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The Application of MIMO to Non-Orthogonal Multiple Access
Zhiguo Ding, Fumiyuki Adachi, H. Vincent Poor
TL;DR
MIMO-NOMA addresses how MIMO can be integrated with NOMA while improving performance relative to orthogonal access. The paper develops precoding, detection, user-pairing, and QoS-oriented power-allocation designs, and reports analytical and simulation-based performance evaluation. Its scope includes fixed power allocation and assumes randomly deployed users.
Problem
The paper examines how MIMO techniques can be applied to NOMA systems and how their performance compares with conventional MIMO-OMA.
Method
The paper proposes MIMO-NOMA precoding and detection matrices, analyzes fixed power allocation, applies user pairing, and develops cognitive-radio-inspired power-allocation choices for QoS constraints.
Results
Analytical and numerical results show better outage performance for MIMO-NOMA than conventional MIMO-OMA, including for users with strong co-channel interference.
Takeaways & Limitations
User pairing and more sophisticated power-allocation coefficients are characterized as ways to improve MIMO-NOMA performance and meet QoS requirements.
Takeaways & Limitations
The analysis assumes users are randomly deployed.
Abstract
from arXiv · showhide
This paper considers the application of multiple-input multiple-output (MIMO) techniques to non-orthogonal multiple access (NOMA) systems. A new design of precoding and detection matrices for MIMO-NOMA is proposed and its performance is analyzed for the case with a fixed set of power allocation coefficients. To further improve the performance gap between MIMO-NOMA and conventional orthogonal multiple access schemes, user pairing is applied to NOMA and its impact on the system performance is characterized. More sophisticated choices of power allocation coefficients are also proposed to meet various quality of service requirements. Finally computer simulation results are provided to facilitate the performance evaluation of MIMO-NOMA and also demonstrate the accuracy of the developed analytical results.
I. INTRODUCTION
The paper applies MIMO techniques to NOMA and proposes new precoding and detection designs. It further studies user pairing and power-allocation choices to characterize performance and QoS trade-offs against MIMO-OMA.
- Motivation: MIMO-NOMA is studied as an extension of NOMA downlink communication, whose power-domain multiplexing seeks to balance throughput and fairness.NOMA allocates more power to users with poorer channel conditions, while conventional opportunistic schemes favor stronger channels.
- Contributions: A new precoding and detection matrix design is proposed for a general downlink where all users participate in NOMA with fixed power-allocation coefficients.Its impact is characterized analytically using outage probabilities and diversity orders.
- Contributions: MIMO-NOMA achieves better outage performance than conventional MIMO-OMA, including for users experiencing strong co-channel interference.The paper supports this conclusion with analytical and numerical results.
- Contributions: User pairing is applied to enlarge the performance gap between MIMO-NOMA and MIMO-OMA, with analytical expressions developed for the average sum-rate gap.The analysis includes an exact expression and a high-SNR approximation.
- User pairing: NOMA pairing favors users with very distinct channel conditions, unlike conventional scheduling, which prefers users with superior channel conditions.This behavior is reported as consistent with findings for single-antenna NOMA.
- Power allocation: Cognitive-radio-inspired power-allocation choices address fixed and dynamic QoS constraints through targeted-rate requirements relative to NOMA or conventional OMA.Analytical results are developed for both constraint types.
II. SYSTEM MODEL WITH FIXED POWER ALLOCATION
The fixed-power system model describes clustered MIMO-NOMA downlink transmission, detection, channel ordering, and successive interference cancellation. Instantaneous channel-dependent power optimization is outside the paper’s scope.
- System configuration: The base station uses M antennas to communicate with users equipped with N antennas, which are randomly grouped into M clusters with K users each.The transmitted signals contain the information-bearing signals of users within each cluster.
- Signal model: The M × M precoding matrix and detection vectors produce each user’s observation from the channel, precoded superposition, and additive Gaussian noise.The NOMA power-allocation coefficient is denoted by α_i,j.
- Ordering and allocation: Users are ordered by effective channel gains, and their NOMA power-allocation coefficients follow the corresponding ordering.The model assumes constant power-allocation coefficients in this section.
- Scope: Instantaneous channel-dependent power-allocation optimization could further improve MIMO-NOMA performance but is beyond the paper’s scope.The fixed-power model therefore does not perform that optimization.
- Detection: Successive interference cancellation requires each intermediate user to decode messages for users with poorer channel conditions before detecting its own message.Successfully decoded messages are removed from the observation when their achievable rate exceeds the targeted data rate.
III. DESIGN OF PRECODING AND DETECTION MATRICES
The paper designs precoding and detection matrices for MIMO-NOMA that remove inter-cluster interference while supporting analytical outage analysis and comparison with MIMO-OMA. The resulting scheme achieves the same diversity gain as conventional MIMO-OMA, while sharing bandwidth among users and improving spectral efficiency.
- Matrix design: The precoding and detection matrices are designed to completely remove inter-cluster interference.
- Matrix design: The detection vector is obtained from the null space of a channel submatrix, requiring N ≥ M for existence.The submatrix has dimension N × (M −1), and the null-space basis contains N −M + 1 left singular vectors.
- CSI requirement: The base station needs only the ordering of users’ effective channel gains, rather than all users’ channel matrices, to implement NOMA.
- Detection: MRC provides one possible choice for the optimization vector z_i,k, which determines the SINRs through the resulting effective channel.
- Performance analysis: Theorem 1 gives exact and high-SNR outage-probability expressions for the ordered users in MIMO-NOMA.
- Comparison with MIMO-OMA: MIMO-NOMA achieves the same diversity gain as conventional MIMO-OMA while allowing all K users in a cluster to share the same bandwidth resource.The shared resource produces better spectral efficiency, including smaller outage probability than conventional NOMA and a larger sum rate in the user-pairing analysis.
IV. THE IMPACT OF USER PAIRING
User pairing reduces NOMA complexity by grouping fewer users for NOMA and using conventional OMA between groups. The achievable gain over MIMO-OMA depends strongly on pairing quality, with larger gains from pairing users with more disparate channel conditions.
- Grouping: User grouping can reduce NOMA complexity by applying NOMA within smaller groups and conventional OMA between groups.
- Pairing model: The analysis focuses on pairing two ordered users per cluster, with the n-th user having better channel conditions than the k-th user.
- Rate-gap analysis: The average sum-rate gap between MIMO-NOMA and conventional MIMO-OMA is characterized analytically for paired users.The expression can replace Monte Carlo simulations, although its integrals and special functions remain complicated.
- Case studies: At high SNR in Case I, the sum-rate gain increases with K when the worst-channel user is paired with the best-channel user.Case I uses n = 1 and k = K.
- Case studies: In Case II, the sum-rate gain diminishes as K increases when the best-channel user is paired with the second-best user.Case II uses n = 1 and k = 2.
- Implication: Careful user pairing is critical for MIMO-NOMA to outperform conventional MIMO-OMA.The analysis also shows that NOMA behaves differently from conventional multiple-access scheduling, where better-channel users are generally favored for throughput.
V. COGNITIVE RADIO INSPIRED MIMO-NOMA
The paper introduces more sophisticated power-allocation choices for MIMO-NOMA to address the tension between system throughput and user fairness. One choice is inspired by cognitive radio networks.
- Power allocation: The paper replaces fixed power-allocation coefficients with more sophisticated choices for selected NOMA user pairs.
- Throughput–fairness dilemma: Choosing α_1,k = 0 maximizes the perspective of overall system throughput by assigning all power to the better-channel user.
- Throughput–fairness dilemma: The same all-power allocation completely ignores user fairness, creating a dilemma in selecting α_1,k.
- Power allocation: The section considers two power-coefficient choices inspired by cognitive radio networks.
A. To meet a fixed QoS requirement
For a fixed QoS target, the cognitive-radio-inspired allocation assigns the poorer-channel user the minimum power needed to meet its requirement and gives the remainder to the better-channel user. The resulting diversity order is characterized, but the better-channel user’s diversity remains constrained by the poorer user’s channel.
- Fixed QoS requirement: A targeted SINR threshold is imposed to ensure the poorer-channel user’s QoS requirement.
- Power allocation: The allocation gives the k-th user the minimum transmission power needed to meet its QoS requirement and assigns the remaining power to the n-th user.
- Diversity analysis: The k-th user can achieve diversity order (N −k + 1)(N −M + 1) under the fixed-QoS design.
- Diversity analysis: The n-th user achieves diversity gain (N −M + 1)(K −k + 1) with the cognitive-radio-inspired coefficient.
- Diversity limitation: The n-th user’s diversity gain is constrained by the k-th user’s channel condition because of the power-allocation design.
- Implications: MIMO-NOMA can achieve a larger diversity order, and MIMO-OFDM can serve more users simultaneously.
B. To meet a dynamic QoS constraint
The dynamic QoS design selects a power allocation coefficient to preserve one user’s rate while enabling NOMA, with an achievable diversity order characterized analytically and tested by simulation.
- B. To meet a dynamic QoS constraint: The coefficient in (39) is chosen as the maximal value satisfying the dynamic QoS constraint.The resulting exact outage-probability expression is difficult to derive because the coefficient is complicated.
- B. To meet a dynamic QoS constraint: The selected coefficient ensures the k-th user has exactly the same outage probability as conventional MIMO-OMA.This follows because the k-th user’s rate should not be reduced by NOMA.
- B. To meet a dynamic QoS constraint: The proposed CR-MIMO-NOMA system achieves a diversity order of (N −M + 1)(K −k + 1) under the dynamic QoS constraint.This order is presented as achievable for the n-th user.
- B. To meet a dynamic QoS constraint: The diversity order in Lemma 4 is only an achievable one, not necessarily the exact diversity order in every rate regime.The paper explicitly qualifies the result as an achievable lower bound.
- B. To meet a dynamic QoS constraint: Simulations show the diversity lower bound is tight for R1,k > 1, while a larger diversity order can occur for 0 ≤R1,k ≤1.The paper attributes the latter possibility to a loose bound used in the proof.
VI. NUMERICAL RESULTS
Numerical studies evaluate fixed- and adaptive-power MIMO-NOMA, user pairing, and cognitive-radio-inspired QoS designs. The simulations support the analytical results and show performance gains over MIMO-OMA under several configurations.
- Simulation configurations: The fixed-power simulations use configurations including four users in two clusters and three clusters with three users each.All users within each cluster participate in NOMA, and Fig. 1 confirms the analytical results of Theorem 1 and Corollary 1.
- Fixed power allocation: MIMO-NOMA achieves better outage performance than MIMO-OMA while both schemes realize the same diversity gain.The simulations also confirm the accuracy of the high-SNR approximation and show different users can have different diversity orders.
- User pairing: The exact average sum-rate-gap expression matches simulations perfectly, while its high-SNR approximation provides a tight bound.The comparison is made between MIMO-NOMA and MIMO-OMA.
- User pairing: Increasing K improves the performance gap when the best and worst users are paired, but diminishes it when the two best users are paired.The difference follows from whether the selected users’ channel conditions become more different or more similar as K increases.
- User pairing: A 2 BPCU sum-rate gap appears with 2 users per group, and the gap doubles with 5 users per group.This result concerns the best-user/worst-user pairing strategy.
- Cognitive-radio-inspired power allocation: For fixed QoS, the achievable diversity order from Lemma 3 is tight, and P(α2 = 1) is a tight lower bound at high SNR.The lower-bound behavior is associated with the event that all power is allocated to the k-th user.
VII. CONCLUSION
The paper proposes and analyzes MIMO-NOMA precoding and detection, then considers user pairing and power allocation to improve performance and satisfy QoS requirements. Simulations support the analytical results, while dynamic low-complexity grouping remains future work.
- A new design of precoding and detection matrices for MIMO-NOMA is proposed and its performance is analyzed.
- User pairing is considered to improve the performance gap between MIMO-NOMA and conventional OMA.
- Cognitive-radio-inspired power allocation coefficients are proposed to meet various QoS requirements.
- Simulation results demonstrate the accuracy of the developed analytical results.
- Users are assumed to be randomly divided into multiple groups, motivating future work on low-complexity dynamic clustering and grouping.
B. Case II with n = 1 and k = 2
This case derives an exact average rate-gap expression and then develops a high-SNR approximation using exponential-integral representations and algebraic simplification.
- The derivation evaluates the factor ϕ(2, φ) after removing integral factors that hinder high-SNR approximation.
- The average rate gap is derived in exact form for Case II with n = 1 and k = 2.
- The exponential integral is simplified through its series representation to obtain the high-SNR average rate-gap approximation.
APPENDIX C
The appendix analyzes outage events for users in a cluster by separating decoding cases, bounding their probabilities, and deriving asymptotic outage expressions.
- Outage events are categorized according to whether a user can decode another user’s message and whether it can decode its own message.
- When a stronger user can decode a weaker user’s message, its decoding condition is linked to the weaker user’s ability to decode its own message.
- The analysis uses SINR conditions, channel-variable constraints, and CDF or incomplete-gamma evaluations to bound outage probabilities.
- High-SNR approximations and upper bounds are obtained for the overall outage probability in the considered cases.