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Optimized Backhaul Compression for Uplink Cloud Radio Access Network

Yuhan Zhou, Wei Yu

arXiv:1304.7509v3cs.IT

TL;DR

Finite backhaul constrains uplink C-RAN performance, motivating efficient compression and quantization design. The paper develops VMAC schemes with Wyner-Ziv or single-user compression, optimizes quantization noise, and shows near-optimal settings with constant-gap capacity results. Simulations indicate significant cellular-network gains from optimized or approximate quantization.

  • Problem

    The paper addresses how to optimize quantization noise levels and backhaul use for uplink C-RAN with finite-capacity backhaul.

  • Method

    It combines compress-and-forward VMAC schemes with alternating convex optimization for VMAC-WZ and backhaul-capacity reformulation for VMAC-SU.

  • Results

    Quantization noise proportional to background noise is near optimal at high SQNR; VMAC-WZ has a constant-gap sum-capacity result, while VMAC-SU has one under diagonal dominance.

  • Takeaways & Limitations

    These results yield efficient quantization-noise and backhaul-allocation algorithms, and simulations show significant performance gains in multicell and heterogeneous networks.

Abstract

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This paper studies the uplink of a cloud radio access network (C-RAN) where the cell sites are connected to a cloud-computing-based central processor (CP) with noiseless backhaul links with finite capacities. We employ a simple compress-and-forward scheme in which the base-stations(BSs) quantize the received signals and send the quantized signals to the CP using either distributed Wyner-Ziv coding or single-user compression. The CP decodes the quantization codewords first, then decodes the user messages as if the remote users and the cloud center form a virtual multiple-access channel (VMAC). This paper formulates the problem of optimizing the quantization noise levels for weighted sum rate maximization under a sum backhaul capacity constraint. We propose an alternating convex optimization approach to find a local optimum solution to the problem efficiently, and more importantly, establish that setting the quantization noise levels to be proportional to the background noise levels is near optimal for sum-rate maximization when the signal-to-quantization-noise ratio (SQNR) is high. In addition, with Wyner-Ziv coding, the approximate quantization noise level is shown to achieve the sum-capacity of the uplink C-RAN model to within a constant gap. With single-user compression, a similar constant-gap result is obtained under a diagonal dominant channel condition. These results lead to an efficient algorithm for allocating the backhaul capacities in C-RAN. The performance of the proposed scheme is evaluated for practical multicell and heterogeneous networks. It is shown that multicell processing with optimized quantization noise levels across the BSs can significantly improve the performance of wireless cellular networks.

I. INTRODUCTION

The paper targets uplink C-RAN optimization under finite backhaul, using lower-complexity compression and decoding schemes while optimizing quantization noise levels. It shows that noise levels proportional to background noise can provide near-optimal performance and constant-gap capacity results.

  • Finite backhaul limits uplink C-RAN, where cloud processing can coordinate multiple cells and mitigate intercell interference.
  • The paper uses compress-and-forward with Wyner-Ziv or single-user compression, followed by successive decoding of quantization codewords and user messages.
  • The main optimization target is jointly selecting base-station quantization noise levels for weighted sum-rate maximization under shared backhaul constraints.
  • An alternating convex optimization algorithm optimizes VMAC-WZ quantization levels, while VMAC-SU is reformulated as a backhaul-capacity allocation problem.
  • In high SQNR, quantization noise proportional to background noise is near optimal; VMAC-WZ achieves sum capacity within a constant gap, and VMAC-SU does so under diagonal dominance.
  • Simulations in multicell and heterogeneous networks show significant gains, with approximate quantization settings realizing much of the optimized performance.

B. The VMAC-WZ Scheme

The VMAC schemes compress correlated base-station observations before cloud decoding. Wyner-Ziv compression exploits this correlation, while single-user compression treats each base station separately under a shared backhaul constraint.

  • Wyner-Ziv compression exploits statistical correlation among base-station observations to use limited backhaul more efficiently than per-link single-user compression.
  • The VMAC-WZ achievable region is defined under a sum-backhaul constraint using quantization-noise covariance Λq and channel-dependent mutual-information bounds.
  • The compression test channel adds independent complex Gaussian quantization noise with level qi to each received signal.
  • Single-user compression omits cross-base-station statistical correlation and quantizes each received signal independently with a vector quantizer.
  • For VMAC-SU, the achievable rate uses I(X(S); Ŷ), while the sum-backhaul constraint is the sum of individual compression costs I(Yi; Ŷi).

III. QUANTIZATION NOISE LEVEL OPTIMIZATION FOR VMAC-WZ

This section formulates weighted sum-rate maximization over quantization noise levels under a sum backhaul constraint and develops an alternating convex optimization approach for VMAC-WZ. The method converges to a stationary point while jointly optimizing quantization levels across BSs.

  • Problem formulation: The VMAC-WZ problem maximizes weighted sum rate over quantization noise levels under a sum backhaul capacity constraint.The quantization levels are represented by the diagonal matrix Λq = diag(qi).
  • Problem formulation: The original objective is convex rather than concave in Λq, making global optimization challenging.This nonconcavity motivates reformulation rather than direct maximization.
  • Comparison: Joint optimization across BSs provides better overall performance than per-BS optimization, while the proposed approach is faster than a related gradient projection method.The comparison attributes the speed difference to joint optimization versus per-BS block coordinate gradient descent.
  • Alternating convex optimization: Reformulating the objective as a double maximization makes the problem convex in either Λq or Σ when the other variable is fixed.This coordinate-wise convexity enables iterative coordinate ascent.
  • Alternating convex optimization: The ACO algorithm alternates convex updates over Λq and Σ, repeatedly optimizing one variable while fixing the other.The algorithm initializes both variables, alternates updates, and repeats until convergence.
  • Convergence and scope: The ACO algorithm produces a nondecreasing objective sequence and converges to a stationary point of the weighted sum-rate problem.The approach is also applicable to MIMO systems with quantization covariance matrices.

C. Optimal Quantization Noise Level at High SQNR

This section analyzes quantization-noise optimization in the high-SQNR regime, where direct ACO implementation can be computationally intensive. It shows that quantization noise proportional to background noise is approximately optimal for sum-rate maximization.

  • Motivation: ACO can be computationally intensive when fading or scheduled users change frequently, motivating a simpler high-SQNR characterization.The approximation targets practical quantization-level selection under changing channel and scheduling conditions.
  • Optimality analysis: The high-SQNR sum-rate problem is nonconvex, so its KKT condition provides a necessary optimality condition rather than a global-optimality guarantee.The derivation uses a Lagrangian with a dual variable for the sum backhaul constraint.
  • High-SQNR result: Under high SQNR, the optimal quantization noise level is proportional to the background noise level.The proportionality factor is selected to satisfy the sum backhaul constraint.
  • High-SQNR result: As the sum backhaul capacity becomes more constrained, the optimal quantization noise level increases; λ = 0 corresponds to infinite backhaul.The parameter λ lies in [0, 1).

D. Sum Capacity to Within a Constant Gap

This section establishes that background-noise-proportional quantization achieves the uplink C-RAN sum capacity within a constant gap. For VMAC-WZ, the gap is one bit per BS per channel use, and the resulting rule supports a simple bisection-based algorithm.

  • Constant-gap result: Background-noise-proportional quantization achieves uplink C-RAN sum capacity within a constant gap independent of the channel matrix and SNRs.The gap depends on the number of BSs.
  • VMAC-WZ guarantee: 1 bit per BS per channel use is the VMAC-WZ gap to sum capacity under a sum backhaul constraint.The result applies when quantization noise levels are proportional to background noise levels.
  • Proof interpretation: The constant-gap proof compares the achievable rate with a cut-set outer bound and retains the guarantee when backhaul capacity is small after constant scaling.The small-backhaul case still has at most a one-bit-per-BS gap.
  • Implementation implication: The proportional-noise rule is advantageous because it does not require adaptation to transmit power, channel matrix, or user schedule.This invariance holds from both the high-SQNR and constant-gap perspectives.
  • Approximate algorithm: A bisection search chooses α so that CWZ(α) = C, then sets qi = ασi.The constraint is enforced with equality because CWZ(α) is monotonic in α.
  • Evaluation: Algorithm 2 performs very close to the optimized scheme in practical channel scenarios.The approximate algorithm uses proportional quantization noise rather than full joint optimization.

IV. OPTIMAL BACKHAUL ALLOCATION FOR VMAC-SU

This section addresses VMAC-SU optimization under a sum backhaul constraint by reformulating quantization-noise selection as backhaul-rate allocation. At high SQNR, the resulting quantization levels are again proportional to background noise levels.

  • Problem reformulation: VMAC-SU weighted sum-rate maximization can be reformulated in terms of rates allocated to independent BS backhaul links.The reformulation exploits the independence of compression at each BS.
  • Problem reformulation: The transformed problem has a polyhedral feasible set with linear constraints, making it easier to solve than direct quantization-noise optimization.A dual method and outer bisection can be used to find a local optimum of the Lagrangian.
  • High-SQNR analysis: The high-SQNR analysis derives approximately optimal backhaul rates and corresponding quantization noise using KKT conditions.The derivation introduces multipliers for nonnegative link rates and the sum backhaul constraint.
  • High-SQNR result: For VMAC-SU, quantization noise proportional to background noise is near optimal for sum-rate maximization at high SQNR.This parallels the VMAC-WZ result.
  • Interpretation: The proportionality follows because multicell decoding nulls intercell interference, leaving combined quantization and background noise as the sum-rate limitation.Under this interpretation, optimal quantization levels depend only on background noise levels.

C. Sum Capacity of Diagonally Dominant Channels

For single-user compression, the paper studies when proportional quantization noise can preserve uplink C-RAN sum capacity under diagonally dominant received covariance. It also notes a stronger near-optimality guarantee when backhaul capacity is large.

  • Sum-capacity guarantee: Proportional quantization noise achieves VMAC sum capacity within a constant gap under the stated covariance condition.The result applies to Wyner-Ziv coding when the received signal covariance supports the paper’s constant-gap argument.
  • Diagonal dominance: The received covariance is often diagonally dominant because users typically associate with their strongest base station.
  • Diagonal dominance: The paper defines κ-strict diagonal dominance through a fixed κ > 1 and matrix-entry inequalities.The definition specifies the structural condition used for the single-user compression theorem.
  • Sum-capacity guarantee: Under κ-strict diagonal dominance, VMAC-SU achieves uplink C-RAN sum capacity within the theorem’s stated per-BS constant gap.The supplied theorem statement identifies the covariance condition and the gap’s units, but not its numerical value.
  • Large-backhaul regime: When C is large, background-level quantization noise gives at most 1 bit gap per channel use per BS to sum capacity.In this regime, VMAC-SU is approximately optimal for the entire uplink C-RAN capacity region.

D. Backhaul Allocation for Heterogeneous Networks

For heterogeneous C-RANs, the paper uses proportional quantization noise within each BS tier and a bisection-based allocation approach, while accounting for tier-specific or coupled backhaul constraints. Simulations evaluate these schemes in multicell and heterogeneous settings, including comparisons of VMAC-WZ and VMAC-SU.

  • Heterogeneous-network allocation: Under high SQNR, quantization noise is near optimal when proportional to background noise, with different proportionality constants allowed across tiers.The paper applies this result to macro- and pico-BSs in heterogeneous networks.
  • Heterogeneous-network allocation: A simultaneous bisection algorithm independently optimizes quantization noise levels, or equivalent backhaul capacities, for each BS tier.The approach exploits VMAC-SU’s tier-wise independence under separate macro- and pico-tier constraints.
  • Coupled backhaul structures: When pico-BSs connect through macro-BSs rather than directly to the CP, macro- and pico-BS backhaul constraints become coupled.For fixed tier backhaul allocations, the approximate algorithm can still optimize the quantization noise levels for each tier.
  • Multicell evaluation: In the multicell evaluation, VMAC-WZ significantly outperforms the baseline, and proportional quantization noise is approximately optimal, especially at larger backhaul capacities.The comparison uses 120Mbps and 270Mbps per macro-cell, with optimized and approximately optimized quantization choices.
  • Multicell evaluation: VMAC-WZ provides significant gains over per-BS successive interference cancellation, while VMAC-SU also benefits from optimized quantization noise levels.The experiments compare user-rate distributions and per-cell sum rates across VMAC variants and allocation strategies.

B. Multi-Tier Heterogeneous Network

The VMAC-SU scheme is evaluated in a two-tier heterogeneous network, where optimized quantization noise levels substantially improve user-rate performance over a baseline and uniform backhaul allocation.

  • Network setup: The evaluation uses 7 wrapped macro-cells, 3 sectors per cell, 3 randomly placed pico-BSs per sector, and 20 users per sector.Users connect to the macro or pico BS with the highest received SNR; the topology contains 7 cells, 3 sectors per cell, and 3 pico-BSs per sector.
  • Network setup: The VMAC-SU evaluation applies 189 Mbps of aggregate backhaul for the 3 macro-BSs and 81 Mbps for the 9 pico-BSs in each cluster.The cluster comprises 3 macro-BSs and 9 pico-BSs within each 3-sector macrocell.
  • Performance: The C-RAN architecture more than doubles the 50-percentile user rate relative to the baseline scheme.The comparison is shown through cumulative distribution plots of user rates.
  • Performance: Optimizing quantization noise levels is important because naive uniform backhaul allocation achieves only half of the potential C-RAN gain.Setting quantization noise proportional to background noise is approximately optimal in this evaluation.
  • Performance: Across the evaluated systems, optimized quantization levels or equivalently optimized backhaul allocations maximize the improvement, while proportional levels remain close to optimal over practical SQNR values.The paper reports this conclusion for its numerical evaluations of multicell and heterogeneous wireless networks.

APPENDIX A PROOF OF THEOREM 2

Theorem 2 is proved by comparing the VMAC-WZ achievable sum rate with a cut-set-like upper bound and choosing quantization noise proportional to background noise. This yields a gap below one bit per BS per channel use.

  • Proof strategy: The proof compares the VMAC-WZ achievable rate with a cut-set-like sum-capacity upper bound consisting of user-to-BS and backhaul cuts.The first term represents the cut from users to BSs, and the second represents the cut across backhaul links.
  • Quantization choice: The quantization noise levels are set proportional to the background noise levels, with α selected according to the backhaul constraint.The proof considers the choice α = 1, corresponding to quantization noise at the background-noise level.
  • Achievability: With α = 1, the sum backhaul constraint is satisfied and the rate R_sum = I(X; Ŷ) is achievable.The proof uses C = I(Y; Ŷ) as an upper bound in the relevant case.
  • Gap bound: The resulting VMAC-WZ gap to sum capacity is always less than 1 bit per BS per channel use.The bound follows by combining the cases in the proof.

APPENDIX B PROOF OF THEOREM 3

Theorem 3 applies the same comparison strategy to VMAC-SU, setting quantization noise proportional to background noise and using diagonal dominance to bound the gap to sum capacity.

  • Diagonal-dominance condition: Under κ-strict diagonal dominance, the matrix used in the gap analysis remains κ-strictly diagonally dominant, enabling a determinant-based bound.The proof invokes Lemma 2 after establishing diagonal dominance of the relevant matrix.
  • Proof strategy: The VMAC-SU proof compares its achievable rate with a cut-set-like upper bound after setting q_i = ασ_i^2.Here α is a positive constant depending on the sum backhaul capacity C.
  • Case analysis: The proof separates the analysis into cases according to the available sum backhaul capacity.One case begins when C is at least log |diag(HKXH^H) + 2diag(σ_i^2)|.
  • Gap bound: The resulting VMAC-SU gap to sum capacity is less than 1 + log κ.This bound holds when quantization noise levels are proportional to the background noise levels.
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