# Context-Aware CSI Tracking and Path Loss Prediction Using Machine Learning
and Dynamical Systems

Anis Hamadouche and Mathini Sellathurai  Anis Hamadouche and Mathini
Sellathurai are with the School of Engineering & Physical Sciences, Heriot-
Watt University, Edinburgh EH14 4AS, UK (e-mail:
{ah225,ah2161,m.sellathurai@}@hw.ac.uk).

###### Abstract

In this paper, we present an advanced model for Channel State Information
(CSI) tracking, leveraging a dynamical system approach to adapt CSI
dynamically based on exogenous contextual information. This methodology allows
for continuous updates to the Channel Knowledge Map (CKM), enhancing
communication reliability and responsiveness in dynamic environments. To
generate realistic and comprehensive datasets for training and evaluation, we
developed a new MATLAB simulator that models radio wave propagation in urban
environments. We address the challenge of real-time CKM adaptation using
online learning of the Koopman operator, a technique that forecasts channel
behavior by exploiting dynamical system properties. Our approach supports
real-time updates with high accuracy and efficiency, as demonstrated by
experiments with varying window sizes for the Koopman Autoencoder model. A
window size of 100 was found to offer the best balance between prediction
accuracy (RMSE: 1.8323 ± 1.1071, MAE: 0.3780 ± 0.2221) and computational
efficiency (training time: 231.1 ± 82.5 ms, prediction time: 109.0 ± 55.7 ms).
Additionally, we introduce a moving window mechanism to address privacy and
security concerns by updating the Koopman operator within the window and
purging input data thereafter, minimizing data retention and storage risks.
This ensures the CKM remains accurate and relevant while maintaining stringent
data privacy standards. Our findings suggest that this approach can
significantly improve the resilience and security of communication systems,
making them highly adaptable to environmental changes without compromising
user privacy.

###### Index Terms:

Channel State Information; Channel Knowledge Map; Dynamic Systems Theory;
Privacy-preserving techniques; Data assimilation; Machine learning; Contextual
information; Wireless communication systems; CSI tracking; Koopman Operator;
cellular-connected UAV; UAV-assisted communication.

##  I INTRODUCTION

Adjusting transmission parameters like power and rate in response to
fluctuating channel conditions can markedly enhance the efficiency of wireless
communication systems, as demonstrated by Knopp and Humblet [3]. Nevertheless,
obtaining perfect channel state information (CSI) is impractical, typically
due to estimation inaccuracies and feedback delays, complicating this
enhancement [6]. Modern transceivers modify their power and coding strategies
based on channel measurements to capitalize on advantageous conditions. Yet,
the use of imprecise CSI can lead to severe channel outages [6].

With recent advancements in antenna array technology, the channel dimension in
wireless communications is poised for substantial expansion. This is
particularly evident in the transition from 5G’s massive MIMO technology to
6G’s XL-MIMO, predicting a rise in the number of base station antennas from
the current average of 64 to potentially thousands. Similarly, on the user
equipment side, advancements in antenna technology and the shift to higher
frequencies suggest a comparable increase. The bandwidth in 6G systems might
extend from the current several hundred MHz to GHz levels, with frequencies
ranging up to the Terahertz band. This expansion could lead to a significant
rise in the number of subcarriers, potentially enhancing the channel dimension
by up to three orders of magnitude. However, factors like higher user mobility
and more complex environments heighten the difficulty and cost of accurately
estimating channel matrices [14].

To tackle these challenges, a novel approach known as the channel knowledge
map (CKM) has been proposed. Unlike a physical environment map, the CKM is
designed to mirror the specific wireless channel characteristics of a
location, containing site-specific data tagged with transmitter and receiver
positions. This database aids in enhancing environmental awareness and could
reduce or eliminate the need for intricate CSI acquisition [13].

However, wireless environments are characterized by their high dynamism, which
causes wireless channels to fluctuate rapidly. In response, it is vital to
refine CKM-based, environment-aware communication strategies that can adeptly
adjust to these environmental shifts. A promising method is to integrate
additional sensory information with machine learning techniques to improve the
stability and accuracy of CKM against environmental variations. Specifically,
the incorporation of advanced sensing and vision technologies—such as radar,
LiDAR, and cameras—provides extra data about the dynamic surroundings,
including tracking the movement and positions of objects and individuals. This
enriched sensory data, when combined with CKM, enables the effective inference
of channel knowledge even in unstable conditions. Moreover, employing physical
maps can assist CKM in discerning between significant and minor environmental
alterations, facilitating timely and relevant updates to the CKM [14].

A critical aspect of maintaining the Channel Knowledge Map (CKM) is its need
for continuous updates as new data arrives. The principal challenge in this
online map reconstruction is to update the CKM efficiently without needing to
recompute the entire map each time new data is incorporated. A viable approach
to this challenge is through the use of incremental learning algorithms, which
facilitate efficient updates by integrating new data into the existing
framework without necessitating a full retraining of the model [5].

Moreover, the use of user location data in environment-aware communications
like CKM poses significant privacy risks. If mismanaged, such data could
reveal sensitive personal information, including details about a user’s daily
routine, whereabouts, and social interactions. Addressing these privacy
concerns is critical before CKM and similar technologies can be practically
and safely implemented. It is essential to develop and integrate robust
privacy-preserving techniques that safeguard user data against unauthorized
access and misuse.

Dynamical systems theory has been effectively used for decades to enhance
modeling capabilities, simplify and clarify models, and improve forecasting
accuracy. Starting in the 1960s, Rudolf Kalman established a robust
mathematical framework [1, 2] that enabled the combination of empirical data
and theoretical models through data assimilation techniques [4, 7]. These
techniques are particularly valuable for applications in forecasting and
control, showcasing the longstanding relationship between streaming data and
dynamical models, spanning nearly seven decades. In contemporary settings, it
is increasingly common to construct dynamical models directly from data using
machine learning approaches [4, 8, 10, 9]. This approach is crucial in complex
systems where traditional first-principles models may be inadequate or when
the appropriate state-space variables are unclear.

In this work, we address the first issue by adapting CKM for dynamic
environments through a dynamical system approach with CSI as a forced dynamics
by an extrogeneous contextual information. This strategy ensures that the
channel knowledge is continuously updated to reflect real-time changes, thus
improving the reliability and effectiveness of communications in highly
dynamic environments. Secondly, we address the second limitation by online-
learning of the Koopman operator. This method leverages the properties of
dynamical systems to learn and predict channel behavior over time, allowing
for real-time updates to the CKM as new environmental data becomes available.
This approach optimizes the efficiency of updates while ensuring that the CKM
remains accurate and reflective of the current state of the environment.
Finally, to mitigate these privacy and security issues, we propose a solution
that involves using a moving window for Koopman operator learning. In this
approach, the Koopman operator is updated within this window, after which the
input data is immediately purged from the system. This method ensures that
personal data is not retained longer than necessary, thereby reducing the risk
of privacy breaches while still allowing for the dynamic updating of the CKM
based on recent and relevant data. This approach strikes a balance between
leveraging real-time data for environmental awareness and maintaining strict
data privacy protocols.

##  II PROPOSED METHOD

In this section, we mathematically describe the channel state information
(CSI) tracking problem as a nonlinear dynamical system and solve it using the
Koopman operator combined with classical Kalman filter.

###  II-A Contextually-forced Nonlinear CSI dynamics

Let $\mathbf{h}_{k}\in\mathbb{C}^{N_{t}\times N_{r}}$ represent the CSI at
sampling instant $k$, which evolves according to a nonlinear dynamical
equation influenced by contextual information
$\mathbf{x}_{k}\in\mathbb{C}^{M}$ (e.g., location of transmitter(s) and
receiver(s), temperature, water denisty, surface material type, permittivity
and conductivity…etc.). The real-time CSI can be expressed as:

$\mathbf{h}_{k+1}=f(\mathbf{h}_{k},\mathbf{x}_{k})+\mathbf{w}_{k}$ (1)

where $f$ is a nonlinear function that incorporates the transmit power, the
channel gain, the antenna gain and fading as functions of contextual
information $\mathbf{x}_{k}\in\mathbb{C}^{M}$, and
$\mathbf{w}_{k}\in\mathbb{C}^{N_{t}\times N_{r}}$ represents the modelling
error, typically modeled as Gaussian noise $\mathcal{N}(0,\mathbf{Q}_{k})$.

Note that different from conventional model-based approaches for which the CSI
tracking are based on specific antenna and static channel models, the proposed
model (1) is applicable to any antenna and channel configuration [11].

Koopman theory suggests that any nonlinear dynamical system can be represented
by a linear operator acting on an infinite-dimensional Hilbert space spanned
by some observable function $\phi$.

In our case, the Koopman operator, $\mathcal{K}$, advances observable
functions that are dependent on the CSI and contextual data

$\phi(\mathbf{h}_{k+1},\mathbf{x}_{k+1})=\phi(f(\mathbf{h}_{k},\mathbf{x}_{k}),\mathbf{x}_{k+1})=\mathcal{K}\phi(\mathbf{h}_{k},\mathbf{x}_{k})$

This equation represents how the observable function $\phi$ evolves as the
system transitions from one CSI state to another, influenced by the contextual
changes.

Koopman theory suggests that within the observable space of nonlinear
dynamical systems with discrete spectra, there exists a finite-dimensional
subspace that is invariant under the Koopman operator. This subspace is termed
Koopman invariant subspaces. In this work we assume that the dimension of this
subspace is $d$. Consequently, within this subspace, the Koopman operator is
represented by a linear matrix ($\mathbf{K}\in\mathbb{R}^{d\times d}$), and
the observable function simplifies to a finite-dimensional map
$\Phi:\mathbb{C}^{N_{t}\times
N_{r}}\times\mathbb{C}^{M}\rightarrow\mathbb{C}^{d}$. Using the known
historical state CSI ${\mathbf{h}}$ and contextual information $\mathbf{x}$
data, the matrix $\mathbf{K}$ can be computed. Figure 1 depicts the context-
driven CSI dynamical system concept in parallel with Koopman operator learning
process:

Figure 1: Context-driven CSI dynamics and Koopman operator learning process.

Let ${z}_{k,i}$ be the observable that is related to the CSI and contextual
information by a nonlinear observable function $\phi_{i}(\cdot)$, for
$i=1,\dots,L$:

${z}_{k,i}=\phi_{i}(\mathbf{h}_{k},\mathbf{x}_{k})$

Let $\Phi:\mathbb{C}^{N_{t}\times
N_{r}}\times\mathbb{C}^{M}\rightarrow\mathbb{C}^{d}$ be a vector of observable
functions (where $d>\max\\{N_{t}\times N_{r},M\\}$), which transforms the
state $\mathbf{h}_{k}$ and contextual information $\mathbf{x}_{k}$ into a
higher-dimensional space
$\mathbf{z}_{k}=\Phi(\mathbf{h}_{k},\mathbf{x}_{k})\in\mathbb{C}^{d}$. This
observable includes nonlinear transformations such as polynomial terms,
interactions, and trigonometric functions [12].

The effectiveness of the Koopman operator approach largely hinges on the
appropriate selection of the observable function $\phi_{i}$. This function
should be complex enough to capture the inherent nonlinearities of the
variables $\mathbf{h}_{k}$ and $\mathbf{x}_{k}$, typically involving
polynomial expansions, interactions, trigonometric transformations, and
potentially other nonlinear forms to ensure a comprehensive representation of
the system’s dynamics in the lifted space. The dimension $d$ of this space
should be sufficiently large to encompass the dynamics, yet it should be
manageable to avoid computational burdens and the risk of overfitting. The
specific form of $\phi_{i}$ might require adjustments based on empirical data
analysis to tailor it to the dynamics observed, ensuring that the function
maintains mathematical properties such as differentiability and boundedness,
which are crucial for the stability and operability of the Koopman operator.

###  II-B Autonomous dynamics

By defining the augmented nonlinear function $\tilde{f}$ on an extended CKM
state space $\tilde{h}=[h^{T},x^{T}]^{T}$, the CKM dynamics are made
autonomous, making them independent of explicit contextual changes within the
state-transition function. This formulation is particularly useful when the
next contextual information $\mathbf{x}_{k+1}$ follows a predictable function
$H(\mathbf{x}_{k})$, or in cases where $\mathbf{x}_{k}$ is prescribed by a
feedback rule, i.e., $\mathbf{x}_{k}=\psi(\mathbf{h}_{k})$.

When contextual information $\mathbf{x}$ changes discretely or according to
specific rules, the Koopman operator simplifies the analysis by focusing on
the transitions under each fixed contextual input, separating these actions
from the CSI dynamics

$\mathcal{K}_{\bar{\mathbf{x}}}\Phi(\mathbf{h}_{k},\bar{\mathbf{x}})=\Phi(f(\mathbf{h}_{k},\bar{x}),\bar{\mathbf{x}})=\Phi(h_{k+1},\bar{\mathbf{x}})$

where $\bar{x}$ represents a constant or discrete contextual action. The
system is analyzed under each possible contextual input separately, allowing
for simpler modeling and optimization, especially when focusing on the timing
of switching between different contextual inputs.

###  II-C Data-driven approximation of Koopman operator

Assume we are using a moving window of size $N$ for learning, which includes
the most recent $N$ observations of $\mathbf{z}_{k}$. Mathematically, the
dataset within the window at any time step $k$ is represented by:

$\mathbf{D}_{k}=\begin{bmatrix}{z}_{k-N+1,1}&\ldots&\ldots&{z}_{k,1}\\\
{z}_{k-N+1,2}&\ldots&\ldots&{z}_{k,2}\\\ \vdots&\vdots&\vdots&\vdots\\\
{z}_{k-N+1,d}&\ldots&\ldots&{z}_{k,d}\end{bmatrix}$

In order to delete data, e.g. for privacy compliance, as new data
$[{z}_{k+1,1},\ldots,{z}_{k+1,L}]^{\top}$ arrives, the training dataset is
updated by removing the oldest data point
$[{z}_{k-N+1,1},\ldots,{z}_{k-N+1,L}]^{\top}$ and including the new data
point:

$\mathbf{D}_{k+1}=\begin{bmatrix}{z}_{k-N+2,1}&\ldots&\ldots&{z}_{k+1,1}\\\
{z}_{k-N+2,2}&\ldots&\ldots&{z}_{k+1,2}\\\ \vdots&\vdots&\vdots&\vdots\\\
{z}_{k-N+2,d}&\ldots&\ldots&{z}_{k+1,d}\end{bmatrix}$

This rolling update method ensures that data older than $N$ observations is
not retained in system memory. Once data exits the window, it is completely
purged from the system, eliminating any trace or residual information. This
approach effectively secures data privacy by ensuring that historical data
beyond the most current $N$ entries is not accessible or recoverable,
mitigating potential privacy breaches or misuse of outdated information.

In order to approximate the Koopman operator from past CSI and contextual
data, let $\mathbf{D}_{k}$ and $\mathbf{D}_{k+1}$ be the stacked data vectors
from observable functions evaluated at data points. Matrix $\mathbf{D}_{k}$
contains the observables evaluated at time $k$, while $\mathbf{D}_{k+1}$
contains the observables evaluated at time $k+1$. The Koopman matrix
$\mathbf{K}_{x}$ is then approximated by solving a least squares problem to
find $\mathbf{K}_{x}$ that best maps $\mathbf{D}_{k}$ to $\mathbf{D}_{k+1}$:

$\mathbf{K}_{x}\approx\mathbf{D}_{k+1}\mathbf{D}_{k}^{\dagger}$

where $\mathbf{D}_{k}^{\dagger}$ is the Moore-Penrose pseudoinverse of
$\mathbf{D}_{k}$.

###  II-D State estimation with Kalman filter

The evolution of $\mathbf{z}_{k}\in\mathbb{C}^{d}$ in the lifted space is
approximated by a linear dynamical system:

$\mathbf{z}_{k+1}\approx\mathbf{K}_{x}\mathbf{z}_{k}$ (2)

where $\mathbf{K}_{x}\in\mathbb{C}^{d\times d}$ is the Koopman matrix that
approximates the Koopman operator $\mathcal{K}$ in a finite dimensional space.

Assume noisy observations

$\tilde{\mathbf{z}}_{k+1}=\mathbf{z}_{k+1}+\mathbf{v}_{k}$

where $\mathbf{v}_{k}\in\mathbb{C}^{d}\sim\mathcal{N}(0,\mathbf{Q}_{k})$
represents the approximation/measurement error, which is assumed to be
Gaussian with zero mean and covariance matrix $\mathbf{Q}_{k}$, we design a
Kalman filter to estimate $\mathbf{z}_{k}$ from noisy observations
$\tilde{\mathbf{z}}_{k}$. This involves predicting and updating steps to
compute the best estimate $\hat{\mathbf{z}}_{k}$ and its error covariance.

To recover the estimated CSI $\hat{\mathbf{h}}_{k}$ from
$\hat{\mathbf{z}}_{k}$, we use the inverse of the observable function,
$\Phi^{-1}$, which might be explicit or approximated using techniques such as
pseudo-inversion or learning-based methods:

$\hat{\mathbf{h}}_{k}=\Phi^{-1}(\hat{\mathbf{z}}_{k}).$
$\hat{\mathbf{h}}_{k+m}=\Phi^{-1}(\hat{\mathbf{z}}_{k+m}).$

In this work we use an Autoencoder architecture to simultaneously learn the
observable function $\Phi$ and it inverse $\Phi^{-1}$ as discussed next.

###  II-E Learning observable functions

The encoder is a function $E:\mathbb{R}^{n}\rightarrow\mathbb{R}^{d}$, where
$n=N_{t}\times N_{r}+M$ is the dimensionality of the state space (involving
both CSI $\mathbf{h}_{k}$ and contextual information $\mathbf{x}_{k}$), and
$d$ is the dimensionality of the latent (observable) space
$\hat{\mathbf{z}}_{k}$. The encoder

$\hat{\mathbf{z}}_{k}=E(\mathbf{h}_{k},\mathbf{x}_{k};\theta_{E})$

approximates the mapping
$\hat{\mathbf{z}}_{k}=\Phi(\mathbf{h}_{k},\mathbf{x}_{k}$ where $\theta_{E}$
are the parameters of the encoder learned during training.

The dynamics of the system in the latent space are modeled by (2).

On the other hand, the decoder is a function
$D:\mathbb{R}^{d}\rightarrow\mathbb{R}^{n}$ that maps from the latent space
back to the original state space, approximating
$(\mathbf{h}_{k},\mathbf{x}_{k})=\Phi^{-1}(\mathbf{z}_{z})$, and is given by

$(\hat{\mathbf{h}}_{k},\hat{\mathbf{x}}_{k})=D(\hat{\mathbf{z}}_{k};\theta_{D})$

where $\theta_{D}$ are the parameters of the decoder learned during training.

The training of the autoencoder involves minimizing a loss function that
includes both reconstruction loss and dynamic consistency loss. The
reconstruction loss ensures the fidelity of the state space reconstruction,
and the dynamic consistency loss ensures that the predicted dynamics via the
Koopman operator aligns with actual future states as represented in the latent
space. The reconstruction loss is given by

$L_{\text{recon}}=\sum_{k}\|(\mathbf{h}_{k},\mathbf{x}_{k})-D(E(\mathbf{h}_{k},\mathbf{x}_{k};\theta_{E});\theta_{D})\|^{2}$

Define the dynamics Consistency Loss by

$L_{\text{dynamic}}=\sum_{k}\|\mathbf{K}_{x}E(\mathbf{h}_{k},\mathbf{x}_{k};\theta_{E})-E(\mathbf{h}_{k+1},\mathbf{x}_{k+1};\theta_{E})\|^{2}$

Combining the above two losses, the total Loss can be formulated as

$L_{\text{total}}=L_{\text{recon}}+\lambda L_{\text{dynamic}}$

where $\lambda$ is a regularization parameter balancing the two components of
the loss. $\lambda$ can be seen as a weight that promotes linearity or prior
information about the linear dynamics in the lifted space.

During training, the objective is to optimize the parameters $\theta_{E}$ and
$\theta_{D}$, to minimize the total loss $L_{\text{total}}$. We use ADAM
solver to adjust $\theta_{E}$ and $\theta_{D}$ iteratively.

After training, the autoencoder can be reconstruct the orginal CKM, i.e., CSI
and contextual information $(\mathbf{h}_{k},\mathbf{x}_{k})$ from the latent
representations to interpret or analyze the dynamics of the system. A
schematic diagram of this process is shown below.

Figure 2: Schematic diagram summarising the process of Koopman mapping and CSI
tracking with learnable observable functions.

By applying this approach, we can transform a challenging nonlinear estimation
problem into a more tractable linear problem in an augmented state space,
leveraging powerful linear system tools such as the Kalman filter for
efficient state tracking. The pseudo-code of the proposed algorithm is
presented in Algorithm 1.

Algorithm 1 Koopman Operator Based CSI Tracking with Nonlinear Dynamics

1:Input: Real-time measurements of contextual information
$\\{\mathbf{x}_{i}\\}_{k-N+1<i\leq k}$, CSI estimates
$\\{\hat{\mathbf{h}}_{i}\\}_{k-N+1<i\leq k}$, modelling and observation noise
covariances $\mathbf{Q}_{k}$ and $\mathbf{R}_{k}$.

2:Initialize:

3:Choose observable functions $\phi_{i}$ for $i=1,\dots,L$,

4:Initialize observable vector:
$\hat{\mathbf{z}}_{k|k}=\Phi(\hat{\mathbf{h}}_{k},\mathbf{x}_{k})$,

5:Initialize covariance $\mathbf{P}_{k|k}$,

6:Estimate Koopman Operator:

7:Choose sliding (moving) window size $N$,

8:Arrange $\\{(\hat{\mathbf{h}}_{i},\mathbf{x}_{i})\\}_{k-N+1<i\leq k}$ data
and form $\mathbf{D}_{k-1}$ and $\mathbf{D}_{k}$.

9:if $\mathbf{x}_{k-N+1}\neq\mathbf{x}_{k-N}$ then

10: Approximate the Koopman operator $\mathbf{K}_{x}$:
$\mathbf{K}_{x}\approx\mathbf{D}_{k}\mathbf{D}_{k-1}^{\dagger}$

11:end if

12:Kalman Filter Initialization:

13:Prediction:

14:Predicted state estimate:
$\hat{\mathbf{z}}_{k+1|k}=\mathbf{K}_{x}\hat{\mathbf{z}}_{k|k}$

15:Predicted CSI estimate:
$\hat{\mathbf{h}}_{k+1|k}=\mathbf{\Phi}^{-1}(\hat{\mathbf{z}}_{k+1|k})$

16:Predicted covariance:
$\mathbf{P}_{k+1|k}=\mathbf{K}_{x}\mathbf{P}_{k|k}\mathbf{K}_{x}^{\top}+\mathbf{Q}_{k}$

17:if $\mathbf{x}_{k+1}$ is available then:

18: ${\mathbf{z}}_{k+1}=\Phi(\hat{\mathbf{h}}_{k+1|k},\mathbf{x}_{k+1})$

19: Update:

20: Innovation:
$\mathbf{\tilde{y}}_{k+1}=\mathbf{{z}}_{k+1}-\hat{\mathbf{z}}_{k+1|k}$

21: Innovation covariance:
$\mathbf{S}_{k+1}=\mathbf{P}_{k+1|k}+\mathbf{R}_{k+1}$

22: Compute Kalman Gain:
$\mathbf{G}_{k+1}=\mathbf{P}_{k+1|k}\mathbf{S}_{k+1}^{-1}$

23: Update estimate:
$\hat{\mathbf{z}}_{k+1|k+1}=\hat{\mathbf{z}}_{k+1|k}+\mathbf{G}_{k+1}\tilde{y}_{k+1}$

24: Update CSI estimate:
$\hat{\mathbf{h}}_{k+1|k+1}=\mathbf{\Phi}^{-1}(\hat{\mathbf{z}}_{k+1|k+1})$

25: Update covariance:
$\mathbf{P}_{k+1|k+1}=(\mathbf{I}-\mathbf{G}_{k})\mathbf{P}_{k+1|k}$

26:end if

27:Output: Estimated channel state $\\{\hat{\mathbf{h}}_{k+1}\\}$

Note that the Koopman matrix $\mathbf{K}_{x}$ in Algorithm 1 only needs to be
re-estimated once a contextual change in $\mathbf{x}$ is detected (lines
9-11).

Also, the Kalman filter in Algorithm 1 alternates between two stages: the
prediction stage (lines 12-14) advances the state until the next planned
observation, and the update stage (lines 18-23) incorporates this observation
to enhance the CSI estimate. However, this alternation is not mandatory. For
instance, if an observation or contextual information is missing for any
reason, the update phase can be omitted and several prediction steps can be
carried out consecutively according to

$z_{m_{2}}\approx\mathbf{K}_{x}^{m_{2}-m_{1}}\mathbf{z}_{m1}$

Similarly, if multiple independent observations are available simultaneously,
several updates can be conducted in succession in the update stage of the
Kalman filter.

##  III EXPERIMENTAL DATA AND RESULTS

The simulation dataset was generated using a custom MATLAB simulator designed
to model radio wave propagation in an urban environment. This setup leverages
Google Maps API for realistic geographical data and MATLAB’s RF toolbox to
simulate environmental and signal propagation characteristics.

In this experiment we consider an UAV-assisted communication in an urban site
with receivers in non-line-of-sight as depicted below. The receiver is
positioned at a specific latitude and longitude (51.50216, -0.01769) in the
simulation environment, with its antenna placed at a height of 1 meter above
the ground.

Details about the simulation model and data acquisition process are deferred
to Appendix -A.

Figure 3: Snapshot 1: UAV BS to UE propagation path using ray tracing (Jubilee
Park, Canary Wharf, London, UK) Figure 4: Snapshot 4: UAV BS to UE propagation
path using ray tracing (Jubilee Park, Canary Wharf, London, UK)

The real-time building, clutter and weather data recordings are depicted
below.

Figure 5: Contextual data acquisition.

The collected data from the MATLAB simulator was then used to train and
evaluate a Koopman Autoencoder model in Python.

In the Python setup, real-time training is performed using a sliding window
approach on the dataset generated by the MATLAB simulator. The data is scaled,
and the Koopman Autoencoder model is initialized. For each window size (50,
100 or 200), the dataset is divided into overlapping windows, where each
window is used to train the model for a fixed number of epochs. Training is
performed on a computational device featuring an Intel(R) Xeon(R) CPU @
2.20GHz with 2 physical cores and 2 logical processors. The system includes
12.67 GB of RAM, with 9.94 GB available.

After training on each window, predictions are made, and performance metrics
such as Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) are
computed. Additionally, training and prediction times are recorded. The
average and standard deviation of these metrics across all windows are
calculated to evaluate the model’s performance comprehensively.

The results of this experiment are summarised in Table I below and the PL
tracking is depicted in Figure 6

Window Size | RMSE | MAE | Training Time (ms) | Prediction Time (ms)
---|---|---|---|---
50 | 3.3124 ± 2.3502 | 0.6884 ± 0.5416 | 131.6 ± 19.8 | 77.8 ± 13.9
100 | 1.8323 ± 1.1071 | 0.3780 ± 0.2221 | 231.1 ± 82.5 | 109.0 ± 55.7
200 | 2.6367 ± 1.3615 | 0.5558 ± 0.2896 | 388.3 ± 68.1 | 103.6 ± 23.1

TABLE I: Performance Metrics for Different Window Sizes Figure 6: Path Loss
prediction and tracking with window size of 50.

Table I presents the performance metrics for the Koopman Autoencoder model
trained with different window sizes (50, 100, 200). The Root Mean Squared
Error (RMSE) and Mean Absolute Error (MAE) are key metrics that assess the
model’s prediction accuracy, while the training and prediction times (measured
in milliseconds) indicate the computational efficiency.

For a window size of 50, the model achieved an RMSE of 3.3124 ± 2.3502 and an
MAE of 0.6884 ± 0.5416, with training and prediction times of 131.6 ± 19.8 ms
and 77.8 ± 13.9 ms, respectively. Increasing the window size to 100 improved
the accuracy, reducing the RMSE to 1.8323 ± 1.1071 and the MAE to 0.3780 ±
0.2221. However, this came with increased computational costs, as the training
time rose to 231.1 ± 82.5 ms and prediction time to 109.0 ± 55.7 ms. For the
largest window size of 200, the RMSE and MAE values were 2.6367 ± 1.3615 and
0.5558 ± 0.2896, respectively, with training and prediction times of 388.3 ±
68.1 ms and 103.6 ± 23.1 ms.

Figure 6 shows PL prediction and tracking with a window size of 50. In this
figure, for a simulation recording of 43 samples, the model demonstrates a
high level of accuracy and adaptivity. Although there are slight fluctuations,
especially during transitions beyond sample index 79, the model tracks the PL
accurately when the PL is steady. This illustrates the model’s capability to
handle real-time data effectively, maintaining precision despite rapid changes
and achieving near-perfect tracking when conditions stabilize.

These results suggest a trade-off between window size and performance. Smaller
window sizes lead to faster training and prediction times but less accurate
predictions. Larger window sizes improve prediction accuracy at the cost of
increased computational time. The window size of 100 strikes a balance,
offering the lowest RMSE and MAE, though with higher variance in computational
times. This indicates that for applications requiring more accurate
predictions, a moderate window size like 100 may be optimal, whereas smaller
window sizes may be preferred for scenarios where computational efficiency is
critical.

##  IV CONCLUSION

In this work, we have developed an advanced model for Channel State
Information (CSI) tracking, leveraging the dynamical system approach with CSI
treated as dynamics influenced by exogenous contextual information, ensuring
real-time adaptation to environmental changes. This method effectively allows
the Channel Knowledge Map (CKM) to be continuously updated, thus enhancing the
reliability and responsiveness of communications in dynamic settings. To
generate the datasets necessary for training and evaluation, we created a new
MATLAB simulator that accurately models radio wave propagation in urban
environments. We tackled the challenge of adapting CKM to these environments
through online learning of the Koopman operator, a technique that exploits the
properties of dynamical systems to forecast channel behavior. This online-
learning approach not only facilitates real-time updates as new environmental
data emerge but also improves the efficiency of these updates, maintaining the
accuracy and relevance of the CKM.

Our experiments demonstrate the effectiveness of this method using varying
window sizes for training the Koopman Autoencoder model.

To address significant privacy and security concerns, we introduced a moving
window for learning the approximate Koopman operator. In this model, the
Koopman operator is updated within this window, and subsequently, the input
data is completely purged from the system. This strategy minimizes data
retention and limits storage, significantly reducing the risk of privacy
breaches while still allowing the CKM to be dynamically updated with recent
and relevant data. By balancing the need for real-time environmental awareness
with stringent data privacy measures, this approach yields resilient and
secure communication systems that adapt swiftly to changes without
compromising user privacy.

###  -A Experimental Setup

This setup leverages Google Maps API for realistic geographical data and
MATLAB’s propagation modeling tools to simulate environmental and signal
propagation characteristics. Below is a detailed description of the simulator
and the conducted experiment.

####  -A1 Overview

The MATLAB simulator initializes with the configuration of a transmitter (TX)
and receiver (RX) located in an urban setting, specifically in Canary Wharf,
London. The TX and RX positions are defined by their geographical coordinates
(latitude and longitude). The transmitter is positioned at a latitude of
51.50375 and a longitude of -0.01843, while the receiver is located at
51.50216 latitude and -0.01769 longitude. These coordinates are then converted
to Cartesian coordinates for further processing.

A `siteviewer` object is used to visualize the buildings in the simulation
area using OpenStreetMap data. The simulation plots various key performance
indicators (KPIs) such as Transmitter Power, Receiver Power, Path Loss,
Temperature, Pressure, Rain Rate, Clutter, Street Width, Mean Clutter Height,
and Standard Deviation of Clutter Height.

####  -A2 Dynamics and propagation model

The TX is simulated to move in small steps from its starting position to an
end position over 1000 steps, updating its latitude and longitude
incrementally. Propagation models are initialized to simulate the effects of
ray tracing, gas absorption, and rain attenuation on the radio waves. The
`raytracing` model accounts for single-bounce reflections and diffractions
using buildings and terrain materials defined as concrete. The gas and rain
models introduce atmospheric effects like temperature, water density, air
pressure, and rain rate.

####  -A3 Contextual data acquisition

Environmental factors such as temperature, water density, and air pressure are
simulated with slight random variations to introduce realistic fluctuations.
Rain rate is also varied randomly. The `raytrace` function is used to compute
the received signal strength at the RX given these propagation conditions.

Buildings around the TX and RX are identified using the Google Maps API, and
their elevations are calculated. The simulator calculates the average street
width and clutter density along the direct path between TX and RX. Clutter
density is determined by the number of buildings intersecting the direct path.

####  -A4 Visualisation

The simulation collects data over 1000 steps, recording metrics such as TX
power, RX power, path loss, temperature, water density, pressure, rain rate,
clutter density, street width, and heights of intersecting buildings. This
data is saved into a CSV file for subsequent analysis.

During each simulation step, the KPIs are plotted in real-time to visualize
the changes dynamically as depicted in Figure 3. This helps in understanding
how different environmental and propagation factors affect the signal strength
and path loss.

###  -B Time complexity

To derive the time complexity of the Koopman Operator Based CSI Tracking
algorithm with respect to the CSI dimension $N_{t}\times N_{r}$ (where $N_{t}$
and $N_{r}$ are the number of transmit and receive antennas, respectively),
the dimension $M$ of the contextual inputs $\mathbf{x}_{k}$, and the window
size $N$, we need to consider several factors.

The CSI vector $\mathbf{h}_{k}$ can be considered to have a dimension of
$N_{t}\times N_{r}$ due to the number of transmit and receive antenna
combinations. The contextual input vector $\mathbf{x}_{k}$ has a dimension of
$M$. The observable function $\Phi$ transforms the combined input of CSI and
contextual data into a higher dimensional space. If $\Phi$ involves linear or
polynomial mappings, the dimension of the lifted space can grow substantially.
Let us denote the dimension of the lifted space as $d$, which can depend non-
linearly on $N_{t}\times N_{r}$ and $M$, potentially as a polynomial function,
e.g., $d=c(N_{t}\times N_{r}+M)^{p}$ for some constants $c$ and $p$, depending
on the complexity of $\Phi$.

The Koopman operator approximation involves estimating matrix $\mathbf{K}_{x}$
based on the data within the moving window. Each entry in $\mathbf{K}_{x}$ is
computed from regression on $d$ dimensional data collected over $N$ data
points within the window. For each update, a regression task is performed,
which typically involves operations on the order of $O(d^{2}N)$ due to matrix
multiplication and inversion processes required for parameter estimation. This
is executed only when there is a contextual change in $\mathbf{x}$.

The Kalman filter steps involve matrix operations on the lifted state space
dimension $d$. Specifically, prediction and update steps involve matrix
multiplications and possibly inversion, contributing a complexity of
$O(d^{3})$ per timestep due to matrix operations in higher dimensions.

Given the above considerations, the time complexity for each update cycle
within the algorithm can be approximated as: $O(d^{2}N)$ for each update
within the moving window of Koopman operator estimation. $O(d^{3})$ for each
Kalman filter update which is dominated by matrix operations in the lifted
space.

If $d$ scales as $c(N_{t}\times N_{r}+M)^{p}$, then the overall complexity per
update cycle, taking into account both Koopman operator estimation and Kalman
filter operations, can be approximated as:

$O\left((N_{t}\times N_{r}+M)^{2p}N+(N_{t}\times N_{r}+M)^{3p}\right)$ (3)

This complexity suggests that the performance of the algorithm is highly
sensitive to the number of antennas (both transmit and receive), the dimension
of contextual information, and the chosen complexity and size of the
observable function $\Phi$, along with the size of the data window $N$. For
practical implementation, considerations must be made regarding the scaling of
$N_{t}$, $N_{r}$, and $M$ to ensure that the computations remain feasible for
real-time applications.

Ignoring the Kalman filtering step we obtain a quadratic time complexity in
the number of antennas and users and linear in terms of data samples:

$O\left((N_{t}\times N_{r}+M)^{2p}N\right).$ (4)

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