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Fronthaul Compression and Transmit Beamforming Optimization for Multi-Antenna Uplink C-RAN

Yuhan Zhou, Wei Yu

arXiv:1604.05001v2cs.ITmath.IT

TL;DR

Uplink C-RAN beamforming must account for finite fronthaul capacity while coordinating transmissions across cells. The paper jointly optimizes beamformers and compression, and reports that a low-complexity separate design performs close to the joint design in the practical SQNR regime.

  • Problem

    The paper addresses joint transmit-beamforming and fronthaul-compression design for uplink C-RAN, where finite fronthaul capacity must be considered in beamforming optimization.

    Source: §I · p. 1, A. Related Work · p. 2

  • Method

    The paper proposes WMMSE-SCA for joint optimization under single-user compression and adapts it to Wyner-Ziv coding, alongside a separate beamforming and per-antenna quantization design.

    Source: §I · p. 2, §VII · p. 12

  • Results

    The separate design performs very close to the optimized joint design in the practical SQNR regime.

    Source: §I · p. 2

  • Takeaways & Limitations

    At high SQNR, the low-complexity separate design is near-optimal for weighted sum-rate maximization.

    Source: §VII · p. 12

Abstract

from arXiv · show

This paper considers the joint fronthaul compression and transmit beamforming design for the uplink cloud radio access network (C-RAN), in which multi-antenna user terminals communicate with a cloud-computing based centralized processor (CP) through multi-antenna base-stations (BSs) serving as relay nodes. A compress-and-forward relaying strategy, named the VMAC scheme, is employed, in which the BSs can either perform single-user compression or Wyner-Ziv coding to quantize the received signals and send the quantization bits to the CP via capacity-limited fronthaul links; the CP performs successive decoding with either successive interference cancellation (SIC) receiver or linear minimum-mean-square-error (MMSE) receiver. Under this setup, this paper investigates the joint optimization of the transmit beamformers at the users and the quantization noise covariance matrices at the BSs for maximizing the network utility. A novel weighted minimum-mean-square-error successive convex approximation (WMMSE-SCA) algorithm is first proposed for maximizing the weighted sum rate under the user transmit power and fronthaul capacity constraints with single-user compression. Assuming a heuristic decompression order, the proposed algorithm is then adapted for optimizing the transmit beamforming and fronthaul compression under Wyner-Ziv coding. This paper also proposes a low-complexity separate design consisting of optimizing transmit beamformers for the Gaussian vector multiple-access channel along with per-antenna quantizers with uniform quantization noise levels across the antennas at each BS. Numerical results show that with optimized beamforming and fronthaul compression, C-RAN can significantly outperform conventional cellular networks. Furthermore, the low complexity separate design already performs very close to the optimized joint design in regime of practical interest.

I. INTRODUCTION

The paper studies uplink C-RAN, where cloud-based coordination can manage inter-cell interference but capacity-limited fronthaul must be accounted for in beamforming and compression. It develops joint and separate designs and compares compression, receiver, and clustering strategies.

I. INTRODUCTION

  • Uplink C-RAN uses cloud coordination across cells to manage inter-cell interference and support network MIMO, which can achieve higher data rates than conventional cellular networks.The paper frames finite fronthaul capacity as an additional constraint on beamforming design.Source: §I · p. 1
  • The paper proposes WMMSE-SCA for joint beamformer and quantization-noise optimization, and a separate design that performs close to the joint design in the practical SQNR regime.The separate design uses Gaussian vector multiple-access channel beamforming and per-antenna quantizers with uniform noise levels at each BS.Source: §I · p. 2

B. Paper Organization and Notation

  • The paper evaluates VMAC with single-user compression and Wyner-Ziv coding, using linear MMSE or SIC receivers, and examines disjoint and user-centric BS clustering.The organization passage also identifies a separate design near-optimal at high SQNR and numerical evaluation on practical multicell and multicluster networks.Source: §I · p. 2, B. Paper Organization and Notation · p. 3

I. INTRODUCTION

  • User-centric clustering significantly outperforms disjoint clustering in uplink C-RAN with optimized beamforming and fronthaul compression.The paper attributes the improvement to effectively eliminating cell edges.Source: §I · p. 2

A. Related Work

  • Earlier fronthaul-compression algorithms optimized quantization-noise covariances with fixed beamformers, whereas this work jointly optimizes both coupled design variables.The paper combines WMMSE beamforming, with successive convex approximation, to find a local optimum.Source: A. Related Work · pp. 2–3

I. INTRODUCTION

  • SIC improves performance more than Wyner-Ziv coding over single-user compression, while single-user compression with SIC captures most of C-RAN's gain.With limited fronthaul, Wyner-Ziv coding can further improve on single-user compression.Source: §I · p. 2, A. Related Work · p. 3

A. System Model

The system model has multi-antenna users send signals through multi-antenna BS relays to a CP over capacity-limited fronthaul. Under VMAC, BSs compress their received signals and the CP successively decodes compression codewords and user messages.

A. System Model

  • The uplink has K multi-antenna users communicating with a CP through L multi-antenna BS relays, with capacity-limited fronthaul links.Each user and BS has M and N antennas, respectively; the fronthaul link to BS ℓ has capacity Cℓ.Source: A. System Model · p. 3
  • VMAC uses single-user compression or Wyner-Ziv coding at the BSs, followed by successive decoding of quantization codewords and then user messages at the CP.Single-user compression uses separate vector quantizers, whereas Wyner-Ziv coding exploits correlation across BS signals.Source: A. System Model · p. 3
  • The model assumes quasi-static channels and perfect CSI at the BSs, with CSI and design feedback overhead ignored because it can be amortized over channel coherence time.The CP selects quantization-noise covariances and user beamformers and feeds them back to the BSs and users.Source: A. System Model · p. 3
  • Each user sends d Gaussian-codebook data streams through a linear transmit beamformer, subject to a per-user power constraint.The transmitted signal is x_k = V_k s_k, with covariance V_k V_k†.Source: A. System Model · p. 3
  • The received signal at each BS is modeled with additive Gaussian noise and Gaussian quantization noise having covariance Qℓ.The quantized signal follows a Gaussian quantization test channel; a covariance Qℓ can equivalently be implemented through eigenvector beamforming followed by independent quantization.Source: A. System Model · pp. 3–4

B. Achievable Rate of the VMAC scheme

  • The CP's achievable user rates are specified for linear MMSE reception and for SIC, while BS compression rates must satisfy the fronthaul capacities.The passages give information-theoretic fronthaul constraints for single-user compression and Wyner-Ziv coding, including Gaussian-input expressions.Source: B. Achievable Rate of the VMAC scheme · p. 4

III. JOINT BEAMFORMING AND COMPRESSION DESIGN UNDER SINGLE-USER COMPRESSION

The section jointly optimizes user beamformers and BS quantization-noise covariances to maximize weighted sum rate under power and fronthaul constraints. It develops WMMSE-SCA by reformulating the objective and approximating the nonconvex objective and fronthaul constraints with convex forms.

A. Weighted Sum Rate Maximization

  • The design maximizes weighted sum rate by jointly choosing beamforming vectors and quantization-noise covariances, which together determine user rates and fronthaul use.User weights represent priorities, and scheduling is assumed to occur before solving the optimization problem.Source: A. Weighted Sum Rate Maximization · p. 4

B. The WMMSE-SCA Algorithm

  • WMMSE-SCA reformulates the weighted sum-rate objective and linearizes the objective and fronthaul rate expressions to form a convex approximation solved successively.The reformulation uses the WMMSE representation with the CP's linear receiver and weight matrix.Source: B. The WMMSE-SCA Algorithm · p. 5
  • The fronthaul constraints are replaced by convex approximations that remain feasible for the original constraints and are equivalent at the specified equality condition.The approximation follows from a log-determinant inequality.Source: B. The WMMSE-SCA Algorithm · p. 5
  • A quadratic regularization term makes the reformulated problem strictly convex in each optimization variable, enabling convex updates for the beamformers and quantization covariances.With other variables fixed, the receiver, weight, and auxiliary variables have specified optimal updates; the beamformer and covariance update is convex.Source: B. The WMMSE-SCA Algorithm · p. 5

C. Convergence and Complexity Analysis

The WMMSE-SCA algorithm produces nondecreasing objective values and converges to a stationary point. Its computational burden is dominated by the convex joint update of beamformers and quantization covariances.

C. Convergence and Complexity Analysis

  • The WMMSE-SCA objective values are nondecreasing, and the algorithm converges to a stationary point of the weighted sum-rate problem.The convergence guarantee holds from any initial point.Source: C. Convergence and Complexity Analysis · p. 6
  • The stationary-point argument can use block coordinate descent theory, while the paper's proof instead applies a successive convex approximation convergence result.The regularization term makes the reformulated objective strongly convex in the beamformers and quantization covariances, guaranteeing Algorithm 1's convergence.Source: C. Convergence and Complexity Analysis · p. 6
  • For the stated network-size ordering, the dominant computational cost is the convex joint optimization of beamformers and quantization covariances in Algorithm 1's Step 5.The overall complexity is tied to the total number of iterations to convergence.Source: C. Convergence and Complexity Analysis · p. 6

IV. JOINT BEAMFORMING AND COMPRESSION OPTIMIZATION UNDER WYNER-ZIV CODING

The section extends joint beamforming and fronthaul-compression optimization to Wyner-Ziv coding, which uses correlation among BS observations. It adapts WMMSE-SCA under a heuristic decompression order because exhaustive search over orders is impractical for large networks.

IV. Joint Beamforming and Compression Optimization under Wyner-Ziv Coding

  • Wyner-Ziv coding exploits correlation among BS observations of the same transmitted signals, motivating joint optimization of beamformers and quantization-noise covariances.The section contrasts this with single-user compression, which processes compression and decompression independently across BSs.Source: §IV · p. 6
  • Exhaustively finding the best decompression order requires searching L! orders, which is impractical for large L.The proposed heuristic decompresses first at BSs with larger fronthaul capacity or lower received signal power, aiming to keep quantization-noise levels small.Source: §IV · p. 6
  • The weighted sum-rate problem under Wyner-Ziv coding is formulated for a fixed heuristic decompression order and includes user weights and receiver-dependent rate terms.The receiver may be linear MMSE or SIC.Source: §IV · p. 6
  • The Wyner-Ziv problem is nonconvex, so the method reformulates it into a form suited to successive convex approximation and applies the WMMSE-SCA procedure.The reformulation addresses a fronthaul constraint containing three log-determinant functions.Source: §IV · p. 7

V. SEPARATE DESIGN OF BEAMFORMING AND COMPRESSION

To reduce the computational burden of joint optimization, the section proposes a separate design that is approximately optimal in the high-SQNR regime under appropriate conditions.

V. Separate Design of Beamforming and Compression

  • The design targets practical use where joint WMMSE-SCA can be computationally intensive, especially under fast fading or frequent changes in scheduled users.The section presents the separate approach as an efficient design for practical uplink C-RAN systems.Source: §V · p. 7
  • The separate design is approximately optimal in the high-SQNR regime under appropriate conditions.It uses user-side beamforming optimized for the Gaussian vector multiple-access channel and per-antenna quantizers with uniform noise levels at each BS.Source: §V · p. 7
  • The design separates transmit beamforming for the Gaussian vector multiple-access channel from per-antenna quantization with uniform noise levels across each BS's antennas.The stated goal is to derive near-optimal beamformers and quantization covariances in the high-SQNR regime.Source: §V · p. 7

A. Quantization Noise Design Under High SQNR

Under high SQNR, uniform quantization-noise levels across each BS’s antennas are nearly optimal, subject to full receive-dimension occupancy and high-rate transmission. The analysis derives this design result from the optimization’s KKT condition.

A. Quantization Noise Design Under High SQNR

  • For the stated high-SQNR regime, the received covariance must be full rank and received signal strength must greatly exceed combined quantization and background noise.The passage identifies K_d = LN as a case meeting the full-rank condition.Source: A. Quantization Noise Design Under… · pp. 7–8
  • The analysis justifies uniform noise levels through the KKT condition for the sum-rate optimization, with fronthaul and transmit-power constraints represented by their dual variables.Source: A. Quantization Noise Design Under… · pp. 7–8
  • Uniform quantization-noise levels across antennas at each BS are nearly optimal under high SQNR, though levels may differ across BSs with background noise and fronthaul constraints.The same conclusion is reported for Wyner-Ziv coding.Source: A. Quantization Noise Design Under… · p. 8
  • The high-SQNR condition requires at least as many scheduled user streams as receive dimensions, with every stream transmitting at high rate.Source: A. Quantization Noise Design Under… · p. 8

B. Beamforming Design Under High SQNR

At high SQNR, the proposed separate beamforming design uses effective-channel eigenmodes and water-filling while ignoring fronthaul limits in beamformer selection. The design obtains approximate beamformers from a rank-relaxed covariance optimization.

B. Beamforming Design Under High SQNR

  • The design whitens combined quantization, background noise, and interference, then selects strong effective-channel eigenmodes and iteratively water-fills power among users.Source: B. Beamforming Design Under High SQNR · p. 8
  • At high SQNR, optimal quantization-noise covariance matrices are proportional to background-noise covariance matrices; equal total streams and system degrees of freedom reasonably contain multiuser interference.Source: B. Beamforming Design Under High SQNR · p. 8
  • The separate design selects user beamformers by optimizing a Gaussian vector multiple-access channel while ignoring fronthaul-capacity limits.Source: B. Beamforming Design Under High SQNR · p. 8
  • Removing covariance rank constraints makes the weighted sum-rate problem convex; the design then uses the relaxed solution’s leading eigenvalues and eigenvectors to form approximate beamformers.Source: B. Beamforming Design Under High SQNR · p. 8
  • With a linear MMSE receiver, dropping rank constraints does not yield a convex problem, so the paper instead applies WMMSE to optimize beamforming vectors.Source: B. Beamforming Design Under High SQNR · p. 8

C. Separate Beamforming and Compression Design

The separate design combines Gaussian-vector-MAC beamforming with per-antenna scalar quantization and uniform noise levels at each BS. It reduces computation, performs well at high SQNR, and trades performance against complexity outside that regime.

C. Separate Beamforming and Compression Design

  • The separate design pairs Gaussian-vector-MAC beamforming with per-antenna scalar quantizers and uniform quantization-noise levels across each BS’s antennas.Source: C. Separate Beamforming and… · p. 8
  • The beamforming procedure solves the Gaussian-vector-MAC problem after removing rank constraints, then uses the relaxed covariance solution’s eigenstructure to construct beamformers.The algorithm specifies the rank-relaxed solve and eigenvalue decomposition.Source: C. Separate Beamforming and… · p. 9
  • For single-user compression, bisection selects each BS’s quantization parameter independently to meet fronthaul capacity; Wyner-Ziv coding selects parameters successively with earlier values fixed.Source: C. Separate Beamforming and… · p. 9
  • The separate design performs very well at high SQNR, while other regimes trade complexity against performance; when high SQNR fails, beamforming should account for fronthaul limits.The joint design is fronthaul-aware and quantizes each BS’s received vector, unlike the separate design’s per-antenna quantization.Source: C. Separate Beamforming and… · p. 9
  • The separate design is significantly less computationally complex, using one convex optimization and bisection instead of iteratively solving a series of convex problems.Source: C. Separate Beamforming and… · p. 9

VI. SIMULATION RESULTS

The simulations compare joint WMMSE-SCA and separate designs across receiver and compression choices in a 19-cell network. SIC gains are substantial, while Wyner-Ziv gains depend on fronthaul capacity and the separate design approaches WMMSE-SCA as capacity increases.

A. Single-Cluster Network

B. Multi-Cluster Network

In the 65-cell multi-cluster evaluation, user-centric clustering outperforms disjoint clustering, and larger clusters and more fronthaul improve rates. The conclusion also reports that SIC accounts for most performance gains and user-centric clustering saves fronthaul.

B. Multi-Cluster Network

  • The 65-cell evaluation uses single-user compression and a linear MMSE receiver because overlapping clusters complicate Wyner-Ziv coding and SIC under fronthaul constraints.The setup includes disjoint and user-centric clustering strategies.Source: B. Multi-Cluster Network · p. 11
  • For disjoint clustering, increasing cluster size from 2 to 6 improves the 50-percentile rate by 60%, and replacing it with user-centric clustering doubles that gain.The comparisons use WMMSE-SCA with single-user compression and linear MMSE reception.Source: B. Multi-Cluster Network · p. 11
  • At cluster size 6 and 110 Mbps per-cell sum rate, user-centric clustering needs 220 Mbps fronthaul versus 360 Mbps for disjoint clustering.The paper describes this as more than 60% improvement on fronthaul requirement.Source: B. Multi-Cluster Network · p. 12, Figure 10 · p. 12
  • User-centric clustering achieves higher per-cell sum rate than disjoint clustering even with 120 Mbps less fronthaul, with the advantage growing at larger cluster sizes.The stated comparison is disjoint clustering at 360 Mbps versus user-centric clustering at 240 Mbps.Source: B. Multi-Cluster Network · p. 12, Figure 11 · p. 12

VII. CONCLUSION

  • The conclusion reports that most performance gains come from SIC at the central receiver, while user-centric clustering significantly outperforms disjoint clustering in fronthaul capacity saving.Source: §VII · p. 12

APPENDIX A PROOF OF THEOREM 1

The appendix establishes convergence of the proposed WMMSE-SCA algorithm by applying a general SCA convergence result. Its assumptions include regularity and strong convexity properties, with a regularization term supporting strong convexity.

APPENDIX A PROOF OF THEOREM 1

  • A regularization term plays a key role in making the convex approximation uniformly strongly convex over the design variables.Source: Appendix A · p. 13
  • The convergence argument requires the listed continuity, differentiability, convexity, majorization, and regularity conditions for the objective and constraints.Source: Appendix A · p. 13
  • Every limit point generated by WMMSE-SCA is a stationary point of problem (11), by the stated SCA convergence result.The proof invokes the convergence result of [32, Theorem 2].Source: Appendix A · p. 13

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