Performance Analysis of Cell-Free Massive MIMO Systems: A Stochastic Geometry Approach

I. INTRODUCTION

• Cell-free massive MIMO jointly serves users through distributed APs, combining macro-diversity with channel hardening and favorable propagation.The APs use local CSI, which limits information exchange between them.Source: A. Prior Work · pp. 1–2

A. Prior Work

• Prior cell-free massive MIMO studies covered topics including precoding, pilot contamination, hardware impairments, user-centric service, and backhaul, but usually assumed uniformly placed APs in analysis.Prior PPP work on cell-free systems examined channel hardening and favorable propagation, not performance analysis.Source: A. Prior Work · p. 2

B. Motivation

• The paper asks how realistic randomness in AP locations affects cell-free massive MIMO performance, since prior analytical work relied on idealized uniform placement.In practice, AP locations can be highly irregular.Source: B. Motivation · p. 2

C. Contributions and Outcomes

• The authors model AP locations with a PPP and derive deterministic-equivalent downlink SINR, coverage probability, and per-user achievable rate, with small cells as a benchmark.They present the first analytical coverage-probability result for PPP-deployed cell-free massive MIMO, to the authors’ knowledge.Source: C. Contributions and Outcomes · pp. 2–3
• Cell-free massive MIMO outperforms small cells in coverage and rate; coverage rises with AP density before saturating, while more users worsen performance less in cell-free systems.The reported mechanisms include favorable propagation, channel hardening, and interference suppression; higher path-loss exponents reduce rate, and users nearer APs affect rate more.Source: C. Contributions and Outcomes · p. 3

E. Notation

• The notation uses bold symbols for vectors and matrices, standard transpose, Hermitian transpose, trace, expectation, and variance operators, and complex Gaussian vectors with specified covariance.The paper lists its notation in Table I.Source: E. Notation · p. 3, A. System Core and APs Arrangement · p. 3

A. System Core and APs Arrangement

• A software-defined controller coordinates APs over a perfect backhaul, and all APs jointly serve K single-antenna users without cells.AP locations follow a two-dimensional homogeneous PPP; the analysis focuses on a typical user at the origin.Source: A. System Core and APs Arrangement · p. 3, B. Channel Model · p. 3

B. Channel Model

• The channel model separates static large-scale path loss from faster small-scale fading and assumes Rayleigh fading with independent complex Gaussian elements.The analysis assumes conditions for channel hardening and favorable propagation; other fading models are left for future work.Source: B. Channel Model · pp. 3–4

III. UPLINK CHANNEL ESTIMATION

• All K users transmit non-orthogonal pilots simultaneously for τtr<K samples, and pilot reuse creates pilot contamination.The paper identifies multi-user interference in the pilot observation as the source of contamination.Source: §III · pp. 4–5
• Each AP projects its received pilot signal onto the user’s pilot and forms a linear MMSE channel estimate using the known AP-user distance.The pilot projection is the stated estimation step.Source: §III · p. 5
• The estimation error is the difference between the channel and its estimate; the two vectors are uncorrelated Gaussian variables with specified variances.The channel and estimated-channel vectors are modeled as complex Gaussian.Source: §III · p. 5

IV. DOWNLINK TRANSMISSION

• The APs use scaled conjugate beamforming, with precoders based on estimated channels and statistical channel-inversion power control.The scaling parameter enforces the transmit-power constraint.Source: §IV · p. 5
• The typical user’s received signal includes the desired signal, multi-user interference, and noise, with imperfect CSIT due to pilot contamination.Users are assumed to know channel statistics rather than instantaneous CSI; the resulting achievable rate is a rigorous bound close to the instantaneous-CSI rate.Source: §IV · pp. 5–6
• The paper derives a deterministic-equivalent downlink SINR conditioned on AP distances, using large-system analysis with K and W growing at a fixed ratio.The authors report that the approximation is tight even for 20 APs serving 10 users, while exact SINR statistics are intractable because of path-loss terms and correlations.Source: §IV · p. 6
• The deterministic SINR is invariant to AP density under single-slope path loss, saturates as antennas per AP increase, and falls with user count and pilot contamination.It also reaches a ceiling at high downlink SNR.Source: §IV · p. 6

V. COVERAGE PROBABILITY

• Theorem 1 lower-bounds downlink coverage for a pilot-contaminated CF network with PPP-distributed APs, single-slope path loss, and conjugate beamforming.Coverage is defined by whether a typical user's downlink SINR exceeds target T; the result is described as the first analytical coverage expression for this setting.Source: §V · p. 6
• Coverage decreases with the number of users, pilot contamination, target SINR T, and path-loss exponent α > 2.As T approaches infinity, coverage approaches zero.Source: §V · p. 6
• At high downlink SNR ρd, coverage saturates in the interference-limited regime; as ρd approaches zero, it approaches zero in the noise-limited regime.Source: §V · p. 6
• Coverage increases with AP density λAP and saturates at large density; a similar pattern holds as antennas per AP increase.These dependencies are represented indirectly through W and are shown numerically.Source: §V · pp. 6–7

VI. ACHIEVABLE RATE

• The rate bound averages over AP distances after averaging the terms in ¯γk over small-scale fading, and its pre-log factor reflects pilot overhead.The bound is presented as a tractable lower bound for a large number of APs.Source: §VI · p. 7
• Theorem 2 gives a lower bound on the downlink average achievable rate per user with conjugate beamforming and PPP-distributed APs.Jensen's inequality is applied to avoid lengthy numerical evaluation of integrals over AP distances.Source: §VI · p. 7
• The rate decreases with the number of users K and pilot contamination, and also decreases with path-loss exponent α.The parameter dependencies are expressed through ˇγk.Source: §VI · p. 7
• The rate increases with antennas per AP N, AP density λAP, and transmit power, but saturates as N becomes large and has a ceiling at high ρd.At high ρd, the rate continues increasing with AP density; similar AP-density behavior is reported for small-cell systems under single-slope path loss.Source: §VI · p. 7

VII. NUMERICAL RESULTS

• The analysis examines coverage probability and achievable rate for PPP-located APs in cell-free massive MIMO, an AP-location model not previously considered in the analysis.The section discusses analytical expressions provided by Theorems 1 and 26.Source: §VII · p. 7
• Small cells associate each independent user with its nearest multi-antenna AP, while the remaining APs act as interferers.The comparison uses M = 4 antennas and K = 1 user per base station, with the current imperfect-CSIT model and no hardware impairments or channel aging.Source: §VII · p. 7
• Both theorems use a single-slope path-loss model, with generalization to multi-slope models identified as ongoing work.The authors state that the theorems could be generalized to more general path-loss models.Source: §VII · p. 7
• Monte Carlo simulations compare analytical expressions with simulated results for CF massive MIMO and small cells.The simulations average over 10^4 channel instances and 10^4 random AP topologies.Source: §VII · p. 8

A. Setup

• The simulation distributes APs at 40 APs/km^2 in a wrapped 1 km × 1 km area, with N = 5 antennas per AP and K = 10 users.The wrapped edges address the finite simulation area's boundary effects; default parameter values are listed in Table II.Source: A. Setup · p. 8, Table II · p. 8

1) Coverage Probability:

• CF massive MIMO provides higher coverage than SCs across target SINR values, and the advantage grows with AP density as coverage increases and then saturates.The discussion attributes the coverage advantage to favorable propagation, channel hardening, and suppression of inter-cell interference; saturation is linked to SINR invariance.Source: 1) Coverage Probability · p. 9, Figure 2 · p. 9
• At λAP = 80 APs/km2, increasing the number of users or shortening uplink training lowers rate, while the CF–SC gap grows with user count.For τtr = 20 samples, the gap rises from 1.6 Mbits/s at K = 10 to 2.1 Mbits/s at K = 55.Source: 1) Coverage Probability · pp. 9–10, Figure 3 · p. 9
• Increasing AP density raises achievable rate in both architectures, with CF massive MIMO outperforming SCs and a widening rate gap at higher density.The reported gap is almost 1.6 Mbits/s at λAP = 40 APs/km2 and almost 2 Mbits/s at λAP = 100 APs/km2.Source: Figure 4 · p. 9, 1) Coverage Probability · p. 10
• Achievable rate decreases monotonically as the path-loss exponent α increases in both architectures; users closer to APs affect rate more strongly.The discussion also reports improved performance with a larger number of APs.Source: 1) Coverage Probability · p. 10, Figure 5 · p. 10

VIII. CONCLUSION

• The paper models AP locations with a PPP and derives tractable closed-form coverage-probability and achievable-rate expressions; it identifies its analysis as the first to provide CF coverage with PPP-distributed APs.Its conclusion reports that CF massive MIMO outmatches SCs in both coverage and rate, with less performance deterioration as user count increases.Source: §VIII · p. 10

APPENDIX A PROOF OF PROPOSITION 1

• The Proposition 1 proof derives deterministic equivalents for the desired signal power and denominator terms, then substitutes them into the rate expression.The proof also checks the chosen precoder scaling.Source: Appendix A · p. 11

APPENDIX B PROOF OF THEOREM 1

• The Theorem 1 proof obtains conditional SINR and coverage expressions before averaging coverage over AP locations.The derivation uses a gamma approximation for W and Alzer’s inequality in the conditional expression.Source: Appendix B · p. 11
• The proof evaluates the intermediate expectations using Jensen’s inequality and independence before substituting them into the expressions completing the theorem.It derives I1 and I2 and then uses the binomial theorem to obtain the final expression.Source: Appendix B · pp. 12–13

APPENDIX C PROOF OF THEOREM 2

• The proof is divided into two subsections.The first provides a more tractable bound than (21); the second derives the PPP-averaged inverse SINR.Source: Appendix C · p. 13
• The first subsection’s bound is intended to allow averaging over a PPP realization of the APs.Source: Appendix C · p. 13
• The second subsection derives the PPP-averaged inverse SINR.Source: Appendix C · p. 13

A. Lower bound of the downlink SE

• The bound in (21) is rewritten using the inverse of γk.Source: A. Lower bound of the downlink SE · p. 13
• Jensen’s inequality is applied after rewriting the bound.Source: A. Lower bound of the downlink SE · p. 13
• The expectation is applied directly to the inverse SINR.Source: A. Lower bound of the downlink SE · p. 13

• The expected inverse SINR is rewritten by expressing each matrix trace as the sum of its entrywise elements.Source: B. Derivation of ˇγk · p. 13
• The expectation is evaluated by conditioning on the AP count in a radius-R ball, applying the law of large numbers, then removing the count conditioning as R → ∞.The ball contains M = Φ(B(o, R)) points and has area |B(o, R)|.Source: B. Derivation of ˇγk · p. 13
• With SINR derived by deterministic-equivalent analysis for M → ∞, the proof applies [48, Lemma 1] and uses EM[M] = λAP|B(o, R)| to obtain ˇγk.Other expectations needed in the lemma were derived in parts in Appendix B.Source: B. Derivation of ˇγk · p. 13