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# Channel Charting for Efficient Precoding in Cell-Free Massive MIMO

Hussein Abbass, Luc Le Magoarou, Joumana Farah, Matthieu Crussière

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Hussein Abbass, Luc Le Magoarou, Joumana Farah, Matthieu Crussière. Channel Charting for Efficient Precoding in Cell-Free Massive MIMO. Asilomar Conference on Signals, Systems, and Computers, Oct 2025, California, United States. hal-05393026

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Distributed under a Creative Commons Attribution 4.0 International License
# Channel Charting for Efficient Precoding in Cell-Free Massive MIMO

Hussein Abbass, Luc Le Magoarou, Joumana Farah, Matthieu Crussière

Univ Rennes, INSA Rennes, CNRS, IETR-UMR 6164, Rennes, France

***Abstract***—Cell-free massive MIMO (CF-MIMO) systems promise enhanced spectral efficiency and uniform service by deploying numerous distributed access points (APs) that jointly serve users. However, centralized precoding, which offers optimal performance by leveraging global channel state information (CSI), imposes significant fronthaul overhead due to the extensive CSI exchange between APs and the central processing unit (CPU). In contrast, distributed precoding reduces fronthaul demands, but undergoes performance degradation due to the lack of coordination between APs. This paper introduces a novel framework that strikes a balance between these two scenarios by employing channel charting (CC) as a lightweight CSI compression mechanism at each AP. CC maps high-dimensional CSI into a low-dimensional pseudo-location space, capturing essential channel characteristics. These pseudo-locations are transmitted to the CPU with minimal overhead, where advanced machine learning models are employed to infer effective downlink precoding vectors, thereby approximating centralized performance while reducing fronthaul signaling. Extensive simulations on realistic synthetic datasets demonstrate that our approach achieves very promising results. This work highlights the potential of integrating CC and machine learning to enhance scalability and efficiency in CF-MIMO networks.

***Index Terms***—Cell-free massive MIMO, Channel charting, Fronthaul compression, Deep learning, Dimensionality reduction.

## I. INTRODUCTION

The exponential growth of mobile data traffic and emerging applications requiring ultra-reliable low-latency communications have exposed fundamental limitations of traditional cellular architectures, particularly cell-edge performance degradation and inter-cell interference.

Cell-free massive MIMO (CF-MIMO) has emerged as a paradigm shift to address these challenges [1], [2], [3], [4]. Unlike conventional cellular systems, CF-MIMO deploys numerous distributed access points (APs) that cooperatively serve users without cell boundaries, achieving macro-diversity gains and uniform service quality [3]. This architecture promises 5-10 fold throughput improvements over traditional small-cell systems through optimal power control [1], [3].

However, CF-MIMO implementation faces a critical trade-off between achieving high performance and maintaining practical feasibility. Centralized processing with global CSI achieves optimal performance through joint precoding across cooperating APs [2], but requires extensive fronthaul capacity for CSI exchange, a burden that scales with the number of antennas and subcarriers. In contrast, distributed processing minimizes fronthaul overhead but sacrifices performance [3]. Recent studies show that conventional fronthaul compression

techniques, which typically focus on quantizing the full-dimensional CSI, can certainly achieve compression ratios [5], [6], [7]. However, these gains are limited by the high initial dimensionality of the CSI. A more significant reduction can be achieved by first decreasing the total number of parameters that need to be transmitted, a dimensionality reduction approach that can be subsequently combined with quantization.

Channel charting (CC) emerges as a transformative solution to this fundamental trade-off [8], [9], [10], [11]. By mapping high-dimensional CSI into low-dimensional pseudo-location spaces [12], through unsupervised learning, CC preserves the essential spatial relationships while achieving significant compression [13]. Recent advances have demonstrated CC's effectiveness for CSI compression in massive MIMO, achieving compression ratios exceeding 1000:1, far surpassing conventional methods. The integration of end-to-end learning architectures has further strengthened CC's capability to directly infer precoders from chart locations [13]. This work extends CC to CF-MIMO, for the first time addressing the unique challenges posed by distributed multi-AP systems, multi-subcarrier scenarios, and inter-AP phase coherence. By leveraging phase-insensitive distance metrics [14] and advanced neural architectures [15], we demonstrate that CC can optimize the performance-complexity gap in CF-MIMO, enabling near-optimal centralized performance with minimal fronthaul overhead.

**Contributions.** The main contributions of this paper are summarized as follows:

* We develop a distributed channel charting framework that enables near-optimal centralized precoding in CF-MIMO systems with minimal fronthaul overhead. Our approach compresses local channel information at distributed APs to reconstruct approximate global CSI at the CPU, effectively optimizing the performance gap between centralized and distributed precoding through an end-to-end learning process tailored for user-centric, multi-subcarrier systems.

* We introduce and compare several methods for fusing multi-subcarrier information into compact chart representations. These methods explore different strategies for aggregating frequency-domain features into low-dimensional embeddings, each striking a distinct balance between compression efficiency and system performance.

* We provide extensive numerical results that confirm the effectiveness of the proposed end-to-end model, vali-
dated using realistic synthetic datasets generated through Sionna [16] and DeepMIMO [17] frameworks.

## II. SYSTEM MODEL

### A. System Description

We consider a user-centric cell-free massive MIMO downlink system that serves K users with L distributed APs, each equipped with N antennas. The total number of transmit antennas is M = LN. The system employs orthogonal frequency division multiplexing (OFDM) over a total bandwidth B with S subcarriers. The set of users served by AP ℓ is denoted by $A_ℓ ⊆ \{1, ..., K\}$, and the set of APs serving user k is denoted by $M_k ⊆ \{1, ..., L\}$, determined based on the initial access and AP selection algorithm proposed in [1]. We further define $S_k = \{i : M_i ∩ M_k ≠ ∅\}$ as the set of users served by partially the same APs as user k.

The frequency-domain channel from AP $ℓ$ to user $k$ across all $S$ subcarriers is represented by $H_{kℓ} ∈ C^{N×S}$, where the $s$-th column $h_{kℓ}^{(s)}$ denotes the channel vector for subcarrier $s$. The global channel matrix for user $k$ is

$$H_k \triangleq \begin{bmatrix} H_{k1} \\ \vdots \\ H_{kL} \end{bmatrix} \in C^{M×S}. \quad (1)$$

We define selection matrices $A_{kℓ} ∈ C^{N×N}$ that indicate whether AP $ℓ$ serves user $k$ ($A_{kℓ} = I_N$ if it does, $0^{N×N}$ otherwise), and aggregate them into a block-diagonal matrix $A_k = \text{diag}\{A_{k1}, ..., A_{kL}\}$ to select the active antenna blocks in the global precoding vectors. The system operates in time division duplexing (TDD) mode, taking advantage of channel reciprocity between uplink and downlink. Channels are estimated at the APs using pilot signals transmitted by the users.

### B. Downlink Signal Model and Power Constraint

The transmitted signal from AP $ℓ$ on subcarrier $s$ is

$$\mathbf{x}_{\ell}^{(s)} = \sum_{i=1}^{K} \mathbf{A}_{i\ell} \mathbf{w}_{i\ell}^{(s)} s_{i}^{(s)}, \quad (2)$$

where $s_i^{(s)}$ is the unit-power data symbol for user $i$ on subcarrier $s$, and $\mathbf{w}_{i\ell}^{(s)} ∈ C^N$ is the precoding vector used by AP $ℓ$ for user $i$ on subcarrier $s$.

The received signal at user $k$ on subcarrier $s$ is

$$y_k^{(s)} = \sum_{\ell=1}^{L} (\mathbf{h}_{k\ell}^{(s)})^H \mathbf{x}_{\ell}^{(s)} + n_k^{(s)}, \quad (3)$$

where $n_k^{(s)} \sim CN(0, \sigma_k^2)$ is an additive Gaussian noise.

A common total power constraint across all APs and subcarriers is imposed:

$$\sum_{s=1}^{S} \sum_{\ell=1}^{L} \mathbb{E}\{\|\mathbf{x}_{\ell}^{(s)}\|^2\} \leq P_{\text{tot}}, \quad (4)$$

where $P_{\text{tot}}$ is the total transmit power budget across all APs, users, and subcarriers.

### C. Centralized and Distributed Precoding

We present the general forms of centralized and distributed precoding strategies, considering power allocation coefficients $p$. In this work, we adopt a uniform power allocation strategy across all users and subcarriers for all evaluated configurations.

a) Centralized minimum mean square error (MMSE) precoding: With full CSI $\{\mathbf{h}_k^{(s)}\}_{k=1,s=1}^{K,S}$ available at the CPU, the centralized MMSE precoding vector for user $k$ on subcarrier $s$ is given by

$$\tilde{\mathbf{w}}_k^{(s)} = \left( \sum_{i \in S_k} \mathbf{A}_k \mathbf{h}_i^{(s)} (\mathbf{h}_i^{(s)})^H \mathbf{A}_k^H + \sigma^2 \mathbf{I}_M \right)^{-1} \mathbf{A}_k \mathbf{h}_k^{(s)}, \quad (5)$$

where $\tilde{\mathbf{w}}_k^{(s)}$ denotes the unnormalized vector. The final normalized vector with power allocation is:

$$\mathbf{w}_k^{(s)} = \sqrt{p_k^{(s)}} \frac{\tilde{\mathbf{w}}_k^{(s)}}{\|\tilde{\mathbf{w}}_k^{(s)}\|}, \quad (6)$$

where $p_k^{(s)}$ is the power allocated to user $k$ on subcarrier $s$, and $\mathbf{h}_k^{(s)} = [(\mathbf{h}_{k1}^{(s)})^T, ..., (\mathbf{h}_{kL}^{(s)})^T]^T \in C^M$ represents the global channel vector for user $k$ on subcarrier $s$.

b) Distributed MMSE Precoding: In the distributed case, each AP $ℓ$ constructs the local precoder for user $k \in A_ℓ$ on subcarrier $s$ as

$$\tilde{\mathbf{w}}_{k\ell}^{(s)} = \left( \sum_{i \in A_\ell} \mathbf{h}_{i\ell}^{(s)} (\mathbf{h}_{i\ell}^{(s)})^H + \sigma^2 \mathbf{I}_N \right)^{-1} \mathbf{A}_{k\ell} \mathbf{h}_{k\ell}^{(s)}, \quad (7)$$

where $\tilde{\mathbf{w}}_{k\ell}^{(s)}$ is the unnormalized precoding vector at AP $ℓ$. The final normalized precoding vector with power allocation becomes:

$$\mathbf{w}_{k\ell}^{(s)} = \sqrt{p_{k\ell}^{(s)}} \frac{\tilde{\mathbf{w}}_{k\ell}^{(s)}}{\|\tilde{\mathbf{w}}_{k\ell}^{(s)}\|}. \quad (8)$$

### D. SINR Expression

The instantaneous SINR of user $k$ on subcarrier $s$ is

$$\text{SINR}_k^{(s)} = \frac{\left| (\mathbf{h}_k^{(s)})^H \mathbf{A}_k \mathbf{w}_k^{(s)} \right|^2}{\sum_{\substack{j=1 \\ j \neq k}}^{K} \left| (\mathbf{h}_k^{(s)})^H \mathbf{A}_j \mathbf{w}_j^{(s)} \right|^2 + \sigma_k^2}, \quad (9)$$

where $\mathbf{w}_k^{(s)} \in C^M$ is the global precoding vector obtained by stacking $\{\mathbf{w}_{k\ell}^{(s)}\}_{\ell \in M_k}$.

### E. Objective

The aim is to design precoding matrices $\mathbf{W}^{(s)} = [\mathbf{w}_1^{(s)}, ..., \mathbf{w}_K^{(s)}] \in C^{M×K}$ for each subcarrier $s$ that maximize the instantaneous downlink sum-rate when the $\mathbf{w}_k^{(s)}$ are computed from decoded CSI vectors reconstructed from compressed chart locations $z$ obtained by channel charting
of the original frequency-domain channel matrices $\mathbf{H}_k$. The optimization problem is formulated as

$$ \max_{\{\mathbf{W}^{(s)}\}_{s=1}^S} \sum_{s=1}^S \sum_{k=1}^K \log_2 (1 + \text{SINR}_k^{(s)}), \quad (10) $$

subject to uniform power allocation across users and subcarriers:

$$ \sum_{s=1}^{S} \|\mathbf{w}_k^{(s)}\|_2^2 = \frac{P_{\text{tot}}}{K}, \quad \forall k \in \{1, \dots, K\}. \quad (11) $$

This constraint is satisfied by setting the power allocation coefficients differently for each architecture to ensure equal total radiated power. For centralized precoding, we set $p_k^{(s)} = \frac{P_{\text{tot}}}{KS}$. For distributed precoding, to account for the multiple APs serving each user, we set:

$$ p_{kl}^{(s)} = \frac{P_{\text{tot}}}{KS|\mathcal{M}_k|}, \quad \forall k, l, s \quad (12) $$

where $|\mathcal{M}_k|$ denotes the number of APs serving user $k$. This normalization guarantees that the total transmit power remains $P_{\text{tot}}$ in both the centralized and distributed scenarios.

### III. METHODOLOGY

Our methodology operates in three stages: APs compress local channels via distributed channel charting, the CPU reconstructs approximate global CSI from aggregated embeddings, and applies centralized precoding using the decoded channels as illustrated in Figure 2.

#### A. Distributed Channel Charting Encoding

Each AP independently compresses its local CSI using CC based on a phase-insensitive distance metric [14]. The local encoder at AP $\ell$ maps the global multi-subcarrier channel $\mathbf{H}_{k\ell}$ to a single low-dimensional embedding:

$$ \mathbf{z}_{k,\ell} = \mathcal{E}_{\ell}(\mathbf{H}_{k\ell}) \in \mathbb{R}^d, \quad d \ll NS, \quad (13) $$

where $\mathcal{E}_{\ell}$ is initialized using ISOMAP [14] applied to a calibration set of $N_c$ multi-subcarrier channel matrices. The calibration stage provides matrices $\mathbf{C}_{\ell} \in \mathbb{C}^{NS \times N_c}$ and $\mathbf{Z}_{\ell} \in \mathbb{R}^{d \times N_c}$, which anchor the local chart, similarly to [13], [15]. The encoding process (see Figure 1) consists of the following operations applied to an unseen vectorized channel $\text{vec}(\mathbf{H}_{k\ell}) \in \mathbb{C}^{NS}$:

1) Correlations with the calibration matrix $\mathbf{C}_{\ell}^{\text{H}}$ columns.

2) Magnitude extraction ($| \cdot |$) for phase-invariant features.

3) Hard thresholding $\mathbf{H}\mathbf{T}_K$ to retain dominant components.

4) $\ell_1$ normalization to standardize scales.

5) Linear mapping to produce the chart embedding.

#### B. Centralized Channel Reconstruction and Precoding

At the CPU, distinct decoders $\{\mathcal{D}_{\ell}\}_{\ell=1}^L$ reconstruct the channel matrices from the chart embeddings (see Figures 1 and 2). For AP $\ell$, the decoder reconstructs:

$$ \mathbf{V}_{k\ell} = \mathcal{D}_{\ell}(\mathbf{z}_{k,\ell}) \in \mathbb{C}^{N \times S}, \quad (14) $$

<figure><img src="image_4.png" /><figcaption>Fig. 1. Encoding and decoding pipeline for AP $\ell$.</figcaption></figure>

where $\mathcal{D}_{\ell}$ employs a random Fourier features (RFF) network followed by a multi-layer perceptron (MLP), as was done in [13], [15], to capture high-frequency channel variations across both spatial and frequency dimensions. The reconstructed global channel for user $k$ is:

$$ \mathbf{V}_k = \begin{bmatrix} \mathbf{V}_{k1} \\ \vdots \\ \mathbf{V}_{kL} \end{bmatrix} \in \mathbb{C}^{M \times S}. \quad (15) $$

The reconstructed global channels $\{\mathbf{V}_k\}_{k=1}^K$ are then fed into a second decoding stage $\mathcal{D}^*$ at the CPU, which performs centralized MMSE precoding. After phase alignment (described in Section III.D), the CPU computes the precoding vectors by substituting the reconstructed channels $\tilde{\mathbf{v}}_k^{(s)}$ for the true channels $\mathbf{h}_k^{(s)}$ in (5) and (6), thereby enabling joint transmission across serving APs with minimal fronthaul overhead.

#### C. End-to-End Training

The term “end-to-end” refers to the joint optimization of all learnable components, the distributed encoders $\{\mathcal{E}_{\ell}\}_{\ell=1}^L$ and decoders $\{\mathcal{D}_{\ell}\}_{\ell=1}^L$, using a single task-oriented loss function that directly targets beamforming performance rather than CSI reconstruction accuracy [18].

The training strategy operates as follows: the decoder corresponding to each AP $\ell$ produces a reconstructed channel $\mathbf{V}_{k\ell}$ from its local embedding $\mathbf{z}_{k\ell}$. These per-AP reconstructions are phase-aligned (as described in Section III.D) and then concatenated across APs to form the global reconstructed channel $\mathbf{V}_k$ for each user $k$. The loss function evaluates the alignment between these concatenated reconstructed channels and the true global channels, with gradients backpropagating through the entire encoder-decoder pipeline.

1) **Beamforming Alignment Loss:** The training loss maximizes the squared cosine similarity between the phase-aligned concatenated normalized reconstructed channels and the true global channel for each user:

$$ \mathcal{L} = 1 - \frac{1}{|\mathcal{B}|} \sum_{k \in \mathcal{B}} \frac{|\text{vec}(\mathbf{V}_k)^{\text{H}} \text{vec}(\mathbf{H}_k)|^2}{\| \text{vec}(\mathbf{H}_k) \|_2^2}, \quad (16) $$

where $\mathbf{H}_k \in \mathbb{C}^{M \times S}$ represents the ground-truth channel and $\mathbf{V}_k \in \mathbb{C}^{M \times S}$ the concatenated reconstructed channel for user $k$ across all $M$ antennas and $S$ subcarriers, with $\text{vec}(\cdot)$ denoting the vectorization operator and $\mathcal{B}$ the training batch. Note that $\mathcal{L}$ is computed using normalized vectors to focus solely on directional alignment; the channel norms are transmitted separately to denormalize $\mathbf{V}_k$ before the final precoding stage $\mathcal{D}^*$.
<figure><img src="image_5.png" /><figcaption>Fig. 2. Proposed channel charting framework for CF-mMIMO precoding.</figcaption></figure>

2) Properties of the Learned Representation: The loss function in (16) is minimized when the reconstructed channels satisfy:

$$ \mathbf{V}_{k}^{*} = \frac{\mathbf{H}_{k}}{\|\mathbf{H}_{k}\|_{F}} e^{j\phi_{k}}, \qquad (17) $$

where $\| \cdot \|_{F}$ denotes the Frobenius norm, and $\phi_{k} = [\phi_{k,1}, \dots, \phi_{k,L}]^{T}$ contains $L$ independent scalar phase rotations corresponding to each AP. This property reveals that the end-to-end learning framework successfully captures directional channel information while maintaining intra-AP phase relationships. However, the $L$ independent phase ambiguities prevent coherent inter-AP beamforming, motivating our phase alignment strategy described in the following subsection. This explicit alignment sets a common reference ground, pushing the optimizer to converge toward solutions with consistent phase ambiguities across APs so that the relative phase relationships between APs are preserved and driving the loss described in (16) further down.

### D. Phase Alignment Strategy for Coherent Joint Transmission

We address the phase ambiguity problem through a lightweight phase alignment mechanism that restores inter-AP coherence with minimal additional overhead.

1) Phase Reference Extraction: At each AP $\ell$ and for each user $k$, a reference phase is extracted that captures the average channel phase across antennas on each subcarrier. Specifically, the following phases are computed based on the original channel matrix $\mathbf{H}_{k\ell} \in \mathbb{C}^{N \times S}$:

$$ \phi_{k\ell}^{(s)} = \angle \left( \frac{1}{N} \sum_{n=1}^{N} h_{k\ell,n}^{(s)} \right), \qquad (18) $$

where $h_{k\ell,n}^{(s)}$ denotes the channel coefficient from antenna $n$ of AP $\ell$ to user $k$ on subcarrier $s$, and $\angle(\cdot)$ extracts the phase. This averaging operation provides a robust phase estimate.

2) Phase-Coherent Reconstruction: After the decoder reconstructs the channel matrices $\mathbf{V}_{k\ell} \in \mathbb{C}^{N \times S}$ at the CPU, a phase correction is applied to restore inter-AP coherence:

$$ \tilde{\mathbf{v}}_{k\ell}^{(s)} = \mathbf{v}_{k\ell}^{(s)} \cdot e^{j\phi_{k\ell}^{(s)}}, \qquad (19) $$

where $\mathbf{v}_{k\ell}^{(s)} \in \mathbb{C}^{N}$ is the $s$-th column of $\mathbf{V}_{k\ell}$. This operation aligns the reconstructed channels with their true phase references, enabling coherent combining across APs. The phase-aligned global channel vector for user $k$ on subcarrier $s$ is then reconstructed as:

$$ \tilde{\mathbf{v}}_{k}^{(s)} = \begin{bmatrix} \tilde{\mathbf{v}}_{k1}^{(s)} \\ \vdots \\ \tilde{\mathbf{v}}_{kL}^{(s)} \end{bmatrix} \in \mathbb{C}^{M}, \qquad (20) $$

which serves after denormalization as the input for linear precoding techniques (e.g., MMSE, zero-forcing, etc.) implemented in the second decoding stage, $\mathcal{D}^{*}$ (see Figure 2), at the CPU.

### E. Fronthaul Overhead Analysis and Compression Trade-offs

The proposed CC framework compresses the frequency-domain CSI of each AP-user pair, originally consisting of $2NS$ real coefficients, into a compact representation transmitted over fronthaul links.

1) Baseline Compression with Full Phase Information: In the baseline configuration, each AP transmits per user:

* a $d$-dimensional chart embedding,

* $S$ phase references to ensure inter-AP coherence,

* one scalar for channel norm to enable proper denormalization at the CPU.

This yields a compression ratio of:

$$ \gamma_{\text{baseline}} = \frac{2NS}{d + S + 1}. \qquad (21) $$

2) Enhanced Compression via Phase Progression Modeling: The compression efficiency can be significantly improved by exploiting the physical relationship between phase and frequency across subcarriers. Rather than transmitting $S$ independent phase values, the AP transmits only a single reference phase $\phi_{k\ell}^{\text{ref}}$ for the central subcarrier. The CPU then reconstructs the remaining phases using this approximation:

$$ \phi_{k\ell}^{(s)} = \phi_{k\ell}^{\text{ref}} \cdot \frac{f^{(s)}}{f^{\text{ref}}}, \qquad (22) $$
<table><caption>TABLE I
PERFORMANCE COMPARISON OF CHANNEL CHARTING STRATEGIES ON SIONNA AND DEEPMIMO DATASETS</caption><thead><tr><th rowspan="2">Method</th><th rowspan="2">Chart Type</th><th rowspan="2">Phase Information</th><th>Compression</th><th colspan="2">Sionna</th><th colspan="2">DeepMIMO</th></tr><tr><th>Ratio</th><th>Median ρ</th><th>Mean ρ</th><th>Median ρ</th><th>Mean ρ</th></tr></thead><tbody><tr><td>True Locations</td><td>Physical coordinates</td><td>Full phase information</td><td>×13.48</td><td>0.85</td><td>0.80</td><td>0.86</td><td>0.81</td></tr><tr><td>Best Beamforming</td><td>One chart location per subcarrier</td><td>Full phase information</td><td>×5.23</td><td>0.94</td><td>0.91</td><td>0.96</td><td>0.92</td></tr><tr><td>Maximum Compression</td><td>Chart location for all subcarriers</td><td>One reference phase</td><td>×64</td><td>0.84</td><td>0.80</td><td>0.85</td><td>0.79</td></tr><tr><td>Hybrid Approach (G=4)</td><td>Chart location per group of subcarriers</td><td>One reference phase per group</td><td>×19.7</td><td>0.90</td><td>0.88</td><td>0.91</td><td>0.89</td></tr><tr><td>Baseline CC</td><td>Chart location for all subcarriers</td><td>Full phase information</td><td>×13.48</td><td>0.86</td><td>0.81</td><td>0.87</td><td>0.81</td></tr></tbody></table>

where $f^{(s)}$ denotes the frequency of subcarrier $s$ and $f^{\text{ref}}$ the reference frequency. With this enhancement, the compression ratio becomes:

$$ \gamma_{\text{enhanced}} = \frac{2NS}{d + 2}, \qquad (23) $$

which corresponds to a substantial improvement, particularly for systems with large $S$.

3) *Compression-Performance Trade-offs:* The proposed framework offers flexibility in balancing compression efficiency and beamforming performance through different chart encoding strategies:

a) *Maximum Compression:* The configuration described above, using a single chart location for all subcarriers with phase progression modeling, achieves the highest compression ratio given by (23).

b) *Best Beamforming:* Alternatively, encoding a separate chart location for each subcarrier while transmitting full phase information maximizes beamforming accuracy at the cost of reduced compression:

$$ \gamma_{\text{per-subcarrier}} = \frac{2NS}{Sd + S + 1} \qquad (24) $$

c) *Hybrid Approach:* A compromise strategy groups the subcarriers into $G$ clusters, each containing $S' = S/G$ subcarriers, with one chart location being constituted per group. Using phase progression modeling within each group, the compression ratio becomes:

$$ \gamma_{\text{hybrid}} = \frac{2NS}{Gd + G + 1} = \frac{2NS'G}{Gd + G + 1} \qquad (25) $$

This hybrid approach enables system designers to effectively manage the trade-off between fronthaul overhead and precoding accuracy by adjusting the grouping parameter $G$.

# IV. SIMULATION RESULTS

## A. Simulation Settings

The proposed channel charting framework is evaluated using two realistic datasets: DeepMIMO [17] with the urban outdoor O1 scenario and Sionna [16] with the Etoile scene in Paris. In both cases, the system operates at a central frequency of 3.5 GHz with a bandwidth of 20 MHz. The cell-free system comprises $L = 10$ distributed APs, each with $N = 8$ antennas, serving users over $S = 16$ subcarriers. For training and

evaluation, we use $N_c = 3k$ calibration channels, $4k$ training channels for model optimization, and $2k$ test channels for performance assessment.

System-level performance is evaluated through Monte Carlo simulations with 1,000 independent realizations. In each realization, $K = 25$ users are randomly selected from the test set. The initial access and AP selection algorithm from [1] is employed, where $\tau_p = 10$ constrains each AP to serve at most 10 users, thereby ensuring system scalability and limiting fronthaul overhead.

For channel charting, the decoded channel vectors serve as proxies for true channels at the CPU, enabling centralized MMSE precoding using (6). *Baseline centralized and baseline distributed* schemes that use the true channels with their respective MMSE formulations are also considered for performance comparison.

## B. Hyperparameters

The embedding space dimension in CC is set to $d = 2$ for all the configurations presented in Table I. For the distributed decoders architecture, i.e., $\{D_l\}_{l=1}^L$, we employ $F = 200$ random Fourier features with bandwidth parameter $\sigma_B$ optimized separately for each dataset as in [13]. The MLP components utilize $T = 128$ hidden units per layer, with 3 layers, providing sufficient capacity to capture complex channel-to-precoder mappings while maintaining computational efficiency.

## C. Performance Analysis

Table I presents the performance comparison of different channel charting strategies on both datasets, evaluated using the cosine similarity metric $\rho$ between predicted and optimal beamforming vectors, defined as:

$$ \rho = \frac{|\mathbf{V}^H \mathbf{H}|}{\|\mathbf{V}\| \cdot \|\mathbf{H}\|} \qquad (26) $$

where $\mathbf{V}$ represents the reconstructed channel vector and $\mathbf{H}$ the ground-truth channel. This phase-insensitive metric quantifies directional alignment, with $\rho = 1$ indicating perfect alignment.

The Best Beamforming configuration achieves the highest performance with median $\rho$ values of 0.94 (Sionna) and 0.96
<figure><img src="image_6.png" /><figcaption>Fig. 3. Compression-performance trade-off at 40 dBm transmit power.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Sum SE performance vs. transmit power budget for the proposed CC framework and baseline approaches.</figcaption></figure>

(DeepMIMO). This superior performance validates that per-subcarrier chart embeddings capture fine-grained frequency-selective channel characteristics essential for accurate precoding. As illustrated in Figure 3, this configuration achieves an average sum spectral efficiency (sum SE) of 24.57 bps/Hz at 5.23× compression, representing 96% of the centralized upper bound performance at a power budget of 40 dBm.

Notably, the *Baseline CC* method outperforms the *True Locations* baseline on both datasets, despite the fact that the *True Locations* approach exploits perfect geometric information by directly using the known physical coordinates of users as input to the decoders {$D_e$}$_{e=1}^L$, thereby bypassing the encoding stage entirely. This approach provides a baseline to assess whether learned chart embeddings offer advantages over direct coordinate-based inference, and demonstrates that learned bidimensional channel charts encode more meaningful information than physical coordinates for beamforming tasks. The superior performance stems from channel charts directly capturing the underlying channel manifold structure, which incorporates both geometric and propagation-specific features that true locations alone cannot represent.

The *Hybrid* approach (with $G = 4$) achieves an excellent balance between compression and performance, with median $\rho$ values of 0.90 (Sionna) and 0.91 (DeepMIMO). As shown in Figure 3, this configuration maintains over 90% of the centralized performance while reducing fronthaul overhead by a factor of 19.7×, achieving 23.4 bps/Hz (91% of the baseline centralized approach).

The *Maximum Compression* configuration maintains reasonable performance while achieving the highest compression ratio of 64×. This validates the effectiveness of the phase progression modeling described in Equation (19) for capturing frequency-dependent phase relationships with minimal overhead. At this compression level, the system still achieves 21.5 bps/Hz (84% of the baseline centralized approach), significantly exceeding the *Baseline Distributed* scheme that requires no fronthaul CSI exchange. Indeed, in the *Baseline Distributed* approach, each AP independently computes local

precoders using only locally estimated channels, following (7) and (8). However, the lack of inter-AP coordination results in this substantial performance gap, yielding a sum SE of only 15 bps/Hz.

It is clear that bidimensional chart locations perform well across all proposed approaches. This holds true under the continuity hypothesis of the underlying location-to-channel mapping, where nearby locations in physical space correspond to similar channel realizations. This latent structure is discovered through the end-to-end training process, enabling accurate precoder inference from compact representations.

Figure 4 examines system performance across a wide range of power budget values from 20 to 55 dBm. All proposed methods consistently outperform the distributed baseline, with the performance gap widening as transmit power increases and the system transitions from noise-limited to interference-limited operation. This behavior is best analyzed by splitting the power range into three distinct operating zones:

* **Noise-Limited Regime (20-30 dBm):** At these low power levels, the system is primarily noise-limited. Consequently, the performance differences between all methods are modest, as thermal noise, rather than inter-user interference, dominates and masks the benefits of coordination.

* **Practical/Transition Regime (30-45 dBm):** As power increases, the system transitions to being interference-limited, and the advantages of coordinated precoding through channel charting become increasingly pronounced. This practical operating range highlights the value of even highly compressed coordination: the *Maximum Compression* (64×) configuration already demonstrates substantial gains of 6-7 bps/Hz over the distributed processing baseline.

* **High-Power, Interference-Limited Regime (at 55 dBm):** The trend culminates at the 55 dBm endpoint, where the system is deep in the interference-limited regime. Here, the performance gap between the central-
ized and distributed baselines reaches its maximum at approximately 13 bps/Hz. At this point, our methods show their full potential: the *Best Beamforming* method captures 11.5 bps/Hz of this gain with only 5.23× compression, while the *Hybrid Approach* achieves a good balance, maintaining strong performance (within 5 bps/Hz of centralized) despite its high 19.7× compression.

Finally, the power scaling analysis reveals that all channel charting methods exhibit superior high-SNR slopes that approach the centralized bound, a characteristic not shared by the distributed baseline. These consistent relative performance trends confirm that the learned representations capture the essential spatial structure and degrees of freedom required for effective interference management, independent of the operating SNR regime, which is a critical requirement for dense cell-free deployments.

## V. CONCLUSION

In this paper, we proposed a channel charting framework for cell-free massive MIMO that compresses CSI via distributed learning while preserving beamforming performance. In this framework, APs map high-dimensional channels to 2D chart locations, transmit compact channel representations with phase references to the CPU, where effective precoders are reconstructed through end-to-end learning. Results on DeepMIMO and Sionna datasets show that the proposed methods can reach 84%–96% of the baseline centralized performance with a compression ratio of up to 64x. This dimensionality reduction can be coupled with subsequent quantization techniques to further increase the overall fronthaul compression, effectively bridging the performance-fronthaul trade-off.

## ACKNOWLEDGMENT

This work was supported by the Agence Nationale de la Recherche (ANR) under the France 2030 program, grant "NF-PERSEUS: ANR-22-PEFT-0004".

## REFERENCES

[1] S. Chen, J. Zhang, E. Björnson, J. Zhang, and B. Ai, "Structured massive access for scalable cell-free massive MIMO systems," *IEEE Journal on Selected Areas in Communications*, vol. 39, no. 4, pp. 1086-1100, Apr. 2021.

[2] E. Björnson and L. Sanguinetti, "Making cell-free massive MIMO competitive with MMSE processing and centralized implementation," *IEEE Transactions on Wireless Communications*, vol. 19, no. 1, pp. 77-90, Jan. 2020.

[3] Ö. T. 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.

[4] J. Farah, C. Ghanem, and E. P. Simon, "Uncoordinated transmissions in uplink IoT cell-free massive MIMO systems based on NOMA," in *31st European Signal Processing Conference (EUSIPCO)*, 2023, pp. 1479-1483.

[5] M. Bashar, H. Q. Ngo, A. G. Burr, D. Maryopi, K. Cumanan, and E. G. Larsson, "On the performance of backhaul constrained cell-free massive MIMO with linear receivers," in *52nd Asilomar Conference on Signals, Systems, and Computers*, Pacific Grove, CA, USA, Nov. 2018, pp. 494-501.

[6] S. Park, A. H. Gokceoglu, L. Wang, and O. Simeone, "Scalable multivariate fronthaul quantization for cell-free massive MIMO," *IEEE Transactions on Signal Processing*, vol. 73, pp. 1658-1673, 2025.

[7] A. Burr, M. Bashar, and D. Maryopi, "Cooperative access networks: Optimum fronthaul quantization in distributed massive MIMO and cloud RAN," in *2018 IEEE 87th Vehicular Technology Conference (VTC Spring)*, Porto, Portugal, 2018, pp. 1-5.

[8] C. Studer, S. Medjkouh, E. Gönültaş, T. Goldstein, and O. Tirkkonen, "Channel charting: Locating users within the radio environment using channel state information," *IEEE Access*, vol. 6, pp. 47682-47698, 2018.

[9] J. Deng, S. Medjkouh, N. Malm, O. Tirkkonen, and C. Studer, "Multipoint channel charting for wireless networks," in *52nd Asilomar Conference on Signals, Systems, and Computers*, Pacific Grove, CA, USA, Oct. 2018, pp. 286-290.

[10] S. Taner, V. Palhares, and C. Studer, "Channel charting in real-world coordinates," in *GLOBECOM 2023 - 2023 IEEE Global Communications Conference*, Kuala Lumpur, Malaysia, 2023, pp. 3940-3946.

[11] T. Yassine, L. Le Magoarou, S. Paquelet, and M. Crussière, "Leveraging triplet loss and nonlinear dimensionality reduction for on-the-fly channel charting," in *2022 IEEE 23rd International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)*, Oulu, Finland, Jul. 2022, pp. 1-5.

[12] T. Yassine, L. Le Magoarou, M. Crussière, and S. Paquelet, "Optimizing multicarrier multiantenna systems for LoS channel charting," *IEEE Transactions on Wireless Communications*, vol. 23, no. 10, pp. 14702-14714, Oct. 2024.

[13] B. Chatelier, V. Corlay, M. Crussière, and L. Le Magoarou, "CSI compression using channel charting," in *2024 58th Asilomar Conference on Signals, Systems, and Computers*, Pacific Grove, CA, USA, 2024, pp. 1660-1664.

[14] L. Le Magoarou, "Efficient channel charting via phase-insensitive distance computation," *IEEE Wireless Communications Letters*, vol. 10, no. 12, pp. 2634-2638, Dec. 2021.

[15] T. Yassine, L. Le Magoarou, S. Paquelet, and M. Crussière, "Model-based deep learning for beam prediction based on a channel chart," in *57th Asilomar Conference on Signals, Systems, and Computers*, 2023, pp. 1636-1640.

[16] J. Hoydis, F. A. Aoudia, A. Vem, A. Ashikhmin, S. Bengtsson, M. Marcus, and A. Keller, "Sionna: An open-source library for next-generation physical layer research," arXiv preprint arXiv:2203.11854, 2022.

[17] A. Alkhateeb, "DeepMIMO: A generic deep learning dataset for millimeter wave and massive MIMO applications," in *Information Theory and Applications Workshop (ITA)*, San Diego, CA, USA, Feb. 2019, pp. 1-8.

[18] C.-K. Wen, W.-T. Shih, and S. Jin, "Deep learning for massive MIMO CSI feedback," *IEEE Wireless Communications Letters*, vol. 7, no. 5, pp. 748-751, Oct. 2018.