PaperlayerView exact revision on arXiv

Source-linked AI summary

Joint Spatial Division and Multiplexing for mm-Wave Channels

Ansuman Adhikary, Ebrahim Al Safadi, Mathew Samimi, Rui Wang, Giuseppe Caire, Theodore S. Rappaport, Andreas F. Molisch

arXiv:1312.2045v3cs.IT

TL;DR

The paper addresses the CSIT burden of FDD mm-Wave massive MIMO under realistic sparse channels with overlapping angular spectra. It extends JSDM with graph-based grouping, greedy selection, and a covariance-only variant, finding that the reduced-complexity scheme performs well in ray-traced and measured outdoor channels.

  • Problem

    FDD massive MIMO requires substantial transmitter-side CSI feedback, while realistic mm-Wave channels include sparse paths and partially overlapping user angular spectra.

  • Method

    The paper formulates user grouping as graph-theoretic optimization, develops greedy algorithms, and proposes covariance-based JSDM using only channel second-order statistics.

  • Results

    The proposed JSDM schemes perform well in ray-traced and measured outdoor channels, with covariance-based JSDM requiring no instantaneous CSIT feedback.

  • Takeaways & Limitations

    JSDM with suitable user selection, and sometimes covariance-based JSDM, is attractive for outdoor small- to medium-range mm-Wave MU-MIMO downlinks.

Abstract

from arXiv · show

Massive MIMO systems are well-suited for mm-Wave communications, as large arrays can be built with reasonable form factors, and the high array gains enable reasonable coverage even for outdoor communications. One of the main obstacles for using such systems in frequency-division duplex mode, namely the high overhead for the feedback of channel state information (CSI) to the transmitter, can be mitigated by the recently proposed JSDM (Joint Spatial Division and Multiplexing) algorithm. In this paper we analyze the performance of this algorithm in some realistic propagation channels that take into account the partial overlap of the angular spectra from different users, as well as the sparsity of mm-Wave channels. We formulate the problem of user grouping for two different objectives, namely maximizing spatial multiplexing, and maximizing total received power, in a graph-theoretic framework. As the resulting problems are numerically difficult, we proposed (sub optimum) greedy algorithms as efficient solution methods. Numerical examples show that the different algorithms may be superior in different settings.We furthermore develop a new, "degenerate" version of JSDM that only requires average CSI at the transmitter, and thus greatly reduces the computational burden. Evaluations in propagation channels obtained from ray tracing results, as well as in measured outdoor channels show that this low-complexity version performs surprisingly well in mm-Wave channels.

I. INTRODUCTION

The paper adapts JSDM to realistic sparse mm-Wave channels, where common scatterers and overlapping angular spectra complicate user separation. It proposes grouping algorithms and a covariance-only variant to reduce CSIT requirements.

  • Motivation: FDD massive MIMO faces substantial transmitter-side CSIT overhead because channel reciprocity is unavailable.JSDM targets this setting by reducing the instantaneous channel information that must be acquired.
  • JSDM principle: JSDM partitions users with similar channel covariances and applies statistical pre-beamforming before instantaneous MU-MIMO precoding.The first stage reduces effective channel dimensionality, while the second stage performs spatial multiplexing.
  • Channel realism: Realistic mm-Wave channels include common scatterers, partial angular-spectrum overlap, and relatively few significant multipath components.These properties depart from the disjoint one-cluster model used in earlier analyses.
  • Contributions: The paper formulates user grouping for spatial multiplexing and received-power objectives, then develops greedy algorithms because the underlying optimization is difficult.The preferred objective can depend on operating SNR when coupled groups are either multiplexed or assigned orthogonal resources.
  • Contributions: Covariance-based JSDM uses only channel second-order statistics and performs well when few users occupy sparse channels with small angular spreads.The paper evaluates the approach using ray-traced and measured outdoor channels, including a 28 GHz urban campaign.

II. SPATIAL CHANEL MODELS

The paper models realistic MIMO propagation through double-directional channels and derives antenna-dependent analytical representations. It emphasizes clustered, sparse mm-Wave propagation and shared far scatterers across users.

  • Modeling framework: The double-directional impulse response is modeled as contributions from discrete multipath components indexed by delay and departure and arrival angles.Diffuse radiation can be represented by continuous angle or delay intervals.
  • Modeling framework: Double-directional models represent propagation independently of antenna structures and support realistic large-scale channel variations.Analytical transfer-function models can then be derived for the specific antenna configuration.
  • Analytical channel: The paper specializes the model to an omnidirectional user antenna and a uniform linear array at the base station.It focuses on the frequency-domain channel matrix under OFDM and block fading assumptions.
  • Scattering structure: The one-cluster model describes local scattering over a limited angular range, while additional far clusters can be visible to multiple users.This distinction motivates models with common scatterers and inter-user spatial coupling.
  • Scattering structure: Mm-Wave propagation exhibits pronounced sparsity because only a small number of reflected or diffuse paths carry significant power.The paper links this behavior to rough surfaces and the reduced importance of paths undergoing many reflections.

III. JOINT SPATIAL-DIVISION AND MULTIPLEXING

JSDM separates user processing into statistical pre-beamforming and instantaneous within-group precoding. Group selection and beam design can suppress inter-group interference, exactly or approximately, subject to spatial-rank limits.

  • System model: JSDM serves users partitioned into groups with statistically independent but identically distributed channels sharing group covariance matrices.The covariance eigenspaces describe the spatial dimensions available to each group.
  • Two-stage precoding: The overall precoder factors as V = BP, with B providing group pre-beamforming and P applying block-diagonal MU-MIMO precoding.This is the defining two-stage structure of the reviewed scheme.
  • Two-stage precoding: The pre-beamformer depends on channel second-order statistics, whereas the within-group precoder uses the instantaneous effective channel.Pre-beamforming therefore reduces the dimension of the instantaneous CSIT problem.
  • Inter-group separation: Group selection and pre-beamforming can enforce exact or approximate elimination of inter-group interference.Approximate block diagonalization retains residual interference when exact spatial separation is unavailable.
  • Design constraints: The number of spatially multiplexed streams is limited by the effective group rank, while per-group processing reduces instantaneous CSIT requirements.Analog implementation can further reduce the number of required RF chains.

A. Application to the one-cluster model

For one-cluster channels, the paper constructs covariance-based group subspaces and pre-beamformers that remove other groups’ spatial components before within-group multiplexing. The common-scatterer example illustrates why shared angular directions must be handled explicitly.

  • One-cluster channel: The one-cluster model represents each user’s paths as many components concentrated around cluster-specific angles and angular spreads.A Gaussian channel vector with covariance derived from those clusters follows from the large-path approximation.
  • Pre-beamforming: For co-located groups, approximate block diagonalization projects each group away from the eigenspaces associated with other groups.The resulting pre-beamformer then applies eigenbeamforming along dominant projected covariance modes.
  • Common scatterers: A common remote scattering cluster couples two otherwise locally scattered user groups through shared angular directions.The example contains local clusters for each group plus a common cluster at angle 0 degrees.
  • Within-group multiplexing: Within-group spatial multiplexing cannot exceed the rank of the equivalent channel after pre-beamforming.The paper uses zero-forcing MU-MIMO precoding when the effective dimension supports multiple streams.

B. Multiple scattering clusters

This section extends JSDM to users whose channels include multiple scattering clusters, including common scatterers that couple user groups. It compares spatial multiplexing with orthogonalization and shows that their relative advantage depends on SNR.

  • Multiple scattering clusters: JSDM is generalized from disjoint covariance subspaces to user groups coupled by common scattering clusters.The common-scatterer model motivates serving groups either simultaneously after nulling shared components or separately while retaining them.
  • Multiple scattering clusters: Multiplexing uses block diagonalization to separate groups on the same transmission resource, whereas orthogonalization assigns groups to different resources and retains all channel eigenmodes.These approaches trade spatial multiplexing against received power from common multipath components.
  • Multiple scattering clusters: 400 BS antennas and 100 users are used in the example, with two equal groups and one cluster common to both groups.The groups have local clusters at -45° and 60°, plus a common cluster at 0°, with angular spread Δ=15°.
  • Multiple scattering clusters: Full CSIT requires 400 training dimensions in the example, while JSDM requires 100, a reduction by 4; a covariance-only variant is then proposed.The covariance-only version is introduced because the remaining JSDM training burden may still be too large for practical scenarios.
  • Multiple scattering clusters: At low SNR, Orthogonalization performs better because the common scatterer contributes additional received power.This advantage comes from using the multipath components associated with the common scatterer.
  • Multiple scattering clusters: At high SNR, Multiplexing performs much better because it serves more users simultaneously, producing a spatial-multiplexing factor of 2 relative to Orthogonalization.Removing the common scatterer reduces received power but enables simultaneous transmission to both groups.

IV. APPLICATION OF JSDM TO HIGHLY DIRECTIONAL CHANNELS

The paper applies JSDM to highly directional mm-Wave channels with multiple narrow-spread scattering clusters. Partial angular overlap creates a user-selection problem that balances separable groups against retained power.

  • IV. APPLICATION OF JSDM TO HIGHLY DIRECTIONAL CHANNELS: Highly directional mm-Wave channels are modeled using multiple scattering clusters with distinct departure angles and narrow angular spreads.Each user covariance occupies subsets of the angular directions resolvable by the BS array.
  • IV. APPLICATION OF JSDM TO HIGHLY DIRECTIONAL CHANNELS: Different users’ angular subsets may overlap on some intervals and remain disjoint on others, generalizing the common-scatterer case.The common-scatterer example is the special case of two groups with three angular intervals, overlapping on one interval.
  • IV. APPLICATION OF JSDM TO HIGHLY DIRECTIONAL CHANNELS: User allocation seeks a tradeoff between spatial multiplexing from separable groups and power gain from combining more multipath components.The resulting allocation problem is combinatorial and can be formulated as an integer program.

A. Channel eigenvalue spectrum and angular occupancy

The channel model represents each user’s angular support through the eigenvalue spectrum of its covariance matrix. Users are represented by occupied angular regions, whose overlaps define a graph for selection.

  • A. Channel eigenvalue spectrum and angular occupancy: For large antenna arrays, the covariance eigenvalue spectrum converges to the discrete-time Fourier transform of the antenna correlation function.The spectrum is denoted ξ_k(f) for user k.
  • A. Channel eigenvalue spectrum and angular occupancy: Discrete multipath channels are represented by quantizing [−1/2, 1/2] into M angular bins of size 1/M.A user occupies a bin when the mapped path angle −D sin θ_kp lies in that bin.
  • A. Channel eigenvalue spectrum and angular occupancy: The support W_k of ξ_k(f) records the angular region occupied by user k.This support is used as the user’s node weight in the graph formulation.
  • A. Channel eigenvalue spectrum and angular occupancy: Users become graph nodes with weights W_k, while an edge connects users whose angular supports overlap.The edge weight is the intersection W_k ∩ W_ℓ.

B. Optimization Problem 1

Optimization Problem 1 selects users by maximizing combined spectral support while removing overlap between selected users. Because the integer program can be computationally difficult, the paper uses a greedy feasible-selection algorithm.

  • B. Optimization Problem 1: The first optimization objective maximizes the total area of the selected users’ combined eigenvalue spectra after eliminating subspace overlap.This objective favors users that contribute non-overlapping angular support.
  • B. Optimization Problem 1: The formulation uses binary selection variables and graph neighborhoods to encode which users and overlapping supports are selected.For x∈{0,1}, xW equals W when selected and ∅ otherwise; N_k contains users adjacent to k.
  • B. Optimization Problem 1: The integer optimization may be computationally complex for real-time use with many users and many angular bins.The paper therefore seeks an easily computable feasible selection rather than relying directly on the exact solution.
  • B. Optimization Problem 1: Greedy Algorithm 1 starts with no selected users and iteratively adds a user only when the objective Q1 increases.At each iteration, it evaluates candidate additions and updates the selected set when improvement is available.
  • B. Optimization Problem 1: The greedy rule selects a user with maximum spectral area and penalizes users whose spectra overlap already selected users.Qualitatively, this implements an orthogonalization preference.
  • B. Optimization Problem 1: The graph-based angular formulation relies on linear arrays; other array geometries require a different mapping and problem formulation.The stated limitation follows from the simple angle-to-interval mapping available for linear arrays.

C. Optimization Problem 2

Optimization Problem 2 maximizes the number of users that can be served simultaneously while ensuring each selected user has a non-overlapping angular interval. Because the integer program is exponentially difficult, the paper uses a linear-complexity greedy heuristic controlled by an overlap threshold.

  • Optimization Problem 2: The optimization maximizes served users subject to each selected user having at least one non-overlapping spectral interval.The constraint ensures that scheduled user nodes retain a non-empty interval not shared with other scheduled users.
  • Optimization Problem 2: The solution represents the largest simultaneously served user set with no common angular-spectrum overlap.This formulation targets spatial multiplexing by removing overlap regions among selected users.
  • Greedy Algorithm 2: Greedy Algorithm 2 initializes an empty selected set, constructs feasible candidates, and stops when no candidate remains.Feasibility is evaluated with the currently selected users and the threshold ϵ.
  • Greedy Algorithm 2: The heuristic selects a feasible user with the fewest edges, while ϵ limits the maximum number of multiplexed users.Alternative heuristics may produce different results.
  • Greedy Algorithm 2: Exhaustive search has complexity O(2^K), whereas the proposed greedy selection has complexity O(K).The greedy method repeatedly adds feasible users until the constraints can no longer be satisfied.
  • Interpretation: For highly directional channels, the algorithm schedules users with at least one non-overlapping angular bin, enabling substantial spatial multiplexing.In this regime, users can be viewed as occupying bins associated with multipath-component arrival angles.

D. Application of JSDM after selection

After user selection, JSDM separates spatial processing into covariance-based pre-beamforming and subsequent user-level transmission, with spatial multiplexing and orthogonalization as the two considered cases. The paper evaluates these schemes using analytical, ray-traced, and measured outdoor channel settings, while noting a limitation of the ray-tracing tool.

  • JSDM with spatial multiplexing: In spatial multiplexing, selected user groups share the same transmission resource and approximate block diagonalization forms covariance-based pre-beamformers.Pre-beamforming spatially separates the selected groups before subsequent transmission processing.
  • JSDM with spatial multiplexing: Covariance-based JSDM may approach full JSDM in small mm-Wave cells when discrete user channels overlap on only a few scattering angles.After selection, each group may consist of a single user, reducing the need for further grouping within the group.
  • JSDM framework: The algorithms depend only on channel covariance matrices, which are constant with frequency under the narrowband assumption.The covariance is independent of multipath delays and can therefore be treated as frequency-invariant for a single OFDM subcarrier.
  • Evaluation settings: The evaluation covers multi-cluster, ray-traced, and measured outdoor channels to examine highly directional channels with common scatterers.The ray-tracing setup models the USC campus, while measurements use 28 GHz outdoor channel data from downtown New York City.
  • Evaluation settings: The ray tracer omits diffuse multipath components, although the authors state that significant multipath components are sufficient for the presented evaluation.Ray-tracing accuracy also depends on environmental data, launched rays, and the number of modeled interactions.

C. JSDM with spatial multiplexing

The spatial-multiplexing evaluation compares greedy and exhaustive user-selection strategies as the number of groups and SNR vary. More groups can increase multiplexing but reduce beamforming gain and increase inter-group interference, so the throughput benefit is not monotonic.

  • Simulation setup: The simulations use 10 random scattering clusters, vary user groups from G = 2 to 5, and set M = 400 antennas.Each group has a unique scatterer plus possibly shared scatterers, and users are spatially multiplexed using ZFBF after approximate block diagonalization.
  • Throughput comparison: Figures 5(a) and 5(b) compare total achievable throughput versus SNR for Greedy Algorithms 1 and 2 and exhaustive search.The comparison evaluates the proposed selection methods against the exhaustive-search counterpart.
  • Throughput comparison: More groups reduce beamforming gain and increase inter-group interference, preventing spatial-multiplexing gains from being fully realized.The interference results from non-perfect block diagonalization.
  • Throughput comparison: Lower ϵ favors more groups but yields lower throughput here, whereas higher ϵ sacrifices spatial dimensions and produces higher throughput.The trade-off reflects the competing effects of multiplexing, beamforming gain, and inter-group interference.
  • Throughput comparison: For G = 5, both greedy algorithms perform well relative to exhaustive search.The paper cites Figures 5(b) and 6(b) as evidence for this comparison.

D. Covariance-based JSDM

Covariance-based JSDM reduces precoding requirements by using only second-order statistics, but its throughput and spatial multiplexing depend strongly on channel structure and user selection. It performs best for highly directional, sparse channels and can lose substantially when spatial multiplexing is unavailable.

  • Covariance-based JSDM: Covariance-based JSDM serves one user per group using only second-order statistics, eliminating instantaneous effective-channel CSIT and simplifying precoder design.This reduction in complexity comes with a considerable achievable-throughput penalty relative to full JSDM in multi-cluster settings.
  • Covariance-based JSDM: In multi-cluster user groups, covariance-based JSDM produces a huge reduction in achievable data rates and spatial multiplexing compared with full JSDM.The comparison is made for G = 5 groups, with sum spectral efficiency versus SNR and spatial multiplexing versus the tuning parameter ϵ.
  • Covariance-based JSDM: With K = 20 users and multiple scattering clusters, only an average of 7 users are served simultaneously because common scattering clusters limit total spatial multiplexing.The reduced throughput is attributed to the lack of spatial multiplexing in covariance-based JSDM; full JSDM can group users spanning the same dimensions using instantaneous CSIT.
  • Covariance-based JSDM: Ray-tracing channels recover spatial multiplexing with covariance-based JSDM when channels are highly directional and contain few multipath components.These channels use disjoint angular-frequency bins, allowing the grouping algorithm to set ϵ = 0.
  • Covariance-based JSDM: Measured-channel evaluations show that user-selection algorithms perform differently by scenario, with Algorithm 2 outperforming Algorithm 1 for BS 3 because of higher spatial multiplexing.For BS 2, both algorithms achieve the same spatial multiplexing.
  • Covariance-based JSDM: The proposed selection algorithms are meaningful when most users concentrate channel energy in a few directions; near-isotropic channels reduce the scheme to one-user or instantaneous-CSIT service.Thus, the method’s favorable behavior is tied to directional propagation rather than guaranteed across propagation environments.

VI. CONCLUSION

The paper applies JSDM to sparse, highly directional mm-Wave channels where users may have partially overlapping angular subspaces. It formulates user selection and angular allocation as conflict-graph optimization problems, evaluates greedy solutions in realistic channels, and proposes covariance-based JSDM using only second-order channel statistics.

  • Partially overlapping eigenspaces from common scattering clusters or similarly directed MPCs make angular-dimension allocation difficult.
  • User selection and angular allocation are formulated as integer-programming problems over a conflict graph, with objectives tied to the desired optimization target.
  • Low-complexity greedy algorithms solve the proposed optimization problems and are evaluated using ray-traced campus channels and measured urban channels.
  • JSDM with good user selection can exploit highly directional mm-Wave statistics for multiuser MIMO downlink in massive MIMO systems.
  • Covariance-based JSDM achieves remarkable spatial multiplexing using only channel second-order statistics, without instantaneous CSIT feedback.
Loading 1312.2045v3…