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A General MIMO Framework for NOMA Downlink and Uplink Transmission Based on Signal Alignment

Zhiguo Ding, Robert Schober, H. Vincent Poor

arXiv:1508.07433v1cs.IT

TL;DR

MIMO-NOMA needs a framework that improves NOMA performance while covering downlink and uplink transmission with randomly deployed users and interferers. The paper uses signal alignment, stochastic geometry, closed-form outage analysis, and multiple power allocation strategies. The proposed framework is more general and offers significant reception-reliability gains, while assuming global CSI that may create substantial training overhead.

  • Problem

    Applying MIMO to NOMA is important for enhancing NOMA performance, but a general framework covering downlink and uplink transmission is needed.

  • Method

    The paper applies signal alignment to MIMO-NOMA and uses stochastic geometry to derive closed-form outage expressions for randomly located users and interferers.

  • Results

    The proposed framework applies to both uplink and downlink transmissions and offers significant gains in reception reliability.

  • Takeaways & Limitations

    Signal alignment provides a general MIMO-NOMA framework for evaluating transmission reliability under random user and interferer locations.

  • Takeaways & Limitations

    The analysis assumes global CSI, which may introduce significant training overhead in practice.

Abstract

from arXiv · show

The application of multiple-input multiple-output (MIMO) techniques to non-orthogonal multiple access (NOMA) systems is important to enhance the performance gains of NOMA. In this paper, a novel MIMO-NOMA framework for downlink and uplink transmission is proposed by applying the concept of signal alignment. By using stochastic geometry, closed-form analytical results are developed to facilitate the performance evaluation of the proposed framework for randomly deployed users and interferers. The impact of different power allocation strategies, such as fixed power allocation and cognitive radio inspired power allocation, on the performance of MIMO-NOMA is also investigated. Computer simulation results are provided to demonstrate the performance of the proposed framework and the accuracy of the developed analytical results.

I. INTRODUCTION

The paper proposes a general MIMO-NOMA framework for downlink and uplink transmission using signal alignment, addressing randomly deployed users and interferers. It develops analytical performance results and studies power allocation, precoding, detection, and diversity gains.

  • Motivation: MIMO provides additional degrees of freedom that can further improve NOMA performance and support serving more users with lower latency and improved fairness.The paper motivates MIMO-NOMA through performance improvement and practical pressure to serve many users.
  • Framework: The proposed framework applies signal alignment to support MIMO-NOMA in both downlink and uplink transmission.The multi-user MIMO-NOMA scenario can be decomposed into separate single-antenna NOMA channels.
  • Power allocation: Two power allocation strategies are studied to address the throughput-fairness tradeoff in NOMA systems.Fixed power allocation targets long-term QoS requirements, while cognitive radio inspired allocation aims to ensure users’ QoS requirements are met.
  • Signal processing: The proposed precoding and detection vector selection exploits excess MIMO degrees of freedom and offers a more general framework than existing MIMO-NOMA work.It is applicable when users have fewer antennas than the base station.
  • Performance: For a scenario where all nodes have M antennas, the proposed scheme achieves diversity order M, compared with diversity gain 1 for the scheme in [13].This comparison is reported as one benefit over the existing scheme.
  • Performance analysis: Exact and asymptotic outage-probability results are developed using stochastic geometry to capture randomly located users and interferers.Outage probability is used as the performance criterion, and diversity order is computed to assess use of channel degrees of freedom.

A. Downlink MIMO-NOMA Transmission

The downlink framework uses signal alignment with precoding and detection design to manage inter-pair interference and transform the MIMO-NOMA channel into parallel NOMA channels.

  • Signal alignment is applied to overcome the nonexistence of nonzero precoding vectors under the original inter-pair interference constraint.Serving fewer user pairs would ensure feasibility but reduce overall system throughput.
  • Detection vectors are designed from the zero-singular-value subspace of an effective channel matrix, with normalized auxiliary vectors.The normalization supports uplink power constraints and tractable performance analysis.
  • The alignment design projects the two users’ channels onto a shared effective channel vector and significantly reduces the inter-pair constraint dimensions.This reduction enables the subsequent zero-forcing precoder construction.
  • The resulting precoding and detection matrices decompose the multi-user MIMO-NOMA downlink into M pairs of single-antenna NOMA channels.Within each pair, the users receive scalar observations and share the same small-scale fading gain with different effective channels.
  • Users in each pair are ordered by distance, allowing power coefficients to follow the NOMA principle with α_m ≤ α_m′.User m′ decodes its message, while user m first decodes user m′’s message using successive interference cancellation.

B. Uplink MIMO-NOMA transmission

The uplink reuses the alignment-based vector design in reciprocal roles, enabling detection of user-pair signals while controlling inter-pair interference.

  • In uplink, users transmit information-bearing messages and the base station applies a detection matrix to its observations.The received model includes co-channel interference and base-station noise.
  • Signal alignment is applied again so the inter-pair interference constraint can be satisfied using the same vector design as in downlink.The downlink detection vectors serve as uplink precoding vectors, while the downlink precoding matrix becomes the uplink detection matrix.
  • The proposed uplink precoding vectors constrain the total transmission power within each user pair.The total transmission power from one user pair is normalized in the uplink model.
  • The resulting matrices decompose the multi-user MIMO-NOMA uplink channel into M orthogonal single-antenna NOMA channels.The successive interference cancellation strategy can then be applied to decode the users’ messages.

III. PERFORMANCE ANALYSIS FOR DOWNLINK MIMO-NOMA TRANSMISSION

The downlink analysis evaluates fixed power allocation using outage-probability expressions and high-SNR approximations under randomly deployed interferers.

  • Two power allocation policies are considered: fixed power allocation and a cognitive-radio-inspired strategy.The fixed-allocation analysis first treats the auxiliary vector choice as random.
  • The correlation between the detection vector and effective channel makes direct evaluation of user m’s outage probability challenging.The analysis therefore focuses on a modified outage-probability expression.
  • For δ ≥ N, the modified probability upper-bounds user m′’s outage probability, while δ = 1 produces a very small difference in Fig. 1.The figure compares the modified and original outage probabilities under the stated simulation parameters.
  • The modified outage probability provides a tight approximation to the outage probability of user m′ and is sufficient to identify the achievable diversity order.For user m′, the derived diversity order is one.
  • At fixed interference power and transmit SNR approaching infinity, the outage probability is approximated by a high-SNR expression involving the interference parameter.The analytical results use closed-form expressions and incomplete Gamma functions.

B. Cognitive Radio Power Allocation

The cognitive-radio-inspired allocation treats user m′ as primary and analyzes whether user m can share its spectrum without degrading the primary user’s outage performance.

  • Under cognitive-radio power allocation, user m′’s outage probability matches that of conventional orthogonal multiple-access systems.The analysis therefore focuses on user m’s outage probability.
  • The cognitive-radio allocation imposes α_m = 0 in the relevant outage condition, so outage for user m′ can occur even when all power is assigned to it.User m′ is treated as the primary user whose target rate must remain satisfied.
  • The outage analysis for user m is restricted to ρ_I = 0 because the effective channel and users’ co-channel interference terms are correlated.The two users experience different but correlated interference, with I_m ≠ I_m′.
  • With ρ_I = 0, the derived high-SNR result shows that user m achieves diversity gain one, with no error floor.This performance is obtained while user m′ experiences the same outage performance as if it used the channel alone.
  • The proposed cognitive-radio NOMA introduces user m to share the spectrum with primary user m′ without performance degradation at the primary user.The result is stated within the assumed decoding strategy, where m′’s message is decoded first at both receivers.

C. Selection of the User Detection Vectors

The paper selects user detection vectors from available null-space degrees of freedom to improve outage performance and diversity in MIMO-NOMA. The resulting analysis covers downlink and uplink transmission, including sum-rate outage behavior.

  • Selection algorithm: The selection algorithm resolves the coupled detection-vector choice across user pairs by exploiting additional null-space degrees of freedom.A pair’s detection vectors affect other pairs through the effective fading matrix and their data rates.
  • Diversity gain: The proposed selection algorithm increases diversity gain from 1 to (2N −M), whereas the scheme in [13] achieves diversity gain 1 for an unordered user.When N = M, the proposed scheme achieves diversity gain M, while the scheme in [13] achieves 1.
  • Uplink transmission: For uplink NOMA, the sum rate is unchanged by the decoding order, enabling sum-rate outage analysis with fixed power allocation.A randomly selected x_m is used to obtain tractable analytical results.
  • Uplink transmission: Successive interference cancellation decodes one user’s message first and then the other, yielding the same NOMA sum rate for either decoding order.The sum-rate outage probability is expressed after considering both decoding orders.
  • Uplink transmission: The uplink sum-rate outage probability has achievable diversity gain 1 when interference is fixed and SNR tends to infinity.The high-SNR approximation uses bounded user distances, causing ζ(d_m,d_m′) to approach zero.

B. Cognitive Radio Power Allocation

The cognitive-radio uplink NOMA design is more complicated because decoding order creates different outage tradeoffs between the two users. Two decoding cases impose different power constraints to protect the required user QoS.

  • Cognitive-radio uplink design: Cognitive-radio uplink NOMA requires separate analysis because the decoding order produces different outage tradeoffs between the two users.The two strategies decode either user m′ or user m first.
  • Case I: Case I imposes a power constraint to guarantee the QoS of user m′ when its message is decoded first.The outage probabilities are derived following the proofs of Lemmas 2 and 3.
  • Case II: Case II imposes a corresponding power constraint to guarantee user m′’s QoS when user m is decoded first.This case is analyzed separately from the decoding strategy in which user m′ is decoded first.

2) Case II:

Case II allocates power while decoding user m first, producing equal outage probabilities for both users and better outage performance for user m than Case I. Numerical studies also validate the analytical framework and show gains from NOMA, precoding, cognitive allocation, and additional user antennas.

  • Case II: Case II chooses power allocation so the two users have identical outage probabilities.The analysis therefore focuses on the outage probability for user m.
  • Case comparison: Case I is preferable when user m′ has a strict QoS requirement, while Case II favors outage performance for user m.The two cases therefore represent different performance tradeoffs between the users.
  • Case II: Case II avoids compensating the large path loss of user m′, allocates more power to user m, and improves user m’s outage performance.Case I instead offers lower outage probability for user m′.
  • Downlink numerical results: The two NOMA schemes achieve larger outage sum rates than the two OMA schemes in the downlink comparison.The comparison uses interference-free transmission and defines outage sum rate as R_m′(1−P_m′)+R_m(1−P_m).
  • Downlink numerical results: Precoding improves outage performance over non-precoded schemes by using base-station degrees of freedom more efficiently.SA-MIMO-NOMA and MIMO-NOMA have similar outage sum-rate performance, while SA-MIMO-NOMA offers better reception reliability at high transmission power.
  • Downlink numerical results: SA-MIMO-NOMA lowers user m’s outage probability relative to precoded MIMO-OMA but increases user m′’s outage probability.The observed degradation for the poorer-channel user is attributed to co-channel interference.
  • Cognitive-radio numerical results: Cognitive-radio NOMA supports user m at a target rate with probability approaching one at high SNR without degrading user m′’s outage performance.OMA cannot admit user m into the channel occupied by user m′.
  • Antenna scaling: Increasing the number of user antennas decreases outage probability and increases outage-curve slope, confirming higher diversity order.More antennas enlarge the null space and provide more detection-vector choices.

VI. CONCLUSIONS

The paper proposes a signal-alignment-based MIMO-NOMA framework for both downlink and uplink transmission, with stochastic-geometry analysis and power-allocation strategies. Compared with existing MIMO-NOMA work, it is more general and offers improved reception reliability.

  • Framework and analysis: The proposed signal-alignment-based MIMO-NOMA framework applies to both downlink and uplink transmission.The framework captures randomly located users and interferers through stochastic-geometry tools.
  • Framework and analysis: Closed-form outage-probability expressions were developed to facilitate performance evaluation.The analysis explicitly captures the random locations of users and interferers.
  • Power allocation: Both fixed and cognitive-radio-inspired opportunistic power-allocation strategies were investigated.The latter strategy is described as more opportunistic than fixed power allocation.
  • Comparison with prior work: Compared with existing MIMO-NOMA work, the framework is more general because it supports both uplink and downlink transmissions.It also applies when users have fewer antennas than the base station.
  • Comparison with prior work: The proposed framework offers a significant performance gain in reception reliability.The conclusion states this comparison at the framework level rather than giving a numerical gain.
  • Limitation: The analysis assumes global CSI, which may introduce significant training overhead in practice.Studying transmission with limited CSI is identified as an important future direction.

APPENDIX A

Appendix A derives outage-probability expressions and high-SNR approximations for users distributed in discs or rings under interference. It uses stationarity, polar-coordinate averaging, and channel-projection properties to obtain bounds and approximations.

  • Outage derivation: Outage probabilities are derived by averaging over user distance and interference distributions.Users are uniformly distributed in the relevant disc or ring, while interference statistics exploit PPP stationarity and Slivnyak’s theorem.
  • Channel projection: Projection-matrix properties preserve a complex Gaussian channel vector after projection by a randomly generated normalized vector.The projection matrix is idempotent, and its eigenvalues are zero or one.
  • Interference modeling: PPP stationarity allows interference at a receiver to be evaluated equivalently at a node located at the origin.The interference contribution is then expressed using distances from interference sources to that origin.
  • High-SNR analysis: At high transmission power, the outage probability is simplified by approximating incomplete Gamma-function terms.The approximation considers transmission power tending to infinity while interference power remains fixed.
  • Bounds and approximation: The approximate outage probability is an upper bound when δ ≥ N.The text states that δ = 1 is sufficient to yield a tight approximation, as shown in Fig. 1.

APPENDIX C

Appendix C decomposes user-m outage into three events involving power allocation, decoding the other user’s message, and decoding its own message. It then evaluates these events using shared fading and user-location distributions.

  • Outage events: Three outage events are considered for user m: zero allocated power, failure to decode user m′, and failure to decode its own message.The events distinguish power allocation from the two decoding outcomes.
  • Outage events: When ᾱ²_m = 0, all power is consumed by user m′ and no power is allocated to user m.This case is designated event E1.
  • Outage events: When ᾱ²_m > 0, user m may fail either to decode user m′’s message or to decode its own message after decoding user m′.These cases are designated E2 and E3, respectively.
  • Channel dependence: The two channel gains h_m and h_m′ share the same small-scale fading.This shared fading is an important observation in the event-probability analysis.
  • Spatial averaging: The event probabilities are calculated using the users’ uniform spatial distributions and polar-coordinate integration.User m is uniformly distributed in disc D1, and its distance is determined by its location.

APPENDIX D

Appendix D derives upper bounds and high-SNR approximations for outage probability under the proposed detection algorithm. The derivation uses independent channel-related quantities and distinguishes optimized from random detection vectors.

  • Outage bounds: The outage probability for user m′ is bounded using the minimum detected signal quantity.The bound compares γ_m,i* with γ_min,i* under the detection vector selected by the algorithm.
  • Outage bounds: The upper-bound derivation uses independence between γ_min,i and γ_min,j.This independence follows from independence between the corresponding channel vectors g_m,i and g_m,j.
  • High-SNR analysis: At high transmission power with fixed interference power, the outage-probability upper bound is approximated asymptotically.The derivation then evaluates the bound using polar coordinates and algebraic sum rearrangement.
  • Detection-vector comparison: The corresponding result for user m is obtained through steps similar to those used for user m′.The appendix explicitly reuses the preceding derivation structure.
  • Detection-vector comparison: For a random detection vector, replacing (2N − M) with 1 yields an upper bound matching Lemma 1 except for an extra factor M.The extra factor is introduced by upper bounding the outage probability.
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