Scalable Distortion-Aware Clustering for Fronthaul-Limited Cell-Free MIMO Networks

Scalable Distortion-Aware Clustering for Fronthaul-Limited Cell-Free MIMO Networks Zehua Li∗ , Raviraj Adve∗, Israfil Bahceci†, Yahia Ahmed† ∗

arXiv:2610.04776v1 [cs.IT] 3 Oct 2026

Dept. of Electrical and Computer Engineering, University of Toronto, Toronto, Canada † Ericsson Canada, Ottawa, Canada Emails: samzehuali.li@mail.utoronto.ca, rsadve@ece.utoronto.ca, {israfil.bahceci, yahia.ahmed}@ericsson.com Abstract—This paper studies distortion-aware clustering for uplink fronthaul-limited cell-free MIMO networks employing maximum ratio combining (MRC). While MRC is appealing for its low-complexity, its performance is limited by interference, fronthaul distortions, and channel imperfections. This motivates clustering strategies that compensate for these impairments through coordination of access points. Since instantaneous smallscale fading-based optimization is not scalable in large systems, and is impractical due to frequent channel variations, we instead optimize an objective depending only on large-scale fading coefficients. To this end, asymptotic analysis is used to derive deterministic equivalent expressions for the average network sum rate, with focus on quantization distortion, leading to a quadraticover-linear objective. Although maximizing such an objective under binary constraints is non-convex, we exploit its structure to develop a polynomial time scheme that attains the global optimum. Numerical results show that the proposed clustering method provides performance gains over improved variants of existing literature, and remains competitive with the small-scale fading-based global optimum obtained via exhaustive search.

I. I NTRODUCTION Cell-free multi-input multi-output (MIMO) networks are envisioned as a key technology for next-generation wireless systems [1]. By coordinating distributed access points (APs) through a central processing unit (CPU), cell-free architectures provide spatial diversity and improved channel conditions by bringing APs closer to users. Centralized processing can also improve interference management, for example, through minimum mean square error (MMSE) combining at the CPU. However, MMSE receivers do not scale well as the network densifies [2], [3], motivating clustering methods where each user is served by only a subset of APs. A low complexity alternative is maximum ratio combining (MRC), which is more scalable but lacks an interference suppression capability. This makes cluster selection crucial, since better clusters can reduce interference and improve user rates [4]. The problem becomes even more challenging in fronthaul-limited networks, where APs must compress their received signals before forwarding them to the CPU, introducing quantization distortion [5], [6]. Since MRC does not explicitly account for interference or fronthaul distortion, clustering for MRC receivers requires a design perspective different from clustering for MMSE receivers. Prior works have separately studied clustering design [4] and fronthaul limitations [7], often optimizing objectives that depend on small-scale fading information. While such ob-

jectives provide accurate performance characterization, they require frequent updates. Practical aspects have also been explored, including learning-based clustering [8] and partial use of channel statistics for fronthaul compression [9]. However, these methods still rely on small-scale fading information and are therefore less scalable under rapidly varying channels. A more scalable approach is to rely only on large-scale fading coefficients (LSFCs), so that optimization is required only at the time scale of LSFC variations. One established method is to use asymptotic analysis to derive deterministic equivalent (DE) expressions for average rates and optimize these approximations instead [10]. While DE-based optimization incurs some performance loss by targeting average rate approximations rather than instantaneous rates, it is more practical because the resulting strategies require far less frequent updates. We emphasize that only clustering is updated at the LSFC time scale, while MRC weights are still formed using small-scale fading realizations. The main focus of this paper is scalable distortion-aware clustering for fronthaul-limited cell-free MIMO networks with MRC receivers. The main contributions are: • We formulate the instantaneous achievable user rate under limited fronthaul and imperfect channel state information. • We derive a DE-based rate expression that accounts for MRC and quantization distortion. This leads to a per-user cluster selection problem in the form of a quadratic-overlinear maximization under binary constraints. Despite this being non-convex, we derive a globally optimal solution. • Through simulations, we validate the DE approximation, verify the LSFC-based global optimality of the proposed clustering scheme, and compare against exhaustive search and enhanced versions of literature benchmarks [4] that account for limited fronthaul. II. S YSTEM M ODEL A. Channel and Signal Model We consider the uplink of a cell-free MIMO network operating in time-division duplex (TDD) mode. A single CPU controls a set of APs denoted by the set R; each AP is equipped with M antennas. The set of single-antenna users (UEs) is denoted as U. We use subscripts r and u to refer to APs and users respectively. We refer to the links between users and APs as access links and between APs and

CPU as fronthaul links. Each fronthaul link has a spectral efficiency (SE) limit of Lr bits/s/Hz. We refer to rate and SE interchangeably since, in our TDD mode, they only differ by a scale factor. For the access channel hru between user u and AP r, only an imperfect channel estimate ĥru is available. From [11], 1/2 (1) hru = ĥru + eru = Ψ1/2 ru gru + Ψ̃ru g̃ru , where eru denotes estimation error, Ψru denotes the channel correlation matrix, and Ψ̃ru denotes the error covariance. gru , g̃ru ∼ CN (0M×1 , IM ) are the small-scale fading and normalized estimation error assumed mutually independent. The user transmission assumes Gaussian signaling and is denoted as xu ∼ CN (0, pu ) where pu is the transmit power. At AP r, the received signal vector yr ∈ CM is given by X X yr = hru xu + nr = (ĥru + eru )xu + nr , (2) u∈U

u∈U

where nr ∼ CN (0M×1 , Σnr ) denotes the additive Gaussian noise. The APs then compress (quantize) the received signals and forward the compressed versions to CPU. This is referred as compress-forward [5] or centralized operation [2]. The compressed signal vector ζr ∈ CM seen at CPU is given by X ζr = Q(yr ) = yr + qr = (ĥru + eru )xu + nr + qr , (3) u∈U

where Q(·) denotes the quantization operation. We assume a Gaussian test channel model [5], [6], [12] with quantization error qr ∼ CN (0M×1 , Σqr ). The covariance Σqr is designed to satisfy the fronthaul limit and will be discussed later. At the CPU, only signals from a subset of APs, called user’s serving cluster, are combined. For each user u, we define Cu ⊆ R as its serving cluster. The estimated symbol yields X H x̂u = wru ζr r∈Cu

= wCHu

X

u′ ∈U

(ĥCu u′ + eCu u′ )xu′ + nCu + qCu

!

,

(4)

where wru ∈ CM is the combiner for user u corresponding to the signals from AP r ∈ Cu . We introduce using Cu as subscripts to denote vertical concatenations of all vectors from the set as wCu , nCu , qCu , ĥCu u′ , eCu u′ ∈ CM|Cu | . For example, ĥCu u′ is the collective estimated channels between cluster Cu T and user u′ , i.e., ĥCu u′ = [· · · , ĥT ru′ , · · · ]r∈Cu . wCu is viewed as the centralized receive combiner used at the CPU to estimate the symbol sent by user u. The collective noise and quantization distortion covariances are ΣnCu = diag{Σnr }r∈Cu and ΣqCu = diag{Σqr }r∈Cu . Similarly, the collective channel correlation matrix is ΨCu u′ = diag{Ψru′ }r∈Cu and similarly for the collective error covariance Ψ̃Cu u′ . The instantaneous effective signal-to-interference-plusquantization-and-noise ratio (SIQNR), similar to [2], [7], is pu |wCHu ĥCu u |2 = γu , (5) SIQNRu = ENu + Intf u P ENu = wCHu (ΣnCu + ΣqCu + u′ ∈U pu′ Ψ̃Cu u′ )wCu , (6) P (7) Intf u = u′ ∈U \u pu′ |wCHu ĥCu u′ |2 , where Intf u denotes the contribution of interference and ENu

is the effective noise that includes quantization distortions and channel uncertainties. Then, an achievable rate in centralized operation is Ru = log2 (1 + SIQNRu ) = log2 (1 + γu ). B. Compression Model The codebook design of the quantizers is equivalent to setting the covariances of quantization noise Σqr [5], [6]. The distortion should be made as small as possible to achieve higher rates where the minimum amount of distortion is limited by fronthaul capacity, given by rate-distortion theory. Similar to [6], [7], the rate (normalized over access bandwidth) required to transmit the signal received at the m-th antenna of AP r over the fronthal channel ! is given by Γq σy2r,m + σq2r,m , (8) lr,m = log2 σq2r,m i hP H , (9) ĥ + Ψ̃ ) + Σ p ( ĥ σy2r,m = ru n u ru r ru u∈U mm

where [·]ij is to denote (i, j)-th entry of a matrix and Γq is the gap to rate-distortion theory. For simplicity, the total fronthaul rate is allocated equally to its M antennas, i.e. lr,m = Lr /M, ∀r. The distortion variance is then Γq σy2 σq2r,m = L /Mr,m , (10) 2 r −1 for each antenna and the distortion covariance for each AP Lr /M − is Σqr = diag{σq2r,m }M m=1 . By defining κr = Γq /(2 D 1) and Kr = κr IM , the distortion covariance can also be alternatively written  Pas H (11) ⊙ Σq r = K D r u∈U pu (ĥru ĥru + Ψ̃ru ) + Σnr ,

with ⊙ denoting the Hadamard product. The superscript-D is used on matrices to remind us that they are diagonal. The intention of AD ⊙ B is to only keep the diagonal of the entrywise multiplication of the two matrices. We choose to use the Gaussian test channel model as (10) provides continuous relationships between rate of quantization and distortion. Other models such as Bussgang decomposition [13] or AQNM [14] can also be extended to our analysis. We remark that the APs may alternatively perform local combining before forwarding to the CPU [2]; however, this makes the distortion in (10) dependent varies with |Cu | or cluster-dependent. The centralized scheme avoids this coupling and yields the tractable clustering formulations in Section IV, while MRC reduces complexity of centralized processing. III. A SYMPTOTIC A NALYSIS OF SIQNR UNDER MRC For this paper, maximum ratio combining (MRC) is used for centralized processing where wCu = ĥCu u . Compared to zero-forcing and MMSE combiner, the data rates achieved when using MRC degrade as it lacks interference management. However, especially in dense networks, it is less complex to implement making it more practical. As a first step of our development, we use the large system approach and derive the asymptotic SIQNR under MRC. More precisely, we approximate the SIQNR of user u by a deterministic equivalent (DE) γuo such that γu −γuo → 0 almost surely, as the system dimensions M and |U| go to infinity.

The approximation is tight asymptotically and yields accurate results even when the dimensions are small [10]. There are some common assumptions to be made for the asymptotic analysis. The system dimension ratio is bounded, i.e. 0 < limM,|U |→∞ M/|U| < ∞, and the correlation matrices Ψru , Ψ̃ru have uniformly bounded spectral norm. The per-antenna compression rates are assumed to be strictly positive, i.e. limM→∞ Lr /M > 0, to avoid the degenerate case of no service. To denote all system dimensions approach infinity with the above assumptions, we use M → ∞ and M→∞ denote a − ao −→ 0 almost surely as a ≍ ao . For each user, let Mu = M |Cu | denote the combining dimension. We define the normalized small-scale fading vector zCu u′ ∼ CN (0Mu ×1 , M1u IMu ) and normalized channels bCu u′ √ √ 1/2 such that ĥCu u′ = Mu ΨCu u′ zCu u′ = Mu bCu u′ . To find the DE expressions for SIQNR under MRC, we introduce the following lemmas. Lemma 1: For each positive integer N , let AN ∈ CN ×N be a random matrix with uniformly bounded spectral norm. Let xN ∈ CN be a random vector of i.i.d. entries with zero mean, variance 1/N , and independent of AN . Then 1 N →∞ xH (12) trAN −→ 0, N An xN − N almost surely. Proof: The proof is given in [10].  Lemma 2: Let N and xN be as in Lemma 1 and let N ×N AD be deterministic matrices with uniformly N , BN ∈ C bounded spectral norm. Further, AD N is diagonal with positive entries and BN is positive semidefinite. Then 1/2 1/2 1/2 1/2 D H xH N BN (AN ⊙ (BN xN xN BN ))BN xN − (13) 2 N →∞ trBN (AD N ⊙ BN ) −→ 0, 2 N almost surely. Proof: The proof is given in Appendix A.  The SIQNR and the corresponding DE is expressed as Sgnlu γu = P4 Nsu + Impf u + Intf u + i=1 QNi,u Sgnlou ≍ o = γuo , (14) P4 Nsu + Impf ou + Intf ou + i=1 QNoi,u 2 Sgnlu = pu |bH Cu u bCu u | 1 P ≍ 2 ( r∈Cu trΨru )2 = Sgnlou , Mu 1 H Nsu = b Σ n bC u Mu Cu u Cu u 1 P ≍ 2 r∈Cu trΨru Σnr = Nsou , Mu 1 X pu′ bH Impf u = Cu u Ψ̃Cu u′ bCu u Mu ′ u ∈U X pu′ P ≍ ( r∈Cu trΨru Ψ̃ru′ ) = Impf ou , 2 M u u′ ∈U X 2 pu′ |bH Intf u = Cu u bCu u′ |

(15)

(16)

(17)

u′ ∈U \u

≍

pu′ P o 2( r∈Cu trΨru Ψru′ ) = Intf u , u′ ∈U \u Mu

P

(18)

1 H b (KD ⊙ ΣnCu )bCu u Mu Cu u Cu
1 P o ≍ 2 r∈Cu trΨru KD (19) r ⊙ Σnr = QN1,u , Mu 1 X D pu′ bH QN2,u = Cu u (KCu ⊙ Ψ̃Cu u′ )bCu u Mu ′ u ∈U X pu′ X o ≍ trΨru (KD r ⊙ Ψ̃ru′ ) = QN2,u , (20) Mu2 ′ u ∈U r∈Cu X H D (21) pu′ bH QN3,u = Cu u (KCu ⊙ bCu u′ bCu u′ )bCu u

QN1,u =

u′ ∈U \u

≍

X

u′ ∈U \u

pu′ X o trΨru (KD r ⊙ Ψru′ ) = QN3,u , Mu2 r∈Cu

H D QN4,u = pu bH Cu ,u (KCu ⊙ (bCu ,u bCu ,u ))bCu ,u

2pu o trΨru (KD (22) r ⊙ Ψru ) = QN4,u , Mu2 where showing (22) requires Lemma 2 and the others require Lemma 1. From the continuous mapping and dominated convergence theorems [10], the DE of user data rate is obtained as Ru ≍ log2 (1 + γuo ) = Ruo and the average rate is E[Ru ] ≍ Ruo . We remark that DEs obtained for the denominator of the SIQNR matches its expectation when MRC is used. This partly aligns with the fact that E[log2 (1 + A/B)] ≈ log2 (1 + E[A]/E[B]) used in the massive MIMO literature [15]. ≍

IV. O PTIMAL DE-BASED C LUSTERING Although, the simple structure of MRC makes it scalable to dense network settings, it becomes crucial to optimize each user’s serving cluster, Cu [4]. MRC not only lacks interference mitigation, but also is not aware of the quantization distortion in the fronthaul-limited setting or the impact of imperfect channel estimates. Hence, an optimized clustering scheme compensates for all aspects through better coordination. Optimizing rates Ru for each channel realization yields good results, but this is impractical since the channels may have changed before optimization finishes. As a practical alternative, we choose to optimize over the interval in which the large-scale fading coefficients (LSFCs) remain constant. More precisely, the DE-based rate Ruo is used as it serves as a good approximation, even in small dimensions [10]. For notational convenience, we assume the case of additive white Gaussian noise (AWGN), Σnr = σ 2 IM , and spatially uncorrelated channels, Ψru = (1 − τ 2 )βrt IM and Ψ̃ru = τ 2 βrt IM , where σ 2 is the noise power, τ is the channel imperfection factor, and βru denotes the LSFC containing pathloss and shadowing. The assumptions are for cleaner presentation and do not fundamentally impact the analysis. Observe that, since the channels and noise across APs are uncorrelated, each component of γuo can be broken up into contributions of each AP; for example, ΨCu u is block diagonal. The AP-UE association index is defined as sru ∈ {0, 1} to represent user’s serving cluster where sru = 1 means r ∈ Cu and vice versa. The DE of SIQNR γuo without pre-determined clusters can be formulated as P ( r∈R aru sru )2 2 fu (Su ) = pu M (1 − τ ) · P , (23) r∈R bru sru

aru = βru ,

(24) 2

2

90

2 ))βru

bru = σ (1 + κr )βru + pu (τ + κr (2 − τ P (25) + u′ ∈U \u pu′ (1 + κr )βru βru′ , where Su is the set of association variables for user u, with fu (Su ) defined as zero for an empty serving cluster. From (23), we note that the user’s rate is only a function of the user’s cluster, i.e., fu is only dependent on Su . Hence, optimizing the network sum data rate is equivalent to optimizing the SIQNR for each user and is formulated as (P1) : max fu (Su ) (26a) Su

s.t. sru ∈ {0, 1}, ∀r ∈ R. (26b) We propose to solve the problem by first relaxing the binary feasible set into its convex hull to obtain (P2) : max fu (Su ) (27a)

MC MC MC

80

Average Sum Rate [bps/Hz]

2

DE, M=4 DE, M=16 DE, M=64

3.4%

70 60

1.7% 3.8%

50 40

2.6% 4.5%

30 6.8% 20

Network Traffic: 25 APs & 12 UEs

10 1

2

3

4

5

Normalized Per-Antenna Compression Rate [bps/Hz]

Su

s.t. 0 ≤ sru ≤ 1, ∀r ∈ R. (27b) Although (P2) is a non-convex problem, the structure can be exploited to find the global optimum described as follows. Theorem 1: Consider a particular quadratic-over-linear ratio maximization problem with box P constraints as ( r ar sr )2 P (28a) max {sr }∀r r b r sr s.t. 0 ≤ sr ≤ 1, ∀r, (28b) where ar , br > 0, ∀r, and the objective defined as zero for the trivial solution when sr = 0, ∀r. Define an ordering metric ρr , br /ar , a permutation π(·) such that ρπ(i−1) ≤ ρπ(i) , and ordered sums P P Ak , ki=1 aπ(i) , Bk , ki=1 bπ(i) . (29) Then, there exists ( a globally optimal solution 1, i ≤ k ∗ , s∗π(i) = (30) 0, i > k ∗ , A2 k = arg max k , K = {1, · · · , R}, (31) k∈K Bk with R , |{sr }∀r |, the size of the variable set. Proof: The proof is given in Appendix B.  Using Theorem 1, a globally optimal solution can be obtained for (P2). For each user, the ordering metric is ρru = bru /aru , the permutation is πu (·), and the ordered sum is Pk Pk (32) Au,k = i=1 aπu (i) , Bu,k = i=1 bπu (i) . ∗

The global optimal ( solution is obtained as 1, i ≤ ku∗ , s∗πu (i)u = 0, i > ku∗ ,

(33)

A2u,k , K = {1, · · · , |R|}. (34) k∈K Bu,k Note that the optimal solution Su∗ of (P2) lies in the feasible set of (P1). Hence, the solution obtained from (33) is also globally optimal for (P1). The reason is that (P2) maximizes a convex objective over a convex constraint set, so a global optimum is attained at an extreme point of the feasible set. These extreme points are exactly the binary feasible points of (P1). However, ku∗ = arg max

Fig. 1: Sum of user data rates, obtained using MC and DE, versus preantenna compression rate, Lr /M , with each user served by |Cu | = 5 nearest APs.

it is important to note that the global optimality is achieved with respect to the DE-based rates, but not to instantaneous rates per channel realization or average rates. There are two aspects to the complexity of the proposed DEbased clustering scheme. First, it only needs to be executed once over each interval during which the LSFCs remain constant, i.e., much less frequently than small-scale channel variations. Second, it attains the global optimum of a combinatorial problem with complexity O(|R| log |R|) for sorting and O(|R|) for searching. V. N UMERICAL R ESULTS AND D ISCUSSIONS In this section, we provide numerical results to compare the average sum data rate and the DE-based approximations as well as to illustrate the effectiveness of the proposed clustering scheme. Some important parameters chosen are: The APs and UEs are uniformly distributed in a square region of 400m per side, with wraparound. The AWGN and spatially uncorrelated channel model is used as in Section IV. A Gaussian quantizer with Γq = 1 and channel imperfection factor of τ = 0.2 is considered. All users transmit at the same power with pu = 23 dBm. Interference is significantly stronger than noise with pathloss being βru = (1 + dru /0.392)−3.76, where dru is the distance, and noise power is σ 2 = −106 dBm. First, we show that the DE-based approximations are accurate by comparing it with the empirical average sum rate. For the average rate, Monte-Carlo (MC) simulation is used and averaged over 1000 different channel realizations and topologies. For the DEs, only topological averages are made as they do not depend on small-scale fading. From Fig. 1, the relative gap between the average sum rates and corresponding DEs decrease as number of antennas at each AP, M , increases; this suggests equivalence in the asymptotics. Even with small M , the DE is quite accurate. More impor-

0.3

Network Traffic: 6 APs & 4 UEs Fronthaul limit: 20 bps/Hz/AP

Network Traffic: 25 APs & 12 UEs Fronthaul limit: 20 bps/Hz/AP

0.07

16 0.25

0.06 0.05

15

14

Probability

0.2

Probability

Average Sum Rate [bps/Hz]

Channel-Based Schemes

0.15

0.04 0.03

0.1 0.02

LSFC-Based Schemes

13

0.05 -Fixed

o

-Optimal -Optimal

Local o -Fixed Local -Fixed Proposed o -Optimal (Rate)

12

1

2

3

4

5

o

-Optimal

0 1

2

3

4

5

Optimized Cluster Size

Proposed o -Optimal (Avg. Size) Exhaustive o -Optimal -Optimal (Rate) -Optimal (Avg. Size)

Network Traffic: 6 APs & 4 UEs Fronthaul limit: 20 bps/Hz/AP

11

0

0.01

6

1

7

13

19

25

Optimized Cluster Size

Fig. 3: Probability mass function for optimized cluster sizes, |Cu∗ |. 6

Predetermined Cluster Size

Fig. 2: Average sum rate, obtained from MC, versus cluster schemes.

tantly, Fig. 1 justifies use of DEs as the optimization objective as it accurately captures the trend of the sum rates. Next, we illustrate the effectiveness of the proposed DEoptimal clustering (γ o -Optimal) by comparing to the following benchmarks: Greedy LSFC-based selection with fixed cluster size (βFixed): Select the |Cu | APs with largest LSFCs. |Cu | is predetermined and the same for all u and denoted by “-Fixed” in the notation. • Greedy local SIQNR-DE-based selection with fixed cluster size (Localγ o-Fixed): Select the APs with top-|Cu | largest DE of local SIQNR, which is a psuedo-quantity as if user u is only processed at AP r and obtained as o γu|r = fu (sru = 1, sr′ u = 0, ∀r′ ∈ R\r) using (23). • Greedy local SIQNR-based selection with fixed cluster size (Localγ-Fixed): Same as Localγ o-Fixed but for every realization. The instantaneous local SIQNR is obtained as γu|r = γr |Cu ={r} using (5). • Optimal clustering that maximizes instantaneous SIQNR for each channel realization (γ-Optimal): Maximizing (5) for every realization using exhaustive search. •

We emphasize that the above benchmarks are strong baselines, as they improve upon existing literature, e.g., [4], by accounting for quantization distortion and channel imperfections. The final benchmark represents the best clustering performance attainable without complexity requirements. In Fig. 2, a relatively small network is considered such that it is possible to execute the exhaustive search. Each AP has M = 4 antennas. The figure compares the performance of our proposed scheme to two LSFC-based benchmarks, which are updated at the same frequency as the proposed method. Due to interference, β-Fixed, a common clustering heuristic when considering MMSE combiners, underperforms when MRC is used. Another heuristic, Localγ o -Fixed, offers better performance but it requires searching for best cluster size and sorting the local SIQNR, which is superseded by

the proposed method as the latter has similar complexity but provides the global optimum under the DE-based objective. Two channel-based benchmarks, that are updated every channel realization which is significantly more frequent, are also compared. The globally optimal clustering approach, γOptimal, is obtained through exhaustive search but is impractical for large networks as it scales exponentially for every channel instance. On the other hand, our proposed scheme is polynomial-time and updated less frequently but only penalized by 7.4% in terms of the average sum rates. Another polynomial-time heuristic, Localγ-Fixed, underperforms despite the fact that it is updated for every smallscale fading. This suggests the importance of dynamic service clusters for each user when using MRC as it implicitly manages the interference without complicating the combiner structure. The advantage is further illustrated in Fig. 3 which shows that the optimized cluster sizes, |Cu∗ |, vary significantly across users. This indicates that spatial diversity is utilized adaptively rather than wasted as under fixed-size clustering. VI. C ONCLUSIONS This paper investigates scalable distortion-aware clustering for uplink fronthaul-limited cell-free networks with imperfect channel knowledge and MRC receivers. By leveraging large system analysis, a deterministic equivalent expression of the average network sum rate, under fronthaul distortions, depending only on the LSFCs is derived. This makes the clustering optimization more practical, since LSFC-based updates are required far less frequently than updates based on small-scale fading. A polynomial time scheme is then developed to solve the resulting DE-based problem, and it attains global clustering optimum when restricted to LSFCs. Although low-complexity MRC suffers from interference and fronthaul distortion, the results show that optimized clustering yields meaningful performance gains even when only LSFCs are available. A PPENDIX A P ROOF OF L EMMA 2

√ 1/2 N [BN xN ]i1 Let ai = [AD N ]ii , di = [BN ]ii , and bi = such that bi ∼ CN (0, di ). Set ξi = ai (|bi |4 − 2d2i ), the gap

between the stochastic expression and the proposed DE is N 1 X ξi . (35) ∆N = 2 N i=1

The gap is zero mean since E[|bi |4 ] = 2d2i and it can be shown that E[ξi2 ] = 20a2i d4i from E[|bi |8 ] = 24d4i . The variance of the gap is bounded by N 1 X E[∆2N ] ≤ 3 20a2i d4i , (36) N i=1 using Cauchy-Schwarz inequality and without requiring independence between bi . Since AD N and BN are uniformly bounded, there exists a finite constant C such that a2i d4i ≤ C, ∀i and 20C E[∆2N ] ≤ . (37) N2 −k Choosing ǫN = N and from inequality,   2Chebyshev’s E ∆N ≤ 20CN 2k−2 . (38) Pr (|∆N | ≥ ǫN ) ≤ ǫ2N The sum of probability of the sequence of events {{|∆N | ≥ ǫN }}∞ N =1 is a converging series by choosing 0 < k < 1/2. The sum is ∞ ∞ X X Pr (|∆N | ≥ ǫN ) ≤ 20CN 2k−2 < ∞, (39) N =1

N =1

is finite. By the Borel-Cantelli lemma [10],   Pr lim sup {|∆N | ≥ ǫN }

= 0,

(40)

N →∞

which suggests |∆N | < ǫN almost surely for sufficiently large N . As N → ∞, the gap goes to zero almost surely which completes the proof.

A PPENDIX B P ROOF OF T HEOREM 1 Define the variable transform xr , ar sr such that 0 ≤ xr ≤ ar , and its sum of all r as U . The problem is equivalent to U2 P (41a) max {xr }∀r ,U r ρr xr s.t. 0 ≤ xr ≤ ar , ∀r, (41b) P (41c) U = r xr . We remark that, for fixed U , the objective (41a) can be maximized greedily by filling up xπ(i) in ascending order of ρπ(i) until the sum reaches U because the denominator is minimized. In other words,  there exists an index k such that  i < k, aπ(i) , ∗ (42) xπ(i) = U − Ak−1 , i = k,   0, i > k. The denominator can be rewritten in terms of U and k as Pk P r ρr xr = i=1 ρπ(i) xπ(i)

= (Bk−1 − ρπ(k) Ak−1 ) + ρπ(k) U. (43) In addition, U and k are related as k decides which window U is located in, i.e., U ∈ [Ak−1 , Ak ]. As a result, the problem

is equivalent to U2 (44a) U,k (Bk−1 − ρπ(k) Ak−1 ) + ρπ(k) U s.t. k ∈ {1, . . . , R}, U ∈ [Ak−1 , Ak ]. (44b) where the empty sum A0 , B0 , 0. Observe that, for fixed k, the objective (44a) is convex so the maximizing points are at extreme points, i.e., U ∗ = Ak−1 or U ∗ = Ak . Hence, the global optimal solution can be expressed as A2 (45) k ∗ = arg max k , K = {1, · · · , R}, k∈K Bk and corresponds to U ∗ = A2k∗ . Plugging in (42) and mapping back to sr , the global optimal solution to the original problem is ( 1, i ≤ k ∗ , ∗ sπ(i) = (46) 0, i > k ∗ , which completes the proof. max

R EFERENCES [1] E. Björnson and L. Sanguinetti, “Scalable cell-free massive MIMO systems,” IEEE Trans. on Comm., vol. 68, no. 7, pp. 4247–4261, 2020. [2] Özlem Tugfe Demir, E. Björnson, and L. Sanguinetti, “Foundations of user-centric cell-free massive MIMO,” Foundations and Trends® in Signal Processing, vol. 14, no. 3-4, pp. 162–472, 2021. [3] Z. Li and R. Adve, “Uplink resource allocation optimization for usercentric cell-free MIMO networks,” IEEE Trans. on Wireless Comm., vol. 23, no. 10, pp. 12622–12637, 2024. [4] A. Jayawardane, R. Senanayake, E. Khordad, and J. Evans, “Optimal power allocation and clustering in cell-free wireless networks,” IEEE Trans. Wireless Commun., vol. 24, no. 5, pp. 3669–3683, 2025. [5] W. Yu, P. Patil, B. Dai, and Y. Zhou, “Cooperative beamforming and resource optimization in C-RANs,” in Cloud Radio Access Networks: Principles, Technologies, and Applications, p. 54–81, Cambridge Univ. Press, 2017. [6] L. Liu, W. Yu, and O. Simeone, “Fronthaul-aware design for cloud radio access networks,” in Key Technologies for 5G Wireless Systems (V. W. S. Wong, R. Schober, D. W. K. Ng, and L.-C. Wang, eds.), p. 48–75, Cambridge University Press, 2017. [7] Z. Li, R. Adve, I. Bahceci, and G. Boudreau, “Fronthaul-aware joint compression and resource allocation for user-centric cell-free networks.” Submitted for publication, 2026. [8] Y. Al-Eryani, M. Akrout, and E. Hossain, “Multiple access in cell-free networks: Outage performance, dynamic clustering, and deep reinforcement learning-based design,” IEEE J. Sel. Areas Commun., vol. 39, no. 4, pp. 1028–1042, 2021. [9] Z. Li, J. Wei, and R. Adve, “Decentralized uplink adaptive compression for cell-free MIMO with limited fronthaul,” in ICC 2025 - IEEE Int. Conf. on Comm., pp. 5121–5126, 2025. [10] S. Wagner, R. Couillet, M. Debbah, and D. T. M. Slock, “Large system analysis of linear precoding in correlated MISO broadcast channels under limited feedback,” IEEE TIT, vol. 58, no. 7, pp. 4509–4537, 2012. [11] H. A. Ammar, R. Adve, S. Shahbazpanahi, G. Boudreau, and K. V. Srinivas, “Downlink resource allocation in multiuser cell-free MIMO networks with user-centric clustering,” IEEE Trans. Wireless Commun., vol. 21, no. 3, pp. 1482–1497, 2022. [12] L. Liu, S. Bi, and R. Zhang, “Joint power control and fronthaul rate allocation for throughput maximization in OFDMA-based cloud radio access network,” IEEE TCOM, vol. 63, no. 11, pp. 4097–4110, 2015. [13] J. Bussgang, Crosscorrelation Functions of Amplitude-Distorted Gaussian Signals. Cambridge, MA, USA: MIT Press, Mar. 1952. [14] J. Max, “Quantizing for minimum distortion,” IRE Transactions on Information Theory, vol. 6, no. 1, pp. 7–12, 1960. [15] Q. Zhang, S. Jin, K.-K. Wong, H. Zhu, and M. Matthaiou, “Power scaling of uplink massive MIMO systems with arbitrary-rank channel means,” IEEE JSTSP, vol. 8, no. 5, pp. 966–981, 2014.

Related documents

• Scalable Distortion-Aware Clustering for Fronthaul-Limited Cell-Free MIMO Networks#1200427
• Selective Depthwise Separable Convolution for Lightweight Joint Source-Channel Coding in Wireless Image Transmission#1297104
• Statistical Ambiguity-Function-Aware Joint Beamforming and Subcarrier Allocation for MIMO-OFDM ISAC under Random Symbols#1357543
• Fluid-Antenna-Assisted Distributed Joint Decoding for Cell-Free Massive MIMO Unsourced Random Access#662077
• Fluid-Antenna-Assisted Distributed Joint Decoding for Cell-Free Massive MIMO Unsourced Random Access#661027