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Cell-Free Millimeter-Wave Massive MIMO Systems with Limited Fronthaul Capacity

Guillem Femenias, Felip Riera-Palou

arXiv:1902.07503v1eess.SP

TL;DR

When users outnumber available orthogonal pilots, max-min pilot assignment is a difficult combinatorial problem. The paper develops a cell-free mmWave framework with hybrid processing and constrained fronthaul, and proposes a lower-complexity pilot strategy. Simulations report that the strategy outperforms random approaches, while also showing diminishing rate gains as fronthaul capacity or AP density increases.

  • Problem

    When K exceeds the number of orthogonal pilots, fully orthogonal assignment is impossible and optimal max-min pilot assignment is computationally unmanageable in many practical network setups.

    Source: §V · p. 8

  • Method

    The paper analyzes cell-free mmWave systems with hybrid precoding and constrained fronthaul, and proposes dissimilarity-based pilot allocation alongside max-min resource-allocation methods.

    Source: B. Aim and contributions · p. 2

  • Results

    DCPA avoids the computational burden of optimal small-scale CSI-based pilot allocation and outperforms pure random and balanced random schemes.

    Source: §VIII · pp. 13–14

  • Takeaways & Limitations

    Increasing fronthaul capacity or AP density improves max-min user rates, but the marginal gains diminish, especially with many active users.

    Source: §VIII · p. 14

Abstract

from arXiv · show

Network densification, massive multiple-input multiple-output (MIMO) and millimeter-wave (mmWave) bands have recently emerged as some of the physical layer enablers for the future generations of wireless communication networks (5G and beyond). Grounded on prior work on sub-6~GHz cell-free massive MIMO architectures, a novel framework for cell-free mmWave massive MIMO systems is introduced that considers the use of low-complexity hybrid precoders/decoders while factors in the impact of using capacity-constrained fronthaul links. A suboptimal pilot allocation strategy is proposed that is grounded on the idea of clustering by dissimilarity. Furthermore, based on mathematically tractable expressions for the per-user achievable rates and the fronthaul capacity consumption, max-min power allocation and fronthaul quantization optimization algorithms are proposed that, combining the use of block coordinate descent methods with sequential linear optimization programs, ensure a uniformly good quality of service over the whole coverage area of the network. Simulation results show that the proposed pilot allocation strategy eludes the computational burden of the optimal small-scale CSI-based scheme while clearly outperforming the classical random pilot allocation approaches. Moreover, they also reveal the various existing trade-offs among the achievable max-min per-user rate, the fronthaul requirements and the optimal hardware complexity (i.e., number of antennas, number of RF chains).

I. INTRODUCTION

Cell-free massive MIMO coordinates distributed access points to serve users across a coverage area, while mmWave and hybrid processing introduce a distinct design setting. This paper addresses cell-free mmWave systems with capacity-constrained fronthaul and develops methods for fair resource allocation and user service.

A. Motivation and previous work

  • Cell-free massive MIMO coordinates distributed access points through a CPU to coherently serve users across the coverage area.The paper motivates mmWave as a less-congested spectrum option and notes that its high propagation losses make large arrays and beamforming relevant.Source: A. Motivation and previous work · pp. 1–2

B. Aim and contributions

  • The framework targets cell-free mmWave systems with hybrid precoders and capacity-constrained fronthaul, addressing DL and UL rates, pilot assignment, power control, and quantization.It uses large-scale channel statistics for high-dimensional RF processing and small-scale estimated CSI for low-dimensional baseband processing; its pilot strategy clusters by dissimilarity, and user selection addresses per-AP RF-chain limits.Source: B. Aim and contributions · p. 2

C. Paper organization and notational remarks

  • The paper organizes its system model, rate and fronthaul analyses, allocation algorithms, and numerical results across Sections II–VIII.Figure 1 depicts sample allocation across large- and short-scale coherence intervals; the section also gives notation conventions.Source: C. Paper organization and notational… · pp. 2–3, Figure 1 · p. 3

II. SYSTEM MODEL

  • The system has M access points and K single-antenna mobile stations, with CPU baseband processing, AP RF processing, and L RF chains per AP.Each AP has N>K antennas and a fully connected phase-shifter architecture; when K>L, an algorithm selects which L users each AP serves. TDD separates uplink training from downlink and uplink payload transmission.Source: §II · p. 3

A. Channel Model

The channel model represents mmWave links through clustered multipath and accounts for distinct link states and changing channel statistics. It uses a modified clustered model to capture mmWave propagation characteristics.

A. Channel Model

  • The model adopts a modified discrete-time narrowband clustered channel model to capture mmWave's high distance-based losses, sparse scattering, and high antenna correlation.The cited basis is the model proposed by Akdeniz et al. and extended by Samimi and Rappaport.Source: A. Channel Model · p. 3
  • Links are modeled as outage, LOS, or NLOS; outage links have infinite propagation loss, while other links use a shadowed path-loss model.Small-scale fading is static within a coherence interval and changes independently between intervals, whereas spatial covariance matrices vary more slowly.Source: A. Channel Model · pp. 3–4
  • Each AP–MS channel is the sum of scattering-cluster path contributions, with path gains and array responses specified by their fading and angular parameters.Cluster central azimuth angles are uniform over [−π, π], elevation angles use the LOS elevation angle, and cluster angular spreads are exponentially distributed.Source: A. Channel Model · p. 4

B. Large-scale training phase

The large-scale training design uses spatial covariance information for RF precoding and combining, reducing the dimension of the equivalent channel. When users exceed an AP's RF-chain capacity, a reverse-delete procedure selects which users it serves.

1) RF precoder/combiner design:

  • The RF precoder/combiner is designed from selected users' spatial channel covariance matrices using constrained statistical eigen beamforming.The analog RF matrix is common to uplink and downlink, and its design uses the dominant covariance eigenvector with phase-only implementation.Source: 1) RF precoder/combiner design · p. 4
  • Including the RF matrix produces a lower-dimensional equivalent channel, which substantially simplifies small-scale training.The equivalent channel dimension is much smaller than the AP antenna-array size.Source: 1) RF precoder/combiner design · p. 4

2) Selection of MSs to beamform from each AP:

  • When K>L, each AP selects L users to beamform to by maximizing the minimum average sum-energy of the equivalent channels across users.An iterative reverse-delete algorithm removes eligible AP–user edges to maximize the minimum average sum energy after removal, stopping when all APs serve L users; it requires M(K−L) iterations.Source: 2) Selection of MSs to beamform from… · pp. 4–5

C. Small-scale training phase

Uplink training lets access points estimate channels, but when users outnumber available pilot samples, pilot reuse causes contamination. The section also describes MMSE channel estimation and its error decomposition.

C. Small-scale training phase

  • All K mobile stations simultaneously transmit τp-sample pilot sequences, and the mth access point receives an LA × τp signal matrix.τp is the uplink training duration, measured in samples per coherence interval.Source: C. Small-scale training phase · p. 5
  • When K > τp, pilot sequences are reused among mobile stations, resulting in pilot contamination.The text notes that training sequences should ideally be mutually orthogonal.Source: C. Small-scale training phase · p. 5

D. Channel estimation

  • The minimum mean square error filter estimates the channel between each active mobile station and access point.The section states that channel estimation is central to massive MIMO and cell-free performance.Source: D. Channel estimation · p. 5
  • The channel is decomposed into its MMSE estimate and an estimation error that is statistically independent of both the channel and its estimate.The estimated channel vector is also stated to have a complex normal distribution.Source: D. Channel estimation · p. 5

E. Downlink payload data transmission

Downlink transmission applies baseband precoding, fronthaul compression, and access-point RF precoding. With ZF precoding, the received signal includes useful signal, imperfect-CSI interference, and quantization and thermal noise.

E. Downlink payload data transmission

  • The downlink signal processing comprises CPU baseband precoding, compression over fronthaul links, and RF precoding at each access point.The CPU-to-access-point functional split uses quantization and unquantization, which introduce distortion.Source: E. Downlink payload data transmission · p. 6
  • The compress-after-precoding model assumes Gaussian quantization noise, supported by large-block lattice codes that can approximate that distribution.The quantization noise vector is assumed to follow a complex normal distribution.Source: E. Downlink payload data transmission · p. 6
  • The power-control coefficients are chosen to satisfy a necessary transmit-power constraint at each access point.The mobile-station symbol vector also has a power constraint, with the maximum average transmit power available at access point m denoted P_m.Source: E. Downlink payload data transmission · p. 6
  • The signal received by each mobile station separates into useful signal, imperfect-CSI interference, and quantization and thermal noise.The interference is attributed to imperfect CSI, including pilot contamination.Source: E. Downlink payload data transmission · p. 6

F. Uplink payload data transmission

In the uplink, access points quantize received signals for forwarding to the CPU, which unquantizes and jointly processes them using ZF detection. The detected signal includes useful signal, imperfect-CSI interference, and quantization and thermal noise.

F. Uplink payload data transmission

  • Access points quantize their received uplink signals and forward them to the CPU, which unquantizes and jointly processes them with baseband combining vectors.The uplink model assumes complex-normal quantization noise.Source: F. Uplink payload data transmission · p. 7
  • The CPU uses a ZF MIMO detection matrix to obtain the vector of detected samples.The uplink signal model includes mobile-station symbols, power-control coefficients, and thermal noise at the access-point RF chains.Source: F. Uplink payload data transmission · pp. 6–7
  • The detected signal consists of useful signal, imperfect-CSI interference, and quantization and thermal noise.The text attributes the interference to imperfect CSI.Source: F. Uplink payload data transmission · p. 7

III. ACHIEVABLE RATES

The section derives achievable downlink and uplink rates by treating interference and noise as effective noise. It gives rate expressions for downlink with ZF precoding and uplink with ZF detection.

III. ACHIEVABLE RATES

  • The rate derivation treats interference and quantization and thermal noise as effective noise, using their uncorrelatedness with the desired signal.The analysis also invokes uncorrelated Gaussian noise as worst case from a capacity perspective.Source: §III · p. 7
  • The section gives achievable downlink and uplink rate expressions for the cell-free system using ZF baseband precoding and ZF MIMO detection, respectively.The uplink expression is stated for any M, N, and K.Source: §III · p. 7
  • The uplink achievable rate is expressed as R_uk = log2(1 + SINR_uk).The theorem specifies this expression for the kth mobile station and any M, N, and K.Source: §III · p. 7

IV. FRONTHAUL CAPACITY CONSUMPTION

The section describes fronthaul-rate bounds for quantized downlink and uplink signals, then presents pilot assignment for cases where orthogonal pilots are insufficient. DCPA uses large-scale propagation fingerprints to reuse pilots among dissimilar users.

IV. FRONTHAUL CAPACITY CONSUMPTION

  • The downlink fronthaul rate is bounded using the quantized signal’s mutual information and differential entropy, with Jensen’s inequality applied to the log-determinant expression.The analysis assumes a Gaussian quantization-noise model and uses the differential entropy of a complex Gaussian vector.Source: §IV · pp. 7–8
  • The uplink quantizes each AP’s received vector by adding quantization noise, then upper-bounds the average rate needed to transfer that vector over its fronthaul link.The derivation follows arguments analogous to the downlink case.Source: §IV · p. 8

V. PILOT ASSIGNMENT

  • When the number of mobile stations exceeds the available orthogonal pilots, fully orthogonal assignment is impossible and pilot-reuse strategies are needed.If the number of stations does not exceed the pilot-sequence length, an optimal assignment can give each station an orthogonal pilot.Source: §V · p. 8
  • Optimal max-min pilot assignment is computationally unmanageable in many practical setups, while random and balanced-random assignment can perform poorly; DCPA is proposed as a suboptimal alternative.DCPA is based on clustering by dissimilarity, aiming to avoid the optimal scheme’s complexity while improving on random baselines.Source: §V · p. 8
  • DCPA represents each user by its vector of average equivalent-channel Frobenius norms across APs, measures fingerprint similarity by cosine similarity, and reuses pilots among users that are far apart.The cosine-similarity values range from 0 for perfect dissimilarity to 1 for an exact match; the assignment aims to reduce residual interuser interference.Source: §V · pp. 8–9

VI. MAX-MIN POWER ALLOCATION AND OPTIMAL

The downlink optimization jointly selects user power-control coefficients and AP quantization-noise variances to maximize the minimum achievable user rate under transmit-power and fronthaul constraints. A block coordinate descent procedure alternates between these variable blocks.

A. Downlink power control and quantization

  • The downlink problem maximizes the minimum user rate by choosing power-control coefficients and AP quantization-noise variances subject to per-AP transmit-power and fronthaul-capacity constraints.The optimization treats the power allocation and quantization-noise variables as interdependent blocks.Source: A. Downlink power control and… · p. 9
  • Block coordinate descent solves the interdependent-variable problem by cyclically optimizing one block while holding the others fixed.The text states that convergence is ensured when each subproblem is exactly solved to its unique optimum.Source: A. Downlink power control and… · p. 9
  • At each iteration, the fronthaul-capacity constraints are met with equality to set the acceptable quantization noise, after which the power-allocation problem is updated.The quantization-noise solution is obtained numerically because it has no closed-form algebraic expression.Source: A. Downlink power control and… · p. 9

B. Uplink power control and quantization

The uplink optimization jointly chooses user power coefficients and AP quantization-noise variances to maximize the minimum uplink rate under power and fronthaul constraints. In simulations, DCPA improves on balanced random assignment when pilots must be reused, while downlink and uplink max-min rates tend to converge as user load grows.

B. Uplink power control and quantization

  • The uplink problem maximizes the minimum achievable user rate by choosing user power-control coefficients and AP quantization-noise variances under user power and AP fronthaul constraints.The formulation is solved using block coordinate descent, with a nonconvex transcendental term approximated using the preceding iteration’s power allocation.Source: B. Uplink power control and quantization · pp. 9–10

VII. NUMERICAL RESULTS

  • The simulations use randomly distributed APs and mobile stations in a square coverage area and a modified clustered channel model, with default parameters summarized in Table I.The shadow-fading spatial-correlation model uses a 50 m decorrelation distance and parameter δ = 0.5.Source: §VII · p. 10, Table I · p. 10

A. Impact of the pilot allocation process

  • DCPA clearly improves achievable max-min user rates over balanced random assignment when the number of active users exceeds the pilot-sequence length; below that threshold, both provide the same performance.Fig. 2 compares average max-min rate per user against active-MS count for pilot strategies with N = 64 antennas, L = 8 RF chains, and 64 bit/s/Hz fronthaul in each direction.Source: Figure 2 · p. 10, A. Impact of the pilot allocation… · pp. 10–11
  • As active-user count grows beyond the pilot length, downlink and uplink max-min rates tend to become similar because residual pilot-contamination interference increasingly dominates quantization and thermal noise.Small differences remain due to differing quantized information conveyed over the fronthaul links and differences in AP and user thermal-noise powers.Source: A. Impact of the pilot allocation… · p. 11

B. Modifying the capacity of the fronthaul links and the RF infrastructure at the APs

The results show diminishing returns from increased fronthaul capacity and AP density, while fronthaul constraints can limit the benefit of additional RF resources. The AP-density results also show higher user rates and less variation at higher AP densities.

B. Modifying the capacity of the fronthaul links and the RF infrastructure at the APs

  • At high fronthaul capacity, more antennas raise max-min rate moderately, while the benefit of more RF chains depends on active-user load.With many active users, pilot-contamination interference dominates; at low loads, quantization noise can make additional RF chains disadvantageous.Source: Figure 4 · p. 11, B. Modifying the capacity of the… · pp. 11–12, Figure 5 · p. 12
  • As fronthaul capacity increases, max-min user rate rises with diminishing returns; under low capacity and heavy load, UL rate can slightly exceed DL rate.At high-capacity fronthaul, DL rate is slightly higher than UL rate across the considered user loads.Source: B. Modifying the capacity of the… · p. 11
  • With fixed users and RF chains, adding antennas can become counterproductive near 24 bit/s/Hz; optimal RF-chain counts are 10 at 64, 4 at 32, and 1 below 24 bit/s/Hz.At 256 bit/s/Hz, added antennas or RF chains are beneficial because quantization noise is negligible in the considered setup.Source: Figure 6 · p. 12, Figure 7 · p. 12, B. Modifying the capacity of the… · pp. 12–13

C. Impact of the density of APs

  • For K = 25, 95%-likely user rate rises from 2.55 to 6.50 bit/s/Hz as AP count grows from 25 to 200, with diminishing returns.The reported values are 4.33 bit/s/Hz at 50 APs and 6.11 bit/s/Hz at 100 APs.Source: Figure 8 · p. 13, C. Impact of the density of APs · p. 13
  • As AP density increases, max-min user-rate dispersion around the median tends to decrease; DL and UL rates are nearly identical at higher user loads.At low user loads, DL max-min rates are slightly higher than UL rates.Source: C. Impact of the density of APs · p. 13

VIII. CONCLUSION

  • DCPA reduces the computational burden of optimal small-scale CSI-based pilot allocation while outperforming pure random and balanced random schemes.The conclusion also notes diminishing returns from increasing fronthaul capacity or AP density, and that quantization noise can compromise gains from additional antennas or RF chains.Source: §VIII · p. 14

APPENDIX A PROOF OF THEOREM 1

The appendix derives downlink and uplink achievable-rate lower bounds by decomposing detected signals into useful, interference, quantization-noise, and thermal-noise terms, then applying a worst-case uncorrelated-noise bound.

APPENDIX A PROOF OF THEOREM 1

  • The downlink proof decomposes the received signal into useful, interuser-interference, and quantization-noise terms, then lower-bounds the achievable rate using the worst-case uncorrelated additive-noise result.The signal decomposition also includes thermal noise; mutual independence is used to establish that the terms are uncorrelated.Source: Appendix A · p. 14

APPENDIX B PROOF OF THEOREM 2

  • The uplink proof decomposes the detected signal into useful, interuser-interference, quantization-noise, and thermal-noise terms, then lower-bounds the achievable rate using the worst-case uncorrelated additive-noise result.The proof states that data symbols, quantization noise, thermal noise, and channel-related coefficients are mutually independent.Source: Appendix B · p. 14
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