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Random Beamforming in Millimeter-Wave NOMA Networks

Z. Ding, P. Fan, H. V. Poor

arXiv:1607.06302v3cs.IT

TL;DR

The paper addresses how to combine NOMA with mmWave communications without requiring complete user CSI at the base station. It develops stochastic-geometry analyses and low-feedback random-beamforming schemes, finding significant gains over mmWave-OMA and accurate analytical results. The schemes include distance-based ordering, one-bit feedback, and aligned-user scheduling for multiple beams.

  • Problem

    NOMA–mmWave coexistence requires transmission methods that limit the system overhead associated with acquiring many users’ CSI.

  • Method

    The paper applies random beamforming and stochastic geometry to mmWave-NOMA, proposing distance-based ordering, one-bit feedback, and aligned-user scheduling for multiple beams.

  • Results

    1 BPCU gain of mmWave-NOMA over mmWave-OMA occurs at Rj = 4 BPCU and 30 dBm transmission power; analytical results are accurate against simulations.

  • Takeaways & Limitations

    The proposed mmWave-NOMA transmission schemes yield significant performance gains over conventional mmWave-OMA while reducing feedback requirements.

Abstract

from arXiv · show

This paper investigates the coexistence between two key enabling technologies for the fifth generation (5G) mobile networks, non-orthogonal multiple access (NOMA) and millimeter-wave (mmWave) communications. Particularly, the application of random beamforming to the addressed mmWave-NOMA scenario is considered in this paper, in order to avoid the requirement that the base station knows all the users' channel state information. Stochastic geometry is used to characterize the performance of the proposed mmWave-NOMA transmission scheme, by using the key features of mmWave systems, e.g., mmWave transmission is highly directional and potential blockages will thin the user distribution. Two random beamforming approaches which can further reduce the system overhead are also proposed to the addressed mmWave-NOMA communication scenario, where their performance is studied by developing analytical results about sum rates and outage probabilities. Simulation results are also provided to demonstrate the performance of the proposed mmWave-NOMA transmission schemes and verify the accuracy of the developed analytical results.

I. INTRODUCTION

The paper studies NOMA coexistence with directional mmWave communications, motivated by correlated channels, dense connectivity, and continuing spectral-efficiency demands. It proposes random-beamforming approaches that reduce CSI feedback and analyzes their performance under mmWave propagation features.

  • NOMA is paired with mmWave communications because directional transmission can create correlated user channels, while dense connectivity and bandwidth-intensive services sustain spectral-efficiency needs.
  • The baseline random-beamforming scheme avoids requiring the base station to know every user’s channel vector, but conventional feedback of all users’ channel gains can remain costly.
  • Directional mmWave transmission schedules users within wedge-shaped sectors aligned with randomly generated beams, reducing the number of users that must report channel-quality information.
  • A low-feedback scheme uses only users’ distances, ordering users by path loss rather than effective channel gains in fast time-varying channels.
  • A one-bit feedback scheme broadcasts a threshold, with analytical results showing that suitable threshold design can preserve full diversity for the selected strong NOMA user.
  • For multiple orthonormal beams, aligned-user scheduling suppresses inter-beam interference, while stochastic geometry models random users and blockage-thinned mmWave propagation.

III. RANDOM BEAMFORMING: A SINGLE-BEAM CASE

The single-beam case uses analog random beamforming and schedules users whose channels align with the beam, reducing feedback while supporting two-user NOMA analysis. The section derives outage and sum-rate expressions relative to OMA under the modeled user distribution.

  • Random beamforming is introduced to reduce the overhead required by CSI-dependent NOMA precoding and beamforming schemes.
  • The analog beamformer changes signal phase while keeping its modulus constant, with the beam direction uniformly distributed between −1 and 1.
  • A. The Application of Random Beamforming to NOMA: The Fejér-kernel representation shows that effective channel gain is large when a user’s channel vector aligns with the randomly generated beam.
  • A. The Application of Random Beamforming to NOMA: Only users in a wedge-shaped sector of central angle 2∆ are scheduled, with ∆→0 used to ensure large effective channel gain.
  • B. The Implementation of NOMA: Users in the sector are ordered by effective channel gains, and two users are paired for NOMA transmission with allocated powers and successive interference cancellation.
  • B. The Implementation of NOMA: The analysis derives outage probabilities and outage sum rates for mmWave-NOMA, while using OMA when only one sector user is available.

C. Characterization of the Sum Rate and Outage Probabilities

The section derives analytical outage probabilities and sum rates for random-beamformed mmWave-NOMA, accounting for blocked-user point-process thinning and ordered channel gains.

  • The treatment notes that K = 1 has very small probability, so alternative designs for this case have little effect on overall sum rate.
  • The analysis first obtains the distribution of unordered channel gains and then uses order statistics to characterize ordered gains.
  • Blockages thin the original homogeneous Poisson point process, producing a new spatial intensity for line-of-sight users.
  • The probability of having K users in the beam sector is derived after accounting for the blockage-induced point-process thinning.
  • Corollary 1 gives conditional outage probabilities for the strong and weak NOMA users.
  • Substituting the conditional outage expressions into the rate formulas yields the mmWave-NOMA and mmWave-OMA sum rates.

D. Asymptotic Performance Analysis

The asymptotic analysis simplifies the outage expressions under a narrow beam-sector assumption and high SNR, exposing the resulting diversity gains.

  • The exact expressions involve double integrals, so the analysis assumes ∆→0 and high SNR to obtain tractable approximations.
  • Lemma 2 approximates the conditional outage probabilities when ∆→0 and at high SNR.
  • The diversity gain available at user k is k for k ∈ {i, j}.
  • The narrow-sector approximation relies on simplifying the Fejér kernel when the angular mismatch is small.
  • The analysis also notes that a small ∆ can leave the sector empty, although dense deployments may still contain multiple users there.

IV. RANDOM BEAMFORMING WITH LIMITED FEEDBACK

With limited feedback, the base station orders users using slowly varying distance information rather than rapidly changing effective channel gains, and the resulting outage and sum-rate expressions are derived.

  • A. With the Distance Information Available at the Base Station: The distance-only scheme uses partial CSI because user distances vary more slowly than channel phases and fading coefficients.
  • A. With the Distance Information Available at the Base Station: When only distance information is available, users are ordered by distance because shorter distance indicates a stronger channel condition.
  • A. With the Distance Information Available at the Base Station: The distributions of ordered distances in the blockage-thinned sector are derived using sector occupancy probabilities and the corresponding CDF and pdf.
  • A. With the Distance Information Available at the Base Station: The distance-only outage expression is numerically evaluable, and narrow-sector and high-SNR approximations are developed for further insight.
  • A. With the Distance Information Available at the Base Station: Lemma 3 gives outage probabilities for the k-th nearest user and an associated outage sum-rate expression under distance-only information.
  • A. With the Distance Information Available at the Base Station: The defined sum rate omits transmission when the i-th nearest user is absent and uses NOMA when only the j-th nearest user is present.
  • A. With the Distance Information Available at the Base Station: A distance-conditioned CDF is not used because conditioning on K converts the Poisson point process into a Bernoulli process, invalidating the cited result.
  • A. With the Distance Information Available at the Base Station: Distance-only ordering yields diversity gain one for all users because fading dynamics are unavailable to the base station.

B. With One-Bit Feedback

The one-bit feedback protocol partitions users by a broadcast channel-gain threshold, then randomly selects NOMA pairs from the resulting groups. Analytical outage expressions account for empty groups and quantify the threshold’s performance impact.

  • Protocol: Each user compares its effective channel gain with broadcast threshold ξ and feeds back one bit, forming groups S1 and S2.Users with gains above ξ send 1; the others send 0.
  • Protocol: The base station randomly pairs users from S1 and S2 when both groups are nonempty, and otherwise selects two users from one group.With one user it uses sole-user service; if both groups are empty, no user is served.
  • Outage analysis: The strong- and weak-user outage probabilities are derived for K≥2 users under the one-bit feedback protocol.The derivation uses conditional channel-gain distributions for users randomly selected from S1 and S2.
  • Outage analysis: The strong-user outage expression combines the CDF for a randomly selected S2 user with the fallback case in which S2 is empty.When S2 is empty, a user from S1 is selected as the strong NOMA user.
  • Outage analysis: The outage sum-rate expression follows directly from the derived outage probabilities, but is omitted for space limitations.The threshold ξ is then studied through its effect on the conditional channel-gain CDFs.

1) The impact of the threshold on FSk|K(˜ηk):

The threshold ξ controls the asymptotic channel distributions and outage behavior. The weak user retains diversity order one for the considered threshold choices, while a suitable SNR-dependent threshold can provide full diversity for the strong user.

  • Threshold impact: The weak user’s diversity gain is always one for the discussed threshold choices, whereas the strong user’s gain can increase with K.Simulation results confirm that the strong-user outage slope increases with K while the weak-user slope remains unchanged.
  • Threshold impact: When ξ is constant, the weak-user outage probability simplifies asymptotically and has diversity order one.The constant threshold is not a function of transmit SNR ρ.
  • Threshold impact: For ξ scaling as ρ^-x with x>0, the weak user’s diversity order remains one.The result holds for the considered positive exponent x.
  • Threshold impact: For constant ξ, the strong user’s diversity order is one because the high-SNR CDF for S2 becomes zero and the fallback probability remains constant.The fallback uses the weak-group distribution when S2 is empty.
  • Threshold impact: The threshold ξ has a significant impact on the achievable diversity gain, and a suitable choice can yield full diversity gain K.The paper explicitly identifies a threshold choice achieving diversity K.

A. System Model and Outage Performance

The multi-beam system uses predefined orthonormal random beams and schedules two users per beam for NOMA. Inter-beam interference makes users’ SINRs depend on multiple beam projections and decoding steps.

  • Beamforming: The base station forms N orthonormal beams, with predefined beamforming vectors known to both the base station and users.The construction supports a hybrid-precoding interpretation with N radio-frequency chains.
  • User scheduling: Users in each beam’s wedge-shaped sector measure their effective channel gains, and two users are scheduled as weak and strong NOMA users.The effective gain on beam m is |h^H_m,k p_m|^2.
  • NOMA transmission: Two users’ messages are superimposed on every beam, and the strong user applies SIC before decoding its own message.SIC succeeds only when the partner’s message meets its target SINR threshold.
  • Outage formulation: With multiple beams, each user’s SINR depends on its intended-beam projection and projections onto other beams.This differs from single-beam scheduling based only on one SINR function.
  • Outage formulation: The strong-user outage probability is the complement of jointly decoding the partner’s message and the strong user’s own message.The corresponding outage expression is developed using the mmWave channel model.

B. Asymptotic Performance Analysis

The asymptotic analysis approximates inter-beam interference under sufficiently separated beams and narrow sectors. These approximations are then used to obtain high-SNR outage expressions and related sum-rate results.

  • Asymptotic approximation: For the first beam, the cross-beam factor is expanded using a Taylor series around the beam’s angular configuration.The analysis assumes the required derivatives of the channel-distribution function exist.
  • Asymptotic approximation: Sufficient beam separation and Δ→0 make cross-beam channel terms approach zero for beams n≥2.These assumptions simplify the interference contributions in the SINR expressions.
  • Asymptotic approximation: The sum of interference terms is approximated using the preceding channel-model relation, enabling high-SNR outage analysis.The resulting approximation is applied first to user i’s outage probability.
  • Asymptotic results: The outage probability for user j is obtained similarly, followed by outage sum-rate and user-outage results using steps analogous to the single-beam analysis.The derivation includes the strong user’s partner-message decoding condition.

VI. NUMERICAL STUDIES

The numerical studies evaluate random-beamforming mmWave-NOMA against mmWave-OMA under perfect CSI, distance-only information, one-bit feedback, and multiple beams. Across these settings, NOMA improves sum rate and outage performance, while threshold selection and directional propagation shape diversity and interference.

  • Perfect CSI: At Rj = 4 BPCU, mmWave-NOMA gains 1 BPCU over mmWave-OMA at 30 dBm, increasing to 5 BPCU when Rj = 6 BPCU.
  • Distance information only: With distance information only, mmWave-NOMA significantly improves sum rate and reduces outage probability relative to OMA.The simulations also demonstrate the accuracy of the analytical results.
  • One-bit feedback: A properly designed one-bit-feedback threshold can preserve full diversity gain despite ordering ambiguity at the base station.Each user reports one bit indicating channel quality, reducing feedback while introducing ambiguity.
  • Threshold impact: The strong user achieves diversity gain K, whereas the weak user’s diversity gain remains one for the discussed choices of ξ.Increasing K increases the slope of the strong user’s outage curve, but not the weak user’s.
  • Multiple beams: With multiple beams, directional propagation suppresses inter-beam interference when ∆ = 0.01, whose central angle is about 4 degrees.The strong NOMA user can remove its partner’s message using SIC but still experiences interference from other beams.
  • Limited versus perfect CSI: Limited-feedback schemes can outperform perfect-CSI scheduling because ordering ambiguity may select users with better channel conditions and hence a larger sum rate.For the stated four-user example, this improvement occurs when transmission power is 20.

VII. CONCLUSIONS

The paper applies stochastic geometry to random-beamforming mmWave-NOMA and proposes two approaches that reduce feedback overhead. It evaluates single- and multiple-beam scenarios using mmWave directionality and blockage-induced user thinning.

  • The study characterizes mmWave-NOMA performance with stochastic geometry, incorporating directional transmission and blockage-induced thinning of the user distribution.
  • Two beamforming approaches are proposed to reduce feedback overhead in mmWave-NOMA networks.
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