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A Mathematical Proof of the Superiority of NOMA Compared to Conventional OMA

Zhiyong Chen, Zhiguo Ding, Xuchu Dai, Rui Zhang

arXiv:1612.01069v1cs.IT

TL;DR

The paper examines whether NOMA's advantage over OMA persists after both systems receive optimal resource allocation. It derives a closed-form NOMA solution using power splitting and proves that NOMA always outperforms conventional OMA under the studied fairness settings.

  • Problem

    Existing work indicates NOMA gains over OMA with fixed allocation, but whether those gains persist under optimal time/frequency/power allocation remains unclear.

  • Method

    The paper compares optimized NOMA and conventional OMA systems, deriving NOMA's closed-form optimum sum rate through power splitting and using rigorous mathematical proofs.

  • Results

    NOMA always achieves a higher optimum sum rate than conventional OMA, including OMA with jointly optimized power and time/frequency allocation.

  • Takeaways & Limitations

    The paper's supported conclusion is that NOMA retains its sum-rate advantage over the considered optimally allocated OMA systems.

Abstract

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While existing works about non-orthogonal multiple access (NOMA) have indicated that NOMA can yield a significant performance gain over orthogonal multiple access (OMA) with fixed resource allocation, it is not clear whether such a performance gain will diminish when optimal resource (Time/Frequency/Power) allocation is carried out. In this paper, the performance comparison between NOMA and conventional OMA systems is investigated, from an optimization point of view. Firstly, by using the idea of power splitting, a closed-form expression for the optimum sum rate of NOMA systems is derived. Then, with rigorous mathematical proofs, we reveal the fact that NOMA can always outperform conventional OMA systems, even if both are equipped with the optimal resource allocation policies. Finally, computer simulations are conducted to validate the accuracy of the analytical results.

I. INTRODUCTION

NOMA is presented as a spectrally efficient alternative to conventional OMA, using superposition coding and successive interference cancellation to let users share resource blocks. The paper situates this approach within prior theoretical and system-level studies of NOMA and its variants.

  • Conventional OMA: Conventional OMA assigns users orthogonal time, frequency, or code resources to avoid multiple-access interference.Examples include FDMA, TDMA, CDMA, and OFDMA across cellular generations.
  • Motivation: NOMA is motivated by the inefficient use of spectrum allocated to users with poor channel conditions.The paper states that NOMA is proposed to further improve spectrum efficiency.
  • NOMA operation: NOMA combines superposition coding at the base station with successive interference cancellation at users.The near user decodes and removes the far user's message before decoding its own.
  • NOMA operation: NOMA allows both users to access all resource blocks, while the near user can decode its message without far-user interference.
  • Related studies: Prior work reports NOMA gains in spectral efficiency, user fairness, ergodic sum rates, and information-theoretic sum rate.Related studies also investigate cooperative NOMA, user pairing, MIMO-NOMA, quasi-degradation, and BER optimization.

B. Contributions

The paper asks whether NOMA's reported advantage over fixed-allocation OMA persists under optimal resource allocation. It formulates fair optimization problems and compares NOMA with two OMA models that differ in time/frequency allocation flexibility.

  • Research question: The central question is whether NOMA's performance gain over fixed-allocation OMA diminishes when time, frequency, and power are optimized.
  • System comparison: OMA-TYPE-I uses optimum power allocation with fixed, equal time/frequency allocation, whereas OMA-TYPE-II jointly optimizes power and time/frequency allocation.
  • Optimization formulation: Both NOMA and OMA optimization problems incorporate a user-fairness minimum rate constraint.The minimum rate is denoted r∗, and achievable rates are defined for each user.
  • Contributions: The paper derives a closed-form optimum NOMA sum rate using the power-splitting method.
  • Contributions: It introduces minimum required power and proves that NOMA requires less power than both OMA-TYPE-I and OMA-TYPE-II.
  • Contributions: Rigorous proofs establish that NOMA's optimum sum rate exceeds both OMA-TYPE-I and OMA-TYPE-II under varied user-fairness considerations.

A. Closed-form Solution of NOMA

The NOMA analysis separates the power needed to satisfy every user's minimum rate from excess power used to maximize sum rate. This yields a closed-form optimum in which excess power is assigned to the strongest-channel user.

  • Theorem 1: Theorem 1 gives the optimum closed-form NOMA solution when the stated feasibility condition involving total power P and minimum rate r∗ holds.
  • Power splitting: Power splitting divides total power into minimum power for minimum-rate transmission and excess power.Allocating minimum powers lets all users achieve the minimum rate; the remaining power is optimized separately.
  • Power splitting: The excess-power optimization is reduced to maximizing the excess sum rate subject to the remaining power budget.
  • Optimal allocation: All excess power is optimally allocated to user 1, the user with the strongest channel condition.The excess rates of the other users are zero under this solution.

B. Solution of OMA-TYPE-I

The OMA-TYPE-I solution also splits power between minimum-rate support and excess power, then applies water-filling to the excess allocation. The resulting comparison proves NOMA's superiority, with equality only under identical channel magnitudes.

  • Theorem 2: Theorem 2 establishes the superiority of NOMA compared with OMA-TYPE-I under the stated feasibility condition.
  • Solution method: The OMA-TYPE-I proof separates minimum-rate power from excess power before optimizing the remaining allocation.
  • Solution method: The excess-power allocation for OMA-TYPE-I is obtained using a water-filling policy.
  • Proof technique: The proof uses the Arithmetic Mean-Geometric Mean inequality and Chebyshev's Sum Inequality to compare the optimized rates.
  • Comparison: NOMA's advantage over OMA-TYPE-I becomes equality only when all channel magnitudes are identical.

C. Solution of OMA-TYPE-II

The OMA-TYPE-II solution separates minimum-rate power from excess power and uses bounds and lemmas to establish feasibility and compare optimized rates.

  • Equality in a key bound occurs only when all channel magnitudes are equal.
  • The analysis splits total power into minimum power for minimum-rate transmission and excess power allocated to increase rates.
  • The OMA-TYPE-II optimization is non-convex, making a closed-form optimum or tight upper bound difficult to obtain directly.
  • A tighter upper bound for the excess-rate optimization is introduced through Lemma 2.
  • Lemma 3 provides a lower bound for the minimum required power, supporting the feasibility analysis of the OMA-TYPE-II problem.
  • Combining Lemmas 2 and 3 completes the proof of the stated OMA-TYPE-II result.

D. Major Results

The paper summarizes analytical results for minimum required power and optimum sum rate under NOMA and two conventional OMA configurations. These results establish closed-form expressions and comparisons among the schemes.

  • Minimum powers for reliable transmission under a minimum-rate constraint are derived for NOMA, OMA-TYPE-I, and OMA-TYPE-II.
  • The relationships among the required minimum powers of NOMA, OMA-TYPE-I, and OMA-TYPE-II are characterized.
  • A closed-form expression for the optimum NOMA sum rate is derived.
  • The optimum sum rates of NOMA, OMA-TYPE-I, and OMA-TYPE-II are compared analytically.

IV. SIMULATION RESULTS

Simulations validate the analytical results using deterministic and Rayleigh fading channels, with outage probability and ergodic sum rate used to compare NOMA and two OMA variants.

  • The simulations validate the correctness of the analytical results.Numerical examples use deterministic channels first, followed by Rayleigh fading channels for additional comparisons.
  • The results cover both deterministic channels and Rayleigh fading channels.
  • Outage probability and ergodic sum rate evaluate NOMA, OMA-TYPE-I, and OMA-TYPE-II.

A. Deterministic Channels

Under deterministic channels, simulations compare required minimum power and optimum sum rates across NOMA, OMA-TYPE-I, and OMA-TYPE-II for unequal and identical channel coefficients.

  • NOMA requires less minimum power than OMA-TYPE-II, which requires less than OMA-TYPE-I when channel coefficients differ.This ordering agrees with the analytical results.
  • When channel coefficients are identical, the three systems have the same required minimum power.
  • NOMA numerical and analytical optimum sum rates match perfectly.The analytical values come from the closed-form expression, while numerical values solve the corresponding optimization problem.
  • With unequal channel coefficients, sum rates follow NOMA above OMA-TYPE-II above OMA-TYPE-I.The comparison uses specified channel realizations and r∗ = 1.
  • With identical channel coefficients, the three sum rates become equal.

B. Rayleigh Fading Channels

Under Rayleigh fading, simulations evaluate outage probability and ergodic sum rate for NOMA, OMA-TYPE-I, and OMA-TYPE-II across different user counts and target rates.

  • At Pr = 10^-1, OMA-TYPE-II gains about 1.5dB over OMA-TYPE-I, while NOMA gains about 2.5dB over OMA-TYPE-II.
  • NOMA’s outage-probability gain becomes larger as the number of users increases.
  • Ergodic sum rates consistently rank NOMA above OMA-TYPE-II above OMA-TYPE-I.The evaluated settings are K = 3, r∗ = 1 and K = 6, r∗ = 2.
  • When outage probabilities tend to zero, ergodic sum rate increases linearly with SNR.In Fig. 7, NOMA gains about 0.3 NPCU over OMA-TYPE-II, and OMA-TYPE-II gains about 0.7 NPCU over OMA-TYPE-I.

V. CONCLUSION

The paper compares optimally allocated NOMA and conventional OMA using closed-form analysis, rigorous proofs, and simulations. It establishes NOMA's superior sum-rate performance under both fixed and jointly optimized OMA resource allocations, including Rayleigh fading channels.

  • The comparison addresses whether NOMA's performance gain over OMA persists when resource allocation is optimized rather than fixed.
  • The closed-form optimum NOMA sum rate and corresponding power allocation policy are derived using power splitting.
  • Rigorous mathematical proof shows that NOMA achieves better sum-rate performance than OMA-TYPE-I with optimum power allocation and equal user time/frequency allocation.
  • NOMA also outperforms OMA-TYPE-II in sum rate when power and time/frequency allocation are jointly optimized.
  • Computer simulations validate the analytical results and demonstrate NOMA's advantages over OMA in practical Rayleigh fading channels.
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