https://orcid.org/0000-0001-6219-6895
Figures
Abstract
In this paper, we propose a new non-orthogonal multiple access (NOMA)-aided Single-Input Multiple-Output (SIMO) multi-hop relay scheme using wirelessly energy harvesting (EH) and Fountain Codes (FCs). Specifically, a single antenna source node employs NOMA to transmit two data streams to two multiple antenna destinations with the assistance of N intermediate multiple antenna Decode-and-Forward (DF) relays. The destinations accumulate received information until sufficient data is gathered to successfully recover the original data. This configuration ensures the SIMO structure is maintained across all hops, as the relays utilize multiple antennas for reception and a single antenna for transmission. To enhance channel capacity at the relay and destination nodes, the selection combining (SC) technique is applied. For data transmission, both the source and relay nodes harvest wireless energy from a power station deployed within the network, follows a time-switching approach. We first derive exact expressions for the end-to-end (e2e) instantaneous signal-to-interference-plus-noise ratios (SINRs) at both destinations, along with their corresponding cumulative distribution functions (CDFs) over Rayleigh fading channels. Then, we derive analytical expressions for the average channel capacity (ACC-e2e) and outage probability (OP-e2e) at each destination and validate their accuracy through Monte Carlo simulations.
Citation: An NH, Tu L-T, Duy TT, Nguyen T-T (2026) On performance evaluation of NOMA-aided SIMO multi-hop schemes using energy harvesting and fountain coding based information accumulation. PLoS One 21(3): e0344332. https://doi.org/10.1371/journal.pone.0344332
Editor: Sushank Chaudhary, Guangdong University of Petrochemical Technology, CHINA
Received: May 20, 2025; Accepted: February 18, 2026; Published: March 27, 2026
Copyright: © 2026 An et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: This research is funded by the Posts and Telecommunications Institute of Technology in 2025.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Multi-hop relay networks [1–3] are commonly applied to self-configuring networks, such as wireless sensor networks, device-to-device communication networks, etc., without infrastructure support (or only partial support). In [4], the authors proposed and evaluated the end-to-end performance of cluster-based multi-hop relaying networks, where each cluster selects one cluster node to receive and forward the received data from the previous cluster, using decode-and-forward
technique. Similar to [4], the authors in [5] studied the e2e performance of multi-hop relay networks using node selection at each hop over Rayleigh fading channels. However, unlike [4], the study in [5] considered the impact of co-channel interference on the e2e performance. In [6], the authors evaluated performance of multi-hop relay networks, in terms of outage probability (OP) of data channels and intercept probability (IP) of eavesdropping channels. In addition, published work [6] investigated the trade-off between OP and IP, as well as the impact of hardware imperfection on the considered performance. In [7], the source and the selected relays at each cluster must adjust their transmit power to reduce the e2e IP below a desired value. Then, an optimal relay selection method at each hop is performed to improve the OP performance, in the presence of correlated data channels and multiple eavesdroppers. In [8], a multi-hop full-duplex relay scheme employing multiple-input multiple-output (MIMO) devices and short-packet communications (SPC) was proposed and analyzed. The scheme in [8] significantly improves the e2e performance in terms of throughput, energy efficiency, reliability and latency. Published work [9] evaluated the e2e block error rate and secrecy throughput of secure multi-hop Internet of Things (IoT) relay networks under the impact of imperfect channel state information (CSI). However, the multi-hop relaying schemes in [1–7] do not consider radio-frequency energy harvesting (RF-EH) as a means of powering the wireless nodes.
The RF-EH technique [10–12] can help address the limited power issue in self-configuring networks, allowing network nodes to harvest energy from wireless signals in their surrounding environment. In time-switching (TS) EH technique [13], the EH phase and the data transmission phase are performed sequentially, i.e., the transmitters first harvest RF energy and then use the harvested energy to transmit their data. In power-switching (PS) EH technique [13], receivers can simultaneously harvest energy and decode information from the received signals of transmitters. In particular, the received signals are divided into two parts: one part is used for EH, while the remaining part is used for data decoding. In [14], the authors designed a multi-hop device-to-device communication model that utilizes RF-EH and operates in an underlay cognitive environment, where the transmit power of secondary transmitters depends on the interference threshold and the energy collected from the power station. The authors of [15,16] extended the work in [14] to a cluster-based cognitive multi-hop scheme with node selection at each cluster. In addition, multiple power station and multi-antenna power station cases were considered in [15] and [16], respectively. In [17], the authors proposed a cooperative-based multi-hop transmission scheme powered by a single power station. Additionally, incremental cooperation at each hop and a partial relay selection method (see [18]) were examined in [17]. The authors of [19] studied a power station-aided multi-hop mobile-to-mobile relay scenario using cooperative communication at each hop, where all nodes were mobile and the channels were cascaded Rayleigh fading. Unlike [14–17] and [19], this paper applies non-orthogonal multiple access (NOMA) to the RF EH–aided multi-hop relaying scheme to improve system throughput, compared with the corresponding scheme without using NOMA.
The NOMA technique [20–22] enables a transmitter to send different data to multiple receivers at the same time, frequency and code. To achieve this, a power allocation method for the transmitted signals is performed at the transmitter, while a successive interference cancellation (SIC) method is performed at the receivers. Consequently, NOMA significantly enhances throughput and multiplexing gain, as compared with the conventional orthogonal multiple access (OMA) techniques. Recently, there have been few published works, in which NOMA has been applied to multi-hop relay networks to improve the e2e system throughput. In [23], the authors evaluated the OP-IP trade-off for secure multi-hop communication networks using NOMA, in the presence of an active eavesdropper. In particular, the source and relay nodes in [23] must reduce the transmit power to ensure that the instantaneous signal-to-noise ratio (SNR) received by the eavesdropper remains below a predetermined threshold. Additionally, the transmitters in [23] use NOMA at each hop to forward two data to a destination node. Published work [24] compared the e2e OP and the e2e throughput of multi-hop relay models using NOMA with those of the corresponding OMA models over Nakagami-m fading channels, considering the impact of imperfect transceiver hardware. In [25], the secrecy outage probability of NOMA-based multi-hop transmission schemes was evaluated through analysis and simulations. However, this paper considers Fountain codes (FCs) and the RF-EH technique, which are not investigated in [23–25].
Fountain codes (FCs), also known as rateless codes [26,27], have gained significant interest because of their ease of implementation and flexibility in varying channel environments. With this coding technique, the transmitter can produce and send an endless sequence of encoded packets. As long as the receiver collects a sufficient number of information (i.e., more than the amount of information originally sent), the original message can be fully reconstructed [26,27]. This characteristic eliminates the dependency on a feedback link and removes the need for receivers to possess channel state information (CSI). In the studies [28,29], several approaches integrating FCs into cooperative relay schemes have been introduced. These schemes have demonstrated improvements in both system capacity and transmission efficiency, outperforming traditional cooperative methods. Reference [30] explored the use of FCs in broadcasting over cognitive radio networks. The findings indicated that the proposed scheme in [30] notably decreased transmission delays in comparison to conventional methods that do not utilize FCs. In [31], the authors introduced a dynamic cooperative transmission algorithm that leverages mutual information accumulation, where the use of FCs allows nodes to mitigate retransmission delays under poor channel conditions. In addition, mutual information accumulation accelerates data collection at the receiver, helping to further minimize transmission delay. The authors of [32] investigated a dual-hop cooperative transmission system powered by RF-EH, where dynamic DF relaying was utilized in combination with FCs. In [33], the authors analyzed decoding failure probability in multi-hop relay networks employing SPC, considering both recoding and FC techniques. Reference [34] proposed an FC-based multi-path routing algorithm designed for low-latency and highly dynamic unmanned aerial vehicle (UAV) networks. In [35], performance of cooperative full-duplex (FD) NOMA relay scheme using FCs and Intelligent Reflective Surface (IRS) was performed.
Unlike the previous works, this paper investigates the performance of single-input multiple-output (SIMO) multi-hop relay systems employing FCs, NOMA and RF-EH. Although the authors of [17] also investigated the multi-hop relaying scheme using FCs and RF-EH, their approach employed cooperative transmission with relay selection at each hop, and did not employ the NOMA technique or the SIMO model. In [23–25], NOMA-based multi-hop relay schemes were proposed. However, these works did not take into account FCs, RF-EH, and SIMO. Different from [31,32], our scheme considers the multi-hop relay scenario which integrates NOMA and RF-EH. While [33] and [34] focus on the FC-based multi-hop relaying schemes operating in SPC and UAV environments, respectively, this paper investigates such schemes in the NOMA, RF-EH, and SIMO environment. Although the published work [36] proposed a joint FC- and NOMA-based relay scheme combining with the advanced techniques such as IRS and FD, it did not consider the multi-hop relaying model or the SIMO model.
In the proposed scheme, a source node uses NOMA to simultaneously transmits two data to two intended destinations, via the help of DF relays. Each destination accumulates received information until it becomes sufficient to reconstruct the desired data. To improve the channel capacity at both the relay and destination nodes, the selection combining (SC) technique is also employed. Furthermore, the source and relay nodes harvest wireless energy from a power station deployed within the network for the data transmission, follows a time-switching approach.
Next, the main contributions of this paper can be listed as follows:
- Firstly, we propose the new NOMA-Aided SIMO Multi-hop Schemes Using RF-EH and FCs, which obtains higher throughput, energy efficiency and reliable communication.
- Secondly, we derive exact closed-form expressions for the e2e instantaneous signal-to-interference-plus-noise ratios (SINRs) at both destinations and their corresponding cumulative distribution functions (CDFs) under Rayleigh fading channels.
- Thirdly, from the derived CDFs of SINRS, we then give analytical formulas for the e2e average channel capacity (ACC-e2e) and outage probability (OP-e2e) at each destination. All the derived formulas are then validated through Monte Carlo simulations.
- Finally, simulation and theoretical results demonstrate the effectiveness of the proposed scheme and provide insights into the impact of key system parameters.
System model
Fig 1 illustrates the system model of NOMA-aided multi-hop relay schemes employing FCs and RF-EH, which includes one transmitter (T0), N relays (), and one power station
. In this figure, the source node
attempts to transmit the data
and
to its destinations U and V, respectively. Assume that data transmission is carried out with the assistance of intermediate DF relays denoted by
. For the transmitters including
and
, they harvest energy from the single-antenna power station
, and utilize the harvested energy to transmit the data. Assume that all the transmitters are equipped with one transmit antenna, while the receivers such as
, U and V have M receive antennas and use the SC technique. It is worth noting that the implementation of SC is simpler than that of maximal ratio combining (MRC), because SC only requires selecting the branch with the highest SNR, while MRC requires a combination of all the branches, leading to higher computational and hardware complexity. Furthermore, the implementation of SIMO schemes is much simpler than that of MIMO schemes. This is because MIMO schemes require perfect CSI at both the transmitter and the receiver, which incurs significant latency and computational overhead. Assume that the e2e transmission time of
and
is one time unit, and hence, the time allocated for each hop (each time slot) is
.
Let denote the channel gain between the transmitter
and the m-th antenna of the receiver
, where m = 1,2,...,M. All channels are assumed to experience quasi-static, frequency-flat Rayleigh fading. As given in [23,24],
is an exponential random variable, and its CDF and probability density function (PDF) are given, respectively as
where with
being distance between A and C,
is a path-loss exponent [23,24].
Considering the time slot, the relay
uses energy collected from the Power Station (B) based on the Time-Switching (TS) protocol to send the data to
. At first,
harvests energy from B during the time
, where
is a predetermined value [18,19]. Then, the energy harvested by
can be given as (In this paper, we use the linear EH model [18,19]).
where is conversion efficiency [18,19], and
is transmit power of the power station (B).
The remaining time is used for the data transmission between
and
, and hence the transmit power of
can be formulated as
where χ = ηα / ( 1 − α ) .
Using NOMA, combines modulated signals of the data
and
as in [36]:
, where
. Additionally, we assume that U is nearer to
than V, we can then set a1 and a2 as in [36]:
. It is noted that the values of a1 and a2 are used at all the hops. We also note adaptive power allocation across hops may further improve the system performance, and extending the proposed model to incorporate such adaptive power allocation is straightforward.
Next, sends p+ to
which decodes
first, and then performs SIC to remove
before decoding
[36–40]. The SINRs obtained at the mth antenna of
for decoding
and
can be formulated, respectively, as in [36–40]:
where .
It is worth noting from (4) that can know the value of the channel gains
through channel estimation, but cannot know the value of
. Therefore,
has to select its receive antenna so that the value of
is highest, i.e.,
where b denotes the selected antenna of .
As a result, the SINRs obtained at for decoding
and
can be expressed, respectively as
Similarly, we can formulate the SINRs obtained at the destination U for decoding and
, respectively as
where .
For the destination V, it directly decodes its data , and the obtained SINR can be given as
where .
Since the channel gains are assumed to be independent and identically distributed random variables, CDFs and PDFs of
can be expressed under the following forms (see [15]):
where ,
and
denotes a binomial coefficient, i.e.,
Due to the DF relaying technique [23,24], we can formulate the SINRs obtained at V and U, respectively as
where
Finally, we can formulate the e2e instantaneous channel capacity obtained at V and U, respectively as in [14,15]:
Let us denote as the required information that U and V need to accumulate in order to recover their desired data. Similar to [28], the required time for U and V to sufficiently receive amount of information
can be calculated, respectively as
Performance analysis
This section analyzes the OP-e2e and ACC-e2e at each destination. The OP-e2e at U(V) represents the probability that the information accumulation time exceeds a maximum delay threshold
, i.e.,
where .
Then, the ACC-e2e obtained at the destination X can be expressed as
where denotes an expected operator.
We now attempt to derive expressions of CDFs of and
.
CDF of and
From (11), we can formulate CDF of as
In (18), can be rewritten from (6) as
It is straightforward from (19) that if x ≥ a2 / a1, then .
We now consider the case where 0 < x < a2 / a1, (19) can be expressed as
where .
Substituting in (1) and
in (9) into (20), we obtain
Utilizing equation (3.324.1) in [41] to solve the integral in (21), we finally obtain
where is the first-order modified Bessel function of the second kind [41].
With the same derivation method, we can obtain CDF as
By substituting (22) and (23) into (18), we obtain an exact closed-form expression for under the condition 0 < x < a2 / a1, as follows:
where
Next, we consider CDF of ; from (12), we have
From (6) and (13), we can formulate in (26) as
We can observe from (27) that if , then
.
In the case where , we can rewrite (27) as
where .
From the result obtained in (22), we can obtain as
where
with
Similarly, we can obtain in (26) as
where
with
From (26)–(34), we can express under the following form:
where
OP-e2e and ACC-e2e
From (15) and (16), the OP-e2e at the destination X can be formulated as
where
Substituting (24) and (35)-(36) into (37), we obtain exact closed-form expressions of the OP-e2e at the destinations V and U, respectively.
From (17), the ACC-e2e obtained at the destination X can be expressed as
Furthermore, using integration by parts, the ACC-e2e in (39) can be computed by using as follows:
It is worth noting from (40) that obtaining a closed-form expression for is impossible. Therefore, we apply the Chebyshev-Gauss quadrature method to approximate it. Indeed, the Chebyshev-Gauss approximation can be stated as in [42]:
where is a designed value [42], and
Applying (41) for (40), we can approximate as follows:
where
The computational cost for evaluating the analytical expressions for relies on the efficient Chebyshev-Gauss Quadrature method, guaranteeing accurate numerical results with a deterministic complexity significantly lower than Monte Carlo simulation.
Numerical results
This section presents simulation results (Sim) to validate the theoretical analysis (Theory). In the simulation setup, the nodes are placed on a straight line as in [6,7,15–17], i.e.,
, while the B, U and V nodes are positioned as
,
and
, where n = 0,1,...,N.
To analyze the impact of the important parameters such as the transmit power of the power station (or the transmit SNR ), the time allocation factor (
), the power allocation coefficient (a1 or a2), and the number of hops (N), values of the remaining parameters are fixed as in Table 1. It is worth noting that the derived formulas in this paper can be applied to any practical values of the parameters.
Fig 2 illustrates the OP-e2e at the destinations U and V as a function of the transmit SNR ( in dB) with different values of M, and with N = 3,
and a1 = 0.25. As observed, the OP-e2e at U and V decreases as
increases due to the increase in transmit power at all the transmitters. It is also seen that the OP performance of V is better than that of U. It is due to the fact that the data
is allocated more transmit power than the data
(i.e.,
). Next, Fig 2 shows that the OP-e2e at both U and V significantly decreases with an increasing number of receiving antennas (M). It is because increasing M also improves the quality of the data channel at each hop. Because the scheme with M = 1 is a SISO (Single Input Single Output) configuration, Fig 2 presents that the proposed SIMO model obtains much better OP performance, as compared with the corresponding SISO one. Finally, we can see the ‘Sim’ and ‘Theory’ results match very well, which validates the correction of the derived expressions of OP-e2e.
Fig 3 investigates the impact of the power allocation factor on the OP performance at the two destinations for various values of a1 with
= 15 (dB), M = 2 and N = 2. Similar to Figs 2, 3 shows that OP-e2e at V is lower than that at U. As shown in Fig 3, the OP-e2e at V (U) increases (decreases) as a1 increases from 0.1 to 0.3. This occurs because increasing a1 corresponds to allocating more (less) transmit power to the data
(
). As a result, the smaller the value of a1, the larger the performance gap between U and V, and vice verse. Fig 3 also illustrates that
effects on the OP-e2e at U and V. Interestingly, there exist optimal values of
at which the OP-e2e at U and V is lowest. For example, with a1 = 0.2, the OP-e2e at U and V is minimized at
and
, respectively. Again, it can be observed from Fig 3 that the ‘Sim’ and ‘Theory’ results are in a good agreement.
Here, this raises the question of what the optimal value of is. Indeed, to determine the optimal value of
and other system parameters, we should aim to minimize the overall OP-e2e of the proposed scheme. One possible approach is to consider the average OP-e2e between nodes U and V, i.e.,
.
In Fig 4, we present the OP-e2e of U and V, and the average OP-e2e as a function of a1, with and 0.75, and with
= 12.5 (dB), M = 3 and N = 4. As observed, the OP-e2e of V (U) increases (decreases) as a1 increases. It is worth noting that there exist certain values of a1 at which the OP-e2e of U and V is the same, which obtains the performance fairness between U and V. It is also seen from Fig 4 that the OP-e2e of U can be lower that of V when a1 takes high values. Next, we can observe that there are optimal values of a1 which minimize the average OP-e2e. In particular, as
and 0.6, the optimal values of a1 are 0.31 and 0.29, respectively.
Fig 5 presents the ACC-e2e at U and V as a function of (in dB) with M = 3,
, a1 = 0.1 and N = 2,3. As observed, V achieves a higher ACC-e2e than U, and ACC-e2e of both destinations increases as
increases. However, as
is sufficiently high, the ACC-e2e of U and V converges to an upper-bound value that is independent of
. Indeed, from (6), (7) and (8), it is straightforward that
Then, from (45), we have
which leads to
It is also seen from Fig 5 that at medium and high values, the ACC-e2e performance of both destinations improves with lower number of hops (N + 1). This is because the time allocated for each hop increases as N decreases, thereby enhancing the channel capacity obtained at each hop. Finally, we can observe that the simulations confirm the accuracy of the analysis.
Fig 6 analyzes the impact of the number of hops on ACC-e2e of U and V when
dB, M = 2 and a1 = 0.2. Since increasing the number of hops (N) reduces the distance between two adjacent nodes
and
, it can be observed from Fig 6 that, as N increases from 1, the ACC values at both users U and V increase. However, as N increases, the time allocated to each hop decreases, leading to a reduction in the instantaneous channel capacity of each hop (see equation (14)). Therefore, it is seen that the ACC values at U and V reduce as N is high. To determine the optimal value of N, we should consider the average ACC-e2e at U and V, i.e.,
. As observed,
with
is highest as N = 2, while
with
reaches its highest value at N = 4. Moreover, in this figure,
with
is higher than that with
.
Fig 7 investigates the impact of the time allocation factor on the ACC-e2e performance as
dB, M = 2 and N = 2. As we can see, the ACC-e2e values vary with changes in
. Notably, with low
, the harvested energy and the transmit power of the transmitters are low, resulting in a low ACC-e2e at both destinations. However, if
is too high, the time allocated for the data transmission is low, which also degrades the ACC-e2e performance. Therefore, as seen from Fig 7, there exist optimal values of
at which ACC-e2e of U and V is highest. For example, with a1 = 0.1, ACC-e2e of U and V is maximum at
and
, respectively. However,
should be designed to maximize the average ACC-e2e
. Indeed, if a1 = 0.1, the optimal value of
is 0.25. Fig 6 also shows that
with a1 = 0.1 is higher than
with a1 = 0.25. However, as a1 = 0.1, the performance gap between U and V becomes higher.
In Fig 8, the impact of the power allocation on the ACC-e2e performance of both destinations is investigated. In this simulation, we set
, M = 3 and N = 3. It is seen from Fig 7 that ACC-e2e of V always decreases with the increasing of a1 because the power allocated to the data
decreases (or the value of a2 decreases). For the destination U, we see that there exists an optimal value of a1 at which ACC-e2e is highest. For example, as
5 dB (
10 dB), the optimal value of a1 is 0.175 (0.125). However, as mentioned in Fig 7, we should design the value of a1 to maximize the average ACC-e2e (
). Indeed, Fig 7 shows that with
5 dB (
10 dB),
achieves its maximum value when
. However, it is worth noting that when a1 is low, the performance gap between U and V is too high.
Conclusion
In this paper, we derived closed-form expressions of the OP-e2e and ACC-e2e for the proposed NOMA-based multi-hop SIMO scheme using FCs and RF-EH over Rayleigh fading channels. These expressions were validated through Monte Carlo simulations. The results demonstrated that the far destination (V) generally outperforms the near user (U), with a significant performance gap when the transmit SNR , number of hops
, the time allocation factor
, and the power allocation factors
are low. Moreover, we proved that at high
region, the ACC-e2e of both destinations converges to an upper-bound value which is independent of
. Additionally, the findings revealed that the OP-e2e and the ACC-e2e can be optimized by appropriately selecting the system parameters such as N,
and a1. Finally, the OP-e2e and ACC-e2e performance can be further improved by increasing the number of receiving antennas at the nodes
.
Our future work will concentrate on jointly optimizing the key parameters—the energy harvesting duration () and the number of relays (N)—to maximize system capacity by effectively managing inherent system trade-offs. Building on this foundation, we will extend the framework to incorporate non-ideal practical constraints, such as synchronization errors and hardware imperfections, and formulate a rigorous multi-objective optimization problem. Finally, we plan to explore advanced applications by integrating the system into heterogeneous networks and leveraging Machine Learning techniques for dynamic resource management.
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