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Local Partial Zero-Forcing Precoding for Cell-Free Massive MIMO

Giovanni Interdonato, Marcus Karlsson, Emil Björnson, Erik G. Larsson

arXiv:1909.01034v3cs.ITeess.SP

TL;DR

Cell-free Massive MIMO needs scalable downlink precoding that manages interference without network-wide instantaneous CSI exchange. The paper proposes distributed PZF and PPZF, derives closed-form SE expressions under estimation errors and pilot contamination, and finds substantial gains over MRT and ZF with performance comparable to RZF. These expressions also enable long-term power-control design applicable to RZF.

  • Problem

    Scalable cell-free downlink precoding must manage interference without the instantaneous CSI exchange and centralized processing required by network-wide coordination.

  • Method

    The paper proposes fully distributed local PZF and PPZF schemes and derives closed-form achievable downlink SE expressions accounting for estimation errors and pilot contamination.

  • Results

    PZF and PPZF substantially outperform MRT and ZF, perform comparably to RZF, and achieve up to 25% per-user SE improvement over FZF and up to 7% additional sum-SE gain with PPZF over PZF.

  • Takeaways & Limitations

    The closed-form SE expressions support optimal long-term power-control strategies, including strategies suitable for RZF despite its unavailable closed-form SE expression.

Abstract

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Cell-free Massive MIMO (multiple-input multiple-output) is a promising distributed network architecture for 5G-and-beyond systems. It guarantees ubiquitous coverage at high spectral efficiency (SE) by leveraging signal co-processing at multiple access points (APs), aggressive spatial user multiplexing and extraordinary macro-diversity gain. In this study, we propose two distributed precoding schemes, referred to as \textit{local partial zero-forcing} (PZF) and \textit{local protective partial zero-forcing} (PPZF), that further improve the spectral efficiency by providing an adaptable trade-off between interference cancelation and boosting of the desired signal, with no additional front-hauling overhead, and implementable by APs with very few antennas. We derive closed-form expressions for the achievable SE under the assumption of independent Rayleigh fading channel, channel estimation error and pilot contamination. PZF and PPZF can substantially outperform maximum ratio transmission and zero-forcing, and their performance is comparable to that achieved by regularized zero-forcing (RZF), which is a benchmark in the downlink. Importantly, these closed-form expressions can be employed to devise optimal (long-term) power control strategies that are also suitable for RZF, whose closed-form expression for the SE is not available.

I. INTRODUCTION

Cell-free Massive MIMO distributes coherent transmission across APs to combine macro-diversity and interference management, but scalable downlink precoding must avoid network-wide instantaneous CSI exchange. This paper proposes fully distributed partial zero-forcing schemes with closed-form SE analysis and power-control optimization.

  • Motivation: Network-wide coordination requires substantial signaling and CSI exchange, creating performance limitations and scalability issues.Centralized ZF additionally requires the CPU to construct precoders and feed them back to APs.
  • Motivation: Cell-free Massive MIMO uses distributed APs to provide coherent combining, macro-diversity gain, and interference management without cell boundaries.APs coordinate through CPUs while potentially serving all users on the same time-frequency resources.
  • Motivation: The downlink schemes studied here are designed for fully distributed, scalable implementation without instantaneous CSI exchange between APs and CPUs.This preserves reduced front-hauling overhead and supports local precoder construction.
  • Contributions: PZF and PPZF provide an adaptable trade-off between interference mitigation and desired-signal power, with no additional front-hauling overhead and few AP antennas.Their design addresses the antenna constraint of local full-pilot ZF, which requires M > τP.
  • Contributions: The paper derives closed-form achievable downlink SE expressions incorporating independent Rayleigh fading, channel-estimation errors, and pilot contamination.These expressions support global max-min fairness power-control optimization under per-AP power constraints.
  • Contributions: PZF and PPZF are quantitatively compared with MRT, FZF, and local RZF under max-min fairness and heuristic channel-dependent power control.The comparison covers both proposed schemes and established distributed or benchmark precoders.

A. Uplink Training

During uplink training, users transmit pilots to all APs, which correlate the received signals and apply MMSE channel estimation. Pilot reuse creates linearly dependent estimates, so APs construct precoders from a reduced full-rank pilot-based channel matrix.

  • Uplink Training: Each user simultaneously transmits a τP-length pilot sequence to all APs during uplink training.The pilot length satisfies τP ≤ K, allowing multiple users to share an orthogonal pilot sequence.
  • Uplink Training: APs correlate received pilots with the corresponding sequences and then perform MMSE channel estimation.The resulting estimate and estimation error are independent Gaussian vectors with variances determined by γl,k and βl,k − γl,k.
  • Uplink Training: Users sharing a pilot have linearly dependent channel estimates at each AP, producing pilot contamination.APs cannot spatially separate linearly dependent channels.
  • Uplink Training: When τP < K, the full channel-estimate matrix is rank-deficient because some estimated channels are parallel.A full-rank matrix is formed from one channel direction per orthogonal pilot.
  • Downlink Data Transmission: Each AP effectively constructs τP precoding vectors, one per orthogonal pilot, and reuses each vector for users sharing that pilot.Constructing the full-rank matrix requires at least one channel estimate per uplink pilot.
  • Downlink Data Transmission: The received downlink signal separates into desired signal, multi-user interference, and receiver noise.Power-control coefficients can exclude an AP from serving a user, thereby forming cooperation clusters.

III. PERFORMANCE ANALYSIS

The paper evaluates downlink spectral efficiency using a hardening-bound lower bound on ergodic capacity. This framework applies across precoding schemes and supports closed-form analysis under independent Rayleigh fading.

  • The achievable downlink SE is obtained from a hardening-bound lower bound on ergodic capacity.The bound treats the effective channel gain as deterministic and the remaining terms as uncorrelated effective noise.
  • SE is expressed as log2(1 + SINRk) bit/s/Hz for UE k.
  • The SE expression is valid regardless of the precoding scheme used.Closed-form expressions are then derived for different schemes under independent Rayleigh fading.
  • The bound assumes infinitely long codewords and may overestimate rates for short codewords.

B. Maximum Ratio Transmission

This material develops local precoding analysis from MRT through full-pilot and partial zero-forcing. PZF locally suppresses interference for strong UEs while retaining MRT for weak UEs, interpolating between FZF and MRT.

  • Maximum Ratio Transmission: MRT provides a closed-form achievable downlink SE under the paper’s channel model.The MRT precoding vector is constructed locally at each AP and inserted into the general SINR expression.
  • Full-pilot Zero-Forcing: Local FZF uses AP-local CSI, avoiding centralized computation, instantaneous-CSI exchange, and precoder feedback.It requires smaller pseudo-inverse matrices than centralized ZF.
  • Full-pilot Zero-Forcing: Local FZF suppresses an AP’s own interference but not interference generated by other APs.Its cancellation capability depends on CSI quality and requires M > τP antennas.
  • Local Partial Zero-Forcing: PZF applies local FZF to strong UEs and MRT to weak UEs, suppressing intra-group interference while managing inter-group interference as in MRT.Co-pilot UEs are grouped together because an AP cannot separate them spatially.
  • Local Partial Zero-Forcing: The proposed PZF scheme has a closed-form ergodic SE under independent Rayleigh fading.
  • Local Partial Zero-Forcing: PZF reduces to FZF when every UE is strong and to MRT when every UE is weak.The corresponding strong-UE pilot count becomes τP for FZF and zero for MRT.

E. Local Protective Partial Zero-Forcing

PPZF enhances PZF by projecting weak-UE MRT signals into the orthogonal complement of the strong-UE channel subspace. It preserves adjustable interference–signal trade-offs while protecting strong UEs from non-coherent interference.

  • PPZF places weak-UE MRT precoders in the orthogonal complement of the strong-UE channel estimates.This construction ensures the weak-UE precoders are orthogonal to strong-UE estimated channels.
  • PPZF has a closed-form ergodic SE characterized by an effective SINR.
  • For strong UEs, PPZF’s non-coherent interference almost vanishes, with a residual contribution from channel-estimation errors.Its array gain is M − τSl for both strong and weak UEs.
  • PPZF provides full interference protection to strong UEs except for pilot contamination while still serving weak UEs.
  • PPZF offers an adjustable balance between interference cancellation and desired-signal boosting through the UE grouping criterion.The grouping must satisfy M > τSl.

F. Local Regularized Zero-Forcing

Local RZF applies regularization to locally estimated channels and provides a simultaneous trade-off between interference suppression and desired-signal boosting. Its achievable SE is evaluated numerically because a closed form is unavailable.

  • RZF simultaneously trades interference suppression against boosting of the intended signal for all UEs.
  • Local RZF constructs each precoding vector from channel estimates collected at the same AP.A regularization matrix is added to the matrix being inverted, with diagonal elements related to each UE’s SNR^-1.
  • With different UE power levels, each AP must construct K RZF precoding vectors.
  • A closed-form achievable-SE expression for RZF is unavailable because the regularization term makes derivation intractable.The paper instead evaluates achievable SE using Monte-Carlo simulations.

IV. POWER CONTROL

The paper formulates max-min fairness power control to maximize the minimum downlink SINR under per-AP power constraints. The resulting problem is convex for fixed target SINR and can be solved efficiently through feasibility checking and bisection.

  • Max-min fairness power control maximizes the lowest user’s downlink SE to provide uniform service across the network.
  • The optimization uses per-AP power constraints and represents the objective through ν, the minimum SINR among users.
  • The power-control formulation covers MRT, FZF, PZF, and PPZF through a common precoding-scheme representation.
  • For fixed ν, the reformulated problem is a second-order cone program because its relevant constraints are second-order cones in the power-control variables.
  • The solution can be obtained by solving the corresponding feasibility problem with bisection, using interior-point methods such as CVX for the convex subproblem.
  • Centralized optimal power control requires exchanging long-term channel statistics and feedback, and its polynomial complexity can challenge scalability for large networks.

B. Distributed Heuristic Channel-Dependent Strategy

The distributed heuristic strategy bases power-control coefficients on local long-term channel statistics, allocating more power to users with stronger channels. The section also evaluates computational complexity and simulation performance under specified propagation, pilot, and network assumptions.

  • B. Distributed Heuristic Channel-Dependent Strategy: The heuristic power-control policy depends exclusively on local long-term channel statistics, providing a scalable alternative to centralized optimization.
  • B. Distributed Heuristic Channel-Dependent Strategy: ρ_l,k ∝ γ_l,k allocates more transmit power to users with larger channel gains, while making APs transmit with full power.
  • B. Distributed Heuristic Channel-Dependent Strategy: The proposed schemes’ computational complexity is evaluated per coherence block, counting complex multiplications and divisions while neglecting additions and subtractions.
  • B. Distributed Heuristic Channel-Dependent Strategy: For M = 16 and τP = 10, Fig. 1 plots normalized computational complexity against τSl for the proposed precoding schemes.
  • B. Distributed Heuristic Channel-Dependent Strategy: PZF and PPZF have lower complexity than FZF because τSl ≤ τP, while PPZF exceeds PZF by 2(τP − τSl)τSlM additional complex multiplications.
  • B. Distributed Heuristic Channel-Dependent Strategy: The simulations collect SE values over 500 random network snapshots using closed-form ergodic-SE lower bounds conditioned on large-scale fading.
  • B. Distributed Heuristic Channel-Dependent Strategy: Unless otherwise stated, simulations use D = 1000 m, σsh = 4 dB, B = 20 MHz, τC = 200 samples, and 200 mW maximum transmit power per AP.
  • B. Distributed Heuristic Channel-Dependent Strategy: The simulation model includes randomly assigned UL pilots with τP < K and large-scale fading incorporating pathloss, shadow fading, and spatial correlations.

B. Performance Evaluation

The evaluation shows that PZF and PPZF balance interference suppression with array gain, outperforming MRT and often matching RZF across power-control and loading conditions. PPZF also supports flexible antenna counts, pilot reuse, AP clustering, and service decisions for weak UEs.

  • CDF-based comparisons: PZF and PPZF outperform MRT and FZF in high-percentile per-user SE, with PZF achieving up to 25% improvement over FZF.FZF loses array gain because nearly all degrees of freedom cancel interference, whereas partial schemes cancel interference among strong UEs while retaining array gain.
  • CDF-based comparisons: PPZF adds up to 7% sum-SE improvement over PZF and matches RZF under the evaluated pilot-contamination setup.Its protective design favors UEs with larger channel gain, while closed-form and Monte-Carlo results closely overlap.
  • CDF-based comparisons: With τP = 10, pilot contamination reduces PZF and PPZF performance, while FZF remains near 3.8 bit/s/Hz/user through increased array gain.Under this setting, FZF, PZF, and PPZF perform similarly and remain better than MRT.
  • Power-control comparisons: Under MMF power control, PPZF improves 95%-likely SE by up to 6-fold over PPZF with HCD power control, and PZF and PPZF are identical and best-performing.The comparison uses L = 100, M = 8, τP = 7, and K = 10.
  • Implementation versatility: PZF, PPZF, and RZF support any AP antenna count, unlike FZF, which requires M > τP; PPZF needs 6 antennas for 3 bit/s/Hz/user versus 8 for FZF.FZF also constrains the number of available orthogonal pilots.
  • UE and AP grouping: The optimal strong-UE threshold is υ = 95%, selecting roughly one-third to one-half of UEs and typically using τS_l = 4–6 degrees of freedom.Larger thresholds sacrifice more array gain than they gain in interference cancellation, whereas smaller thresholds can leave intolerable interference.
  • UE and AP grouping: The optimal AP-clustering threshold is around κ = 95%, corresponding to 15–20 serving APs per UE on average; under this clustering, PZF and PPZF perform almost identically.Dropping some UEs before local strong/weak grouping reduces weak-UE interference.
  • Weak-UE service: Serving weak UEs provides substantial 95%-likely SE gains when K/M or τP/M is large, because good-channel UEs may otherwise receive no service.The comparison is between PPZF and PPZF without MRT transmission to weak UEs.

VII. CONCLUSION

The paper proposes distributed PZF and PPZF schemes that improve cell-free Massive MIMO spectral efficiency while avoiding additional front-hauling overhead and supporting APs with few antennas. Their performance is comparable to RZF, and the derived closed forms support long-term power control, including for RZF.

  • PZF and PPZF significantly improve spectral efficiency compared with traditional MRT and ZF precoding.
  • PPZF can outperform PZF in practical scenarios by providing full interference protection to users with better channel conditions.
  • PZF and PPZF perform as well as RZF, the downlink benchmark.
  • The closed-form spectral-efficiency expressions enable optimal power-control strategies that are also suitable for RZF.
  • Future work should examine the schemes in LoS-dominated sparse-scattering environments and under blockage effects.

APPENDIX

The appendix derives closed-form terms for achievable spectral efficiency by decomposing desired-signal and interference expectations under the proposed and baseline precoders. The derivations use independence, zero-mean properties, user-group membership, and substitutions into the main spectral-efficiency expressions.

  • The appendix computes the numerator and denominator terms of the achievable spectral-efficiency expressions in closed form.Intermediate results are substituted into the governing expressions to obtain the stated formulas.
  • Cross-expectation terms vanish when channel estimation errors are independent of the relevant zero-forcing precoders.
  • Interference calculations split according to whether users belong to pilot-sharing, strong, or weak UE groups.The cases determine which terms remain in the decomposed expectations.
  • The derivations rely on zero-mean and independence properties of channels, estimation errors, estimates, and MRT or PZF precoding vectors.
  • The appendix combines intermediate identities and substitutions to obtain the final closed-form expressions for the considered cases.

C. Proof of Corollary 3

The proof of Corollary 3 modifies the MRT precoder by projecting it onto a subspace orthogonal to selected channel directions. The calculation uses a central complex Wishart result under an antenna-dimension condition.

  • The MRT precoding vector is projected onto the M − τS_l dimensional subspace orthogonal to the column space of H̄_l E_S_l.
  • The projection provides the structural basis for the corollary’s treatment of users in the weak set and their pilot-sharing relationships.
  • The remaining calculations follow the methodology established for the preceding corollary.
  • A central complex Wishart identity is applied when M ≥ τS_l + 1.
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