Ergodic Capacity of NOMA-Based Overlay Cognitive Integrated Satellite-UAV-Terrestrial Networks

I.   Introduction

With the rapid development of wireless communication, the next generation wireless communication network must have the ability of full coverage and high rate [1]. However, owing to the low economic benefits and terrain constraints, the seamless coverage of wireless communication is far from being realized, especially in remote areas and navigation [2], [3]. Therefore, introducing satellites into the existing wireless communication network is considered to be a promising method to solve the above problem. Besides, the satellite-terrestrial network (STN) is considered to be a very significant part of the next generation wireless communication network. On the other hand, unmanned aerial vehicle (UAV) communication is an effective supplement for the satellite-terrestrial network due to its flexible operation, rapid deployment, and low-cost [4]-[6], which can eliminate the impacts of obstacles and shadow effects in the satellite-terrestrial network. At the same time, the static allocation mode of the spectrum makes the limited spectrum resources difficult to meet the growing demand. Notably, cognitive radio (CR) and non-orthogonal multiple access (NOMA) are two effective technologies that can enhance spectrum utilization respectively. Hence, this motivates us to introduce the cognitive radio and the non-orthogonal multiple access into our considered system to improve the transmission performance [7]-[9].

1   Related works

Nowadays, many academics and industry experts have carried out researches on STN [10]. The authors in [11] considered a land mobile satellite (LMS) with hardware impairments and interference where the satellite was utilized to be a relay to aid the communication of two terrestrial users, and the system performance was discussed. In [12], a STN with user fairness scheduling was considered, and the exact outage probability (OP) and ergodic capacity (EC) of the system were derived. Methods to improve the security and reliability of the physical layer for STN were investigated in [13], and a beamforming scheme was developed to enhance the effective achievable rate. In [14], the authors investigated the adaptive transmission schemes of STN under the condition of meeting the spectrum and power efficiency demands, and achievable channel capacity was derived. In [15], the content-based caching scheme was applied in STN, and the closed-form expression of OP was obtained. In [16], the authors investigated the performance of uplink STN with multi-relays, where the impacts of co-channel interference (CCI) as well as hardware impairments were considered.

With the unprecedented increase of UAV applications, UAV-assisted communication has been extensively employed in plenty of temporary events and natural disasters [17], [18]. The authors in [19] discussed several key enabling technologies of UAV communication over the millimeter-wave frequency band, and the challenges in this field are summarized. In [20], the authors maximized the achievable rate of millimeter-wave network based on a full-duplex UAV relay by jointly optimizing the UAV’s position, beamforming, and power allocation. Two beamforming schemes based on total power and per-antenna power constraints were developed to maximize the energy efficiency (EE) of the integrated satellite-unmanned aerial vehicle-terrestrial network (IS-UAV-TN) in [17]. The authors in [21] investigated UAV-based STN with rate-splitting multiple access (RSMA), in which the sum-rate of the system was maximized. In [22], the authors optimized the EE and secure transmission of the similarity model in cognitive satellite-terrestrial networks. The authors in [5] discussed the performance of the UAV-assisted multi-relay communication network, in which the hardware impairments were considered, and the achievable sum rate was derived. In [8], the author investigated the performance of UAV-based STN with cache to reduce latency and improve file update rate. The authors in [23] studied mobile edge computing (MEC)-assisted UAV network, in which the energy consumption was optimized under allocation latency requirements and resource constraints. In [24], resources assignment of the considered UAV network was optimized with the help of the game theory, and the authors designed a multi-agent reinforcement learning framework to search for the optimal strategy. To find the resource management policy of a great quantity of UAVs, the potential game, mean-field game, Stackelberg game, graphical game, and coalition game were exploited in [25].

CR is considered as a promising way to break through the bottleneck of spectrum shortage, and the integration of CR and STN is worth exploring [26]. The authors in [26] discussed the performance of an overlay cognitive STN with secondary network selection (SNS), partial and opportunistic SNS schemes were proposed to select the proper secondary network. An EE and spectral efficiency (SE) tradeoff scheme was designed to optimize resource allocation for a cognitive satellite-vehicle network in [27]. In [28], the authors developed an adaptive transmission scheme for cognitive STN, where EE was maximized under symbol error rate (SER) constraints. The authors in [29] designed a two layer iterative beamforming scheme to take advantage of interference to improve physical layer security for cognitive STN. In [30], a beamforming scheme utilizing artificial noise and cooperative interference was adopted to improve the security of the cognitive STN in the presence of unknown eavesdroppers.

In addition, the NOMA scheme can improve the spectrum efficiency of large-scale users. The achievable rate of NOMA-based millimeter-wave communication system was maximized by optimizing power and beamforming in [31]. In [32], the authors investigated the optimization scheme of user pairing and power allocation in the NOMA network. Besides, the influences of NOMA on STN have been studied in many existing works [33], [34]. The authors in [35] investigated the NOMA-based STN with bandwidth compression (BC) to achieve non-orthogonal signals in the frequency domain and power domain, where iterative successive interference cancellation (SIC) and symmetrical coding were proposed to eliminate internal interference and reduce error probability. The authors in [36] investigated the impacts of imperfect SIC on NOMA-based STN, the exact and asymptotic OP were obtained. In [37], the influences of hardware impairments on the security of STN were investigated, and secrecy outage probability of colluding case and non-colluding case were derived. In [38] and [39], the authors investigated NOMA-based cognitive STNs, in which CR technology was applied between the primary network and secondary network, and the NOMA scheme was adopted in the process of transmitting signals to primary users.

2   Our contributions

On the foundation, an overlay cognitive integrated satellite-UAV-terrestrial network (CIS-UAV-TN) with NOMA scheme is considered in our paper. As far as we know, no similar work has been published.

In particular, the major contributions are summed up as below:

1) First, considering the importance of STN and the flexibility of UAV, we consider a novel CIS-UAV-TN with NOMA scheme, in which the secondary network accesses the authorized spectrum of the primary network in overlay mode, and acts as a relay to assist the signal transmission of the primary network.

2) Second, probability density functions (PDFs) and cumulative distribution functions (CDFs) of shadowed-Rician (SR) fading as well as Nakagami-m fading, and Meijer-G functions are utilized to derive the closed-form expressions of EC for both the primary network and secondary network.

3) Third, simulation results are provided to demonstrate the validity of the mathematical derivations and indicate the influences of critical system parameters on transmission performance. In addition, we compare the orthogonal multiple access (OMA)-based as well as direct transmission schemes with our considered scheme to verify the superiority of our system.

II.   System Model and Problem Formulation

1   System model

As shown in Fig.1, we consider an NOMA-based CIS-UAV-TN under overlay mode, where exists a satellite-terrestrial primary network and a UAV-terrestrial-terrestrial network^①. In the primary network, the satellite ($S$) communicates with $K$ corresponding users $\left\{ {{U_i}} \right\}_{n = 1}^K$, the NOMA scheme is applied in the transmission. Moreover, the secondary network is consist of a UAV ($R$) and its receiver ($D$). It is assumed that all nodes are equipped with a single antenna and work in half-duplex mode^②. Recalling the overlay mode, the secondary transmitter plays the role of the relay to forward the signal from $S$ to $U_i$. As a reward, the authorized spectrum of the primary network can be utilized by the secondary network to transmit its signal. To reduce the complexity of receivers, we divide $K$ NOMA users into $1/2$ two-user groups [26], [38], thus we assume $U_1$ and $U_2$ compose a two-user group^③. Hence, the primary signal can be transmitted by $S$-$U_i$ ($i=1,2$) link and $S$-$R$-$U_i$ link. Besides, the perfect channel state information (CSI) is assumed to be available, which can be realized by the training data and feedback from the receivers^④ [40], [41].

Figure  1.  System model.

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The SR fading and Nakagami-m fading are utilized to model the satellite-UAV as well as satellite-terrestrial links and UAV-terrestrial links respectively^⑤. To make the expression more concise, $g_{su_i}$, $g_{sr}$, $g_{sd}$, $h_{ru_i}$ and $h_{rd}$ are defined to represent the shadowing coefficients of $S$-$U_i$ link, $S$-$R$ link, $S\text{-}D$ link, $R$-$U_i$ link, and $R\text{-}D$ link respectively.

Furthermore, all receivers in our considered system suffer additive white Gaussian noise (AWGN) with ${\cal C}{\cal N}\left( {0,{\sigma ^2}} \right)$.

2   Channel model

From [42], the PDF of $g_{sc}$ can be expressed as

where $c = \{ {u_i},r,d\}$, ${\alpha _{sc}} \buildrel \Delta \over = \frac{1}{{2{b_{sc}}}}{\left( {\frac{{2{b_{sc}}{m_{sc}}}}{{2{b_{sc}}{m_{sc}} + {\Omega _{sc}}}}} \right)^{{m_{sc}}}}$, ${\beta _{sc}} \buildrel \Delta \over = \frac{1}{{2{b_{sc}}}}$, and ${\delta _{sc}} \buildrel \Delta \over = \frac{{{\Omega _{sc}}}}{{2{b_{sc}}\left( {2{b_{sc}}{m_{sc}} + {\Omega _{sc}}} \right)}}$. ${m_{sc}}>0$ stands for the Nakagami-m parameter, $2{b_{sc}}$ is the average power of the multi-path part, and ${\Omega _{sc}}$ is the average power of line-of-sight (LOS) part.

If ${m_{sc}}$ is an integer, according to the formula No.07.20.03.009.01 and formula No.07.02.03.0014.01 in [43], $_1{F_1}\left( {{m_{sc}};1;{\delta _{sc}}x} \right)$ can be rewritten as

In formula (2), ${\left( . \right)_n}$ stands for the Pochhammer symbol (see [44, p.xliii]).

By substituting (2) into (1), we can get

Considering the practical propagation impacts, the channel coefficients are expressed as

where $V_{sc}$ represents radio propagation loss, which consists of free space loss and radiation pattern, it can be expressed as

where $\lambda$ stands for the carrier wavelength, $k$ represents the Boltzman constant, $d_{sc}$ denotes the distance from the source to the destination, $B$ is the carrier bandwidth, $T$ denotes the noise temperature. Moreover, ${G_{t,{sc}}}$ and ${G_{r,{sc}}}$ are the antenna gain of transmitter and receiver with

where $G_{\max }$ is the maximum of beam gain, ${{J_n}}$ represents the $n$-order first-kind Bessel function, and $u = 2.70123\frac{{\sin \theta }}{{\sin {\theta _{3{\rm{dB}}}}}}$ with $\theta$ denotes the angle from the beam center to receiver location with regard to the $S$, $\theta _{3{\rm{dB}}}$ represents the 3 dB angle.

Due to ${\gamma _{sc}} = {{\bar \gamma }_s}{\left| {{h_{sc}}} \right|^2}$, the PDF of $\gamma_{sc}$ is given by

where $\zeta \left( n \right) = \frac{{{{\left( { - {\delta _{sc}}} \right)}^n}{{\left( {1 - {m_{sc}}} \right)}_n}}}{{{{\left( {n!} \right)}^2}{{\left( {{{\bar \gamma }_s}V_{sc}^2} \right)}^{n + 1}}}}$, ${\Delta _{sc}} = \frac{{{\beta _{sc}} - {\delta _{sc}}}}{{{{\bar \gamma }_s}}}$.

With the help of formula No.3.351.2 in [44], the CDF of $\gamma_{sc}$ is represented as

Besides, according to [26], the PDF and CDF of $\gamma_{rj}$ are respectively represented by

and

where ${\Xi _{rj}} = \frac{{{m_{rj}}}}{{{\Omega _{rj}}{{\bar \gamma }_r}}}$, ${\Omega _{rj}}$ denotes the average power, and ${m_{rj}}$ strands for the fading severity parameter.

3   Problem formulation

As shown in Fig.1, there are two phases in the whole transmission process. In the first phase, $S$ communicates with $U_i$, $R$, and $D$, where the superposition coding technique (SCT) is applied to integrate the different signals to multiple users. Hence, we can get the primary signal

where $s_i$ stands for the excepted signal of $U_i$ with $E\left( {{{\left| {{s_i}} \right|}^2}} \right) = 1$, $P_S$ denotes the transmit power of $S$, and $\beta_i$ is the power assignment factor of $U_i$ with $\beta_1 +\beta_2 =1$. According to the characteristics of the NOMA scheme, the transmitter assigns less power to the user with better channel quality. Without loss of genera, we assume ${\left| {{h_{s{u_1}}}} \right|^2} < {\left| {{h_{s{u_2}}}} \right|^2}$, thus we can get $\beta_1 > \beta_2$. Besides, we can obtain the primary signals at $R$, $D$, $U_i$ as

where $k= \{ r, d, {u_i}\}$, $n_k$ denotes AWGN.

In the second phase, the decode-and-forward (DF) protocol is adopted at $R$ to transmit the expected signal of $U_i$. At the same time, the desired signal of $D$ is sent by $R$, thus the combined signal is given by

where $P_{R}$ stands for the transmit power of $R$, $s_r$ denotes the desired signal of $D$, $0<\mu<1$ is the power allocation factor.

Hence, the signals received by $D$ and $U_i$ is given by

where $j= \{ {u_i},d\}$ and $n_{rj}$ is AWGN^⑥.

After that, we can get the signal-to-interference plus noise ratio (SINR) of direct satellite (DS) as well as $R$-assisted links. To DS links, $s_1$ is decoded first with treating $s_1$ as CCI, and the SINR of $s_1$ at $U_1$ can be represented as

where ${\gamma _{s{u_1}}} = {\bar \gamma _s}{\left| {{h_{s{u_1}}}} \right|^2}$ and ${\bar \gamma _s} = \frac{{{P_S}}}{{{\sigma ^2}}}$.

Then, the SIC is executed at $U_2$ to obtain its expected signal^⑦. First, $s_1$ is decoded and removed from the combined signal, thus the SINR of $s_1$ at $U_2$ is given by

where ${\gamma _{s{u_2}}} = {\bar \gamma _s}{\left| {{h_{s{u_2}}}} \right|^2}$.

Next, $s_2$ can be obtained only with AWGN, and the SINR of $s_2$ at $U_2$ is represented as

Moreover, $R$ and $U_i$ perform a similar decoding process. Hence, we can obtain the SINR of $s_i$ at $R$ as

and

where ${\gamma _{sr}} = {\bar \gamma _s}{\left| {{h_{sr}}} \right|^2}$.

In addition, the SINR of $s_i$ at $U_i$ are given by

and

where ${\tau _1} = {\beta _2}\mu + 1 - \mu$, ${\tau _2} = 1 - \mu$, ${\gamma _{r{u_1}}} = {\bar \gamma _r}{\left| {{h_{r{u_1}}}} \right|^2}$, ${\gamma _{r{u_2}}} = {\bar \gamma _r}{\left| {{h_{r{u_2}}}} \right|^2}$, ${\bar \gamma _r} = \frac{{{P_R}}}{{{\sigma ^2}}}$.

By the same way, the SINR at $D$ can be expressed as

where ${\gamma _{rd}} = {\bar \gamma _r}{\left| {{h_{rd}}} \right|^2}$.

III.   EC of the CIS-UAV-TN

EC is a significant indicator to measure the performance of the wireless system. It represents the time average of the maximum information rate in all fading states between the transmitter and receiver. In this section, we derive the exact expressions of EC for both the primary network and secondary network.

1   EC of the primary network

In general, we define the EC of $R$-assisted link as the minimal EC of the two phases [45]. Therefore, we can represent the EC of the primary network as

where $E{C_{DS}}$ denotes EC of the DS links, and $E{C_R} = \min \left( {E{C_{SR}}, E{C_{RU}}} \right)$ stands for EC of the $R$-assisted link with $E{C_{SR}}$ being EC of $S$-$R$ link and $E{C_{RU}}$ being that of $R$-$U_i$ link.

First, we derive the expression of $E{C_{DS}}$. According to the definition of EC, $E{C_{DS}}$ is given by

By taking (15), (16), and (17) into (25), we can get

Let ${\beta _2}{\gamma _{s{u_1}}} = z$, by utilizing the probability transformation formula $\left( {t = Ax \to {f_t}\left( t \right) = \frac{1}{A}{f_x}\left( {\frac{x}{A}} \right)} \right)$, the PDF of $z$ can be obtained as

Then, formula (26) is re-written as

By substituting $\ln \left( {1 + x} \right) = G_{2,2}^{1,2}\Big[ {\left. x \right|\begin{array}{*{20}{c}} {1,1}\\ {1,0} \end{array}} \Big]$, (7), and (27) into (28), we can get (29) as following, where $G_{p,q}^{m,n}\left( {\left. x \right|\begin{array}{*{20}{c}} {{a_1},}&{ \ldots ,}&{{a_p}}\\ {{b_1}}&{ \ldots ,}&{{b_q}} \end{array}} \right)$ is Meijer-G function (see formula No.9.301 in [33]).

With the help of the formula No.2.24.3.1 in [46], the exact expression of $E{C_{DS}}$ can be obtained as (30).

To solve the situation that ${{\Delta _{s{u_i}}}}=0$, by applying the formula No.8.2.2.14 in [46]:

$E{C_{DS}}$ can be re-written as (32).

By the similar derivation, $E{C_{SR}}$ and $E{C_{RU}}$ can be obtained as (33) and (34), respectively.

Noting that $1/2$ is necessary because of DF protocol.

2   EC of the secondary network

The EC of the secondary network is expressed as

With the similar method of obtaining the EC of the primary network, we can get the exact $E{C_{SN}}$ as

IV.   Numerical Results

In this section, simulations are conducted to demonstrate the validity of our derivation. The simulation tool is MATLAB 2019. In general, we set $\overline \gamma_{s} = \overline \gamma_{r}=\overline \gamma$, and other system parameters and channel parameters are shown in Table 1 and Table 2 [38], [31].

Table  1.  System parameters

Parameter name	Parameter value
Satellite	GEO
$f$	2 GHz
$\theta _{3{\rm{dB}}}$	$0.8^\circ$
$G_{{\rm{max}}}$	48 dB
${G_{r,p}}$	4 dB
$B$	15 MHz
$k$	$1.38 \times {10^{ - 23}}$ J/K
$T$	$300^\circ$
$\sigma^2$	1
$\Omega_{rj}$	1

 | Show Table

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Table  2.  Channel parameters

Shadowing	$m$	$b$	$\Omega$
Infrequent light shadowing (ILS)	10	0.158	1.29
Average shadowing (AS)	5	0.251	0.279
Frequent heavy shadowing (FHS)	1	0.063	0.0007

 | Show Table

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First, we can clearly see that the theoretical analysis coincides with Monte Carlo (MC) simulations, which testifies the correctness of our analysis. Besides, our considered system has higher EC than only the DS system. It proves our considered system has better transmission capability.

Fig.2 shows the EC of primary network with different $\overline \gamma$, we set $m_{rj}=2$, $\beta_1=0.67$, $\mu=0.92$. It is clear that the EC of the primary network increase with $\overline \gamma$. This is because the enhancement of SNR can improve transmission performance.

Figure  2.  The EC of primary network with different $\overline \gamma$.

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Fig.3 illustrates the EC of primary network with different $\mu$ with setting $m_{rj}=2$, $\beta_1=0.67$, $\overline \gamma=30\;\text{dB}$. It can be found that our proposed system has better performance with the increase of $\mu$, it is due to the fact that the raise of $\mu$ indicates that $R$ assigns more power to transmit the signal of the primary network, which will lead to the enhancement of the SNR. Furthermore, the EC of only the DS transmission scheme is uncorrelated with $\mu$. It can be explained by the reason that the transmission in the first phase may not be affected by $\mu$. Besides, our proposed system has better performance under ILS and worse performance under FHS.

Figure  3.  The EC of primary network with different $\mu$.

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Fig.4 plots the EC of primary network with different $\beta_1$ with $m_{rj}=2$, $\mu=0.92$, $\overline \gamma=30\;\text{dB}$. Transmission performance of both our considered scheme and only DS transmission scheme perform better with large $\beta_1$, which shows that assigning more power to users with worse channel quality leads to the improvement of EC. Notably, this phenomenon is different from the properties of OP for the NOMA system, which has an optimal power allocation factor less than 1. It is due to the fact that large $\beta_1$ will lead to the outage of $U_2$.

Figure  4.  The EC of primary network with different $\beta_1$.

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Fig.5 depicts the EC of primary network with NOMA and OMA, we set $m_{rj}=2$, $\beta_1=0.67$, $\mu=0.92$. We can easily find that the OMA-based system performs worse than our considered NOMA-based system. Besides, with the improvement of average SNR, the gaps of curves with different channel conditions become larger, or our proposed system has more obvious advantages than the traditional OMA system.

Figure  5.  The EC of primary network with different NOMA and OMA.

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Fig.6 depicts the EC of secondary network with different $\overline \gamma$ with $\beta_1=0.67$ and FHS. The transmission performance of secondary network enhances with $\overline \gamma$. Besides, the larger $m_{rj}$ means the better channel quality, which leads to the enhancement of system performance. Contrary to the primary network, diminishing $\mu$ will improve the EC of the secondary network. Hence, $R$ can balance the performance of the primary network and secondary network by adjusting $\mu$.

Figure  6.  The EC of secondary network with different $\overline \gamma$.

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V.   Conclusions

In our paper, an NOMA-based CIS-UAV-TN work in overlay mode was studied, in which the primary signal was forwarded by the secondary transmitter to enhance the EC of the primary network, and the secondary network obtained the opportunity to access its spectrum. Specifically, with the utilization of the statistical expressions of SR fading and Nakagami-m fading, the exact EC of both the two networks were derived. Simulation results were provided, which indicated that the transmission performance of our considered system can be enhanced by adjusting the power allocation factor and power assignment factor.