of users and the number of base-station antennas.The conclusion reports this scaling for the considered massive-MIMO setting.
  • Multi-cell systems: A larger cell size is preferred by NOMA in both single-cell and multi-cell systems, due to enhanced near-far gain and reduced inter-cell interference, respectively.The paper reports that simulations verify the analytical results and corresponding insights.
  • Assumptions and future work: The analysis assumes perfect CSI and error-propagation-free SIC, while future work will consider imperfect CSI and SIC error propagation.The paper notes that perfect CSI is difficult to acquire because of channel-estimation errors, feedback delays, and quantization errors.

APPENDIX

The appendix proves asymptotic results for multi-antenna NOMA by constructing a virtual system, bounding its capacity, and analyzing channel directions and MRC-SIC rates. The bounds converge as the number of users grows.

  • Capacity bounds: A virtual K-user M × M MIMO system is constructed as an upper-bound model for the original uplink K-user 1 × M MIMO system.The virtual system gives each user M parallel subchannels with identical gain ∥h_k∥, and the original system is recovered by choosing the precoder u_k = h_k.
  • Capacity bounds: The virtual system’s capacity upper bound and the MRC-SIC achievable-rate lower bound converge asymptotically as K →∞.The result holds for any given power-allocation strategy p = [p_1, ..., p_K].
  • MRC-SIC analysis: The MRC-SIC receiver is used to derive achievable user rates and an achievable sum-rate for the K-user 1 × M MIMO-NOMA system.The analysis assumes user power allocation p and no error propagation during the decoding process.
  • Channel statistics: Each normalized channel direction e_k is uniformly distributed on the unit sphere and is independent of its channel gain ∥h_k∥.The appendix also characterizes the mean and covariance of e_k and the projection between users’ channel directions.
  • Asymptotic rates: For large K, the random inter-user-interference term is approximated by a deterministic value, enabling asymptotic individual-rate and sum-rate expressions.The appendix obtains these expressions from the MRC-SIC achievable-rate formulation and related channel-statistics results.

B. Proof of Theorem 2

The proof of Theorem 2 derives asymptotic NOMA and OMA rates under equal resource allocation and channel hardening. It compares the resulting sum-rates to establish the massive-MIMO ESG behavior.

  • Massive-MIMO NOMA: For large K, the random inter-user-interference term is approximated by a deterministic quantity under equal resource allocation.This approximation is substituted into the individual-rate and ergodic-sum-rate expressions for massive-MIMO NOMA.
  • Channel hardening: As M →∞, channel hardening averages out small-scale fading, leaving the channel gain mainly determined by large-scale fading.The resulting individual rate becomes deterministic when both K →∞ and M →∞.
  • NOMA–OMA comparison: The proof derives asymptotic individual and ergodic sum-rates for both massive-MIMO NOMA and massive-MIMO OMA.The NOMA and OMA expressions use channel-hardening approximations for the considered equal-resource-allocation setting.
  • NOMA–OMA comparison: The resulting asymptotic comparison yields the massive-MIMO NOMA ESG expression through the derived rate formulas.The proof concludes after substituting the asymptotic expressions into the target result.
Loading 1903.01683v1…