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Joint Precoding and Multivariate Backhaul Compression for the Downlink of Cloud Radio Access Networks

Seok-Hwan Park, Osvaldo Simeone, Onur Sahin, Shlomo Shamai

arXiv:1304.3179v1cs.IT

TL;DR

Cloud radio access networks must jointly address downlink precoding and compression over finite-capacity backhaul links, where BS signals are conventionally compressed independently. The paper optimizes precoding and correlated quantization noise through multivariate compression, and reports that the joint strategy outperforms independent compression and separate design, especially under large power or inter-cell gains and tight backhaul constraints.

  • Problem

    Finite-capacity backhaul links constrain cloud radio access networks, while conventional approaches independently compress signals for different BSs instead of controlling their joint quantization-noise effect.

  • Method

    The paper jointly optimizes the precoding matrix and quantization-noise correlation under power and backhaul constraints, using an iterative MM algorithm and a successive MMSE-estimation/per-BS-compression architecture.

  • Results

    The proposed multivariate-compression joint strategy outperforms independent compression and separate precoding/compression, especially with large transmit power or inter-cell channel gain and limited backhaul.

  • Takeaways & Limitations

    Correlating quantization noises across BSs provides a downlink compression strategy whose reported numerical performance exceeds conventional independent-compression and separate-design approaches.

Abstract

from arXiv · show

This work studies the joint design of precoding and backhaul compression strategies for the downlink of cloud radio access networks. In these systems, a central encoder is connected to multiple multi-antenna base stations (BSs) via finite-capacity backhaul links. At the central encoder, precoding is followed by compression in order to produce the rate-limited bit streams delivered to each BS over the corresponding backhaul link. In current state-of-the-art approaches, the signals intended for different BSs are compressed independently. In contrast, this work proposes to leverage joint compression, also referred to as multivariate compression, of the signals of different BSs in order to better control the effect of the additive quantization noises at the mobile stations (MSs). The problem of maximizing the weighted sum-rate with respect to both the precoding matrix and the joint correlation matrix of the quantization noises is formulated subject to power and backhaul capacity constraints. An iterative algorithm is proposed that achieves a stationary point of the problem. Moreover, in order to enable the practical implementation of multivariate compression across BSs, a novel architecture is proposed based on successive steps of minimum mean-squared error (MMSE) estimation and per-BS compression. Robust design with respect to imperfect channel state information is also discussed. From numerical results, it is confirmed that the proposed joint precoding and compression strategy outperforms conventional approaches based on the separate design of precoding and compression or independent compression across the BSs.

I. INTRODUCTION

Cloud radio access networks centralize BS encoding and rely on finite-capacity backhaul links, creating compression constraints. This work proposes jointly designed precoding and multivariate compression to control quantization noise and improve downlink performance.

  • Cloud radio access networks migrate BS encoding and decoding to a central unit, with BSs acting as soft relays over baseband-only backhaul links.
  • Finite-capacity backhaul links are a major implementation impairment, especially for pico/femto-BSs and BSs using wireless backhaul.
  • In the downlink, the central encoder jointly encodes MS messages, then independently compresses BS-specific baseband signals before backhaul delivery.
  • A. Contributions: The proposed approach jointly designs precoding and compression while correlating quantization noises across BSs to limit their effect at MSs.
  • A. Contributions: Multivariate compression is implemented through successive MMSE estimation and per-BS compression rather than joint compression across all BSs.
  • A. Contributions: The formulation maximizes weighted sum-rate under power and backhaul constraints, with an iterative algorithm reaching a stationary point and numerical advantages over conventional designs.

II. SYSTEM MODEL

The system has a central encoder serving multiple multi-antenna MSs through distributed multi-antenna BSs over finite-capacity backhaul links. Precoding controls interference, while compression creates the rate-limited BS signals.

  • A central encoder communicates with multiple MSs through distributed BSs that may be macro-BSs or pico/femto-BSs, with MSs distributed across cells.
  • Each BS connects to the central encoder through a digital backhaul link with finite capacity C_i bits per channel use.
  • Each MS message is separately channel-encoded into a Gaussian-coded symbol vector, and the central encoder processes all symbol vectors through precoding and compression.
  • Precoding controls interference among data streams, whereas compression produces the rate-limited bit streams delivered to the BSs.
  • Each BS reconstructs a multi-antenna baseband vector for transmission, while the channel model assumes flat fading and additive Gaussian noise.
  • The central encoder is assumed to know global channel matrices, each MS knows its own channel, and imperfect central-encoder CSI is treated separately.

III. PRELIMINARIES

The preliminaries contrast conventional single-source compression with multivariate compression, where multiple correlated reconstructions are selected jointly under coupled rate conditions.

  • Conventional compression: Conventional compression maps an i.i.d. source sequence to one reconstruction sequence using a conditional test channel and a rate-R codebook.
  • Conventional compression: The compressor selects a codeword jointly typical with the source, and successful recovery becomes arbitrarily likely as blocklength grows under the required rate condition.
  • Multivariate compression: Multivariate compression seeks multiple codewords jointly typical with the source according to a joint conditional test channel.
  • Multivariate compression: Each reconstruction is indexed separately and sent to its corresponding decompression unit, while codeword selection is performed jointly at the compression unit.
  • Multivariate compression: A sufficient condition guarantees jointly typical codeword tuples with probability arbitrarily close to one for sufficiently large blocklength and suitable codebook rates.
  • Multivariate compression: The multivariate rate inequalities are coupled because the reconstructions must exhibit the joint correlation specified by the test channel.

IV. PROPOSED APPROACH AND PROBLEM DEFINITION

The paper proposes multivariate compression and joint precoding-compression for cloud-RAN downlinks, then formulates the corresponding design under power and backhaul constraints.

  • IV. PROPOSED APPROACH AND PROBLEM DEFINITION: The proposed approach uses multivariate compression for backhaul signals and jointly designs the precoding matrix with compression.The section presents these as the core components of the proposed strategy.
  • A. Encoding Operation at the Central Encoder: Precoding linearly maps the encoded signals into per-BS precoded data streams for interference management across mobile stations and among data streams.The precoding matrix A is partitioned into components corresponding to the mobile stations.
  • A. Encoding Operation at the Central Encoder: Each precoded BS stream is compressed before transmission over its finite-capacity backhaul link, and each BS forwards its compressed signal.The BSs need quantization codebooks but need not know the channel codebooks or central precoding matrix.
  • A. Encoding Operation at the Central Encoder: Compression is modeled with Gaussian test channels whose joint quantization-noise covariance contains diagonal and cross-BS blocks.The off-diagonal blocks define correlations between quantization noises associated with different BSs.
  • A. Encoding Operation at the Central Encoder: Correlated compression is proposed instead of independent BS compression to better control the additive quantization noise affecting mobile stations.Independent compression corresponds to zero cross-BS covariance blocks.
  • A. Encoding Operation at the Central Encoder: The precoding matrix and quantization covariance may be designed separately or jointly, with conventional ZF or MMSE precoding available as separate-design choices.DPC is also considered as a possible nonlinear precoding method, with its scaling based on quantization noise levels.

B. Multivariate Backhaul Compression

Multivariate compression introduces correlations among BS quantization noises and requires joint compression of their precoded signals, with additional backhaul constraints.

  • B. Multivariate Backhaul Compression: Reliable compression assumes that each BS is informed about the quantization codebook used by the central encoder.This assumption is stated for the backhaul communication model.
  • B. Multivariate Backhaul Compression: The proposed scheme introduces nonzero cross-BS quantization-noise covariance to control quantization noise at the mobile stations.This differs from conventional independent compression, which sets cross-BS covariance blocks to zero.
  • B. Multivariate Backhaul Compression: Correlated quantization noises require joint rather than independent compression of the precoded signals from different BSs.This family of strategies is referred to as multivariate compression.
  • B. Multivariate Backhaul Compression: The Gaussian test channel provides sufficient conditions for reliably delivering all compressed signals to the BSs over the backhaul links.The conditions apply across subsets of BSs.
  • B. Multivariate Backhaul Compression: Compared with uncorrelated compression, correlated quantization noises impose additional constraints on the backhaul-link capacities.The comparison follows from the respective multivariate and independent-compression conditions.

C. Weighted Sum-Rate Maximization

The weighted sum-rate problem jointly optimizes precoding and correlated quantization-noise covariance under backhaul and power constraints. Multivariate compression can be implemented through successive MMSE estimation and per-BS compression, though corner-point conditions may restrict feasible backhaul allocations.

  • The formulation optimizes the precoding matrix A and compression-noise covariance Ω subject to multivariate-compression backhaul and transmit-power constraints.
  • The optimization is non-convex in both the weighted sum-rate objective and the multivariate-compression backhaul functions.
  • Multivariate compression coordinates quantization noises across BSs, while joint compression of all precoded signals would otherwise be required in principle.
  • Successive Estimation-Compression Architecture: The proposed architecture successively estimates and compresses each BS signal in a fixed order, using MMSE estimation followed by per-BS compression with independent quantization noise in each test channel.
  • Successive Estimation-Compression Architecture: The backhaul-capacity region is a contrapolymatroid whose corner points correspond to BS permutations and have equality on nested subsets.
  • Successive Estimation-Compression Architecture: For any correlation matrix Ω, the successive architecture realizes multivariate compression at a corner point, but its corner-point conditions are more restrictive than the full backhaul region.

V. JOINT DESIGN OF PRECODING AND COMPRESSION

The paper jointly optimizes precoding and compression, while also considering a suboptimal alternative that fixes precoding and optimizes only the compression covariance.

  • The proposed design jointly optimizes the precoding matrix A and compression covariance Ω by solving the weighted sum-rate problem.
  • A suboptimal strategy fixes precoding using ZF, MMSE, or weighted sum-rate-maximizing techniques while optimizing only Ω.

A. MM Algorithm

The joint design is addressed by reformulating the problem in covariance variables and applying a majorization-minimization procedure to its difference-of-convex structure. The resulting algorithm converges to a stationary point and can use successive compression when corner-point conditions hold.

  • MM Algorithm: The optimization is non-convex and is treated as a difference-of-convex problem after defining R_k = A_kA_k†.
  • MM Algorithm: The majorization-minimization algorithm linearizes non-convex objective and constraint terms, producing a sequence of convex problems.
  • MM Algorithm: The MM algorithm converges to a stationary point of the original non-convex problem.
  • Implementation: The corner-point test checks which nested-subset backhaul inequalities hold with equality for a candidate permutation.
  • Implementation: Given a solution satisfying corner-point conditions for a BS permutation, the simpler successive estimation-compression architecture can replace joint compression.

C. Independent Quantization

The paper also treats independent quantization and robust design under imperfect channel information. Worst-case uncertainty models preserve the MM framework through equivalent channel substitutions or finite reformulations.

  • Independent Quantization: Independent quantization is obtained by setting the off-diagonal compression-covariance blocks Ω_i,j to zero, while the MM algorithm remains applicable.
  • Robust Design: Robust design jointly considers precoding and compression under uncertainty in the channel matrices using deterministic worst-case optimization.
  • Robust Design: The uncertainty analysis uses singular-value and ellipsoidal uncertainty models, with the latter incorporating covariance information about CSI errors.
  • Singular Value Uncertainty Model: Under the singular-value model, the robust problem is equivalent to the nominal problem with H_k replaced by (1 − ε_k)Ĥ_k.
  • Ellipsoidal Uncertainty Model: The ellipsoidal model converts infinitely many uncertainty constraints into finitely many linear constraints through the S-procedure and auxiliary variables β_k.
  • Ellipsoidal Uncertainty Model: The resulting robust problem remains a difference-of-convex problem, and a related MM algorithm is guaranteed to converge to a stationary point.

VI. SEPARATE DESIGN OF PRECODING AND COMPRESSION

The separate-design approach fixes the precoder using a standard criterion and then optimizes the compression covariance independently, while accounting for the power consumed by compression noise. Feasibility depends on selecting the power offset factors appropriately.

  • The precoding matrix A is fixed beforehand using ZF, MMSE, or weighted sum-rate maximizing precoding that neglects compression noise.
  • The compression covariance Ω is then optimized separately to maximize the weighted sum-rate.
  • The precoder uses a reduced power constraint γ_iP_i because each transmitted BS signal also contains compression noise.
  • With A fixed, the remaining optimization over Ω is a difference-of-convex problem handled by an MM algorithm.
  • The compression-covariance problem may be infeasible when the power-offset factors γ_i are too large, so they can be selected through search strategies such as bisection.

VII. NUMERICAL RESULTS

Numerical experiments compare multivariate compression and joint precoding-compression design with independent compression and separate design across Wyner and fading models. The proposed strategies gain most when backhaul, transmit power, or inter-cell coupling makes compression important.

  • Experimental setup: The experiments evaluate sum-rate in three-cell systems with one active MS per cell, equal BS power constraints P, and equal backhaul capacities C.
  • A. Wyner Model: Multivariate compression significantly outperforms independent compression for both linear and DPC precoding in the circular Wyner model.
  • A. Wyner Model: RCoF remains most effective at moderate backhaul C, while multivariate compression compensates for much of the rate loss of standard DPC in the low-backhaul regime.
  • A. Wyner Model: Increasing γ generally improves separate-design sum-rate, but performance degrades beyond a threshold because compression-covariance optimization becomes more likely infeasible.
  • B. General Fading Model: The gain from multivariate compression and joint design becomes more pronounced at larger transmit power, and only the proposed joint design approaches the cutset bound as power increases.
  • B. General Fading Model: With limited backhaul at high SNR, overall performance is determined more by the compression strategy than by the precoding method.
  • B. General Fading Model: When backhaul capacity exceeds 5 bit/c.u., separate design with multivariate compression outperforms joint design with independent quantization.
  • B. General Fading Model: For small inter-cell channel gain α, multivariate compression offers little advantage because MSs do not receive quantization noise superposed from multiple BSs.

VIII. CONCLUSIONS

The paper jointly designs precoding and multivariate backhaul compression to control correlated quantization noise under power and backhaul constraints. Numerical results show gains over independent compression and separate design, especially when transmit power or inter-cell channel gain is large and backhaul is limiting.

  • The proposed approach correlates quantization noises across BSs through multivariate compression to control their effect at the MSs.
  • The weighted sum-rate is optimized over precoding and the joint quantization-noise correlation matrix subject to power and backhaul constraints.
  • An iterative MM algorithm is proposed that achieves a stationary point of the formulated optimization problem.
  • A practical implementation uses successive per-BS estimation-compression steps rather than requiring joint compression of all BS signals.
  • Numerical results confirm that joint precoding and multivariate compression outperform independent compression and separate design, particularly at larger transmit power or inter-cell channel gain under finite-capacity backhaul.
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