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Citation: Mirbadin, A.; Zaraki, A.
Partial Path Overlapping Mitigation:
An Initial Stage for Joint Detection
and Decoding in Multipath Channels
Using the Sum–Product Algorithm.
Appl. Sci. 2024,14, 9175. https://
doi.org/10.3390/app14209175
Academic Editor: Christos Bouras
Received: 8 September 2024
Revised: 30 September 2024
Accepted: 8 October 2024
Published: 10 October 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
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Attribution (CC BY) license (https://
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4.0/).
applied
sciences
Article
Partial Path Overlapping Mitigation: An Initial Stage for Joint
Detection and Decoding in Multipath Channels Using the
Sum–Product Algorithm
Anoush Mirbadin 1,*,† and Abolfazl Zaraki 2,†
1Department of Engineering, SELTA Digital Platforms Company, 29010 Cadeo, Italy
2School of Physics, Engineering and Computer Science (SPECS), University of Hertfordshire,
Hatfield AL10 9AB, UK; a.zaraki@herts.ac.uk
*Correspondence: anoush.mirbadin@dplatforms.it
† These authors contributed equally to this work.
Abstract: This paper addresses the problem of mitigating unknown partial path overlaps in commu-
nication systems. This study demonstrates that by utilizing the front-end insight of communication
systems along with the sum–product algorithm applied to factor graphs, it is possible not only to
track these overlapping components accurately, but also to detect all multipath channel impairments
simultaneously. The proposed methodology involves discretizing channel parameters, such as chan-
nel paths and attenuation coefficients, to ensure the most accurate computation of means of Gaussian
observations. These parameters are modeled as Bernoulli random variables with priors set to 0.5.
A notable aspect of the algorithm is its integration of the received signal power into the calculation
of noise variance, which is critical for its performance. To further reduce the receiver complexity, a
novel implementation strategy, based on provided pre-defined look up tables (LOTs) to the reciver, is
introduced. The simulation results, covering both distributed and concentrated pilot scenarios, reveal
that the algorithm performs almost equally under both conditions and surpasses the established
upper bound in performance.
Keywords: joint; detection; decoding; multipath
1. Introduction
The initial approaches to reduce the effect of multipath impairment are broadly catego-
rized into three main groups: diversity, multicarrier transmission, and their combinations.
Diversity considerably reduces the probability of simultaneous fade on several replicas of
the same information signal. Various diversity transmission techniques for fading envi-
ronments have been extensively studied in [
1
–
5
]. In multicarrier modulation schemes like
orthogonal frequency division multiplexing (OFDM), employing longer symbol durations
effectively reduces the intersymbol interference (ISI), which can further be mitigated by
utilizing cyclic prefixes. The effectiveness of OFDM and its application in combating ISI
channels has been well investigated in [
6
]. Additionally, the combination of multicar-
rier modulation and diversity techniques has also been well addressed in the literature,
e.g., [7,8].
Later, with the emergence of machine learning algorithms like expectation propagation
(EP), new attempts to mitigate multipath phenomena in both time and frequency domains
have been pursued [
9
,
10
]. However, EP has its drawbacks, such as numerical instabili-
ties when performing distribution divisions [
11
]. As a result, our focus shifts toward the
fundamental aspects of wireless communications. More recently, AI-assisted algorithms
have been proposed to handle multipath channels [
12
]. In the first two studies, the chan-
nel parameters, such as the number of paths, path gains, and delays, are known as the
receiver. To provide additional clarity, we will briefly outline the algorithm proposed in [
9
].
Appl. Sci. 2024,14, 9175. https://doi.org/10.3390/app14209175 https://www.mdpi.com/journal/applsci
Appl. Sci. 2024,14, 9175 2 of 10
The receiver utilizes an equalizer based on EP. This equalizer employs the decoder feedback
to execute the EP. Specifically, the posteriors generated by the decoder are mapped into
Gaussian messages, with their means and variances dependent on channel impairments,
such as channel taps. A similar methodology is applied in the frequency domain in [
10
].
In the last work, the number of channel taps is fixed, and it is assumed that the channel
taps follow a Gaussian distribution with an exponential power profile, but the channel is
not dense over a block of the received signal.
The main contribution of this paper is the design of a receiver that, instead of de-
pending on the low-pass equivalent of the channels, is based on front-end communication
systems. This design allows for the use of the sum–product algorithm (SPA) to simultane-
ously track all multipath channel parameters. In addition, a discretization and modeling
technique for unknown channel parameters is introduced, where the parameters are rep-
resented as Bernoulli random variables. The SPA is then used to track the unknown
parameters. Notably, the receiver does not require prior knowledge of the channel param-
eters, such as the number of paths or Gaussian channel taps. Moreover, extending this
algorithm to the general multipath channels is straightforward.
This paper introduces a novel approach for performing joint partial path overlapping
detection and channel decoding symbol by symbol in a dense partial path overlapping over
a block of received symbols, utilizing the SPA on factor graphs. By focusing on front-end
communication systems, we can track all unknown channel parameters through SPA on
factor graphs. While this approach, i.e., the front-end receiver design using SPA, was
initially proposed by the authors of this paper in [
13
], its advantages quickly gained the
attention of researchers [
14
], where it has been named as discrete time models. In our
study, we discretize channel parameters such as channel paths and attenuation coefficients,
both in the channel low-pass equivalent (for illustrative purposes) and in the front-end
strategy. These parameters are modeled as Bernoulli random variables with priors set
to 0.5, denoted as path existence (PE) and attenuation existence (AE). Given a vector of
observations, we factorize the joint posterior distribution of channel model parameters
and codeword bits. SPA is then applied to the corresponding factor graph to compute the
marginals of the desired random variables. The exact implementation of SPA results in a
significant computational complexity. To reduce this considerably, we propose the use of
pre-defined look-up tables (LOTs) at the receiver. The dimension of LOTs depends on the
constellation symbols’ dimensionality and the discretization levels of unknown channel
parameters. Average signal power at the receiver determines the noise variance. However,
when we compensate the reflected components in the means of Gaussain observations,
we should remove the power of those components from the noise variance. This can
also be accomplished through the provided LOTs for the receiver. We found that the
constellation size dictates the quantity of estimated patterns of reflected rays, with only
one occurring at each time epoch, similar to additive noise. Therefore, the receiver relies
on making hard decisions based on the provided posteriors by the decoder to select the
most suitable one among them. Performance evaluations are conducted through bit error
rate (BER) curves in both distributed and concentrated pilots scenarios. As none of the
channel parameters are known to the receiver, previous works, such as [
9
], were unable to
effectively track these parameters. This limitation highlights the need to modify existing
algorithms. For this reason, we introduce a relative solution as the upper bound of the
system performance. The relative solution can also serve as a comparison benchmark in
this context. Our simulation results prove that by increasing receiver’ iterations, the BER
curves differ from the relative solution considerably.
The paper is organized as follows: Section 2introduces the discretization strategy for
channel parameters, followed by the system model in Section 3. Section 4discusses the
corresponding factor graph and its SPA equations. Receiver implementations are detailed
in Section 5, with performance analysis provided in Section 6. Finally, conclusions are
drawn in Section 7.
Appl. Sci. 2024,14, 9175 3 of 10
2. Discretization Strategy
Consider the general model for a
L
-Path multipath channel (Equations (13.1)–(5)
of [15], Equations (3.3)–(3.6) of [16])
h(τ;t) =
L−1
∑
i=0
αi(t)exp(−j2πfcτi(t)) expj2πvicos ϕi(t)
c0
fctδ(τ−τi(t)) (1)
in which
αi(t)
is the real-valued gain of
i
-th path,
exp(−j
2
πfcτi(t))
is the phase shift (often,
these two terms are considered as one term, called the complex attenuation factor),
fc
is the
carrier frequency,
τi(t)
is the propagation delay of
i
-th path,
vi
is the velocity associated
with the
i
-th path,
ϕi(t)
is the angle of arrival of the wave (for the sake of simplicity, its
value is assumed zero throughout this article), and
c0
is the speed of light. We consider that
the channel remains constant during each symbol period. This implies that the channel
can change symbol by symbol, i.e.,
L
changes symbol by symbol. In this project, we model
the path gain in (1) by the exponential function as
αi(t) = exp(−τi(t)ai)
in which
ai
is the
attenuation coefficient. It is assumed that
τi
is a Weibull random variable. The parameter
of the Weibull distribution is adjusted such that its behavior closes a semi-exponential
function. We assume that each
τi(t)
is smaller than the symbol duration
T
. This ensures
partial paths overlapping across the successive symbol, and guarantees dense overlapping
environment over a block of received symbols. At the receiver side, the following channel
model is employed, i.e., all channel hypotheses are considered.
h(τ;t) =
N−1
∑
i=0
PEiexp
−iTN
N′
∑
i′=1
AEi′i′a′
N′
exp(−j2πfciTN)
×exp
j2π
N′′
∑
i′′ =1
VEi′′ i′′ VN′′
c0
fct
δ(τ−iTN)
(2)
where
TN=T
N
is the delay resolution (
T
is the symbol duration),
a′
N′=1
N′
is the attenuation
resolution, and
VN′′ =Vm
N′′
is the velocity resolution (in which
Vm
is the maximum possible
velocity).
PE
,
AE
, and
VE
are Bernoulli random variables representing path, attenuation,
and velocity existences, respectively. Their prior probabilities are set to 0.5. If
N
,
N′
, and
N′′
go to infinity and we have exactly the knowledge of
PE
,
AE
, and
VE
, the error modeling
goes to zero. It is important to highlight that the low-pass equivalent equations mentioned
above primarily represent our discretization strategy. Our simulations are rooted in our
front-end approach.
3. System Model
We start this research from the simplest case, i.e., baseband communications, to avoid
missing all aspects of this research field. We consider a baseband communication system
in which symbol duration is much smaller than the coherent time of the channel, and the
Doppler effect is negligible. This assumption is unavoidable unless otherwise the baseband
pulses will be broadened at the receiver due to the Doppler effect, and a high amount of
interference corrupts the communication.
Consider a modern baseband communication system. The transmitter encodes data
bits using a low-density parity-check (LDPC) encoder. Pilot symbols are inserted to enhance
the performance of the receiver. This is followed by a binary pulse amplitude (B-PAM)
modulator. The coded modulation symbols
c= (c0
,
c1
, ...,
cK−1)
are pulsed-shaped using
the root-raised cosine filter and then transmitted over an additive white Gaussian noise
(AWGN) channel impaired by path overlapping.
Appl. Sci. 2024,14, 9175 4 of 10
4. Factor Graph
We consider a scenario where a maximum of
N
potential paths are considered during
each time epoch, with
N−
1 of these paths susceptible to interference to the successive
symbol. Additionally, we account for a maximum of
N′
discretization levels for attenuation
coefficients. To illustrate, let us examine a channel model with
N=
2. Here, at each
time epoch, three paths emerge: the potential interfering path from the preceding symbol;
the delayed version of the current symbol, encompassing segments of both the current and
preceding symbols; and the direct line-of-sight (LOS) signal. Each of these paths undergoes
processing via the receiver-matched filter to calculate the conditional Gaussian observation
mean. To facilitate this, we conduct offline computations of the receiver-matched filter
for all possible paths, constructing a LOT to simplify the message-passing algorithm.
Consequently, four values, denoted as
βk
,
γk
,
ζk
, and
ψk
, where
k
denotes the current time
epoch, are acquired at the output of the receiver-matched filter. From the system model,
the observation at time kcan be determined by
rk=βk+γk+ζk+ψk+nk,(k=0, 1, ..., K−1)(3)
The value
rk
is acquired post receiver-matched filtering. The sequence
{nk}
comprises
independent and identically distributed (i.i.d.) Gaussian noise components, denoted as
nk∼ N (nk
; 0,
σ2)
.
ψk
results from matched filtering of the LOS component.
βk
arises from
matched filtering a segment of the preceding symbol at time
k−
1, representing the delayed
version of the symbol at time
k−
1.
γk
stems from matched filtering a segment of the
preceding symbol at time
k
.
ζk
is obtained through matched filtering a segment of the
current symbol at time epoch
k
. The computation of these signals, considering Equation (2),
is summarized as
ψk=[ckp(n)]NTN
n=0∗p∗(−n)|n=NTN(4)
in which
p(n)
is the pulse shaping filter (here it is the root-raised cosine one).
NTN
is the
symbol duration.
βk=exp−lTNl′a′(N′)[ck−1p(n)]NTN
n=NTN−lTN∗p∗(−n)|n=NTN(5)
where land l′are detected channel parameters for the non-LOS path at time epoch k−1.
γk=exp−l′′ TNl′′′ a′(N′)[ck−1p(n)]NTN−l′′ TN
n=0∗p∗(−n)|n=NTN(6)
ζk=exp−l′′ TNl′′′ a′(N′)[ckp(n)]NTN
n=NTN−l′′ TN
∗p∗(−n)|n=NTN(7)
where
l′′
and
l′′′
are the detected channel parameters for the non-LOS path at time epoch
k
.
The receiver can be implemented through factorizing the joint posterior distribution
of data bits and channel parameters, given the observation samples after the matched filter
and applying the SPA to the corresponding factor graph.
p(b,PE,AE|r)∝P(b)P(PE)P(AE)(r|b,PE,AE)∝
K−1
∏
k=0
prk|ck,ck−1,PEk,PEk−1,AEk,AEk−1N−1
∏
k′=0
P(PEk
k′)
N′
∏
k′′ =1
P(AEk′
k′′ )×
N−1
∏
k′′′ =1
P(PEk−1
k′′′ )
N′
∏
k′′′′ =1
P(AEk′′
k′′′′ )T[c=ȷc(b)]
(8)
where
T[
.
]
is the code indicator function that is equal to 1 if
c
is the codeword corresponding
to
b
, and to 0 otherwise. The detector estimates different patterns for paths at each time,
conditioned on
ck−1
, with only one occurring at any given moment. This concept will be
Appl. Sci. 2024,14, 9175 5 of 10
further explained in the following section. The corresponding factor graph is depicted in
Figure 1. One can conclude by considering the mentioned example.
prk|ck,ck−1,PEk,PEk−1,AEk,AEk−1∝exp(−(rk−(βk+γk+ζk))2
2σ2)(9)
Code Check Nodes
ck−1ck
Pdk−1
pk−1
uk−1(ck−1)Pk
dk(ck)
Pk−1
dk(ck−1)
Pk
uk−1(ck−1)
prk−1|ck−1, ck−2,PE k−1,P Ek−2,AEk−1,AE k−2prk|ck, ck−1,P Ek,P Ek−1,AE k,AEk−1
... ...
PE k−1
0PE k−1
1PE k−1
N−1PE k
0
PE k
1
PE k
N−1
P(PE k−1
0)P(PE k−1
1)P(PE k−1
N−1)P(PE k
0)P(PE k
1)
P(PE k
N−1)
......
pk−1
uk−1(PE k−1
1)
pk
dk−1(PE k−1
1)
...
...
...
AEN−1
1AEN−1
2AEN−1
N′
AE0
1AE0
2AE0
N′...
...
... AE0
1
AE0
1
AE0
N′
AEN−1
1
AEN−1
2
AEN−1
N′
P(AE0
1)P(AE0
2)
P(AE0
N′)
P(AEN−1
1)P(AEN−1
2)
P(AEN−1
N′)
P(AEN−1
1)P(AEN−1
2)P(AEN−1
N′)
P(AE0
1)P(AE0
2)P(AE0
N′)
pk−1
uk−1(AEN−1
1)
pk
dk−1(AEN−1
1)
Figure 1. Factor graph representation of Equation (8). The receiver contains two detection algorithms,
each corresponding to a specific unknown channel parameter. The variable node
PEk
i
represents the
i-th path existence at time k, while AEk
irepresents the i-th attenuation existence at the same time.
5. Implementation of the Receiver
The iterative receiver consists of two algorithms to detect
PE
s and
AE
s. This is
followed by hard decisions on variable nodes. The detection is performed bit by bit.
This detection process iterates, and after several iterations, LDPC decoding is executed.
The entire process is reiterated a few times, followed by error rate computations at the
receiver. During forward recursion, backward messages are not available. To compute the
forward message at time epoch
k
, we assume that the decoder and detector have made
their decisions on
ck−1
,
PEk−1
, and
AEk−1
at time
k−
1, thus, they are known. Now, we
compute the messages
pk
uk(ck)
and
pk+1
uk(PEk)pk+1
uk(AEk)
. To achieve this, we calculate the
following messages by applying the SPA on the corresponding factor graph.
pk
uk(ck) =
N−1
∏
n=0
∑
PEk
n
P(PEk
n)
∑
AEn
1
P(AEn
1)... ∑
AEn
N′′
P(AEn
N′′ )
prk|ck,PEk,AEk(10)
pk+1
uk(PEk
i) = ∑
ck
Pk
dk(ck)
N−1
∏
n=0
n=i
∑
PEk
n
P(PEk
n)
∑
AEn
1
P(AEn
1)... ∑
AEn
N′′
P(AEn
N′′ )
×
∑
AEi
1
P(AEi
1)... ∑
AEi
N′′
P(AEi
N′′ )
P(PEk
n)prk|ck,PEk,AEk
(11)
Appl. Sci. 2024,14, 9175 6 of 10
pk+1
uk(AEj
i) = ∑
ck
Pk
dk(ck)
N′
∏
n=1
n=i
∑
AEj
n
P(AEj
n)
∑
PEk
0
P(PEk
0)... ∑
PEk
N−1
P(PEk
N−1)
×
N′
∏
m=1
m=j
∑
AEm
0
P(AEm
0)... ∑
AEm
N′
P(AEm
N′)
P(AEj
n)prk|ck,PEk,AEk
(12)
Provided the fact that
PE
and
AE
are Bernoulli random variables and only one (
PE,AE
)
pair occurs at every time epoch, the above equations form a Bahl–Cocke–Jelinek–Raviv
(BCJR) algorithm. In this case,
pk+1
uk(PEk
i)
and
pk+1
uk(AEj
i)
are equal. This is followed by
hard decisions to find the indices of (
PE
,
AE
) pair. A similar algorithm is employed to
compute the backward messages. It should be noted that at every time epoch, we make
hard decisions on
Pdk−1
and
Pdk+1
to determine the symbols
ck−1
and
ck+1
, respectively.
Finally, the bit log-likelihood ratios (bitLLRs) are computed by
puk(ck) = pk+1
uk(ck)pk
uk(ck)(13)
As implementing the SPA directly on such a factor graph is intricate, we propose
a simplified implementation strategy to streamline the process and mitigate complexity
issues. To realize Equations (10)–(12), we introduce four LOTs, each corresponding one-
to-one with possible combinations of
ck−1ck
, as per Equations (4) to (7). These LOTs have
dimensions
N×N′
. Consequently, potential Gaussian observation means are computed
prior to the receiver computations. Thus, the algorithm simplifies to:
• At every time epoch, the receiver performs matrix subtractions, rkminus LOTs.
•
Receiver finds the indices of the minimum matrices members of the detector’s up-
ward messages.
It is evident by considering the interference of the current symbol on many successive
symbols only the number of LOTs increases. The main algorithm could still be implemented
through matrix subtractions and finding the indices of the minimum value at every time.
To conclude this section, it is essential to highlight three key aspects of this research.
Firstly, as per the aforementioned process, the receiver provides overlapping patterns at
each time epoch, with the number of patterns equating to the constellation size. Only one
pattern occurs at a time. We employ feedback from the decoder to select the pattern corre-
sponding to the constellation symbol with the highest probability of occurrence. Secondly,
the average signal power at the receiver determines the noise variance. However, when
compensating for reflected components in the means of the Gaussian observations, it is
crucial to remove the power of those components from the noise variance. We simplify this
adjustment using the LOTs provided for the receiver. Finally, the Gaussian multiplication
is employed, whenever it is needed, in the message passing algorithm.
6. Simulation Results
Computer simulations were carried out to evaluate the proposed algorithm’s perfor-
mance in multipath scenarios. To this end, we draw the BER versus the signal-to-noise ratio
(SNR) curves. We consider a
(
3, 6
)
-regular LDPC code of length 4000. The modulation for-
mat is B-PAM. In the distributed pilots, one pilot is inserted in every 20 successive symbols,
while in the concentrated one a group of 100 pilots is inserted before and after the LDPC
Packet. In a baseband communication system, pulse shaping was carried out using a raised
cosine filter with a filter span of 10 per symbol, samples of 16 per symbol, i.e., dividing the
symbol duration to 16 (16 possible
PE
s), and the roll-off factor of 0.5. We assumed that the
symbol duration is much smaller than the coherent time of the channel; thus, no Doppler
effect (otherwise, the baseband pulses will be broadened at the receiver, due to the Doppler
effect). We also considered the transmitter and receiver filter delays in our simulations.
Attenuation coefficients were randomly chosen from a set of 100 numbers in the range of
[
0
:
99
]×1
100
. Please confirm this revision. To simplify the algorithm, we assume that LOS
Appl. Sci. 2024,14, 9175 7 of 10
always exists. The delay profile of the channel follows a Weibull distribution with a scale
parameter of 1 and a shape parameter of 0.4. We consider code rate in the computation
of noise variance at the receiver. We also set the minimum error packets to 100, detector
iterations to 10, maximum number of decoder iterations to 200, and global iterations of the
receiver to 5 and 30. We used lookup tables to compute the means of Gaussian likelihoods
according to the explained instructions in this paper. Figures 2–5show the simulation
results for the distributed and concentrated pilots. The curve labelled by “LDPC” was
obtained through pulse shaping, matched filtering and LDPC decoding in an AWGN
channel. The name of the rest of the curves is based on their multipath numbers and global
iterations of the receiver, e.g., 3-JDD Glob. It. 5 represents joint detection and decoding in
a scenario involving three unknown overlapping paths and 5 overall receiver iterations.
The relative solution is obtained through 200 iterations of the decoder and single iteration
of the receiver where the observations are computed by
exp−(rk−ck)2
2σ2
.
σ2
is inversely
proportional to the average power of the received signal. The relative solution can be used
to compare the performance of various algorithms in the multipath scenario. The more an
algorithm outperforms the relative solution, the more accurate it is. As the receiver lacks
knowledge of the channel parameters, previous methods have been unable to track them,
requiring changes to existing algorithms. In response, we present a relative solution that
establishes an upper bound for system performance. For simplicity, the simulation assumes
that only the number of interfering paths remains fixed at each time epoch.
The results indicate that the performances of distributed and concentrated pilots
are nearly identical under the same conditions. In the scenario with two overlapping
paths, the proposed algorithm experiences a degradation of 0.6 dB compared with LDPC,
and in the scenario with three overlapping paths, this degradation increases to 1 dB.
However, in every case, the proposed algorithm outperforms the relative solution by 0.2 dB.
Although the difference between 5 and 30 iterations of the receiver is negligible for two
overlapping paths, as the number of unknown paths increases, the difference becomes
more significant.
1 1.2 1.4 1.6 1.8 2 2.2 2.4
Eb/N0
10-4
10-3
10-2
10-1
BER
LDPC
2-JDD Glob. It. 30
2-JDD Glob. It. 5
Relative Solution
Figure 2. 2-JDD in the distributed pilots, including the lower and upper bounds (LDPC and rela-
tive solution).
Appl. Sci. 2024,14, 9175 8 of 10
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
Eb/N0
10-5
10-4
10-3
10-2
10-1
BER
LDPC
3-JDD Glob. It. 30
3-JDD Glob. It 5
Relative Solution
Figure 3. 3-JDD in the distributed pilots pilots, including the lower and upper bounds (LDPC and
relative solution).
1 1.2 1.4 1.6 1.8 2 2.2 2.4
Eb/N0
10-4
10-3
10-2
10-1
BER
LDPC
2-JDD Glob. It. 30
2-JDD Glob. It. 5
Relative Solution
Figure 4. 2-JDD in the concentrated pilots, including the lower and upper bounds (LDPC and
relative solution).
Appl. Sci. 2024,14, 9175 9 of 10
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
Eb/N0
10-4
10-3
10-2
10-1
BER
LDPC
3-JDD Glob. It. 30
3-JDD Glob. It. 5
Relative Solution
Figure 5. 3-JDD in the concentrated pilots, including the lower and upper bounds (LDPC and
relative solution).
We compute the complexity of the proposed algorithms in terms of the number of
accesses to the LOTs, the number of operations (additions and multiplications) between
two real arguments, and the number of comparisons to find the minimum value of a matrix
at each time epoch. The results are summarized in Table 1.
Table 1. Computational Load at each time epoch for B-PAM Modulation.
Operations LUT Accesses Comparisons
(4N2N′+8N′+4)(L−1)6N(L−1) + 4N2N′
2−NN′+N+2N′−6
7. Conclusions and Future Work
In this paper, the tracking of all multipath parameters using SPA was made possible
through the front-end consideration of the communication systems. This process was fol-
lowed by discretizing channel parameters and assigning them Bernoulli random variables.
The primary limitation of the algorithm is its complexity in detection of path patterns when
the number of interfering paths increases significantly. The use of quantum computing
algorithms for path pattern detection could offer a potential solution to this challenge.
In future work, we aim to compute the optimal discretization levels.
Author Contributions: Conceptualization, A.M. and A.Z.; methodology, A.M. and A.Z.; software,
A.M. and A.Z.; validation, A.M. and A.Z.; formal analysis, A.M. and A.Z.; investigation, A.M. and
A.Z.; resources, A.M. and A.Z.; data curation, A.M. and A.Z.; writing—original draft preparation, A.M.
and A.Z.; writing—review and editing, A.M. and A.Z.; visualization, A.M. and A.Z.; supervision,
A.M. and A.Z.; project administration, A.M. and A.Z. All authors have read and agreed to the
published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Appl. Sci. 2024,14, 9175 10 of 10
Informed Consent Statement: Not applicable.
Data Availability Statement: The original contributions presented in the study are included in the
article, further inquiries can be directed to the corresponding author.
Conflicts of Interest: Author Anoush Mirbadin was employed by the SELTA Digital Platforms
Company. The remaining authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential conflict of interest.
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