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Capacity Comparison between MIMO-NOMA and MIMO-OMA with Multiple Users in a Cluster
Ming Zeng, Animesh Yadav, Octavia A. Dobre, Georgios I. Tsiropoulos, H. Vincent Poor
TL;DR
The paper investigates MIMO-NOMA when multiple users share a cluster, addressing the need for analytical comparison with MIMO-OMA and principled user admission. It proves MIMO-NOMA’s superiority in sum capacities, establishes a sum-rate/user-count tradeoff, and proposes an admission scheme with optimality under equal SINR thresholds.
Problem
The performance of MIMO-NOMA with multiple users grouped in one cluster, including its comparison with MIMO-OMA and user-admission tradeoffs, is insufficiently addressed.
Method
The paper analytically compares MIMO-NOMA and MIMO-OMA, studies how sum rate changes with admitted-user count, and develops a SINR-threshold-based user admission scheme.
Results
MIMO-NOMA outperforms MIMO-OMA in sum channel capacity and ergodic sum capacity; adding users lowers sum rate, while the admission scheme is optimal for equal SINR thresholds.
Takeaways & Limitations
Cluster design must balance sum rate against the number of admitted users, and the proposed scheme provides optimal joint performance when users have equal SINR thresholds.
Abstract
from arXiv · showhide
In this paper, the performance of multiple-input multiple-output non-orthogonal multiple access (MIMO-NOMA) is investigated when multiple users are grouped into a cluster. The superiority of MIMO-NOMA over MIMO orthogonal multiple access (MIMO-OMA) in terms of both sum channel capacity and ergodic sum capacity is proved analytically. Furthermore, it is demonstrated that the more users are admitted to a cluster, the lower is the achieved sum rate, which illustrates the tradeoff between the sum rate and maximum number of admitted users. On this basis, a user admission scheme is proposed, which is optimal in terms of both sum rate and number of admitted users when the signal-to-interference-plus-noise ratio thresholds of the users are equal. When these thresholds are different, the proposed scheme still achieves good performance in balancing both criteria. Moreover, under certain conditions,it maximizes the number of admitted users. In addition, the complexity of the proposed scheme is linear to the number of users per cluster. Simulation results verify the superiority of MIMO-NOMA over MIMO-OMA in terms of both sum rate and user fairness, as well as the effectiveness of the proposed user admission scheme.
I. INTRODUCTION
The paper addresses the underexplored multiple-user-per-cluster setting by analytically comparing MIMO-NOMA with MIMO-OMA and studying user admission. It proves capacity advantages for MIMO-NOMA, identifies a sum-rate/user-count tradeoff, and proposes a low-complexity admission scheme.
- Research gap: Most existing MIMO-NOMA studies consider two users per cluster, leaving multiple-user clustering less studied.The paper motivates investigating multiple users per cluster to serve more users simultaneously.
- Research objectives: The paper analytically compares MIMO-NOMA and MIMO-OMA in terms of sum channel capacity and ergodic sum capacity with multiple users in a cluster.This extends beyond prior work that mainly provided simulations or considered limited cluster sizes.
- User admission: As more users are admitted to a cluster, the achieved sum rate decreases, creating a tradeoff between sum rate and the number of admitted users.The paper studies this variation as a basis for user admission design.
- User admission: The proposed user admission scheme is optimal for equal SINR thresholds, balances sum rate and admitted-user count otherwise, and has linear complexity.Under certain conditions, it also maximizes the number of admitted users.
- Capacity comparison: MIMO-NOMA outperforms MIMO-OMA in both sum channel capacity and ergodic sum capacity for clusters containing multiple users.For any MIMO-OMA power split, a larger sum rate can be achieved by assigning the same power coefficient to MIMO-NOMA.
- Capacity comparison: MIMO-NOMA achieves higher user fairness than MIMO-OMA when clusters contain two or three users.The comparison is supported by numerical results.
II. SYSTEM MODEL
The paper models a downlink multiuser MIMO system with ML users randomly grouped into M clusters, each containing L users. MIMO-NOMA uses shared time-frequency resources with power-domain multiplexing, while detection and SIC suppress inter- and intra-cluster interference.
- System configuration: The base station has M antennas and serves ML users, each equipped with N antennas, through M clusters of L users.Channels are modeled as quasi-static i.i.d. fading, with Hm,l representing the channel matrix and nm,l the additive white Gaussian noise vector.
- Interference management: The precoding matrix P and user detection vectors vm,l are designed to remove interference from other clusters.The detection-vector constraint is constructed from the channel columns outside the user’s own cluster.
- Detection-vector construction: The detection vector is feasible when each user has at least as many antennas as the base station.The construction uses singular vectors of the inter-cluster channel matrix and a maximum-ratio-combining vector.
- MIMO-NOMA transmission: MIMO-NOMA superposes users’ signals on the same frequency and time resources while assigning different power coefficients.The coefficients for the L users in each cluster sum to one, and total transmit power is normalized to one.
- Successive interference cancellation: Successive interference cancellation at user (m, l) removes signals from users with worse channel gains.The paper’s lemma states that interference from every user (m, k), k ∈ {l + 1, ..., L}, can be removed at user (m, l).
B. MIMO-OMA
MIMO-OMA divides the available degrees of freedom among users in each cluster while using the same power coefficients as the MIMO-NOMA comparison. Its two-user sum-rate bound extends to an upper bound for clusters with L users.
- Power allocation: The MIMO-OMA comparison uses the same power coefficients allocated to the L users per cluster as in MIMO-NOMA.This provides the common power split used for the capacity comparison.
- Orthogonal resource allocation: MIMO-OMA assigns each user a fraction λm,l of the time or frequency degrees of freedom, with the fractions summing across the L users.The users are orthogonalized in time or frequency rather than sharing the same resources.
- Two-user bound: The sum rate for two users in a cluster is bounded by Lemma 2, with equality under the stated allocation condition.The lemma supplies the basis for extending the bound to clusters containing L users.
- L-user sum-rate bound: Theorem 1 upper-bounds the sum rate in the mth cluster for L users and specifies when equality holds.Once power coefficients are fixed, the degrees of freedom can be allocated optimally to attain the maximum cluster sum rate.
III. CAPACITY COMPARISON BETWEEN MIMO-NOMA AND MIMO-OMA
The paper compares instantaneous sum rates and sum channel capacities of MIMO-NOMA and MIMO-OMA for clusters with multiple users. Analytically, MIMO-NOMA strictly outperforms MIMO-OMA under any instantaneous channel gains and power split.
- Bounds: The lower bound of the MIMO-NOMA sum rate is established for the multiple-user setting.The proof combines the MIMO-OMA upper bound with the MIMO-NOMA lower bound.
- Sum-rate comparison: MIMO-NOMA achieves a larger sum rate than MIMO-OMA for any power split by assigning MIMO-OMA’s split unchanged to MIMO-NOMA.When the MIMO-OMA split is optimal, MIMO-NOMA also achieves a larger sum channel capacity.
- Sum channel capacity: MIMO-NOMA also achieves a larger sum channel capacity than MIMO-OMA when both schemes use the same power split.The comparison follows from the MIMO-OMA upper bound and the corresponding MIMO-NOMA rate expression.
- Multiple-user result: MIMO-NOMA strictly outperforms MIMO-OMA in sum rate with multiple users per cluster, regardless of instantaneous channel gains and power split.The result is stated without a constraint on the power-split values.
- Two-user gap: For two users, the MIMO-NOMA–MIMO-OMA sum-rate gap is maximized by a specified power-allocation coefficient.The maximizing coefficient is determined only by the first user’s channel, and the resulting maximum gap is obtained by substitution.
B. Ergodic Sum Capacity
The ergodic analysis extends the instantaneous capacity comparison to fading channels. MIMO-NOMA achieves a larger ergodic sum rate and ergodic sum capacity than MIMO-OMA, including under the MIMO-OMA-optimal power split.
- Ergodic sum rate: For any MIMO-OMA power split, MIMO-NOMA achieves a larger ergodic sum rate using the same split.The conclusion follows by taking expectations of the instantaneous sum-rate inequality.
- Ergodic capacity: When the power split is optimal for MIMO-OMA, MIMO-NOMA achieves a larger ergodic sum capacity.This is the ergodic counterpart of the instantaneous sum channel capacity result.
- Channel-distribution scope: The ergodic-capacity conclusions hold regardless of the distribution of Hm,l.The distribution-independence statement accompanies the expectation-based comparison.
- Overall comparison: MIMO-NOMA strictly outperforms MIMO-OMA in both sum channel capacity and ergodic sum capacity with multiple users per cluster.The paper summarizes both capacity conclusions as holding in the multiple-user setting.
IV. USER ADMISSION
The user-admission problem examines how cluster size affects capacity and receiver complexity. Because more admitted users lower sum rate while complicating SIC, the section motivates balancing capacity against admission count.
- MIMO-NOMA analytically outperforms MIMO-OMA in sum rate and ergodic sum rate, even with multiple users per cluster.
- Increasing cluster membership complicates successive interference cancellation at the receiver, limiting the practical number of users per cluster.
- The section therefore studies how sum rate changes with the number of admitted users and motivates a tradeoff between sum rate and admission count.
A. Sum Rate versus Number of Users
This section compares MIMO-NOMA sum rates when a cluster contains l versus l + 1 users. The analysis concludes that admitting more users lowers the sum rate, motivating explicit admission control.
- The sum-rate difference Λ = S(l+1) − S(l) is analyzed using ordered effective channels and the power allocations for l and l + 1 users.
- The derivation establishes Λ ≤ 0 by separately bounding the terms contributing to the sum-rate difference.
- Consequently, admitting more users produces a lower sum rate, creating a tradeoff between rate and the number of admitted users.
B. Proposed User Admission Scheme
The proposed scheme admits users sequentially, allocates power to satisfy their SINR thresholds, and stops when the remaining power cannot support the next user. Its optimality depends on threshold conditions.
- Users are processed sequentially in channel order, with each power coefficient computed from the preceding allocation and the user’s SINR requirement.
- Admission stops when the cumulative allocation leaves insufficient power to satisfy the next user’s SINR threshold; later users receive zero power.
- The proposed scheme maximizes admitted users when each admitted-user threshold is no greater than each nonadmitted-user threshold.
- With equal SINR thresholds, the scheme is optimal for both admitted-user count and sum rate.
- With different thresholds, it balances the two objectives; it maximizes admission under the stated conditions, but may sacrifice sum-rate optimality.
- The scheme has computational complexity O(L), linear in the number of users per cluster.
V. NUMERICAL RESULTS
Simulations compare MIMO-NOMA and MIMO-OMA across power allocation, transmit power, fairness, and admission settings. They confirm higher NOMA rates and fairness, declining rate with larger clusters, and effective admission control.
- Sum rate: 2.04 bps/Hz is the maximum observed gap between MIMO-NOMA and MIMO-OMA in the three-user power-split comparison.
- Fairness: MIMO-NOMA has better Jain’s fairness index than MIMO-OMA for both two- and three-user clusters.
- Transmit power: MIMO-NOMA provides higher sum rate and ergodic sum rate than MIMO-OMA, while the two-user case exceeds the three-user case.
- User admission: The proposed admission algorithm matches exhaustive search in both sum rate and admitted-user count across the tested target SINRs.
VI. CONCLUSION
The paper analytically compares MIMO-NOMA and MIMO-OMA with multiple users per cluster, then develops user admission results and validates them through simulations.
- MIMO-NOMA outperforms MIMO-OMA in both sum channel capacity and ergodic sum capacity.
- The optimal power coefficient maximizing the sum-rate gap is derived for clusters containing two users.
- MIMO-NOMA dominates MIMO-OMA in user fairness for clusters with two or three users.
- Admitting more users to one cluster lowers the achieved sum rate, creating a tradeoff with the number of admitted users.
- The proposed admission scheme is optimal for both sum rate and admitted-user count when users have equal SINR thresholds.
- With unequal SINR thresholds, the scheme balances both criteria, has linear complexity in users per cluster, and is validated by simulations.
APPENDIX I PROOF OF LEMMA 1
The appendix proves Lemma 1 using the SIC feasibility constraint and mathematical induction to establish the relevant sum-rate relationship.
- The receiver constraint ensures user (m, l) can remove interference from users with worse channel gains.
- Ordering users by effective channel gains guarantees the SIC condition at receiver (m, l).
- The equality condition is identified after the theorem's inequality is established.
- The proof applies mathematical induction from the first user through the cases of L1, L1 + 1, and L users.
- The induction establishes a lower bound for the MIMO-NOMA sum rate as all L users are included.
APPENDIX IV PROOF OF THEOREM 3
Theorem 3 is proved by contradiction: the proposed admission scheme requires no more power for the same users and cannot be surpassed in admitted-user count.
- The proof compares the proposed scheme with an alternate scheme admitting the same l users in descending channel-gain order.
- Therefore, no other scheme can admit more users than the proposed scheme.
- For the same number of admitted users, the proposed scheme requires minimum total power.
- An alternate scheme admitting an extra user would imply that the proposed scheme can also admit that user.
- This implication contradicts the assumption that only l users can be admitted by the proposed scheme.