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Sparse Beamforming and User-Centric Clustering for Downlink Cloud Radio Access Network

Binbin Dai, Wei Yu

arXiv:1410.5020v1cs.IT

TL;DR

The paper addresses network utility maximization in downlink C-RAN when per-BS backhaul capacities constrain user-centric cooperative transmission. It uses sparse beamforming, reweighted ℓ1 approximations, and generalized WMMSE for dynamic or static clustering, and reports substantial gains over naive clustering schemes.

  • Problem

    Downlink C-RAN utility maximization must account explicitly for per-BS backhaul limits, whose consumption depends on both continuous rates and discrete user associations.

  • Method

    The paper formulates clustering through sparse beamforming, approximates weighted ℓ0 backhaul constraints with reweighted ℓ1 constraints, and applies generalized WMMSE to dynamic and static designs.

  • Results

    Dynamic clustering significantly improves performance over naive clustering, while heuristic static clustering achieves a substantial portion of the dynamic scheme’s gain.

  • Takeaways & Limitations

    Explicit per-BS backhaul constraints enable more efficient backhaul utilization and greater flexibility in selecting user-specific cluster sizes.

  • Takeaways & Limitations

    The generalized WMMSE procedure is heuristic and targets a local stationary point rather than the globally optimal weighted sum rate.

Abstract

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This paper considers a downlink cloud radio access network (C-RAN) in which all the base-stations (BSs) are connected to a central computing cloud via digital backhaul links with finite capacities. Each user is associated with a user-centric cluster of BSs; the central processor shares the user's data with the BSs in the cluster, which then cooperatively serve the user through joint beamforming. Under this setup, this paper investigates the user scheduling, BS clustering and beamforming design problem from a network utility maximization perspective. Differing from previous works, this paper explicitly considers the per-BS backhaul capacity constraints. We formulate the network utility maximization problem for the downlink C-RAN under two different models depending on whether the BS clustering for each user is dynamic or static over different user scheduling time slots. In the former case, the user-centric BS cluster is dynamically optimized for each scheduled user along with the beamforming vector in each time-frequency slot, while in the latter case the user-centric BS cluster is fixed for each user and we jointly optimize the user scheduling and the beamforming vector to account for the backhaul constraints. In both cases, the nonconvex per-BS backhaul constraints are approximated using the reweighted l1-norm technique. This approximation allows us to reformulate the per-BS backhaul constraints into weighted per-BS power constraints and solve the weighted sum rate maximization problem through a generalized weighted minimum mean square error approach. This paper shows that the proposed dynamic clustering algorithm can achieve significant performance gain over existing naive clustering schemes. This paper also proposes two heuristic static clustering schemes that can already achieve a substantial portion of the gain.

I. INTRODUCTION

The paper formulates downlink C-RAN network utility maximization with explicit per-BS backhaul limits, jointly addressing scheduling, user-centric clustering, and beamforming. It develops sparse-beamforming and generalized WMMSE methods for dynamic and static clustering, reporting gains over naive schemes.

  • I. INTRODUCTION: The proposed dynamic clustering significantly outperforms naive channel-strength clustering, while heuristic static schemes achieve a substantial portion of that gain.The paper studies both clustering modes and their scheduling, clustering, and beamforming designs.
  • I. INTRODUCTION: Finite-capacity backhaul links make downlink C-RAN utility maximization a mixed continuous-discrete problem because consumption depends on rates and associated-user counts.Existing approaches often fix clusters or penalize backhaul consumption instead of explicitly enforcing per-BS constraints.
  • I. INTRODUCTION: The paper explicitly models per-BS backhaul constraints and targets more efficient backhaul utilization within the network utility maximization framework.The authors report that this formulation can significantly improve network utility.
  • I. INTRODUCTION: Dynamic clustering represents each user’s serving cluster through a sparse beamforming vector and approximates the resulting ℓ0-norm problem using reweighted ℓ1-norm optimization.Nonzero beamforming entries identify the BSs serving that user.
  • I. INTRODUCTION: A generalized WMMSE approach jointly optimizes user scheduling and beamforming under either dynamic or fixed BS clustering.The approach addresses weighted sum rate maximization with explicit backhaul constraints.

C. Paper Organization and Notations

The paper organizes its analysis around dynamic and fixed clustering designs, using a network-wide sparse beamformer to unify scheduling, clustering, and transmission. The system model includes cooperative serving, per-BS power and backhaul limits, and data-sharing assumptions.

  • C. Paper Organization and Notations: Section III develops joint scheduling, beamforming, and dynamic clustering algorithms, while Section IV treats fixed clustering and proposes two heuristic static schemes.The paper then evaluates these designs numerically before presenting conclusions.
  • C. Paper Organization and Notations: The downlink C-RAN has L BSs with M antennas each, K users, and per-BS backhaul capacities C_l; the CP sends each user’s data to a selected cooperative BS cluster.Users are served through joint beamforming.
  • C. Paper Organization and Notations: The network-wide beamforming vector w_k concatenates each BS-to-user block, with zero blocks excluding BSs from user k’s serving cluster.Because users use relatively few BSs, w_k is group sparse.
  • C. Paper Organization and Notations: The received signal model uses channel matrix H_k, noise vector n_k, and linear beamforming across all transmit antennas.The noise is modeled as circularly symmetric complex Gaussian with covariance σ^2I.
  • C. Paper Organization and Notations: Scheduling is encoded by whether w_k is zero, so user scheduling, BS clustering, and beamforming become one sparse-beamformer design task.A scheduled user has nonzero rate and nonzero beamformer.
  • C. Paper Organization and Notations: Backhaul consumption accumulates the rates of users served by each BS, with dynamic clusters controlled jointly and fixed clusters controlled through scheduling and beamforming.The model accounts only for user-data sharing and ignores CSI and beamforming-coefficient backhaul.

III. UTILITY MAXIMIZATION WITH DYNAMIC BS CLUSTERING

The dynamic-clustering formulation jointly designs group-sparse beamforming, user scheduling, and clustering for weighted sum-rate maximization under explicit per-BS backhaul constraints.

  • Dynamic clustering: The algorithm designs a group-sparse beamforming vector for each user in every scheduling slot.Nonzero beamforming entries identify the user’s serving BS cluster.
  • Dynamic clustering: The optimization variables are the users’ beamforming vectors, whose sparsity controls BS-user associations.This links cluster selection directly to the beamforming design.
  • Optimization formulation: The objective is weighted sum-rate maximization, with user priority weights that can be updated using proportional fairness.The achievable rate depends on the sparse beamforming vectors.
  • Optimization formulation: Dynamic clustering treats each user’s serving cluster as an optimization variable in each scheduling slot.The formulation includes both per-BS transmit-power and per-BS backhaul-capacity constraints.

B. Proposed Algorithm

The proposed algorithm approximates discrete BS-cluster indicators with reweighted l1 penalties, converts backhaul constraints into weighted power constraints, and solves the resulting problem iteratively using generalized WMMSE.

  • Problem difficulty: The conventional weighted sum-rate problem is nonconvex, and the mixed discrete-continuous backhaul constraint makes global optimization challenging.The paper therefore focuses on heuristic algorithms targeting a local optimum.
  • Backhaul approximation: The paper represents each scalar indicator as an l0-norm and approximates the resulting constraint using a convex reweighted l1-norm.This extends reweighted l1 approximation from l0 objectives to l0 constraints.
  • Backhaul approximation: The approximated backhaul constraint acts as a weighted per-BS power constraint after fixing the rate from the previous iteration.This removes the rate’s simultaneous appearance in the objective and constraint during each update.
  • WMMSE solution: The approximated problem is reformulated as an equivalent WMMSE problem and solved by block coordinate descent over MSE weights, receivers, and beamformers.Each block subproblem is convex with respect to its individual optimization variables.
  • Algorithm 1: Algorithm 1 updates receivers, MSE weights, beamformers, rates, and reweighting parameters in a single iterative loop.The beamformer update is obtained by solving a QCQP, while rates and weights are refreshed afterward.

C. Complexity Analysis

The per-iteration cost is dominated by transmit-beamformer optimization, especially solving the equivalent second-order cone program with an interior-point method.

  • Per-iteration complexity: O(K^2LMN) is the complexity of covariance and MSE-related computations in the receiver and rate-update steps.The same order applies to achievable-rate computation.
  • Per-iteration complexity: O(K) is the additional complexity for updating all MSE weights.This update follows the MSE computation in Step 1.
  • Per-iteration complexity: O((KLM)^3.5) is the approximate interior-point complexity for solving the SOCP beamformer subproblem.The equivalent SOCP has KLM variables, making beamformer design the main per-iteration cost.

D. Heuristic Complexity Reduction Techniques

Two heuristic techniques reduce Algorithm 1’s per-iteration burden by shrinking the candidate BS set and the user set considered during optimization.

  • Heuristic reductions: Iterative link removal reduces the number of potential transmit antennas serving each user.It removes candidate BS-user links whose transmit power becomes sufficiently small during iterations.
  • Heuristic reductions: Iterative user pool shrinking decreases the total number of users considered in each iteration.Together with link removal, it improves Algorithm 1’s efficiency.

1) Iterative Link Removal:

The paper adapts sparse beamforming and generalized WMMSE methods to jointly optimize scheduling and beamforming under fixed BS clusters and per-BS backhaul limits. Scheduling is implicit: users with nonzero beamformers are selected, improving performance at added computational and CSI-overhead costs.

  • Problem formulation: Fixed-cluster design jointly determines scheduled users and beamformers while satisfying per-BS power and backhaul constraints.Each user’s beamformer contains only the BSs in its predetermined serving cluster.
  • Sparse approximation: The reweighted ℓ1-norm approximation converts the fixed-clustering backhaul problem into an iteratively solvable weighted formulation.The approximation fixes the previous-iteration user rate, replaces indicator functions, and updates user-level weights βk.
  • Iterative optimization: Algorithm 2 extends generalized WMMSE to fixed clusters by alternating MMSE receivers, MSE weights, beamformers, achievable rates, and βk updates until convergence.The beamformer subproblem is solved as a QCQP under fixed receivers and MSE weights.
  • Implicit scheduling: Unlike conventional preselection, both algorithms optimize beamformers for all users and schedule only users receiving nonzero beamforming vectors.The paper reports better performance from implicit scheduling, but at a complexity cost.
  • Practical limitation: Global CSI at the central processor is required for scheduling and beamforming, potentially creating large channel-estimation overhead left for future study.The impact of this overhead under the paper’s realistic model is described as nontrivial.

B. Proposed Static Clustering Algorithms

The paper proposes heuristic static clustering methods after optimizing scheduling and beamforming for a given fixed clustering. These methods balance users’ channel conditions against each BS’s limited radio and backhaul resources.

  • Motivation: Static clustering is optimized heuristically after joint scheduling and beamforming are handled for any given fixed clustering.The clustering strategies are designed to control traffic load while respecting the C-RAN setting in which users are served by BS clusters.
  • User-centric clustering: Each user is served by an individually selected, potentially overlapping BS subset rather than a single BS or a disjoint network partition.This user-centric design differs from traditional single-BS association and disjoint clustering.
  • Load balancing: Choosing an equal number of strongest BSs can imbalance traffic, particularly when macro-BSs have substantially higher transmit power than pico-BSs.The proposed heuristics therefore account for both channel conditions and BS resource limits.
  • Proposed schemes: The two proposed static schemes use either BS-specific maximum supported-user limits or biased received signal strength.The biasing approach generalizes SINR-bias ideas to form static user-centric clusters.

1) Maximum Loading Based Static Clustering:

The maximum-loading scheme forms candidate clusters from near-strongest BSs, then negotiates associations while limiting how many users each BS accepts. The candidate threshold adapts cluster size to user location, while BS limits prevent overload.

  • Candidate-cluster construction: Candidate cluster Ck contains BSs whose received signal strength is within η1 of the strongest BS for user k.Received signal strength is defined using maximum transmit power compensated by path loss, without antenna beamforming gain.
  • Location-dependent cluster size: Cell-edge users can obtain larger candidate clusters because they observe more nearby BSs with similar signal strengths.Cell-center users generally need fewer serving BSs because their channel conditions are already favorable.
  • Association procedure: The heuristic lets each user request candidate BSs from strongest to weakest while each BS accepts at most Kl,max users.The procedure repeats multi-round negotiation until BS capacity or users’ candidate lists are exhausted.
  • Joint control: Using Kl,max and Ck together is important because either BS-only limits or candidate lists alone can leave cell-edge users unserved or overload high-power BSs.The paper presents their combination as one practical way to jointly control traffic load and user channel conditions.

2) Biased Signal Strength Based Static Clustering:

The biased-signal-strength scheme controls static user-centric clusters by adding BS-specific received-signal-strength biases, encouraging users near overloaded BSs to connect with underloaded BSs.

  • Biased cluster formation: The scheme assigns each BS a received signal strength bias ζl and forms each user’s serving cluster from biased signal strengths.Higher bias values for underloaded BSs prompt nearby users to connect with those BSs instead.

V. SIMULATION RESULTS

Simulations evaluate the proposed algorithms in a wrapped-around two-tier heterogeneous network under varied per-tier backhaul constraints. Dynamic clustering narrows serving candidates, reduces computation, improves user-rate performance, and uses backhaul more efficiently, while convergence lacks a rigorous proof.

  • Simulation setup: Simulations use a 7-cell wrapped-around two-tier heterogeneous network with equal macro-BS and pico-BS backhaul constraints within each tier.The algorithms retain the same power constraints while varying (Cmacro, Cpico).
  • Dynamic BS Clustering: After around 20 iterations, only the first and third strongest BSs maintain reasonable power for user 3, forming its serving cluster.With a −100 dBm/Hz threshold, candidate BSs reduce to four after iteration 5 and to the first, third, and fourth strongest BSs after iteration 8.
  • Dynamic BS Clustering: Per-iteration execution time drops from around 170 seconds to about 5 seconds within 20 iterations as candidate clusters and the user pool shrink.At iteration 20, the average candidate cluster size is 1.96 and 54 users remain in the scheduling pool.
  • Dynamic BS Clustering: Dynamic Clustering (690, 107) Mbps outperforms Strongest 3 BSs (1197, 173) Mbps while requiring about 60% of the backhaul.Dynamic clustering concentrates backhaul consumption near its constraint and adapts clusters to the available backhaul budget, balancing traffic.
  • Dynamic BS Clustering: Algorithm 1 with dynamic clustering converges in roughly 40–50 iterations across the evaluated backhaul settings, without a rigorous theoretical convergence proof.The convergence behavior is assessed through simulated log-utility evolutions.

B. Static BS Clustering

The static clustering evaluation shows that cluster size must match available backhaul, while the proposed heuristic schemes retain much of dynamic clustering’s benefit.

  • Adding a third strongest BS improves network utility while maintaining the same backhaul consumption by improving low-rate users at high-rate users’ expense.
  • Adding a fourth strongest BS reduces performance below the two-BS baseline because backhaul-overloaded BSs schedule only low-rate users.
  • The proposed maximum-loading and biased-signal-strength static clustering schemes have similar performance under (Cmacro, Cpico) = (690, 107) Mbps.
  • At the 50th percentile user rate, both proposed static schemes improve performance by around 26% over the baseline, versus 41.9% for dynamic clustering.
  • The paper concludes that explicit per-BS backhaul constraints enable significant gains from dynamic clustering, while static schemes achieve a substantial portion of that gain.
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