R2M

    raw 2nd moment
CSI

    channel state information
UE

    user equipment
EM

    electromagnetic
UL

    uplink
BS

    base station
TDD

    time division duplex
FDD

    frequency division duplex
ECC

    error-correcting code
MLD

    maximum likelihood decoding
HDD

    hard decision decoding
IF

    intermediate frequency
RF

    radio frequency
SDD

    soft decision decoding
NND

    neural network decoding
CNN

    convolutional neural network
ML

    maximum likelihood
GPU

    graphical processing unit
BP

    belief propagation
LTE

    Long Term Evolution
BER

    bit error rate
SNR

    signal-to-noise-ratio
ReLU

    rectified linear unit
BPSK

    binary phase shift keying
QPSK

    quadrature phase shift keying
AWGN

    additive white Gaussian noise
MSE

    mean squared error
LLR

    log-likelihood ratio
MAP

    maximum a posteriori
NVE

    normalized validation error
BCE

    binary cross-entropy
CE

    cross-entropy
BLER

    block error rate
SQR

    signal-to-quantisation-noise-ratio
MIMO

    multiple-input multiple-output
mMIMO

    massive multiple-input multiple-output
OFDM

    orthogonal frequency division multiplex
RF

    radio frequency
LOS

    line of sight
NLoS

    non-line of sight
NMSE

    normalized mean squared error
CFO

    carrier frequency offset
SFO

    sampling frequency offset
IPS

    indoor positioning system
TRIPS

    time-reversal IPS
RSSI

    received signal strength indicator
MIMO

    multiple-input multiple-output
ENoB

    effective number of bits
AGC

    automatic gain control
ADC

    analog to digital converter
ADCs

    analog to digital converters
FB

    front bandpass
FPGA

    field programmable gate array
JSDM

    Joint Spatial Division and Multiplexing
NN

    neural network
IF

    intermediate frequency
LoS

    line-of-sight
NLoS

    non-line-of-sight
DSP

    digital signal processing
AFE

    analog front end
SQNR

    signal-to-quantisation-noise-ratio
SINR

    signal-to-interference-noise-ratio
ENoB

    effective number of bits
PCB

    printed circuit board
EVM

    error vector mangnitude
CDF

    cumulative distribution function
MRC

    maximum ratio combining
MRP

    maximum ratio precoding
MRT

    maximum ratio transmission
DeepL

    deep-learning
DL

    downlink
SISO

    single-input single-output
SGD

    stochastic gradient descent
CP

    cyclic prefix
MISO

    Multiple Input Single Output
LMMSE

    linear minimum mean square error
ZF

    zero forcing
USRP

    universal software radio peripheral
RNN

    recurrent neural network
GRU

    gated recurrent unit
LSTM

    long short-term memory
NTM

    neural turing machine
DNC

    differentiable neural computer
TCN

    temporal convolutional network
FCL

    fully connected layer
MANN

    memory augmented neural network
RNN

    recurrent neural network
DNN

    deep neural network
FIR

    finite impulse response
BPTT

    back-propagation through time
GAN

    generative adversarial network
ELU

    exponential linear unit
tanh

    hyperbolic tangent
BICM

    bit-interleaved coded modulation
OTA

    over-the-air
IM

    intensity modulation
DD

    direct detection
RL

    reinforcement learning
SDR

    software-defined radio
WGAN

    Wasserstein generative adversarial network
BMD

    bit-metric decoding
BMI

    bit-wise mutual information
LDPC

    low-density parity-check
IDD

    iterative demapping and decoding
JSD

    Jensen-Shannon divergence
MMSE

    minimum mean square error
FFT

    fast Fourier transform
DFT

    discrete Fourier transform
IFFT

    inverse fast Fourier transform
QAM

    quadrature amplitude modulation
EMD

    earth mover’s distance
TDL

    tapped delay line
KL

    Kullback-Leibler
PRACH

    physical random access channel
URLLC

    ultra-reliable low-latency communication
ANOMA

    asynchronous non-orthogonal multiple access
FEC

    forward error correction
PAPR

    peak-to-average power ratio
APP

    a posteriori probability
COTS

    commercial off-the-shelf
PLL

    phase locked loop
STO

    sampling time offset
SFO

    sampling frequency offset
CFO

    carrier frequency offset
CPO

    carrier phase offset
CSI

    channel state information
GNSS

    global navigation satellite system
ELAA

    extremely large aperture array
UE

    user equipment
DICHASUS

     Distributed Channel Sounder by University of Stuttgart
JCaS

    Joint Communication and Sensing
CT

    continuity
TW

    trustworthiness
KS

    Kruskal’s stress
PCA

    principal component analysis
AoA

    angle of arrival
RD

    Rajski’s distance
ADP

    angle-delay profile
MPC

    multipath component
CIR

    channel impulse response
MAE

    mean absolute error
MDS

    multidimensional scaling
t-SNE

    t-distributed stochastic neighbor embedding
SM

    Sammon’s mapping
CIRA

    channel impulse response amplitude
CS

    cosine similarity
FCF

    forward charting function
CMD

    correlation matrix distance

# Angle-Delay Profile-Based and Timestamp-Aided Dissimilarity Metrics for
Channel Charting

Phillip Stephan*  —, , Florian Euchner*  —, , Stephan ten Brink*  — This work
is supported by the German Federal Ministry of Education and Research (BMBF)
within the projects Open6GHub (grant no. 16KISK019) and KOMSENS-6G (grant no.
16KISK113).

###### Abstract

Channel charting is a self-supervised learning technique whose objective is to
reconstruct a map of the radio environment, called channel chart, by taking
advantage of similarity relationships in high-dimensional channel state
information. We provide an overview of processing steps and evaluation methods
for channel charting and propose a novel dissimilarity metric that takes into
account angular-domain information as well as a novel deep learning-based
metric. Furthermore, we suggest a method to fuse dissimilarity metrics such
that both the time at which channels were measured as well as similarities in
channel state information can be taken into consideration while learning a
channel chart. By applying both classical and deep learning-based manifold
learning to a dataset containing sub-6 GHz distributed massive MIMO channel
measurements, we show that our metrics outperform previously proposed
dissimilarity measures. The results indicate that the new metrics improve
channel charting performance, even under non-line-of-sight conditions.

###### Index Terms:

Channel charting, machine learning, localization, massive MIMO, manifold
learning.

##  I Introduction

Spatial multiplexing techniques like massive multiple-input multiple-output
(mMIMO) are indispensable for future wide-area wireless telecommunication
standards to fulfill the ever-increasing throughput demands with the limited
available sub-$6\,\mathrm{GHz}$ electromagnetic (EM) spectrum. The large
number of base station (BS) antennas required for mMIMO also enable
observation of the channel with high spatial resolution, facilitating
additional applications including Joint Communication and Sensing (JCaS) and
localization services. These applications make use of channel state
information (CSI), which has to be collected at the BS for communication
purposes, regardless. Classical source localization techniques such as, e.g.,
triangulation or multilateration, assume a channel model with line-of-sight
(LoS) conditions. They tend to be inaccurate if strong multipath components
disturb the direct LoS component and fail entirely under non-line-of-sight
(NLoS) conditions. CSI fingerprinting-based localization methods, on the other
hand, do not make any model assumptions and have been applied successfully,
even in more complex environments [1] [2] [3]. However, these machine learning
methods require accurate ground truth position labels, which are generally
unavailable in practical systems. Channel charting, on the other hand, learns
a usually two- or three-dimensional map of the radio environment, the so-
called channel chart, in a self-supervised manner [4], i.e., without relying
on position labels. While most of the early work on channel charting
concentrated on reconstructing local geometrical features of the radio
environment [4], later work increasingly placed focus on also learning its
global topology [5] [6]. The environment reconstruction is expressed as a
manifold learning problem, relying on the assumption that in a sufficiently
static environment, measured CSI data is mainly a function of the user
equipment (UE) position and therefore lies on a low-dimensional manifold in a
high-dimensional CSI feature space. Some applications of channel charting,
like user localization, require global spatial consistency. For others, such
as radio resource management tasks like handover prediction [7] and beam
tracking [8], the accurate representation of local spatial relationships is
sufficient. In this work, we aim at an accurate reconstruction of both local
geometry and global topology of the environment, with a focus on comparing
different novel and established dissimilarity metrics.

###  I-A Related Work

In their seminal work on channel charting [4], C. Studer et al. suggest
multiple manifold learning methods for obtaining a low-dimensional
representation of high-dimensional CSI: principal component analysis (PCA)
[9], Sammon’s mapping (SM) [10] and a deep learning approach using an
autoencoder [11]. In contrast to the other two methods, SM does not directly
process features derived from CSI, but relies on a matrix of pairwise
dissmilarities between any two datapoints, forming the basis of what we call
_dissmilarity metric-based_ channel charting, which is the focus of this work.
Their paper also coins the term _ forward charting function (FCF)_, denoting
the mapping from CSI feature space to low-dimensional channel chart, and
establishes a distinction between two types of FCFs, namely _parametric_ and
_nonparametric_ ones. Parametric FCFs (like aforementioned autoencoder), once
learned, can map previously unseen datapoints onto an existing channel chart
given the additional sample’s CSI features. Nonparametric FCFs (like SM), on
the other hand, cannot trivially predict channel chart positions for points
outside their training set. This makes makes parametric (i.e., usually deep
learning-based) methods the preferred way to perform channel charting for many
practical applications. Parametric dissimilarity metric-based channel charting
was only enabled later by the proposals to employ a Siamese neural network
structure [5] to parameterize SM or to use a triplet neural network based on
triplet loss [12].

Besides the choice of manifold learning technique, the definition of the
dissimilarity metric is crucial for the success of dissimilarity metric-based
channel charting. The first paper on channel charting [4] proposes a
dissimilarity metric based on the Frobenius distance between special matrices
derived from CSI. Later papers propose different metrics, some of them
geometrically motivated and mathematically derived, others devised in a more
ad-hoc manner based on empirical results. Notable CSI dissimilarity metric
proposals include a metric derived from superresolution angular-domain
information [13], a correlation matrix distance (CMD)-based metric [14], a
cosine similarity (CS)-based metric [15], a metric making use of time domain
CSI that we call channel impulse response amplitude (CIRA) [16] and, more
recently, an angle-delay profile (ADP)-based metric [17]. In addition, [16]
highlights that it is often sufficient to be able to compute dissimilarities
that are indicative of local distances and that global distances, which are
then called _geodesic distances_ , can be computed as a combination of many
local distances. In [12] and in our previous work [18], CSI-based
dissimilarities were avoided entirely by using side information, namely the
absolute difference in datapoint timestamps, as a metric.

While most research in channel charting is carried out on simulated channels
[4][5][13][14][15][17], some authors compute channel charts for real-world
measurement datasets [12][16][18]. Some previous research [13][19][20][21]
specifically considers the case of _multipoint_ channel charting, where CSI is
available from multiple distributed BSs.

###  I-B Contributions and Outline

After listing some limitations of our work and defining some terminology and
notations, we first review our system model and present a short overview of
the principles of dissimilarity metric-based channel charting in Section II.
In Section III, we then introduce several commonly used dissimilarity metrics
and explain our own novel metrics. Our proposals include exploiting the ADP,
learning the dissimilarity metric with a neural network and fusing information
from multiple existing metrics. Building on well-known classical and deep
learning-based manifold learning methods summarized in Sections IV and V,
respectively, and using the customary channel charting evaluation metrics
given in Section VI, we compare all proposed dissimilarity metrics on a real-
world dataset in Section VII. Additionally, we demonstrate that, with our
novel metrics, we can get good performance even if a large portion of the
dataset experiences NLoS propagation conditions. Finally, in Section VIII, we
draw conclusions from the observed results and suggest some open questions for
future work.

###  I-C Limitations

To be able to provide an extensive performance comparison of many different
dissimilarity metrics and manifold learning techniques, we had to constrain
our evaluations to just one measurement dataset. Out of the dissimilarity
metrics from literature, we had to limit ourselves to those that perform best
on our dataset. Like most current channel charting work, we assume that the
environment is mostly static over the duration of the data capture.

###  I-D Terminology

In the context of channel charting, _distributed mMIMO_ can be seen as an
extension of _multipoint_ channel charting. Whereas the latter only assumes
the availability of CSI at multiple BSs, the former also ensures some
additional synchronization between them. In our case, we use datasets with
phase synchronization across all antennas and will therefore refer to
“distributed mMIMO”, even though many of our results also apply to multipoint
channel charting.

The notion of computing _dissimilarity metrics_ between datapoints is common
in channel charting literature. Various terms like _distance_ , _similarity_
or _dissimilarity metric_ have been proposed to refer to this measure, but it
is yet unclear which will prevail. We chose to use the expression
_dissimilarity metric_ , since it conveys the distinction between physical
distances to dissimilarities in CSI space. Note that _metric_ is used in a
colloquial sense here, as not all _dissimilarity metrics_ fulfill the
requirements for a metric in the sense of a metric space.

###  I-E Notations

We use boldface uppercase letters for matrices and tensors, and comma-
separated subscripts for indexing. For example $\mathbf{H}_{i,j}$ refers to
the entry in the $i$-th row and $j$-th column of the matrix $\mathbf{H}$.
Boldface lowercase letters are reserved for vectors.

##  II Overview: Principles of Channel Charting

CollectCSI
DatasetComputeDissimilarityMatrixPerformManifoldLearningChannelChart Figure 1:
The two central steps in dissimilarity metric-based channel charting

In a sufficiently static environment and neglecting hardware impairments and
noise, CSI, which is estimated at the BS for communication purposes, is
primarily determined by the location and orientation of the UE. This is the
premise of channel charting, which assumes that CSI in a suitable feature
space lies on a single low-dimensional manifold. The steps involved in
dissimilarity metric-based channel charting are illustrated in Fig. 1: First,
having collected a large CSI dataset at the BS, a _dissimilarity matrix_
containing the dissimilarities between any two datapoints in the dataset is
computed using a particular dissimilarity metric. Then, manifold learning is
applied to obtain the FCF. The success of the second step is highly dependent
on the first step, namely the choice of dissimilarity metric. Ideally, the
computed dissimilarities reflect the physical distances between datapoints to
a large extent. The low-dimensional representation should minimize the
discrepancy between the point-to-point distances in the channel chart and the
respective distances in the dissimilarity matrix. A suitable representation
can be found with either classical or deep learning-based manifold learning
methods. The latter implement parametric FCFs which can predict channel chart
positions based on CSI input features. These input features are computed from
the raw estimated CSI in a _feature engineering_ step.

###  II-A System and Data Model

In the following, we describe an abstract system model that we will employ to
define dissimilarity metrics and manifold learning techniques. Refer to
Section VII-A for information on the particular dataset and system used for
our analyses.

We consider a mMIMO BS consisting of $B$ distributed antenna arrays equipped
with $M$ antennas each, and a UE with a single antenna. A dataset consisting
of $L$ datapoints measured at discrete time instances $l=1,\ldots,L$ is
collected, where one datapoint consists of the following parameters:

  * •

Channel coefficients between the UE and all $B\times M$ BS antennas for
$N_{\mathrm{sub}}$ orthogonal frequency division multiplex (OFDM) subcarriers,
collected in the frequency-domain CSI tensor
$\mathbf{H}^{(l)}\in\mathbb{C}^{B\times M\times N_{\mathrm{sub}}}$. We assume
that $N_{\mathrm{sub}}$ is even and that the subcarrier channel coefficients
in $\mathbf{H}$ are ordered such that the coefficients
$\mathbf{H}^{(l)}_{b,m,1}$ correspond to the subcarrier at the center of the
bandwidth (customary for discrete Fourier transform (DFT) frequency bins).

  * •

Corresponding UE positions $\mathbf{x}^{(l)}\in\mathbb{R}^{D}$, measured by
some reference localization system, with $D$ being the spatial dimension (we
assume $D=2$). This information may be used for evaluation purposes.

  * •

A timestamp $t^{(l)}\in\mathbb{R}$, measured in seconds, indicating the exact
(millisecond-precision) time the CSI and position measurements were taken.

Hence, the complete dataset may be written as a set of 3-tuples
$\left(\mathbf{H}^{(l)},\mathbf{x}^{(l)},t^{(l)}\right)$ of cardinality $L$:

$\text{Dataset:}~{}\mathcal{S}=\left\\{(\mathbf{H}^{(l)},\mathbf{x}^{(l)},t^{(l)})\right\\}_{l=1,\ldots,L}$

For some processing steps, it may be beneficial to use a time-domain CSI
representation $\tilde{\mathbf{H}}^{(l)}\in\mathbb{C}^{B\times M\times
N_{\mathrm{sub}}}$. This representation is obtained by applying the unitary
inverse discrete Fourier transform to $\mathbf{H}^{(l)}$ along the frequency
axis:

$\tilde{\mathbf{H}}_{b,m,\tau}^{(l)}=\frac{1}{\sqrt{N_{\mathrm{sub}}}}\sum_{n=1}^{N_{\mathrm{sub}}}\mathrm{e}^{2\pi\mathrm{j}\cdot\nicefrac{{1}}{{N_{\mathrm{sub}}}}\cdot(n-1)\cdot\left(\tau-\nicefrac{{N_{\mathrm{sub}}}}{{2}}-1\right)}~{}\mathbf{H}_{b,m,n}^{(l)}$
(1)

Note that $\tau$ in Eq. (1) is shifted such that $\tau=1$ corresponds to the
earliest and $\tau=N_{\mathrm{sub}}$ corresponds to the last time tap.

###  II-B Manifold Learning and Dissimilarity Metrics

Manifold learning finds a mapping from the CSI tensors
$\mathbf{H}^{(l)}\in\mathbb{C}^{B\times M\times N_{\mathrm{sub}}}$ to channel
chart positions $\mathbf{z}^{(l)}\in\mathbb{R}^{D^{\prime}}$, expressed by the
FCF

$\mathcal{C}:\mathbb{C}^{B\times M\times
N_{\mathrm{sub}}}\rightarrow\mathbb{R}^{D^{\prime}},$

with $D^{\prime}$ being the dimensionality of the channel chart. Subsequently,
we assume $D^{\prime}=D=2$ and only consider manifold learning techniques
which are based on dissimilarity metrics. A good FCF ensures that the pairwise
Euclidean distance between any two mapped positions in the channel chart
$\mathbf{z}^{(i)},\mathbf{z}^{(j)}\in\mathbb{R}^{D^{\prime}}$ is approximately
proportional to the distances between the corresponding true UE positions
$\mathbf{x}^{(i)},\mathbf{x}^{(j)}$:

$\lVert\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\rVert_{2}\propto\lVert\mathbf{x}^{(i)}-\mathbf{x}^{(j)}\rVert_{2},\quad\forall
i,j\in\\{1,\ldots,L\\}.$

In case of dissimilarity metric-based channel charting, dissimilarities
computed based on a dissimilarity metric $d$ are considered while learning the
FCF. Ideally, these dissimilarity metric should also fulfill

$d_{i,j}\propto\lVert\mathbf{x}^{(i)}-\mathbf{x}^{(j)}\rVert_{2},\quad\forall
i,j\in\\{1,\ldots,L\\},$ (2)

with $d_{i,j}$ being the computed dissimilarity between the datapoints $i$ and
$j$. Dissimilarities are calculated based on information that is available to
the BS, such as the estimated CSI [4] [16] [15] or timestamp information [12].
A more detailed overview of different dissimilarity metrics for channel
charting is given in Section III. The pairwise dissimilarities between all
samples in the dataset are collected in the dissimilarity matrix

$\mathbf{D}_{\mathrm{pw}}=\begin{pmatrix}d_{1,1}&\cdots&d_{1,L}\\\
\vdots&\ddots&\vdots\\\ d_{L,1}&\cdots&d_{L,L}\end{pmatrix},$

which is generally symmetric ($d_{i,j}=d_{j,i}$) and fulfills $d_{i,j}\geq
0\;\;\forall i,j$ and $d_{i,i}=0\;\;\forall i$.

###  II-C Feature Engineering

For deep learning-based techniques, we propose to use CSI features based on
sample autocorrelations of the time-domain CSI matrices
$\tilde{\mathbf{H}}^{(l)}$. These sample autocorrelations are computed for
every pair $b_{1}$, $b_{2}$ of antenna array indices, for any pair $m_{1}$,
$m_{2}$ of antenna indices and for every tap index $\tau$. This way, the
feature vector also contains sample correlations of antennas that are part of
different antenna arrays. To this end, we define a set of index tuples
$\mathcal{J}$ as the cartesian product

$\mathcal{J}=\left\\{1,\ldots,B\right\\}^{2}\times\left\\{1,\ldots,M\right\\}^{2}\times\left\\{\tau_{\mathrm{min}},\ldots,\tau_{\mathrm{max}}\right\\}.$

Note that only time tap indices $\tau$ from $\tau_{\mathrm{min}}$ to
$\tau_{\mathrm{max}}$ are taken into account, where
$1\ll\tau_{\mathrm{min}}<\nicefrac{{N_{\mathrm{sub}}}}{{2}}<\tau_{\mathrm{max}}\ll
N_{\mathrm{sub}}$, i.e., only a few channel impulse response (CIR) taps
containing the LoS path (if it exists) as well as the first few multipath
components are even considered. The choice of $\tau_{\mathrm{min}}$ and
$\tau_{\mathrm{max}}$ depends on the expected maximum delay spread. Since OFDM
is usually operated such that the number of subcarriers $N_{\mathrm{sub}}$ is
high enough to ensure sufficiently frequency-flat subcarriers despite the
channel’s frequency selectivity, the considered interval of CIR taps
$\tau_{\mathrm{min}},\ldots,\tau_{\mathrm{max}}$ is small compared to the
overall number of taps $N_{\mathrm{sub}}$.

For each combination of indices, we compute a sample autocorrelation and
collect the autocorrelations in the vector
$\mathbf{c}^{(l)}\in\mathbb{C}^{B^{2}M^{2}(\tau_{\mathrm{max}}-\tau_{\mathrm{min}}+1)}$:

$\mathbf{c}^{(l)}=\left(\tilde{\mathbf{H}}^{(l)}_{b_{1}m_{1}\tau}\left(\tilde{\mathbf{H}}^{(l)}_{b_{2}m_{2}\tau}\right)^{*}\right)_{(b_{1},b_{2},m_{1},m_{2},\tau)\in\mathcal{J}}$

Since we use a neural network implementation that can only handle real-valued
numbers, we provide $\mathbf{c}^{(l)}$ to the network in a vectorized form
with real and imaginary part separated, yielding the feature vectors

$\mathbf{f}^{(l)}=\begin{pmatrix}\mathrm{Re}\left\\{\mathbf{c}^{(l)}\right\\}&\mathrm{Im}\left\\{\mathbf{c}^{(l)}\right\\}\end{pmatrix}\in\mathbb{C}^{2B^{2}M^{2}(\tau_{\mathrm{max}}-\tau_{\mathrm{min}}+1)}.$

Intuitively, this means that the neural network is provided information about
relative phases and powers between any two antennas in the system. Obviously,
our choice of feature engineering produces very large input vectors
$\mathbf{f}^{(l)}$. The feature engineering stage may be further improved,
e.g., by compressing useful information in a more sparse set of features.

Dissimilarity MetricsSide information-basedTimestampEuc: Eucl.DistanceG-
EucCSI-basedCS: CosineSimilarityG-CSCIRA: CIRAmplitudeG-CIRAADPG-ADPDL:
Dissim.LearningG-DLGeodesic Versions:FusedG-Fused

Figure 2: Overview of dissimilarity metrics compared in this paper. Metrics
marked with a star ( ) are newly introduced. Metrics marked with a red circle
( ) use ground truth positioning data and are only meant as baselines, not for
practical implementation of a channel charting system. Note that we are
considering a metric that fuses information from both ADP and timestamp
metrics here, but, in general, any set of metrics may be fused into one.

##  III Dissimilarity Metrics

A variety of dissimilarity metrics that compute pseudo-distances between
measured datapoints have been proposed for channel charting. The choice of
metric depends on properties of the available CSI data (e.g., number and
distribution of antennas, bandwidth, phase-coherence) and the availability of
side information. In general, dissimilarity metrics can be grouped into side
information-based and CSI-based metrics, as illustrated in Fig. 2. Side
information-based methods assume that some additional information is available
at the BS in addition to wireless channel estimates. Other dissimilarity
metrics are computed solely based on measured CSI. We compare the suitability
of different CSI-based dissimilarity metrics from literature, namely a CS-
based metric [15] and a CIRA-based metric [16], to our newly developed metric,
which we refer to as ADP dissimilarity. In addition, we describe an approach
where a deep neural network (DNN) learns a suitable dissimilarity metric
between measured CSI, which we call dissimilarity metric learning, and we
propose a method to combine information from multiple metrics.

###  III-A Euclidean Distance (Reference System)

Under the assumption that some reference system (e.g., a global navigation
satellite system (GNSS)) provides “ground truth” position labels
$\\{\mathbf{x}^{(l)}\\}_{l=1,\ldots,L}$, the Euclidean distance

$d_{\mathrm{Euc},i,j}=\lVert\mathbf{x}^{(i)}-\mathbf{x}^{(j)}\rVert_{2}$

between these labels can be used directly as the dissimilarity metric.
Obviously, this metric is unsuitable for most practical uses of channel
charting, where only channel measurements at the BS are available, but is
nevertheless included here as a baseline for benchmarking subsequent
processing steps.

###  III-B Absolute Time Difference

As first suggested in [12], timestamps may also reveal some information about
spatial relationships between datapoints: Since the velocity of a UE is
bounded, two channel measurements between BS and the same UE which are
measured in quick succession are likely to be also spatially close to each
other. It is reasonable to assume that timestamps are available alongside CSI
measurements in practical systems.

The timestamp-based dissimilarity metric is defined as the absolute time
difference between two measurements:

$d_{\mathrm{time},i,j}=\lvert t^{(i)}-t^{(j)}\rvert$ (3)

Obviously, this dissimilarity metric fails to predict meaningful distances for
large absolute time differences, but its usefulness has been demonstrated
before [12] [18] and it can be especially powerful in combination with CSI-
based metrics.

###  III-C Channel Impulse Response Amplitude (CIRA)

A dissimilarity metric based on the CIRA was introduced in [16]. An important
advantage of this metric is that phase synchronization between receivers is
not required, since it only operates on CIR amplitudes, not phases. However,
the metric assumes a setup with distributed receivers and is less suitable for
setups with one single or few mMIMO arrays. We can apply the CIRA metric to
our time-domain CIR tensors $\tilde{\mathbf{H}}^{(l)}$ as follows:

$d_{\mathrm{CIRA},i,j}=\sum\limits_{b=1}^{B}\sum\limits_{m=1}^{M}\sum\limits_{\tau=\tau_{\mathrm{min}}}^{\tau_{\mathrm{max}}}\left|\left|\tilde{\mathbf{H}}_{b,m,\tau}^{(i)}\right|-\left|\tilde{\mathbf{H}}_{b,m,\tau}^{(j)}\right|\right|$
(4)

As previously explained in Section II-C, $\tau_{\mathrm{min}}$ and
$\tau_{\mathrm{max}}$ should be chosen appropriately.

###  III-D Cosine Similarity (CS)

The use of a CS-based metric for channel charting was first proposed in [15],
which assumes a narrowband system with just one mMIMO antenna array at the BS.
In [15], the dissimilarity between two channel coefficient vectors
$\mathbf{h}^{(i)}\in\mathbb{C}^{M}$ and $\mathbf{h}^{(j)}\in\mathbb{C}^{M}$,
where $M$ is the number of antennas in the mMIMO array, is defined as

$d_{\mathrm{Magoarou},i,j}=\sqrt{2-2\frac{\left|\left(\mathbf{h}^{(i)}\right)^{\mathrm{H}}\mathbf{h}^{(j)}\right|}{\left\lVert\mathbf{h}^{(i)}\right\rVert_{2}\left\lVert\mathbf{h}^{(j)}\right\rVert_{2}}}.$

Empirically we determined that a slightly modified version of the metric
considering the squared CS, which can be interpreted as a normalized power
value, performs better with respect to all evaluation metrics:

$d_{\mathrm{CS-
vec},i,j}=1-\frac{\left|\left(\mathbf{h}^{(i)}\right)^{\mathrm{H}}\mathbf{h}^{(j)}\right|^{2}}{\left\lVert\mathbf{h}^{(i)}\right\rVert_{2}^{2}\left\lVert\mathbf{h}^{(j)}\right\rVert_{2}^{2}}$

Note that $0\leq d_{\mathrm{CS-vec},i,j}\leq 1$, with $d_{\mathrm{CS-
vec},i,j}=0$ if and only if $\mathbf{h}^{(i)}$ and $\mathbf{h}^{(j)}$ are
linearly dependent. We generalize this metric to wideband OFDM channels with
$N_{\mathrm{sub}}$ subcarriers and distributed antenna setups with $B$ antenna
arrays. We achieve this by computing the summation over the dissimilarities
calculated separately for each antenna array and each subcarrier:

$d_{\mathrm{CS},i,j}=\sum_{b=1}^{B}\sum_{n=1}^{N_{\mathrm{sub}}}\left(1-\frac{\left\lvert\sum_{m=1}^{M}\left({\mathbf{H}}_{b,m,n}^{(i)}\right)^{*}{\mathbf{H}}_{b,m,n}^{(j)}\right\rvert^{2}}{\left(\sum_{m=1}^{M}\left\lvert{\mathbf{H}}_{b,m,n}^{(i)}\right\rvert^{2}\right)\left(\sum_{m=1}^{M}\left\lvert{\mathbf{H}}_{b,m,n}^{(j)}\right\rvert^{2}\right)}\right)$
(5)

###  III-E Angle-Delay Profile (ADP)

Taking inspiration from the CIRA metric in Eq. (4), we propose an enhanced
version of a CS-based metric by exploiting the sparsity of time-domain CSI.
Most signal power in time domain is concentrated in the LoS path (if it
exists) and a few strong multipath components. As opposed to Eq. (5), which
computes cosine similarities for frequency-domain CSI data, we suggest to
perform the summation over time-domain CSI matrices $\tilde{\mathbf{H}}$ for
taps $\tau=\tau_{\mathrm{min}},\ldots,\tau_{\mathrm{max}}$, where
$\tau_{\mathrm{min}}$ and $\tau_{\mathrm{max}}$ must be adapted to the radio
environment such that all relevant MPCs are accounted for:

$\begin{split}d_{\mathrm{ADP},i,j}=&\sum_{b=1}^{B}\sum_{\tau=\tau_{\mathrm{min}}}^{\tau_{\mathrm{max}}}\left(1-\frac{\left\lvert\sum_{m=1}^{M}\left(\tilde{\mathbf{H}}_{b,m,\tau}^{(i)}\right)^{*}\tilde{\mathbf{H}}_{b,m,\tau}^{(j)}\right\rvert^{2}}{\left(\sum_{m=1}^{M}\left\lvert\tilde{\mathbf{H}}_{b,m,\tau}^{(i)}\right\rvert^{2}\right)\left(\sum_{m=1}^{M}\left\lvert\tilde{\mathbf{H}}_{b,m,\tau}^{(j)}\right\rvert^{2}\right)}\right)\end{split}$
(6)

Eq. (6) can be interpreted as using the angular-domain similarity in terms of
power measured across multiple time-domain taps and MPCs, hence we suggest the
name ADP for our novel metric. The restriction to
$\tau_{\mathrm{min}}\leq\tau\leq\tau_{\mathrm{max}}$ excludes taps with weak
signal and strong noise contributions (refer to Section II-C). In contrast to
previously proposed angular domain metrics [13] [22], we deliberately forgo
the use of superresolution techniques or the computation of earth mover’s
distances, which makes computation practical for large datasets.

###  III-F Dissimilarity Metric Learning (DL)

Feature Engineering$\tilde{\mathbf{H}}^{(i)}$$\tilde{\mathbf{H}}^{(j)}$DNN
$\mathcal{D}_{\Theta}$Loss Function
$\mathcal{L}_{\mathrm{DL}}$$\mathbf{f}^{(i)}$$\mathbf{f}^{(j)}$$d_{\mathrm{time},i,j}=\left|t^{(i)}-t^{(j)}\right|$$t^{(i)}$$t^{(j)}$
Figure 3: Dissimilarity Metric Learning: The DNN $\mathcal{D}_{\Theta}$ is
trained to estimate the absolute time difference between samples $i$, $j$
based on their CSI.

The previous dissimilarity metrics were explicitly defined and motivated based
on assumptions grounded in the movement of UEs or in EM wave propagation.
However, modeling wave propagation is complicated, which makes it difficult to
formulate dissimilarity metrics that adapt to all environments. In an entirely
different approach, we suggest to not only use neural networks for manifold
learning, but to use them to learn the dissimilarity metric itself. The
architecture of the dissimilarity metric learning process is shown Fig. 3.

The challenge for this approach is the definition of a suitable loss function
that does not rely on the “ground truth” UE positions $\mathbf{x}^{(l)}$,
which must not be used. Instead, we suggest to train a DNN
$\mathcal{D}_{\Theta}$ to estimate the absolute time difference
$d_{\mathrm{time},i,j}$ (as defined in Eq. (3)) between pairs of CSI features
$\mathbf{f}^{(i)},\mathbf{f}^{(j)}$ (as defined in Section II-C). The hope is
that the predictions of $\mathcal{D}_{\Theta}$ may also be a useful indicator
of physical distances (compare Section III-B). As loss function, we employ a
modified form of normalized mean squared error (NMSE):

$\mathcal{L}_{\mathrm{DL}}=\sum\limits_{i,j}\left(\frac{\mathcal{D}_{\Theta}\left(\mathbf{f}^{(i)},\mathbf{f}^{(j)}\right)-d_{\mathrm{time},i,j}}{d_{\mathrm{time},i,j}+\beta}\right)^{2}$
(7)

In contrast to mean squared error (MSE), NMSE places a higher importance on
predicting small time differences correctly by normalizing the error to
$d_{\mathrm{time},i,j}$. The additional hyperparameter $\beta>0$ in Eq. (7)
has the following impact on the behavior of $\mathcal{L}_{\mathrm{DL}}$: For
$d_{\mathrm{time,i,j}}\ll\beta$, Eq. (7) behaves more like the regular MSE
loss function, i.e., normalization is applied less aggressively to very small
time differences. For $d_{\mathrm{time,i,j}}\gg\beta$, Eq. (7) approaches an
NMSE loss function, implying that prediction errors for large true time
differences $d_{\mathrm{time},i,j}$ are weighted low. Intuitively, this makes
sense since we cannot expect $\mathcal{D}_{\Theta}$ to generate accurate
distance predictions for large time differences due to unknown UE
trajectories.

For training the DNN, we only select pairs of datapoints $i,j$ for which
$d_{\mathrm{time,i,j}}\leq\alpha$, i.e., for which the absolute time
difference is smaller than some threshold $\alpha$. The DNN
$\mathcal{D}_{\Theta}$ consists of four hidden layers with 1024, 512, 256 and
128 neurons each, all of them with ReLU activation. In our experiments, we
choose $\beta=1\,\mathrm{s}$ and $\alpha=500\,\mathrm{s}$. Since subsequent
processing steps may assume perfectly symmetric dissimilarity matrices, we
symmetrize the metric during inference to obtain

$d_{\mathrm{DL},i,j}=\frac{1}{2}\left(\mathcal{D}_{\Theta}(\mathbf{f}^{(i)},\mathbf{f}^{(j)})+\mathcal{D}_{\Theta}(\mathbf{f}^{(j)},\mathbf{f}^{(i)})\right).$

###  III-G Fused Dissimilarity Metrics

Observing some pathological situations, illustrated in Fig. 5, leads to the
realization that different metrics may fail or perform well under different
sets of circumstances: If channel conditions change dramatically over a short
spatial distance, as is the case in Fig. 4(a), the channel charting algorithm
may fail to place neighboring datapoints in proximity to each other in the
channel chart if using a CSI-based dissimilarity metric. On the other hand, if
a UE takes the $\Omega$-shaped trajectory shown in Fig. 4(b), any timestamp-
based metric will fail to correctly indicate the closeness of the two turning
points.

(a) Vastly different channel conditions in physical proximity: CSI-based
metrics fail

(b) UE locations are close in space, but far away in time: Timestamp-based
metric fails

Figure 4: Pathological cases for CSI\- or timestamp-based dissimilarity
metrics

ABDissimilarity $d_{\mathrm{ADP},\mathrm{A,B}}$: InaccurateGeodesic
dissimilarity $d_{\mathrm{G-ADP},\mathrm{A,B}}$ Figure 5: Illustration of
geodesic dissimilaritiy

These two examples, which can occur in real-world datasets, lead us to suggest
_fused_ dissimilarity metrics that combine information from multiple sources.
A fused dissimilarity metric $d_{\mathrm{fuse}}$ is computed separately for
every combination of two datapoints $i$, $j$ and takes into account two or
more other dissimilarity metrics $d_{\mathrm{A}}$, $d_{\mathrm{B}}$, …:

$d_{\mathrm{fuse},i,j}=f_{\mathrm{fuse}}(\gamma_{\mathrm{A}}d_{\mathrm{A},i,j},\gamma_{\mathrm{B}}d_{\mathrm{B},i,j},\ldots)$
(8)

The function $f_{\mathrm{fuse}}$ in Eq. (8) defines how the combined metric is
calculated. For example, if the dissimilarity metrics are prone to sometimes
overestimating the true distance, but immune from underestimating, then
$f_{\mathrm{fuse}}$ may simply choose the minimum of all metrics. The
constants $\gamma_{\mathrm{A}}$, $\gamma_{\mathrm{B}}$, $\ldots$ in Eq. (8)
are scaling factors which may be required to scale all metrics to a common
reference. For instance, all metrics may be scaled such that they approximate
the true distance in meters. In practice, these scaling factors can often be
estimated from the dataset itself. As will be shown in Section VII, the fusion
of a CSI-based metric like $d_{\mathrm{ADP}}$ with $d_{\mathrm{time}}$ as a
timestamp-based dissimilarity metric is of particular interest and leads to
significant performance improvements.

###  III-H Global Geodesic Dissimilarities

As [16] points out, dissimilarity metrics are most reliable if the true
distance between two datapoints is small. On the other hand, if two datapoints
are separated widely in physical space, CSI-based dissimilarity metrics are
rarely indicative of the true physical distance. A channel chart learned
directly from CSI-based dissimilarity metrics without any further processing
will usually preserve local neighborhood relationships, but fail at capturing
the global topology of the environment. As in [16] and inspired by the idea
behind Isomap [23], we compute _geodesic dissimilarities_ , which more
accurately predict true distances between far-away datapoints. The basic
concept is illustrated in Fig. 5: The dissimilarity between two far-away
points A and B is computed as the sum of smaller dissimilarities (which are
more accurate) between intermediate points that are closer together.

In practice, geodesic distances are computed as follows: First, we find the
pairwise dissimilarities between any two datapoints in the dataset using one
of the aforementioned metrics and collect them in the dissimilarity matrix
$\mathbf{D}_{\mathrm{pw}}\in\mathbb{R}^{L\times L}$. If, as explained in
Section III-G, a fused dissimilarity metric shall be used,
$\mathbf{D}_{\mathrm{pw}}$ must contain the fused dissimilarities.
$\mathbf{D}_{\mathrm{pw}}$ is interpreted as a matrix containing the weights
of a graph, and we find its $k$-nearest neighbor graph $G_{k\mathrm{-NN}}$,
where $k$ is a tunable hyperparameter that we choose as $k=20$. We then apply
a shortest path algorithm, such as Dijkstra’s algorithm [24] to
$G_{k\mathrm{-NN}}$. The length of the computed shortest paths between any two
nodes of $G_{k\mathrm{-NN}}$ are the geodesic dissimilarities, which we
collect in the geodesic dissimilarity matrix
$\mathbf{D}_{\mathrm{geo}}\in\mathbb{R}^{L\times L}$. In our notation,
geodesic versions of pairwise dissimilarity metrics are denoted by a “G-”
prefix (for example, $d_{\mathrm{CIRA}}$ becomes $d_{\mathrm{G-CIRA}}$).

##  IV Classical Manifold Learning

Manifold LearningClassicalMDS /IsomapSammon’sMappingt-SNEDeep Learning-
basedSiameseNetworkTripletNetwork

Figure 6: Overview of classical and deep learning-based manifold learning
methods

$\mathbf{f}^{(i)}$$\mathbf{f}^{(j)}$$d_{i,j}$DNN $\mathcal{C}_{\Theta}$DNN
$\mathcal{C}_{\Theta}$ shared weights $\Theta$ Loss
$\mathcal{L}_{\mathrm{Siamese}}$$\mathbf{z}^{(i)}$$\mathbf{z}^{(j)}$

Figure 7: Siamese neural network structure

Classical manifold learning techniques formulate optimization problems, where
the objective is to find a low-dimensional representation, which, according to
some cost function, minimizes the discrepancy in point-to-point distances
between channel chart and dissimilarity matrix.

###  IV-A Multidimensional Scaling and Isomap

Multidimensional scaling (MDS) [25] places points
$\\{\mathbf{z}^{(l)}\\}_{l=1}^{L}$ in the channel chart such that the mean
squared distance between point-to-point distances in the channel chart
$\left\lVert\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\right\rVert_{2}$ and the
dissimilarities $d_{i,j}$ in the dissimilarity matrix is minimized, which is
achieved by

$\min_{\\{\mathbf{z}^{(l)}\\}_{l=1}^{L}}\sum\nolimits_{i=1}^{L-1}\sum\nolimits_{j=i+1}^{L}\left(d_{i,j}-\lVert\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\rVert_{2}\right)^{2}.$
(9)

This non-convex optimization problem is solved with gradient descent. It is
not guaranteed that the global optimum of the objective function is found. Any
orthogonal transform of the optimized point set
$\\{\mathbf{z}^{(l)}\\}_{l=1}^{L}$ is also an optimum. multidimensional
scaling (MDS) applied to a geodesic dissimilarity matrix is known as Isomap.

###  IV-B Sammon’s Mapping

SM [10] is a variation of MDS which emphasizes the importance of small point-
to-point dissimilarities. This is achieved by adding a weighting factor to the
objective function:

$\min_{\\{\mathbf{z}^{(l)}\\}_{l=1}^{L}}\sum\nolimits_{i=1}^{L-1}\sum\nolimits_{j=i+1}^{L}\frac{1}{d_{i,j}}\left(d_{i,j}-\lVert\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\rVert_{2}\right)^{2}$
(10)

###  IV-C t-distributed Stochastic Neighbor Embedding

t-distributed stochastic neighbor embedding (t-SNE) [26] is another commonly
used manifold learning technique. The algorithm starts by computing what are
called conditional probabilities for the distances in the dissimilarity
matrix:

$p_{j|i}=\frac{\exp{\left(-d_{i,j}^{2}/2\sigma_{i}^{2}\right)}}{\sum_{k\neq
i}\exp{\left(-d_{i,k}^{2}/2\sigma_{i}^{2}\right)}}\quad\text{and}\quad
p_{i|i}=0$ (11)

In Eq. (11), $\sigma_{i}$ is a hyperparameter. For the computation of
$\sigma_{i}$, we refer to [26, Section 2] and note that our values of
$\sigma_{i}$ are based on a “perplexity” of 120, which was empirically found
to deliver good results. t-SNE then defines symmetrized conditional
probabilities as $p_{ij}=\frac{p_{j|i}+p_{i|j}}{2L}$ and defines pairwise
similarities in the channel chart as

$q_{ij}=\frac{\left(1+\lVert\mathbf{z}_{i}-\mathbf{z}_{j}\rVert_{2}^{2}\right)^{-1}}{\sum_{k}\sum_{\ell\neq
k}\left(1+\lVert\mathbf{z}_{k}-\mathbf{z}_{\ell}\rVert_{2}^{2}\right)^{-1}}\quad\text{and}\quad
q_{ii}=0.$

The location of channel chart points $\\{\mathbf{z}^{(l)}\\}_{l=1}^{L}$ is
then determined by minimizing the Kullback-Leibler divergence:

$\min_{\\{\mathbf{z}^{(l)}\\}_{l=1}^{L}}\sum_{i\neq
j}p_{ij}\log\frac{p_{ij}}{q_{ij}}$

##  V Deep Learning-Based Manifold Learning

In the case of deep-learning based manifold learning methods, the FCF is
realized as a DNN $\mathcal{C}_{\Theta}$, which maps some CSI feature vector
$\mathbf{f}^{(l)}$ (see Section II-C) to the estimated corresponding position
in the channel chart $\mathbf{z}^{(l)}$, i.e.,

$\mathbf{z}^{(l)}=\mathcal{C}_{\Theta}\left(\mathbf{f}^{(l)}\right),$

with $\Theta$ being the set of trainable parameters. The challenge is that
labels for $\mathbf{z}^{(l)}$ are unavailable, hence $\mathcal{C}_{\Theta}$
cannot be trained in a supervised manner. Instead, during the training step,
it has been proposed to embed $\mathcal{C}_{\Theta}$ into either a _Siamese_
[5] or a _triplet_ [12] neural network, as explained in the following. For
inference, this embedding is no longer required.

###  V-A Siamese Neural Networks

The structure of a Siamese neural network, as illustrated in Fig. 7, enables
the use of objective functions known from classical manifold learning
techniques as loss function. During training, two weight-sharing DNNs
$\mathcal{C}_{\Theta}$ map two CSI feature vectors
$\mathbf{f}^{(i)},\mathbf{f}^{(j)}$ to channel chart positions
$\mathbf{z}^{(i)},\mathbf{z}^{(j)}$ at once. Then, the weights are updated
based on a loss function $\mathcal{L}_{\mathrm{Siamese}}$ that takes into
account $\mathbf{z}^{(i)},\mathbf{z}^{(j)}$ and the dissimilarity metric
$d_{i,j}$ from the dissimilarity matrix. Here, we assume that the MDS cost
function is used as loss function for training:

$\mathcal{L}_{\mathrm{Siamese}}=\sum\nolimits_{i=1}^{N-1}\sum\nolimits_{j=i+1}^{N}\left(d_{i,j}-\|\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\|_{2}\right)^{2}.$
(12)

As for MDS, this ensures that the DNN learns to place the estimated points
such that their distance in the channel chart match with the corresponding
computed dissimilarities. In that sense, the Siamese neural network can be
seen as a parametric version of MDS. However, other loss functions such as,
e.g., Sammon’s loss, could also be used [5].

###  V-B Triplet Neural Networks

Triplet SelectionDNN $\mathcal{C}_{\Theta}$DNN $\mathcal{C}_{\Theta}$DNN
$\mathcal{C}_{\Theta}$$\mathbf{D}_{\mathrm{pw}}$ or
$\mathbf{D}_{\mathrm{geo}}$ shared weights $\Theta$ Loss
$\mathcal{L}_{\mathrm{Triplet}}$$\mathbf{f}^{(j)}$$\mathbf{f}^{(i)}$$\mathbf{f}^{(k)}$$\mathbf{z}^{(j)}$$\mathbf{z}^{(i)}$$\mathbf{z}^{(k)}$

Figure 8: Triplet neural network structure

Another approach is to embed the DNN $\mathcal{C}_{\Theta}$ into a triplet
neural network [27], as depicted in Fig. 8. Here, the loss function does not
require information from the dissimilarity matrix. Instead, information about
dissimilarities is incorporated into _triplet selection_. This triplet
selection step selects 3-tuples $(i,j,k)\in\\{1,\ldots,L\\}^{3}$ of
datapoints, where the datapoint with index $i$ is called the _anchor_ sample,
datapoint $j$ is the _close_ sample and datapoint $k$ is the _far_ sample. In
our case, we choose the anchor sample $i$ randomly among all datapoints in the
dataset with uniform probability. Then, we define the set
$\mathcal{N}_{q}^{(i)}$ of suitable close samples for the anchor sample $i$ as

$\mathcal{N}_{q}^{(i)}=\left\\{l~{}|~{}d_{i,l}<d_{\mathrm{thresh},q}^{(i)}\right\\},$

where $d_{\mathrm{thresh},q}^{(i)}$ is the $q$-quantile of all dissimilarities
$\\{d_{i,l}\\}_{l=1,\ldots,L}$. In other words, $\mathcal{N}_{q}^{(i)}$
contains the $qL$ nearest neighbors of datapoint $i$ with respect to the
dissimilarities.

The parameter $q$ is a hyperparameter that may be adjusted during training. We
suggest large values for $q$ at first (e.g., $q=0.2$) and to reduce $q$ over
the training epochs (e.g., to $q=0.02$). This way, the neural network first
learns the global and then the local structure of the manifold. Next, the
index of the close sample $j$ is picked randomly from $\mathcal{N}_{q}^{(i)}$,
and the index of the far sample $k$ is picked randomly from
$\\{1,\ldots,L\\}\setminus\mathcal{N}_{q}^{(i)}$, in both cases with uniform
probability. This process is repeated until the desired number of triplets
$(i,j,k)$ has been collected in the set $\mathcal{T}$. Ideally, each triplet
should fulfill

$\lVert\mathbf{x}^{(i)}-\mathbf{x}^{(j)}\rVert_{2}<\lVert\mathbf{x}^{(i)}-\mathbf{x}^{(k)}\rVert_{2}\quad\text{with}\quad(i,j,k)\in\mathcal{T}.$
(13)

That is, in physical space, the close sample should be nearer to the anchor
sample than the far sample. Since triplet selection guarantees that
$d_{i,j}<d_{i,k}$, Eq. (13) should be fulfilled for most triplets as long as a
good dissimilarity metric is chosen.

The three weight-sharing DNNs estimate channel chart positions
$\mathbf{z}^{(i)}$, $\mathbf{z}^{(j)}$ and $\mathbf{z}^{(k)}$ based on the CSI
feature vectors $\mathbf{f}^{(i)}$, $\mathbf{f}^{(j)}$ and $\mathbf{f}^{(k)}$.
The triplet loss $\mathcal{L}_{\mathrm{Triplet}}$ [27] then makes use of the
knowledge that $\mathbf{z}^{(j)}$ should be closer to $\mathbf{z}^{(i)}$ than
$\mathbf{z}^{(k)}$:

$\mathcal{L}_{\mathrm{Triplet}}=\sum_{\left(i,j,k\right)\in\mathcal{T}}\frac{\max\left\\{\|\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\|_{2}-\|\mathbf{z}^{(i)}-\mathbf{z}^{(k)}\|_{2}+M,0\right\\}}{\mathcal{|T|}},$
(14)

In Eq. (14), $M$ is yet another hyperparameter called _margin_ that we choose
to be $M=1$. In contrast to Siamese networks, triplet neural networks only
require qualitative information about dissimilarities: It is sufficient to
know that the close sample is closer to the anchor than the far sample.
Therefore, triplet neural networks only require a monotonically positive
relationship between true distances
$\lVert\mathbf{x}^{(i)}-\mathbf{x}^{(j)}\rVert_{2}$ and dissimilarities
$d_{i,j}$, but not necessarily the proportional relationship stipulated in Eq.
(2). This is a key advantage for metrics that do not approximate the
proportional relationship (e.g., $d_{\mathrm{time}}$), but can also be a
disadvantage if good absolute dissimilarities are available, but the
information is not used.

##  VI Evaluation Metrics

The quality of a channel chart can be subjectively judged by comparing the
learned chart to the map of true locations. However, for a more objective
assessment, well-known evaluation metrics from manifold learning like
continuity (CT), trustworthiness (TW) [28] and Kruskal’s stress (KS) [25] are
commonly applied. While CT and TW indicate how well local neighborhood
relationships are reflected in the channel chart, KS measures to what extent
the global structure is preserved.

Instead of applying these metrics to the resulting channel chart, and thereby
evaluating dissimilarity metric and manifold learning technique jointly, CT,
TW and KS may also be used to evaluate dissimilarity metrics separately [4].
We also consider Rajski’s distance (RD) [29] for dissimilarity metrics and,
for channel charts, we compute the mean absolute error (MAE) between the
ground truth positions and the channel chart positions assuming an optimal
affine coordinate transform.

###  VI-A Continuity and Trustworthiness

CT and TW compare neighborhood relationships in physical space to neighborhood
relationships in the representation space. Let $r_{l,i}$ denote the rank of
the neighbor with index $i$ among all neighbors of datapoint $l$ in physical
space. That is, $r_{l,i}=1$ if $i$ is the closest neighbor of $l$, $r_{l,i}=2$
if it is the second closest neighbor and so on. We define the set of $K$
nearest neighbors of $l$ in physical space as
$\mathcal{V}_{K}^{(l)}=\left\\{i~{}|~{}r_{l,i}\leq K\right\\}$.

In a similar way, let $\hat{r}_{l,i}$ denote the rank of the neighbor with
index $i$ among all neighbors of datapoint $l$ in the representation space.
When evaluating CT / TW of a dissimilarity metric, this rank is calculated
over the entries in the dissimilarity matrix, i.e., over
$\left\\{d_{l,i}\right\\}_{i=1,\ldots,L}$. If CT / TW are applied to a channel
chart, the rank is calculated over the Euclidean distances of the channel
chart locations, i.e., over
$\left\\{\lVert\mathbf{z}^{(l)}-\mathbf{z}^{(i)}\rVert_{2}\right\\}_{i=1,\ldots,L}$.
We define the set of $K$ nearest neighbors of $l$ in the representation space
as $\mathcal{U}_{K}^{(l)}=\left\\{i~{}|~{}\hat{r}_{l,i}\leq K\right\\}$. CT is
then defined as

$\mathrm{CT}\left(K\right)=1-\frac{2}{LK\left(2L-3K-1\right)}\sum^{L}_{l=1}\sum_{i\in\mathcal{V}_{K}^{(l)}}\max\left\\{0,\hat{r}_{l,i}-K\right\\},$

and, intuitively speaking, indicates whether the nearest neighbors in physical
space are among those of the representation space. Analogously, TW is defined
as

$\mathrm{TW}\left(K\right)=1-\frac{2}{LK\left(2L-3K-1\right)}\sum^{L}_{l=1}\sum_{i\in\mathcal{U}_{K}^{(l)}}\max\left\\{0,r_{l,i}-K\right\\},$

and indicates whether the neighbors in representation space are among those in
physical space, or, conversely, if the representation contains false
neighbors. The normalization factor ensures that both CT and TW are in the
range $\left[0,1\right]$, where values close to $1$ are better. As already
proposed in [4], we choose a neighborhood size of $K=0.05L$.

###  VI-B Kruskal’s Stress

KS [25] is a performance measure for the preservation of the global structure
of the channel chart. It compares pairwise distances between points in the
original space to those between points in the representation space:

$\begin{split}\mathrm{KS}=\sqrt{\frac{\sum_{i=1}^{L-1}\sum_{j=i+1}^{L}\left(\|\mathbf{x}^{(i)}-\mathbf{x}^{(j)}\|_{2}-\beta\|\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\|_{2}\right)^{2}}{\sum_{i=1}^{L-1}\sum_{j=i+1}^{L}\|\mathbf{x}^{(i)}-\mathbf{x}^{(j)}\|_{2}^{2}}}\\\
\text{with}\quad\beta=\frac{\sum_{i=1}^{L-1}\sum_{j=i+1}^{L}\|\mathbf{x}^{(i)}-\mathbf{x}^{(j)}\|_{2}\|\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\|_{2}}{\sum_{i=1}^{L-1}\sum_{j=i+1}^{L}\|\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\|_{2}^{2}}.\end{split}$

The scaling factor $\beta$ ensures that the value of KS is bounded to the
range $\left[0,1\right]$, where $0$ indicates the best preservation of the
global structure. If KS is evaluated for a dissimilarity metric (as opposed to
a channel chart), $\|\mathbf{z}^{(i)}-\mathbf{z}^{(j)}\|_{2}$ is replaced by
the corresponding dissimilarity $d_{i,j}$.

###  VI-C Rajski’s Distance

RD, which was proposed for use in channel charting in [22], is an indicator
for the mutual information between the true distances and the dissimilarities.
To this end, all pairwise true distances
$\lVert\mathbf{x}^{(l)}-\mathbf{x}^{(i)}\rVert_{2}$, as well as all
dissimilarities $d_{l,i}$ ($l,i\in\\{1,\ldots,L\\}$) are quantized into $100$
bins each such that they can be interpreted as distributions modeled by the
random variables $V$ and $Q$, respectively. RD is then defined as [29]

$\begin{split}\mathrm{RD}\left(V,Q\right)=1-\frac{I\left(V,Q\right)}{H\left(V,Q\right)},\quad\text{for}\quad
H\left(V,Q\right)\neq 0,\\\ \text{with}\quad I\left(V,Q\right)=\sum_{v\in
V,q\in
Q}P_{V,Q}\left(v,q\right)\mathrm{log}_{2}\frac{P_{V,Q}\left(v,q\right)}{P_{V}\left(v\right)P_{Q}\left(q\right)}\end{split}$

denoting the mutual information between the distributions of $V$ and $Q$,
where $P_{V,Q}\left(v,q\right)$ is the joint probability distribution of $V$
and $Q$ and $P_{V}\left(v\right)$ and $P_{Q}\left(q\right)$ are the respective
marginal distributions. The joint entropy of $V$ and $Q$ is given by

$H\left(V,Q\right)=-\sum_{v\in V,q\in
Q}P_{V,Q}\left(v,q\right)\mathrm{log}_{2}P_{V,Q}\left(v,q\right).$

RD is between 0 and 1, and 0 indicates the best performance.

###  VI-D Mean Absolute Error

A learned channel chart should reconstruct both local and global geometry, but
may be rotated, scaled, flipped and / or translated compared to the true map.
As previously proposed by Stahlke et al. [16], we find the optimal affine
transformation from channel chart to the true locations and compute the MAE
under the assumption that this transformation has been found and applied. The
resulting MAE thereby provides a lower bound to the absolute localization
error. We anticipate that it will be possible to achieve performance close to
this bound by applying proposals like [21]. To find the optimal affine
transformation, we solve the least squares problem

$(\mathbf{\hat{A}},\mathbf{\hat{b}})=\operatorname*{arg\,min}\limits_{(\mathbf{A},\mathbf{b})}\sum_{l=1}^{L}\lVert\mathbf{A}\mathbf{z}^{(l)}+\mathbf{b}-\mathbf{x}^{(l)}\rVert_{2}^{2}$

and compute the MAE as
$\mathrm{MAE}=\frac{1}{L}\sum_{l=1}^{L}\lVert\mathbf{\hat{A}}\mathbf{z}^{(l)}+\mathbf{\hat{b}}-\mathbf{x}^{(l)}\rVert_{2}.$

##  VII Experimental Results

###  VII-A DICHASUS: Channel Measurement Datasets

(a)

$-10$$-6$$-2$$2$$-18$$-16$$-14$$-12$$-10$$-8$$-6$$-4$$-2$Coordinate
$x~{}[\mathrm{m}]$Coordinate $y~{}[\mathrm{m}]$ (b)

$-10$$-6$$-2$$2$$-14$$-12$$-10$$-8$$-6$$-4$$-2$Coordinate
$x~{}[\mathrm{m}]$Coordinate $y~{}[\mathrm{m}]$ (c)

Figure 9: Information about the industrial environment the measurement was
conducted in. The figure shows (a) a photograph of the environment, (b) a top
view map and (c) a scatter plot of colorized “ground truth” positions of
datapoints in $\mathcal{S}_{\mathrm{full}}$, also in top view. The antenna
arrays in the map are drawn to scale as black rectangles and their viewing
direction is indicated by the green sectors.

All dissimilarity metrics and manifold learning techniques were tested on a
dataset generated by _ Distributed Channel Sounder by University of Stuttgart
(DICHASUS)_, our distributed mMIMO channel sounder, whose architecture is
thoroughly described in [30]. In brief, DICHASUS measures the propagation
channel between a single transmitter and many receive antennas. It achieves
long-term phase-coherence, even if antennas are distributed over a wide area.
DICHASUS provides large datasets containing frequency-domain CSI tensors
$\mathbf{H}^{(l)}$, alongside side information like timestamps and accurate
information about the positions of all antennas.

The dataset chosen for the following analyses is called _dichasus-cf0x_ [31],
and was captured in an industrial environment with $B=4$ separate antenna
arrays made up of $M=2\times 4$ antennas each. $N_{\mathrm{sub}}=1024$ OFDM
channel coefficients were measured at a carrier frequency of
$1.272\,\mathrm{GHz}$ and with a channel bandwidth of $50\,\mathrm{MHz}$. The
single dipole transmit antenna is mounted on top of a robot, which travels
along a set of trajectories inside a defined, L-shaped area, with an overall
size of approximately $14\,\mathrm{m}\times 14\,\mathrm{m}$. A photo and a top
view map of the environment are shown in Fig. 9. A large metal container is
located at the inner corner of the L-shape, blocking the LoS. Every position
in the measurement area has at least one LoS path to one of the four arrays,
and some positions in the dataset even have a LoS path to all arrays.

To reduce the total amount of data, we choose three subsets of _dichasus-cf0x_
called _dichasus-cf02_ , _dichasus-cf03_ and _dichasus-cf04_ and then collect
only every 5th datapoint into a smaller dataset that we call
$\mathcal{S}_{\mathrm{full}}$, now containing
$\left|\mathcal{S}_{\mathrm{full}}\right|=16797$ datapoints. The true
datapoint positions $\mathbf{x}^{(l)}$ are shown in Fig. 9. The points have
been colorized and the datapoints will retain their color even as the FCF maps
them to a position in the channel chart. This allows for a visual evaluation
of the generated chart: If the global topology is preserved, a similar color
gradient should appear in the chart.

$504$$506$$508$$510$$512$$514$$516$$518$$520$$522$$524$$526$$0$$1$$2$$3$$\tau_{\mathrm{min}}$$\tau_{\mathrm{max}}$$\tau$$|\tilde{\mathbf{H}}_{1,m,\tau}^{(l)}|$$m=1$$m=2$$m=3$$m=4$$m=5$$m=6$$m=7$$m=8$
Figure 10: Examplary time-domain CSI amplitudes
$|\tilde{\mathbf{H}}_{1,m,\tau}^{(l)}|$ at some time instant $l$. Each line
corresponds to one antenna $m$ at antenna array $b=1$ .

Exemplary CSI data measured by DICHASUS is visualized in Fig. 10, which shows
the CIRs of all antennas in antenna array $b=1$. Note that phase information
is also available, but not displayed in the figure. It is easy to see that
much of the received signal power is concentrated within a relatively short
interval of taps (area shaded red in Fig. 10), leading us to choose
$\tau_{\mathrm{min}}=507$ and $\tau_{\mathrm{max}}=520$ for all subsequent
analyses.

###  VII-B Fusion of $d_{\mathrm{ADP}}$ and $d_{\mathrm{time}}$

As described in Section III-G, multiple dissimilarity metrics may be combined
into a fused metric. We choose to fuse $d_{\mathrm{ADP}}$ as a CSI-based
dissimilarity metric and $d_{\mathrm{time}}$ as a side information-based
metric and show that using the fused metric $d_{\mathrm{fuse}}$ significantly
improves the channel chart. In _dichasus-cf0x_ , the robot either moves at
approximately constant velocity $v_{\mathrm{robot}}$, or remains at a
standstill. The combination of $d_{\mathrm{ADP}}$ and $d_{\mathrm{time}}$ is
based on a simple assumption: For short time intervals
$d_{\mathrm{time}}<t_{\mathrm{thresh}}$, we assume that the transmitter either
moves along an approximately straight line (due to its inertia) with velocity
$v_{\mathrm{robot}}$ or does not move at all. For such short intervals, if the
robot is moving, the time-based metric $d_{\mathrm{time}}$ predicts the true
distance between points much better than any CSI-based metric could.
Therefore, we propose to fuse $d_{\mathrm{time}}$ and $d_{\mathrm{ADP}}$ by
taking their minimum after appropriate scaling of $d_{\mathrm{time}}$:

$d_{\mathrm{fuse},i,j}=\mathrm{min}\left\\{d_{\mathrm{ADP},i,j},\gamma\cdot
d_{\mathrm{time},i,j}\right\\}$ (15)

This way, whenever $d_{\mathrm{time},i,j}$ is sufficiently small, it predicts
the dissimilarity between points $i$ and $j$. The challenge remains to find
the scaling factor $\gamma$ in Eq. (15): To determine $\gamma$, we need to
observe the distribution of ratios
$\nicefrac{{d_{\mathrm{ADP},i,j}}}{{d_{\mathrm{time},i,j}}}$ in Fig. 11, there
shown for all pairs of datapoints $i$, $j$ in $\mathcal{S}_{\mathrm{full}}$
with $d_{\mathrm{time},i,j}<t_{\mathrm{thresh}}=2\,\mathrm{s}$. The diagram
clearly shows a bimodal distribution for the ratios, where the right mode can
be attributed to the datapoints where the robot is moving, and the left mode
can be attributed to the cases where it remains static. We choose a scaling
factor that separates the two modes, e.g., $\gamma=12$, such that $\gamma
d_{\mathrm{time},i,j}<d_{\mathrm{ADP},i,j}$ if the robot is moving. For other
real-world datasets, the assumption of constant transmitter velocity may not
hold. In that case, it might be necessary to estimate the relative velocities
of transmitters over time and between transmitters based on CSI data. This
remains a topic for future research that we do not consider within the scope
of this work. After computing the pairwise fused metric
$d_{\mathrm{fuse},i,j}$, we find geodesic distances as explained in Section
III-H to obtain $d_{\mathrm{G-fuse},i,j}$.

$0$$5$$10$$15$$20$$25$$30$$0$$2{,}000$$4{,}000$$\gamma$Rely
on$d_{\mathrm{ADP},i,j}$Rely
on$d_{\mathrm{time},i,j}$$\nicefrac{{d_{\mathrm{ADP},i,j}}}{{d_{\mathrm{time},i,j}}}$Occurences
for$d_{\mathrm{time},i,j}<t_{\mathrm{thresh}}$ Figure 11: Histogram depiciting
the distribution of the ratios
$\nicefrac{{d_{\mathrm{ADP},i,j}}}{{d_{\mathrm{time},i,j}}}$ for over all
pairs of datapoints $i,j$ in $\mathcal{S}_{\mathrm{full}}$ with
$d_{\mathrm{time},i,j}<t_{\mathrm{thresh}}$.

###  VII-C Evaluation of Dissimilarity Metrics

TABLE I: Evaluation of dissimilarity metrics on $\mathcal{S}_{\mathrm{full}}$. Dissimilarity |  CT $\uparrow$ |  TW $\uparrow$ |  KS $\downarrow$ |  RD $\downarrow$
---|---|---|---|---
$d_{\mathrm{Euc}}$ | $1.0000$ | $1.0000$ | $0.0000$ | $0.0000$
$d_{\mathrm{G-Euc}}$ | $0.9999$ | $1.0000$ | $0.0231$ | $0.3617$
$d_{\mathrm{CS}}$ | $0.9529$ | $0.9542$ | $0.4281$ | $0.9595$
$d_{\mathrm{G-CS}}$ | $0.9799$ | $0.9774$ | $0.2676$ | $0.9112$
$d_{\mathrm{CIRA}}$ | $0.9278$ | $0.9477$ | $0.3173$ | $0.9406$
$d_{\mathrm{G-CIRA}}$ | $0.9685$ | $0.9684$ | $0.2197$ | $0.8972$
$d_{\mathrm{DL}}$ | $0.9762$ | $0.9742$ | $0.4267$ | $0.9499$
$d_{\mathrm{G-DL}}$ | $0.9800$ | $0.9772$ | $0.2250$ | $0.8953$
$d_{\mathrm{ADP}}$ | $0.9653$ | $0.9689$ | $0.4350$ | $0.9431$
$d_{\mathrm{G-ADP}}$ | $0.9885$ | $0.9866$ | $0.1738$ | $0.8384$
$d_{\mathrm{fuse}}$ | $0.9655$ | $0.9691$ | $0.4347$ | $0.9429$
$d_{\mathrm{G-fuse}}$ | $\mathbf{0.9957}$ | $\mathbf{0.9948}$ | $\mathbf{0.1098}$ | $\mathbf{0.7619}$

In Tab. I, all dissimilarity metrics presented in Section III, as well as
previously described fused dissimilarity metric $d_{\mathrm{fuse},i,j}$ are
evaluated using CT, TW, KS and RD. The true distance $d_{\mathrm{Euc},i,j}$ is
only included for reference and it obviously achieves the optimal performance
metric in every category. In all tables, results based on true distances are
only included for comparison and are shaded in grey. The results indicate that
our newly proposed dissimilarity metrics $d_{\mathrm{ADP}}$ and
$d_{\mathrm{DL}}$ outperform the baseline dissimilarities $d_{\mathrm{CS}}$
and $d_{\mathrm{CIRA}}$ regarding most performance metrics. Overall, the
geodesic fused dissimilarity $d_{\mathrm{G-fuse},i,j}$ is the best-performing
one.

In general, it is apparent that geodesic dissimilarities are better than the
respective non-geodesic versions regarding CT, TW, KS and RD. This is also
illustrated by Fig. 12, which, inspired by [16], plots $d_{\mathrm{ADP},i,j}$
and $d_{\mathrm{G-ADP},i,j}$ as a function of the Euclidean distance
$d_{\mathrm{Euc},i,j}$. Unsurprisingly, the geodesic dissimilarity
approximates the Euclidean distance better (down to a scaling factor) than the
non-geodesic dissimilarity.

$0$$2$$4$$6$$8$$10$$12$$14$$16$$0$$0.2$$0.4$$0.6$$0.8$$1$Euclidean Distance
[m]$d_{\mathrm{Euc}}/d_{\mathrm{Euc,max}}$$d_{\mathrm{ADP}}/d_{\mathrm{ADP,max}}$$d_{\mathrm{G-ADP}}/d_{\mathrm{G-ADP,max}}$
Figure 12: Dissimilarity metrics $d_{\mathrm{ADP}}$ and $d_{\mathrm{G-ADP}}$
normalized to a range from 0 to 1 as a function of the true distance. To
generate the diagram, the dissimilarities are grouped into 100 bins according
to the true distances. The lines represent the medians of these bins, and the
shaded areas represent the range from the 25th to the 75th percentile. See
also [16, Fig. 6 and 7].

###  VII-D Channel Charting Results

All channel charts in this section are learned on the training set
$\mathcal{S}_{\mathrm{full}}$, as specified in Section VII-A. While classical
manifold learning methods directly produce channel charts, deep learning-based
(parametric) methods are evaluated on a different test set
$\mathcal{S}_{\mathrm{full}}^{\prime}$. This test set also contains
$\left|\mathcal{S}_{\mathrm{full}}^{\prime}\right|=16797$ datapoints measured
in the same area, and is generated by selecting a different subset of points
from _dichasus-cf02_ , _dichasus-cf03_ and _dichasus-cf04_. This ensures that
deep learning-based methods actually work with previously unseen feature
vectors, and, in a performance comparison, puts them at a slight disadvantage
compared to classical methods.

In our experiments, the DNN $\mathcal{C}_{\Theta}$ consists of six hidden
layers with 1024, 512, 512, 256, 128, 64 neurons and ReLU activation each, and
an output layer with 2 neurons and linear activation. Batch normalization is
applied between the layers to enhance training performance.

Tab. II shows the channel charting performance for different classical
methods. The results for the distance $d_{\mathrm{Euc}}$ serve as benchmark
for the respective manifold learning technique. As expected, the channel
charts learned by SM often represent the local geometry better than those
learned by MDS (higher CT / TW). Overall, the best performance is achieved by
Isomap, which applies MDS to geodesic distances. As already concluded in
Section VII-C, the newly introduced dissimilarity metrics, and especially the
fused metric appear to be most suitable, as they result in the best channel
charts. It is remarkable that the t-SNE algorithm produces only slightly worse
channel charts than Isomap without relying on geodesic distances.

The performance metrics for deep learning-based methods are shown in Tab. III.
Unsurprisingly, the Siamese network produces better channel charts for most
geodesic dissimilarity metrics, whereas the triplet neural network performs
better for non-geodesic metrics. This can be explained by the fact that the
triplet neural network only uses qualitative information about dissimilarity,
and does not require the metrics to be proportional to true distances, whereas
the Siamese network can also make use of absolute dissimilarity information.

The channel chart produced by a triplet neural network using purely time-based
triplet selection ($d_{\mathrm{time}}$) is surprisingly good with respect to
all metrics, but is still outperformed by the Siamese network: The best
overall performance is achieved by the Siamese network in combination with
$d_{\mathrm{G-fuse}}$.

After the optimal affine coordinate transformation has been applied as in
Section VI-D, the cumulative distribution function (CDF) of the absolute
localization errors can be visualized. Various such CDFs, belonging to channel
charts produced with different dissimilarities and manifold learning
techniques, are depicted in Fig. 13. Fig. 14 shows several channel charts
obtained using different dissimilarity metrics and manifold learning
techniques. The channel charts can be directly compared to the true map in
Fig. 9(c). The first row of Fig. 14 shows the channel charts produced with the
baseline metrics $d_{\mathrm{G-CIRA}}$ and $d_{\mathrm{G-CS}}$ as well as a
chart learned from non-geodesic $d_{\mathrm{ADP}}$ by Siamese network. It also
contains Fig. 14(d), which is the surprisingly good channel chart obtained
through time-based triplet selection, i.e., the combination of a triplet
network with $d_{\mathrm{time}}$. The second row shows channel charts for
$d_{\mathrm{G-ADP}}$, demonstrating that using the geodesic version of the ADP
metric greatly improves performance. What is more, the fused metric
$d_{\mathrm{G-fuse}}$, here with Isomap or a Siamese network, generates even
better channel charts.

$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$4.5$$5$$5.5$$6$$0$$0.2$$0.4$$0.6$$0.8$$1$Absolute
Localization Error [m]Siamese, $d_{\mathrm{G-CS}}$Siamese,
$d_{\mathrm{G-CIRA}}$Triplet, $d_{\mathrm{time}}$Siamese,
$d_{\mathrm{G-DL}}$Siamese, $d_{\mathrm{G-ADP}}$Siamese, $d_{\mathrm{G-fuse}}$
Figure 13: CDF of absolute localization error for different manifold learning
methods and dissimilarity metrics, computed on $\mathcal{S}_{\mathrm{full}}$.

$-600$$-400$$-200$$0$$200$$400$$600$$800$$-600$$-400$$-200$$0$$200$$400$$600$$800$Latent
variable $x$Latent variable $y$ (a) Siamese, $d_{\mathrm{G-CIRA}}$,
$\mathcal{S}_{\mathrm{full}}$

$-6$$-4$$-2$$0$$2$$4$$6$$-10$$-8$$-6$$-4$$-2$$0$$2$$4$$6$$8$$10$Latent
variable $x$Latent variable $y$ (b) Siamese, $d_{\mathrm{G-CS}}$,
$\mathcal{S}_{\mathrm{full}}$

$-30$$-20$$-10$$0$$10$$20$$30$$-30$$-20$$-10$$0$$10$$20$$30$Latent variable
$x$Latent variable $y$ (c) Siamese, $d_{\mathrm{ADP}}$,
$\mathcal{S}_{\mathrm{full}}$

$-8$$-6$$-4$$-2$$0$$2$$4$$6$$8$$10$$-10$$-5$$0$$5$$10$Latent variable
$x$Latent variable $y$ (d) Triplet, $d_{\mathrm{time}}$,
$\mathcal{S}_{\mathrm{full}}$

$-200$$-150$$-100$$-50$$0$$50$$100$$150$$200$$-300$$-200$$-100$$0$$100$$200$Latent
variable $x$Latent variable $y$ (e) Siamese, $d_{\mathrm{G-ADP}}$,
$\mathcal{S}_{\mathrm{full}}$

$-6$$-4$$-2$$0$$2$$4$$6$$-6$$-4$$-2$$0$$2$$4$$6$Latent variable $x$Latent
variable $y$ (f) Triplet, $d_{\mathrm{G-ADP}}$, $\mathcal{S}_{\mathrm{full}}$

$-200$$-100$$0$$100$$200$$-300$$-200$$-100$$0$$100$$200$$300$Latent variable
$x$Latent variable $y$ (g) Isomap, $d_{\mathrm{G-fuse}}$,
$\mathcal{S}_{\mathrm{full}}$

$-300$$-200$$-100$$0$$100$$200$$-300$$-200$$-100$$0$$100$$200$Latent variable
$x$Latent variable $y$ (h) Siamese, $d_{\mathrm{G-fuse}}$,
$\mathcal{S}_{\mathrm{full}}$

$-60$$-40$$-20$$0$$20$$40$$-40$$-20$$0$$20$$40$$60$$80$Latent variable
$x$Latent variable $y$ (i) Siamese, $d_{\mathrm{G-ADP}}$,
$\mathcal{S}_{\mathrm{NLoS}}$

$-20$$-15$$-10$$-5$$0$$5$$10$$15$$20$$-25$$-20$$-15$$-10$$-5$$0$$5$$10$Latent
variable $x$Latent variable $y$ (j) Siamese, $d_{\mathrm{G-DL}}$,
$\mathcal{S}_{\mathrm{NLoS}}$

$-10$$-8$$-6$$-4$$-2$$0$$2$$4$$6$$8$$10$$-10$$-5$$0$$5$$10$Latent variable
$x$Latent variable $y$ (k) Triplet, $d_{\mathrm{time}}$,
$\mathcal{S}_{\mathrm{NLoS}}$

$-80$$-60$$-40$$-20$$0$$20$$40$$60$$80$$-100$$-50$$0$$50$$100$Latent variable
$x$Latent variable $y$ (l) Siamese, $d_{\mathrm{G-fuse}}$,
$\mathcal{S}_{\mathrm{NLoS}}$

Figure 14: Comparison of channel charts generated by various combinations of manifold learning technique, dissimilarity metric and dataset (see captions). TABLE II: Performance of Classical Methods on $\mathcal{S}_{\mathrm{full}}$. Method | Metric |  CT $\uparrow$ |  TW $\uparrow$ |  KS $\downarrow$ |  MAE $\downarrow$ | Fig.
---|---|---|---|---|---|---
MDS | $d_{\mathrm{Euc}}$ | $1.0000$ | $1.0000$ | $0.0001$ | $0.000\,\mathrm{m}$ |
MDS | $d_{\mathrm{CS}}$ | $0.9106$ | $0.8888$ | $0.3673$ | $1.913\,\mathrm{m}$ |
MDS | $d_{\mathrm{CIRA}}$ | $0.9281$ | $0.9200$ | $0.3400$ | $1.861\,\mathrm{m}$ |
MDS | $d_{\mathrm{DL}}$ | $0.9289$ | $0.9011$ | $0.4015$ | $2.349\,\mathrm{m}$ |
MDS | $d_{\mathrm{ADP}}$ | $0.8902$ | $0.9007$ | $0.3437$ | $2.037\,\mathrm{m}$ |
MDS | $d_{\mathrm{fuse}}$ | $0.9254$ | $0.9255$ | $0.3262$ | $1.782\,\mathrm{m}$ |
Isomap | $d_{\mathrm{G-CS}}$ | $0.9712$ | $0.9721$ | $0.2557$ | $1.533\,\mathrm{m}$ |
Isomap | $d_{\mathrm{G-CIRA}}$ | $0.9690$ | $0.9656$ | $0.2340$ | $1.185\,\mathrm{m}$ |
Isomap | $d_{\mathrm{G-DL}}$ | $0.9689$ | $0.9617$ | $0.2237$ | $1.292\,\mathrm{m}$ |
Isomap | $d_{\mathrm{G-ADP}}$ | $0.9893$ | $0.9878$ | $0.1371$ | $0.705\,\mathrm{m}$ |
Isomap | $d_{\mathrm{G-fuse}}$ | $\mathbf{0.9956}$ | $\mathbf{0.9960}$ | $\mathbf{0.0859}$ | $\mathbf{0.438\,m}$ | 14(g)
SM | $d_{\mathrm{Euc}}$ | $0.9833$ | $0.9339$ | $0.1853$ | $1.274\,\mathrm{m}$ |
SM | $d_{\mathrm{CS}}$ | $0.9311$ | $0.8984$ | $0.3482$ | $1.877\,\mathrm{m}$ |
SM | $d_{\mathrm{CIRA}}$ | $0.9124$ | $0.9167$ | $0.3242$ | $1.787\,\mathrm{m}$ |
SM | $d_{\mathrm{DL}}$ | $0.9231$ | $0.8572$ | $0.4249$ | $2.454\,\mathrm{m}$ |
SM | $d_{\mathrm{ADP}}$ | $0.9493$ | $0.9383$ | $0.3159$ | $1.518\,\mathrm{m}$ |
SM | $d_{\mathrm{fuse}}$ | $0.9529$ | $0.9413$ | $0.3153$ | $1.505\,\mathrm{m}$ |
t-SNE | $d_{\mathrm{Euc}}$ | $0.9980$ | $0.9969$ | $0.1025$ | $0.894\,\mathrm{m}$ |
t-SNE | $d_{\mathrm{CS}}$ | $0.9811$ | $0.9819$ | $0.1800$ | $1.030\,\mathrm{m}$ |
t-SNE | $d_{\mathrm{CIRA}}$ | $0.9579$ | $0.9514$ | $0.2392$ | $1.261\,\mathrm{m}$ |
t-SNE | $d_{\mathrm{DL}}$ | $0.9793$ | $0.9778$ | $0.1637$ | $0.914\,\mathrm{m}$ |
t-SNE | $d_{\mathrm{ADP}}$ | $0.9851$ | $0.9874$ | $0.1776$ | $0.775\,\mathrm{m}$ |
t-SNE | $d_{\mathrm{fuse}}$ | $0.9888$ | $0.9902$ | $0.1509$ | $0.697\,\mathrm{m}$ |
TABLE III: Performance of Deep Learning-based Methods on $\mathcal{S}_{\mathrm{full}}$ Method | Metric |  CT $\uparrow$ |  TW $\uparrow$ |  KS $\downarrow$ |  MAE $\downarrow$ | Fig.
---|---|---|---|---|---|---
Siamese | $d_{\mathrm{Euc}}$ | $0.9907$ | $0.9902$ | $0.0801$ | $0.364\,\mathrm{m}$ |
Siamese | $d_{\mathrm{CS}}$ | $0.8928$ | $0.8869$ | $0.3394$ | $1.712\,\mathrm{m}$ |
Siamese | $d_{\mathrm{CIRA}}$ | $0.9195$ | $0.9040$ | $0.3344$ | $1.926\,\mathrm{m}$ |
Siamese | $d_{\mathrm{DL}}$ | $0.8982$ | $0.8681$ | $0.4020$ | $2.368\,\mathrm{m}$ |
Siamese | $d_{\mathrm{ADP}}$ | $0.8882$ | $0.8845$ | $0.3496$ | $1.817\,\mathrm{m}$ | 14(c)
Siamese | $d_{\mathrm{fuse}}$ | $0.8907$ | $0.8845$ | $0.3466$ | $1.771\,\mathrm{m}$ |
Siamese | $d_{\mathrm{G-CS}}$ | $0.9710$ | $0.9718$ | $0.2517$ | $1.519\,\mathrm{m}$ | 14(b)
Siamese | $d_{\mathrm{G-CIRA}}$ | $0.9524$ | $0.9466$ | $0.2266$ | $1.216\,\mathrm{m}$ | 14(a)
Siamese | $d_{\mathrm{G-DL}}$ | $0.9638$ | $0.9530$ | $0.2269$ | $1.250\,\mathrm{m}$ |
Siamese | $d_{\mathrm{G-ADP}}$ | $0.9807$ | $0.9798$ | $0.1515$ | $0.783\,\mathrm{m}$ | 14(e)
Siamese | $d_{\mathrm{G-fuse}}$ | $\mathbf{0.9879}$ | $\mathbf{0.9880}$ | $\mathbf{0.1067}$ | $\mathbf{0.542\,m}$ | 14(h)
Triplet | $d_{\mathrm{Euc}}$ | $0.9825$ | $0.9803$ | $0.1338$ | $0.671\,\mathrm{m}$ |
Triplet | $d_{\mathrm{CS}}$ | $0.9470$ | $0.9362$ | $0.3011$ | $1.675\,\mathrm{m}$ |
Triplet | $d_{\mathrm{CIRA}}$ | $0.9299$ | $0.9265$ | $0.2770$ | $1.429\,\mathrm{m}$ |
Triplet | $d_{\mathrm{DL}}$ | $0.9487$ | $0.9246$ | $0.3198$ | $2.178\,\mathrm{m}$ |
Triplet | $d_{\mathrm{ADP}}$ | $0.9656$ | $0.9594$ | $0.2810$ | $1.420\,\mathrm{m}$ |
Triplet | $d_{\mathrm{fuse}}$ | $0.9657$ | $0.9601$ | $0.2781$ | $1.438\,\mathrm{m}$ |
Triplet | $d_{\mathrm{time}}$ | $0.9823$ | $0.9822$ | $0.2155$ | $0.621\,\mathrm{m}$ | 14(d)
Triplet | $d_{\mathrm{G-CS}}$ | $0.9702$ | $0.9669$ | $0.1733$ | $0.951\,\mathrm{m}$ |
Triplet | $d_{\mathrm{G-CIRA}}$ | $0.9533$ | $0.9430$ | $0.2215$ | $1.239\,\mathrm{m}$ |
Triplet | $d_{\mathrm{G-DL}}$ | $0.9640$ | $0.9538$ | $0.2241$ | $1.199\,\mathrm{m}$ |
Triplet | $d_{\mathrm{G-ADP}}$ | $0.9769$ | $0.9743$ | $0.1506$ | $0.747\,\mathrm{m}$ | 14(f)
Triplet | $d_{\mathrm{G-fuse}}$ | $0.9805$ | $0.9785$ | $0.1429$ | $0.640\,\mathrm{m}$ |

###  VII-E Results for significant NLoS propagation

To figure out whether our metrics $d_{\mathrm{G-DL}}$, $d_{\mathrm{G-ADP}}$
and $d_{\mathrm{G-fuse}}$ are useful for environments with significant NLoS
propagation, we define a dataset $\mathcal{S}_{\mathrm{NLoS}}$, which we
obtain by removing CSI data for antenna arrays $b=2$ and $b=4$ from
$\mathcal{S}_{\mathrm{full}}$. This way, the metal container is blocking the
LoS path for a significant share of the area (compare Fig. 9(b)). Tab. IV
shows the performance of the respective dissimilarities for this scenario. All
metrics perform at least slightly worse than in the LoS scenario (compare Tab.
I), but the CSI-based dissimilarity metrics suffer most from the NLoS
conditions. $d_{\mathrm{CS}}$ and $d_{\mathrm{CIRA}}$ are especially hard hit,
but the performance metrics for $d_{\mathrm{ADP}}$ also deteriorate. Fusing
$d_{\mathrm{ADP}}$ with $d_{\mathrm{time}}$ and using geodesic dissimilarities
significantly improves performance, as indicated by the analyses for
$d_{\mathrm{G-fuse}}$. This underlines the importance of taking into account
timestamp information, especially in NLoS conditions. The results for Isomap,
Siamese networks and triplet networks on the NLoS dataset are listed in Tab.
V. As expected, the channel charts produced with metrics that exploit
timestamp information perform best.

This claim is supported by the final row of Fig. 14: The global geometry is
distorted for the channel chart produced with $d_{\mathrm{G-ADP}}$, whereas
the channel charts using timestamp-aided dissimilarity metrics
($d_{\mathrm{G-DL}}$, $d_{\mathrm{time}}$ and $d_{\mathrm{G-fuse}}$) are
globally more meaningful. The best localization performance is achieved by the
Siamese network combined with the fused dissimilarity $d_{\mathrm{G-fuse}}$,
where a mean localization error of $0.672\,\mathrm{m}$ is reached. As before,
CDFs of localization errors for different metrics and techniques are shown in
Fig. 15.

$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$4.5$$5$$5.5$$6$$6.5$$7$$0$$0.2$$0.4$$0.6$$0.8$$1$Absolute Localization Error [m]Triplet, $d_{\mathrm{time}}$Siamese, $d_{\mathrm{G-DL}}$Siamese, $d_{\mathrm{G-ADP}}$Siamese, $d_{\mathrm{G-fuse}}$ Figure 15: CDF of absolute localization error for different dissimilarity metrics and manifold learning techniques computed on $\mathcal{S}_{\mathrm{NLoS}}$. TABLE IV: Evaluation of dissimilarity metrics on $\mathcal{S}_{\mathrm{NLoS}}$ Distance |  CT $\uparrow$ |  TW $\uparrow$ |  KS $\downarrow$ |  RD $\downarrow$
---|---|---|---|---
$d_{\mathrm{CS}}$ | $0.8362$ | $0.8221$ | $0.4338$ | $0.9781$
$d_{\mathrm{G-CS}}$ | $0.8285$ | $0.8141$ | $0.4349$ | $0.9752$
$d_{\mathrm{CIRA}}$ | $0.7599$ | $0.8546$ | $0.4174$ | $0.9691$
$d_{\mathrm{G-CIRA}}$ | $0.6025$ | $0.6066$ | $0.4693$ | $0.9869$
$d_{\mathrm{DL}}$ | $0.9649$ | $0.9497$ | $0.4450$ | $0.9611$
$d_{\mathrm{G-DL}}$ | $0.9679$ | $0.9508$ | $0.3210$ | $0.9295$
$d_{\mathrm{ADP}}$ | $0.9072$ | $0.9027$ | $0.4297$ | $0.9671$
$d_{\mathrm{G-ADP}}$ | $0.9516$ | $0.9299$ | $0.3521$ | $0.9490$
$d_{\mathrm{fuse}}$ | $0.9081$ | $0.9038$ | $0.4294$ | $0.9668$
$d_{\mathrm{G-fuse}}$ | $\mathbf{0.9823}$ | $\mathbf{0.9699}$ | $\mathbf{0.2204}$ | $\mathbf{0.8889}$
TABLE V: Channel Charting Performance on $\mathcal{S}_{\mathrm{NLoS}}$. Method | Metric |  CT $\uparrow$ |  TW $\uparrow$ |  KS $\downarrow$ |  MAE $\downarrow$ | Fig.
---|---|---|---|---|---|---
Isomap | $d_{\mathrm{G-DL}}$ | $0.9623$ | $0.9485$ | $0.3300$ | $1.955\,\mathrm{m}$ |
Isomap | $d_{\mathrm{G-ADP}}$ | $0.9387$ | $0.8904$ | $0.3759$ | $2.298\,\mathrm{m}$ |
Isomap | $d_{\mathrm{G-fuse}}$ | $\mathbf{0.9778}$ | $\mathbf{0.9664}$ | $\mathbf{0.2041}$ | $\mathbf{0.935\,m}$ |
Siamese | $d_{\mathrm{Euc}}$ | $0.9866$ | $0.9855$ | $0.0961$ | $0.419\,\mathrm{m}$ |
Siamese | $d_{\mathrm{G-DL}}$ | $0.9581$ | $0.9420$ | $0.3233$ | $1.297\,\mathrm{m}$ | 14(j)
Siamese | $d_{\mathrm{G-ADP}}$ | $0.9352$ | $0.8864$ | $0.3760$ | $2.291\,\mathrm{m}$ | 14(i)
Siamese | $d_{\mathrm{G-fuse}}$ | $0.9656$ | $0.9512$ | $0.2150$ | $1.042\,\mathrm{m}$ | 14(l)
Triplet | $d_{\mathrm{Euc}}$ | $0.9769$ | $0.9728$ | $0.1486$ | $0.701\,\mathrm{m}$ |
Triplet | $d_{\mathrm{G-DL}}$ | $0.9591$ | $0.9391$ | $0.2743$ | $1.867\,\mathrm{m}$ |
Triplet | $d_{\mathrm{G-ADP}}$ | $0.9477$ | $0.9162$ | $0.3894$ | $2.292\,\mathrm{m}$ |
Triplet | $d_{\mathrm{G-fuse}}$ | $0.9633$ | $0.9423$ | $0.2768$ | $1.323\,\mathrm{m}$ |
Triplet | $d_{\mathrm{time}}$ | $0.9704$ | $0.9576$ | $0.2795$ | $1.341\,\mathrm{m}$ | 14(k)

##  VIII Conclusion

With our ADP-based metric, dissimilarity learning and the concept of fused
metrics, we propose three novel metrics for channel charting. An extensive
study on a publicly available dataset demonstrates that we are capable of
reconstructing the global geometry of the environment, outperforming
previously proposed metrics. Using our fused metric and a Siamese neural
network, we achieve a mean absolute localization error of $0.542\,\mathrm{m}$
on the whole dataset when assuming an ideal coordinate transformation from
channel chart to spatial coordinates. With a mean absolute error of
$1.042\,\mathrm{m}$, performance is acceptable even for the NLoS dataset with
two missing antenna arrays. Improving methods to find the ideal transformation
from channel chart to spatial coordinates, enabling absolute localization, is
one potential topic for future research. This work highlights the significance
of the dissimilarity metric, illustrating that the quality of dissimilarities
is at least as, if not more impactful than the applied manifold learning
technique. We encourage others to experiment on our publicly available dataset
and to propose even better metrics or techniques.

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  *[mMIMO]: massive multiple-input multiple-output
  *[EM]: electromagnetic
  *[BS]: base station
  *[JCaS]: Joint Communication and Sensing
  *[CSI]: channel state information
  *[LoS]: line-of-sight
  *[NLoS]: non-line-of-sight
  *[UE]: user equipment
  *[PCA]: principal component analysis
  *[SM]: Sammon’s mapping
  *[FCF]: forward charting function
  *[FCFs]: forward charting function
  *[CMD]: correlation matrix distance
  *[CS]: cosine similarity
  *[CIRA]: channel impulse response amplitude
  *[ADP]: angle-delay profile
  *[BSs]: base station
  *[ mMIMO]: massive multiple-input multiple-output
  *[OFDM]: orthogonal frequency division multiplex
  *[DFT]: discrete Fourier transform
  *[CIR]: channel impulse response
  *[DNN]: deep neural network
  *[GNSS]: global navigation satellite system
  *[MPCs]: multipath component
  *[UEs]: user equipment
  *[NMSE]: normalized mean squared error
  *[MSE]: mean squared error
  *[MDS]: multidimensional scaling
  *[t-SNE]: t-distributed stochastic neighbor embedding
  *[DNNs]: deep neural network
  *[CT]: continuity
  *[TW]: trustworthiness
  *[KS]: Kruskal’s stress
  *[RD]: Rajski’s distance
  *[MAE]: mean absolute error
  *[DICHASUS]: Distributed Channel Sounder by University of Stuttgart
  *[CIRs]: channel impulse response
  *[CDF]: cumulative distribution function
  *[CDFs]: cumulative distribution function