Foundations of User-Centric Cell-Free Massive MIMO

Introduction and Motivation

• Cellular networks produce severe within-cell rate disparities: a common 40 dB SNR difference between cell center and edge can correspond to 80 Mbit/s versus below 1 Mbit/s.Cell-edge users also suffer neighboring-cell interference, and their performance can dominate perceived network-wide service quality because they require more airtime for the same services.Source: Introduction and Motivation · pp. 9, 11
• The motivation is to replace cellular peak-rate optimization with uniformly high data rates across nearly all locations, since advanced Massive MIMO raises rates but does not remove cell-level rate variation.The monograph therefore frames cell-free networking as a clean-slate architecture for ubiquitous high-rate data services.Source: Introduction and Motivation · pp. 11–12

1.1 Cell-Free Networks

• The proposed cell-free architecture consists of geographically distributed APs jointly serving UEs through CPUs connected by fronthaul, with CSI, physical-layer signals, received data, and phase synchronization shared as needed.Each UE can be served by a dynamically changing subset of APs, potentially across CPUs, while a Master AP appoints the CPU responsible for associated uplink decoding and downlink encoding.Source: §1.1 · pp. 12, 13, 15
• Cell-free operation achieves almost uniformly high downlink rates, reaching 52–80 Mbit/s across the coverage area by jointly transmitting from surrounding APs and alleviating inter-cell interference.The comparison uses 64 APs on the same grid as a cellular network, where cellular operation exhibits large rate variations.Source: Figure 1.5 · p. 16, §1.1 · p. 17

1.2 Historical Background

• Further densification eventually suffers from increasing inter-cell interference, especially in ultra-dense networks, motivating cell-free designs that move beyond cellular capacity limits.Early distributed wireless concepts already proposed serving each UE with its nearest or preferred set of distributed antennas rather than a predefined cell.Source: §1.2 · pp. 18, 20
• Historically, cell-free networking converts cellular interference channels into multiaccess uplinks and broadcast downlinks, allowing distributed antennas to combine observations or design transmissions so interference becomes useful rather than unusable.Joint uplink decoding can improve SNR, while downlink cooperation can use precoding such as zero-forcing to avoid interference.Source: §1.2 · pp. 18, 20

1.2.1 Towards Standardization in 4G

• CoMP joint processing supports centralized CPU-based and decentralized AP-based architectures, but practical gains depend on limited-backhaul algorithms and, in pure TDD, channel reciprocity that avoids sharing downlink CSI.Centralized architectures send AP information to a CPU, while decentralized architectures retain local CSI at APs but still require data sharing; FDD/TDD standardization constraints can make signaling costs erase gains.Source: §1.2 · p. 21, §1.2.1 · p. 22, User-Centric Clustering · p. 24

User-Centric Clustering

• User-centric clustering lets each UE select preferred APs with overlapping cooperation sets, guaranteeing interference control irrespective of location, whereas disjoint network-centric clusters leave interference at cluster edges.The monograph adopts dynamic cooperation clustering for user-centric operation; well-designed user-centric clustering contains network-centric clustering as a special case.Source: Network-Centric Clustering · pp. 22–23, User-Centric Clustering · p. 24

Benefits

• Massive MIMO provides channel hardening and favorable propagation, but efficient operation still requires active interference suppression such as MMSE processing rather than relying on maximum-ratio processing alone.The many-antenna regime offers spatial diversity and near-orthogonal channels, yet advanced linear processing is needed to exploit it efficiently.Source: Benefits · pp. 25–26, Background · p. 28

Limitations

• Conventional cellular Massive MIMO can increase rates but retains inter-cell interference and large service variations, while compact planar arrays provide poorer spatial resolution than the wide ULAs often assumed in the literature.Large differences in UE distance further motivate a deployment architecture that delivers more uniform service quality across the coverage area.Source: Limitations · pp. 26–27

Background

• Cell-free networks originally use TDD pilots for local CSI and distributed MR processing, while semi-centralized or centralized processing can improve data rates by using statistical CSI or moving processing to the CPU at additional fronthaul cost.More recent results show that MMSE processing can substantially outperform MR and that distributed APs must actively suppress interference even when favorable propagation exists.Source: Background · pp. 27–28

The Roots of Cell-Free Massive MIMO

• The first cell-free studies served every UE from every AP, after which CoMP-inspired user-centric clustering and decentralized scalable clustering frameworks were introduced to reduce the impracticality of universal cooperation.These developments connect cell-free networking to earlier CoMP and ultra-dense-network clustering concepts.Source: User-Centric Clustering · p. 24, The Roots of Cell-Free Massive MIMO · p. 28
• Cell-free Massive MIMO is defined narrowly as the intersection of Cellular Massive MIMO, CoMP joint transmission, and ultra-dense networks, with a user-centric focus on scalable implementation and comparisons to cellular Massive MIMO and small cells.The monograph uses this intersection-based definition rather than treating conventional Massive MIMO, CoMP, and ultra-dense networks merely as special cases.Source: The Roots of Cell-Free Massive MIMO · pp. 29–30, Figure 1.8 · p. 29

Cell-Free Setup

• With 64 antennas in each compared setup, cell-free combining coherently accumulates received power from all APs and therefore always exceeds small-cell SNR, although its advantage is small when one AP dominates.Single-cell Massive MIMO SNR scales with the antenna count, whereas the cell-free-versus-Massive-MIMO comparison depends on UE location because it compares distributed and co-located channel norms.Source: §1.3 · p. 30, Massive MIMO Setup · p. 33, Cell-Free Setup · p. 34

Numerical Comparison

• Cell-free achieves 24.5 dB or higher at 95% of UE locations versus 6.5 dB for single-cell Massive MIMO, while outperforming small cells by 4 dB at that percentile.Cell-free has smaller SNR variations because UEs are generally closer to cooperating APs; its advantage over small cells becomes minor with one UE but grows with multiple UEs.Source: Numerical Comparison · p. 35, §1.4 · p. 44

Massive MIMO Setup

• In single-cell Massive MIMO, the maximum uplink SINR of UE k is given by (1.17), using a receive combining vector at the co-located array.The setup models the channel from UE k to the AP and applies combining to estimate its signal amid concurrent transmissions.Source: Massive MIMO Setup · pp. 36–37

Cellular Setup With Small Cells

• In small-cell networks, each UE selects the AP with the largest SINR, although interference can make that AP different from the one with the largest SNR and serve multiple UEs.Each AP has a single-AP detection perspective, so inter-cell interference is not jointly suppressed.Source: Cellular Setup With Small Cells · p. 38

Cell-Free Setup

• Cell-free jointly detects all UEs across L cooperating APs with an optimal combining vector, which is at least as good as selecting only the AP with highest local SINR.Comparing cell-free with small cells is direct through this suboptimal selection, while comparison with single-cell Massive MIMO requires simulations because their channel vectors differ.Source: Cell-Free Setup · pp. 38–39

1.3.2 Benefit 2: Better Ability to Manage Interference

• With K = 8 UEs, cell-free greatly outperforms small cells because joint AP processing manages interference that single-antenna APs cannot suppress.Massive MIMO and cell-free are barely affected by interference, whereas small-cell performance shifts far lower; cell-free has only a tiny lower tail when two UEs nearly coincide.Source: Numerical Comparison · pp. 39–40, §1.4 · p. 44

1.3.3 Benefit 3: Coherent Transmission Increases the SNR

• Coherent transmission raises SNR even when weaker-channel APs share fixed total power, producing local signal amplification rather than angular beams.Distributed focusing is strong within a region smaller than a wavelength; signals focused several wavelengths apart have small mutual interference.Source: §1.3.3 · pp. 41–42, §1.4 · p. 44

1.4. SUMMARY OF THE KEY POINTS IN SECTION 1

• Cellular architecture gives UEs widely varying SNRs, whereas cell-free surrounds each UE with cooperating APs and serves it through its nearest APs.The cell-free architecture is user-centric and uses relatively simple AP hardware cooperating within each UE’s area of influence.Source: §1.4 · p. 44
• Cell-free Massive MIMO combines Massive MIMO, ultra-dense networks, and coordinated multipoint processing, with novelty in co-designing them for scalable user-centric operation.Its three principal benefits are smaller SNR variations, joint interference management, and coherent transmission that can include weaker-channel APs.Source: §1.4 · p. 44

2.1 Definition of Cell-Free Massive MIMO

• A network of L distributed APs, each with N antennas, jointly serves K single-antenna UEs, with M = NL total antennas and fronthaul-connected CPUs coordinating APs.Each UE communicates with a selected AP subset; multi-antenna UEs are outside this monograph’s main analysis.Source: §2.1 · p. 46, Figure 2.1 · p. 46
• The intended regime is L ≫ K and M ≫ K, providing spatial degrees of freedom for separation, although the framework and signal-processing methods apply for any L, K, and N.Conventional Cellular Massive MIMO and small-cell networks arise as special cases, enabling comparisons within the same framework.Source: §2.1 · p. 47

2.2 User-Centric Dynamic Cooperation Clustering

• Restricting service to APs with non-negligible received power preserves performance while reducing fronthaul signaling and computational complexity.Only the selected APs receive each UE’s downlink data, return corresponding uplink estimates, and process signals for served UEs.Source: §2.2 · pp. 47–48
• Dynamic cooperation clustering assigns each UE k an adaptable AP set M_k based on location, service requirements, and interference, producing overlapping user-centric clusters rather than disjoint cells.The clusters are assumed fixed and known during data transmission because their variations occur over longer intervals than channel fading.Source: §2.2 · p. 48

2.3.1 Block Fading and Coherence Blocks

• The block-fading model treats each coherence block as time-invariant and frequency-flat, with independent channel realizations across blocks; τ_c = T_cB_c symbols occupy each block.The monograph uses this abstraction to analyze one block at a time while neglecting practical multicarrier details.Source: §2.3.1 · pp. 50–51
• Coherence blocks may contain τ_c = 200 symbols outdoors or τ_c = 250 000 indoors, while practical OFDM channels vary across subcarriers and correlate between adjacent blocks.The outdoor example uses T_c = 1 ms and B_c = 200 kHz; the indoor example uses T_c = 50 ms and B_c = 5 MHz.Source: §2.3.1 · pp. 52–53

2.3.2 Time-Division Duplex Protocol

• The synchronized TDD protocol uses uplink pilots to estimate reciprocal channels, enabling coherent uplink and downlink processing without explicit downlink pilots.The coherence block is partitioned into pilots, uplink data, and downlink data under τ_c = τ_p + τ_u + τ_d.Source: §2.3.2 · pp. 54–55, Figure 2.4 · p. 54

2.3.3 Uplink System Model / 2.3.4 Downlink System Model

• In the uplink, APs receive superposed UE signals and apply CSI-based combining only for served UEs; in the downlink, each AP transmits a precoded superposition for its served UEs.Effective combining and precoding vectors are set to zero for AP–UE pairs outside the corresponding cooperation cluster.Source: §2.3.3 · p. 56, §2.3.4 · p. 57

2.3.4 Downlink System Model

• The downlink model separates desired signal, inter-user interference, and noise, while precoding-vector selection depends on channel estimates and AP cooperation.The general downlink expression is retained for performance analysis, although an alternative form can be obtained from the effective transmit precoding vectors.Source: §2.3.4 · p. 58

2.4 Network Scalability

• Scalability requires channel estimation, data reception and transmission, fronthaul signaling, and power allocation to remain practically implementable as K →∞.Cell-free networks face this challenge because APs can be directly or indirectly involved in serving many UEs, while their computational and fronthaul resources are finite.Source: §2.4 · p. 59

2.4.1 Example of Unscalable Network Operation

• Serving all K UEs at one AP is unscalable because channel-estimation complexity and stored channel estimates grow at least linearly with K.As K →∞, the required complexity approaches infinity and any finite-sized memory module becomes insufficient.Source: §2.4.1 · p. 60, channel estimation · p. 60

2.4.2 A Sufficient Condition for Scalability

• Finite |D_l| makes the first three scalability tasks feasible, while local power decisions satisfying Lemma 2.2 ensure all scalability conditions as K →∞.Each AP computes signal-processing quantities and exchanges fronthaul data only for its served UEs; transmit powers must depend only on local information, with each UE’s power selected by one serving AP.Source: §2.4.2 · pp. 61–62

2.5 Channel Modeling

• Correlated Rayleigh fading is analytically tractable but can inadequately represent few-path or strong-LoS channels, for which finite-path or Rician models are alternatives not covered here.Using the same spatial correlation matrices, correlated Rayleigh fading can serve as a worst-case approximation of these more structured channels.Source: §2.5 · p. 64, §2.5.3 · p. 68

2.5.1 Correlated Rayleigh Fading

• The monograph uses correlated Rayleigh fading, where R_kl captures pathloss, shadowing, antenna gains, and spatial correlation, and assumes independently distributed channels across APs.The collective correlation matrix is block diagonal, and the correlation matrices are assumed available wherever needed.Source: §2.5.1 · p. 65

2.5.2 Large-Scale Fading Model for Urban Deployments

• The simulation model places APs 10 meters above densely populated urban UE locations and uses a 3GPP Urban Microcell large-scale fading model with distance-dependent and correlated shadow fading.These assumptions apply to simulations; the analytical results apply to Rayleigh-fading propagation environments more generally.Source: §2.5.2 · pp. 65–66

2.5.3 Local Scattering Model for Spatial Correlation

• Spatial correlation is modeled through antenna-array geometry and multipath angular distributions, using a half-wavelength-spaced ULA and a Gaussian local-scattering model in simulations.The local-scattering model centers multipath around the AP–UE direction and characterizes angular variation through azimuth and elevation ASDs.Source: §2.5.3 · pp. 66–67
• With σφ = σθ = 5°, the first four eigenvalues contribute 99.99% of the total, while the largest contributes 80%; at 20°, the largest-to-smallest eigenvalue ratio remains 425.Increasing angular standard deviation reduces eigenvalue spread, but substantial spatial correlation remains relative to uncorrelated Rayleigh fading.Source: §2.5.3 · pp. 68–69, Figure 2.6 · p. 69

2.6 Channel Hardening and Favorable Propagation

• Cell-free channel hardening and favorable propagation depend on spatial correlation, total antennas M = LN, large-scale-fading variation across APs, and power allocation.These properties are analyzed for multi-AP service but are not required for the monograph’s other results.Source: §2.6 · p. 70

2.6.1 Channel Hardening

• Channel hardening occurs when the combined effective channel from serving APs has small variance around its mean, making it approximately deterministic across coherence blocks.The formal definition combines the serving AP set and fixed downlink power-allocation coefficients under MR precoding.Source: §2.6.1 · p. 71, Definition and Sufficient Condition… · pp. 71–72

Definition and Sufficient Condition for Channel Hardening

• Equal uncorrelated spatial eigenvalues maximize channel hardening, whereas spatial correlation slows convergence by requiring more antennas for the same hardening degree.The hardening condition is approximately achieved when expression (2.24) is close to zero.Source: Definition and Sufficient Condition… · p. 73

Impact of the Number and Geographical Distribution of APs

• With equal AP contributions, channel hardening depends on total AP antennas, but unequal large-scale fading generally requires more antennas than a co-located cellular deployment.Power allocation can equalize AP contributions and improve hardening, though allocating more power to nearby APs may provide higher total received power.Source: Impact of the Number and Geographical… · p. 74
• For 64 distributed APs, 8–10 antennas per AP produce hardening comparable to a 64-antenna cellular reference, while single-antenna APs remain substantially worse.At N = 5, the median reaches the reference but half of realizations are substantially larger; N = 8 or 10 requires an order-of-magnitude more total antennas.Source: Impact of the Number and Geographical… · pp. 75–76, Figure 2.7 · p. 75

Takeaways

• Cell-free networks generally have less channel hardening than cellular systems with the same number of serving antennas, although channel hardening remains useful but not mandatory.It helps capacity lower bounds approach true capacity and motivates average rather than coherence-block power optimization.Source: Takeaways · p. 76

2.6.2 Favorable Propagation

• Favorable propagation makes inter-user interference negligible under MR processing when effective interfering channels vanish relative to desired channels, but it is not necessary because interference-suppressing processing can compensate.Channel hardening and favorable propagation are distinct: a UE may exhibit one without the other, and different UE pairs may differ.Source: §2.6.2 · p. 76, Definition and Sufficient Condition… · pp. 77–78, Takeaways · p. 81

Definition and Sufficient Condition for Favorable Propagation

• Favorable propagation is strongest when UEs have very different spatial directivity, whereas nearly identical spatial correlation matrices produce the weakest condition.The sufficient condition evaluates the normalized variance of the effective interfering channel; values near zero indicate approximate favorable propagation.Source: Definition and Sufficient Condition… · p. 78

Impact of the Geographical Distribution of APs

• Favorable propagation improves with antennas and favorable AP geometry: it may occur with N = 1 for distant serving APs, but closely spaced UEs may need around N = 10.Power allocation can reduce interference by lowering power on APs with large cross-channel fading, while distant UEs can still show substantial realization-to-realization variation.Source: Impact of the Geographical… · pp. 79–80, Figure 2.8 · p. 80

3.1 Estimation Theory for Gaussian Variables

• Gaussian MMSE estimation provides closed-form channel estimates and error correlations under independent Gaussian noise or interference.For x ∼ NC(0, R) observed through a known pilot with Gaussian disturbance, the MMSE estimate and estimation-error correlation matrix are given in closed form; the estimate and error are independent.Source: §3.1 · pp. 84–85

3.2 Capacity Bounds and Spectral Efficiency

• Achievable spectral efficiency is measured in bit/s/Hz and represents reliably transmitted information per complex-valued sample, while channel capacity is the maximum achievable spectral efficiency.Information rate equals spectral efficiency multiplied by bandwidth B.Source: §3.2 · p. 85

3.2.1 Capacity Bounds for Deterministic Channels

• For deterministic AWGN channels, every spectral efficiency up to capacity is achievable with Gaussian signaling, and practical operation can approach capacity using long finite blocks.The capacity-achieving input is x ∼ NC(0, p), and arbitrarily low error probability is approached asymptotically.Source: §3.2.1 · pp. 86–87
• With interference, treating uncorrelated non-Gaussian interference as independent Gaussian noise yields a convenient capacity lower bound that is practically achievable through an effective SINR.The bound is generally below capacity under strong interference but is provably optimal in the low-interference regime and can use AWGN-designed codes.Source: §3.2.1 · pp. 87–88

3.2.2 Capacity Bounds for Random Channels

• For random block-fading channels, ergodic capacity averages the logarithmic rate over channel realizations and is achieved with Gaussian signaling when the receiver knows the realization.The random-channel capacity differs from deterministic capacity by placing an expectation in front of the logarithm.Source: §3.2.2 · p. 89
• For random channels with interference, treating interference as noise produces an achievable ergodic spectral-efficiency lower bound expressed through an effective instantaneous SINR.The expectation is over the channel and interference-variance variables, and fading-AWGN codes can practically achieve the bound; channel-estimation uncertainty can also be included as interference.Source: §3.2.2 · pp. 90–91

3.3 Maximization of Rayleigh Quotients

• Maximizing a standard Rayleigh quotient yields maximum-ratio combining, with the optimal combiner proportional to the channel vector v = h.This result follows from the Cauchy–Schwarz inequality and maximizes SNR for independent identically distributed noise.Source: §3.3 · p. 92
• For colored noise with positive definite covariance B, the generalized Rayleigh quotient is maximized by v = B−1h or v = (hhH + B)−1h.The generalized formulation extends matched filtering to receive combining under correlated noise.Source: §3.3 · p. 92

3.3. MAXIMIZATION OF RAYLEIGH QUOTIENTS

• The generalized Rayleigh quotient is maximized by v = cB−1h for any c ≠ 0, a result reused for uplink receiver optimization.The solution is nonunique because any nonzero scaling of B−1h achieves the same maximum.Source: §3.3 · p. 93

3.4 Optimization Algorithms for Utility Maximization

• Achievable SEs form a feasible SE region whose outer boundary represents tradeoffs, summarized through max-min fairness or sum SE scalarization.Max-min fairness equalizes users’ SEs, whereas sum SE maximizes total transmitted bits at the cost of potentially unequal user rates.Source: §3.4 · pp. 95–96, Figure 3.2 · p. 96

3.4.1 Max-Min SE Fairness

• Max-min SE fairness is reformulated with an auxiliary minimum-SINR variable and solved by bisection over feasibility problems under linear power constraints.To improve convergence in large cell-free networks, the feasibility subproblem additionally minimizes total power, encouraging equal SINRs while preserving optimality.Source: §3.4.1 · pp. 97–98
• When the SINR satisfies the stated monotonicity and scalability conditions, a fixed-point algorithm can solve the max-min fairness problem and converge to its optimum.At the optimum, all UEs have positive power and at least one power constraint is active.Source: §3.4.1 · p. 100

3.4.2 Sum SE Maximization

• Sum-SE maximization is generally nonconvex, so weighted MMSE reformulation with block coordinate descent provides a practical method converging to a stationary point and local optimum.The reformulation is globally equivalent to the original problem because the optimal weight is d_k = 1/e_k = 1 + SINR_k.Source: §3.4.2 · pp. 101–102
• 3.4.2 Sum SE Maximization requires optimal convex power-allocation subproblems for block coordinate descent to converge to a local optimum.The power update is performed while other variables remain fixed, and standard convex-optimization algorithms can obtain the optimum for the considered problems.Source: §3.4.2 · pp. 103–104

3.5 Summary of the Key Points in Section 3

• Section 3 provides MMSE channel estimation, achievable spectral-efficiency lower bounds, generalized-Rayleigh-quotient maximization, and max-min or sum-SE utility optimization algorithms.The capacity lower bounds apply to deterministic and fading channels, are achievable with AWGN channel codes, and support later spectral-efficiency analyses; Algorithms 3.1 and 3.3 solve the max-min fairness and sum-SE utility problems, respectively.Source: §3.5 · p. 105

4.1 Uplink Pilot Transmission

• Uplink pilots let serving APs estimate channels once per coherence block, either locally or at the CPU, while independent channel vectors permit separate AP-side estimation without loss of optimality.Only τp mutually orthogonal pilots are available, so practical networks with K ≫ τc must reuse pilots among multiple UEs.Source: §4.1 · pp. 107–108

4.2 MMSE Channel Estimation

• The MMSE estimator is optimal with full channel-statistical knowledge, producing independent Gaussian estimates and errors while minimizing mean-square estimation error.The estimator requires correlation matrices for the desired channel and pilot-sharing UEs, which can be precomputed and exchanged with negligible fronthaul signaling because channel statistics are constant during transmission.Source: §4.2 · pp. 109–111
• MMSE channel-estimation complexity remains scalable as K →∞ when each AP’s served-UE set remains finite, despite matrix operations proportional to N^2.This scalability condition aligns with the monograph’s sufficient condition for scalable cell-free operation.Source: §4.2 · pp. 110–111

4.2.1 Normalized MSE

• Pilot processing yields an effective SNR enhanced by the pilot-processing gain τp, while normalized MSE ranges from 0 for perfect estimation to 1 when only noise and interference are received.For τp = 10 pilot samples, the effective SNR is 10 dB above the nominal single-sample SNR; increasing pilot length or concentrating pilot power can improve it.Source: §4.2.1 · pp. 111–113

4.2.2 Pilot Contamination

• Pilot-sharing interference increases NMSE, and with spatially correlated channels, estimation improves when interfering UEs have substantially different spatial correlation properties.With uncorrelated Rayleigh fading, multiple antennas provide neither an NMSE advantage nor disadvantage relative to the single-antenna expression.Source: §4.2.2 · p. 113
• Pilot contamination also makes channel estimates of statistically independent pilot-sharing UEs correlated, complicating interference mitigation in both uplink and downlink.Under uncorrelated Rayleigh fading, estimates sharing a pilot become fully correlated up to a scaling factor,.Source: §4.2.2 · p. 114

4.2.3 MMSE Estimation of the Collective Channel

• Scalable cell-free processing estimates only the serving AP components of each UE’s collective channel, and its collective NMSE normalizes summed MSEs by summed channel gains rather than averaging individual NMSEs.Pilot-sharing UEs increase the collective NMSE and create correlation between their respective channel estimates.Source: §4.2.3 · pp. 115–116

4.3.1 Impact of the Cell-Free Architecture

• Serving more than one AP does not necessarily improve collective-channel estimation: with unequal pilot effective SNRs, the exact NMSE exceeds the dominant-AP approximation, and the best single AP can yield the lowest NMSE.The approximation is exact when only one AP serves the UE or all serving APs have equal pilot effective SNRs.Source: §4.3.1 · p. 117, Figure 4.1 · p. 117
• Cell-free combining can produce higher data-detection SNR than small-cell and Cellular Massive MIMO setups despite lower average estimation quality, through coherent processing across multiple APs.Small-cell networks achieve better average NMSE because they select the AP with the best channel, but this does not imply higher communication performance for cell-free operation.Source: §4.3.1 · pp. 118–120, Figure 4.3 · p. 120

4.3.2 Impact of Pilot Contamination

• Pilot contamination increases NMSE as the interfering pilot strengthens, but is negligible when the interference is 10 dB below the noise, making nearby interfering UEs the main concern.Pilots should preferably differ among UEs served by the same AP, and longer pilots can limit contamination when assigned properly, although the exact distance thresholds depend on transmit power, bandwidth, and propagation model.Source: §4.3.2 · pp. 121–123

Impact of Spatial Correlation on Estimation Accuracy

• For a fixed average channel gain, uncorrelated fading produces the largest NMSE, whereas rank-one spatial correlation produces the smallest because stronger correlation provides more exploitable channel structure.The NMSE depends on the eigenvalues of the correlation matrix, not its eigenvectors.Source: Impact of Spatial Correlation on… · pp. 124–125
• Spatially correlated channels can achieve an NMSE one order of magnitude below the uncorrelated case, but this benefit is essentially lost when the azimuth ASD reaches 40° or more.Increasing AP antennas reduces NMSE, although gains become marginal for N ≥8 under the studied correlation model.Source: Figure 4.6 · p. 125, Impact of Spatial Correlation on… · p. 126, Figure 4.7 · p. 126

Impact of Spatial Correlation on Pilot Contamination

• Spatially orthogonal UE correlation matrices, R1R2 = 0N×N, completely eliminate pilot-contamination effects in theory; assigning pilots to minimize tr(R1R2) is the practical rule-of-thumb.When UE eigenspaces are well separated, the NMSE remains below 0.1 even when the interfering pilot has the same strength as the desired signal.Source: Impact of Spatial Correlation on Pilot… · p. 127
• Spatial correlation always lowers or matches NMSE relative to uncorrelated fading under pilot contamination, including when desired and interfering UEs have identical angles.The improvement is strongest when pilot-sharing UEs have sufficiently different spatial correlation characteristics.Source: Impact of Spatial Correlation on Pilot… · p. 128, Figure 4.8 · p. 128

4.4 Pilot Assignment and Dynamic Cooperation Cluster Formation

• Pilot assignment should use spatial correlation matrices rather than geometry, but exhaustive optimization is exponential, making a locally executed greedy algorithm practically necessary.The matrices capture average channel gains, including shadow fading, and spatial characteristics; the greedy method avoids assigning closely spaced, similarly strong UEs the same pilot.Source: §4.4 · p. 129

An Algorithm for Joint Pilot Assignment and Cluster Formation

• The joint algorithm assigns initial orthogonal pilots, then assigns each remaining UE the pilot with minimum neighboring-AP contamination and forms clusters by selecting the strongest UE per pilot at each AP.This local rule is practical but can be suboptimal for multi-antenna APs that could spatially separate same-pilot UEs.Source: §4.4 · pp. 129–130, An Algorithm for Joint Pilot… · p. 130
• Dynamic initial access lets each UE select a Master AP, receive a locally preferred pilot, and trigger neighboring APs to update service clusters, while guaranteeing service may require accepting pilot contamination.In the ultra-dense regime L ≫ K, an AP rarely has the strongest channel to more than τp UEs, but the risk is not eliminated.Source: An Algorithm for Joint Pilot… · pp. 132–133, §4.5 · p. 135

5.1 Centralized Uplink Operation

• Centralized uplink operation gathers pilot and data signals from serving APs at the CPU for channel estimation, receive combining, and data detection.The CPU is a logical entity that can be divided across geographically distributed edge-cloud processors, with each UE associated with a nearby processor.Source: Uplink Operation · p. 136, §5.1 · p. 137

5.1.1 Spectral Efficiency With Centralized Operation

• The centralized achievable-SE analysis uses partial MMSE channel estimates and treats the unknown desired-channel component as additional interference in an effective-SINR capacity lower bound.Theorem 5.1 applies to multi-antenna APs, arbitrary correlated fading, and arbitrary dynamic cooperation clusters; its expectation can be evaluated by Monte Carlo sampling.Source: §5.1 · p. 139, §5.1.1 · pp. 140–141

5.1.2 An Alternative Spectral Efficiency Expression

• Theorem 5.2 provides a simpler use-and-then-forget achievable-SE bound applicable with any channel estimator and fading model, but it is expected to be smaller than SE(ul,c).Unlike Theorem 5.1, this bound has no expectation outside the logarithm because it treats the channel as deterministic for detection.Source: §5.1.2 · p. 142

5.1.3 Optimal Receive Combining

• The MMSE combining vector maximizes the instantaneous SINR and minimizes conditional data-detection MSE by combining the desired-channel estimate with spatial whitening of interference and noise.Its matrix inverse balances desired-signal strength against interference suppression, while only serving AP dimensions affect the computation.Source: §5.1.3 · pp. 143–144
• Although optimal centralized MMSE combining depends computationally on K and is therefore unscalable, its matrix operations involve only the serving APs for each UE.Reordering APs exposes an upper block of dimension N|M_k| × N|M_k|, so the computation depends on |M_k| rather than the total number of APs.Source: §5.1.3 · pp. 145–146

5.1.4 Scalable Combining Schemes for Centralized Operation

• Scalable centralized combining schemes are introduced with computational complexity per UE independent of K, beginning with maximum-ratio combining.These alternatives address the scalability limitation of optimal centralized MMSE combining.Source: §5.1.4 · p. 148
• Centralized combining includes low-complexity MR, interference-aware P-MMSE, and P-RZF schemes whose scalability depends on restricting processing to serving and strongly interfering UEs.MR is scalable but can perform poorly under interference; P-MMSE approximates optimal MMSE, while P-RZF trades interference suppression against desired-signal preservation.Source: §5.1.4 · p. 148, Partial MMSE Combining · p. 148, Partial Regularized Zero-Forcing… · p. 151

Partial MMSE Combining

• P-MMSE remains scalable because its matrix size and computation are independent of K, with |S_k| bounded by (τ_p − 1)|M_k| + 1.P-MMSE coincides with MMSE only when UE k is served by all APs; additional interfering UEs can be included in S_k without violating scalability.Source: Partial MMSE Combining · p. 149

Partial Regularized Zero-Forcing Combining

• P-RZF replaces P-MMSE’s N|M_k| × N|M_k| inverse with an |S_k| × |S_k| regularized inverse, reducing complexity while potentially lowering spectral efficiency.The regularization balances interference suppression with preservation of desired-signal power, and the resulting complexity is independent of K.Source: Partial Regularized Zero-Forcing… · p. 151

5.1.5 Fronthaul Signaling Load for Centralized Operation

• Centralized fronthaul requires (τ_p + τ_u)NL complex scalars per coherence block, irrespective of the CPU combining scheme and independent of K.This count assumes equal precision for all scalars and uses τ_pN as an upper bound for pilot signaling.Source: §5.1.5 · pp. 151–152, Table 5.2 · p. 152

5.2 Distributed Uplink Operation

• Distributed uplink operation moves channel estimation and local combining to the APs, then sends local data estimates to the CPU for coherent fusion and final detection.The architecture targets L ≫ K deployments, limits CPU dependence on the number of APs, and reduces centralized fronthaul signaling and quantization distortion.Source: §5.2 · pp. 153–154

5.2.1 Spectral Efficiency and Optimal LSFD Weights

• The distributed achievable SE is expressed through an effective SINR for arbitrary local combining and LSFD weights, while optimal LSFD maximizes that SINR from channel statistics.L-MMSE locally minimizes conditional MSE but is not scalable because its complexity grows with K; optimal LSFD is likewise unscalable because its required statistical parameters grow with K.Source: §5.2.1 · pp. 156–158
• The UatF-based spectral-efficiency bound is a state-of-the-art capacity lower bound, but may underestimate achievable SE when channel hardening is weak with few antennas per AP.The bound is appropriate because APs combine locally while the CPU detects data without channel knowledge; its tightness is evaluated later.Source: §5.2.1 · p. 159

5.2.2 Scalable Schemes for Distributed Operation

• Distributed L-MMSE combining and optimal LSFD are not scalable because complexity and fronthaul signaling increase with the number of UEs, motivating suboptimal scalable alternatives.The scalable alternatives are designed from the corresponding preferable distributed schemes.Source: §5.2.2 · pp. 159–160

Scalable Local Receive Combining

• Local MR combining is computationally simple and analytically tractable, while LP-MMSE remains scalable as K →∞ and reduces per-UE complexity by inverting an N × N matrix.MR requires no additional combining computations and yields closed-form SE; LP-MMSE expectations generally require numerical Monte Carlo evaluation.Source: Scalable Local Receive Combining · pp. 160, 162, 164
• With N = 1, the effective SINR separates coherent desired-signal combining, non-coherent interference, pilot-contamination interference, and noise across the serving APs.The signal combines weighted channel-estimate variances coherently, whereas ordinary interference adds powers without squaring; pilot contamination has a coherent form.Source: Scalable Local Receive Combining · pp. 161–162

Scalable Large-Scale Fading Decoding

• No-LSFD fusion is communication-light but can degrade SE because it weights all APs equally, whereas n-opt LSFD limits interference consideration to overlapping serving clusters and has K-independent fronthaul.The n-opt LSFD scheme requires fewer statistical parameters than optimal LSFD, while local MR with unit LSFD reproduces centralized MR but may perform poorly under interference.Source: Scalable Local Receive Combining · p. 162, Scalable Large-Scale Fading Decoding · pp. 164–165

5.2.3 Fronthaul Signaling Load for Distributed Operation

• Combining n-opt LSFD with any scalable local combiner makes uplink distributed operation completely scalable, whereas optimal LSFD requires (3K + 1) /2 statistical scalars per served UE.The local data estimates require τu|Dl| complex scalars per coherence block and do not grow with K under the stated pilot assignment.Source: §5.2.3 · pp. 165–166

5.3 Running Example

• The running example models a 1 km × 1 km wrap-around network with M = LN = 400 antennas, either L = 400, N = 1 or L = 100, N = 4, under 20 MHz bandwidth.It uses τc = 200 samples, 100 mW maximum UE uplink power, 200 mW maximum AP downlink power, and −94 dBm receiver noise power.Source: §5.3 · p. 167, Table 5.4 · p. 168

Cellular Network

• Cellular benchmarks treat each UE as served by one AP and use optimal, non-scalable L-MMSE combining to represent the best conceivable small-cell implementation.The same formulas also cover Cellular Massive MIMO with fewer APs and more antennas per AP than the corresponding cell-free network.Source: Cellular Network · pp. 169–170

5.4.1 Performance With Centralized Operation

• Centralized P-MMSE with a DCC achieves nearly the same SE as unscalable MMSE serving all APs, while P-RZF is slightly worse and MR is substantially worse.This makes properly designed scalable DCC processing nearly as effective as the optimal unscalable solution.Source: §5.4.1 · p. 175, Figure 5.4 · p. 176
• With centralized operation, many single-antenna APs provide better low-tail SE than fewer multi-antenna APs because spreading antennas yields macro-diversity gain.The difference is especially large at the lower end of the CDF, and P-RZF suffers more from correlated estimation errors with multi-antenna APs.Source: §5.4.1 · pp. 175, 177

Tightness of the SE Expressions

• Centralized SE bounds are tight: genie-aided SE is only slightly higher than the original expression, while the UatF bound is almost negligibly lower.Both Theorem 5.1 and Theorem 5.2 therefore represent practically achievable centralized performance.Source: Tightness of the SE Expressions · p. 177, Figure 5.5 · p. 178

5.4.2 Performance With Distributed Operation

• Distributed LP-MMSE with scalable n-opt LSFD and subset AP selection achieves essentially the same SE as locally optimal L-MMSE with optimal LSFD, whereas MR performs substantially worse.The scalable implementation requires all three simplifications—subset AP participation, n-opt LSFD, and LP-MMSE combining—without meaningful performance loss.Source: §5.4.2 · pp. 177–178, Figure 5.6 · p. 179
• Distributed SE curves are more spread with N = 4 than N = 1, while multi-antenna APs improve channel estimation through spatial correlation but can worsen low-tail performance.Compared with centralized operation, distributed processing benefits more unevenly from local arrays because APs suppress interference locally rather than jointly at the CPU.Source: §5.4.2 · pp. 178, 180

Tightness of the SE Expressions

• For distributed operation, the SE gap to genie-aided detection is reasonably small with LP-MMSE but substantial with MR because MR creates larger desired-signal and interference-power variations.The gap can be reduced by normalizing MR as explained in [2, Sec. 4.2.1] and.Source: Tightness of the SE Expressions · p. 180, Figure 5.7 · p. 181

Benefits of LSFD

• LSFD substantially improves SE for all UEs, while user-centric AP selection causes only minor LP-MMSE degradation relative to L-MMSE serving all APs.The results identify LSFD as essential; LP-MMSE with LSFD is more effective than MR, whose low SE is only partially addressable through power control.Source: Benefits of LSFD · pp. 181–182, Figure 5.8 · p. 182

5.4.3 Comparison Between Cell-Free and Cellular Networks

• The evaluation uses equal total antennas, LN = 400, identical UE locations and pilot assignments, and compares user-centric cell-free operation with Cellular Massive MIMO and small cells.The cellular baseline uses four 100-antenna APs, while cell-free and small-cell setups use either 400 single-antenna APs or 100 four-antenna APs.Source: §5.4.3 · pp. 182, 183, 185, Table 5.6 · p. 183, Figure 5.9 · p. 184
• Centralized cell-free operation with P-MMSE combining achieves the largest SEs, while distributed cell-free operation improves 90%-likely SE by 86% over small-cell and 55% over Cellular Massive MIMO for L = 400, N = 1.For L = 100 and N = 4, distributed cell-free operation improves 90%-likely SE by 24% over small-cell and 90% over Cellular Massive MIMO; centralized operation remains best overall and cell-free operation provides more uniform service.Source: §5.4.3 · pp. 185–186

5.4.4 Impact of Spatial Correlation

• Spatial correlation improves average SE for centralized operation but degrades it for distributed and small-cell systems, with distributed cell-free and small-cell performance nearly equal under highly correlated channels.Centralized processing exploits correlation to separate UEs across many antennas, whereas local combining mainly experiences its negative effects; cell-free benefits more as correlation decreases.Source: §5.4.4 · pp. 186–187, Figure 5.10 · p. 187

5.4.5 Impact of the Pilot Sequence Length

• Increasing pilot length initially improves uplink SE by reducing pilot contamination and expanding AP cooperation, but SE eventually declines from the pre-log penalty, with saturation at τp = 20 for centralized, τp = 10 for distributed, and τp = 5 for small-cell operation.The distributed and centralized gains saturate when added estimation quality no longer compensates for fewer uplink-data symbols, while small-cell interference suppression benefits little from longer pilots.Source: §5.4.5 · pp. 187–188, Figure 5.11 · p. 188

Fronthaul Signaling

• Scalable distributed DCC keeps data fronthaul from growing with K, whereas all-UE cooperation and opt LSFD produce non-scalable signaling; distributed fronthaul is nevertheless larger than centralized fronthaul.Centralized operation has UE-independent per-block fronthaul, while n-opt LSFD with DCC scales because each AP serves at most τp UEs.Source: Fronthaul Signaling · pp. 189, 191, Figure 5.12 · p. 190

Computational Complexity

• Scalable DCC can reduce centralized computational complexity as K grows, and distributed LP-MMSE and MR complexity does not grow with K, unlike unscalable distributed L-MMSE schemes.Distributed processing requires substantially fewer multiplications overall; P-RZF can be useful for small K, but its complexity increases with K relative to P-MMSE.Source: Computational Complexity · pp. 191, 193, Figure 5.13 · p. 192

5.5. SUMMARY OF THE KEY POINTS IN SECTION 5

• Centralized uplink processing forwards all AP signals to the CPU for channel estimation and combining, while distributed processing combines locally and forwards weighted local estimates through LSFD.MMSE maximizes centralized SE but is non-scalable; P-MMSE, P-RZF, and MR are scalable alternatives, while distributed LP-MMSE and MR replace non-scalable L-MMSE and optimal LSFD.Source: §5.5 · p. 194
• Scalable operation achieves negligible SE loss compared with serving all APs and UEs while substantially reducing computational complexity; centralized processing provides higher SE and less fronthaul, whereas distributed processing has lower complexity.With a fixed total number of AP antennas, many single-antenna APs favor centralized operation and weak users, while fewer multi-antenna APs provide local interference suppression for strong users in distributed operation.Source: §5.5 · p. 195

Downlink Operation

• Downlink operation is organized around centralized CPU precoding and distributed AP-local precoding, with spectral-efficiency analysis and scheme comparisons covering both implementations.In centralized operation, the CPU performs channel estimation, data encoding, and precoding; in distributed operation, APs perform local processing while the CPU only encodes data.Source: Downlink Operation · p. 196, Figure 6.2 · p. 200

6.1 Centralized Downlink Operation

• Centralized precoding lets the CPU coordinate AP signals so interference contributions can cancel across APs, a capability unavailable when APs independently select their transmissions.The CPU can adjust transmit powers and phases so that a + b + c ≈0, whereas distributed operation limits interference by making individual contributions small.Source: §6.1 · pp. 198–199, Figure 6.1 · p. 199

6.1.1 Spectral Efficiency With Centralized Operation

• The centralized achievable-SE expression applies to arbitrary precoding and cooperation-cluster selection, treating the effective channel through its average and an effective SINR with a downlink-data pre-log factor.The result can be evaluated for any channel distribution and precoder using Monte Carlo averages over channel realizations.Source: §6.1.1 · pp. 202–203

6.1.2 Centralized Transmit Precoding

• Centralized downlink precoding is jointly coupled across UEs, so there is no per-UE closed-form optimum; uplink-downlink duality instead motivates using scaled uplink combining directions for downlink precoding.The duality result makes the uplink effective SINRs achievable in the downlink, but downlink power is optimized separately because duality-based allocation can violate per-AP constraints or underuse available downlink power.Source: §6.1.2 · pp. 203–205

MMSE Precoding

• MMSE precoding balances desired-signal strength against interference and is nearly optimal under flexible downlink power allocation, but its computational cost grows with the number of UEs and is therefore not scalable.MMSE precoding is theoretically optimal only under the duality-specified power allocation and otherwise is described as nearly optimal.Source: MMSE Precoding · pp. 206–207

P-MMSE and P-RZF Precoding

• P-MMSE and P-RZF precoding are introduced as scalable downlink counterparts of the corresponding scalable uplink combining schemes.Their construction is motivated by uplink-downlink duality and uses the centralized precoding-vector formulation.Source: §6.1.2 · p. 206, P-MMSE and P-RZF Precoding · p. 207
• P-MMSE and P-RZF reduce centralized precoding complexity by restricting processing to relevant user subsets; P-RZF additionally neglects estimation-error correlation, while matching uplink and downlink choices avoids extra computation.With a properly designed DCC, P-MMSE can have a small performance difference from MMSE because the selected subset captures the relevant AP influence.Source: P-MMSE and P-RZF Precoding · pp. 207–208

6.1.3 Fronthaul Signaling Load for Centralized Operation

• Centralized downlink fronthaul requires τdN complex scalars per AP per coherence block, independent of K and the precoding scheme, making it scalable.The CPU performs all signal processing, so APs need not know the precoding scheme or number of served UEs.Source: §6.1.3 · pp. 208–209, Table 6.1 · p. 209

6.2. Distributed Downlink Operation

• Distributed operation has the same received-signal structure and achievable-SE characterization as centralized operation, but precoding is selected locally from each AP’s channel estimates.Power allocation provides the implicit second-stage coordination across APs.Source: §6.2.1 · p. 210, §6.2 · p. 210

6.2.2 Local Transmit Precoding

• Local transmit precoding separates rapidly changing directivity, recomputed every coherence block from local estimates, from statistically based power allocation maintained over many blocks.Uplink-downlink duality motivates reusing normalized local combining vectors, so identical uplink and downlink vectors incur no extra computation.Source: §6.2.2 · pp. 211–212

L-MMSE Precoding

• L-MMSE precoding is expected to provide the highest distributed SE, but each AP suppresses only its own interference using N spatial degrees of freedom, unlike centralized cooperation across N|Mk| antennas.L-MMSE is not scalable because its computation includes a summation over all UEs.Source: L-MMSE Precoding · pp. 212–213

MR Precoding

• MR precoding maximizes desired received power and yields closed-form effective-SINR expressions, with coherent combining across serving APs producing a power gain.Its interference includes non-coherent terms and pilot-contamination-induced coherent interference; performance depends on channel-estimation quality and favorable propagation.Source: MR Precoding · pp. 213–216

LP-MMSE Precoding

• LP-MMSE provides a scalable approximation to L-MMSE by considering only the UEs served by each AP and requires no additional downlink computation when paired with the same uplink scheme.It approximately matches L-MMSE when an AP serves all UEs in its area of influence.Source: LP-MMSE Precoding · p. 216

6.2.3 Fronthaul Signaling Load for Distributed Operation

• Distributed downlink sends each AP τd|Dl| complex scalars per coherence block, so fronthaul scales with the number of served UEs rather than antennas.Only data encoding remains at the CPU; local AP processors compute channel estimates, select precoders, and form transmitted signals.Source: §6.2 · p. 209, §6.2.3 · p. 217, Table 6.2 · p. 217

6.3.2 Performance With Centralized Operation

• In centralized operation, restricting service through DCC improves SE over serving with all APs, while MMSE, P-MMSE, and P-RZF perform almost identically.The scalable P-MMSE and P-RZF schemes therefore retain nearly the MMSE performance without its scalability limitation.Source: §6.3.2 · p. 220, Figure 6.3 · p. 221
• With a fixed total number of antennas, centralized operation prefers many single-antenna APs, whereas distributed operation benefits from fewer multi-antenna APs using local interference suppression.Distributed precoding has reduced interference-suppression capability because each AP can suppress only the interference it generates.Source: §6.3.2 · p. 220, §6.4 · pp. 231–232

Tightness of the SE Expression

• The achievable SE expressions closely match genie-aided results for centralized P-MMSE and P-RZF and distributed LP-MMSE, but MR has a substantial gap caused by large signal and interference-power variations.For MR, refined downlink channel estimation schemes or tighter capacity bounds are needed.Source: Tightness of the SE Expression · pp. 222, 223, 225, 226, Figure 6.4 · p. 222

6.3.3 Performance With Distributed Operation

• In distributed operation, DCC with L-MMSE or LP-MMSE substantially outperforms all-AP L-MMSE because it avoids wasting power on weakly connected UEs and reduces interference.LP-MMSE achieves roughly the same SE as unscalable L-MMSE, whereas MR performs almost equally poorly across the evaluated scenarios.Source: §6.3.3 · p. 223, Figure 6.5 · p. 224

6.3.4 Impact of the Number of UEs

• Increasing UEs from K = 20 to K = 100 causes no more than a two-fold reduction in SE per UE, while sum SE increases almost linearly with K.Centralized schemes outperform distributed schemes, P-MMSE slightly exceeds P-RZF, and LP-MMSE substantially exceeds MR in the evaluated setup.Source: §6.3.4 · pp. 226, 229, Figure 6.8 · p. 228

6.3.5 Impact of Spatial Correlation

• Spatial correlation degrades distributed-operation SE but improves centralized-operation SE, with performance approaching the uncorrelated reference as ASD increases.Under correlation, the P-MMSE–P-RZF gap is slightly larger because P-MMSE exploits estimation-error correlation matrices.Source: §6.3.5 · pp. 229–230, Figure 6.9 · p. 230

6.4 Summary of the Key Points in Section 6

• The downlink supports centralized CPU-selected precoding and distributed AP-local precoding, with scalable implementations motivated by uplink-downlink duality and coherent reception at desired UEs.Centralized operation enables joint interference suppression across APs, while distributed operation uses only locally available channel estimates.Source: §6.4 · pp. 231–232
• Scalable P-MMSE, P-RZF, and LP-MMSE precoding achieve almost the same downlink SE as their unscalable counterparts, with substantially lower complexity.This holds across the centralized and distributed running-example comparisons.Source: §6.4 · p. 232

Spatial Resource Allocation

• Spatial resource allocation covers transmit powers, pilot signals, cooperation clusters, and fronthaul resources, with network-wide power optimizations serving as theoretical benchmarks because they are not scalable.Scalable heuristic power allocation alternatives and comparisons are presented elsewhere in the section, while pilot assignment, DCC selection, and fronthaul constraints are also treated as resource-allocation topics.Source: Spatial Resource Allocation · p. 233

7.1 Transmit Power Optimization

• Transmit-power optimization maximizes max-min fairness or sum SE using fixed-point and weighted-MMSE algorithms based on channel statistics and combining or precoding choices, without coherence-block adaptation.The same solution can be reused while channel statistics remain unchanged, but the optimal powers differ across network operations because their effective parameters differ.Source: §7.1 · p. 234, §7.1.1 · p. 236

7.1.1 Uplink Power Optimization

• Uplink power optimization uses a common UatF-based SINR structure for centralized and distributed operation, enabling algorithms for max-min SE fairness and sum SE maximization under individual UE power constraints.The algorithms are optimized for the more conservative bounds in Theorems 5.2 and 5.4, but can also be used with Theorem 5.1; centralized and distributed cases differ through their SINR parameters.Source: §7.1.1 · pp. 234–236

Uplink Max-Min SE Fairness

• The uplink max-min fixed-point algorithm converges to the optimal solution, equalizes all UEs’ SEs, and makes at least one UE use pmax, but its complexity grows with K and is not scalable.Under strictly positive desired-signal, interference, and noise coefficients, the algorithm converges quickly through iterative closed-form updates.Source: Uplink Max-Min SE Fairness · pp. 236–237

Uplink Sum SE Maximization

• The uplink sum-SE block-coordinate descent algorithm uses closed-form updates, converges quickly to a local optimum, and is not scalable because its complexity grows with K.Different initializations may produce different solutions, and sum-SE optimization can assign zero power and zero SE to some weak-channel UEs, although this did not occur in the running example.Source: Uplink Sum SE Maximization · p. 238

7.1.2 Downlink Power Optimization

• Downlink power allocation differs between centralized and distributed operation because centralized precoding vectors are coupled across APs, whereas distributed precoding vectors are not.Centralized operation optimizes K UE-level coefficients, while distributed operation optimizes one coefficient per serving AP–UE pair, making the latter more complex.Source: §7.1.2 · pp. 239–240

Optimized Centralized Downlink Power Allocation

• Centralized downlink max-min optimization uses a fixed-point algorithm that converges quickly to equal UE SEs but is not scalable, while sum-SE optimization uses weighted-MMSE block coordinate descent with convex QCQP subproblems solved by a convex solver.Both problems have per-AP power constraints; the sum-SE problem is nonconvex and yields a local optimum, and its subproblem generally lacks a closed-form solution because multiple constraints share variables.Source: Optimized Centralized Downlink Power… · pp. 242–243
• Centralized optimized downlink comparisons use the general-purpose solver SDPT3 through CVX, prioritizing faster code development over dedicated-solver runtime efficiency.Dedicated solvers exploit problem structure for efficiency, whereas general-purpose solvers abstract implementation details and accelerate development.Source: Optimized Centralized Downlink Power… · p. 243

Optimized Distributed Downlink Power Allocation

• Distributed downlink max-min fairness is reformulated with square-root power variables and solved by bisection over a second-order cone program, while sum SE admits a weighted-MMSE block-coordinate descent method.The square-root variables preserve a one-to-one correspondence with the original powers; the resulting max-min subproblem is convex, and each sum-SE iteration solves a convex quadratically constrained quadratic program.Source: Optimized Distributed Downlink Power… · pp. 246–247

7.2 Scalable Distributed Power Optimization

• Network-wide power optimization becomes unscalable as K grows, motivating distributed and heuristic schemes based on local CSI and limited device coordination.The complexity of optimizing K SINRs with at least K variables grows unboundedly with K; the challenge is finding heuristics that retain reasonably good spectral efficiency.Source: §7.2 · pp. 250–251

7.2.1 Scalable Uplink Power Control

• Fractional uplink power control sets each UE’s power from the aggregate channel gain to its serving APs, with υ = 0 giving full power and υ = −1 approximating max-min fairness.Values υ ∈ [−1, 0] trade fairness against sum SE, while υ > 0 favors UEs with good channel conditions; channel-estimation errors can also be incorporated into the gain measure.Source: §7.2.1 · p. 252
• A scalable uplink variant restricts denominator maximization to UEs partially served by the same APs, using a set whose size does not grow unboundedly with K.This localizes interference-related computations to mostly nearby UEs, avoiding the original denominator’s dependence on all K UEs; the parameter υ can be selected by comparing per-UE SE CDFs.Source: §7.2.1 · pp. 252–253

7.2.2 Scalable Centralized Downlink Power Allocation

• Scalable centralized downlink allocation preserves common serving-AP power directions, using equal or channel-gain-proportional powers with υ ∈ [−1, 1] and an additional κ ∈ [0, 1] scaling factor.υ > 0 favors high-gain UEs and υ < 0 favors fairness; the κ-based construction satisfies every AP power constraint while keeping computational complexity independent of K, although the basic normalization can be conservative.Source: §7.2.2 · pp. 254–255

7.2.3 Scalable Distributed Downlink Power Allocation

• The scalable distributed downlink heuristic allocates each AP’s maximum power among its served UEs using channel-statistics weights, with υ ∈ [0, 1] favoring stronger channels and ensuring non-zero power per serving AP.Its per-AP complexity remains scalable because each AP serves a bounded number of UEs; υ = 0.5 was used in the running-example evaluation.Source: §7.2.3 · pp. 257–259

7.3.1 Comparison of Uplink Power Control Schemes

• Sum-SE maximization generally provides the preferred uplink performance, while fractional power control with υ = −0.5 offers a scalable fairness improvement for the worst UEs.Full-power transmission matches the sum-SE solution, whereas max-min fairness yields only marginal lower-tail gains at substantially lower SE for most UEs.Source: §7.3.1 · pp. 260, 262, Figure 7.1 · p. 261

7.3.2 Comparison of Downlink Power Allocation Schemes

• In centralized downlink operation, sum-SE maximization is preferable overall, while fractional power allocation with υ = −0.5 and κ = 0.5 provides a strong fairness–sum-SE tradeoff.Fractional allocation gives most UEs higher SE than sum-SE maximization and a 90% likely SE better than max-min fairness; its 90% likely SE is around 1.5 bit/s/Hz higher than network-wide equal allocation.Source: §7.3.2 · pp. 263, 265, Figure 7.2 · p. 264
• In distributed downlink operation, max-min fairness improves SE for the 15% worst-conditioned UEs, whereas υ = 0.5 fractional allocation nearly approaches sum-SE maximization and outperforms per-AP equal power.The υ = −0.5 heuristic performs poorly, while υ = 0.5 reduces interference by concentrating power at each UE’s strongest serving AP.Source: §7.3.2 · pp. 265–266, Figure 7.3 · p. 265

7.4.1 Utility-Based Pilot Assignment

• Pilot assignment should avoid assigning the same pilot to closely located or similarly exposed UEs, but utility-based optimization is combinatorial with exponential exhaustive-search complexity in K.Greedy, clustering, tabu-search, and Hungarian-algorithm approaches provide suboptimal alternatives, while channel gains and shadow fading can make location alone unreliable.Source: §7.4 · p. 267, §7.4.1 · pp. 268–269

7.5. SELECTION OF DYNAMIC COOPERATION CLUSTERS

• For scalable cell-free networks with L ≫ K, sparse pilot reuse makes coherent interference less dominant, so pilot assignment should rely on local interactions between UEs and neighboring APs rather than network-wide optimization.Network-wide utility methods become impractical as K grows, motivating locally implemented structured assignment schemes.Source: §7.4.1 · p. 269, §7.5 · p. 270, Scalable Pilot Assignment · p. 270

Scalable Pilot Assignment

• Pilot assignment selects, for each UE, the pilot with least measured interference at a neighboring AP, yielding good results despite not being optimal.The assignment aims to reuse each pilot sparsely in space; further scalable algorithms, potentially using machine learning, are needed.Source: Scalable Pilot Assignment · p. 270

7.5. SELECTION OF DYNAMIC COOPERATION CLUSTERS

• Dynamic cluster selection can use large-scale fading thresholds, strongest-channel APs, or a received-power threshold such as δ = 0.95.The received-power criterion selects the strongest-channel AP subset contributing more than 95% of the total received downlink power.Source: §7.5 · pp. 271–272

7.5 Selection of Dynamic Cooperation Clusters

• Cluster formation must account for per-AP processing, fronthaul, and pilot-contamination limits, including the practical rule that an AP serves at most one UE per pilot.Pilot assignment can determine clusters by having each AP serve the strongest-large-scale-fading UE among those sharing each pilot.Source: §7.5 · p. 272
• Dynamic cooperation clusters achieve scalable operation with small spectral-efficiency loss relative to serving all UEs by all APs, at least when L ≫ K.One-UE-at-a-time approaches are scalable, whereas jointly considering all UEs becomes unscalable as K →∞.Source: §7.5 · p. 273, §7.7 · p. 277

7.6 Implementation Constraints

• Practical scalability requires finite signal-processing complexity and fronthaul signaling per AP and UE as K →∞, while low-cost hardware introduces distortion from non-linearities, mismatches, quantization, and phase noise.More detailed hardware-impairment models remain an open research direction.Source: §7.6 · pp. 273–274, §7.6.1 · p. 273

7.6.2 Quantization of Fronthaul Signaling

• Fronthaul compression creates a rate-distortion tradeoff and makes processing location consequential: local channel estimation or detection is more accurate than processing compressed signals centrally.Centralized systems can choose between estimate-and-quantify and quantify-and-estimate protocols.Source: §7.6.2 · p. 274

7.6.3 Radio Stripes & Other Constrained Fronthaul Architectures

• Constrained fronthaul architectures can reduce cabling and support neighboring-AP co-processing, sequential MMSE combining, and relaxed synchronization within user-centric clusters.Coherent transmission still requires APs to be synchronized in time and frequency, although distant APs serving non-overlapping UE sets need less synchronization.Source: §7.6.3 · pp. 274–275, §7.6.4 · p. 275, §7.7 · p. 277

7.7. SUMMARY OF THE KEY POINTS IN SECTION 7

• In uplink, full transmit power maximizes sum SE, while centralized and distributed downlink fractional power allocation provide practical fairness-performance tradeoffs.A negative exponent improves most centralized-downlink UEs relative to sum-SE maximization; a positive exponent substantially outperforms distributed per-AP equal power, with a relatively small gap to sum-SE maximization.Source: §7.7 · pp. 276–277

Mathematical Notation

• Mathematical notation distinguishes matrices, column vectors, scalars, and sets through boldface, italic, and calligraphic letter styles.Upper-case boldface denotes matrices, lower-case boldface denotes column vectors, italic letters denote scalars, and calligraphic letters denote sets.Source: Mathematical Notation · p. 280
• The notation defines complex- and real-valued matrix sets, with C^N and R^N abbreviating complex and real vectors.C^(N×M) and R^(N×M) denote complex- and real-valued N × M matrices, while C^N and R^N are short forms for C^(N×1) and R^(N×1).Source: Mathematical Notation · p. 280
• Set-membership, subset, vector-element, matrix-element, column-selection, and diagonal-matrix operators are specified for subsequent use.The conventions include x ∈ S, x ∉ S, property-defined subsets, [x]_i, [X]_ij, [X]_:,1, and diag(·).Source: Mathematical Notation · p. 280

438 Notation and Abbreviations

• Matrix notation covers block diagonals, conjugation, transposition, inversion, square roots, traces, norms, and identity or all-ones/all-zeros matrices.It also defines real and complex Gaussian distributions, expectations, variances, and common scalar functions.Source: 438 Notation and Abbreviations · p. 281

Abbreviations

• The abbreviation list standardizes terms for system components, channel information, processing methods, cooperation, power control, and performance metrics.Examples include AP, CSI, DCC, CoMP, MIMO, MMSE, MR, SE, SINR, and UE.Source: Abbreviations · pp. 281–283

Matrix inversion lemma

• The matrix inversion lemma states that a block-matrix identity holds when all involved inverses exist.It applies to matrices A, B, C, and D with the dimensions specified in the lemma.Source: Useful Lemmas · p. 284

Trace positivity

• For positive semidefinite A and positive definite B, tr(AB) > 0 when A has at least one positive eigenvalue.The proof uses the eigendecomposition A = UΛU^H and the fact that B is positive definite.Source: Useful Lemmas · p. 284

Lemma B.4

• Lemma B.4 establishes an inequality for positive semidefinite matrices A and C and a positive definite matrix B.The proof decomposes A through its positive eigenvalues and applies the matrix inversion lemma; equality holds when CB^-1A = 0_N×N.Source: 442 Useful Lemmas · p. 285

Lemma B.5

• Lemma B.5 gives a moment identity for a ∼ NC(0_N, A) under an arbitrary deterministic matrix B.The proof represents a as A^1/2w with w ∼ NC(0_N, I_N), expands indexed sums, uses E{|w_n|^2}=1 and E{|w_n|^4}=2, and rewrites the result with traces.Source: 442 Useful Lemmas · pp. 285–286

• The appendix provides proofs for lemmas, theorems, and corollaries that were too lengthy to include in the main text.Source: Collection of Proofs · p. 287

• The section presents proofs from Section 4.Source: §C.1 · p. 287

C.1.1 Proof of Lemma 4.2

• For λn ≥0, D2NMSE (λ) is negative definite because it is diagonal with strictly negative entries.The proof reduces to verifying negative definiteness of the Hessian.Source: §C.1.1 · p. 287

Proof setup

• NMSE(λ) − NMSE(λ′) depends only on the summation terms in (4.29) for n ∈ {r−1, r}.This follows because all elements of λ and λ′ are identical except the (r−1)th and rth entries.Source: §C.1.2 · p. 287

Algebraic reduction

• The proof introduces x = λr−1, y = λr, and c = σ2_ul/(ητp) to simplify the NMSE difference.The resulting algebraic expression is rewritten through successive cancellations of common numerator terms.Source: §C.1.2 · p. 288

Conclusion

• The final inequality follows from x > 0, y > 0, and c > 0 after canceling the common terms in the intermediate expressions.The cancellations include (x2 + y2)c(x + y + c) and (x + y)2xy.Source: §C.1.2 · pp. 287–288

• The section presents the proofs from Section 5.Source: §C.2 · p. 288

C.2.1 Proof of Theorem 5.1

• Theorem 5.1 follows by modeling the processed signal as a discrete memoryless interference channel and verifying conditional zero-mean, variance, and uncorrelated-input requirements.The resulting variance equals the denominator of (5.5), and the uplink capacity bound includes the data fraction τu/τc.Source: §C.2.1 · pp. 288–289

C.2.2 Proof of Theorem 5.2

• Theorem 5.2 is proved by separating the deterministic average channel from additive interference-plus-noise and applying Lemma 3.3 after establishing zero mean and uncorrelated desired-signal interference.Independence across UEs and AP noise contributions yields the theorem’s spectral-efficiency expression, with the τu/τc uplink-data fraction.Source: §C.2.2 · pp. 289–290

C.2.3 Proof of Theorem 5.4

• Theorem 5.4 uses the CPU’s average effective channel as a deterministic channel, models the remainder as uncorrelated interference-plus-noise, and applies Lemma 3.3.The resulting spectral-efficiency lower bound accounts for the τu/τc fraction of samples used for uplink data transmission.Source: §C.2.3 · pp. 290–291

C.2.4 Proof of Corollary 5.6

• Corollary 5.6 is obtained by directly computing the expectations in (5.26) using Lemma B.2, channel independence, and the estimate–error decomposition.The derivation produces (5.33)–(5.35) and (5.34), and remains valid when Rkl is non-invertible because the inverse only shortens notation.Source: §C.2.4 · pp. 291–292

C.3 Proofs from Section 6

• The document reports below the proofs from Section 6.Source: §C.3 · p. 292

C.3.1 Proof of Theorem 6.1

• Theorem 6.1 follows by treating the UE-known average effective channel as deterministic, proving uncorrelated interference, and applying Lemma 3.3.The downlink capacity lower bound accounts for the τd/τc fraction of samples used for downlink data transmission.Source: §C.3.1 · pp. 292–293

C.3.2 Proof of Theorem 6.2

• The downlink SINR constraints are expressed as dl = (Γ −Σ)ρ, and a feasible power vector exists when Γ −Σ is invertible, which holds when the uplink power vector is feasible.The total transmit-power condition follows by direct computation.Source: §C.3.2 · p. 294