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Performance Analysis of MR-NOMA with Arbitrary Number of Users and Symbol Rates

Zainab Khader, Arafat Al-Dweik, Emad Alsusa

arXiv:2610.03266v1cs.ITmath.IT

TL;DR

MR-NOMA had not been analyzed for BER across arbitrary symbol durations and user counts, despite heterogeneous wireless-service requirements. This paper derives conditional and average BER analyses and formulates power allocation as an average-BER minimization problem. Its results report a BER advantage over SR-NOMA, while the generalized analysis for more than three users assumes perfect SIC for users beyond the first.

  • Problem

    MR-NOMA research was scarce, and BER performance for arbitrary symbol durations and user counts had not previously been investigated.

    Source: B. Motivation and Contributions · p. 2

  • Method

    The paper derives BER expressions for MR-NOMA over Nakagami-m fading and formulates average-BER minimization to optimize power allocation.

    Source: A. Power Allocation · p. 8, §VIII · p. 11

  • Results

    At BER 10^-5, U1's SNR gains over SR-NOMA are approximately 12.3 dB for T = [8, 4, 1] and 10.4 dB for T = [6, 3, 1].

    Source: §VII · p. 9

  • Takeaways & Limitations

    MR-NOMA offers greater flexibility in user pairing while maintaining reliable BER performance.

    Source: §VIII · pp. 11–12

    For more than three users, the remaining users' BER expressions assume perfect SIC because imperfect-SIC analysis is mathematically intractable.

    Source: §VIII · p. 11

Abstract

from arXiv · show

This paper analytically investigates the bit error rate (BER) performance of downlink multi-rate (MR)-non-orthogonal multiple access (NOMA) over Nakagami-m fading channels for an arbitrary number of users and symbol rates. The analysis is first developed for a three-user case, then generalized to systems with more than three users, yielding an exact BER expression for the first detected user and an accurate closed-form approximations for the remaining users under the perfect-successive interference cancellation (SIC) assumption. These results are further unified into a general average-BER expression over Nakagami-m fading for arbitrary user numbers and symbol-rate configurations. A symbol-pattern-dependent signal-to-interference plus noise ratio (SINR) analysis further shows how the symbol-duration diversity enables inherent interference cancellation, explaining the BER advantage of MR-NOMA over conventional single-symbol rate (SR)-NOMA. An average BER minimization problem is also formulated and numerically solved to optimize the users' power allocation, with simulation results corroborating the analytical findings. Overall, MR-NOMA is shown to offer greater flexibility in user pairing while maintaining reliable BER performance and accommodating diverse quality-of-service (QoS) requirements.

I. Introduction

Heterogeneous wireless-service requirements motivate access schemes that support different user rates, while MR-NOMA research had not yet established BER analysis and power optimization for arbitrary symbol durations and user counts. This paper develops those analyses and compares MR-NOMA with conventional SR-NOMA.

A. Related Work

  • Single-carrier MR-NOMA represents mixed numerology through time-domain symbol-rate heterogeneity; prior work introduced it, analyzed two-user BER, and studied Nakagami-m average BER and sum-rate optimization.The paper frames its work as extending earlier two-user BER analyses and Nakagami-m average-BER work.Source: A. Related Work · p. 2

B. Motivation and Contributions

  • MR-NOMA BER for arbitrary symbol durations and user counts, and its power-coefficient optimization, had not previously been investigated.The cited motivation identifies these as gaps in a relatively new research area.Source: B. Motivation and Contributions · p. 2
  • For N = 3, the analysis derives exact BER for the first and second detected users at arbitrary symbol rates, plus a closed-form third-user expression for T = [4, 2, 1].This is described as the first nontrivial extension beyond the two-user case in.Source: B. Motivation and Contributions · p. 2
  • For N > 3, the analysis gives exact BER for U1 and accurate closed-form approximations for the remaining users under perfect SIC, then generalizes average BER over Nakagami-m fading.The average-BER expression covers arbitrary user numbers and symbol-rate configurations.Source: B. Motivation and Contributions · p. 2
  • The paper compares MR-NOMA with SR-NOMA, analyzes symbol-duration diversity and inherent interference cancellation, and formulates average-BER minimization for power allocation.Its analytical and simulation results report a BER advantage for MR-NOMA over SR-NOMA.Source: B. Motivation and Contributions · p. 2

D. Paper Organization

The paper models a downlink MR-NOMA system whose users have different symbol durations and power coefficients, then describes its channel and detection models. It considers SIC as a lower-complexity receiver alternative to joint multiuser maximum-likelihood detection.

A. Transmitted Signal Model

  • The system model assigns users normalized symbol durations and power coefficients, with longer-duration, farther users receiving higher power under the stated ordering.The minimum duration is normalized to one, and users' durations are selected according to their QoS requirements.Source: A. Transmitted Signal Model · p. 3
  • Each user's data symbols occupy successive samples according to its duration, and the transmitted MR-NOMA signal combines the users' power-weighted symbols.The model uses BPSK symbols, with total transmit power normalized to unity.Source: A. Transmitted Signal Model · p. 3

B. Received Signal Model

  • Under quasi-static fading and a uniform sampling period, each user's received sequence includes the channel and independent AWGN samples.The channel and noise model is specified at each user receiver.Source: B. Received Signal Model · p. 3

C. Receiver Design

  • The paper considers joint multiuser maximum-likelihood detection but uses SIC because joint detection has high complexity, particularly with many users.The text states that SIC and joint detection offer equivalent BER for downlink MR- and SR-NOMA,.Source: C. Receiver Design · p. 3

D. SIC Receiver

  • SIC detects the highest-power user's symbols first, then recursively cancels higher-power interference before detecting later users with single-user MLD.For users with multiple symbols in the observation window, the received samples are fragmented into vectors for detection.Source: D. SIC Receiver · pp. 3–4

III. Conditional BER Analysis (N = 3)

For the three-user case, the conditional BER analysis constructs U1's effective constellation by combining oversampled observations and accounting for transmitted symbol patterns. It develops the BPSK analysis and extends its structure to QPSK.

III. Conditional BER Analysis (N = 3)

  • The U1 analysis derives BER by enumerating transmitted cases, first developing the approach for BPSK and then generalizing it to QPSK.The derivation uses the N = 2 results in, analyzes N = 3, and uses induction for arbitrary N; the N = 4 case is omitted for space.Source: §III · p. 4

A. Conditional BER Analysis of U1

  • For T = [4, 2, 1] with BPSK users, sample combining yields 30 distinct accumulated signal values, with repeated transmitted sequences mapping to the same xΣ.The figure groups these values by U1, U2, and U3 contributions and indicates their multiplicities.Source: A. Conditional BER Analysis of U1 · p. 4
  • The case enumeration maps each transmitted case to its symbol sequence and resulting signal amplitude, enabling the conditional U1 error probability to be averaged across equiprobable cases.The example has 128 possible transmitted cases.Source: A. Conditional BER Analysis of U1 · pp. 4–5
  • BPSK sign symmetry and equiprobable case groups reduce C1–C64 to 15 distinct terms for the conditional BER expression at T = [4, 2, 1].The result accounts for multiplicities and symmetric counterparts; the QPSK expressions retain the same general structure with a modulation-dependent modification.Source: A. Conditional BER Analysis of U1 · pp. 5–6

B. Conditional BER Analysis of U2

The U2 conditional BER analysis accounts for errors in detecting U1 before applying SIC and detecting U2's symbols. It extends the case-based expressions to arbitrary symbol durations by induction.

B. Conditional BER Analysis of U2

  • U2 detection first detects U1, applies SIC, and then uses MLD; U2's two symbols are detected independently and have equal error probability.Thus, the derivation needs to analyze only one of the two U2 symbols.Source: B. Conditional BER Analysis of U2 · p. 6
  • For each transmitted case, the U2 error calculation accounts for the outcome of detecting U1 and the resulting signal after SIC.The example evaluates the U1 error probability at U2's channel condition before analyzing the remaining U2 symbol.Source: B. Conditional BER Analysis of U2 · p. 6
  • The U2 conditional BER expressions are generalized to arbitrary symbol durations by induction, with the overall BER formed by combining the relevant conditional error terms.The paper gives the N = 3, T = [4, 2, 1] result and its arbitrary-duration generalization as sums of the corresponding terms.Source: B. Conditional BER Analysis of U2 · p. 6

C. Conditional BER Analysis of U3

For three users, U3’s BER depends on both SIC outcomes; its conditional BER analysis applies the chain rule and averages over possible transmission cases.

C. Conditional BER Analysis of U3

  • U3 performs two SIC operations, so its BER depends on the outcomes of both operations.Because U3’s symbols have identical error probability and are detected independently, the analysis considers only its first symbol.Source: C. Conditional BER Analysis of U3 · p. 6
  • PB3 is derived by averaging over all possible data-transmission cases after applying the chain rule.The analysis defines an error event for the first symbol and considers the possible scenarios for the preceding SIC outcomes.Source: C. Conditional BER Analysis of U3 · pp. 6–7
  • U3’s conditional error probabilities are equivalent to U2’s derivations, with G2,2 replaced by G3,2 in Pr(Er2|[∇1,Cj]) as specified for (27) and subsequent formulae.Source: C. Conditional BER Analysis of U3 · pp. 6–7

A. BER Analysis of U1

A recurring combinatorial pattern in the superposed constellation supports generalizing U1’s BER analysis beyond three users, while the expression’s term count grows rapidly with user number.

A. BER Analysis of U1

  • Identical signal-point multiplicities follow Pascal’s-triangle binomial coefficients as user number or symbol-duration ratios increase.This regularity is observed from induction on N = 2 and the N = 3 cases.Source: A. BER Analysis of U1 · p. 7
  • The recurring constellation pattern lets the analysis generalize U1’s BER expression to N > 3 without case-by-case derivations.The multiplicity of identical signal points follows Pascal’s triangle as the number of users or symbol-duration ratios increases.Source: A. BER Analysis of U1 · p. 7
  • For N ≥ 4, setting αN, αN−1, and αN−2 to zero yields accurate BER approximations because αN ≪ αN−3.The number of terms otherwise grows exponentially with N; the paper notes that N is typically small for user pairing and power optimization.Source: A. BER Analysis of U1 · p. 7

B. BER Analysis of Um for m > 1

For more than three users, the analysis adopts perfect SIC to keep BER derivations tractable and extends the framework to a V-BLAST MIMO configuration using zero-forcing equalization.

B. BER Analysis of Um for m > 1

  • For N > 3, perfect SIC keeps the analysis tractable because imperfect-SIC BER expressions involve too many cases when m > 2.The resulting BER expressions closely approach exact BER, unlike conventional SR-NOMA, and the generalized expression covers arbitrary users and symbol durations.Source: B. BER Analysis of Um for m > 1 · p. 7
  • The BER analysis extends to V-BLAST MIMO with K transmit/receive antennas and zero-forcing equalization.The channel is assumed fixed over T1.Source: B. BER Analysis of Um for m > 1 · pp. 7–8
  • After zero-forcing, the noise PDF matches that of the SISO model’s noise divided by the channel, so the SISO probability analysis applies to this MIMO configuration.The post-ZF noise is non-Gaussian.Source: B. BER Analysis of Um for m > 1 · p. 8

V. Average BER Analysis in Nakagami Fading

The paper extends average-BER analysis to arbitrary-user MR-NOMA over Nakagami-m fading, a model adopted for its flexibility in representing fading conditions.

V. Average BER Analysis in Nakagami Fading

  • The work develops a closed-form average-BER expression for MR-NOMA over Nakagami-m fading with an arbitrary number of users.Earlier average-BER analysis covered N = 2; Nakagami-m is adopted for its flexibility in representing fading conditions.Source: §V · p. 8
  • For integer m, the average-BER expression is evaluated using the single-Q-function form of the conditional BER expression.The passage states that the integral is then evaluated to obtain the expression.Source: §V · p. 8

A. Power Allocation

Reliable SIC requires avoiding sub-constellation overlap, but that condition alone does not guarantee a target BER. The section formulates average-BER power optimization and motivates search-based solutions because the nonlinear problem is analytically intractable.

A. Power Allocation

  • The critical constellation point must remain on the positive side of the real and imaginary axes; symmetry makes checking one of four quadrant points sufficient.This avoids sub-constellation overlap, a condition needed for reliable SIC detection.Source: A. Power Allocation · p. 8
  • Power allocation can minimize average BER because preventing sub-constellation overlap is necessary for reliable SIC but does not guarantee a specified BER.The section therefore uses derived BER expressions to formulate the optimization objective.Source: A. Power Allocation · p. 8
  • The constraints order user powers, ensure reliable SIC, and normalize total power to unity; the problem remains analytically intractable because it is multivariate and nonlinear.The power ordering makes each user's allocation inversely proportional to its average channel gain.Source: A. Power Allocation · p. 8
  • A power deviation of ±0.05 from the optimum results in only 0.9% BER degradation, supporting near-optimal search over a limited power-allocation grid.The passage describes MR-NOMA as generally insensitive to deviations from optimal power assignment.Source: A. Power Allocation · p. 8
  • Figure 2 shows N = 3 with T = [4, 2, 1] and compares allocations α = [0.56, 0.32, 0.12] and α = [0.8, 0.15, 0.05].These are example constellation diagrams for two power allocations.Source: Figure 2 · p. 9

B. Inherent Interference Cancellation

For N = 2 with BPSK and T = [2, 1], MR-NOMA's SINR depends on the transmitted symbol pattern of U2. The analysis finds inherent interference cancellation and an SINR advantage over SR-NOMA, especially at high SNR.

B. Inherent Interference Cancellation

VII. Numerical Results

The numerical results compare MR-NOMA with SR-NOMA across symbol durations and user counts, and evaluate power allocation. The conclusion reports BER analysis for arbitrary user counts and symbol durations while identifying higher-order modulation as an open analytical challenge.

VII. Numerical Results

  • At PB = 10^-5, U1 gains approximately 12.3 dB for T = [8, 4, 1] and 10.4 dB for T = [6, 3, 1] over SR-NOMA.The conditional-BER comparison covers imperfect SIC with BPSK and QPSK; longer symbol intervals increase symbol energy.Source: §VII · p. 9, Figure 3 · p. 10
  • At Pb ≈ 2.8 × 10^-2, MR-NOMA requires 21.5 dB versus 24 dB for SR-NOMA, with at most 0.3 dB spread across four MR duration settings.The average-BER comparison uses QPSK over Nakagami-m fading with m = 2.Source: §VII · p. 10, Figure 4 · p. 11
  • For N = 4 and N = 5, simulations closely match analytical average-BER curves under perfect SIC; for N = 4, agreement improves with symbol energy.For U4, BER curves are nearly identical across duration settings; for N = 5, close agreement holds for all users.Source: §VII · p. 10, Figure 5 · p. 11
  • As the search step Δα approaches 10^-1, optimized BER increases, especially at 30 dB; the rise is steeper for SR-NOMA than MR-NOMA.At fine resolutions, the optimized BER remains nearly unchanged across SNRs.Source: §VII · pp. 10–11, Figure 6 · p. 11
  • At 15 dB, optimized average BER falls from 0.1 for T = [1, 1, 1] to 0.022 for T = [8, 4, 1], and the minimum decreases with SNR.The study also reports a deeper BER valley and lower plateau for MR than SR in the three-user power-allocation surface.Source: Table II · p. 10, §VII · p. 11, Figure 7 · p. 12

VIII. conclusion and Future Work

  • The analysis covers arbitrary user counts and symbol durations, but higher-order modulation remains an open problem because its symbol and interference combinations make direct analysis impractical.For more than three users, BER expressions for users other than the first assume perfect SIC.Source: §VIII · pp. 11–12

Appendix A

The appendix decomposes U3’s BER into four conditional cases based on whether U1 and U2 are correctly detected. It also points readers to parameter tables and MATLAB code for computing the expressions.

Appendix B BER Expressions for User 3

  • U3’s BER expression in (39) is composed of four cases covering the correct or erroneous detection of U1 and U2.The cases include both users detected correctly, either user detected erroneously, and both users detected erroneously.Source: Appendix B · p. 13
  • Tables III and IV list the j parameters used in expressions (60)–(63).The MATLAB code is identified as a resource for computing the BER expressions.Source: Appendix B · p. 13, Table IV · p. 13
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