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Joint Spatial Division and Multiplexing

Ansuman Adhikary, Junyoung Nam, Jae-Young Ahn, Giuseppe Caire

arXiv:1209.1402v2cs.IT

TL;DR

The paper addresses how to obtain massive-MIMO-like downlink gains in FDD systems without full-dimensional CSIT. It proposes JSDM, combining covariance-based pre-beamforming with reduced-dimensional instantaneous precoding, and establishes optimality conditions, large-ULA DFT designs, and 3D-array extensions with attractive spectral-efficiency performance.

  • Problem

    FDD systems cannot exploit uplink/downlink reciprocity, while full-dimensional downlink training and CSIT feedback become costly for large antenna arrays.

  • Method

    JSDM groups users by channel-covariance eigenspaces, applies covariance-dependent pre-beamforming, and then precodes the resulting reduced-dimensional effective channels.

  • Results

    JSDM has no loss of optimality under a covariance-eigenvector condition; for large ULAs, DFT pre-beamforming approaches this condition using coarse AoA-support information.

  • Takeaways & Limitations

    The design makes large-antenna FDD operation potentially suitable by reducing training and CSIT-feedback overhead, while extending to three-dimensional rectangular arrays.

Abstract

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We propose Joint Spatial Division and Multiplexing (JSDM), an approach to multiuser MIMO downlink that exploits the structure of the correlation of the channel vectors in order to allow for a large number of antennas at the base station while requiring reduced-dimensional Channel State Information at the Transmitter (CSIT). This allows for significant savings both in the downlink training and in the CSIT feedback from the user terminals to the base station, thus making the use of a large number of base station antennas potentially suitable also for Frequency Division Duplexing (FDD) systems, for which uplink/downlink channel reciprocity cannot be exploited. JSDM forms the multiuser MIMO downlink precoder by concatenating a pre-beamforming matrix, which depends only on the channel second-order statistics, with a classical multiuser precoder, based on the instantaneous knowledge of the resulting reduced dimensional effective channels. We prove a simple condition under which JSDM incurs no loss of optimality with respect to the full CSIT case. For linear uniformly spaced arrays, we show that such condition is closely approached when the number of antennas is large. For this case, we use Szego asymptotic theory of large Toeplitz matrices to design a DFT-based pre-beamforming scheme requiring only coarse information about the users angles of arrival and angular spread. Finally, we extend these ideas to the case of a two-dimensional base station antenna array, with 3-dimensional beamforming, including multiple beams in the elevation angle direction. We provide guidelines for the pre-beamforming optimization and calculate the system spectral efficiency under proportional fairness and maxmin fairness criteria, showing extremely attractive performance. Our numerical results are obtained via an asymptotic random matrix theory tool known as deterministic equivalent approximation.

I. INTRODUCTION

JSDM targets massive-MIMO-like gains in FDD systems by exploiting channel-covariance structure to reduce training and feedback dimensions. It combines covariance-dependent pre-beamforming with instantaneous precoding of reduced-dimensional effective channels, with asymptotic and multidimensional array designs.

  • Motivation: FDD MU-MIMO overhead becomes problematic as antenna count and Doppler increase, because training, estimation, prediction, and CSIT feedback reduce multiplexing gain.The prediction penalty is quantified by max{1 −2BdTs, 0}, while large systems make downlink training and uplink feedback significant bottlenecks.
  • JSDM approach: JSDM partitions users with similar covariance eigenspaces and concatenates covariance-based pre-beamforming with MU-MIMO precoding for the effective channel.The pre-beamformer exploits linear independence among groups’ dominant covariance eigenmodes to reduce inter-group interference.
  • JSDM approach: Reduced-dimensional effective channels greatly lower downlink training and uplink feedback overhead, making large-antenna FDD operation attractive.The pre-beamforming stage requires only channel covariance information, which can be tracked with small protocol overhead.
  • Optimality and design: JSDM incurs no loss of optimality relative to full CSIT under a condition on covariance eigenvectors; otherwise, pre-beamforming and regularized zero-forcing designs are examined.The paper also specializes the design to ULAs and analyzes the resulting effective-channel precoding.
  • ULA design: For large ULAs with non-overlapping AoA supports, Szego-based analysis supports DFT pre-beamforming with good performance and effective dimensionality reduction.Selecting blocks of unitary DFT columns requires only coarse AoA-support information rather than accurate covariance estimation.
  • 3D extension: The approach extends to rectangular arrays and three-dimensional beamforming, including multiple elevation-direction beams and guidelines for pre-beamforming optimization.The paper considers simultaneous service of angularly separated user groups in different annular regions.

II. CHANNEL MODEL

The paper models a narrowband FDD multiuser MIMO downlink using correlated channels generated by user scattering geometry, then exploits covariance structure for two-stage JSDM precoding and reduced-dimensional CSIT. Users are grouped by covariance similarity, with pre-beamforming based on channel statistics and instantaneous multiuser precoding applied to effective channels.

  • Channel model: The system is a single-cell, frequency-flat FDD downlink with an M-antenna base station serving K single-antenna user terminals.The received signal is modeled as y = H Hx + z, with x = Vd and Gaussian receiver noise.
  • Channel model: In the one-ring model, a user at azimuth angle θ has angular spread ∆≈arctan(r/s) generated by scatterers of radius r.The channel covariance is determined by the received planar-wave power distribution and antenna positions.
  • JSDM precoding: JSDM partitions users with similar covariance structure into groups and uses two-stage precoding, V = BP, with pre-beamforming B followed by MU-MIMO precoding P.The pre-beamforming dimension b satisfies b ≥ S, where S is the number of transmitted streams.
  • Reduced-dimensional CSIT: JSDM reduces effective-channel dimensionality and can provide multiplexing gains with reduced channel training and CSIT feedback.Per-group processing estimates and feeds back only diagonal effective-channel blocks, using block-diagonal P.
  • Design assumptions: The dominant-eigenmode count r⋆ is a design parameter controlling tolerated signal power outside the selected subspace, so choosing r⋆ equal to the full rank can create a dimensionality bottleneck.Channel covariance matrices are assumed known, while their estimation and tracking are outside the paper’s scope.
  • JSDM precoding: The pre-beamforming matrix depends only on channel second-order statistics or directional information, while P uses the instantaneous reduced-dimensional effective channel.This separates slowly varying covariance information from instantaneous channel knowledge.
  • Inter-group interference: Exact block diagonalization is possible when each group’s covariance subspace is not contained in the span of the other groups’ subspaces.The group can achieve Sg interference-free streams under the stated subspace condition, whereas omitted weak eigenmodes may cause residual interference.

IV. JSDM WITH EIGEN-BEAMFORMING

Under the tall-unitary condition, eigen-beamforming lets JSDM with per-group processing achieve the full-CSIT sum capacity, and under the same condition it can achieve the whole capacity region. The paper also gives scheduling and approximate block-diagonalization alternatives when the condition is restrictive.

  • Eigen-beamforming: Theorem 1 states that JSDM with per-group processing achieves the same sum capacity as the corresponding full-CSIT MU-MIMO downlink when the group eigenvector matrix is tall unitary.The pre-beamforming blocks use the groups’ covariance eigenspaces, producing decoupled channels under the stated orthogonality condition.
  • Eigen-beamforming: Under the tall-unitary condition, JSDM achieves the whole capacity region, not only the sum capacity.The argument applies the determinant identity to the partial sum-rate bounds for every user subset.
  • Scheduling: A practical scheduler groups users with approximately identical eigenspaces, partitions groups into tall-unitary sets, and schedules each set simultaneously.Time-frequency sharing is used across sets.
  • Scheduling: JSDM is not generally optimal because satisfying the tall-unitary condition may require serving fewer groups in parallel, while finding optimal partitions is nontrivial.The paper leaves optimal group-partition design beyond its scope.
  • Alternatives: When the tall-unitary condition is too restrictive, the pre-beamforming matrix can depend on all group eigenspaces to achieve exact or approximate block diagonalization.This provides an alternative to eigen-beamforming when multiplexing requirements prevent the orthogonality condition.

B. Block diagonalization

The paper constructs pre-beamforming matrices by nulling dominant interference subspaces and matching the remaining dimensions to dominant projected-channel eigenmodes. It then derives deterministic-equivalent SINR and spectral-efficiency expressions for linear precoding under symmetric system assumptions.

  • Design: For each group, the proposed block combines projection away from other groups’ dominant subspaces with eigen-beamforming along the dominant eigenmodes of the projected covariance.The resulting block is orthogonal to dominant interference directions and matched to the group’s projected channel covariance.
  • Block diagonalization: Exact block diagonalization requires each pre-beamforming block to lie in the orthogonal complement of the other groups’ dominant eigenspaces.The necessary condition is Span(B_g) ⊆ Span^⊥({U_g′: g′ ≠ g}).
  • Block diagonalization: If the interference-free subspace is too small, the number of streams must be reduced or approximate block diagonalization must retain only dominant eigenmodes.Exact block diagonalization can also be impossible when the other groups span the full antenna space.
  • Design: The design parameters, including effective ranks, pre-beamformer dimensions, and streams per group, must be optimized for the system configuration and spectral efficiency.The paper emphasizes a nontrivial tradeoff rather than fixing one universally optimal choice.
  • Performance analysis: The analysis uses deterministic equivalents to approximate SINR and achievable spectral efficiency for JGP and PGP linear precoding as system dimensions grow proportionally.The per-user rate is obtained by inserting the deterministic SINR into log(1 + γ).
  • System model: The symmetric analysis assumes equal users, streams, and pre-beamforming dimensions per group, with users selected independently of their instantaneous channel realizations.Only pre-scheduled users provide instantaneous CSIT in the simplified scheduling model.

B. JSDM with per-group processing

For per-group processing, the paper derives deterministic-equivalent SINR expressions and evaluates them using a uniform circular array. The results show that the effective rank must retain all strong covariance eigenmodes without unnecessarily including negligible ones.

  • Analysis: The PGP analysis derives the user SINR and deterministic-equivalent quantities for regularized zero-forcing precoding under asymptotically growing system dimensions.The per-group precoder uses each group’s effective channel covariance and a group-specific power normalization.
  • Numerical setup: The numerical setup uses a uniform circular array with M = 100 antennas, six symmetric groups, five streams per group, and 30 served users.Each group has azimuth spread 15°; the covariance rank is r = 21, with effective rank r⋆ = 11 because half the nonzero eigenvalues are very small.
  • Validation: The deterministic-equivalent approximations are compared with finite-dimensional Monte Carlo simulations for the JSDM spectral-efficiency curves.The figure distinguishes simulated solid squares from dotted-x deterministic-equivalent results.
  • Effective-rank selection: Choosing r⋆ large enough to include all significant eigenmodes avoids noticeable interference limitation across a wide SNR range.With r⋆ = 12, the curves eventually flatten only at extremely large SNR, despite the covariance rank being r = 21.
  • Effective-rank selection: The effective rank should include the strongest eigenmodes but exclude near-zero modes, because using r⋆ = r can create a dimensionality bottleneck without reducing interference.Since r⋆G ≤ M, excessive rank can force fewer groups to be served in parallel.

VI. DOWNLINK TRAINING AND NOISY CSIT

The noisy-CSIT analysis estimates reduced-dimensional effective channels from common downlink training and feeds the MMSE estimates back for precoder computation. Its asymptotic validity requires the training dimension to remain below the coherence-block length by a meaningful margin.

  • CSIT assumptions: The analysis assumes ideal, delay-free CSIT feedback and focuses on downlink channel-estimation error and dimensionality reduction.At high SNR, feedback error is treated as negligible relative to downlink estimation error when the feedback exponent exceeds one.
  • Training and estimation: With PGP, unitary training sequences are sent over the b′ virtual pre-beamforming inputs, so the training phase spans b′ symbols.Users apply linear MMSE estimation to their effective channels, and the estimates are fed back to the base station.
  • Training overhead: The downlink-training overhead scales spectral efficiency by max{1 − b′/T, 0} when b′ training symbols occupy a coherence block of T symbols.Only the remaining T − b′ symbols carry downlink data.
  • Training and estimation: The scheme uses a common scaled unitary training matrix across all groups, with postmultiplication isolating the effective-channel observations before MMSE estimation.The processing removes the training matrix while transforming the noise accordingly.
  • Asymptotic analysis: The deterministic-equivalent noisy-CSIT analysis provides an asymptotically accurate rate approximation when M, r⋆, S, and b grow linearly together.Under the stated scaling, the approximation error converges to zero almost surely as M → ∞.
  • Scope boundary: Because the training penalty vanishes when b′ ≥ T, the meaningful operating regime requires b′ to be large but still significantly smaller than T.The paper models this through the crowding factor τ = b′/T and assumes T grows linearly with M.

A. Results with downlink channel estimation

Downlink training and noisy CSIT make the choice of effective rank, streams, and pre-beamforming dimension central to JSDM performance. The reported results show non-monotonic training trade-offs and SNR-dependent operating choices.

  • Results with downlink channel estimation: JSDM performance with noisy CSIT is evaluated using downlink training and MMSE channel estimation, with b′ optimized for each S′, r⋆, and SNR.
  • Results with downlink channel estimation: Sum spectral efficiency including channel estimation is not monotonically increasing with b′.Larger b′ improves effective-channel conditioning but increases the dimensionality cost of downlink training, producing a non-trivial optimum.
  • Results with downlink channel estimation: For a fixed operating SNR, the optimal S′ and b′ are approximately linearly related for both RZFBF and ZFBF.The slope of this relationship is used to characterize the optimized stream and beam dimensions.
  • Results with downlink channel estimation: At low SNR, RZFBF favors S′ = b′, whereas ZFBF serves fewer users than b′ and approaches the RZFBF slope as SNR increases.
  • Uniform linear arrays: For large ULAs, Toeplitz channel-correlation matrices are approximated by circulant matrices whose eigenvectors form a unitary DFT matrix.This asymptotic structure supports replacing covariance eigenvectors with selected DFT columns for pre-beamforming.

A. Approximating the channel eigenspace

For large ULAs, the channel-covariance eigenspace can be approximated through the spectral support of its Toeplitz correlation matrix. Disjoint angular supports yield approximately orthogonal group eigenspaces.

  • The covariance eigenspace can be approximated by DFT columns whose angular frequencies lie within the support S of the spectral density.
  • The asymptotic eigenvalue distribution has a zero-eigenvalue mass of 1 − ρ when the spectral support length satisfies ρ < 1.
  • For each user group, the dominant covariance eigenmodes are approximated by a DFT submatrix selected from its angular-frequency support.
  • Groups with disjoint angular-frequency supports produce an approximately tall-unitary concatenation of their DFT and covariance eigenspaces.
  • With common angular spread, disjoint spectra are guaranteed when the groups’ AoA intervals [θg − ∆, θg + ∆] do not overlap.

B. DFT pre-beamforming

DFT pre-beamforming uses large-ULA asymptotics to implement JSDM from coarse angular information, while 3D extensions separate users by elevation regions and azimuthal groups. The design balances orthogonality, training dimensions, coverage, and computational tractability.

  • DFT pre-beamforming: DFT pre-beamforming selects groups with nearly identical AoA intervals and non-overlapping intervals across groups.
  • DFT pre-beamforming: The approach requires only coarse AoA-interval information rather than an accurate estimate of each user’s channel covariance matrix.
  • DFT pre-beamforming: Up to 20 dB SNR, DFT pre-beamforming performs close to schemes with full CSIT.
  • 3D pre-beamforming: A separable 3D scheme forms elevation beams for concentric annular regions and applies azimuthal precoding within each region.
  • 3D pre-beamforming: For JSDM with JGP or PGP, power-allocation optimization is non-convex and lacks a computationally efficient solution.
  • 3D pre-beamforming: Pattern design leaves gaps between group footprints to preserve near-orthogonality and limit inter-group interference under PGP.
  • 3D pre-beamforming: Higher-dimensional vertical pre-beamforming is conceptually straightforward but not very useful in typical practical scenarios.

A. Results with 3D pre-beamforming

The 3D JSDM evaluation partitions a sector into eight annular regions and combines vertical block diagonalization with within-region JSDM precoding. Under ideal CSIT, the study evaluates DFT and approximate-BD pre-beamforming with RZFBF and ZFBF under proportional and max-min fairness.

  • System setup: The setup uses one 600 m hexagonal-cell sector with a rectangular array of M = 200 antennas at height h = 50 m and N = 300 scattering-ring groups.The sector is divided into eight concentric regions at distances 60l m, for l ∈ {1, . . . , 8}.
  • Fairness and optimization: The spectral-efficiency evaluation uses equal power, optimized stream counts, and RZFBF or ZFBF with DFT or approximate-BD pre-beamforming.The resulting regional efficiencies are shown in Fig. 10, while aggregate PFS and max-min results are reported in Table I.
  • System setup: As regions move farther from the base station, more user groups can be accommodated in each annular region because the horizontal angular spread decreases.The horizontal covariance uses a region-dependent angular spread, and the number of accommodable groups increases with distance.
  • Vertical pre-beamforming: A narrow elevation viewpoint requires many vertical antennas to make the eigenmodes of all annular regions orthogonal.For finite antenna dimensions, maximally separated region subsets are partitioned into patterns and scheduled across time-frequency slots.
  • Vertical pre-beamforming: Vertical block diagonalization makes inter-region interference exactly zero, while each region uses approximate-BD or DFT pre-beamforming with per-group processing.Within each annular region, the resulting effective channels are processed using JSDM with PGP.
  • Fairness and optimization: The network utility allocates time-frequency fractions across patterns and evaluates proportional fairness and max-min fairness scheduling.For proportional fairness, equal pattern fractions are optimal; max-min fairness uses the corresponding pattern-rate balancing solution.

IX. CONCLUDING REMARKS

The concluding remarks establish JSDM as a reduced-dimensional MU-MIMO approach based on channel covariance structure and extend it from linear arrays to 3D rectangular-array beamforming. The paper reports high spectral efficiency under ideal CSIT, quantifies noisy-CSIT degradation, and identifies unresolved covariance-eigenspace and group-formation problems.

  • JSDM principle: JSDM separates user groups spatially with covariance-dependent pre-beamforming and multiplexes users within each group using instantaneous effective-channel precoding.The effective channel is reduced-dimensional, especially under per-group processing.
  • JSDM principle: When the groups’ covariance eigenvectors form a tall unitary matrix, JSDM with PGP achieves the capacity of the underlying MU-MIMO channel with full instantaneous CSIT.For large linear uniform arrays, Szegő asymptotics show that this condition is closely approached.
  • Linear-array design: A DFT pre-beamforming matrix can use a subset of unitary DFT columns, with coarse AoA-range information sufficient when groups’ azimuth ranges do not overlap.Accurate channel-covariance estimation is not required under these assumptions.
  • 3D extension: The 3D extension uses rectangular arrays, vertical pre-beamforming, and concentric annular regions serving groups separated by azimuth angle within each region.The reported evaluation uses a large rectangular array mounted on a tall building under representative propagation conditions.
  • Performance: Under ideal CSIT, the massive 3D JSDM scenario achieves spectral efficiencies of about 1000 bit/s/Hz per sector under multiple fairness criteria and pre-beamforming techniques.The results cover proportional and max-min fairness and both DFT and approximate-BD approaches.
  • Performance: At an SNR around 20 dB, noisy CSIT causes an approximately 30% loss relative to ideal CSIT, leaving about 700 bit/s/Hz attainable in the massive 3D scenario.The paper also provides deterministic-equivalent SINR formulas for efficient performance calculation without lengthy Monte Carlo simulation.
  • Open problems: Future work includes user-group formation and estimation of each user’s dominant covariance eigenspace from noisy received-signal samples.High-dimensional covariance matrices make conventional sample-covariance estimation unsuitable, motivating more sophisticated techniques.

APPENDIX A

Appendix A gives deterministic-equivalent SINR formulas for JSDM with PGP, noisy CSIT, and regularized or unregularized zero-forcing precoding. The formulas apply to fixed pre-beamforming matrices, including approximate BD and DFT designs.

  • Deterministic equivalents: The appendix provides fixed-point equations for deterministic-equivalent SINR approximations under PGP with noisy CSIT and RZFBF or ZFBF.These approximations support efficient system-performance calculations for the considered linear precoders.
  • Scope: The formulas hold for arbitrary pre-beamforming matrices that are fixed independently of instantaneous channel realizations, including approximate BD and DFT.The appendix assumes group parameters and equal power per stream.
  • RZFBF: For RZFBF, the precoding matrix is constructed from channel estimates and includes a regularization term before SINR evaluation.The appendix defines the associated power normalization and user-level SINR expressions.
  • ZFBF: Setting α = 0 reduces the regularized precoder to the zero-forcing precoder, for which the appendix gives corresponding SINR and deterministic-equivalent expressions.The zero-forcing case is treated as a specialization of the regularized formulation.

APPENDIX B GENERAL FORMULA FOR S(ξ)

Appendix B derives a general expression for S(ξ) for Toeplitz covariance matrices while accounting for angular intervals that cross the ±π/2 boundaries. The result is organized into four angular-range cases and recovers the standard Bessel-transform form in the full limiting range.

  • Angular cases: The angular limits are partitioned into four cases according to whether θ − ∆ and θ + ∆ cross the ±π/2 boundaries.The cases cover non-crossing intervals, intervals crossing both boundaries, and intervals crossing either boundary.
  • Angular cases: Using the Dirac-delta property, the appendix writes S(ξ) separately for the identified angular cases.The resulting expressions preserve the dependence on the applicable angular-range limits.
  • Limiting cases: When the angular interval remains within −π/2 to π/2, the general formula reduces to the earlier expression.Taking the limits from −π to π recovers the Fourier transform of the Bessel J0 function used for isotropic correlated Rayleigh fading.
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