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Towards the quantitative characterisation of piglets' robustness to weaning: a modelling approach .08

Gut organoids

Genetics, Microbiota, Health team : GeMS • Obtained from embryonic cells, iPSC, and adult stem cells from any organ

• Near physiological models to dissect complex traits into molecular phenotypes in a reproducible manner Pigs as biomedical models

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Cutaneous melanoma in Man

Genomic classification of cutaneous melanoma

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Melanoma regression in MeLiM

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  et al Animal, 2019 Comparison of the weight dynamics as predicted by the unperturbed and the perturbed (Gompertz-Makeham) models o different BW measures of the individual piglet relative to days from weaning __ the predicted response of the unperturbed growth model ---the perturbed growth model response

  Pig tissues: a, b: ileum organoid, c: original tissue staining d: colon organoid, e: duodemun organoid, f: lung organoid 19/11/2019

  : (dacarbazine) low response rates and low survival 3. New therapeutic strategies : Gene targeted therapies : Kinase inhibitors a. BRAF inhibitors (vemurafenib, dabrafenib, sorafenib) b. MEK inhibitors (trametinib) rapid response, last 6-9 months, need other therapies side effects: arthralgia, fatigue, nausea, rashes, photosensitivity, keratoacanthoma,pyrexia Stimulation of immune responses: Immune checkpoint blockers Benefit only 15% of patients but no available biomarker to identify them a. Anti Cytotoxic T-lymphocyte-associated antigen-4 (CTLA-4) antibody (ipilimumab) b. Programmed-death-1 (PD1) (nivolumab) and its ligand (PD1L) (pembrolizumab) Slow but long term remission up to 5 years side effects: toxicity to the bowel, lung and liver with potential life-threatening colitis, pneumonitis, hepatitis, need high doses steroids, fatigue, skin rashes, vitiligo .016 1. Excision surgery 2. Chemotherapies : (dacarbazine) low response rates and low survival 3. New therapeutic strategies : Gene targeted therapies : Kinase inhibitors a. BRAF inhibitors (vemurafenib, dabrafenib, sorafenib) b. MEK inhibitors (trametinib) rapid response, last 6-9 months, need other therapies side effects: arthralgia, fatigue, nausea, rashes, photosensitivity, keratoacanthoma,pyrexia Stimulation of immune responses: Immune checkpoint blockers Benefit only 15% of patients but no available biomarker to identify them a. Anti Cytotoxic T-lymphocyte-associated antigen-4 (CTLA-4) antibody (ipilimumab) b. Programmed-death-1 (PD1) (nivolumab) and its ligand (PD1L) (pembrolizumab)Slow but long term remission up to 5 years side effects: toxicity to the bowel, lung and liver with potential life-threatening colitis, pneumonitis, hepatitis, need high doses steroids, fatigue, skin rashes, vitiligo share 42 genes not found in mice Mice & humans share 14 genes not found in pigs Frequency of gross protein structural preservation between human and pig for immune related genes is nearly twine that of mouse to human and pig to mouse(Dawson et al, 2016)



  Longitudinal transcriptomic studies: regression relates to melanoma cell's cell cycle arrest, & an immune reaction  Senescence is detected by patches along tumor regression  Regression is partly the result of replicative potential loss in MeLiM  Regression goes along with depigmentation (CD4 locus specific haplotype)  Blood antibodies against porcine melanocytic and melanoma antigens appear during regression Regression is partly the result of immunoreactivity in MeLiM 19/11/2019

UV Athens Depigmentation on MeLiM with melanoma regression No depigmentation High Moderate 80% 45% 60% 37%

Rambow et al, Neoplasia 2008 19/11/2019 .037 Giorgia Egidy /-Follow up of clinical observations on hair, skin, eyes depigmentation in minipigs resembling vitiligo in humans incidence Blanc et al, Immunogenetics 2017 19/11/2019

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UV Athens CD4 alleles may loose antibody binding

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Stages I & II : no metastases III & IV : metastases detected

Acknowledgments :

Master students : Leyla Ahmadova Samuel Badji Kévin Désirabel, Sihem Hamine Sergiu Raileanu Ameni Sdiri

Genomic classification of cutaneous melanoma

The Cancer Genome Atlas Network, Cell 161: 1681, 2015 Pathways altered in melanoma: Multiple linear canonical cascades converge to consolidate autonomous cell proliferation. Cancer regression would gain from reactivation of apoptosis and senescence.

Melanoma development in MeLiM

Giorgia Egidy /UV Athens Foci at confluent growth 100µm.


Augustin Ernoult
email: augustin.ernoult@inria.fr.

Jérémy Cabaret

Juliette Chabassier

Openwind: a software to simulate wind instruments, as a tool for acoustic teachers

Keywords: musical acoustics, simulation, graphical user interface

INTRODUCTION

Openwind (for open Wind INstrument Design) software is an open-source (GPL-v3) python library available for free: https://openwind.inria.fr [START_REF] Openwind | Open Wind INstrument Design[END_REF]. It aims at solving state-of-the-art models in musical acoustics for wind instruments with accurate, robust and efficient techniques from scientific computing science and numerical analysis. It allows the direct computation of the acoustic response in the frequency domain (input impedance) of a given instrument (knowing its main bore and side holes geometrical parameters), or of the sound of this instrument when coupled to different types of oscillators (reed, lips). It also relies on inversion techniques used in the field of geophysics (Full Waveform Inversion, [START_REF] Virieux | An overview of fullwaveform inversion in exploration geophysics[END_REF]) to perform reconstructions from measurements and investigate some sensitivity characteristics of the instruments.

Initially, the purpose of this library was to propose a collaborative tool for researchers from musical acoustic community, as still currently, the usual practice is to reimplement in each laboratory, similar models with small individual specificities. A second objective was to provide a virtual workshop to manufacturers. As these users are often not familiar with coding, a graphical user interface (GUI) has been developed and is freely available online (https://demo-openwind.inria.fr).

In addition to these two communities, some acoustics teachers took advantage of these tools to build educational activities around the acoustic of the musical instruments. For example Morrison and Worland [START_REF] Morrison | Calculating the acoustic input impedance of a simplified brass instrument as an educational laboratory activity[END_REF] designed an educational laboratory activity on the acoustic of brass instruments using Openwind library to illustrates the link between the geometry of the instrument and its acoustic properties. The activity presented this time has been designed in the context of the training of instrument makers in ITEMM. This uses the GUI to help the future manufacturers to understand the relation between the geometry of an instrument, the distribution of the energy along its main bore and its frequencies of resonance.

After a short presentation of the features of the python library and the GUI in sections 2 and 3, the educational activity addressed to the future instrument makers of ITEMM is presented in section 4.

THE PYTHON LIBRARY

What is behind

Finite Elements Method (FEM) is used in Openwind to perform the spatial discretisation of the one-dimensional (1D) pipe equations [START_REF] Tournemenne | A comparison of a one-dimensional finite element method and the transfer matrix method for the computation of wind music instrument impedance[END_REF]: telegraphers' equations with visco-thermal effects, or their approximations [START_REF] Zwikker | Sound absorbing materials[END_REF][START_REF] Chabassier | Viscothermal models for wind musical instruments[END_REF][START_REF] Thibault | Dissipative timedomain one-dimensional model for viscothermal acoustic propagation in wind instruments[END_REF]Chap.5]. The presence of side holes, valves (pistons) are modeled by using pipe junctions acoustic acoustic reactances and inertances to model the distribution of the acoustic energy between the pipes [START_REF] Lefebvre | Characterization of woodwind instrument toneholes with the finite element method[END_REF]. The radiation of each opening is modeled through radiation impedance [START_REF] Silva | Approximation formulae for the acoustic radiation impedance of a cylindrical pipe[END_REF]. Several radiation conditions can be used (infinite flanged, unflanged, semi-open hole, etc). With some constraints on the radiation models, a technique computing directly the modal parameters from the FEM discretisation [START_REF] Chabassier | Direct computation of modal parameters for musical wind instruments[END_REF] has been also implemented. It gives directly access to the frequency, the quality-factor and the magnitude of the impedance resonances.

The FEM constitutes a variational approximation of the pipe equations, suitable for energy consistent timestepping strategies allowing the coupling with various elements such as nonlinear oscillators, pipe junctions or radiation conditions (see [START_REF] Thibault | Time-domain simulation of a dissipative reed instrument[END_REF]). The FEM generalizes the method used in [START_REF] Bilbao | Direct Simulation of Reed Wind Instruments[END_REF] where a low order approximation is performed. A one-degree-of-freedom oscillator modeling the reed or lips has been implemented in Openwind, allowing the simulation of the sound of brass or reed instruments. This necessitates to estimate the values of the oscillator parameters adapted to the studied instrument, which can be challenging in itself.

Moreover, the FEM formulation provides an efficient framework for the computation of the gradient of the acoustic field with respect to design parameters, as the bore radius or the side holes parameters (see [START_REF] Ernoult | Full waveform inversion for bore reconstruction of woodwind-like instruments[END_REF]). This gradient can be used to perform bore reconstruction (estimate the geometry of the instrument from acoustic measurement) by designing optimization problem. This necessitates a good first guess of the geometry to reach the expected local minimum. This has been used, for example, to guess the inner geometry of an historical trumpet [START_REF] Baarsel | Understand and predict acoustic properties of heritage instruments: the case of a Besson trumpet of the Musée de la Musique of Paris[END_REF].

How to use it?

The python library is divided in several modules dealing with the three main features describe above (computations in frequency domain, time domain and inversion), the interpretation of the geometry files, the meshing of this geometry, the physical models (approximation, inclusion or not of several effect), etc. The organization of the library and the detail of each class, method, etc is described in the online documentation available from the main website [START_REF] Openwind | Open Wind INstrument Design[END_REF].

This documentation provides also basic tutorials and several examples ("how to") to help get started with the software. For example, one file describes how to tune the temperature along the main bore of the instrument (uniform or not). Computations can be performed very simply with high level methods, without the necessity to fully understand the articulation between all the modules of the library. For example, the input impedance of a 50 cm long conical pipe, with inner radius from 7 mm to 15 mm can be computed and plotted with the following lines: # perform the computation res = ImpedanceComputation(freq, geom) res.plot_impedance() # plot the curve

THE GRAPHICAL USER INTERFACE

For people who are not familiar with python coding, a GUI has been developed giving access to some features of Openwind (Fig. 1). It is freely available online: https: //demo-openwind.inria.fr [START_REF] Openwind | Open Wind INstrument Design[END_REF]. Currently only the frequency domain simulations can be performed with this interface.

In a first part, the geometry of the instruments can be specified. It is divided in three tabs for the main bore, the holes and valves (pistons) and the fingering chart. Some examples can be loaded to illustrate how to fill these forms. The geometry can be saved locally and re-upload later if necessary. In a second part, different options can be chosen before to perform the impedance computation: the fingering applied, the temperature, the frequency range, the radiation conditions, etc. The resulting impedance is then plotted (Fig. 1). In a last part the resonances characteristics are displayed (magnitude, Q-factor, frequency and the corresponding pitch). For each of them, the acoustic fields (pressure and flow) along the main bore are also plotted. All these results can be downloaded and saved locally.

A TEACHING ACTIVITY FOR INSTRUMENT MAKERS

As presented in [START_REF] Morrison | Calculating the acoustic input impedance of a simplified brass instrument as an educational laboratory activity[END_REF], this library can be used to illustrate some acoustical concepts in college or university courses.

Here a different approach is presented, as it stands for the musical instruments making context. The students of ITEMM institute learn instruments making and especially wind instruments. They are familiar with some basic concepts of acoustics like the input impedance and its relation with the geometric parameters of the instruments. Among the common problems in wind instrument repair, making missing parts without information about them can be complex. It is a matter of determining the right geometry to give the right pitch and other characteristics. The virtual prototyping enabled by Openwind, and in particular the online version, allows multiple tests to be performed before finding the right geometry without having to manufacture the actual part or instrument. This virtual trial and error process saves a lot of time and materials, which is important for their future professional activity.

Two activities using Openwind are proposed in the teaching of the ITEMM. For the first one, an initial geometry made of a conical part and a cylindrical part is given. The instruction is to modify this instrument so that its resonances are as harmonic as possible (integer ratio between the resonance frequencies). To do this, the students calculate the impedance of the initial geometry, analyze the resonance frequencies and the ratio between them. They then modify the proportion between the conical part and the cylindrical part until they obtain the desired result.

The goal of the second activity is to tune a specific frequency of resonance of a given fingering without detuning the other resonances. To do this, students must determine the position and size of a side hole. The presence of a side hole, even if it is closed, can affect the frequency of other resonances, and thus the pitch of other notes. To avoid this, the teacher explains that a local modification of the diameter of a hole does not affect the frequency of a given mode (and thus the pitch of the note) if its position corresponds to a pressure node of this mode, which can be easily checked with the GUI. The "acoustic fields" part of the graphical interface, which displays the evolution of the pressure in the instrument, and thus the position of the nodes for all resonances, is very useful here to see which modes can be altered by the modification of the hole geometry.

In a future work with students, it will be interesting to develop other way to solve problems using an hybrid ap-proach with the bore reconstruction module of Openwind (taking into account experimental impedance curves) and the method described earlier.

CONCLUSION AND PERSPECTIVE

In its current state, the Openwind library and the GUI can already be used in teaching activities. To our knowledge, it has been used mainly to teach the acoustics of musical instruments at universities and in training courses for instrument makers, but also in secondary schools for student projects. We believe that this toolbox could also be useful for more general courses in acoustics or physics, for example to illustrate the concept of mode and resonance, or their dependence with the geometry or the temperature.

The development of both the library and the graphic interface still continues. Openwind is used every day by the team to carry out the research activities on wind instruments and new functionalities are released regularly. We would like to propose soon tutorial videos both for the python library and the GUI. If some features are missing for a specific use, suggestions can be made to the team. As Openwind is an open source software, everybody can also join the Gitlab project available from the main website [START_REF] Openwind | Open Wind INstrument Design[END_REF] and becomes a collaborator to help developing the software.

  import numpy from openwind import ImpedanceComputation # define freq. axis and pipe geometry freq = numpy.arange(50, 5001, 1) geom = [[0, 0.5, 7e-3, 15e-3, 'cone']]

Figure 1 .

 1 Figure 1. Screenshot of the GUI, freely available online.

Turin, Italy • 11 th -15 th September 2023 • Politecnico di Torino


Optimizing Multicarrier Multiantenna Systems for

LoS Channel Charting Taha Yassine, Luc Le Magoarou, Matthieu Crussière, Stéphane Paquelet

Abstract-Channel charting (CC) consists in learning a mapping between the space of raw channel observations, made available from pilot-based channel estimation in multicarrier multiantenna system, and a low-dimensional space where close points correspond to channels of user equipments (UEs) close spatially. Among the different methods of learning this mapping, some rely on a distance measure between channel vectors. Such a distance should reliably reflect the local spatial neighborhoods of the UEs. The recently proposed phase-insensitive (PI) distance exhibits good properties in this regards, but suffers from ambiguities due to both its periodic and oscillatory aspects, making users far away from each other appear closer in some cases. In this paper, a thorough theoretical analysis of the said distance and its limitations is provided, giving insights on how they can be mitigated. Guidelines for designing systems capable of learning quality charts are consequently derived. Experimental validation is then conducted on synthetic and realistic data in different scenarios.

Index Terms-channel charting, dimensionality reduction, MIMO signal processing, machine learning.

I. INTRODUCTION

The next generation of cellular network (6G) standards will most likely be marked by a profound integration and utilization of artificial intelligence and in particular machine learning (ML) methods [START_REF] Tang | Future intelligent and secure vehicular network toward 6g: Machine-learning approaches[END_REF], [START_REF] Ali | 6g white paper on machine learning in wireless communication networks[END_REF], [START_REF] Miltiadis | Pervasive artificial intelligence in next generation wireless: The hexa-x project perspective[END_REF], [START_REF] Shi | Machine learning for large-scale optimization in 6g wireless networks[END_REF], [START_REF] Merluzzi | The hexa-x project vision on artificial intelligence and machine learningdriven communication and computation co-design for 6g[END_REF]. Indeed, their ability to learn complex patterns from data makes them suitable for many applications where traditional model-based approaches rapidly reach their limit, especially with the advent of massive multiple input multiple output (MIMO) systems where very large bandwidths and a large number of antennas are considered.

User positioning, which will certainly play an important role in future wireless communication systems, is one such example where ML is expected to bring improvements over traditional approaches. Although dedicated positioning systems like global navigation satellite systems (GNSS) are capable of accurately locating users, they require dedicated hardware, and integration in traditional communication systems is not always feasible nor desired [START_REF] Studer | Channel charting: Locating users within the radio environment using channel state information[END_REF]. They are also known for their bad performance in indoor scenarios, making them not as versatile as needed. Pure radio-based approaches where the existing communication systems are used for user positioning may thus become more attractive. Such approaches are often model-based where they rely on angle-of-arrival Taha Yassine and Stéphane Paquelet are with b<>com, Rennes, France (email: {taha.yassine ; stephane.paquelet}@b-com.com).

Luc Le Magoarou and Matthieu Crussière are with INSA Rennes, CNRS, IETR -UMR 6164, F-35000, Rennes, France (email: {luc.le-magoarou ; matthieu.crussiere}@insa-rennes.fr).

(AoA), time-difference-of-arrival (TDoA) and round-trip-time (RTT) measurements [START_REF] Dwivedi | Positioning in 5g networks[END_REF]. These rather basic models work well for simple scenarios, e.g., line-of-sight (LoS) propagation, but struggle when tested on complex environments, e.g., nonline-of-sight (NLoS) propagation. They constitute a great step towards accurate and precise positioning but still fall behind in terms of the performance requirements of the next generations of wireless systems.

The today widespread adoption of massive MIMO systems and large bandwidths means higher resolvability in both the temporal and angular domains. The channel, being dependent almost exclusively on the transmit position, is then expected to contain enough information to retrieve it. Nonetheless, handcrafting algorithms capable of this becomes unrealistic as propagation scenarios become more complex. On the other hand, data-driven methods were successfully exploited to learn to locate UEs with great precision [START_REF] Wang | Deepfi: Deep learning for indoor fingerprinting using channel state information[END_REF], [START_REF] Vieira | Deep convolutional neural networks for massive mimo fingerprint-based positioning[END_REF], [START_REF] Decurninge | Csi-based outdoor localization for massive mimo: Experiments with a learning approach[END_REF], [START_REF] Ferrand | Dnn-based localization from channel estimates: Feature design and experimental results[END_REF]. However, these methods come with their own drawbacks as they usually require a database of labelled data often hard to acquire. This becomes even more challenging as a database needs to be provided for every new environment to train the model. This being said, although many components of a communication system may benefit from the knowledge of UEs' locations, most of them would only require a relative positioning of the users.

Recently, channel charting (CC) [START_REF] Studer | Channel charting: Locating users within the radio environment using channel state information[END_REF], [START_REF] Ferrand | Tripletbased wireless channel charting: Architecture and experiments[END_REF], [START_REF] Rappaport | Improving channel charting using a split triplet loss and an inertial regularizer[END_REF] has emerged as an unsupervised alternative to user positioning, relying solely on channel data to pseudo-locate UEs relatively to each other. It consists in learning a mapping between channel measurements and a low dimensional space, giving rise to a chart. Points on the chart are akin to pseudo-locations in that they preserve the structure of small spatial neighborhoods rather than the absolute locations. Although less informative than absolute locations, they often convey enough information for some location-based applications such as SNR prediction [START_REF] Kazemi | Snr prediction in cellular systems based on channel charting[END_REF], beam management [START_REF] Ponnada | Best beam prediction in non-standalone mm wave systems[END_REF], pilot allocation [START_REF] Ribeiro | Channel charting for pilot reuse in mmtc with spatially correlated mimo channels[END_REF] and precoding [START_REF] Magoarou | Channel charting based beamforming[END_REF] among others (see [START_REF] Ferrand | Wireless channel charting: Theory, practice, and applications[END_REF] for a recent survey on the topic).

Among CC methods, some explicitly rely on a distance measure between channels to build the chart [START_REF] Agostini | Channel charting: an euclidean distance matrix completion perspective[END_REF], [START_REF] Magoarou | Efficient channel charting via phase-insensitive distance computation[END_REF], [START_REF] Al-Tous | Channel charting with angle-delay-power-profile features and earth-mover distance[END_REF], [START_REF] Stahlke | Indoor localization with robust global channel charting: A timedistance-based approach[END_REF]. For them to be successful, the considered channel distance must allow to reliably detect if any two channels measurements were taken close to each other or not in a given area. In other words, the spatial neighbourhoods within the considered area should be identifiable by the channel distance measure. One can then raise the following two questions: What distance is the most adequate for identifiability? And how can a multicarrier MIMO system be calibrated and parameterized to guarantee it? Contributions. In this paper, MIMO OFDM systems are considered for the task of CC in LoS conditions. First, the notion of identifiability of an area by a channel distance is rigorously defined and its importance for CC is discussed. Then, the phase-insensitive (PI) distance introduced in [START_REF] Magoarou | Efficient channel charting via phase-insensitive distance computation[END_REF] and used for CC in [START_REF] Magoarou | Channel charting based beamforming[END_REF], [START_REF] Yassine | Leveraging triplet loss and nonlinear dimensionality reduction for on-the-fly channel charting[END_REF] is thoroughly analyzed, highlighting its dependency on system parameters such as antenna positions and subcarrier distribution. Conditions to achieve identifiability of an area that link it to these system parameters using the PI distance are given. Special system configurations easing identifiability and consequently particularly suited for CC are proposed. In particular, the following practical guidelines are provided:

1) A relationship between the radial size of the area to chart and the subcarrier spacing is given in order to avoid longrange ambiguities (Proposition 1). 2) An appropriate threshold on the considered channel distance is given in order to avoid short-range ambiguities (Proposition 2).

3) The advantage of using uniform circular arrays (UCA) instead of ULAs for CC is shown mathematically (Sec. IV). 4) A relationship between the radius of the UCA and the bandwidth is given to minimize warping effects on charts (Proposition 4). Finally, experiments are conducted on both synthetic and realistic channels, validating all the theoretical findings of the study. Related work. The initial approaches to CC relied on extracting useful features from channel measurements before effectively learning a chart. In the seminal paper [START_REF] Studer | Channel charting: Locating users within the radio environment using channel state information[END_REF], features derived from the raw 2 nd moment of channel vectors are used as inputs to manifold learning methods. Subsequent work use this same strategy to preprocess the data before feeding it to relevant CC methods, typically dimensionality reduction algorithms based on pairwise distances in the feature space. Alternatively, some proposed methods [START_REF] Ferrand | Tripletbased wireless channel charting: Architecture and experiments[END_REF], [START_REF] Lei | Siamese neural networks for wireless positioning and channel charting[END_REF], [START_REF] Rappaport | Improving channel charting using a split triplet loss and an inertial regularizer[END_REF], [START_REF] Agostini | Not-too-deep channel charting (n2d-cc)[END_REF] successfully learn channel charts of good quality by relying on deep learning. Neural networks of different natures and structures (e.g. CNNs, auto-encoders, siamese networks, triplet networks) would take the handcrafted features as input and try to implicitly learn the underlying spatial distribution of the channel vectors in order to be able to project them on the chart.

Some work in particular consider the problem from a different angle and frame it as one of finding a channel distance that reflects the spatial properties that should be preserved in the learned chart. Indeed, as observed in [START_REF] Magoarou | Efficient channel charting via phase-insensitive distance computation[END_REF], the Euclidean distance is not suitable for channel vectors. The proposed PI distance eliminates the effect of the global phase, making it insensitive to small scale fading and more adequate to CC. In [START_REF] Ponnada | Best beam prediction in non-standalone mm wave systems[END_REF], the log-Euclidean distance between covariance matrices is used to produce charts. In [START_REF] Agostini | Channel charting: an euclidean distance matrix completion perspective[END_REF], a correlation matrix distance is introduced to rule out channels with large azimuthal separation from the construction of a neighborhood graph, the edges of which are computed relying on the feature distance originally introduced in [START_REF] Studer | Channel charting: Locating users within the radio environment using channel state information[END_REF]. In [START_REF] Stahlke | Indoor localization with robust global channel charting: A timedistance-based approach[END_REF], a multi-anchor time of flight (ToF) based distance is derived and exploited to construct the neighborhood graph. The geodesic distance resulting from it shows great correlation with the spatial Euclidean distance. In [START_REF] Al-Tous | Channel charting with angle-delay-power-profile features and earth-mover distance[END_REF], an earth mover's distance (EMD) applied to super-resolution channel features is introduced and shown to outperform other considered covariance distances (i.e. distances defined on the space of covariance matrices, such as the log-Euclidean distance uses in [START_REF] Ponnada | Best beam prediction in non-standalone mm wave systems[END_REF]).

This paper is (to the best of the authors' knowledge) the first to propose system configurations explicitly targeted at easing CC. To do so, the PI channel distance is considered [START_REF] Magoarou | Efficient channel charting via phase-insensitive distance computation[END_REF], but others could have been also considered.

Although this paper doesn't explore the communication capabilities of the designed system, it could nonetheless be framed as an ISAC problem [START_REF] Wymeersch | Integration of communication and sensing in 6g: a joint industrial and academic perspective[END_REF]. Indeed, this paper focuses on adapting an existing communication system to CC, which can be seen as a form of sensing. Similarly to [START_REF] Mateos-Ramos | Model-driven end-to-end learning for integrated sensing and communication[END_REF], the approach presented here could be easily extended to account for a tradeoff between communication and CC.

II. PROBLEM FORMULATION

In this section, the considered setting is introduced, the CC task is briefly presented and the notion of identifiability is properly defined and discussed.

A. System and channel model

Let us consider a base station (BS) equipped with N a antennas, located at positions p 1 , . . . , p Na relative to the barycenter of the antennas, which is considered as the origin. The BS communicates with single antenna UEs over N s evenly spaced OFDM subcarriers f 1 , . . . , f Ns . Let ∆ f denote the subcarrier spacing, B N s ∆ f denote the total bandwidth, f c 1 Ns i f i the central frequency and λ c fc the central wavelength.

In order to make the theoretical analysis of the paper clear and interpretable, let us consider a two dimensional (2D) configuration in which the BS antennas and the users are at the same height (all in the same plane) and free space propagation. In that case, the channels have a single LoS path, so that the uplink channel between an user at location x = (r, θ) (in polar coordinates) and the nth antenna of the BS on the mth subcarrier can be expressed as

h mn ( x) = h mn (r, θ) = 1 r e -j 2π λ (r fm fc -pn. u(θ)) , (1)
where u(θ) = (cos(θ), sin(θ)) (in cartesian coordinates) is the unit vector pointing towards the location x. The complete channel considering all BS antennas and subcarrier can then conveniently be expressed in vector form as

h( x) = √ N a N s r e -j2π r λ f (r) ⊗ a(θ) ∈ C NaNs , (2)
where f (r) and a(θ) are the frequency signature vector and the steering vector respectively, defined as

f (r) 1 √ N s e -j2π r c (f1-fc) , . . . , e -j2π r c (f Ns -fc) , (3)
and

a(θ) 1 √ N a (e j 2π λ p1. u(θ) , . . . , e j 2π λ p Na . u(θ) ) . ( 4
)
The latter is in generic form and encompasses any 2D antenna array configurations. In particular, for a uniform linear array (ULA), p n is given as (0, (n -1)∆ r λ) where ∆ r is the normalized antenna separation. Similarly, for a uniform circular array (UCA), although less common, p n is given as (R cos( 2π(n-1)

Na

), R sin( 2π(n-1)

Na

)) where R is the radius of the array.

B. Channel charting

Let us denote M = N a N s the considered channel dimension. CC aims at learning a function F that maps the channel vectors of UEs in an area of interest A to a low-dimensional space of dimension D M ,

F : C M → R D h → F(h) = z,
where z is the chart location associated to h. CC methods are assessed on their ability to preserve spatial neighborhoods, that is, how well the original spatial distances match the distances on the learned chart for close points

x i -x j 2 ≈ z i -z j 2 . ( 5
)
The ability to preserve the global geometry can also be considered. Performance measures such as continuity (CT), trustworthiness (TW) and Kruskal stress (KS) are used to assess the quality of such charts (see [START_REF] Lei | Siamese neural networks for wireless positioning and channel charting[END_REF] for a formal definition). They are calibrated using a database of channel measures at different locations

{h i } Nc i=1 = {h( x i )} Nc i=1
, N c being the number of measured channels (training samples), and the two notations can be used indifferently, depending on the wish for brevity or explicitness.

This paper focuses on charting methods based on a distance measure d : C M ×C M → R + between pairs of channel vectors (or features derived from it) that tries to match the Euclidean distances between the real locations as closely as possible such that

x i -x j 2 ≈ d(h i , h j ) 2 . ( 6
)
Such a distance is then used to produce the chart. In general, a distance matrix between all pairs of training channels is computed:

D ∈ R Nc×Nc , with d ij = d(h i , h j ),
and subsequently used as the input to a dimensionality reduction algorithm, such as Isomap or EDMC + ADMM [START_REF] Agostini | Channel charting: an euclidean distance matrix completion perspective[END_REF], capable of producing the final chart.

C. Identifiability

Using an appropriate distance measure is necessary to build good charts. In particular, the distance should allow to detect spatial neighborhoods. In order to make this statement more precise, let us define the identifiability of a set of channel observations by a distance.

Definition 1 (Strong Identifiability). A set of channel observations and their corresponding locations S ⊂ R 2 are said to be strongly identifiable by the channel distance

d iff ∀ x, y, z ∈ S d(h( y), h( x)) > d(h( z), h( x)) ⇔ y -x| 2 > z -x| 2
This definition is described as strong because it amounts to finding a channel distance bijective with the Euclidean distance. Since achieving this is often challenging, a weak version of the problem is presented.

Definition 2 (Weak Identifiability). A set of channel observations and their corresponding locations S ⊂ R 2 is said to be weakly identifiable by the channel distance

d iff ∀ x, y, z ∈ S d(h( y), h( x)) > d(h( z), h( x)) ⇒ |r y -r x | > |r z -r x | or |θ y -θ x | > |θ z -θ x |
This weak version of identifiability is defined in terms of the polar coordinates to better suit the mathematical model underlying the propagation channels. Obviously, strong identifiability implies weak identifiability, so that weak identifiability is necessary for strong identifiability.

For a CC method based on a distance d to perform well on an area A, identifiability is an important pre-requisite. Indeed, it is impossible for the method to preserve spatial neighborhoods (which is its objective) if the channel distance measure it is based on already fails to do so. In the subsequent parts of the paper, the distance measure introduced in [START_REF] Magoarou | Efficient channel charting via phase-insensitive distance computation[END_REF] is analyzed, leading to the statement of system setting guidelines and a thresholding procedure in order to favor weak identifiability. For two UEs communicating with the BS with corresponding channel vectors h i and h j , this distance is defined as

d (h i , h j ) 2 -2s (h i , h j ), (7)
where s (h i , h j )

|h H i hj | hi 2
hj 2 is a similarity measure. The first step of the analysis of the distance is to notice that it can nicely be decomposed into a radial and an angular term. Indeed, when injecting (2) one can highlight the following decomposition:

|h H i h j | h i 2 h j 2 = (f (r i ) ⊗ a(θ i )) H (f (r j ) ⊗ a(θ j )) (a) = f (r i ) H ⊗ a(θ i ) H (f (r j ) ⊗ a(θ j )) (b) = f (r i ) H f (r j ) ⊗ a(θ i ) H a(θ j ) = f (r i ) H f (r j ) f (ri,rj ) × a(θ i ) H a(θ j ) ā(θi,θj ) , (8)
where (a) and (b) are because of the transpose property and the mixed-product property of the Kronecker product respectively. This leads to a nice expression of s (h i , h j ) as the product of two terms: f (r i , r j ) and ā(θ i , θ j ). This decomposition in terms of radial and angular component explains why weak identifiability is studied here. Even if such distance measure is phase insensitive, it is still subject to ambiguities that prohibit the establishment of unique local neighborhoods. Indeed, when moving away from a given reference point in an arbitrary direction, the value of the distance oscillates, meaning that points on that line appear successively farther then closer then farther again with varying amplitudes, creating aliases. In particular, these ambiguities arise from two phenomena: a long-range one and a shortrange one. Combined, they create false neighborhoods that affect the trustworthiness of the measure. On one hand, the long-range ambiguities are caused by the periodic nature of the function, meaning that local neighborhoods are duplicated indefinitely in a periodic manner. A simple solution to go around this problem is to only consider users in a area smaller or equal to one period of the function. System parameters could consequently be adjusted to obtain an area of the desired size. One the other hand, the short-range ambiguities are caused by the oscillatory nature of the function inside one period. Fortunately, the amplitudes of the oscillations decrease monotonically, meaning that applying a threshold to only keep the main lobe is feasible and would allow the elimination of the remaining aliases. These considerations are made mathematically rigorous in the next section assuming a ULA for the antenna array.

III. LIMITATIONS OF CHANNEL CHARTING WITH A ULA

In the classical case of a base station equipped with a ULA, it is possible to explicit further the terms of the decomposition [START_REF] Wang | Deepfi: Deep learning for indoor fingerprinting using channel state information[END_REF]. This allows to highlight the limitations of ULAs for channel charting. Note that uniform planar arrays (UPAs) could almost similarly be analyzed as an extension to the 3D case but are not considered here.

Before proceeding, a review of some classical results is necessary as proposed in the following Lemmas.

Lemma 1. In an OFDM system with evenly spaced subcarriers, the radial term is of the form

f (r i , r j ) = D Ns 2πB(r i -r j ) c , (9)
where D Ns is the Dirichlet kernel defined as

D N (x) = 1 N sin( x 2 ) sin( x 2N ) . It is periodic of period D f = c
∆ f provided that the absolute delays of both channel observations (i.e., τ i and τ j ) are known. Additionally, it contains a main lobe of width

L f = 2c B centered at r i = r j . Proof. See Appendix A.
Lemma 2. With a ULA, the angular term is of the form

ā(θ i , θ j ) = |D Na (2π∆ r N a (Θ i -Θ j ))| , ( 10
)
where Θ = sin θ. It is of period

D ā = 1 ∆r w.r.t. Θ. It contains a main lobe centered at Θ i = Θ j of width 2 ∆rNa . In addition, it is symmetrical w.r.t. θ = π 2 [π]. Proof. See Appendix B.
When taken w.r.t. θ, the sin in (10) makes the periodic pattern of varying width in the intervals [-π 2 , π 2 ] + πZ. When the pattern is of width π, it covers the whole interval, leading to the following corollary:

∀θ i , θ j ∈ [-π 2 , π 2 ]. For a given θ i ∈ [-π 2 , π 2
], the main lobe extends from arcsin(sin θ i -1

Na∆r ) to arcsin(sin

θ i + 1 Na∆r ) when -1 + 1 Na∆r ≤ sin(θ i ) ≤ 1 -1
Na∆r , otherwise it is split in two. For θ i = 0, its width is equal to L a = 2 arcsin( 1Na∆r ) (in rad).

Proof. The proof follows from Lem. 2 where the arcsin is applied to the width of the main lobe to convert it to the angular domain.

In the case where the main lobe is split in two:

1) if sin(θ i ) ≤ -1+ 1
Na∆r , one part extends over the span of [-π 2 , arcsin(sin θ i + 1

Na∆r )] and the other over the span of [arcsin(sin

( π 2 -θ i ) -1 Na∆r ), π 2 ]; 2) if 1 -1 Na∆r ≤ sin(θ i ), one part extends over the span of [-π 2 , arcsin(sin(-π 2 + θ i ) + 1
Na∆r )] and the other over the span of [arcsin(sin

θ i -1 Na∆r ), π 2
]. In the remainder of this paper, ULAs with half a wavelength separated antennas are considered.

Fig. 1 show a 2D color plot of (8) along with plots of f and ā for some arbitrary system parameters and for a reference user with associated channel h ref and position (r ref , θ ref ). It highlights the limitations of the current form of the distance measure that is, from its point of view, 2 UEs far away from each other can appear closer due to its periodic and oscillatory nature. Specifically, for a given reference UE, a UE A located on a local minimum of s will be considered farther than a UE B located on a local maximum even if A is closer than B in reality (i.e. according to the Euclidean distance between the UEs' locations).

To deal with the ambiguities mentioned above, system parameters should be chosen guided by the results of Lemma 1 and Corollary 2.1 with regards to Definition 2. Proposition 1. Necessary identifiability condition for ULA. A necessary condition for UEs in an area A to be weakly identifiable by d , is:

1) the radial size R of A should satisfy R ≤ c( 1 ∆ f -1 B ), 2) for a given θ ∈ [arcsin(- 1
Na ), arcsin( 1 Na )], the angular spread of A should remain in the interval [arcsin(sin(θ) + Na-1 Na ), arcsin(sin(θ) -Na-1 Na )],

Proof. Considering that s is the product of two terms parametrized by orthogonal parameters, the area of identifiable channels is characterized by deriving ranges on these parameters independently. Essentially, this amounts to determining intervals on r i , r j and θ i , θ j where only the central main lobes of f and ā, respectively, remain. Since the period of f is

D f = c ∆ f
, it is straightforward to conclude that for a given r representing the radial center of the area, r i and r j should remain in the range [r -(

D f 2 - L f 4 ), r + ( D f 2 - L f 4 )].
Half the width of the main lobes (i.e., L f

2 ) is subtracted at the extremities to leave them out of the area and avoid resulting ambiguities. This results in an area of radial width of 2 × ( Proposition 1 is illustrated in Fig. 2. Notice how the width of the area is composed of two terms: c ∆ f and c B . When the number of subcarriers N s is high enough, the second one becomes negligible compared to the first, meaning that the radial size of the area is mainly controlled by the subcarrier spacing ∆ f . On the other hand, when the number of antennas is high enough, the angular range of the area tends to [-π 2 , π 2 ], meaning that it covers the whole area in front of the ULA. With this in mind, for UEs in an area of radial width R, the subcarrier spacing should be fixed so that ∆ f c R . Another important remark is that an identifiable area, by definition, should not cross the line of the ULA.

D f 2 - L f 4 ) = c( 1 ∆ f -1 B ).
Proposition 1 is necessary because it is required that any channel set identifiable by d according to Definition 2 is included in a well defined region controlled by system parameters. It is defined as the smallest region where a single main lobe appears wherever the reference user is taken, effectively eliminating long-range ambiguities. However, it is not sufficient because short-range ambiguities still subsist. Fig. 2: The identifiable area's outline. Any region A of any shape included in there (e.g., the red region) verifies the necessary condition for identifiability. The green patch represents the identifiable neighborhood of the channel at its center. The orange segment is its radial axis and the yellow arc is its angular axis. The blue segment represents the ULA at the location of the BS. r is the radial center of the area and θ is its angular center. Proposition 2. Sufficient identifiability condition for ULA. A sufficient condition for UEs in an area A to be weakly identifiable by d is, in addition to verifying the necessary identifiability condition, for their similarity value s to be above or equal to a threshold t s max(|D Ns (3π)|, |D Na (3π)|).

Proof. Given a reference channel h ref in the identifiable area, to cut out the secondary lobes of s (h ref , .), only values above their maximum should be retained for the distance measure. Consequently, corresponding thresholds t f and t ā are derived for the frequency and angular terms respectively. The thresholds are deduced from the amplitude of the second highest extremum (because of the absolute value) of the Dirichlet kernel. The first one is given as

t f |D Ns (3π)| = 1 Ns sin( 3π 2Ns ) .
Similarly, the second threshold is given as t ā |D Na (3π)| = 1 Na sin( 3π 2Na ) . Since s = f × ā and knowing that 0 ≤ f , ā ≤ 1, a sufficient condition for both terms to be above their thresholds is to impose s (h ref , .) ≥ t s max(t f , t ā). The set of channels verifying this condition, called the identifiable neighborhood of h ref , is a subset of the set of all identifiable channels.

For ∀ x in the identifiable region and ∀ y, z in its identifiable neighborhood

d (h( y), h( x)) > d (h( z), h( x)) (11) ⇔ | f (h( y), h( x))| × |ā(h( y), h( x))| > (12) | f (h( z), h( x))| × |ā(h( z), h( x))| (13) ⇒ | f (h( y), h( x))| > | f (h( z), h( x))| (14) or |ā(h( y), h( x))| > |ā(h( z), h( x))| (15)
Since the identifiable neighborhood corresponds to the main lobes of f and ā, this means that they are strictly monotonic in that interval w.r.t. their parameter r and θ respectively. As a consequence,

d (h( y), h( x)) > d (h( z), h( x)) ⇒ |r y -r x | > |r z -r x | or |θ y -θ x | > |θ z -θ x |.
Each identifiable channel has its own identifiable neighborhood. In particular, the spatial region associated to it has a shape that resembles an ellipse with a radial linear axis and an angular curved one, corresponding to the radial and angular lobes respectively, as shown on Fig. 2. Consequently, any channel which location belongs to that region is in the identifiable neighborhood of the reference channel. This results in a neighborhood graph that can be exploited by other methods to construct the low-dimensional channel chart. A consequence of the thresholding operation involved in the construction of this graph is that the main lobes' widths are changed. The new width of the main lobe of the frequency component, noted L f , is the solution to the equation D Ns ( πBL f c ) = t s . It can be computed through the inversion of D Ns in the interval [0, 2π] using classical root finding algorithms. A similar method can be used to derive the new width of the main lobe of the angular component, noted L ā (in rad). It is given this time as the difference between the two roots of the equation

|D Na (2π∆ r N a (sin θ 0 -sin θ))| = t s in the interval [-π 2 , π 2 ]
. Hence, the radial axis of an identifiable neighborhood is of length L f while its angular axis is of length L ā × r ref .

Although the necessary and sufficient conditions guarantee the weak identifiability of d * in the case of a ULA, it can still suffer from the varying geometrical size of the identifiable neighborhoods. Indeed, the width of the main lobe of ā(θ ref , .) is not constant and depends on θ i . In addition, a ULA is limited to the half-space in front of it because of the symmetry at θ = π 2 [π] (the axis of the ULA) causing ambiguities. The next section dives into how using a UCA can help mitigate these limitations.

IV. UCA ARE BETTER THAN ULA FOR CHANNEL

CHARTING

In this section, the use of UCAs as a replacement for ULAs is explored. It is shown that UCAs are better for charting than ULAs because they do not exhibit axial symmetry and their angular resolution is constant (does not depend on the azimuth), which allows to derive a relation between the bandwidth and the radius of the UCA that limits the warping behavior of CC. Lemma 3. With a UCA, a good approximation of ā is of the form

ã(θ i , θ j ) = J 0 4π λ R sin θ i -θ j 2 , ( 16
)
where J 0 is the Bessel integral of order 0. It contains a single main lobe in the interval [-π, π] centered at θ i = θ j of width L ã = 4 arcsin(min( λ 4πR × j 0,1 , 1)) (in rad) where j 0,1 = 2.4048 is the first root of the Bessel integral of order 0 [START_REF] Weisstein | Bessel function zeros[END_REF]. Proof. See Appendix C.

Based on this approximation, a new distance function is defined,

d (h i , h j ) 2 -2s (h i , h j ), (17)
where s (h i , h j ) = f (r i , r j ) × ã(θ i , θ j ). Fig. 3 show a 2D color plot of (8) along with plots of f and ã for some arbitrary system parameters and for a reference user with associated channel h ref and location (r ref , θ ref ). Contrary to s , this time only one angular main lobe appears.

Proposition 1 . Necessary identifiability condition for UCA. For UEs in an area A to be weakly identifiable by

d , its radial size R should satisfy R ≤ c( 1 ∆ f -1 B ).
Proof. The first part of the proof is the same as the one involving f in the proof of Proposition 1.

Regarding ã, the function contains a single main lobe in its domain of definition according to Lem. 3 and is thus not subject to long-range ambiguities that can be caused by the period nature of a a function.

Proposition 1 is illustrated Fig. 4. It highlights the fact that with a UCA, the identifiable area is no longer angularly restricted and is extended to include the whole angular domain (i.e., [-π, π]) with no ambiguities. Proposition 2 . Sufficient identifiability condition for UCA. A sufficient condition for UEs in an area A to be weakly identifiable by d is, in addition to verifying the necessary identifiability condition and for N s > 2, for their similarity value s to be above or equal to a threshold t s 0.403. Fig. 4: The identifiable area's outline. Any region A of any shape included in there (e.g., the red region) verifies the necessary condition for identifiability. The green patch represents the identifiable neighborhood of the channel at its center. The orange segment is its radial axis and the yellow arc is its angular axis. The blue circle represents the UCA at the location of the BS. r is the radial center of the area.

Proof. The proof follows the same logic as that of Proposition 2 with the difference that this time the threshold of the angular term is changed and is derived from the amplitude of the second highest extremum of the Bessel integral. It is straightforwardly given as t ã |J 0 (j 0,1 )| = 0.403 where j 0,1 = 3.8317 is the first root of the derivative of the Bessel integral of order 0 [START_REF] Weisstein | Bessel function zeros[END_REF]. The global threshold is given as t s max(t f , t ã). Finally, knowing that t f is strictly decreasing with N s , and that t f < t ã for N s > 2, it follows that t s = t ã for N s > 2.

This time again, thresholding has an impact on the widths of the main lobes. Since t s has now a constant value, a closedform expression of L f can be derived through inversion of

D Ns ( πBL f c ) = t s.
Although the root depends on N s , its actual value varies negligibly with it and is thus derived for N s → ∞ as an acceptable approximation. As a result, the width of the radial main lobe after thresholding is given by L f = c×4.238 πB . Similarly, the new width of ã (in rad), noted L ã, is given as the solution to the equation J 0 ( 4π λ R sin

L ã 4 ) = t s.
It can be computed through the inversion of J 0 in the interval [0, j 0,1 ]. It's value is given by L ā = 4 arcsin( λ 4πR × 1.692). Proposition 3. Constant angular spread. When using the thresholded channel distance, the UCA (as opposed to the ULA) yields neighborhoods whose angular spread do not depend on the azimuth.

Proof. The proof directly follow from the fact that L ā, which represents the angular spread of the obtained neighborhoods, doesn't depend on θ. This is a very interesting property since it implies that the angular distance perceived by the system is independent of the azimuth, thus avoiding warping in the obtained channel chart, as illustrated in Sec. V-B. However, the size of the resulting identifiable neighborhoods are still not constant everywhere. To ensure that the distance reflects the true geometrical neighborhoods without distortion, the identifiable neighborhoods should be as round as possible. This is measured by the ratio of both their axis, i.e. γ

L f L ã ×rref ≈ c Brref 1.06 π arcsin( λ 4πR .1.692) .
Proposition 4. Round neighborhoods. For a BS equipped with a UCA of radius R, a bandwidth B and an area considered for charting whose center is at a distance r 0 from the BS, the neighborhoods identified by the thresholded PI distance are round at the center of the area if and only if

1 = γ ≈ c Br0 1.06 π arcsin( λ 4πR .1.692) .
Proof. Since an identifiable neighborhood has the shape of an ellipse approximately, it is round when its axes' lengths, L f and L ã × r 0 , are equal. As a consequence, γ = 1.

This result allows to determine the optimal bandwidth B as a function of the distance r 0 and the radius R as B opt = c r0 1.06 π arcsin( λ 4πR .1.692) . It also allows to determine the optimal radius R as a function of the distance r 0 and the bandwidth B as R opt = λ.1.692 4π sin c Br 0 1.06 π . Fig. 5 shows the shape of the identifiable neighborhoods when the reference user is at the inner edge of the identifiable area, at it's center and at it's outer edge, from left to right respectively. Because of distortion, users A and B will be considered at the same distance from the reference user as measured by [START_REF] Wang | Deepfi: Deep learning for indoor fingerprinting using channel state information[END_REF], although it's only true for the center identifiable neighborhood. Furthermore, its area can be approximated with the area of the circle of diameter L f as πL 2 f 4 , as shown on the figure. Since for better performance a minimum number of neighboring users k min should be located inside the area defined by the patch for each reference user, the minimum density of users is given as 4kmin πL 2 f . In reality, this density should be a bit higher to account for the distortion of the neighborhood going towards the inner part of the identifiable area. In practice, the number of neighbors inside of the identifiable neighborhoods will vary for each reference user, but there should be at least one to connect it to the neighborhood graph constructed by Isomap.

V. EXPERIMENTS

In this section, the theoretical claims and system setting guidelines of Sec. III and Sec. IV are empirically validated both on synthetic and realistic channel data.

A. Practical influence of system parameters

A first simple experiment is considered, in which the effect of system parameters on the obtained channel chart is evidenced, showing the practical importance of the theoretical claims of the previous sections. The base scenario consists of a BS equipped with a UCA of 64 antennas and of radius R = 42 cm operating at f c = 3 GHz. It transmits over a bandwidth of B = 10 MHz and N s = 16 subcarriers. A total of N c = 2838 channel observations of UEs located in front of the BS is generated according to [START_REF] Ali | 6g white paper on machine learning in wireless communication networks[END_REF]. These parameters are used to compute and characterize the identifiable area, the identifiable neighborhoods as well as the optimal distance for the radial center r 0 according to the analysis of the previous sections. The identifiable area's width is of 422 m (Proposition 1 ) and its radial center should be placed at a distance of 296 m from the BS to limit neighborhood distortion (Proposition 4). Fig. 6 shows a visualization of the distance between the UE at the center of the area (i.e., r ref = r 0 ) and the rest of the UEs before and after thresholding. The base scenario is tested against 3 variants:

1) not applying the threshold; 2) halved width of the identifiable area by doubling the subcarrier spacing ∆ f and keeping the same bandwidth so that some users end up outside of it; 3) non-round neighborhoods at the center by reducing the bandwidth B. Fig. 7 shows the learned charts for the base scenario as well as for each variant. The word "b<>com" helps visualize how the global structure of the locations is preserved on it. The chart obtained with the first variant looks disorganized. Although long-range ambiguities are eliminated, short-range ones remain as no threshold is applied, causing a degradation in performance in particular in terms of TW and KS. The chart obtained with the second variant is warped as the parts of the considered area that are outside of the identifiable area are confused with other parts inside of it because of long-range ambiguities. As a consequence, TW and KS are negatively impacted. This shows the importance of limiting UEs to the identifiable zone. The chart obtained with the last variant is flattened as a consequence of the non-round identifiable neighborhoods, meaning that distances are perceived differently on the radial and angular directions. While there is little impact on TW and CT, KS is highly impacted because the global structure of the chart is not preserved. Finally, the chart obtained in the base scenario has the best overall performance and, although not perfect in terms of the global structure, represents a pretty faithful representation of the locations of the UEs.

B. ULA vs. UCA

Another experiment is considered, this time to further empirically assess the advantage of a UCA over a ULA, in order to validate the theoretical claims of Sec. IV. A BS placed at the center of a map communicates with UEs uniformly distributed around it. The same system parameters as in the base scenario of the previous section are considered, with the difference that in one case a ULA of 16 antennas is considered, while in the other case a UCA of 64 antennas is considered. The difference in the number of antennas is to obtain lobes of comparable radial sizes in both cases. In addition, two areas that include the UEs' positions are considered: one that is identifiable according to the UCA and one that is identifiable according to

Reduced BW

Fig. 7: The GT UEs' locations and the learned charts corresponding to the base scenario and its three variants. the ULA also. The second one is included in the first one by definition. Note that the communication aspect of the system is ignored. Fig. 8 shows the ground truth UEs' locations as well as the learned charts in each case. In the case of the red area, only points going from yellow to pink appear on the chart associated with the ULA, which represent locations on the right of the map. This is because they overlap with the locations on its left. The channel distance considers that points on the right coincide with points on the right due to its axial symmetry w.r.t. the vertical axis on the figure. Isomap then proceed to learn a chart that places the wrongly matched points at the same locations. On the other hand, the chart produced in the case of the UCA successfully reflects the circular structure of the locations. The TW score in particular being low in the case of the ULA indicates that a lot of false neighborhoods are introduced in the chart which is not the case in the case of the UCA, confirming the observations. In the case of the orange area, both charts appear to be of good quality. However, upon closer inspection, it appears that the chart associated with the UCA is of better quality as indicated by the lower (i.e., better) KS score. Indeed, the points appear to be distributed uniformly whereas in the case the ULA, their density seems to increase the more pink they get. This can be explained by the angular width of the identifiable neighborhoods being dependent on the azimuth in the case of the ULA.

C. Realistic scenario

In this experiment, the presented CC approach is tested on a more realistic scenario using the Sionna ray tracing engine [START_REF] Hoydis | Sionna RT: Differentiable Ray Tracing for Radio Propagation Modeling[END_REF] on the "etoile" scene, which is a 3D reconstruction of the area around the Arc De Triomphe in Paris. The BS is placed at the center of the scene and above the arc. It is equipped with a UCA of 64 elements and a radius of 20 cm. It communicates at a frequency of 3 GHz and over a bandwidth of 7 MHz. A total of N c = 3383 UEs are distributed around it and chosen so that they meet the necessary identifiability condition of Proposition 1 , effectively avoiding long-range ambiguities, and only the LoS path is considered. Note that although the theory developed in this paper considers the BS and UEs to be on the same plane, this scenario however considers the more realistic case where the BS is at a height of 70 m while the UEs are 1.5 m above the ground. Fig. 9 shows the ground truth (GT) locations and the charts generated by Isomap using three different distances. First, the Euclidean distance is used to showcase the capabilities of Isomap with quasiperfect knowledge and to serve as a gold standard. In particular, a neighborhood graph is constructed where neighbors are limited to points present in a radius L f 2 of each node. This amounts to restricting the knowledge to identifiable neighborhoods according to d as defined in Proposition 2 . The resulting chart represents what can be achieved with a strongly identifiable distance (see Definition 1), had one been available. The next chart is generated by constructing the neighborhood graph based on the raw PI channel distance, including its shortrange ambiguities (i.e., not applying the threshold so that the sufficient identifiability condition of Proposition 2 is not met). The locations on the chart look warped and spatial neighborhoods are hardly distinguishable one from another. The final chart is similarly generated by constructing the neighborhood graph based on the channel distance but this time applying the threshold (meeting the sufficient condition of Proposition 2 ) and effectively restricting neighborhoods to identifiable ones, which has the effect of eliminating the shortrange ambiguities. This results in a much better chart that closely resemble the GT. As expected, thresholding allows to greatly improve TW and KS of the obtained chart, which is perfectly in line with the results of Sec. V-A.

VI. CONCLUSION AND PERSPECTIVES

In this paper, a theoretical framework for the identifiability of UEs from their channel measurements is introduced in the perspective of adequately setting the parameters of a MIMO OFDM system for channel charting purpose. To that end, a PI channel distance is analyzed, highlighting its short and longrange ambiguities. A practical method for mitigating these limitations and eliminating ambiguities is then proposed by translating the introduced identifiability rules into guidance in the design of the MIMO OFDM systems. The PI distance has been applied to CC on a MIMO OFDM system and the conducted experiments show a great improvement in the UEs' neighborhood consistency as long as the established designing rules are applied. In addition, the advantage of using UCAs instead of ULAs for CC is demonstrated and validated through simulations. All the achieved results show a very good concordance with the theory and make the proposed system design method very promising.

In future work, the proposed method should be tested on more realistic scenarios (e.g., NLoS channels) and real-world measured data. Finally, it should be considered and extended in the context of ISAC systems with the goal of achieving a trade-off between communication and sensing. With the help of basic trigonometric identities, it can be shown that cos(ω + θ jθ i )cos(ω) = 2 sin θj -θi 2 cos(ω + atan2(sin(θ jθ i ), cos(θ jθ i ) -1)), leading to = 1 2π 2π 0 expj 4π λ R sin θ jθ i 2 cos ω + atan2(sin(θ jθ i ), cos(θ jθ i ) -1) dω .

Subsituting γ for ω +atan2(sin(θ j -θ i ), cos(θ j -θ i )-1)+ π 2 , and with dγ = dω we have This leads to a convenient way of expressing ã as it can be written as

ã(θ i , θ j ) = J 0 4π λ R sin θ j -θ i 2 . ( 22
)
The validity of the integral form as an approximation was empirically verified and holds well when a great number of antennas N a is considered.

Corollary 2 . 1 .Fig. 1 :

 211 Fig. 1: The left figure shows a plot of the similarity s (h ref , h) between the reference user (green cross and lines) and the rest of the points on the map. The BS is placed at its center. The top right figure shows a plot of f (r ref , r) and the bottom right one shows a plot of ā(θ ref , θ). User A (blue cross and lines) is spatially closer to the reference user than user B (red cross/lines), but the distance measure makes the latter appear closer than the former.

  Regarding ā, consider[START_REF] Decurninge | Csi-based outdoor localization for massive mimo: Experiments with a learning approach[END_REF] where the period is D ā = 1 ∆r w.r.t. Θ. For a given θ representing the angular center of the area where sin(θ) = Θ, Θ i and Θ j should remain in the range[Θ-( Dā 2 -Lā 4 ), Θ+( Dā 2 -Lā 4 )] = [Θ-Na-1 Na , Θ+ Na-1 Na ] (with ∆ r =12 ). When converted to the angular domain, the angular range is given by [arcsin(sin(θ) -Na-1 Na ), arcsin(sin(θ) +Na-1Na )]. To respect the definition domain of arcsin, θ should remain in the interval [arcsin(-1 Na ), arcsin( 1 Na )].

Fig. 3 :

 3 Fig. 3: The left figure shows a plot of the similarity s (h ref , h) between the reference user (green cross) and the rest of the points on the map. The BS is placed at its center. The top right figure shows a plot f (r ref , r) and the bottom right one shows a plot of ã(θ ref , θ).

Fig. 5 :

 5 Fig. 5: Visualization of the distance measure for identifiable neighborhoods at (a) the inner, (b) center and (c) outer edge of the identifiable area as pictured in Fig. 4. The circle outline helps visualize how their shapes deviate from being perfectly round. UEs A and B are considered at the same distance from the point of view of the PI distance.

Fig. 6 :

 6 Fig.6: The computed PI distance between a reference UE (red point) and the rest of the UEs before (left) and after (right) thresholding. The greyed points represent the points that are outside the identifiable neighborhood and thus discarded.

  .000, CT: 1.000, KS: 0.163. Base TW: 0.932, CT: 0.993, KS: 0.370. No threshold TW: 0.850, CT: 0.998, KS: 0.252. Increased ∆ f TW: 0.999, CT: 0.999, KS: 0.221.

  GT TW: 0.685, CT: 0.991, KS: 0.537. ULA Red area TW: 0.996, CT: 0.997, KS: 0.301. Orange area TW: 0.971, CT: 0.996, KS: 0.107. UCA TW: 1.000, CT: 1.000, KS: 0.116.

Fig. 8 :

 8 Fig. 8: The GT UEs' locations and the learned charts corresponding to the use of a ULA vs. a UCA for two different areas (red and orange).

Fig. 9 :

 9 Fig.9: The GT UEs' locations for the realistic scenario and the learned charts corresponding to the use of the Euclidean distance (gold standard), the raw PI distance and the thresholded PI distance.

  for the discrete array case. Considering the continuousaperture case by setting N a → ∞ leads toã(θ i , θ j ) γ(cos θ icos θ j ) + sin γ(sin θ isin θ j ) dγ θ i )cos(γθ j ) dγ .Substituting ω for γθ j , and with dω = dγ we have + θ jθ i )cos(ω) dω , with ∆θ = θ jθ i . Since cos is 2π periodic, we have + θ jθ i )cos(ω) dω .

2 e -j 4π λ R sin θ j -θ i 2 sin

 22 (θj -θi),cos(θj -θi)-1)+ π 2 atan2(sin(θj -θi),cos(θj -θi)-1)+ π γ dγ .Finally, exploiting once again the 2π-periodic property of the sine, we have λ R sin θ j -θ i 2 sin γ dγ .

APPENDIX A PROOF OF LEMMA 1

The frequency term can be expressed as

where (a) is obtained since

, and (b) comes from the computation of the geometric series sum.

APPENDIX B PROOF OF LEMMA 2

The angular term can be expressed as

In particular, for a ULA and using the adequate steering vectors ( 4)

APPENDIX C PROOF OF LEMMA 3

Considering a UCA and using the adequate steering vectors (4), ā(θ i , θ j ) becomes