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Design of 5G Full Dimension Massive MIMO Systems

Qurrat-Ul-Ain Nadeem, Abla Kammoun, Mérouane Debbah, and Mohamed-Slim Alouini

arXiv:1612.03993v2cs.ITmath.IT

TL;DR

Massive MIMO deployment is constrained by base-station form factor, while compact FD-MIMO arrays also make spatial correlation important to model. The paper develops a 2D active-array and channel-correlation framework and designs elevation beamforming methods using users’ quasi-static channel correlations. Simulations report gains over existing elevation beamforming methods, with user-group-specific beamforming improving performance further.

  • Problem

    Base-station form factors limit large linear arrays, and compact FD-MIMO arrays increase spatial correlation that elevation-beamforming design must account for.

    Source: §I · p. 2

  • Method

    The paper derives element and port spatial correlation functions and uses users’ quasi-static channel correlation matrices to design elevation beamforming weights.

    Source: §I · p. 2

  • Results

    Simulations show significant gains for single downtilt beamforming over existing elevation beamforming methods, with user-group-specific beamforming improving performance further.

    Source: §I · p. 3

  • Takeaways & Limitations

    The 2D active antenna system provides adaptive electronic beam control in elevation and azimuth and has potential to boost spectral efficiency.

    Source: §VII · p. 13

Abstract

from arXiv · show

Massive multiple-input-multiple-output (MIMO) transmission is a promising technology to improve the capacity and reliability of wireless systems. However, the number of antennas that can be equipped at a base station (BS) is limited by the BS form factor, posing a challenge to the deployment of massive linear arrays. To cope with this limitation, this work discusses Full Dimension MIMO (FD-MIMO), which is currently an active area of research and standardization in the 3rd Generation Partnership Project (3GPP) for evolution towards fifth generation (5G) cellular systems. FD-MIMO utilizes an active antenna system (AAS) with a 2D planar array structure, which provides the ability of adaptive electronic beamforming in the 3D space. This paper presents the design of the AAS and the ongoing efforts in the 3GPP to develop the corresponding 3D channel model. Compact structure of large-scale antenna arrays drastically increases the spatial correlation in FD-MIMO systems. In order to account for its effects, the generalized spatial correlation functions for channels constituted by individual antenna elements and overall antenna ports in the AAS are derived. Exploiting the quasi-static channel covariance matrices of the users, the problem of determining the optimal downtilt weight vector for antenna ports, which maximizes the minimum signal-to-interference ratio of a multi-user multiple-input-single-output system, is formulated as a fractional optimization problem. A quasi-optimal solution is obtained through the application of semi-definite relaxation and Dinkelbach's method. Finally, the user-group specific elevation beamforming scenario is devised, which offers significant performance gains as confirmed through simulations. These results have direct application in the analysis of 5G FD-MIMO systems.

I. INTRODUCTION

FD-MIMO uses a 2D active antenna array to provide adaptive beam control across elevation and azimuth. This section introduces its array architecture and channel modeling, then outlines spatial-correlation-aware elevation beamforming methods.

I. INTRODUCTION

  • FD-MIMO’s 2D planar array supports adaptive beam control in elevation and azimuth, while antenna ports group physical elements and enable vertical-pattern adjustment.The paper presents this architecture as a practical response to base-station form-factor limits on large linear arrays.Source: §I · pp. 1–2
  • The paper derives spatial correlation functions for individual antenna elements in the 2D array and for antenna ports, accounting for the array’s compact structure.The element-level derivation uses spherical harmonic expansion; port correlation depends on element correlation and downtilt weights.Source: §I · p. 2
  • For multi-user MISO, the paper formulates common downtilt-weight selection as maximizing minimum SIR and obtains a quasi-optimal solution using semidefinite relaxation and Dinkelbach’s method.The approach uses users’ quasi-static channel correlation matrices.Source: §I · p. 2
  • The user-group-specific scenario partitions users by quasi-static channel correlation and serves each group through a subset of ports using quasi-optimal downtilt weights.The paper notes that the method is motivated by a user-grouping approach based on channel correlation matrices.Source: §I · p. 2
  • Simulations show that single downtilt beamforming significantly outperforms existing elevation beamforming methods, and user-group-specific beamforming improves performance further.The user-group scenario assigns subsets of antenna ports quasi-optimal downtilt weights based on users’ quasi-static channel correlation matrices.Source: §I · pp. 2–3

II. ANTENNA CONFIGURATION AND 3D CHANNEL MODEL

  • The paper introduces the 2D active antenna array and outlines the corresponding FD-MIMO channel model.The section positions the array as a way to exploit the elevation domain for performance optimization.Source: §II · p. 3

A. Active Antenna Array for FD-MIMO

  • Each antenna port groups vertically arranged elements that carry the same signal, with element weights controlling the port’s downtilt.The ports are placed at equidistant horizontal positions to form the planar array.Source: A. Active Antenna Array for FD-MIMO · p. 3

B. Antenna Element Approach towards 3D Channel Modeling

  • The channel of an antenna port is a weighted sum of the channels of its constituent elements, whose radiation patterns and array responses determine the overall pattern.Earlier 3D channel-model studies considered overall ports and used approximate port radiation patterns.Source: B. Antenna Element Approach towards 3D… · pp. 3–4

III. SPATIAL CORRELATION FUNCTION

The paper derives analytical spatial correlation functions for 3D channels in a 2D active antenna array, covering arbitrary antenna patterns and angular distributions. The element-level derivation uses spherical harmonic expansion and power-spectrum coefficients.

A. Spatial Correlation Function for Antenna Elements

  • The derivation expands each element's array response using spherical harmonics of plane waves, extending the analysis from a linear to a 2D array.The resulting expression uses Fourier-series coefficients of the power azimuth and elevation spectra.Source: §III · p. 4, A. Spatial Correlation Function for… · pp. 4–5
  • Theorem 1 gives the 3D element-channel spatial correlation function for arbitrary antenna patterns and angular distributions in a 2D uniform planar array.It covers azimuth angles φ ∈ [−π, π] and elevation angles θ ∈ [0, π].Source: A. Spatial Correlation Function for… · p. 5
  • Under the stated OFDM assumptions, the analytical element-channel correlation expression is identical across all V subcarriers.The passage assumes i.i.d. path gains and angles across paths.Source: A. Spatial Correlation Function for… · p. 5

B. Spatial Correlation Function for Antenna Ports

The antenna-port correlation matrix is formed from element-level correlations and the downtilt weights applied to each port. Simulations validate the SCF and show that correlation decreases as the number of elements per port increases.

B. Spatial Correlation Function for Antenna Ports

IV. PERFORMANCE OPTIMIZATION OF FD-MIMO SYSTEMS

This section optimizes downtilt weight vectors for single-user and multi-user FD-MISO downlinks. It uses exact radiation-pattern expressions and a correlated Rayleigh channel model.

IV. PERFORMANCE OPTIMIZATION OF FD-MIMO SYSTEMS

  • The optimization targets downlink SNR in single-user MISO and SIR in multi-user MISO settings using exact radiation-pattern expressions.The expressions depend on NE, dz, and the downtilt weights.Source: §IV · p. 7
  • Unlike existing 3D beamforming approaches based on approximate port radiation patterns, this work uses exact radiation-pattern expressions.The exact expressions depend on array size, element spacing, and downtilt weights.Source: §IV · p. 7
  • The analysis models the user channel as correlated Rayleigh fading with covariance RBS derived from the antenna-element correlation matrix and port weights.The model assumes a single isotropic receive antenna element.Source: §IV · p. 7

2 BSz, (32)

The single-user MISO analysis optimizes antenna-port downtilt weight vectors to maximize downlink SNR under a per-port power constraint. It also adopts a covariance-based channel model to make the optimization tractable.

2 BSz, (32)

  • The covariance-based model uses i.i.d. zero-mean, unit-variance complex Gaussian samples and user covariance RBS; it is reported to resemble the ray-tracing model in rich scattering.The model enables use of random matrix theory to derive weight solutions.Source: §2 · p. 7

A. Elevation Beamforming in a Single-User MISO System

  • The single-user problem maximizes downlink SNR over each port's NE × 1 weight vector, with unit-norm weights bounding each port's total power.Direct angle optimization is intractable because downtilt angle depends nonlinearly on the weight vector; the paper instead optimizes its entries directly.Source: §2 · p. 7, A. Elevation Beamforming in a… · p. 8
  • In the large-array regime, the optimal single-user weights are eigenvectors associated with the maximum eigenvalue of the port's element-correlation block.The block is the NE × NE diagonal submatrix of RE for that port.Source: A. Elevation Beamforming in a… · p. 8

B. Elevation Beamforming in a Multi-User MISO System

The multi-user MISO downlink uses linear precoding, with MRT selected for its low complexity and robustness. The analysis derives a large-system deterministic SINR approximation using user-specific channel correlation matrices.

B. Elevation Beamforming in a Multi-User MISO System

  • Each user’s channel correlation matrix is constructed from the antenna-element correlation matrix and the antenna-port downtilt weights.The element correlations depend on the user’s mean elevation and azimuth angles of departure.Source: B. Elevation Beamforming in a… · p. 8
  • MRT is used for digital precoding because it combines low computational complexity, robustness, and high asymptotic performance.The precoding vector is the conjugate of the channel vector.Source: B. Elevation Beamforming in a… · p. 8
  • For a multi-user MISO downlink with non-cooperating single-antenna users, the paper derives a deterministic SINR approximation in the large-system regime.The proposition assumes user channel correlation matrices satisfy Assumption A-1.Source: B. Elevation Beamforming in a… · p. 9
  • The asymptotic SINR expression depends on the downtilt weight vectors assigned to antenna ports and their cross-correlation blocks.The port-pair blocks represent cross-correlations between the elements of two ports.Source: B. Elevation Beamforming in a… · p. 9

1) Problem Formulation:

The section formulates elevation beamforming as a max-min SIR problem for an interference-limited system. A common downtilt vector across ports makes a tractable semidefinite relaxation possible.

1) Problem Formulation:

  • The max-min SIR is chosen for interference-limited systems as a balance among throughput, user fairness, and computational complexity.The objective uses a deterministic SIR approximation.Source: 1) Problem Formulation · p. 9
  • Jointly optimizing distinct downtilt vectors across antenna ports is harder than multicast beamforming because the objective contains products of port-weight matrices.The passage describes the related multicast problem as NP-hard and solved quasi-optimally using semidefinite relaxation.Source: 1) Problem Formulation · p. 9
  • Assuming all ports use the same downtilt vector removes the product terms and enables a tractable semidefinite relaxation of the single-downtilt problem.The relaxed matrix is positive semidefinite with unit trace.Source: 1) Problem Formulation · p. 9

2) Optimization Technique and Solution:

The single-downtilt problem is solved through semidefinite relaxation and generalized Dinkelbach optimization. The resulting relaxed matrix may require post-processing to recover a quasi-optimal weight vector.

2) Optimization Technique and Solution:

  • Generalized Dinkelbach optimization applies after reformulation into ratios with affine numerators, convex denominators, and convex constraints.The passage states that fractional programming is efficient under these properties.Source: 2) Optimization Technique and Solution · pp. 9–10
  • The relaxed problem’s global optimum may have rank greater than one, so eigenvector approximation or randomization is used to extract a quasi-optimal beamforming vector.Randomization samples vectors from a complex Gaussian distribution with covariance given by the relaxed solution.Source: 2) Optimization Technique and Solution · p. 10
  • The SDB design separates elevation beamforming based on users’ channel correlations from MRT precoding on the reduced-dimensional effective channels.The dimension falls from NBS NE to NBS, reducing feedback overhead because antenna ports are fewer than antenna elements.Source: 2) Optimization Technique and Solution · p. 10
  • The method assumes perfect knowledge of user channel correlation matrices, which the authors say can be learned and tracked because they vary more slowly than Rayleigh fading.The matrices are computed from locally estimated angular spread and users’ LoS azimuth and elevation angles.Source: 2) Optimization Technique and Solution · p. 10

V. USER GROUP SPECIFIC ELEVATION BEAMFORMING

User-group-specific elevation beamforming addresses the loss from using one downtilt vector for all users. It assigns group-specific beams using channel-correlation information and corresponding antenna-port subsets.

A. Motivation

  • UG-SDB partitions users by their channel correlation matrices and serves each group with a subset of ports using a group-optimized downtilt vector.Different groups therefore receive different elevation beams, with MRT used within each group.Source: A. Motivation · pp. 10–11
  • Grouping narrows the user statistics used to optimize each elevation beam, because each vector is designed for fewer co-located users.This is the stated rationale for reducing the loss of the single-downtilt assumption.Source: A. Motivation · pp. 10–11
  • The grouping approach draws on fixed quantization in prior user-grouping work, which outperformed K-means in the cited simulations.That prior approach motivated its counterpart for FD-MIMO.Source: A. Motivation · p. 11

B. Method

  • The method forms groups with disjoint elevation angular supports, assigns users by chordal distance between their correlation eigenspaces and group subspaces, and allocates port subsets to groups.Each group’s ports use a downtilt vector optimized from that group’s users’ correlation matrices.Source: B. Method · p. 11

VI. RESULTS AND DISCUSSIONS

The section evaluates downtilt-weight design in single-user MISO and reports that Theorem 2 outperforms preset tilts. Its deterministic capacity approximation also tracks Monte Carlo results well.

A. Performance of Theorem 2 in the Single User MISO Setting

  • Theorem 2’s optimized downtilt weights yield higher user capacity than weights for predefined electrical downtilt angles.The study uses an outdoor cell-edge user and reports that deterministic capacity from the SNR approximation matches Monte Carlo capacity quite well.Source: A. Performance of Theorem 2 in the… · p. 11

B. Performance Comparison of SDB in the Multi-User MISO Setting

In multi-user MISO simulations, SDB improves performance over fixed-angle tilting, while user-group-specific beamforming adds further gains. Grouping benefits depend on group count and user distribution.

B. Performance Comparison of SDB in the Multi-User MISO Setting

C. Performance of User Group Specific Single Downtilt Beamforming Scenario

VII. CONCLUSION

  • The conclusion formulates common-weight multi-user minimum-SIR optimization using SDR and Dinkelbach’s method, then extends the design to user-group-specific beamforming.Source: §VII · p. 13

APPENDIX A PROOF OF THEOREM 2

The appendices support the single-user weight-vector result with a quadratic reformulation and eigenvector solution, and establish a convergence result for the multi-user expression.

APPENDIX A PROOF OF THEOREM 2

  • Theorem 2 follows by expressing tr(hhH) as a quadratic term in z to rewrite the SNR; in the large (NBS, NE) regime, (P1) has constraints ||ws||2 = 1 for s = 1, . . . , NBS and the stated eigenvector solution.Source: Appendix A · p. 13, §2 · p. 14

2 BSz. (67)

  • The single-user optimization is expressed in the large-(NBS, NE) regime with unit-norm port weights.Source: §2 · p. 14

APPENDIX B PROOF OF PROPOSITION 1

  • The appendix applies the continuous mapping theorem and covariance-matrix conditions to establish convergence for the multi-user expression.The proof also uses trace identities to obtain the numerator and denominator convergence results.Source: Appendix B · p. 14
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