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Randomly-Directional Beamforming in Millimeter-Wave Multi-User MISO Downlink

Gilwon Lee, Youngchul Sung, Junyeong Seo

arXiv:1412.1665v2cs.IT

TL;DR

The paper asks how randomly-directional beamforming can provide useful MU-MISO performance in highly directional mm-wave channels, where channel estimation is difficult. It analyzes RDB asymptotically under the UR-LoS model and finds a user-to-antenna transition point, with proper multi-beam scheduling enabling sum-rate scaling arbitrarily close to linear in antenna count.

  • Problem

    The paper addresses limited understanding of MU gain and the required user population for effective RDB in highly directional mm-wave MU-MISO systems.

  • Method

    The paper uses asymptotic analysis of RDB under the UR-LoS channel model, including multi-beam transmission and user scheduling.

  • Results

    The analysis identifies a transition point in users relative to antenna elements and shows that proper scheduling can yield sum-rate scaling arbitrarily close to linear in antenna count.

  • Takeaways & Limitations

    Effective mm-wave MU-MISO random beamforming choices depend on antenna-array size and user count.

Abstract

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In this paper, randomly-directional beamforming (RDB) is considered for millimeter-wave (mmwave) multi-user (MU) multiple-input single-output (MISO) downlink systems. By using asymptotic techniques, the performance of RDB and the MU gain in mm-wave MISO are analyzed based on the uniform random line-of-sight (UR-LoS) channel model suitable for highly directional mm-wave radio propagation channels. It is shown that there exists a transition point on the number of users relative to the number of antenna elements for non-trivial performance of the RDB scheme, and furthermore sum rate scaling arbitrarily close to linear scaling with respect to the number of antenna elements can be achieved under the UR-LoS channel model by opportunistic random beamforming with proper user scheduling if the number of users increases linearly with respect to the number of antenna elements. The provided results yield insights into the most effective beamforming and scheduling choices for mm-wave MU-MISO in various operating conditions. Simulation results validate our analysis based on asymptotic techniques for finite cases.

I. INTRODUCTION

The paper studies randomly-directional beamforming for highly directional mm-wave MU-MISO systems under the UR-LoS channel model, asking how many users are needed for useful performance. Asymptotic analysis identifies operating transitions and shows that proper beam and user scheduling can approach linear sum-rate scaling with antenna count.

  • I. INTRODUCTION: The study addresses limited prior MU analysis for mm-wave systems and the need to determine how many users make simple RDB effective.Existing channel-estimation work focused mainly on single-user mm-wave MIMO, while prior MU-gain work largely concerned rich-scattering models.
  • I. INTRODUCTION: The UR-LoS model captures highly directional mm-wave propagation with a single line-of-sight path having random direction and gain per user.It is adopted because it is analytically tractable while representing highly directional propagation.
  • I. INTRODUCTION: The paper rigorously analyzes RDB, MU gain, and user scheduling under the UR-LoS model with a large ULA at the base station.The analysis uses an asymptotic regime in which the number of antenna elements tends to infinity.
  • I. INTRODUCTION: For single-beam RDB, 2^q−1 of the perfect-CSI exact-beamforming rate is achieved asymptotically when K = c_uM^q, with M marking a transition point.The scheme transmits one random beam and selects the user with maximum received signal power.
  • I. INTRODUCTION: With S = c_bM^ℓ equi-spaced beams and best-beam, best-user selection, multi-beam single-user RDB achieves 2^(q+ℓ)−1 of the perfect-CSI rate.This result holds asymptotically for K = c_uM^q and S = c_bM^ℓ.
  • I. INTRODUCTION: Multi-beam RDB with proper user scheduling can achieve sum-rate scaling arbitrarily close to linear in antenna elements when users increase linearly with antenna count.The paper uses these results to suggest beamforming and scheduling choices based on antenna-array size and user count through the fractional rate order.

II. SYSTEM MODEL AND PRELIMINARIES

The system models a single-cell mm-wave MU-MISO downlink with a ULA and uses a UR-LoS channel containing a dominant line-of-sight path with uniformly distributed user directions.

  • The downlink has an M-antenna ULA base station communicating with K single-antenna users under a transmit-power constraint and additive Gaussian noise.
  • A. Channel Model: The channel comprises a line-of-sight component and few single-bounce multipath components represented through complex gains, normalized directions, and an array steering vector.
  • A. Channel Model: The critically sampled array uses antenna spacing tied to wavelength, and the steering vector is unit norm with normalization by M included in the channel model.
  • A. Channel Model: Because NLoS paths are typically 20 dB weaker than the LoS component, the model neglects NLoS components and retains only the LoS path.
  • A. Channel Model: The UR-LoS model assumes independent uniformly distributed normalized directions and has expected channel power E{||h_k||^2}=M.

B. Review of Opportunistic Random Beamforming in Rich Scattering Environments

The paper contrasts conventional random beamforming for rich scattering with randomly-directional beamforming for highly directional mm-wave channels, where beam randomness lies in direction.

  • Conventional RBF constructs random orthonormal beams, trains users on each beam, collects SINR feedback, and assigns each beam to its highest-SINR user.
  • In rich-scattering MIMO, RBF sum rate scales linearly with antenna count when M grows no faster than log K, but not when M grows faster.
  • Mm-wave systems require large arrays and highly directional beamforming to compensate for large path loss.
  • Under UR-LoS propagation, the analysis progresses from one random beam and one selected user to multiple asymptotically orthogonal beams and multiple users.
  • RDB randomizes beam direction rather than beam vectors, and with one beam its rate is insignificant for a single user but can become asymptotically good with sufficiently many users.
  • The Fejér-kernel response vanishes for fixed beam and user directions as M grows, but remains nonzero when their directional separation is sufficiently small.

IV. ASYMPTOTIC ANALYSIS OF THE RDB RATE: THE SINGLE BEAM CASE

The single-beam analysis identifies how user growth relative to antenna growth determines whether RDB has nontrivial asymptotic performance.

  • The proof bounds the single-beam rate through the distribution of Z_k=M|a(θ_k)^Ha(ϑ)|^2 and its tail probabilities.
  • For K=M^q with q∈(0,1), the single-beam RDB rate is bounded by log(1+M^(2q−1−ε)) and log(1+M^(2q−1+ε)).
  • The performance transition occurs at q=1/2: RDB is nontrivial for q∈(1/2,1) and trivial for q∈(0,1/2).
  • The RDB-to-exact-beamforming rate ratio is 2q−1 under both fixed and Gaussian LoS gains.
  • This ratio approaches one arbitrarily closely when the user count grows almost linearly with M, so RDB can nearly match perfect-CSI beamforming.

V. ASYMPTOTIC ANALYSIS OF THE RDB RATE: THE MULTIPLE BEAM CASE

The multiple-beam analysis studies equi-spaced randomly directional beams, asymptotic orthogonality, and user selection across multiple training beams.

  • The scheme uses S randomly directional beams that are equi-spaced in normalized angle and transmitted sequentially during training.
  • Users identify the corresponding training interval, and the beam-direction offset is randomly generated over the normalized angle domain.
  • The equi-spaced beams become asymptotically orthogonal, and the analysis considers the regime S=o(M).
  • The analysis proceeds from single-user selection based on multiple training beams to multiple-user selection based on multiple beams.

A. The Single User Selection Case

With multiple training beams and single-user selection, RDB performance depends on the balance between users and beams. Increasing training beams can approach the optimal rate when users are scarce, but requires more training time.

  • User selection: The single-user method selects the user reporting the maximum received power among the training beams after training.The received-power maximum is evaluated over the corresponding beam reports.
  • Asymptotic rate bounds: Theorem 3 gives asymptotic bounds for the single-user-selected rate when K = M^q and S = M^ℓ with ℓ + q < 1.The bounds scale through the exponent 2q + 2ℓ − 1, subject to the stated conditions.
  • Training-beam trade-off: The achievable fraction can approach one by jointly increasing the number of users and training beams.When users are scarce, multiple training beams enhance RDB performance; for q = 0, the optimal rate is approached as ℓ increases toward one.
  • Training overhead: Multiple training beams require more training time even though their two rate contributions are not distinguishable during data transmission.The comparison is stated for the rate during the data-transmission period.

B. The Multiple User Selection Case: Multiplexing Gain

Multiple-user selection assigns users to multiple random beams and analyzes the resulting per-user and sum-rate scaling. Under the UR-LoS model, proper scheduling can provide multiplexing gains approaching linear antenna scaling.

  • Scheduling method: The proposed scheduling method is essentially random beamforming with randomly directional beams and proper user selection.The scheduled users transmit independent data streams on the selected beams.
  • Beam matching: The scheduled users are selected for asymptotically orthonormal random beams, with each beam associated with a segment of normalized angle space.When q > ℓ and 2q − 1 − ℓ > 0, many users are available in each segment for beam matching.
  • Asymptotic bounds: For K = M^q and S = M^ℓ, Theorems 4 and 5 bound the selected users’ per-user rates under fixed total or fixed per-user power.Theorems 4 and 5 analyze fixed total transmit power and fixed per-user power, respectively.
  • Multiplexing gain: With multiple random beams and user scheduling, the sum rate can scale arbitrarily close to linearly with the number of antennas.The construction uses ℓ close to one and suitable q, yielding M^(1−δ1) log(M^(−δ1+2δ2+ε)).
  • Channel-model comparison: The UR-LoS model permits linear sum-rate scaling with users increasing linearly in antennas, unlike the Rayleigh model’s exponential-user requirement.The paper attributes the difference to the channels’ degrees of freedom and high propagation directivity.

C. Performance comparison: The fractional rate order

The fractional rate order compares how the three RDB strategies scale with antenna count as the user count grows as M^q. The preferred strategy changes at q = 1/2.

  • Strategy transition: The multi-beam multiple-user strategy has the largest FRO for q ∈ (1/2, 1), while the multi-beam single-user strategy has the largest FRO for q ∈ (0, 1/2).The single-user strategy is best when users are scarce, whereas scheduled multi-user beams are best when users are sufficient.
  • Transition point: The transition point reflects user scarcity under UR-LoS, differing from the Rayleigh model’s exponential-user transition.Under Rayleigh fading, the transition is K = Θ(exp(η2M)), equivalently M = Θ(log K).
  • Few-user regime: For q ∈ (0, 1/2), multi-beam single-user selection is best and channel estimation of the user’s propagation angle is important.This regime corresponds to relatively few users in the cell.
  • Many-user regime: For q ∈ (1/2, 1), equi-spaced random beams with an arbitrary angle offset and user scheduling are sufficient for good performance and linear sum-rate scaling.Downlink channel estimation is less important in this sufficient-user regime.

VI. NUMERICAL RESULTS

The numerical-results section validates the preceding asymptotic analysis using finite-system simulations. Results are averaged over channel realizations with unit transmit power.

  • Validation: The numerical results validate the asymptotic analysis in finite cases.The section introduces numerical results specifically for this validation.
  • Simulation setup: The reported expectations are averaged over 5000 channel realizations.This averaging is used for the numerical evaluations.
  • Simulation setup: The simulations set the transmit power to Pt = 1.The unit-power setting is used throughout the reported numerical results.

A. The Single Beam Case

The single-beam RDB rate exhibits a transition with user scaling: below the threshold it becomes non-growing or trivial, while above it increases with antenna count. Finite simulations broadly follow the asymptotic theorems, with slow convergence causing gaps.

  • A. The Single Beam Case: The simulated ratio of RDB rate to perfect-CSI rate gradually approaches the theoretical line 2q − 1 as M increases.Finite-sample gaps remain because convergence is slow.
  • A. The Single Beam Case: For q below 0.5, the RDB rate decreases as M increases, whereas it nearly remains unchanged at q = 0.5.
  • A. The Single Beam Case: For q > 0.5, the RDB rate increases with M, consistent with logarithmic scaling predicted by Theorem 2.The rate curve is linear because the x-axis is logarithmic.

B. The Multiple Beam Case

The multiple-beam results characterize how user scheduling and beam allocation determine rate scaling. Proper multi-beam scheduling can achieve sum-rate scaling arbitrarily close to linear in M when the user population grows linearly with M.

  • B. The Multiple Beam Case: The simulated multiple-beam rate ratios roughly match the theoretical lines across the tested scheduling and channel cases.
  • B. The Multiple Beam Case: For q = 0.3, the multiple-beam single-user-selection rate increases with M when ℓ > 0.2 and decreases when ℓ < 0.2.These regimes correspond to q + ℓ > 0.5 and q + ℓ < 0.5, respectively.
  • B. The Multiple Beam Case: The per-user rate increases with M when ℓ < 0.4 and decreases when ℓ > 0.4, as predicted by Theorem 4.The corresponding condition is 2q − 1 − ℓ > 0 or < 0.
  • B. The Multiple Beam Case: Sum-rate scaling arbitrarily close to linear in M is achievable with equi-spaced multiple beams and proper user scheduling when the number of users increases linearly with M.
  • B. The Multiple Beam Case: The paper compares three RDB beamforming and scheduling schemes using a fractional rate order to identify effective choices across operating conditions.
  • B. The Multiple Beam Case: Simulation results validate the asymptotic analysis for finite cases under the simplified UR-LoS channel model.The model captures high propagation directivity, but extension to a general channel model remains open.

APPENDIX A DISTRIBUTION OF ϑ −θk

The appendix establishes the distributional and asymptotic bounds used in the RDB analysis. It relates random angular differences and cone occupancy to rate bounds and the transition in performance.

  • APPENDIX A DISTRIBUTION OF ϑ −θk: The angular difference variable ˜θ is uniformly distributed on [−1, 1] under the periodicity argument.
  • APPENDIX A DISTRIBUTION OF ϑ −θk: The expected logarithmic term is bounded between expressions involving M^(2q−1−ε) and M^(2q−1+ε), enabling the asymptotic RDB-rate analysis.
  • APPENDIX A DISTRIBUTION OF ϑ −θk: Theorem 2 bounds M R1 between (2q − 1 − ε) log M and (2q − 1 + ε) log M + log log M.
  • APPENDIX A DISTRIBUTION OF ϑ −θk: For 2q − 1 < 0, the logarithmic term is approximated using log(1 + x) → x as x → 0, yielding a vanishing-scale regime.
  • APPENDIX A DISTRIBUTION OF ϑ −θk: A cone around each coordinate axis captures channel vectors with a probability derived from channel concentration and the χ2(2) distribution of each coordinate magnitude.
  • APPENDIX A DISTRIBUTION OF ϑ −θk: The probability that at least one of K channel vectors lies in a cone provides the physical intuition for asymptotically good RDB performance.
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