Performance Analysis of Spatial Modulation Aided UAV Communication Systems in Cooperative Relay Networks

I.   Introduction

Driven by the escalating communication demand, the fifth generation (5G) and beyond wireless communication networks aim to offer higher data rates and quality of service (QoS) in crowded areas, improve the coverage, and reduce the blind spots of current networks. To meet the above requirements, unmanned aerial vehicles (UAVs)-enabled technology is emerging and has attracted much attention recently. Owing to the facts that UAV has controllable mobility and deployment flexibility, it can provide reliable and cost-effective wireless communication links for many practical scenarios [1], [2]. Such UAV systems enable several applications, an important application of UAV system is aerial base station, the UAVs can establish line-of-sight (LOS) communication links by adjusting their altitude to enhance the transmission reliability [3], [4]. Besides, in device-centric Internet-of-things (IoT) scenarios, UAV can provide ubiquitous and seamless connectivity of billions of smart devices as well as enhance capacity of communication networks [5], [6]. However, there are a number of challenges that need to be resolved before UAV systems can be truly applied in real-world scenarios, such as channel modeling, deployment [7], [8], trajectory optimization [9]-[11], and cellular network planning [12] and performance analysis [13]. In particular, in order to evaluate the system performance and investigate the impact of different parameters on the system, a fundamental performance analysis is necessary. Hence, in this paper, we will analyze the performance of the UAV-based system in terms of two key QoS metrics, i.e. bit error rate (BER) and diversity gain. Such performance analysis can reveal the inherent nature of the system, which is of great significance to the design of UAV-based system.

Multiple-input multiple-output (MIMO) system is another promising technique to boost the data rates of 5G networks. However, the multiple antennas employ the same number of radio frequency (RF) chain, which brings about power dissipation and design complexity. In [14], worst-case robust transmission for the traditional point-to-point MIMO system is studied with a subgradient projection based algorithm, but antenna synchronization for spatial multiplexing systems is a significant issue. To tackle these problems, spatial modulation (SM) is proposed in [15] as a beneficial modulation technique, which can be an effective approach in realizing the promises of next-generation communication network and remain low design complexity at the same time. SM activates single transmit antenna only at each transmission slot and exploits the antenna index to transmit the additional information, which leads to the increased data rate and energy efficiency. SM has been extensively studied in the past decade, authors in [16] investigated the BER performance of the SM system. An approximated BER expression was derived according to the union bound proposed in [17], where optimal maximum likelihood (ML) detector was employed. In order to reduce the complexity of detection, a novel SM detector was introduced in [18]. This suboptimal SM detector decoupled the antenna index and symbol estimation processes, and based on this, an analytical BER bound was deduced, but the obtained expression is only applicable for the case that antenna and symbol detection are independent. Further, authors in [19] introduced an improved union bound framework, and resultant obtained BER approximation is more accurate than conventional union bound of BER.

The mass machine-type communication in 5G requires the technical support of the wireless sensor network, where cooperative communication is one way to improve the network performance, efficiency and flexibility of beyond 5G network [20]. It is well known that the UAV system is difficult to achieve a robust and efficient network, so as one of key use case for the IoT, wireless communication with the assist of UAV relay has been studied. In [21], space-time block coding using index shift keying scheme was introduced into UAV-enabled relay systems, based on the Doppler frequency, coding rate, and modulation throughput, the authors conceived three adaptivity designs to better accommodate the high dynamics of the UAV systems. In terms of performance analysis, the authors in [22] investigated the outage probability (OP), the symbol error rate (SER), and average capacity performance of the UAV relaying systems over $\kappa-\mu$ fading channels, both amplify-and-forward (AF) and decode-and-forward (DF) protocols are considered, and the corresponding analytical expressions were derived in the form of infinite series representation. In order to reduce the power consumption, in [23], energy harvesting technology was utilized in UAV relaying systems, the system BER and OP over Nakagami fading channels were respectively deduced. The research in [24] focuses on the physical-layer security with simultaneous wireless information and power transfer, the secrecy OP and average secrecy rate of the system were, respectively, derived.

According to the analysis above, there exists some literature considering the UAV relay assisted systems from many perspectives, but none of the research has been dedicated to the UAV relaying systems employing SM scheme for transmission and analyzed the corresponding BER performance. Against this background, we propose an SM-aided UAV relaying system, which employs multiple transmit antennas to improve the spectral efficiency of the system, but only single RF chain is used. The burden of signal processing on the UAV is significantly reduced when SM is used at the transmitter because the UAV does not need to decouple multiple data streams. Moreover, the high levels of mobility and adaptability of the UAV are conducive to improving the quality of the wireless communication environment. Furthermore, compared with the conventional single transmit antenna aided systems, the SM based systems are capable of achieving higher data rates. The reason is that additional information is conveyed by the index of the active antenna besides constellation symbols. For the system under consideration, we analyze the performance of the SM-aided UAV relaying systems, and some theoretical BER expressions and diversity gain are derived for the performance evaluation. The main contributions of this paper are summarized as follows:

1) By using SM scheme in the UAV relaying network, an SM-aided UAV (SM-UAV) relaying system is presented. In terms of this system model, we investigate the BER performance over Nakagami-$m$ fading channels, and derive the probability density function (PDF) and moment generating function (MGF) of the effective signal-to-noise ratio (SNR). Then, with these results, the error probabilities of constellation symbol detection and antenna index detection as well as the error probability of joint symbol and antenna index detections, which constitute the average BER, are further deduced. Based on these error probabilities, a closed-form approximate expression of average BER of the system is obtained. For integer $m$, we also derive the simplified BER expression for performance evaluation by removing the calculation of the complicated hypergeometric function.

2) According to the obtained approximate average BER, an asymptotic BER is deduced by the asymptotic performance analysis of the system at high SNR, and the resultant closed-form asymptotic BER expression is achieved. This asymptotic BER can agree well with corresponding simulations for large SNR. Based on the asymptotic BER, the diversity gain is analyzed. As a result, the diversity order is obtained, which depends on the size relationship between the Nakagami fading parameter and one.

3) By means of the analytical and simulation results, the error performance of SM-UAV system is characterized. The results show that the presented theoretical analysis and BER expressions are all effective, having the values close to the corresponding simulations. Thus, the error performance of the system can be effectively evaluated in theory. Moreover, to assess the performance, the impacts of the modulation size, numbers of transmit and receive antennas, Nakagmai fading parameter, UAV position, and the distance between the ground terminals on the performance of SM-UAV are also analyzed.

Notations: In this paper, $(\cdot)^{\mathrm{H}}$, $(\cdot)^{\mathrm{T}}$, and $(\cdot)^{-1}$ stand for the conjugate transpose, transpose and inverse operators, respectively. $\mathbb{C}^{m \times n}$ denotes a $m \times n$ complex matrix. Vectors and matrices are represented by boldface lower-case and upper-case symbols, respectively. ${{\boldsymbol{I}}}_n$ is an identity matrix with $n$ rows and $n$ columns, and $g^i$ denotes the $i$-${\mathrm{th}}$ entry of a vector $\boldsymbol{g}$. Besides, notation $\| \cdot \|$ means the Frobenius norm. Finally, ${\cal{CN}}(0,N_0)$ is used to denote the complex Gaussian distribution with zero mean and variance $N_0$.

II.   The Proposed Method

We consider an SM-aided dual-hop UAV relaying network, as shown in Figure 1, where SM is employed for transmission, and a ground terminal (GT) located at source (S) node is intended to communicates with another GT at destination (D) node via intermediate UAV relay (R) that provides ground-to-air (G2A) and air-to-ground (A2G) links. Due to the obstacles, the direct link between S and D does not exist. For the SM-UAV system in consideration, $N_t$ and $N_r$ antennas are deployed at S and D, respectively, and the AF UAV relay is half duplex and equipped with single antenna.

Figure  1.  SM-aided UAV relaying network.

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The channels from S to R and R to D are expressed by ${{{\boldsymbol{h}}}}_{\rm{sr}}^{\mathrm{T}} \in {\mathbb{C}^{1 \times {N_t}}}$ and ${{\boldsymbol{h}}_{\rm{rd}}} \in {\mathbb{C}^{{N_r} \times 1}}$, which are modeled as Nakagami-$m$ fading channels with fading parameters $m_{\rm{sr}}$ and $m_{\rm{rd}}$ $(m_{\rm{sr}}, m_{\rm{rd}}\geq0.5)$, respectively, where sr represents the links between S and R, and rd represents the links between R and D.

We assume that the path loss coefficients for the G2A and A2G links are denoted as ${\mathrm{P}}{\mathrm{l}}_{\rm{sr}}$ and ${\mathrm{P}}{\mathrm{l}}_{\rm{rd}}$, respectively, inspired from [25], the large-scale fading $L_{pq}$ (for $pq \in \{{\mathrm{sr}},{\mathrm{rd}}\}$) can be given by

where $\delta_{\mathrm{sr}\left(\mathrm{rd}\right)}= \left\| W_{\mathrm{U}}-W_{\mathrm{S}\left(\mathrm{D}\right)} \right\|$ is linear distance between two nodes, $W_{\mathrm{S}}$ and $W_{\mathrm{D}}$ are the coordinate points of S and D, respectively, as shown in Figure 1. A UAV hovers in the sky and is located at $W\mathrm{_U}(d_{\rm{sr}},H)$ (see Figure 1), and $H$ is the altitude. $a$ represents path loss exponent and $b_{pq}$ is constant determined by the application scenario, more specifically, $a$ is set as 2 and $b_{pq}$ is expressed as [25], [26]

where $C = {\eta _{{\mathrm{LOS}}}} - {\eta _{{\mathrm{NLOS}}}}$, $B = 10\lg {\left( {4\pi f/v} \right)^2} + {\eta _{{\mathrm{NLOS}}}}$ and ${\theta _{pq}} = \frac{{180}}{\pi }\arctan \left( {\frac{H}{{{d_{pq}}}}} \right)$ represents the elevation angle between UAV and ground node S or D. Besides, $f$ is the carrier frequency and $v$ denotes the speed of light, $a'$, $b'$, $\eta_{{\mathrm{LOS}}}$, and $\eta_{{\mathrm{NLOS}}}$ are depending on the environment, where $\eta_{\xi}$ ($\xi \in \{{\mathrm{LOS}}, {\mathrm{NLOS}}\}$) is the mean value of the excessive path loss [26], and NLOS denotes the non-line-of-sight.

The transmission process of AF UAV relaying systems is divided into two phases. In the first phase, the source transmits the signal to the relay, and the corresponding received signal at R is given by

where $P_{\mathrm{s}}$ is the transmission power of the GT at S, the noise $z_{\rm{sr}}\sim{\cal{CN}}(0,\sigma^2)$, and $\sigma^2$ is the noise power. ${{\boldsymbol{x}}_{i,n}}$ denotes the SM signal and it can be expressed as

in which the subscript $i\ (i=1,2,\ldots,N_t)$ represents that the $i$-${\mathrm{th}}$ transmit antenna is activated for transmission, $s_n\ (n=1,2,\ldots,M)$ indicates the transmit signal which is the $n$-${\mathrm{th}}$ symbol of the $M$-ary amplitude-phase modulation constellation, and the constellation signals satisfy the power constraint $\mathbb{E}\left\{ {|{s_n}|^2} \right\} = 1$. The $M$-ary quadrature amplitude modulation (M-QAM) is employed for the performance analysis, and similar analytical method can be applied to the $M$-ary phase shift keying modulation (M-PSK) modulation.

In the second phase, the UAV relay amplifies and transfers the signal received in the first phase to the GT at D, then the received signal at D is given by

where $A = \sqrt{P_{\mathrm{r}}/(P_{\mathrm{s}}L_{\rm{sr}}+\sigma^2)}$ is the amplification factor, $P_{\mathrm{r}}$ is transmission power of the UAV relay. The total power is $P_{\mathrm{t}} = P_{\mathrm{s}} + P_{\mathrm{r}}$, and it satisfies $P_{\mathrm{s}} = c_1P_{\mathrm{t}}$ and $P_{\mathrm{r}} = c_2 P_{\mathrm{t}}$ in which $c_1$ and $c_2$ are power allocation coefficients ranging from 0 to 1. For analytical convenience, we define the average SNR as $\rho$ = $P_t/\sigma^2$ in this paper. ${\boldsymbol z}_{\rm{rd}}$ is the noise vecor, and its element obeys the complex Gaussian distribution with zero-mean and variance $\sigma^2$. $h_{\rm{sr}}^i$ is the $i$-${\mathrm{th}}$ entry of $\boldsymbol{h}_{\rm{sr}}$. Let ${\tilde{\boldsymbol{z}}} = A\sqrt {{L_{\rm{rd}}}} {{\boldsymbol{h}}_{\rm{rd}}}{z_{\rm{sr}}} + {{\boldsymbol{z}}_{\rm{rd}}}$, then its covariance is calculated by ${{{\varSigma }}_{{\tilde{\boldsymbol{z}}}}} = {A^2}{L_{\rm{rd}}}\sigma^2{{\boldsymbol{h}}_{\rm{rd}}}{\boldsymbol{h}}_{\rm{rd}}^{\mathrm{H}}+ \sigma^2{{\boldsymbol{I}}_{{N_{\mathrm{r}}}}}$.

According to the Woodbury identity, the inverse of ${{{\varSigma }}_{{\tilde{\boldsymbol{z}}}}}$ can be given by

With (6), the received signal ${{\boldsymbol{y}}}_{\rm{rd}}$ can be whitened by multiplying ${{\varSigma }}_{{\tilde{\boldsymbol{z}}}}^{ -1/2}$. Thus we have

After whitening the noise, the element of ${{\varSigma }}_{{\tilde{\boldsymbol{z}}}}^{{{ - 1} / 2}}{\tilde{\boldsymbol{z}}}$ is distributed as ${\cal{CN}}\; (0,1)$. Using (7), the optimal detection algorithm based on ML principle is written as

With (8), conditioned on the channel information $\{{{\boldsymbol{h}}_{\rm{sr}}},{{\boldsymbol{h}}_{\rm{rd}}}\}$, the conditional pairwise error probability (CPEP) can be given by (9).

where ${\gamma _{\rm{sr}}} = {P_{\mathrm{s}}}{L_{\rm{sr}}}{\left| {h_{\rm{sr}}^i{s_n} - h_{\rm{sr}}^j{s_q}} \right|^2}{\rm{/ (2\sigma^2)}}$, ${\gamma _{\rm{rd}}} = {P_{\mathrm{r}}}{L_{\rm{rd}}}{\left\| {{{\boldsymbol{h}}_{\rm{rd}}}} \right\|^2}/ \sigma^2$, ${\gamma _{{\mathrm{srd}}}} = {\gamma _{\rm{rd}}}{\gamma _{\rm{sr}}}{\rm{/}} \left( {{\gamma _{\rm{rd}}} + G} \right)$, and $G = {P_{\mathrm{s}}}{L_{\rm{sr}}}/\sigma^{2} + 1$. Apparently, the overall average BER of the system hinges on the detection errors of antenna indices and transmitted symbols, so we can divide the detection errors into three cases: due to the modulation symbol detection $(i = j, n\neq q)$, due to the antenna index detection $(i\neq j, n = q)$, and due to the joint detection $(i\neq j, n\neq q)$. The error probabilities of these three cases will be analyzed in next section.

III.   Performance Analysis of SM-UAV System

In this section, we will give the error performance analysis of the SM-aided AF UAV relaying system, an approximate closed-form expression of average BER will be derived. According to [19], the average BER of the SM system can be tightly upper bounded as

where $\overline{{\mathrm{BER}}}_{{\mathrm{sig}}}$ denotes the bit error probability of the transmitted symbol detection given the correct antenna index, $\overline{{\mathrm{BER}}}_{{\mathrm{spa}}}$ indicates the bit error probability of transmit antenna index detection given the correct symbol, and $\overline{{\mathrm{BER}}}_{{\mathrm{joint}}}$ denotes the bit error probability of the event that both antenna index and transmitted symbol are wrongly detected.

Considering that (10) can approximate the actual BER of SM system well, we use it as the approximate BER in this paper. In order to obtain the approximate average BER, we will derive the $\overline{{\mathrm{BER}}}_{{\mathrm{sig}}}$, $\overline{{\mathrm{BER}}}_{{\mathrm{spa}}}$, and $\overline{{\mathrm{BER}}}_{{\mathrm{joint}}}$ in the sequel.

1   Arbitrary Nakagami fading parameter ${\boldsymbol{m}}$ case

We firstly analyze the first term ${\overline {{\mathrm{BER}}} _{{\mathrm{sig}}}}$ in (10). Assuming that antenna index is given and using (7), the received SNR can be obtained as

where ${\gamma _{{\mathrm{sr}}_{{{1}}}}} = {P_{\mathrm{s}}}{L_{\rm{sr}}}{\left| {h_{\rm{sr}}^i} \right|^2}\sigma^{-2}$. Since the channel link A2G follows the Nakagami-$m$ distribution, we can obtain that $f_{\left| {h_{\rm{sr}}^i} \right|}(x) = \frac{2{m_{\rm{sr}}}^{m_{\rm{sr}}}{x ^{2{m_{\rm{sr}}} - 1}}}{{\varGamma ({{m_{\rm{sr}}}})}}\exp \left( - {{m_{\rm{sr}}}x^2 } \right)$. Then, with this result, the PDFs of $\gamma_{{\mathrm{sr}}_1}$ can be calculated as

where ${G_{\rm{sr}}} = {P_{\mathrm{s}}}{L_{\rm{sr}}}/(2\sigma^2)$, ${\bar \gamma _{{\mathrm{sr}}_{{\rm{1}}}}} = 2{G_{\rm{sr}}}$, and $\varGamma(\cdot)$ is Gamma function. Similarly, the PDF of $\gamma_{\rm{rd}}$ can be derived as

where ${G_{\rm{rd}}} = {P_{\mathrm{r}}} L_{\rm{rd}}/\sigma^2$.

When M-QAM is employed, the BER expression in terms of received SNR under additive white Gaussian noise (AWGN) channel is given by (14).

where ${\mathrm{erfc}}(\cdot)$ is complementary error function, parameters $\pi\left(M\right),\mu_a,\ \mathrm{and}\ \nu_a$ are uniquely determined by modulation size $M$ [27], and the BER expression of BPSK under AWGN channel can also be written in this form. Based on this, ${\overline {{\mathrm{BER}}} _{{\mathrm{sig}}}}$ can be obtained by averaging $\mathrm{BER}(\gamma)$ over the PDF of $\gamma_{{\mathrm{srd}}_1}$, i.e., (15).

where ${{\cal{M}}_{{\gamma _{{\mathrm{sr}}_{{\rm{1}}}}}}}( \cdot )$ is the MGF of $\gamma_{{\mathrm{sr}}_1}$, and the integral ${\cal{I}}\left( {{\gamma _{\rm{rd}}}} \right) = \frac{2}{\pi } \times \int_0^{{\pi / 2}} {{\cal{M}}_{{\gamma _{{\mathrm{sr}}_{{\rm{1}}}}}}}$ $\left( { - \frac{{{\nu _a}{\gamma _{\rm{rd}}}}}{{\left( {{\gamma _{\rm{rd}}} + G} \right){{\sin }^2}\theta }}} \right) {\mathrm{d}}\theta$ in (15) can be computed as (16), where $F\left( { \cdot ,\; \cdot \;;\; \cdot \;;} \right)$ is Gauss hypergeometric function.

Utilizing the Gauss-Laguerre quadrature $f(x){{\mathrm{e}}^{ - x}} \approx \sum\nolimits_{k = 1}^{{N_p}} {{\omega _k}f\left( {{t_k}} \right)}$, ${\overline {{\mathrm{BER}}} _{{{\mathrm{sig}}}}}$ in (15) can be rewritten as

in which $N_p$ is the order of Laguerre polynomial, $\omega_k, t_k$ can be obtained by referring to Table (25.9) in [28].

In the following, $\overline{\mathrm{BER}}_{\mathrm{joint}}$ will be calculated. If both the detected modulation symbol and antenna index are different from the sent SM symbol $(i\neq j, n\neq q)$, then we have: ${\gamma _{{\mathrm{sr}}_{2}}} = {G_{\rm{sr}}}{\left| {h_{\rm{sr}}^i{s_n} - h_{\rm{sr}}^j{s_q}} \right|^2}$, ${\gamma _{{\mathrm{srd}}_{2}}} = {\gamma _{\rm{rd}}}{\gamma _{{\mathrm{sr}}_{2}}}{{/}} \left( {{\gamma _{\rm{rd}}} + G} \right)$. Define that $h_{\rm{sr}}^i = {\beta _i}{{\mathrm{e}}^{\ell {\varphi _i}}}$, for $i = 1,2,\ldots,N_t$, $\beta_i$ is assumed to follow Nakagami-$m$ distribution, and ${\varphi _i}$ is assumed to be independent and uniformly distributed in $[0,2\pi)$ and ${\ell ^2} = -1$. Similarly, ${s_n} = {\alpha _n}{{\mathrm{e}}^{\ell {\psi_n}}},n = 1, 2,\ldots, M$, $\alpha_n$ and ${{\psi_n}}$ represent the amplitude and phase of the transmitted symbol, respectively. By exploiting the MGF-based approach and referring to [29], the MGF of $\gamma_{{\mathrm{sr}}_2}$ can be given by

where ${f_{{\beta _i},{\beta _j}}}\left( {{\xi _i},{\xi _j}} \right)$ is the joint PDF of the channel envelopes $\beta_i$ and $\beta_j$, since $\beta_i$ and $\beta_j$ are independent of each other, so the joint PDF can be expressed as

Besides, using the Euler’s formula, the conditional MGF ${{\cal{M}}_{{\gamma _{{\mathrm{sr}}_{2}}}}}\left( {s\left| {{\beta _i},{\beta _j}} \right.} \right)$ can be obtained as (20), which is shown at the top of the next page, where $I_0(\cdot)$ is the modified Bessel function.

By substituting (19) and (20) into (18), ${{\cal{M}}_{{\gamma _{{\mathrm{sr}}_{2}}}}}\left( s \right)$ can be written as (21).

Further, the above integral $J\left( {s,{\xi _i}} \right)$ can be computed by using (6.643.2) in [30], i.e.,

where $\varPhi \left( { \cdot , \cdot ; \cdot } \right)$ is the confluent hypergeometric function [30]. Substituting (22) into (21) and using (7.621.4) in [30], a closed-form expression of the ${{\cal{M}}_{{\gamma _{{\mathrm{sr}}_{2}}}}}\left( s \right)$ can be obtained as (23).

According to (23), utilizing the Gauss-Laguerre quadrature again, the MGF of $\gamma_{{\mathrm{srd}}_2}$ can be calculated as

Then, the average pairwise error probability (APEP) of the transmitted signal ${\boldsymbol{x}}_{i,n}$ and the detected SM signal ${\boldsymbol{x}}_{j,q}$ can be written as (25).

where ${\phi _u} = 0.5\cos \left[ {\left( {2u - 1} \right)\pi /2{N_u}} \right]$, $N_u$ is the order of Chebyshev polynomial [28]. By exploiting (25), $\overline{{\mathrm{BER}}}_{{\mathrm{joint}}}$ in (10) can be calculated as (26), which is shown at the middle of the this page, where ${N\left( {{{\boldsymbol{x}}_{i,n}} \to {{\boldsymbol{x}}_{j,q}}} \right)}$ represents the number of bits in error between the transmitted signal ${\boldsymbol{x}}_{i,n}$ and the estimated signal ${\boldsymbol{x}}_{j,q}$.

Finally, we compute the $\overline{{\mathrm{BER}}}_{{\mathrm{spa}}}$ in (10) when the transmitted signal $s_n$ is known $(i\neq j, \;n = q)$. Conditioned on the $s_n$, ${\gamma _{{\mathrm{sr}}_{3}}} = {G_{\rm{sr}}}{\left| {{s_n}\left( {h_{\rm{sr}}^i - h_{\rm{sr}}^j} \right)} \right|^2}$, ${\gamma _{{\mathrm{srd}}_{3}}} = {\gamma _{\rm{rd}}}{\gamma _{{\mathrm{sr}}_{3}}}{\rm{/}}$ $\left( {{\gamma _{\rm{rd}}} + G} \right)$ and it should be noted that ${\overline{\mathrm{BER}}_{{\mathrm{spa}}}}$ is a special case of ${ \overline{\mathrm{BER}}_{{\mathrm{joint}}}}$, where $\alpha_q = \alpha_n$. By replacing $\alpha_q$ by $\alpha_n$ in (23), it can be shown that the MGF of $\gamma_{{\mathrm{sr}}_3}$ is given by

Using some algebraic manipulations similar to (24) and (25), the ${\mathrm{PEP}}\left( {{{\boldsymbol{x}}_{i,n}} \to {{\boldsymbol{x}}_{j,m}}} \right)$, which indicates the probability of detecting the $j$-${\mathrm{th}}$ transmit antenna when the $i$-${\mathrm{th}}$ antenna is actually transmitting, can be calculated as

With (28), the ${\overline{\mathrm{BER}}_{{\mathrm{spa}}}}$ in (10) can be attained as

By plugging (17), (26), (28), and (29) into (10), we can obtain the average BER as (30).

Equation (30) is an approximate closed-form expression of average BER of SM-UAV system, and will have the value close to the simulation.

2   Integer ${\boldsymbol{m}}$ case

In the above subsection, the approximation of average BER is derived based on the arbitrary Nakagami fading parameters $m_{\rm{sr}}$ and $m_{\rm{rd}}$, and thus the computational complexity is relatively higher since the complicated Gauss hypergeometric functions need to be calculated. For this reason, considering the integer values of $m_{\rm{sr}}$ and $m_{\rm{rd}}$ in this subsection, we will simplify the calculation of approximate average BER.

When $m_{\rm{sr}}$ and $m_{\rm{rd}}$ are integers, the integral ${\cal{I}}\left( {{\gamma _{\rm{rd}}}} \right)$ in (15) is reduced to (31), where the superscript “INT” represents the expression for integer fading parameters $m$. Substituting (31) into (17), the ${{\overline {\mathrm{BER}}}^{{\mathrm{INT}}}_{{\mathrm{sig}}}}$ can be attained.

In what follows, we will analyze the ${\overline{\mathrm{BER}}_{{\mathrm{joint}}}}$ for integer $m_{\rm{sr}}$.

For this case, the Gauss hypergeometric function in (23) can be simplified as (32).

where

By plugging (32) into (23), the MGF of $\gamma_{{\mathrm{sr}}_2}$ for integer $m_{\rm{sr}}$ is given by (34).

Substituting (34) into (24) and (26), we can obtain the ${\overline{\mathrm{BER}}_{{\mathrm{joint}}}}$ for integer $m_{\rm{sr}}$, i.e., ${\overline{\mathrm{BER}}_{{\mathrm{joint}}}^{{\mathrm{INT}}}}$. For (34), when $\alpha_q = \alpha_n$, the MGF of $\gamma_{{\mathrm{sr}}_3}$ for integer $m_{\rm{sr}}$ can be written as

Substituting (35) into (28) and (29), the $\overline{\mathrm{{BER}}}_{{\mathrm{spa}}}$ for integer $m_{\rm{sr}}$, i.e., $\overline{\mathrm{{BER}}} _{{\mathrm{spa}}}^{{\mathrm{INT}}}$ is achieved. With the obtained ${\overline {\mathrm{{BER}}}}^{{\mathrm{INT}}}_{{\mathrm{sig}}}$, ${\overline{\mathrm{BER}}_{{\mathrm{joint}}}^{{\mathrm{INT}}}}$ and $\overline{\mathrm{{BER}}}_{{\mathrm{spa}}}^{{\mathrm{INT}}}$, using (30), the approximate average BER for integer $m_{\rm{sr}}$ is given by

Compared to (30), the computation of (36) will be relatively simpler because the complicated hypergeometric functions in the former do not need to be calculated. Moreover, when Nakagami fading parameters are the integers, they will be identical.

IV.   Asymptotic BER and Diversity Gain

As shown in (17), the value of $\overline {\mathrm{{BER}}} _{{\mathrm{sig}}}$ will depend on the integral ${\cal{I}}\left( {{G_{\rm{rd}}}{t_n}/{m_{\rm{rd}}}} \right)$. With (16), the integral is given by (37).

where ${G_{\rm{sr}}} = {P_{\mathrm{s}}}{L_{\rm{sr}}}/(2\sigma^2)$, $G = 2G_{\rm{sr}}+1$, ${\bar \gamma _{{\mathrm{sr}}_{{\rm{1}}}}} = 2{G_{\rm{sr}}}$, and ${G_{\rm{rd}}} = {P_{\mathrm{r}}}{L_{\rm{rd}}}/\sigma^2$.

When the SNR $\rho$ is large, i.e., $P_{\mathrm{t}}/\sigma^2$ is large, and correspondingly, $P_{\mathrm{s}}/\sigma^2$ and $P_{\mathrm{r}}/\sigma^2$ are also large. So under this case, the value of $F$ function in (37) will be approximated as 1 in terms of the definition of Gauss hypergeometric function. Based on this, the integral ${\cal{I}}\left( {{G_{\rm{rd}}}{t_n}/{m_{\rm{rd}}}} \right)$ at high SNR is changed to (38).

Substituting (38) into (17) yields

where the superscript ASY represents asymptotics. This is an asymptotic expression of $\overline {\mathrm{{BER} }}_{{\mathrm{sig}}}$ at high SNR. It is well known that the diversity gain is defined as the slope of the average BER curve versus the SNR at high SNR regions in log-log scale. Based on this, with (39), the diversity gain of $\overline {\mathrm{{BER} }}_{{\mathrm{sig}}}^{{{\mathrm{ASY}}}}$ is

In the following, we will derive the asymptotic values of $\overline {\mathrm{{BER} }}_{{\mathrm{joint}}}$ and $\overline {\mathrm{{BER} }}_{{\mathrm{spa}}}$, respectively. From formulas (23)–(26), we can see that the term ${{\cal{M}}_{{\gamma _{{\mathrm{sr}}_{2}}}}}\left( \cdot \right)$ will dominate the values of $\overline {\mathrm{{BER} }} _{{\mathrm{joint}}}$. Using (13.4.1) in [28], the approximated value of ${{\cal{M}}_{{\gamma _{{\mathrm{sr}}_{2}}}}}\left( s \right)$ at high SNR can be expressed as

Substituting (41) into (24), the MGF of $\gamma_{{\mathrm{srd}}_2}$ at high SNR can be approximated as

By plugging (42) into (26), an asymptotic expression of the $\overline{\mathrm{{BER}}}_{{\mathrm{joint}}}^{{\mathrm{ASY}}}$ can be attained, i.e.,

Similarly, we can deduce the asymptotic expression of the $\overline{\mathrm{{BER}}}_{{\mathrm{spa}}}^{{\mathrm{ASY}}}$ at high SNR, i.e.,

where ${{\cal{M}}_{{\gamma _{{\mathrm{sr}}_{3}}}}^{{\mathrm{ASY}}}}(\cdot)$ is given by

With (43) and (44), the diversity gains of $\overline{\mathrm{{BER}}}_{{\mathrm{joint}}}^{{\mathrm{ASY}}}$ and $\overline{\mathrm{{BER}}}_{{\mathrm{spa}}}^{{\mathrm{ASY}}}$ can be evaluated as

According to (10), using (39), (43), and (44), the asymptotically approximate expression of the average BER can be obtained as

Equation (47) can match the simulated BER well at high SNR. With this equation, using (40) and (46), the diversity gain of the system is given by

When the fading parameter $m_{\rm{sr}}$ is integer, in terms of (48), the system will achieve the diversity gain of $G_d = 1$ since $m_{\rm{sr}}\ge1$ under this case.

V.   Simulation Results

In this section, we will provide the simulation results to validate the theoretical analysis, and evaluate the BER performance of the SM-aided UAV relaying system over Nakagami-$m$ fading channels for different scenarios. The Monte Carlo method is employed for simulation realization, and Gray mapping of bits to symbol is used. In simulations, if not specified, the parameter setting defaults to $m = m_{\rm{sr}} = m_{\rm{rd}} = 2$, the orders of numerical quadrature are set as $N_p = 20$ and $N_u = 5$, $c_1 = c_2 = 0.5$. We assume that the nodes locations in Figure 1 are shown as $d_{\rm{sr}} = d_{\rm{rd}} = 100$ m, and $H = 15$ m. Additionally, the parameters associated with large scale fading $\eta_{{\mathrm{LOS}}} = 0.1$ dB, $\eta_{{\mathrm{NLOS}}} = 21$ dB, $a' = 5.0018$, $b' = 0.3511$, $v= 3\times 10^8$ m/s, $f=2$ GHz, and $\sigma^2 =-100$ dBm [25]. The simulation results are obtained by $10^6$ channel realizations, and are shown in Figures 2-7.

Figure  2.  BER performance of the system with different modulation orders.

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Figure  3.  BER performance of the system with different receive numbers.

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Figure  4.  BER performance of the system with different Nakagami fading parameters.

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Figure  5.  BER performance versus the distance between S node and D node $d$.

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Figure  6.  BER performance of the system with different power allocation coefficients.

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In Figure 2, we plot the average BER of the system with different modulation orders, where BPSK and 4QAM (QPSK) are employed for modulation, the numbers of transmit and receive antennas are set as $N_{\mathrm{t}} = N_{\mathrm{r}} = 2$. It is observed that the average BERs get smaller for the systems with higher SNR, namely the system performance becomes better as the SNR increases, as expected. Moreover, it is found that the simulated BERs are very close to the corresponding theoretical values, which indicates that the theoretical analysis is valid, and the system BER can be effectively evaluated by the derived theoretical expression. Besides, due to reduction of the Euclidean distance, the BER performance will deteriorate with the increase of modulation order. Namely, the system with 4QAM has worse performance than that with BPSK.

Figure 3 shows the average BER of the system for different numbers of receive antennas, where $N_{\mathrm{t}} = 4$, $N_{\mathrm{r}} = 1, 2$, and BPSK is adopted for modulation. As illustrated in Figure 3, the theoretical average BERs still match the corresponding simulated values well, which further shows the effectiveness of the theoretical analysis. Moreover, the system with $N_{\mathrm{r}} = 2$ has superior performance over that with $N_{\mathrm{r}} = 1$, which means the BER performance can be improved by increasing $N_{\mathrm{r}}$. It is also observed that the diversity gain is independent of the receive antenna number, since the BER curves have the same slop.

Figure 4 illustrates the BER performance of the system with different Nakagami fading parameters, where $m = m_{\rm{sr}} = m_{\rm{rd}} = 0.6, 1, 2.5$, $N_{\mathrm{t }}= N_{\mathrm{r}} = 2$, and $m = 1$ corresponds to the Rayleigh fading channel. The modulation is 4QAM. It is shown in Figure 4 that the asymptotic average BERs have the values close to the corresponding simulated ones, especially at high SNR. Moreover, the performance becomes better as the fading parameter $m$ increases, that is, the system with $m = 2.5$ has superior performance over that with $m = 1$, and the system with $m = 1$ has superior performance over that with $m = 0,6$, this is because the fading severity decreases as the Nakagami parameter $m$ increases. In other words, the less severe fading conditions will obviously improve the system performance. Furthermore, from Figure 4, we can see that the diversity gain depends on the fading parameter $m_{\rm{sr}}$. It is found that as $m_{\rm{sr}} \geq 1$, the BER curves have the same diversity gain of 1, but as $m_{\rm{sr}}<1$, the achievable diversity gain is reduced to $m_{\rm{sr}}$. The above results agree with the diversity gain analysis in Section IV, and thus our theoretical analysis is effective.

Figure 5 depicts the impact of the distance between S and D ($d$) on the BER performance of the system, where the distance $d = d_{\rm{sr}}+d_{\rm{rd}}$, SNR = 20 dB, and the numbers of transmit and receive antennas are $N_{\mathrm{t}} = N_{\mathrm{r}} = 2$, and 4QAM is used for modulation. The BER performances under three cases of $d_{\rm{sr}}/d =$0, 0.5, 1 are compared. From Figure 5, we can see that with the increase of the distance $d$, the case of $d_{\rm{sr}}/d = 0.5$ exhibits superior performance over the other two cases since the equal powers are allocated for node S and node R, especially for small $d$. Besides, when $d$ increases, the BER performance will degrade because of large path loss.

In Figure 6, we plot the BER performance of the system with different power allocation coefficients, where $N_{\mathrm{t}} = N_{\mathrm{r}} = 2$, $c_1 = c_2 = 0.5$, $c_1 = 0.1$, $c_2 = 0.9$, and $c_1 = 0.9$, $c_2 = 0.1$ are considered. 4QAM is employed for modulation. As illustrated in Figure 6, different power allocation coefficients $\{c_1, c_2\}$ will affect the BER performance. It is shown that the BER performance with equal power allocation (i.e., $c_1 = c_2 = 0.5$) has better performance than that with the other two power allocation cases (i.e., $c_1 = 0.1$, $c_2 = 0.9$ and $c_1 = 0.9$, $c_2 = 0.1$). This is because the optimal $c_1$ and $c_2$ tend to be equal under the case of $d_{\rm{sr}}$ equal to $d_{\rm{rd}}$. Thus, superior performance is achieved.

In Figure 7, we evaluate the impact of altitude of UAV on the BER performance of the system under different SNRs. From Figure 7, it is found that the BER is firstly decreased and then increased with the increase of the altitude. Thus, an optimal altitude exists and it brings about the best performance. As shown in Figure 7, the optimal performance is obtained when altitude of UAV $H$ is about 40 m. This is because with the increase of $H$, the air-to-ground link becomes closer to line of sight, and corresponding BER performance will be better. However, as $H$ becomes larger, the path loss is also increased greatly and will be the dominant factor to affect the performance. As a result, the BER performance starts to decrease. Besides, with the increase of SNR, the BER performance will become better, as expected.

Figure  7.  BER versus the altitude of UAV $H$.

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VI.   Conclusions

We have studied the BER performance and diversity gain of the UAV relaying system with SM over Nakagami-$m$ fading channels. Based on the performance analysis, the closed-form expressions of $\overline {\mathrm{{BER}}}_{{\mathrm{sig}}}$, $\overline {\mathrm{{BER}}}_{{\mathrm{spa}}}$ and $\overline {\mathrm{{BER}}}_{{\mathrm{joint}}}$ are, respectively, derived. Using these results, the approximate average BER of the SM-UAV system is attained, and it can obtain the value close to the simulated BER. For the integer $m$, we also derive the simple BER expression to simplify the calculation of average BER, and avoid using the complicated Gauss hypergeometric functions. Then, with the obtained approximate BER, we also deduce the asymptotic BER to assess the asymptotic performance of the system under large SNR. With the asymptotic BER, the diversity gain of the system is analyzed. As a result, the diversity order of $\min\{m_{\rm{sr}},1\}$ is obtained for arbitrary $m$ and of 1 for integer $m$, respectively. Simulation results illustrate the effectiveness of the theoretical BER analysis. Thus, these theoretical expressions can work well and provide good performance evaluation for the system in theory, and avoid the conventional requirements for Monte Carlo simulation. Besides, the influence of the modulation order $M$, the numbers of transmit antenna and receive antenna, fading parameter $m$, the position of UAV and the distance between ground terminals on the system performance are also analyzed. It is found that increasing $N_{\mathrm{r}}$ and/or $m$ and/or decreasing $M$ can bring about the improvement of BER performance. Besides, with the increase of the distance between two ground terminals, the UAV had better be placed in the middle position, and resultant superior performance can be attained.