DSRC

    dedicated short-range communications
C-ITS

    cooperative intelligent transport system
RSU

    road side unit
Bi

    bi-directional
TR

    throughput
TDL

    tapped delay line
SRNN

    Simple RNN
ITS

    Intelligent Transportation Systems
IEEE

    Institute of Electrical and Electronics Engineers
WAVE

    Wireless Access in Vehicular Environment
V2V

    vehicle-to-vehicle
V2I

    vehicle-to-infrastructure
CCH

    control channel
SCH

    service channels
STS

    short training symbols
LTS

    long training symbols
SS

    signal symbol
SoA

    state-of-the-art
DPA

    data-pilot aided
STA

    spectral temporal averaging
EL

    ensemble learning
FL

    frame length
CDP

    constructed data pilots
TRFI

    time-domain reliable test frequency domain interpolation
MMSE-VP

    minimum mean square error using virtual pilots
iCDP

    improved CDP
SBS

    symbol-by-symbol
FBF

    frame-by-frame
E-TRFI

    Enhanced TRFI
SR-CNN

    super resolution CNN
DN-CNN

    denoising CNN
RBF

    radial basis function
CNN

    convolutional neural network
TS-ChannelNet

    temporal spectral ChannelNet
WSSUS

    wide-sense stationary uncorrelated scattering
TDR

    transmission data rate
LSTM

    long short-term memory
ALS

    accurate LS
SLS

    simple LS
ChannelNet

    channel network
ADD-TT

    average decision-directed with time truncation
WI

    weighted interpolation
DD

    decision-directed
SR-ConvLSTM

    super resolution convolutional long short-term memory
RS

    reliable subcarriers
URS

    unreliable subcarriers
AE-DNN

    auto-encoder deep neural network
AE

    auto-encoder
T-DFT

    truncated discrete Fourier transform
TA-TDFT

    temporal averaging T-DFT
TA

    time averaging
PDP

    power delay profile
1G

    first generation
2G

    second generation
3G

    third generation
3GPP

    Third Generation Partnership Project
4G

    fourth generation
5G

    fifth generation
802.11

    IEEE 802.11 specifications
A/D

    analog-to-digital
ADC

    analog-to-digital
AM

    amplitude modulation
AP

    access point
AR

    augmented reality
ASIC

    application-specific integrated circuit
ASIP

    Application Specific Integrated Processors
AWGN

    additive white Gaussian noise
BCJR

    Bahl, Cocke, Jelinek and Raviv
BER

    bit error rate
BFDM

    bi-orthogonal frequency division multiplexing
BPSK

    binary phase shift keying
BS

    base stations
CA

    carrier aggregation
CAF

    cyclic autocorrelation function
Car-2-x

    car-to-car and car-to-infrastructure communication
CAZAC

    constant amplitude zero autocorrelation waveform
CB-FMT

    cyclic block filtered multitone
CCDF

    complementary cumulative density function
CDF

    cumulative density function
CDMA

    code-division multiple access
CFO

    carrier frequency offset
CIR

    channel impulse response
CM

    complex multiplication
COFDM

    coded-OFDM
CoMP

    coordinated multi point
COQAM

    cyclic OQAM
CP

    cyclic prefix
CR

    cognitive radio
CRC

    cyclic redundancy check
CRLB

    Cramér-Rao lower bound
CS

    cyclic suffix
CSI

    channel state information
CSMA

    carrier-sense multiple access
CWCU

    component-wise conditionally unbiased
D/A

    digital-to-analog
D2D

    device-to-device
DAC

    digital-to-analog
DC

    direct current
DFE

    decision feedback equalizer
DFT

    discrete Fourier transform
DL

    deep learning
DMT

    discrete multitone
DNN

    deep neural network
FNN

    feed-forward neural network
DSA

    dynamic spectrum access
DSL

    digital subscriber line
DSP

    digital signal processor
DTFT

    discrete-time Fourier transform
DVB

    digital video broadcasting
DVB-T

    terrestrial digital video broadcasting
DWMT

    discrete wavelet multi tone
DZT

    discrete Zak transform
E2E

    end-to-end
eNodeB

    evolved node b base station
E-SNR

    effective signal-to-noise ratio
EVD

    eigenvalue decomposition
FBMC

    filter bank multicarrier
FD

    frequency-domain
FDD

    frequency-division duplexing
FDE

    frequency domain equalization
FDM

    frequency division multiplex
FDMA

    frequency-division multiple access
FEC

    forward error correction
FER

    frame error rate
FFT

    fast Fourier transform
FIR

    finite impulse response
FM

    frequency modulation
FMT

    filtered multi tone
FO

    frequency offset
F-OFDM

    filtered-OFDM
FPGA

    field programmable gate array
FSC

    frequency selective channel
FS-OQAM-GFDM

    frequency-shift OQAM-GFDM
FT

    Fourier transform
FTD

    fractional time delay
FTN

    faster-than-Nyquist signaling
GFDM

    generalized frequency division multiplexing
GFDMA

    generalized frequency division multiple access
GMC-CDM

    generalized multicarrier code-division multiplexing
GNSS

    global navigation satellite system
GS

    guard symbols
GSM

    Groupe Spécial Mobile
GUI

    graphical user interface
H2H

    human-to-human
H2M

    human-to-machine
HTC

    human type communication
I

    in-phase
i.i.d.

    independent and identically distributed
IB

    in-band
IBI

    inter-block interference
IC

    interference cancellation
ICI

    inter-carrier interference
ICT

    information and communication technologies
ICV

    information coefficient vector
IDFT

    inverse discrete Fourier transform
IDMA

    interleave division multiple access
IEEE

    institute of electrical and electronics engineers
IF

    intermediate frequency
IFFT

    inverse fast Fourier transform
IoT

    Internet of Things
IOTA

    isotropic orthogonal transform algorithm
IP

    internet protocole
IP-core

    intellectual property core
ISDB-T

    terrestrial integrated services digital broadcasting
ISDN

    integrated services digital network
ISI

    inter-symbol interference
ITU

    International Telecommunication Union
IUI

    inter-user interference
LAN

    local area netwrok
LLR

    log-likelihood ratio
LMMSE

    linear minimum mean square error
LNA

    low noise amplifier
LO

    local oscillator
LOS

    line-of-sight
LP

    low-pass
LPF

    low-pass filter
LS

    least squares
LTE

    Long Term Evolution
LTE-A

    LTE-Advanced
LTIV

    linear time invariant
LTV

    linear time-variant
LUT

    lookup table
M2M

    machine-to-machine
MA

    multiple access
MAC

    multiple access control
MAP

    maximum a posteriori
MC

    multicarrier
MCA

    multicarrier access
MCM

    multicarrier modulation
MCS

    modulation coding scheme
MF

    matched filter
MF-SIC

    matched filter with successive interference cancellation
MIMO

    multiple-input, multiple-output
MISO

    multiple-input single-output
ML

    machien learning
MLD

    maximum likelihood detection
MLE

    maximum likelihood estimator
MMSE

    minimum mean squared error
MRC

    maximum ratio combining
MS

    mobile stations
MSE

    mean squared error
MSK

    Minimum-shift keying
MSSS

    mean-square signal separation
MTC

    machine type communication
MU

    multi user
MVUE

    minimum variance unbiased estimator
NEF

    noise enhancement factor
NLOS

    non-line-of-sight
NMSE

    normalized mean-squared error
NOMA

    non-orthogonal multiple access
NPR

    near-perfect reconstruction
NRZ

    non-return-to-zero
OFDM

    orthogonal frequency division multiplexing
OFDMA

    orthogonal frequency division multiple access
OOB

    out-of-band
OQAM

    offset quadrature amplitude modulation
OQPSK

    offset quadrature phase shift keying
OTFS

    orthogonal time frequency space
PA

    power amplifier
PAM

    pulse amplitude modulation
PAPR

    peak-to-average power ratio
PC-CC

    parallel concatenated convolutional code
PCP

    pseudo-circular pre/post-amble
PD

    probability of detection
pdf

    probability density function
PDF

    probability distribution function
PFA

    probability of false alarm
PHY

    physical layer
PIC

    parallel interference cancellation
PLC

    power line communication
PMF

    probability mass function
PN

    pseudo noise
ppm

    parts per million
PRB

    physical resource block
PRB

    physical resource block
PSD

    power spectral density
Q

    quadrature-phase
QAM

    quadrature amplitude modulation
QoS

    quality of service
QPSK

    quadrature phase shift keying
R/W

    read-or-write
RAM

    random-access memmory
RAN

    radio access network
RAT

    radio access technologies
RC

    raised cosine
RF

    radio frequency
rms

    root mean square
RRC

    root raised cosine
RW

    read-and-write
SC

    single-carrier
SCA

    single-carrier access
SC-FDE

    single-carrier with frequency domain equalization
SC-FDM

    single-carrier frequency division multiplexing
SC-FDMA

    single-carrier frequency division multiple access
SD

    sphere decoding
SDD

    space-division duplexing
SDMA

    space division multiple access
SDR

    software-defined radio
SDW

    software-defined waveform
SEFDM

    spectrally efficient frequency division multiplexing
SE-FDM

    spectrally efficient frequency division multiplexing
SER

    symbol error rate
SIC

    successive interference cancellation
SINR

    signal-to-interference-plus-noise ratio
SIR

    signal-to-interference ratio
SISO

    single-input, single-output
SMS

    Short Message Service
SNR

    signal-to-noise ratio
STC

    space-time coding
STFT

    short-time Fourier transform
STO

    symbol time offset
SU

    single user
AI

    artificial intelligence
SVD

    singular value decomposition
TD

    time-domain
TDD

    time-division duplexing
TDMA

    time-division multiple access
TFL

    time-frequency localization
TO

    time offset
TS-OQAM-GFDM

    time-shifted OQAM-GFDM
XAI

    explainable artificial intelligence
UE

    user equipment
VTV-US

    Vehicle-to-Vehicle urban scenario
VTI-US

    Vehicle-to-Infrastructure urban scenario
UFMC

    universally filtered multicarrier
UL

    uplink
US

    uncorrelated scattering
USB

    universal serial bus
UW

    unique word
VLC

    visible light communications
VR

    virtual reality
WCP

    windowing and CP
ANN

    artificial neural network
GRU

    gated recurrent unit
RNN

    recurrent neural network
WHT

    Walsh-Hadamard transform
WiMAX

    worldwide interoperability for microwave access
WLAN

    wireless local area network
W-OFDM

    windowed-OFDM
WOLA

    windowing and overlapping
WSS

    wide-sense stationary
ZCT

    Zadoff-Chu transform
ZF

    zero-forcing
ZMCSCG

    zero-mean circularly-symmetric complex Gaussian
ZP

    zero-padding
ZT

    zero-tail

# Explainable AI for Enhancing Efficiency of DL-based Channel Estimation

Abdul Karim Gizzini, Yahia Medjahdi, Ali J. Ghandour, Laurent Clavier  Abdul
Karim Gizzini, Yahia Medjahdi, and Laurent Clavier are with the Centre for
Digital Systems, IMT Nord Europe, Institut Mines-Télécom, University of Lille,
France (e-mail: {abdul-karim.gizzini, yahia.medjahdi,laurent.clavier}@imt-
nord-europe.fr). Ali J. Ghandour is with the National Center for Remote
Sensing - CNRS, Lebanon (e-mail: aghandour@cnrs.edu.lb).

###### Abstract

The support of artificial intelligence (AI) based decision-making is a key
element in future 6G networks, where the concept of native AI will be
introduced. Moreover, AI is widely employed in different critical applications
such as autonomous driving and medical diagnosis. In such applications, using
AI as black-box models is risky and challenging. Hence, it is crucial to
understand and trust the decisions taken by these models. Tackling this issue
can be achieved by developing explainable AI (XAI) schemes that aim to explain
the logic behind the black-box model behavior, and thus, ensure its efficient
and safe deployment. Recently, we proposed a novel perturbation-based XAI-
CHEST framework that is oriented toward channel estimation in wireless
communications. The core idea of the XAI-CHEST framework is to identify the
relevant model inputs by inducing high noise on the irrelevant ones. This
manuscript provides the detailed theoretical foundations of the XAI-CHEST
framework. In particular, we derive the analytical expressions of the XAI-
CHEST loss functions and the noise threshold fine-tuning optimization problem.
Hence the designed XAI-CHEST delivers a smart input feature selection
methodology that can further improve the overall performance while optimizing
the architecture of the employed model. Simulation results show that the XAI-
CHEST framework provides valid interpretations, where it offers an improved
bit error rate performance while reducing the required computational
complexity in comparison to the classical DL-based channel estimation.

###### Index Terms:

6G, AI, XAI, channel estimation

##  I Introduction

Artificial intelligence (AI) is expected to play a crucial role in the overall
design of future networks. In particular, 6G will transform the classical
Internet of Things (IoT) to “connected intelligence”, by leveraging the power
of AI to connect billions of devices and systems worldwide [1]. This concept
is defined as native (AI) which is a key element that differentiates 6G
networks from the previous wireless networks. In native AI, distributed AI
will be embedded within the functionality of all layers [2] to support demands
for high data rates and low latency-critical applications.

Generally speaking, the AI-enabled intelligent architecture for 6G networks
defines several layers including the intelligent sensing layer [3]. It is
worth mentioning that robust environment monitoring and data detection are of
great interest in 6G smart applications like autonomous driving [4]. Note that
ensuring the reliability of the intelligent sensing layer is highly impacted
by the channel estimation accuracy since a precisely estimated channel
response influences the follow-up equalization and decoding operations at the
receiver, therefore, it affects the sensing accuracy [5]. In this context,
channel estimation is one of the major physical (PHY) layer issues due to the
doubly-selective nature of the channel in mobile applications. Conventional
channel estimation schemes such as least squares (LS) and linear minimum mean
square error (LMMSE) can not be practical in real-case scenarios. This is
because the LS channel estimation scheme ignores the presence of noise in the
estimation process and requires the transmission of a large number of pilots
which decreases the transmission data rate. Whereas, the LMMSE channel
estimator provides good performance assuming the prior knowledge of the
channel and noise statistics in addition to its high computational complexity.

###  I-A DL-based Channel Estimation

Recently, deep learning (DL) has been employed in the PHY layer of wireless
communications [6], including channel estimation [7, 8, 9, 10], due to its
ability in providing good performance-complexity trade-offs. Among different
DL networks, feed-forward neural networks have been widely used as a post-
processing unit following conventional channel estimators. In [11], the
authors proposed an end-to-end FNN-based scheme for channel estimation and
signal detection, where it directly detects the received bits from the
received signal. The proposed FNN model consists of $3$ hidden layers with
$500$, $250$, and $120$ neurons, respectively. We note that in this scheme the
FNN model is trained to predict 16 bits only, hence, several concatenated
models are needed according to the total number of transmitted bits. Using the
same FNN model proposed in [11], the authors in [12] proposed an FNN-based
channel estimation scheme that is used to predict the channel response using
the received signal, received pilots, and previously estimated channel.
Simulation results show that using the previously estimated channel as an FNN
input improves the channel estimation accuracy. Another FNN-based channel
estimation scheme has been proposed in [13], where LS channel estimation is
first applied and combined with the previously estimated channel to be fed as
an input to a $3$ hidden layer FNN consisting of $512$, $256$, and $128$
neurons, respectively. As reported in [13], employing LS as an FNN input
improves the channel estimation accuracy and provides a comparable performance
to the LMMSE channel estimation scheme.

To further improve the performance while preserving low computational
complexity, the authors in [7, 8, 9] tried a different strategy that is based
on improving the conventional channel estimation accuracy and employing low
complex FNN models as post-processing units. In [7], the authors proposed an
FNN-based channel estimation scheme that applies data-pilot aided (DPA)
channel estimation on top of LS channel estimation. After that a $3$ hidden
layer FNN consisting of $40$, $20$, and $40$ neurons, respectively is
utilized. Simulation results reveal that improving the initial channel
estimation allows the use of a low-complex FNN model while recording a
significant performance improvement in comparison to the conventional channel
estimation schemes. Similarly in [8] and [9] the authors proposed two FNN-
based channel estimation schemes that employ spectral temporal averaging (STA)
[14] and time-domain reliable test frequency domain interpolation (TRFI) [15]
channel estimation before the FNN model which consist of $3$ hidden layers
with $15$ neurons each. STA-FNN and TRFI-FNN outperform the DPA-FNN [7] while
recording a substantial computational complexity decrease.

In addition to FNN models, recurrent neural network (RNN) and convolutional
neural network (CNN) models have been also used within the channel estimation
by also trying to combine several inputs such as the received signal, pilots,
and initially estimated channel. RNN-based channel estimation schemes [16, 17,
18] can provide better channel tracking capability in comparison to the FNN-
based channel estimation. Whereas, CNN-based channel estimation [19, 20] is
used in the frame-by-frame channel estimation, where the previous, current,
and future pilots are employed in the channel estimation for each received
signal. Thus, improving the channel estimation accuracy with the cost of a
higher computational complexity as well as inducing high processing time in
comparison to the RNN and FNN-based channel estimation. We note that in this
work we focus on the FNN-based channel estimation since we are targeting low-
complex low-latency DL-based solutions.

Figure 1: XAI categories, concepts, and taxonomy.

There exist three main issues concerning the discussed channel estimation
schemes which can be defined as follows: (i) Identifying the relevant inputs:
as previously discussed, the majority of DL-based channel estimation schemes
use a combination of information as an input to the utilized DL model without
any clear criteria. Hence, is there a way to better select the DL model
inputs? (ii) High computational complexity: to guarantee good performance,
highly complex DL architectures are employed. However, motivated by the fact
that low-complex architectures are required in low-latency applications, so is
there a real need for such high-complex architectures? (iii) Trustworthiness:
Despite the good generalization and performance abilities offered by different
DL-based channel estimation schemes, they lack trustworthiness since they are
considered as “black box” models. Consequently, researchers and industrial
leaders are not able to trust the employment of these models in real-case
sensitive applications [21]. Therefore, is there a way to provide
interpretability to the decision-making strategy employed DL black box models?

The mentioned issues can be tackled by developing explainable artificial
intelligence (XAI) schemes that provide a reasonable explanation of the
decisions taken by black-box models. Thus, ensuring the transparency of the
employed models by transforming them from black-box into white-box models that
can be safely employed in practice. In the following paragraphs, we will
discuss the main XAI concepts, taxonomy, and deployment within the wireless
communication research domain.

###  I-B XAI Main Concepts, Categories, and Taxonomy

XAI defines four main concepts as shown in Figure 1: (i) Interpretability: is
based on the model design and it refers to how much the black-box model can be
understood by humans. For example, decision tree models have good
interpretability since a human can easily understand their logic. ii)
Explainability: is the ability to clarify and justify a particular prediction
performed by the black-box model. Hence, it aims to clarify the internal
functioning of the employed model. (iii) Trustworthiness: is the ability to
make professionals feel confident in the decisions taken by the black-box
model. (iv) Causality: is related to the generalization ability of the black-
box model, where models should be able to detect cause-effect relations and
adapt to environmental changes.

Generally speaking, XAI methods can be divided into two main categories [22]:
(i) Perturbation-based or gradient-free methods, where the concept is to
perturb input features by masking or altering their values and record the
effect of these changes on the model performance. (ii) Gradient-based methods
where the gradients of the output are calculated with respect to the input via
back-propagation and used to estimate importance scores of the input features.
Moreover, in terms of the provided explanations, the XAI methods can be
further classified into [23]:

  * •

Model-agnostic vs model-specific: Model-agnostic XAI schemes are independent
of the internal architecture of the black-box model including the weights and
the hidden layers. Whereas, model-specific schemes depend on a specific model
like FNN or CNN and can not be generalized to any other model. Therefore,
model-agnostic schemes are characterized by their high flexibility and can be
used despite the type of the considered model.

  * •

Local vs global: Local XAI schemes are those that generate explanations for a
group of samples, thus, they are highly dependent on the utilized dataset. In
contrast, global XAI schemes generate explanations that are related more to
the model behavior.

  * •

Pre-model, in-model, and post-model strategies: Knowing that the XAI schemes
can be applied throughout the entire development pipeline. Hence,
interpretability can be acquired in three main phases. Pre-modeling
explainability is used to define the useful features of the dataset for a
better representation. Hence, pre-modeling aims to perform exploratory data
analysis, explainable feature engineering, and dataset description. In
contrast, in-model explainability is to develop inherently explainable models
instead of generating black box models. Finally, the post-model explainability
method extracts explanations that are dependent on the model predictions.

###  I-C XAI for Wireless Communications

Wireless communications are still in the early stages of using XAI. The
majority of related works in the literature are surveys and reviews about the
guidelines and importance of using XAI in wireless communications. In [24] the
authors provided a review of the core concepts of XAI including definitions
and possible performance vs. explainability trade-offs. They mainly focused on
reviewing the recent DL-based schemes for the PHY and MAC layers and specified
the explainability level of the studied schemes which is in general low. In
[25] the authors proposed a novel XAI knowledge-powered framework for network
automation that effectively adapts to the dynamic changes of complex
communication systems as well as provides a human-understandable explanation.
The proposed XAI scheme aims to explain the decision-making for automated path
selection within the network.

The deployment of XAI in the open radio access network (O-RAN) was recently
surveyed in [26], where the authors performed a comprehensive survey on the
use of XAI to design a trustworthy and explainable O-RAN architecture.
Moreover, an explainable machine learning operations (MLOps) for streamlined
automation-native 6G networks has been proposed in [27, 28], where Shapley
additive explanations (SHAP) XAI scheme is employed to assign the features
importance [29]. We note that SHAP XAI scheme has been also employed for
short-term resource reservation in 5G networks [30] and energy-efficient
resource allocation, where the problem becomes more challenging [31, 32]. It
is worth mentioning that, the majority of DL-based resource allocation schemes
are based on deep reinforcement learning (DRL) where SHAP assigns importance
to the features used by the DRL agent at each state. These features could be
the number of active antennas, utilized bandwidth, number of connected users,
and the average data rate. We note that SHAP is useful in such applications,
however, it can not be efficiently used in channel estimation due to the
absence of predefined features and the high dimensionality of the DL input
vector.

In addition to network optimization and resource allocation, XAI has been
employed also in internet-of-things (IoT) networks. In [33] the authors
presented a comprehensive survey on XAI solutions for IoT systems including
the state-of-the-art past and ongoing research activities. In particular, they
focused on the XAI for IoT adaptive solutions using several architectures
based on 5G services, cloud services, and big data management. In [34] the
authors proposed a novel model-agnostic XAI scheme denoted as transparency
relying upon statistical theory (TRUST) for numerical applications. They
further tested the proposed TRUST scheme on cybersecurity of the industrial
IoT (IIoT). Simulation results show that TRUST scheme outperforms the local
interpretable model-agnostic explanations (LIME) scheme [35] in terms of
performance, speed, and explainability.

###  I-D Motivation and Contributions

To the best of our knowledge, the methodology of deploying XAI schemes in PHY
layer applications, specifically, channel estimation is still unclear. Noting
that the proposed XAI-based schemes for network optimization [27], resource
allocation [30], and secured IoT [33] can not be adapted to the PHY layer
applications because in such applications there are no clear discriminative
features within the model inputs. In this context, this paper aims to design a
novel XAI framework for FNN-based channel estimation denoted as XAI-CHEST.
This framework is based on a perturbation-based model-agnostic global pre-
model methodology that jointly performs the channel estimation task and
provides the corresponding interpretability. The XAI-CHEST concept has been
partially proposed in [36], where the key idea is to provide the
interpretability of black box models by inducing noise on the model input
while preserving accuracy. The model inputs are then classified into relevant
and irrelevant sets based on the induced noise. The contributions of this work
can be summarized as follows:

  * •

Establishing the theoretical foundations of the XAI-CHEST framework where the
detailed loss functions are formulated.

  * •

Deriving the analytical expression and the corresponding simulations of the
noise threshold optimization to select the best threshold used in filtering
the relevant model inputs.

  * •

Showing that using only relevant inputs instead of the full set improves the
performance of the considered DL-based channel estimators.

  * •

Optimizing the architecture of the considered DL-based channel estimator where
minimizing the relevant model inputs resulted in a reduction of the model’s
hidden layers while preserving performance levels.

The remainder of this paper is organized as follows: Section II presents the
system model in addition to the DL-based channel estimators to be interpreted.
Section III illustrates the designed XAI-CHEST framework as well as the noise
threshold fine-tuning optimization problem. In Section IV, the performance of
the designed XAI-CHEST framework in terms of bit error rate (BER) is analyzed
considering several criteria. Finally, Section V concludes the manuscript.

Notations: Throughout the paper, vectors are defined with lowercase bold
symbols $\bm{s}$. The ($i,k$) element of ${\bm{s}}$ is represented by
${\bm{s}}_{i}[k]$, where $i$ and $k$ stand for the OFDM symbol and the
subcarrier indices, respectively. We note that the full OFDM symbol
${\bm{s}}_{i}\in\mathbb{C}^{K\times 1}$ includes
${\bm{s}}_{i,d}\in\mathbb{C}^{K_{d}\times 1}$ data symbols and
${\bm{s}}_{i,p}\in\mathbb{C}^{K_{p}\times 1}$ pilots.

##  II System Model

This section illustrates the considered generic system model in addition to
the considered DL-based channel estimation scheme to be interpreted.

###  II-A OFDM transceiver

Figure 2: OFDM transmitter-receiver block diagram.

In this work, we use an orthogonal frequency division multiplexing
(OFDM)-based transmission with non-linear radio frequency (RF) represented by
the high power amplifier (HPA) at the OFDM transmitter. As shown in Figure 2,
the first operation on the transmitter side is the binary bits generation.
Generated bits are scrambled in order to randomize the bits pattern, which may
contain long streams of $1$s or $0$s. The scrambled bits are then passed to
the encoder, which introduces some redundancy into the bits stream. This
redundancy is used for error correction that allows the receiver to combat the
effects of the channel, hence reliable communications can be achieved.

Bits interleaving is used to cope with the channel noise such as burst errors
or fading. The interleaver rearranges input bits such that consecutive bits
are split among different blocks. This can be done using a permutation process
that ensures that adjacent bits are modulated onto non-adjacent subcarriers
and thus allows better error correction at the receiver. After that, the
interleaved bits are mapped according to the employed modulation technique,
i.e., BPSK, QPSK, 16QAM, 64QAM, etc. Bits mapping operation is followed by
constructing the OFDM symbols to be transmitted. The data symbols and pilots
are mapped to the active subcarriers and passed to the IFFT block to generate
the time-domain OFDM symbols and followed by inserting the cyclic prefix (CP).
Finally, the CP-OFDM symbol is subjected to the impacts of HPA non-linear
distortion as well as the channel and the additive white Gaussian noise (AWGN)
noise.

At the receiver side, the CP is removed and the FFT applied to the received
symbol. Channel estimation and equalization are performed where the equalized
data are de-mapped to obtain the encoded bits. Afterwards, deinterleaving,
decoding, and descrambling are performed to obtain the detected bits. We note
that the employed system model is based on the IEEE 802.11p standard [37].

###  II-B Signal Model

Consider a frame consisting of $I$ OFDM symbols. The $i$-th transmitted
frequency-domain OFDM symbol ${\bm{s}}_{i}\in\mathbb{C}^{K\times 1}$, can be
expressed as:

${\bm{s}}_{i}[k]=\left\\{\begin{array}[]{ll}{\bm{s}}_{{i,d}}[k],&\quad
k\in{\mathcal{K}}_{\text{d}}\\\ {\bm{s}}_{{i,p}}[k],&\quad
k\in{\mathcal{K}}_{\text{p}}\\\ 0,&\quad k\in{\mathcal{K}}_{\text{n}}\\\
\end{array}\right.$ (1)

where $0\leq k\leq K-1$ denotes the subcarrier index. We note that
$K_{\text{on}}$ useful subcarriers are used where $K_{\text{on}}=K_{p}+K_{d}$.
${\bm{s}}_{{i,p}}\in\mathbb{C}^{K_{p}\times 1}$ and
${\bm{s}}_{{i,d}}\in\mathbb{C}^{K_{d}\times 1}$ represent the allocated pilot
symbols and the modulated data symbols at a set of subcarriers denoted
${\mathcal{K}}_{\text{p}}$ and ${\mathcal{K}}_{\text{d}}$, respectively.
$K_{n}$ $=K-K_{\text{on}}$ denotes the null guard band subcarriers.
$K_{\text{cp}}$ samples are added to the time-domain OFDM symbol resulting in
${\bm{x}}_{{i}}\in\mathbb{C}^{K+K_{\text{cp}}\times 1}$ which is then passed
to the HPA. According to the Bussgang theorem [38], the HPA output
${\bm{u}}^{\prime}_{{i}}\in\mathbb{C}^{K+K_{\text{cp}}\times 1}$ can be
expressed as follows:

$\begin{split}{\bm{u}}^{\prime}_{{i}}&=\rho{\bm{x}}_{i}+{\bm{z}}^{\prime}_{i},\end{split}$
(2)

where $\rho$ and ${\bm{z}}^{\prime}_{i}$ refer to the complex gain and the
non-linear distortion (NLD), respectively. After that $\rho$ is compensated at
the transmitter and ${\bm{u}}^{\prime}_{{i}}$ can be rewritten as:

$\begin{split}{\bm{u}}_{{i}}=\frac{{\bm{u}}^{\prime}_{{i}}}{\rho}&={\bm{x}}_{i}+{{\bm{z}}}_{i},\end{split}$
(3)

where ${{\bm{z}}_{i}}=\frac{{\bm{z}}^{\prime}_{i}}{\rho}$ denotes the
remaining NLD of the HPA. The received frequency-domain OFDM symbol
${\bm{y}}_{{i}}\in\mathbb{C}^{K_{\text{on}}\times 1}$ is expressed as follows:

$\begin{split}{\bm{y}}_{{i}}[k]&={\bm{h}}_{i}[k]\tilde{\bm{u}}_{i}[k]+{\bm{e}}_{i}[k]+\tilde{\bm{v}}_{i}[k],\end{split}$
(4)

where

$\tilde{\bm{u}}_{i}[k]=\left\\{\begin{array}[]{ll}{\bm{s}}_{{i}}[k],&\quad\text{linear
RF}\\\ {\bm{s}}_{{i}}[k]+\tilde{\bm{z}}_{{i}}[k],&\quad\text{non-linear RF}\\\
\end{array}\right.$ (5)

${\bm{h}}_{i}\in\mathbb{C}^{K_{\text{on}}\times 1}$,
${\tilde{\bm{v}}}_{i}\in\mathbb{C}^{K_{\text{on}}\times 1}$, and
${\bm{e}}_{i}\in\mathbb{C}^{K_{\text{on}}\times 1}$ refer to the frequency-
time response of the doubly-selective channel, the AWGN at the $i$-th OFDM,
and the Doppler-induced inter-carrier interference, respectively.
${\bm{e}}_{i}$ can be expressed as:

$\begin{split}{\bm{e}}_{i}[k]&=\frac{1}{{{K}}}\sum_{\begin{subarray}{c}q=0\\\
q\neq
k\end{subarray}}^{{K}-1}\sum_{n=0}^{{K}-1}{\bm{h}}_{i}[q,n]e^{-j2\pi\frac{n(k-q)}{{K}}}{\bm{s}}_{i}[q].\end{split}$
(6)

The full derivation of ${\bm{e}}_{i}$ can be found in [39]. We note that the
objective is to obtain the channel frequency response estimate denoted by
$\hat{{\bm{h}}}_{i}\in\mathbb{C}^{K_{\text{on}}\times 1}$.

###  II-C DL-based Channel Estimation

Conventional channel estimation depends highly on environment conditions. In
frequency-selective slow fading channels, the preamble-based channel
estimation is sufficient, since the communication system encounters only muti-
path fading and the channel is not changing over time. However, in double
selective channels, the impact of Doppler interference is added to the
communication system. Thus, the estimated channel at the beginning of the
frame, i.e., the preambles, becomes outdated and channel tracking becomes more
challenging, especially, in high mobility scenarios. To cope with this
challenge, pilot subcarriers are allocated within a transmitted OFDM symbol to
allow better channel tracking over time, where several conventional channel
estimation and tracking schemes are proposed in the literature. In order to
further improve the conventional channel estimation accuracy, DL models are
applied as post-processing on top of conventional channel estimators. In this
work, we considered the STA-FNN channel estimator [8] as a case study, where
we used the optimized XAI-CHEST framework to provide the corresponding
reasonable interpretations.

Conventional STA channel estimation scheme [14] is based on the DPA estimation
where the demapped data subcarriers of the previously received OFDM symbol are
used to estimate the channel for the current OFDM symbol such that:

${\bm{d}}_{i}=\mathfrak{D}\big{(}\frac{{\bm{y}}_{i}}{\hat{{\bm{h}}}_{\text{DPA}_{i-1}}}\big{)},\leavevmode\nobreak\
\hat{{\bm{h}}}_{\text{DPA}_{0}}=\hat{{\bm{h}}}_{\text{LS}},$ (7)

where $\mathfrak{D}(.)$ refers to the demapping operation to the nearest
constellation point according to the employed modulation order.
$\hat{{\bm{h}}}_{\text{LS}}$ stands for the LS estimated channel at the
received preambles. Thereafter, the DPA channel estimates are updated in the
following manner:

$\hat{{\bm{h}}}_{\text{DPA}_{i}}=\frac{{\bm{y}}_{i}}{{\bm{d}}_{i}}.$ (8)

After that, frequency-domain averaging is applied where the DPA estimated
channel at each subcarrier is updated as follows:

$\hat{{\bm{h}}}_{\text{FD}_{i}}[k]=\sum_{\lambda=-\beta}^{\lambda=\beta}\omega_{\lambda}\hat{{\bm{h}}}_{\text{DPA}_{i}}[k+\lambda],\leavevmode\nobreak\
\omega_{\lambda}=\frac{1}{2\beta+1}.$ (9)

Finally, time-domain averaging is employed to reduce the AWGN noise impact
such that:

$\hat{{\bm{h}}}_{\text{STA}_{i}}=(1-\frac{1}{\alpha})\hat{{\bm{h}}}_{\text{STA}_{i-1}}+\frac{1}{\alpha}\hat{{\bm{h}}}_{\text{FD}_{i}}.$
(10)

We note that conventional STA channel estimation performs well in the low
signal-to-noise ratio (SNR) region. However, it suffers from a considerable
error floor in high SNR regions due to the large DPA demapping error resulting
from (7) and the fixed frequency and time averaging coefficients
$\alpha=\beta=2$ in (9) and (10), respectively. Therefore, the conventional
STA channel estimation scheme is not practical in real-case scenarios due to
the high doubly-selective channel variations. As a workaround, a $3$ hidden
layer FNN consisting of $15$ neurons per layer is utilized as a nonlinear
post-processing unit following STA [8]. STA-FNN can better capture the time-
frequency correlations of the channel samples, in addition to correcting the
conventional STA estimation error.

##  III XAI-CHEST Framework Design

Figure 3: Block diagram of the XAI-CHEST framework. The first step is to train
the $U$ model and freeze its parameters $\theta_{U}$. After that, the N model
is trained where the objective is to boost the growth of
$\bm{b}_{\Phi_{i}}^{\prime}$ while preserving the same performance of the pre-
trained $U$ model (14). Finally, the subcarriers are filtered based on
$\bm{b}_{\Phi_{i}}^{\prime}$, where higher noise weight signifies that the
corresponding subcarrier is irrelevant (red colors). In contrast, low noise
weights mean that the subcarriers are relevant (blue colors). We recall that
in this work we are considering the STA-FNN channel estimation scheme, hence,
$\Phi=\text{STA}$.

Providing external interpretability of the black-box model used for channel
estimation could be achieved through classifying the model’s input into
relevant and irrelevant by employing a perturbation-based methodology. The
main intuition is that if a subcarrier is relevant for the decision-making of
a trained black box model, then adding noise with high weight to this
subcarrier would negatively impact the accuracy of the model. Whereas, if the
subcarrier is not contributing to the decision-making of the model, then
whatever the induced noise is, the channel estimation accuracy will be
preserved. Therefore, it is expected that considering only the relevant
subcarriers as model inputs would improve channel estimation performance in
comparison to the case where the full subcarriers are given to the model.
Moreover, by reducing the model input size, the employed architecture could be
further optimized resulting in significantly decreasing the overall
computational complexity. Hence, by using the XAI-CHEST framework, we can
obtain a reasonable interpretation of the model decision-making methodology,
improve the channel estimation performance as well as reduce the required
computational complexity.

###  III-A Methodology

Let $U$ be the black-box utility model with parameters $\theta_{U}$. In
general, the $U$ model refers to the channel estimation model, where
$\hat{{\bm{h}}}^{\prime}_{\Phi_{i}}\in^{2K_{\text{on}}\times 1}$ and
$\hat{{\bm{h}}}^{(\text{U})}_{\Phi_{i}}\in^{2K_{\text{on}}\times 1}$ denote
the conventional estimated channel that is given as an input to the $U$ model
and the corresponding output, respectively. We note that the size of
$\hat{{\bm{h}}}^{\prime}_{\Phi_{i}}$ is $2K_{\text{on}}$ since the
conventional estimated channel is converted from complex to real domain before
further processing by stacking the real and imaginary values vertically in one
vector. $\Phi$ refers to the employed conventional channel estimation scheme.
The objective is to provide a reasonable interpretation of the behavior of the
$U$ model. Besides the $U$ model, we define the interpretability noise model
$N$, with parameters $\theta_{N}$, whose purpose is to compute the weight of
the noise induced to each subcarrier within the $U$ input vector. The key idea
is that the induced noise weights of the $N$ model should not impact the
accuracy of the $U$ model. This could be achieved by customizing the loss
function of the $N$ model that will adjust the induced noise while
simultaneously maximizing the performance of the $U$ model. We note that the
$U$ model is trained before the XAI processing of the $N$ model, i.e., the
weights of the $U$ model are frozen. Moreover, the $U$ and $N$ models have the
same FNN architecture.

Figure 4: Normalized restricted loss function.

Let $\hat{{\bm{h}}}^{\prime}_{\Phi_{i}}$ be the input of the interpretability
$N$ model. The role of the $N$ model is to find a mask
$\bm{b}^{\prime}_{\Phi_{i}}\in\mathbb{R}^{2K_{\text{on}}\times 1}$ that can be
represented as follows:

$\bm{b}^{\prime}_{\Phi_{i}}=N(\hat{{\bm{h}}}^{\prime}_{\Phi_{i}},\theta_{N}),$
(11)

where
$\bm{b}^{\prime}_{\Phi_{i}}=(b^{\prime}_{\Phi_{i}}[1],b^{\prime}_{\Phi_{i}}[2],...,b^{\prime}_{\Phi_{i}}[2K_{\text{on}}])\in[0,1]^{{{2K_{\text{on}}}}}$
determines the weight (standard deviation) of the noise applied to each
element in $\hat{{\bm{h}}}^{\prime}_{\Phi_{i}}$. We note that the scaling of
$\bm{b}^{\prime}_{\Phi_{i}}$ is achieved using the sigmoid activation
function. After that, the generated noise weight mask
$\bm{b}^{\prime}_{\Phi_{i}}$ is first multiplied by a random noise
$\epsilon\sim\mathcal{N}(0,1)$ sampled from the standard normal distribution,
the resultant is added to the conventional estimated channel vector, such
that:

$\hat{{\bm{h}}}^{\prime\prime}_{\Phi_{i}}=\hat{{\bm{h}}}^{\prime}_{\Phi_{i}}+\bm{b}_{\Phi_{i}}^{\prime}\epsilon.$
(12)

After that, $\hat{{\bm{h}}}^{\prime\prime}_{\Phi_{i}}$ is fed as input to the
$U$ model, such that:

$\hat{{\bm{h}}}^{(\text{U})}_{\Phi_{i}}=U(\hat{{\bm{h}}}^{\prime\prime}_{\Phi_{i}},\theta_{U}).$
(13)

The customized loss function of the $N$ model can be expressed as follows:

$\pazocal{L}_{N}=\min_{\theta_{N}}\big{(}\pazocal{L}_{U}-\lambda\pazocal{L}_{X}\big{)},$
(14)

$\pazocal{L}_{U}$ denotes the loss unction of the $U$ model when
$\hat{{\bm{h}}}^{\prime\prime}_{\Phi_{i}}$ is fed as an input. Hence,
$\pazocal{L}_{U}$ can be expressed as:

$\pazocal{L}_{U}=\frac{1}{N_{tr}}.\sum_{i=1}^{N_{tr}}\left\|{{\bm{h}}}_{{i}}-\hat{{\bm{h}}}^{(\text{U})}_{\Phi_{i}}\right\|^{2},$
(15)

where ${{\bm{h}}}_{{i}}$ refers to the true channel and $N_{tr}$ is the number
of training samples. Moreover, the induced noise is controlled by
$\pazocal{L}_{X}$ that can be written as:

$\pazocal{L}_{X}=\frac{1}{2K_{\text{on}}}.\sum_{j=1}^{2K_{\text{on}}}\log(b^{\prime}_{\Phi_{i}}[j])$
(16)

We would like to mention that the objective of $\pazocal{L}_{N}$ is to keep
the loss of the $U$ model as low as possible. In other words, minimizing the
added noise by the $N$ model while maximizing the generated
$\bm{b}^{\prime}_{\Phi_{i}}$. It is worth mentioning that the interpretability
measure aims to find a maximum number of low-significant elements. Hence, the
term $-\pazocal{L}_{X}$ gives a negative value, which will get closer to zero
when more weights are close to one, meaning that our $N$ model finds more
irrelevant elements and can better highlight the significant features. We note
that $\lambda$ is a parameter that allows to give more or less weight to the
interpretability measure. The problem is similar to a minimization on
$\pazocal{L}_{U}$ with a constraint on $\pazocal{L}_{X}$ ($\lambda$ being a
Lagrange multiplier) or simply a regularization term.

Finally, in the testing phase, $\bm{b}_{\Phi_{i}}^{\prime}$ is scaled back to
$\bm{b}_{\Phi_{i}}\in\mathbb{R}^{K_{\text{on}}\times 1}$, where the noise
weight of the real and imaginary parts for each subcarrier are averaged. The
motivation behind this averaging lies in the fact that it is noticed during
the training phase that the interpretability model produces almost the same
noise weight for the real and imaginary parts of each subcarrier within
$\hat{{\bm{h}}}^{\prime}_{\Phi_{i}}$. The block diagram of the XAI-CHEST
framework and the $N$ model training procedure are illustrated in Figure 3 and
algorithm 1, respectively. We note that
$\hat{{\bm{H}}}^{\prime}_{\Phi}\in\mathbb{R}^{2K_{\text{on}}\times
I_{\text{tr}}}$ and ${{\bm{H}}}\in\mathbb{R}^{2K_{\text{on}}\times
I_{\text{tr}}}$ denote the training dataset pairs of the conventional
estimated channels and the true ones, where $I_{\text{tr}}$ is the size of the
training dataset.

Algorithm 1 $N$ model training

0: Conventional estimated channel: $\hat{{\bm{H}}}^{\prime}_{\Phi}$, true
channel: ${{\bm{H}}}$, learning rate: $\eta$, trained $U$ model with
parameters $\theta_{U}$

0: Trained N model with parameters $\theta_{N}$

while not converged do

for $\hat{{\bm{h}}}^{\prime}_{\Phi_{i}}\in\hat{{\bm{H}}}^{\prime}_{\Phi_{i}}$,
${{\bm{h}}}_{{i}}\in{{\bm{H}}}_{{i}}$ do

$\bm{b}_{\Phi_{i}}^{\prime}$ $\leftarrow$
$N(\hat{{\bm{h}}}^{\prime}_{\Phi_{i}},\theta_{N})$

$\epsilon$ $\leftarrow$ $\mathcal{N}(0,1)$

$\hat{{\bm{h}}}^{\prime\prime}_{\Phi_{i}}$ $\leftarrow$
$\hat{{\bm{h}}}^{\prime}_{\Phi_{i}}+\bm{b}_{\Phi_{i}}^{\prime}\epsilon$

$\hat{{\bm{h}}}^{(\text{U})}_{\Phi_{i}}$ $\leftarrow$
$U(\hat{{\bm{h}}}^{\prime\prime}_{\Phi_{i}},\theta_{U})$

$\pazocal{L}_{U}$ $\leftarrow$
$\text{MSE}\big{(}{{\bm{h}}}_{{i}}-\hat{{\bm{h}}}^{(\text{U})}_{\Phi_{i}}\big{)}$

$\pazocal{L}_{X}$ $\leftarrow$
$\text{mean}\big{(}\log(\bm{b}^{\prime}_{\Phi_{i}})\big{)}$

$\pazocal{L}_{N}$ $\leftarrow$ $\pazocal{L}_{U}$ $-\lambda\pazocal{L}_{X}$

$\theta_{N}$ $\leftarrow$
$\theta_{N}+\eta\frac{\partial\pazocal{L}_{N}}{\partial\theta_{N}}$

end for

end while

Table I: Characteristics of the employed channel models following Jake’s Doppler spectrum. Channel model | Average path gains [dB] | Path delays [ns]
---|---|---
VTV-EX |  | [0, 0, 0, -6.3, -6.3, -25.1, -25.1, -25.1, -22.7, -2.27, -22.7]
---
| [0, 1, 2, 100, 101, 200, 201, 202, 300, 301, 302]
---
VTV-SDWW |  | [0, 0, -11.2, -11.2, -19, -21.9, -25.3, -25.3, -24.4,-28, -26.1, -26.1]
---
| [0, 1, 100, 101, 200, 300, 400, 401, 500, 600, 700, 701]
---

###  III-B Noise Weight Threshold Fine Tuning

After accomplishing the $N$ model training, the fine-tuning of the noise
weight threshold denoted by $\gamma$ is essential for classifying the model
inputs into relevant and irrelevant. This could be formulated as an
optimization problem, where the objective is to select the best input
combination that minimizes the mean squared error (MSE) between the
corresponding estimated channel by the $U$ model and the true channel. We note
that the fine-tuning optimization problem is subjected to improving or
preserving the BER in comparison to the case where the full subcarriers are
given to the $U$ model.

Let $\Omega$ be the generic input given to the $U$ model and $\Psi$ be the
optimized model input. The considered fine-tuning optimization problem can be
mathematically expressed as:

$\displaystyle\min_{\Psi,\theta_{U}}$
$\displaystyle\pazocal{L}_{U}=\frac{1}{N_{tr}}.\sum_{i=1}^{N_{tr}}\big{(}{\tilde{\bm{h}}}_{{i}}-U(\Omega=\Psi,\theta_{U})\big{)}^{2}$
(17) s.t.
$\displaystyle\text{BER}(\Omega=\Psi)\leq\text{BER}(\Omega=\hat{\tilde{\bm{h}}}^{\prime\prime}_{\text{STA}_{i}})$

We note that the defined optimization problem in (17) is not convex. The non-
convexity can be shown by the line restriction method illustrated in Lemma 1
[40].

######  Lemma 1.

Restriction of a convex function to a line
A function $f$: $\mathbb{R}^{n}\rightarrow\mathbb{R}$ is convex, if and only
if
$\forall x\in\operatorname{dom}f$ and $\forall v\in\mathbb{R}^{n}$, the
function $g=f(x+tv)$ is convex on
$\operatorname{dom}g=\\{t\in\mathbb{R}\;|\;x+tv\in\operatorname{dom}f\\}$

Lemma 1 is based on the line restriction method to prove the convexity of the
considered function. In this context, the initial loss function
$\pazocal{L}_{U}$ is reduced to the restricted loss function denoted as
$g(t)=\pazocal{L}_{U}(\theta_{U}+tv)$, where $v$ and $t$ denote the randomly
selected slice and the step size, respectively. Figure 4 shows the $g(t)$
where we can see numerous local minima signifying the non-convexity of our
optimization function expressed in (17). We note that in the next section, we
provided a heuristic solution of (17), where the BER vs noise weight threshold
is analyzed and the best threshold is selected according to the lowest
recorded BER among all the considered thresholds.

##  IV Simulation Results

(a) Noise weight distribution (b) LFS channel model - QPSK modulation  (c) BER
performance vs. noise threshold

Figure 5: Fine-tuning of the noise threshold $\gamma$ considering the HFS
channel model and QPSK modulation.

This section illustrates the performance evaluation of the proposed XAI-CHEST
framework, where BER performance of STA-FNN channel estimation scheme is
analyzed taking into consideration full, relevant, and irrelevant subcarriers.
First of all, we start with the noise weight threshold fine-tuning, where the
simulation-based solution of (17) is provided. After that, the performance
evaluation is performed according to several criteria including the (i)
modulation order, (ii) frequency selectivity of the channel, (iii) training
SNR, and (iv) conventional channel estimation accuracy. Finally, a detailed
computational complexity analysis is discussed where we show that further
significant reduction in the overall computational complexity can be achieved
by employing only the relevant subcarriers identified by the proposed XAI-
CHEST framework.

We note that the considered channel models [41] as shown in Table I: (i) Low-
frequency selectivity (LFS), where VTV Expressway (VTV-EX) scenario is
employed. (ii) High-frequency selectivity (HFS), where VTV Expressway Same
Direction with Wall (VTV-SDWW) scenario is considered. In both scenarios,
Doppler frequency $f_{d}=1000$ Hz is considered. Both the $U$ and $N$ models
are trained using a $100,000$ OFDM symbols dataset, splitted into $80\%$
training, and $20\%$ testing. ADAM optimizer is used with a learning rate
$lr=0.001$ with batch size equals $128$ for $500$ epoch. Simulation parameters
are based on the IEEE 802.11p standard [5], where the comb pilot allocation is
used so that $K_{p}=4$, $K_{d}=48$, $K_{n}=12$, and $I=50$. Table II
summarizes the simulation parameters considered in this work. Finally, we note
that the STA channel estimation is considered as an initial estimation prior
to the FNN processing. Hence, $\Phi=\text{STA}$, unless stated otherwise.

Table II: Parameters of the studied STA-FNN channel estimation scheme. Parameter | Values
---|---
STA-FNN (Hidden layers; Neurons per layer) | (3;15-15-15)
Activation function | ReLU
Number of epochs | 500
Training samples | 800000
Testing samples | 200000
Batch size | 128
Optimizer | ADAM
Loss function | MSE
Learning rate | 0.001
Training SNR | 40 dB

###  IV-A Noise weight threshold fine-tuning

The first step of solving (17) is to analyze the distribution of the noise
weight vector $\bm{b}^{\prime}_{\text{STA}_{i}}$ over the input subcarriers.
Both the $U$ and $N$ models are trained using the HFS channel model with QPSK
modulation and $40$ dB training SNR. We note that we train the models on SNR =
$40$ dB since the models can learn the channel better when the training is
performed at a high SNR value because the impact of the channel is higher than
the impact of the AWGN noise in this SNR range [42]. Owing to the robust
generalization properties of DL, trained networks can still estimate the
channel even if the AWGN noise increases, i.e., at low SNR values.

Figure 5(a) shows the distribution of $\bm{b}^{\prime}_{\text{STA}_{i}}$. We
notice that the majority of subscribers are distributed more towards zero.
This signifies that the model is not sure if the subcarriers can be neglected
or not. It is worth mentioning that the pilot subcarriers are assigned the
lowest noise weight, i.e., $0.1$ which reveals that the $U$ model is not able
to neglect the estimated channels at the pilots, and considering them is
crucial for high estimation accuracy. This is consistent with the channel
estimation rules, where the channel estimates at the pilots are very close to
the ideal channel estimation.

Selecting the optimized $U$ model input $\Psi$ is determined by choosing the
best noise weight threshold $\gamma$ which is mainly responsible for
classifying the subcarriers into relevant and irrelevant. To select the
optimal $\gamma$, we simulated the BER considering all possible values, i.e,
$\gamma=[0.1,0.2,0.3,...,0.8]$. In each case, we trained the $U$ model
considering both the relevant and irrelevant subcarrier combinations.

As shown in Figure 5(b), we can notice that considering $\gamma=0.4$ gives the
best BER performance among other thresholds. Therefore, the STA-FNN model
needs only $|\Psi|=28$ subcarriers out of the full set, i.e, $|\Psi|=52$ in
order to provide the best possible performance in the considered HFS channel
model. On the contrary, all the irrelevant subcarrier combinations are worse
than the full case in terms of BER performance. In other words, considering
$|\Psi|=48$ which corresponds to excluding only the four pilot subcarriers is
not enough to preserve the BER performance of the full case.

(a) HFS channel model - QPSK modulation (b) HFS channel model - 16QAM
modulation (c) HFS channel model - 64QAM modulation

Figure 6: BER considering HFS channel model employing QPSK, 16QAM, and 64QAM,
respectively. We note that the highlighted gray and blue areas refer to the
BER performance when considering the irrelevant and relevant subcarriers,
respectively.To provide better reading, the best and worst relevant and
irrelevant BER performances are only shown, where the regions between the
irrelevant-full and full-relevant are highlighted in gray and blue,
respectively.

(a) Noise weight distribution (b) BER performance vs. noise threshold

Figure 7: Noise distribution and BER threshold analysis for HFS channel model
employing QPSK, 16QAM, and 64QAM, respectively. We note that R, IR, and Full
in (b) refer to employing only the selected relevant, irrelevant, and Full
subcarriers as model inputs, respectively.

Figure 5(c) shows the BER in terms of $\gamma$ considering SNR = $40$ dB.
Again, increasing $\Psi$ is beneficial until reaching $\gamma=0.4$, where the
BER performance degrades. This signifies that in complicated scenarios as the
case in employing the HFS channel model, the proposed perturbation-based XAI
scheme can smartly filter out the relevant model inputs which maximize the its
performance.

###  IV-B Impact of modulation order

In this section, we further investigate the impact of the employed modulation
order on the noise weight distribution considering also the HFS channel model.
Figure 6 shows the BER performance of employing the HFS channel model using
QPSK, 16QAM, and 64QAM, respectively. In general, the BER performance degrades
as the modulation order increases. This degradation is mainly due to the
impact of the dominant multi-path fading in addition to the DPA remapping
error. Moreover, in this scenario, employing only the four pilot subcarriers
performs almost similarly to the full case. To improve further the BER
performance, more relevant subcarriers are needed. Therefore, when the
environment becomes more challenging, the channel variation increases among
the subcarriers, thus, the noise distribution is shifted towards zero
signifying the need for more relevant subcarriers.

(a) LFS channel model - QPSK modulation (b) LFS channel model - 16QAM
modulation (c) LFS channel model - 64QAM modulation

Figure 8: BER considering LFS channel model employing QPSK, 16QAM, and 64QAM,
respectively. We note that the highlighted gray and blue areas refer to the
BER performance when considering the irrelevant and relevant subcarriers,
respectively.

(a) Noise weight distribution (b) BER performance vs. noise threshold

Figure 9: Noise distribution and BER threshold analysis for LFS channel model
employing QPSK, 16QAM, and 64QAM, respectively.

However, we can notice that for higher the modulation order, the number of
neglected subcarriers increases. This is because the conventional STA channel
estimates at these subcarriers are so noisy, so the model neglects them. It is
worth mentioning that, the model neglects subcarriers due to two main reasons:
(i) LFS: The channel variation among the subcarriers is slow, so few
subcarriers are required to accurately estimate the channel, as we will
discuss in the next Section. (i) HFS: Here the channel variation is
significant among the subcarriers, thus, the $U$ model should consider more
relevant subcarriers to guarantee good channel estimation accuracy. However,
this is subject to the condition where the conventional estimated channel at
the considered subcarriers is useful and not so noisy. Therefore, in the HFS
channel model, more relevant subcarriers are needed and this is shown in
generally shifting the noise weight distribution towards zero. However, for
higher modulation orders, mainly 64QAM, the neglected subcarriers are huge due
to the bad channel estimation quality at these subcarriers. Hence, avoiding
them is useful to guarantee BER performance. We note that the four pilot
subcarriers are assigned the lowest noise weight for all the modulation
orders. Therefore, the $U$ model is able to classify the pilots as the most
relevant subcarriers regardless of the channel’s high selectivity and the
employed modulation order.

Figure 7(b) shows the BER in terms of $\gamma$ considering SNR = $40$ dB using
the HFS channel model. We can notice that ($\gamma=0.5$, $|\Psi|=20$), and
($\gamma=0.5$, $|\Psi|=11$) are the best options corresponding to the 16QAM,
and 64QAM modulation orders, respectively. Again we can notice that the
functionality of the interpretability model in classifying the subcarriers
into relevant and irrelevant is based on the context, i.e., the employed
modulation order in this case.

(a) Noise weight distribution (b) BER performance vs. noise threshold (c) HFS
channel model - IBO = 2dB

Figure 10: BER considering HFS channel model employing QPSK and IBO = 2 dB. We
note that the highlighted gray and blue areas refer to the BER performance
when considering the irrelevant and relevant subcarriers, respectively.

###  IV-C Impact of channel frequency selectivity

In this section, we will investigate the performance evaluation using the same
methodology of Section IV-B but considering the LFS channel model. Figure 8
shows the BER results of employing QPSK, 16QAM, and 64QAM modulation orders,
respectively. We can notice a significant performance degradation as the
modulation order increases which is expected. The nice thing lies in employing
the pilot subcarriers only, where the corresponding BER performance improves
in comparison to the full case. In other words, the BER performance
improvement of employing the pilots in comparison to the full case for 64QAM
modulation is higher than that for 16QAM and QPSK modulations, respectively.
This is because applying the frequency and time domain averaging in the
conventional STA channel estimation is no longer reliable due to high
demapping error resulting from the DPA channel estimation (7) that is applied
prior to the STA estimation. Similarly to the discussion in Section IV-A,
employing more relevant and irrelevant subcarriers leads to a BER performance
degradation in both cases where in the relevant case, the BER performance is
approaching the full case, while in the irrelevant case, the performance is
going off the full case.

Figure 9(a) illustrates the noise weight distribution of training the models
using different modulation orders. We can notice that distribution is shifted
towards one, where the majority of subcarriers are assigned noise weight equal
to one. This signifies that these subcarriers are not important for the
decision-making methodology of the $U$ model. This is because, in the LFS
channel model, the channel presents a smooth variation over the subcarriers,
thus, the STA-FNN model needs few subcarriers to accomplish the channel
estimation task. Moreover, as the modulation order increases, the noise weight
distribution becomes wider, where more subcarriers are assigned more weights.
For example, in the 64QM modulation order, it seems that the model needs more
subcarriers to preserve good performance, thus the number of subcarriers that
are assigned noise weight = 1 decreases. Moreover, in all cases, the model is
able to classify pilots as the most relevant subcarriers by assigning them the
lowest noise weight regardless of the employed modulation order. The BER vs
the noise weight for the considered modulation orders is shown in Figure 9(b)
where we can notice that considering only the pilots in the LFS channel model
is enough, and there is no need to consider any other subcarriers. On the
contrary, all the irrelevant subcarrier combinations are worse than the full
case in terms of BER performance. Hence, the absence of the four pilots leads
to performance degradation even if the other $|\Psi|=48$ subcarriers are
considered.

###  IV-D Impact of RF non-linear distortion

In order to further analyze the impact of HPA-induced nonlinearities, we
employ QPSK modulation and IBO = 2 dB in the HFS channel models. Figure 10
shows the noise weight distribution as well as the BER analysis. It can be
noticed that only 2 pilots are assigned the lowest noise weight in comparison
to 4 in the linear case. This ensures that the HPA-induced nonlinearities
contribute to confusing the subcarrier filtering procedure. However, a slight
BER rate performance improvement can be achieved by employing $|\Psi|=27$ for
$\gamma=0.5$. Therefore, similar insights can be concluded as the linear case
where the proposed perturbation based XAI framework is able to filter out the
relevant subcarriers while preserving the BER performance when using the full
subscribers as an input to the $U$ model.

(a) Noise weight distribution (b) LFS channel model - QPSK modulation

Figure 11: Noise distribution and BER performance considering the LFS channel
model and QPSK modulation employing different training SNRs.

###  IV-E Impact of training SNR

The sensitivity of the $U$ model training considering different SNR values is
analyzed in this section. Figure 11(a) shows the noise distribution when
considering several training SNRs employing the LFS channel model and QPSK
modulation order. Starting by training SNR = $0-5$ dB, we can see that the
pilot subcarriers are assigned $0.2$ noise weight and the distribution is
flattened along the entire range. This reveals that even though the pilots
have accurate channel estimates, due to the dominant impact of AWGN noise, the
$U$ model is not able to assign the lowest noise weight to the pilot
subcarriers. It is worth mentioning that when training on SNR = $10$ dB, the
model starts to identify the pilot subcarriers as the most relevant
subcarriers by assigning to two pilots the lowest noise weight, i.e., $0.1$.
Moreover, as the training SNR increases, the noise distribution is shifted
more towards one, signifying that the model is better identifying the relevant
and irrelevant subcarriers. Figure 11(b) shows the BER performance when the
$U$ model is trained on a specific SNR and tested on the entire SNR range. We
can notice that training on higher SNR gives better performance than training
on the lower SNR due to the fact the AWGN noise is negligible at high SNRs,
thus the $U$ model can learn more efficiently the channel. In addition, the
trained model on high SNR can perform well when tested on lower SNRs due to
the generalization ability of FNN networks. In conclusion, training on low SNR
values leads to a limited performance improvement over the conventional STA
channel estimation. Whereas training on high SNR allows the smart feature
selection resulting in optimizing the $U$ model input size, as well as
significantly improving the BER performance in comparison to the conventional
STA channel estimation.

(a) Noise weight distribution (b) HFS channel model - QPSK modulation

Figure 12: Noise distribution and BER performance of the DPA-FNN, STA-FNN, and
TRFI-FNN channel estimation schemes.

###  IV-F Impact of conventional channel estimation accuracy

To further analyze the impact of the conventional channel estimation which is
implemented prior to the FNN processing on the noise weight distribution, we
considered in this section the DPA-FNN [7] and TRFI-FNN [9] channel estimation
schemes in addition to STA-FNN. We note that we consider the HFS channel model
with QPSK modulation in this analysis since it is more challenging.

As we can see from Figure 11(b), the conventional TRFI channel estimation
outperforms the STA channel estimation in the high SNR region. This is due to
the cubic interpolation employed on top of the DPA channel estimation in the
TRFI scheme. Similar behavior can be seen with respect to the TRFI-FNN and
STA-FNN channel estimators, where the TRFI-FNN also outperforms the STA-FNN in
the high SNR region.

Motivated by the fact that the conventional TRFI is better than the
conventional STA channel estimation, it is expected that the TRFI-FNN model
may neglect more subcarriers than the STA-FNN channel estimation scheme. This
is shown in Figure 11(a), where we can notice that even though we consider the
HFS channel model, the noise distribution of the HFS channel model is still
shifted towards one, signifying that the TRFI-FNN requires less relevant
subcarriers than the STA-FNN channel estimation scheme in order to preserve
the BER performance as the full case, where $|\Psi|=52$. Similar behavior is
recorded for the DPA-FNN channel estimation scheme, where the distribution is
also shifted towards one in a close manner as the TRFI-FNN scheme. This is
because the conventional TRFI scheme slightly outperforms the DPA channel
estimation in the considered scenario. However, the STA-FNN and TRFI-FNN
channel estimation schemes outperform the DPA-FNN due to the averaging
operations and cubic interpolation employed in the conventional STA and TRFI
schemes, respectively.

In this context, we can conclude that as the accuracy of the conventional
channel estimation increases, the number of selected important subcarriers
decreases, where STA-FNN requires $|\Psi|=28$ relevant subcarriers, which are
greater than the relevant subcarriers required by the TRFI-FNN, i.e.,
$|\Psi|=16$. This means that the $N$ model is able to induce more noise to the
TRFI-FNN input, whereas less noise is induced to the STA-FNN input since it is
already noisy.

Table III: FLOPS of the optimized FNN architecture employed in the STA-FNN channel estimation scheme. FNN Architecture | Full (15-15-15) | Relevant (15-15-15) | Relevant (15-15) | Relevant (15) | Relevant (10) | Relevant (5)
---|---|---|---|---|---|---
FLOPS | 7.52 K | 4.64 K | 4.13 K | 3.62 K | 2.48 K | 1.34 K

###  IV-G Computational complexity reduction

This section aims to investigate the possibility of optimizing the U model
architecture following selecting the most relevant subcarriers so that the BER
performance improvement as well as reducing the computational complexity can
be achieved. In this context, we considered the same simulation parameters as
Section IV-A, where the pilot subcarriers are fed to the U model with
different architectures. The objective is to reduce the computational
complexity of the classical STA-FNN model ($15-15-15$) while preserving the
BER performance of the best relevant case, i.e., employing only the pilots in
the LFS channel model.

Figure 13 shows the BER performance of different STA-FNN architectures. We can
notice that the FNN architecture could be reduced up to one hidden layer with
$15$ neurons while preserving the best possible performance. Moreover,
decreasing the number of neurons within this architecture to $10$ performs the
same as the classical STA-FNN channel estimation scheme, i.e., considering the
full subcarriers as inputs with the ($15-15-15$) FNN architecture. However,
employing shallow FNN architecture with $5$ neurons is not useful at all,
where a significant performance degradation is recorded in comparison to the
classical STA-FNN architecture.

Figure 13: STA-FNN BER performance using different optimized FNN
architectures.

The computational complexity of the employed FNNs is computed in terms of the
number of FLOPS111We note that the number of FLOPS are calculated using the
pytorch-OpCounter package [43]. required by each FNN architecture, as shown in
Table III. Employing the same FNN architecture as the classical STA-FNN one
but using the pilot subcarriers as input reduces the complexity by around
$1.5\times$ times in comparison to the classical STA-FNN channel estimation
scheme. However, further complexity reduction can be recorded by employing a
shallow FNN with $15$ neurons, where $2\times$ times can be achieved in
comparison to the classical STA-FNN channel estimation scheme. We would like
to mention that similar BER performance can be guaranteed as the classical
STA-FNN channel estimator by feeding the four pilots to a shallow FNN
architecture with $10$ neurons, where $3\times$ times less computational
complexity is required. Finally, we would like to mention that the proposed
XAI-CHEST framework resolves the main issues related to the black box DL
models by providing interpretability to the model behavior, performance
improvement, and computational complexity reduction by selecting the relevant
model inputs and optimizing the architecture of the employed FNN model.

##  V Conclusion

Ensuring the transparency and trustworthiness of AI is a crucial need for its
efficient and safe deployment in critical applications. In this paper we
designed a novel XAI-CHEST framework that provides the interpretability of the
FNN models employed in the channel estimation application. The XAI-CHEST
framework aims to classify the black-box model inputs into relevant and
irrelevant inputs by using a perturbation-based methodology. We developed the
theoretical foundations of the XAI-CHEST framework by formalizing the related
loss functions. Moreover, the noise threshold fine-tuning optimization problem
has been analytically derived. Extensive simulations have been conducted where
the results reveal that a trustworthy, optimized, and low-complexity channel
estimation scheme can be designed by selecting only the relevant inputs. As a
future perspective, the gradient-based XAI for wireless communication will be
investigated, where we will study the possibility of providing robust
interpretation by using the internal architecture of the FNN model.

## References

  * [1] C.-X. Wang, X. You, X. Gao, X. Zhu, Z. Li, C. Zhang, H. Wang, Y. Huang, Y. Chen, H. Haas, J. S. Thompson, E. G. Larsson, M. D. Renzo, W. Tong, P. Zhu, X. Shen, H. V. Poor, and L. Hanzo, “On the Road to 6G: Visions, Requirements, Key Technologies, and Testbeds,” _IEEE Communications Surveys & Tutorials_, vol. 25, no. 2, pp. 905–974, 2023.
  * [2] G. Liu, Y. Huang, N. Li, J. Dong, J. Jin, Q. Wang, and N. Li, “Vision, requirements and network architecture of 6G mobile network beyond 2030,” _China Communications_ , vol. 17, no. 9, pp. 92–104, 2020.
  * [3] H. Yang, A. Alphones, Z. Xiong, D. Niyato, J. Zhao, and K. Wu, “Artificial-Intelligence-Enabled Intelligent 6G Networks,” _IEEE Network_ , vol. 34, no. 6, pp. 272–280, 2020.
  * [4] M. I. Ashraf, Chen-Feng Liu, M. Bennis, and W. Saad, “Towards Low-Latency and Ultra-Reliable Vehicle-to-Vehicle Communication,” in _2017 European Conference on Networks and Communications (EuCNC)_ , 2017, pp. 1–5.
  * [5] A. K. Gizzini and M. Chafii, “A Survey on Deep Learning Based Channel Estimation in Doubly Dispersive Environments,” _IEEE Access_ , vol. 10, pp. 70 595–70 619, 2022.
  * [6] H. Huang, S. Guo, G. Gui, Z. Yang, J. Zhang, H. Sari, and F. Adachi, “Deep Learning for Physical-Layer 5G Wireless Techniques: Opportunities, Challenges and Solutions,” _IEEE Wireless Communications_ , vol. 27, no. 1, pp. 214–222, 2020.
  * [7] S. Han, Y. Oh, and C. Song, “A Deep Learning Based Channel Estimation Scheme for IEEE 802.11p Systems,” in _IEEE International Conference on Communications (ICC)_ , 2019, pp. 1–6.
  * [8] A. K. Gizzini, M. Chafii, A. Nimr, and G. Fettweis, “Deep Learning Based Channel Estimation Schemes for IEEE 802.11p Standard,” _IEEE Access_ , vol. 8, pp. 113 751–113 765, 2020.
  * [9] A. K. Gizzini, M. Chafii, A. Nimr, and G. Fettweis, “Joint TRFI and Deep Learning for Vehicular Channel Estimation,” in _IEEE GLOBECOM 2020_ , Taipei, Taiwan, Dec. 2020.
  * [10] A. Karim Gizzini, M. Chafii, A. Nimr, R. M. Shubair, and G. Fettweis, “CNN Aided Weighted Interpolation for Channel Estimation in Vehicular Communications,” _IEEE Transactions on Vehicular Technology_ , vol. 70, no. 12, pp. 12 796–12 811, 2021.
  * [11] H. Ye, G. Y. Li, and B. Juang, “Power of Deep Learning for Channel Estimation and Signal Detection in OFDM Systems,” _IEEE Wireless Communications Letters_ , vol. 7, no. 1, pp. 114–117, 2018.
  * [12] X. Ma, H. Ye, and Y. Li, “Learning Assisted Estimation for Time- Varying Channels,” in _2018 15th International Symposium on Wireless Communication Systems (ISWCS)_ , 2018, pp. 1–5.
  * [13] Y. Yang, F. Gao, X. Ma, and S. Zhang, “Deep Learning-Based Channel Estimation for Doubly Selective Fading Channels,” _IEEE Access_ , vol. 7, pp. 36 579–36 589, 2019.
  * [14] J. A. Fernandez, K. Borries, L. Cheng, B. V. K. Vijaya Kumar, D. D. Stancil, and F. Bai, “Performance of the 802.11p Physical Layer in Vehicle-to-Vehicle Environments,” _IEEE Transactions on Vehicular Technology_ , vol. 61, no. 1, pp. 3–14, 2012.
  * [15] Yoon-Kyeong Kim, Jang-Mi Oh, Yoo-Ho Shin, and Cheol Mun, “Time and Frequency Domain Channel Estimation Scheme for IEEE 802.11p,” in _17th International IEEE Conference on Intelligent Transportation Systems (ITSC)_ , 2014, pp. 1085–1090.
  * [16] A. K. Gizzini and M. Chafii, “RNN Based Channel Estimation in Doubly Selective Environments,” _IEEE Transactions on Machine Learning in Communications and Networking_ , vol. 2, pp. 1–18, 2024.
  * [17] J. Pan, H. Shan, R. Li, Y. Wu, W. Wu, and T. Q. S. Quek, “Channel Estimation Based on Deep Learning in Vehicle-to-Everything Environments,” _IEEE Communications Letters_ , vol. 25, no. 6, pp. 1891–1895, 2021.
  * [18] A. K. Gizzini and M. Chafii, “Deep Learning Based Channel Estimation in High Mobility Communications Using Bi-RNN Networks,” in _ICC 2023 - IEEE International Conference on Communications_ , 2023, pp. 2607–2612.
  * [19] L. Li, H. Chen, H.-H. Chang, and L. Liu, “Deep Residual Learning Meets OFDM Channel Estimation,” _IEEE Wireless Communications Letters_ , vol. 9, no. 5, pp. 615–618, 2020.
  * [20] D. Luan and J. S. Thompson, “Channelformer: Attention Based Neural Solution for Wireless Channel Estimation and Effective Online Training,” _IEEE Transactions on Wireless Communications_ , vol. 22, no. 10, pp. 6562–6577, 2023.
  * [21] D. Kaur, S. Uslu, A. Durresi, S. Badve, and M. Dundar, “Trustworthy explainability acceptance: A new metric to measure the trustworthiness of interpretable ai medical diagnostic systems,” in _Complex, Intelligent and Software Intensive Systems_ , L. Barolli, K. Yim, and T. Enokido, Eds. Cham: Springer International Publishing, 2021, pp. 35–46.
  * [22] I. E. Nielsen, D. Dera, G. Rasool, R. P. Ramachandran, and N. C. Bouaynaya, “Robust Explainability: A Tutorial on Gradient-based Attribution Methods for Deep Neural Networks,” _IEEE Signal Processing Magazine_ , vol. 39, no. 4, pp. 73–84, 2022.
  * [23] V. Arya, R. K. Bellamy, P.-Y. Chen, A. Dhurandhar, M. Hind, S. C. Hoffman, S. Houde, Q. V. Liao, R. Luss, A. Mojsilović _et al._ , “One explanation does not fit all: A toolkit and taxonomy of ai explainability techniques,” _arXiv preprint arXiv:1909.03012_ , 2019.
  * [24] W. Guo, “Explainable Artificial Intelligence for 6G: Improving Trust between Human and Machine,” _IEEE Communications Magazine_ , vol. 58, no. 6, pp. 39–45, 2020.
  * [25] Y. Wu, G. Lin, and J. Ge, “Knowledge-Powered Explainable Artificial Intelligence for Network Automation toward 6G,” _IEEE Network_ , vol. 36, no. 3, pp. 16–23, 2022.
  * [26] B. Brik, H. Chergui, L. Zanzi, F. Devoti, A. Ksentini, M. S. Siddiqui, X. Costa-Pérez, and C. Verikoukis, “A Survey on Explainable AI for 6G O-RAN: Architecture, Use Cases, Challenges and Research Directions,” 2023.
  * [27] F. Rezazadeh, H. Chergui, L. Alonso, and C. Verikoukis, “SliceOps: Explainable MLOps for Streamlined Automation-Native 6G Networks,” 2023.
  * [28] F. Rezazadeh, H. Chergui, and J. Mangues-Bafalluy, “Explanation-Guided Deep Reinforcement Learning for Trustworthy 6G RAN Slicing,” in _2023 IEEE International Conference on Communications Workshops (ICC Workshops)_ , 2023, pp. 1026–1031.
  * [29] S. M. Lundberg and S.-I. Lee, “A unified approach to interpreting model predictions,” _Advances in neural information processing systems_ , vol. 30, 2017.
  * [30] P. Barnard, I. Macaluso, N. Marchetti, and L. A. DaSilva, “Resource Reservation in Sliced Networks: An Explainable Artificial Intelligence (XAI) Approach,” in _ICC 2022 - IEEE International Conference on Communications_ , 2022, pp. 1530–1535.
  * [31] A.-D. Marcu, S. K. Gowtam Peesapati, J. Moysen Cortes, S. Imtiaz, and J. Gross, “Explainable Artificial Intelligence for Energy-Efficient Radio Resource Management,” in _2023 IEEE Wireless Communications and Networking Conference (WCNC)_ , 2023, pp. 1–6.
  * [32] N. Khan, S. Coleri, A. Abdallah, A. Celik, and A. M. Eltawil, “Explainable and Robust Artificial Intelligence for Trustworthy Resource Management in 6G Networks,” _IEEE Communications Magazine_ , pp. 1–7, 2023.
  * [33] S. K. Jagatheesaperumal, Q.-V. Pham, R. Ruby, Z. Yang, C. Xu, and Z. Zhang, “Explainable AI Over the Internet of Things (IoT): Overview, State-of-the-Art and Future Directions,” _IEEE Open Journal of the Communications Society_ , vol. 3, pp. 2106–2136, 2022.
  * [34] M. Zolanvari, Z. Yang, K. Khan, R. Jain, and N. Meskin, “TRUST XAI: Model-Agnostic Explanations for AI With a Case Study on IIoT Security,” _IEEE Internet of Things Journal_ , vol. 10, no. 4, pp. 2967–2978, 2023.
  * [35] M. T. Ribeiro, S. Singh, and C. Guestrin, “Why Should I Trust You? Explaining the Predictions of Any Classifier,” in _Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining_ , 2016, pp. 1135–1144.
  * [36] A. K. Gizzini, Y. Medjahdi, A. J. Ghandour, and L. Clavier, “Towards Explainable AI for Channel Estimation in Wireless Communications,” _IEEE Transactions on Vehicular Technology_ , pp. 1–6, 2023.
  * [37] A. Abdelgader and L. Wu, “The Physical Layer of the IEEE 802.11 p WAVE Communication Standard: The Specifications and Challenges,” in _The Physical Layer of the IEEE 802.11 p WAVE Communication Standard: The Specifications and Challenges_ , vol. 2, 10 2014.
  * [38] J. J. Bussgang, “Crosscorrelation functions of amplitude-distorted gaussian signals,” 1952.
  * [39] A. K. Gizzini and M. Chafii, “Low complex methods for robust channel estimation in doubly dispersive environments,” _IEEE Access_ , vol. 10, pp. 34 321–34 339, 2022.
  * [40] S. P. Boyd and L. Vandenberghef, _Convex optimization_. Cambridge university press, 2004.
  * [41] I. Sen and D. W. Matolak, “Vehicle–Vehicle Channel Models for the 5-GHz Band,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 9, no. 2, pp. 235–245, 2008.
  * [42] A. K. Gizzini, M. Chafii, A. Nimr, and G. Fettweis, “Enhancing Least Square Channel Estimation Using Deep Learning,” in _2020 IEEE 91st Vehicular Technology Conference (VTC2020-Spring)_ , 2020, pp. 1–5.
  * [43] V. Sovrasov. (2018-2023) ptflops: a Flops Counting Tool for Neural Networks in Pytorch Framework. [Online]. Available: https://github.com/sovrasov/flops-counter.pytorch

  *[OFDM]: orthogonal frequency division multiplexing
  *[CP]: cyclic prefix
  *[LS]: least squares
  *[LMMSE]: linear minimum mean square error
  *[DL]: deep learning
  *[FNN]: feed-forward neural network
  *[DPA]: data-pilot aided
  *[STA]: spectral temporal averaging
  *[TRFI]: time-domain reliable test frequency domain interpolation
  *[RNN]: recurrent neural network
  *[CNN]: convolutional neural network
  *[XAI]: explainable artificial intelligence
  *[BER]: bit error rate
  *[AWGN]: additive white Gaussian noise
  *[SNR]: signal-to-noise ratio
  *[MSE]: mean squared error


* Lindhard  _et al._ (1963) J Lindhard, V Nielsen, M Scharff, and P V Thomsen, “Integral equations governing radiation effects. (notes on atomic collisions, iii),” Kgl. Danske Videnskab., Selskab. Mat. Fys. Medd. 1804.0945933 (1963).
  * Cadeddu  _et al._ (2021c) M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, and E. Picciau, “Muon and electron g-2 and proton and cesium weak charges implications on dark Zd models,” Phys. Rev. D 104, 011701 (2021c), arXiv:2104.03280 [hep-ph] .
  * Safronova  _et al._ (2018) M.S. Safronova, D. Budker, D. DeMille, Derek F. Jackson Kimball, A. Derevianko, and Charles W. Clark, “Search for new physics with atoms and molecules,” Reviews of Modern Physics 90 (2018), 10.1103/revmodphys.90.025008.
  * Corona  _et al._ (2022b) M. Atzori Corona, M. Cadeddu, N. Cargioli, P. Finelli, and M. Vorabbi, “Incorporating the weak mixing angle dependence to reconcile the neutron skin measurement on ${}^{208}\mathrm{Pb}$ by prex-ii,” Phys. Rev. C 105, 055503 (2022b).
  * Tanabashi  _et al._ (2018) M. Tanabashi _et al._ (Particle Data Group), “Review of Particle Physics,” Phys. Rev. D98, 030001 (2018).
  * Erler and Ramsey-Musolf (2005) Jens Erler and Michael J. Ramsey-Musolf, “The Weak mixing angle at low energies,” Phys. Rev. D72, 073003 (2005), arXiv:hep-ph/0409169 [hep-ph] .
  * Erler and Ferro-Hernández (2018) Jens Erler and Rodolfo Ferro-Hernández, “Weak Mixing Angle in the Thomson Limit,” JHEP 03, 196 (2018), arXiv:1712.09146 [hep-ph] .
  * Wood  _et al._ (1997) C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner, and Carl E. Wieman, “Measurement of parity nonconservation and an anapole moment in cesium,” Science 275, 1759–1763 (1997).
  * Dzuba  _et al._ (2012) V. A. Dzuba, J. C. Berengut, V. V. Flambaum, and B. Roberts, “Revisiting parity non-conservation in cesium,” Phys. Rev. Lett. 109, 203003 (2012), arXiv:1207.5864 [hep-ph] .
  * Anthony  _et al._ (2005) P. L. Anthony _et al._ (SLAC E158), “Precision measurement of the weak mixing angle in Moller scattering,” Phys. Rev. Lett. 95, 081601 (2005), hep-ex/0504049 [hep-ex] .
  * Wang  _et al._ (2014) D. Wang _et al._ (PVDIS), “Measurement of parity violation in electron–quark scattering,” Nature 506, 67–70 (2014).
  * Androic  _et al._ (2018) D. Androic _et al._ (Qweak), “Precision measurement of the weak charge of the proton,” Nature 557, 207–211 (2018).
  * Aristizabal Sierra  _et al._ (2022) D. Aristizabal Sierra, V. De Romeri, and D. K. Papoulias, “Consequences of the Dresden-II reactor data for the weak mixing angle and new physics,” (2022), arXiv:2203.02414 [hep-ph] .
  * Deniz  _et al._ (2010) M. Deniz _et al._ (TEXONO), “Measurement of Neutrino-Electron Scattering Cross-Section with a CsI(Tl) Scintillating Crystal Array at the Kuo-Sheng Nuclear Power Reactor,” Phys. Rev. D81, 072001 (2010), arXiv:0911.1597 [hep-ex] .
  * Ahrens  _et al._ (1990) L.A. Ahrens, S.H. Aronson, P.L. Connolly, B.G. Gibbard, M.J. Murtagh, _et al._ , “Determination of electroweak parameters from the elastic scattering of muon-neutrinos and anti-neutrinos on electrons,” Phys. Rev. D41, 3297–3316 (1990).
  * Hirsch  _et al._ (2003) Martin Hirsch, Enrico Nardi, and Diego Restrepo, “Bounds on the tau and muon neutrino vector and axial vector charge radius,” Phys. Rev. D 67, 033005 (2003), arXiv:hep-ph/0210137 .
  * Vidyakin  _et al._ (1992) G. S. Vidyakin, V. N. Vyrodov, I. I. Gurevich, Yu. V. Kozlov, V. P. Martemyanov, S. V. Sukhotin, V. G. Tarasenkov, E. V. Turbin, and S. Kh. Khakhimov, “Limitations on the magnetic moment and charge radius of the electron-anti-neutrino,” JETP Lett. 55, 206–210 (1992).
  * Allen  _et al._ (1993) R. C. Allen, H. H. Chen, P. J. Doe, R. Hausammann, W. P. Lee, X. Q. Lu, H. J. Mahler, M. E. Potter, K. C. Wang, T. J. Bowles, R. L. Burman, R. D. Carlini, D. R. F. Cochran, J. S. Frank, E. Piasetzky, V. D. Sandberg, D. A. Krakauer, and R. L. Talaga, “Study of electron-neutrino—electron elastic scattering at lampf,” Phys. Rev. D 47, 11–28 (1993).
  * Auerbach  _et al._ (2001) L. B. Auerbach _et al._ (LSND), “Measurement of electron-neutrino electron elastic scattering,” Phys. Rev. D63, 112001 (2001), hep-ex/0101039 .
  * Vilain  _et al._ (1995) P. Vilain _et al._ (CHARM-II), “Experimental study of electromagnetic properties of the muon-neutrino in neutrino - electron scattering,” Phys. Lett. B 345, 115–118 (1995).
  * Collar  _et al._ (2019) J. I. Collar, A. R. L. Kavner, and C. M. Lewis, “Response of CsI[Na] to Nuclear Recoils: Impact on Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS),” Phys. Rev. D 100, 033003 (2019), arXiv:1907.04828 [nucl-ex] .
  * Gninenko  _et al._ (2007) S. N. Gninenko, N. V. Krasnikov, and A. Rubbia, “Search for millicharged particles in reactor neutrino experiments: A Probe of the PVLAS anomaly,” Phys. Rev. D 75, 075014 (2007), arXiv:hep-ph/0612203 .
  * Li  _et al._ (2003) H. B. Li _et al._ (TEXONO), “Limit on the electron neutrino magnetic moment from the Kuo-Sheng reactor neutrino experiment,” Phys. Rev. Lett. 90, 131802 (2003), arXiv:hep-ex/0212003 .
  * Studenikin (2014) Alexander Studenikin, “New bounds on neutrino electric millicharge from limits on neutrino magnetic moment,” Europhys.Lett. 107, 21001 (2014), arXiv:1302.1168 [hep-ph] .
  * Beda  _et al._ (2012) A.G. Beda, V.B. Brudanin, V.G. Egorov, D.V. Medvedev, V.S. Pogosov, _et al._ , “The results of search for the neutrino magnetic moment in GEMMA experiment,” Adv.High Energy Phys. 2012, 350150 (2012).
  * Chen  _et al._ (2014b) Jiunn-Wei Chen, Hsin-Chang Chi, Hau-Bin Li, C. P. Liu, Lakhwinder Singh, Henry T. Wong, Chih-Liang Wu, and Chih-Pan Wu (TEXONO), “Constraints on millicharged neutrinos via analysis of data from atomic ionizations with germanium detectors at sub-keV sensitivities,” Phys. Rev. D90, 011301 (2014b), arXiv:1405.7168 [hep-ph] .
  * Wong  _et al._ (2007) H. T. Wong _et al._ (TEXONO), “A Search of Neutrino Magnetic Moments with a High-Purity Germanium Detector at the Kuo-Sheng Nuclear Power Station,” Phys. Rev. D 75, 012001 (2007), arXiv:hep-ex/0605006 .
  * Das  _et al._ (2020) Arindam Das, Diptimoy Ghosh, Carlo Giunti, and Arun Thalapillil, “Neutrino charge constraints from scattering to the weak gravity conjecture to neutron stars,” Phys. Rev. D 102, 115009 (2020), arXiv:2005.12304 [hep-ph] .
  * Abe  _et al._ (2020) K. Abe _et al._ (XMASS), “Search for exotic neutrino-electron interactions using solar neutrinos in XMASS-I,” Phys. Lett. B 809, 135741 (2020), arXiv:2005.11891 [hep-ex] .
  * Daraktchieva  _et al._ (2005) Z. Daraktchieva _et al._ (MUNU), “Final results on the neutrino magnetic moment from the MUNU experiment,” Phys. Lett. B 615, 153–159 (2005), arXiv:hep-ex/0502037 .
  * Liao  _et al._ (2022) Jiajun Liao, Hongkai Liu, and Danny Marfatia, “Implications of the first evidence for coherent elastic scattering of reactor neutrinos,” (2022), arXiv:2202.10622 [hep-ph] .
  * Alekseev  _et al._ (2022) I. Alekseev _et al._ (nuGeN), “First results of the nuGeN experiment on coherent elastic neutrino-nucleus scattering,” (2022), arXiv:2205.04305 [nucl-ex] .