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Channel Hardening and Favorable Propagation in Cell-Free Massive MIMO with Stochastic Geometry

Zheng Chen, Emil Björnson

arXiv:1710.00395v2cs.ITcs.NI

TL;DR

The paper asks whether channel hardening and favorable propagation can be trusted in CF Massive MIMO with realistic stochastic AP deployments. It analyzes channel distributions and these properties under varying AP antenna configurations and pathloss models, finding that hardening is generally weak and favorable propagation depends strongly on user separation and propagation conditions. Consequently, achievable-rate analyses and resource allocation should not assume these properties universally, while some practical implementation issues remain open.

  • Problem

    It is unclear whether cellular Massive MIMO capacity bounds relying on channel hardening and favorable propagation remain valid for CF networks with spatially distributed, strongly correlated channels.

  • Method

    The paper uses stochastic geometry to analyze channel distributions and hardening and propagation criteria for random AP deployments with different antennas per AP and pathloss models.

  • Results

    Channel hardening is generally weak, while favorable propagation is more likely with spatially separated users, higher antenna density, or a smaller pathloss exponent.

  • Takeaways & Limitations

    Achievable-rate expressions and resource allocation schemes should work without relying on channel hardening and favorable propagation in CF Massive MIMO.

  • Takeaways & Limitations

    The distributed nature of CF Massive MIMO leaves scheduling, power control, pilot allocation, system information broadcast, and random access as open implementation issues.

Abstract

from arXiv · show

Cell-Free (CF) Massive MIMO is an alternative topology for future wireless networks, where a large number of single-antenna access points (APs) are distributed over the coverage area. There are no cells but all users are jointly served by the APs using network MIMO methods. Prior works have claimed that CF Massive MIMO inherits the basic properties of cellular Massive MIMO, namely channel hardening and favorable propagation. In this paper, we evaluate if one can rely on these properties when having a realistic stochastic AP deployment. Our results show that channel hardening only appears in special cases, for example, when the pathloss exponent is small. However, by using 5--10 antennas per AP, instead of one, we can substantially improve the hardening. Only spatially well-separated users will exhibit favorable propagation, but when adding more antennas and/or reducing the pathloss exponent, it becomes more likely for favorable propagation to occur. The conclusion is that we cannot rely on channel hardening and favorable propagation when analyzing and designing CF Massive MIMO networks, but we need to use achievable rate expressions and resource allocation schemes that work well also in the absence of these properties. Some options are reviewed in this paper.

I. INTRODUCTION

CF Massive MIMO jointly serves users through geographically distributed APs, but it raises practical questions about scalable operation and whether cellular Massive MIMO properties remain reliable. This paper studies those questions under stochastic AP deployments and varying antenna configurations and propagation models.

  • Motivation: Network MIMO lets distributed APs jointly serve users, turning interference into useful signals but creating challenges for scalable channel acquisition and data sharing.Local CSI, TDD-based channel estimation, and user-centric clustering are presented as practical approaches to these challenges.
  • CF Massive MIMO: CF Massive MIMO uses many geographically distributed APs to jointly serve fewer users, extending network MIMO with capacity analysis for practical pilot allocation and imperfect CSI.Unlike cellular Massive MIMO, CF systems distribute antennas geographically rather than concentrating them in one array per cell.
  • Research gap: Cellular Massive MIMO benefits from channel hardening and favorable propagation, but it is unclear whether conventional capacity bounds relying on these properties are suitable for CF networks.The paper identifies this uncertainty because prior channel models were designed for co-located arrays, whereas CF channels are strongly spatially correlated.
  • Research gap: Distributed CF networks make scheduling, power control, pilot allocation, system information broadcast, and random access nontrivial because these functions cannot rely on per-cell implementation.Understanding channel hardening and favorable propagation is therefore relevant to simplifying these distributed resource-allocation tasks.
  • Research questions: The study asks whether the two properties occur with single-antenna APs, how antenna placement affects them, which practical factors matter, and which cellular capacity bounds apply.The methodology models AP locations using a stochastic point process and evaluates the resulting channel properties and capacity bounds.
  • Methodology: The analysis uses stochastic AP deployments, conditions on fixed AP locations, and then averages the percentage or probability of users satisfying channel criteria across random networks.It considers different antennas per AP and non-singular single-slope and multi-slope pathloss models.
  • System model: The system model places APs according to a homogeneous PPP, equips each AP with N ≥1 antennas, and connects all APs to a central processing unit through backhaul.The channel model uses independent Rayleigh fading and distance-dependent pathloss, with antennas co-located in groups of N at each AP.

A. Main Advantage of CF Massive MIMO

CF Massive MIMO’s macro-diversity comes from reducing user-to-nearest-AP distances, but its channel-gain distribution remains shaped by AP geometry and small-scale fading. At fixed antenna density, using fewer antennas per AP yields more uniform coverage, while larger per-AP arrays increase variance.

  • Macro-diversity: Macro-diversity reduces the distance between users and their nearest APs, improving the channel-gain distribution for randomly located users.The channel gain is the effective scalar-channel gain under maximum-ratio precoding or combining.
  • Channel-gain statistics: For pathloss exponent 1 < α < 2, the mean and variance of ∥g_k∥2 increase unboundedly as the network region grows.
  • Antenna-density trade-off: At fixed antenna density µ = Nλ_A, the average channel gain is unchanged by the split between AP density and antennas per AP, while variance grows with N.The variance is proportional to (N + 1)µ.
  • Coverage distribution: At the 95%-likely point, N = 1 achieves a 12 dB higher ∥g_k∥2 value than N = 100 when antenna density is fixed.The larger N is, the longer the distribution tail becomes because Var[∥g_k∥2] is proportional to (N + 1).
  • Coverage distribution: Higher AP density reduces the chance that a randomly located user is far from all nearby APs, supporting more uniform coverage.The paper contrasts N = 1 with high AP density against N = 100 with low AP density.

B. Conditional Channel Distribution at Fixed Location

At a fixed user location, CF channel-gain statistics are conditioned on the deployed AP distances, unlike the spatially averaged distribution. With single-antenna APs, unequal pathloss coefficients produce a typically long-tailed Hypoexponential distribution, motivating a spatial channel-hardening measure.

  • Fixed-location conditioning: Once APs are deployed, a fixed user experiences time-varying small-scale fading but fixed large-scale fading determined by AP distances.
  • Single-antenna APs: For N = 1, the conditional channel-gain distribution is Hypo(l(r_1)^−1, …, l(r_L)^−1), typically long-tailed when AP pathloss coefficients differ.This follows from summing exponentially distributed small-scale fading terms with distinct coefficients.
  • Multi-antenna APs: With N antennas per AP, each AP’s channel gain follows Gamma(N, l(r_i)), and the total gain is a sum of independent Gamma variables with different scale parameters.
  • Comparison with cellular Massive MIMO: Cellular Massive MIMO instead has a Gamma(M, β) channel-gain distribution that becomes relatively concentrated around its mean as M increases.The CF and cellular distributions highlight different channel statistics.
  • Channel-hardening measure: The paper defines channel hardening for arbitrary user locations through a spatially averaged measure based on channel-gain variation under a threshold θ.The probability is taken over network realizations that generate different distance vectors.

A. Necessary Conditions for Channel Hardening

Channel hardening depends strongly on propagation conditions and antenna deployment. Increasing antennas per AP always helps, while increasing AP density alone is effective mainly for small pathloss exponents.

  • Increasing the number of antennas per AP always improves the channel-hardening measure, regardless of AP density.The measure increases with N because N appears in its denominator.
  • The exact distribution of Xch is analytically intractable because Y1 and Y2 are strongly correlated, so the paper studies an asymptotically related measure.This surrogate supports analytical insight into AP-density effects without extensive numerical simulation.
  • For α = 3.76 and N = 1, the simulated hardening CDF changes little with AP density until λA = 10^5/km2 (0.1/m2).Densities above 10^3/km2 are described as probably practically unreasonable.
  • Including log-normal shadow fading does not change the paper’s conclusions about channel hardening.The curves for σsh = {0, 5, 10} dB almost overlap in the reported simulation.
  • For α ≤ 2, channel hardening becomes more likely as AP density and network size increase.This includes free-space propagation and indoor near-field propagation cases.
  • With one antenna per AP, increasing AP density does not produce channel hardening for typical pathloss exponents and AP densities.The condition may require approximately 1 AP/m2, which is impractical.

B. More Antennas on Few APs or More APs with Few Antennas?

With fixed total antenna density, concentrating antennas on fewer APs yields stronger channel hardening than distributing them across more APs. The benefit is strongest when moving from one antenna to five or ten per AP, but reduces macro diversity.

  • For fixed antenna density, more antennas on fewer APs provide stronger channel hardening than more APs with fewer antennas.This conclusion follows from the hardening expression and applies across the considered pathloss regimes.
  • The stronger hardening from concentrating antennas comes at the cost of less macro diversity.
  • The fixed-density comparison assumes uncorrelated fading among antennas on each AP; spatial correlation slightly reduces hardening, although more antennas remain beneficial.
  • The largest improvements occur when increasing from N = 1 to N = 5 or N = 10 antennas per AP.The reported fixed-density evaluation uses µ = NλA = 10^3/km2 (10^-3/m2).
  • Using 5–10 antennas per AP can achieve reasonably strong channel hardening within the CF Massive MIMO scope.A smaller pathloss exponent further reduces the required number of antennas per AP.

C. Multi-Slope Pathloss Model

The three-slope pathloss model predicts faster channel hardening because nearby APs experience a small pathloss exponent. Increasing antennas per AP also improves hardening.

  • C. Multi-Slope Pathloss Model: Larger d0 and d1 make the hardening measure approach zero faster as λA increases.This reflects the increasing number of APs operating with small pathloss exponents.
  • C. Multi-Slope Pathloss Model: Adding more antennas per AP improves channel hardening for both fixed λA and fixed total antenna density µ = NλA.
  • C. Multi-Slope Pathloss Model: Fig. 6 compares the CDF of Xch across AP densities for N = 1 and N = 10 antennas per AP.The experiment uses d0 = 10 m, d1 = 50 m, and total antenna densities µ = {500, 1000, 2000}/km2.
  • C. Multi-Slope Pathloss Model: With the three-slope model, nearby small pathloss exponents make channel-gain variance decline rapidly relative to the mean as AP density increases.

IV. FAVORABLE PROPAGATION

This section defines favorable propagation through asymptotic channel orthogonality and introduces a finite-network orthogonality metric and its threshold probability.

  • IV. FAVORABLE PROPAGATION: Favorable propagation means that channel vectors to different users become orthogonal, allowing each user performance comparable to being alone.
  • IV. FAVORABLE PROPAGATION: In CF Massive MIMO, unequal large-scale fading across distributed antennas creates spatial channel correlation unlike co-located arrays.
  • IV. FAVORABLE PROPAGATION: As L grows, the variance of channel orthogonality decreases proportionally to 1/L, establishing asymptotically favorable propagation.
  • IV. FAVORABLE PROPAGATION: For finite L, the analysis uses Xfp and the probability pγ that random user pairs have Xfp no larger than threshold γ.

A. Impact of Antenna Density on the Channel Orthogonality

Increasing AP density or antennas per AP improves channel orthogonality, while shadow fading does not materially change these antenna-density trends.

  • A. Impact of Antenna Density on the Channel Orthogonality: Increasing either λA or N helps user channels offer more favorable propagation because Xfp decreases with each factor.For fixed L, Xfp is inversely proportional to N; with increasing L, it scales roughly inversely with λA.
  • A. Impact of Antenna Density on the Channel Orthogonality: Fig. 7 validates that increasing AP density and increasing antennas per AP improve channel orthogonality.The comparison uses λA = {500, 100}/km2 and N = {1, 5}.
  • A. Impact of Antenna Density on the Channel Orthogonality: Including shadow fading does not substantially affect the distribution of X′fp or the conclusions about antenna-density effects.

B. More Antennas on Few APs or More APs with Few Antennas?

The balance between antennas per AP and AP density is not analytically resolved at fixed total antenna density, but user separation and smaller pathloss favor orthogonality.

  • B. More Antennas on Few APs or More APs with Few Antennas?: When total antenna density is fixed, it is analytically difficult to determine whether more antennas on fewer APs outperform more APs with fewer antennas.
  • B. More Antennas on Few APs or More APs with Few Antennas?: At fixed total antenna density µ, increasing N does not necessarily increase or decrease pγ, although N ≥ 20 makes Xfp very small.Sufficiently large N therefore helps channels become asymptotically orthogonal.
  • B. More Antennas on Few APs or More APs with Few Antennas?: Larger inter-user distances make channel vectors more likely to be nearly orthogonal because no AP is close to both users.
  • B. More Antennas on Few APs or More APs with Few Antennas?: Smaller pathloss exponents improve channel orthogonality, including in the analyzed single-slope setting.Fig. 10 compares α ∈ {2, 3, 4} with N = 1 and inter-user distance 70 m.
  • B. More Antennas on Few APs or More APs with Few Antennas?: Theorem 3 identifies larger AP density, more antennas per AP, smaller pathloss, and greater user separation as favorable-propagation factors.

V. CONCLUSIONS ON CAPACITY BOUNDS FOR CELL-FREE MASSIVE MIMO

CF Massive MIMO exhibits limited channel hardening, making hardening-based capacity bounds potentially loose. General rate expressions better capture achievable performance, while multiple antennas per AP reduce the bound gap but may sacrifice macro-diversity.

  • CF Massive MIMO exhibits little channel hardening compared with cellular Massive MIMO.
  • Capacity bounds relying on channel hardening can be very loose for CF systems.
  • The general rate expression does not rely on channel hardening and is preferred for accurately predicting achievable performance.
  • The general expression nearly matches perfect-CSI rates, whereas UatF is much looser and can yield almost twice the rate gap for some users.
  • N = 5 antennas per AP reduces the UatF-to-perfect-CSI gap, but fewer multi-antenna APs reduce average rates because of lost macro-diversity.

B. Downlink Achievable Rate

Downlink rate expressions that rely on channel hardening can substantially differ from perfect-CSI performance. Estimating the instantaneous effective downlink channel provides a general bound whose estimation penalty vanishes with longer observation.

  • The general downlink expression lets users estimate their instantaneous effective channel without explicit downlink pilots.
  • The general bound contains a perfect-CSI term and an imperfect-channel-estimation penalty.
  • The estimation penalty vanishes as τd →∞, making the bound suitable when channels change slowly.
  • UatF downlink rates have a substantial gap from perfect-CSI rates, while the general rate lies between them.
  • Further downlink rate-expression development is needed to fully understand achievable CF Massive MIMO downlink performance.

APPENDIX

The appendix applies Campbell’s theorem to derive channel-gain moments for a homogeneous Poisson AP deployment. It notes a special pathloss case that is not pursued because it is unlikely in practice.

  • Campbell’s theorem provides the starting point for analyzing sums over the stationary homogeneous AP Poisson point process.
  • For a finite two-dimensional network region, the homogeneous Poisson model yields expressions involving the region radius ρ.
  • The pathloss model l(r) = min(1, r^-α) and Gamma-distributed antenna gains determine the relevant expectation terms.
  • The α = 2 case produces a logarithmic expression involving N, λA, and ρ.
  • The appendix derives the variance using Campbell’s theorem and excludes α = 1 because it is unlikely in real propagation environments.
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