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A joint precoding framework for wideband reconfigurable intelligent surface-aided cell-free network

Zijian Zhang, Linglong Dai

arXiv:2002.03744v3eess.SPcs.ITcs.PFeess.SY

TL;DR

The paper addresses the cost and power burden of improving cell-free network capacity with additional BSs. It proposes a wideband RIS-aided architecture and an alternating joint precoding framework, with simulations showing significantly higher capacity than conventional cell-free networking and a practical limited-CSI extension with little performance loss.

  • Problem

    Further capacity improvement in cell-free networks requires more BSs, increasing cost and power consumption, while RISs offer a low-cost, energy-efficient alternative.

  • Method

    The paper formulates wideband joint BS and RIS precoding for weighted-sum-rate maximization and solves it through alternating optimization after decoupling the subproblems with Lagrangian dual reformulation and MCQT.

  • Results

    RISs significantly improve cell-free network capacity, while the two-timescale extension achieves efficient improvement with limited CSI and little performance loss.

  • Takeaways & Limitations

    The joint precoding framework provides a general capacity-maximization solution covering most existing RIS-aided scenarios as special cases.

Abstract

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Thanks to the strong ability against the inter-cell interference, cell-free network is considered as a promising technique to improve network capacity. However, further capacity improvement requires to deploy more base stations (BSs) with high cost and power consumption. To address this issue, inspired by the recently developed reconfigurable intelligent surface (RIS) technique, we propose the concept of RIS-aided cell-free network to improve the capacity with low cost and power consumption. The key idea is to replace some of the required BSs by low-cost and energy-efficient RISs. Then, in a wideband RIS-aided cell-free network, we formulate the problem of joint precoding design at BSs and RISs to maximize the network capacity. Due to the non-convexity and high complexity of the formulated problem, we develop an alternating optimization framework to solve this challenging problem. In particular, we decouple this problem via fractional programming, and solve the subproblems alternatively. Note that most of the scenarios considered in existing works are special cases of the general scenario studied in this paper, and the proposed joint precoding framework can serve as a general solution to maximize the capacity in most existing RIS-aided scenarios. Finally, simulation results demonstrate that, compared with the conventional cell-free network, the network capacity under the proposed scheme can be improved significantly.

I. INTRODUCTION

Cell-free networks mitigate inter-cell interference through cooperative, user-centric service, but scaling them with more BSs increases cost and power consumption. This paper introduces RIS-aided cell-free networking and a joint active-passive precoding framework for wideband capacity optimization.

  • Motivation: Cell-centric networks face inter-cell interference that limits capacity improvement, especially as cell density increases.The interference is described as an inherent bottleneck of the cell-centric paradigm.
  • Motivation: Cell-free networks have all distributed BSs jointly serve all users without cell boundaries, effectively alleviating inter-cell interference.This user-centric cooperation is presented as a way to increase network capacity.
  • Motivation: Adding more distributed BSs can further improve cell-free capacity but requires high cost and power consumption.RISs are proposed as a low-cost, energy-efficient alternative based on passive reflecting elements.
  • Proposed concept: The proposed RIS-aided cell-free network replaces some required BSs with RISs while all BSs and RISs cooperatively serve all users.The paper presents this as its first attempt to introduce RISs into cell-free networks.
  • Problem formulation: For a general wideband setting, the paper formulates joint BS and RIS precoding to maximize users' weighted sum-rate under BS transmit-power and RIS phase-shift constraints.The scenario includes multiple antennas, BSs, RISs, users, and carriers, making many existing RIS-aided settings special cases.
  • Solution framework: An alternating optimization framework decouples active and passive precoding into QCQP subproblems using Lagrangian dual reformulation and MCQT, then solves them alternatively.Under fully known CSI, the system weighted sum-rate converges to a feasible solution; a two-timescale extension reduces required CSI from the long-term perspective.

B. Transmitters

The wideband system jointly models BS precoding, RIS-assisted channels, received signals, and WSR maximization under BS power and RIS reflection constraints. The resulting non-convex joint design is challenging because it couples BS precoders with RIS phase shifts.

  • System model: Multiple synchronized BSs cooperatively serve multi-antenna users across subcarriers in the wideband RIS-aided cell-free network.The network also includes distributed RISs controlled by the CPU or BSs.
  • Channel model: Each BS-user channel combines a direct BS-user link with cascaded BS-RIS-user links whose reflection is controlled by RIS phase-shift matrices.Signals reflected by two or more RISs are ignored because multiple hops incur large path loss.
  • Receivers: The received signal at each user is the superposition of transmissions from all BSs, comprising desired signal, multiuser interference, and AWGN.The noise has zero mean and covariance Ξk,p = σ2IU.
  • Problem formulation: The design maximizes weighted sum-rate subject to BS transmit-power and RIS reflection-coefficient constraints.User weights ηk and per-subcarrier rates Rk,p determine the weighted sum-rate.
  • Problem formulation: Jointly optimizing the RIS phase-shift matrix Θ and BS precoding vector W is difficult because the objective is non-convex and complex.The paper therefore introduces a fractional-programming-based joint precoding framework.

III. PROPOSED JOINT PRECODING FRAMEWORK

The proposed framework solves the WSR optimization through alternating updates of auxiliary variables, BS active precoding, and RIS passive precoding. It assumes fully known CSI and returns feasible joint designs after convergence.

  • Framework overview: The original WSR problem is divided into three subproblems covering auxiliary-variable updates, active BS precoding, and passive RIS precoding.Detailed algorithms are provided in three corresponding subsections.
  • Framework overview: The framework assumes that the CPU fully acquires network CSI before jointly designing BS and RIS precoding for each small timescale.The outputs are Wopt and Θopt for the considered timescale.
  • Alternating optimization: Algorithm 1 initializes W and Θ, alternately updates ρ, ξ, W, and Θ, and stops when the WSR converges.It returns Wopt, Θopt, and Rsum.
  • Alternating optimization: The method first solves ρ, then updates ξ and W for active precoding, and finally updates the RIS variables and Θ for passive precoding.These updates are performed within the alternating optimization framework.

B. Fix (Θ, W) and solve ρopt

With Θ and W fixed where appropriate, the framework uses Lagrangian dual reformulation and MCQT to decouple the high-dimensional fractional objective before solving active precoding.

  • Auxiliary-variable update: Introducing auxiliary variable ρ through Lagrangian dual reformulation makes the original WSR problem equivalent to a reformulated problem.The optimal ρ is obtained by setting the derivative with respect to ρk,p to zero.
  • Active precoding: The resulting active precoding problem is a QCQP with positive-semidefinite matrices and can be optimally solved by methods such as ADMM.The paper also provides an inversion-free PDS-based feasible solution for Wopt.
  • Active precoding: The active precoding subproblem remains difficult because RIS-aided channels create high-dimensional matrix-form fractions that common scalar fractional-programming methods cannot simply relax.This non-convexity arises in the active BS precoding objective.
  • Active precoding: MCQT reformulates the active subproblem by introducing ξ, allowing ξ and W to be updated alternately.This converts the difficult fractional structure into a form suitable for separate updates.
  • Complexity: ADMM can be computationally expensive because it requires inversion of a high-dimensional matrix A in cell-free channels.The high dimension may prevent practical precoding design, motivating the PDS alternative.

D. Passive precoding: fix (ρ, W) and solve Θopt

The passive precoding stage applies MCQT to the RIS design, producing a convex QCQP that can be solved with ADMM or an inversion-free PDS alternative. Large RISs make matrix inversion costly.

  • Passive precoding: With ρ and W fixed, the RIS precoding task is reformulated as a passive precoding subproblem over the RIS coefficients.The formulation uses θ to represent the RIS phase-shift variables.
  • Passive precoding: MCQT introduces auxiliary variables and separates passive precoding into alternating updates of those variables and Θ.The auxiliary variables are optimized with Θ fixed, then Θ is optimized with the auxiliary variables fixed.
  • Passive precoding: The simplified passive subproblem has a convex objective because Λ is positive semidefinite and a convex constraint defined by |θr,j| ≤ 1.It can therefore be solved using ADMM.
  • Complexity: ADMM for passive precoding requires inversion of Λ, whose complexity becomes high when the RIS element number N is large.A PDS-based method is provided to reduce this complexity and obtain Θopt.

IV. TWO-TIMESCALE EXTENSION OF THE PROPOSED JOINT PRECODING FRAMEWORK

The two-timescale extension reduces frequent CSI acquisition by matching users with strong RIS links over large timescales and using only matched channels during small timescales.

  • Frequent acquisition of all RIS-aided channels is unrealistic because RISs introduce high-dimensional channels, especially in dense networks.
  • The extension matches each user with several well-performing RISs at a large-timescale boundary, then uses matched user-RIS channels for later joint precoding.
  • RISs far from users can be temporarily ignored because they contribute little to system capacity and users move within limited large-timescale scopes.
  • The scheme reduces CSI overhead but incurs performance loss from incomplete CSI, reported as limited in later simulations.
  • The matching problem selects user-RIS pairs through binary indicators, with each user matched to at most Rmatch RISs.

C. User-RIS matching: LCR-based method

The user-RIS matching problem is optimized jointly with active and passive precoding, using LCR to obtain a tractable substitute for an NP-hard binary quadratic program.

  • With the matching vector fixed, the matching problem has the same form as the original precoding subproblem and can be inserted into the alternating algorithm.
  • The virtual sum-rate serves as a matching bonus function that favors stronger user-RIS links while suppressing low-contribution unmatched pairs.
  • The equivalent matching formulation is a zero-one quadratic program with linear constraints and is NP-hard.
  • Brute-force search is practical only for small K and R; dense networks create a search space of 2^KR candidates.
  • LCR replaces uu^T with an auxiliary matrix U, relaxes the rank-one constraint, and converts the problem into a solvable SDP.

V. FRAMEWORK SUPPLEMENTS

The framework is extended beyond ideal RIS hardware to continuous-phase and discrete low-resolution phase constraints, requiring specialized or approximate passive-precoding solutions.

  • Practical RIS hardware is non-ideal, so the framework distinguishes ideal, continuous-phase, and discrete low-resolution phase-shift cases.
  • The ideal phase-shift case F1 yields a convex passive-precoding subproblem that can be solved directly.
  • The continuous-phase case F2 is non-convex because of the constant-modulus constraint and can be solved with MM or CCM algorithms.
  • For discrete phase shifts F3, the method relaxes to F2, solves for θopt, and projects it onto the discrete feasible set.
  • The projection produces a sub-optimal phase matrix Θsub in F3.

B. Algorithm convergency

The framework has strict convergence in ideal and continuous-phase cases, while discrete-phase approximation and matching relaxation weaken guarantees; simulations evaluate complexity and RIS-aided capacity.

  • B. Algorithm convergency: For F1 and F2, every iterative update is monotonic, yielding strict convergence to a feasible solution.
  • B. Algorithm convergency: For F3, convergence cannot be proved strictly because phase projection may violate monotonicity, while matching relaxation also makes global convergence unpredictable.
  • B. Algorithm convergency: The main computational burden comes from iterative QCQP and SDP solutions for W, Θ, and u, rather than closed-form auxiliary-variable updates.
  • A. Simulation setup: The simulation setup uses five BSs, two RISs, and four users, with RISs placed on distant building surfaces to create extra reflection links.

B. Weighted sum-rate of the RIS-aided cell-free network

The proposed RIS-aided cell-free network is evaluated through weighted sum-rate under multiple deployment, phase-shift, convergence, and CSI-error conditions. RIS deployment and passive precoding improve performance, while the framework converges quickly and remains robust to CSI errors.

  • Performance comparison: RIS deployment substantially increases weighted sum-rate and extends signal coverage compared with the conventional cell-free network without RIS.The RIS-aided curves show peaks near users approaching RISs, whereas the conventional scheme does not.
  • Performance comparison: Random RIS phase shifts provide limited gain because they cannot accurately direct reflected signals toward users.This comparison demonstrates the necessity of passive precoding at the RISs.
  • Performance comparison: The two-timescale scheme incurs about 10% average performance loss relative to the ideal RIS case while using fewer RIS-aided channels.The ideal case uses all RIS-aided channels, whereas the two-timescale scheme uses only channels from matched user-RIS pairs.
  • Convergence: The joint precoding framework converges within 15 iterations when the convergence error is no more than 1%.The discrete phase-shift cases converge within 10 iterations, while schemes without RIS precoding converge within 5 iterations.
  • CSI robustness: 5% performance loss occurs at δ = 0.1 and 20% at δ = 0.3 relative to perfect CSI in the ideal RIS case.Performance loss increases as the CSI error parameter δ grows, while the authors characterize the scheme as strongly robust to CSI error.

E. The impact of key system parameters

The experiments examine how transmit power, RIS elements, user-RIS matching, and BS-RIS deployment affect weighted sum-rate and energy efficiency. Results show parameter-dependent gains and a performance–overhead or capacity–power trade-off.

  • WSR against BS transmit power: RIS gains are significant mainly at moderate BS transmit power, from 0 dBm to 20 dBm, but negligible at −20 dBm or 30 dBm.At low power, reflected signals are weak; at high power, BSs favor direct BS-user beams over BS-RIS-user beams.
  • WSR against RIS elements: Weighted sum-rate increases as the number of RIS elements rises, while low-resolution phase-shift approximation loss becomes larger.For N = 100, the 1-bit phase-shift case has about 14% approximation loss relative to the ideal case.
  • WSR with matched user-RIS pairs: When each user can match at most three RISs, average capacity loss is about 5% while using no more than 12 RIS-aided channels.This reduces CSI-acquisition overhead, and performance nearly matches ideal joint precoding from L = 60 m to L = 100 m.
  • Trade-off between BSs and RISs: 0.138 bit/s/Hz/W is the maximum reported energy efficiency at B = 7 and R = 9.With B = 7 and R = 21, energy efficiency decreases to 0.09 bit/s/Hz/W, a 34.8% loss.
  • Trade-off between BSs and RISs: Increasing RIS count can eventually reduce energy efficiency because capacity gains no longer offset additional RIS power consumption.The study therefore identifies BS–RIS count selection as an energy-efficiency trade-off.
  • Conclusions and future works: The framework targets higher capacity with low cost and power consumption, and simulations report higher capacity than conventional cell-free networks.The considered wideband scenario includes multiple antennas, BSs, RISs, users, and carriers, with many existing scenarios as special cases.
  • Conclusions and future works: Future work includes energy efficiency, BS transmit power, user fairness, active RIS hardware, and large intelligent surfaces.The conclusion identifies these as open or not-yet-addressed directions.

APPENDIX A PDS-BASED METHOD FOR SOLVING SUBPROBLEM (24)

Appendix A applies a primal-dual subgradient method to solve the active-precoding subproblem. Iterative updates of the precoder and Lagrange multipliers avoid high-dimensional matrix inversion.

  • PDS-based solution: The active-precoding subproblem is solved by introducing Lagrange multipliers and applying primal-dual subgradient updates.The updates jointly optimize W and λ until convergence.
  • PDS-based solution: The vector F(W) represents BS transmit-power constraint functions, with each component comparing quadratic precoder power against Pb,max.The constraint activity determines the corresponding update direction.
  • Computational benefit: The method obtains Wopt without inversion of the high-dimensional matrix A.This is the stated computational advantage of the iterative formulation.
  • Computational complexity: The overall PDS-based complexity depends on the convergence iteration count Ia and includes terms scaling with B, M, P, and K.The appendix gives the component update complexities and the resulting overall order.

APPENDIX B PDS-BASED METHOD FOR SOLVING SUBPROBLEM (35)

Appendix B solves the passive-precoding subproblem with a corresponding primal-dual subgradient procedure. Iterative updates of RIS variables and multipliers avoid inversion of a high-dimensional matrix.

  • PDS-based solution: The passive-precoding subproblem is solved by introducing multipliers χ and applying iterative updates to θ and χ.The variables are optimized simultaneously until convergence to obtain Θopt.
  • PDS-based solution: The update direction uses the RIS constraint functions represented through G(Θ) and quadratic forms involving θ.The sign of each constraint expression determines the corresponding direction.
  • Computational benefit: The procedure obtains Θopt without inversion of the high-dimensional matrix Λ.This is the appendix’s stated computational benefit.
  • Computational complexity: The overall passive-precoding complexity depends on the convergence iteration count Ip and the RIS dimension RN.The appendix separately states the complexities of updating Θ and χ.
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