On Secure NOMA-Aided Semi-Grant-Free Systems^††thanks: Manuscript received.

Ultra-reliable low latency communications (URLLC) and massive machine-type communications (mMTC) are the two most important scenarios for the next internet of things (IoT). URLLC focuses on mission-critical applications wherein unprecedented levels of reliability and latency are of the utmost importance in the fifth generation and it’s beyond [1]. In contrast, mMTC aspires to connect a vast number of intelligent devices to the Internet. The user initiates the traditional grant-based (GB) access scheme with an access request to the base station (BS) in long term evolution. The BS responds by allocating an access grant through a four-step handshake procedure strategy. Once the BS grants the access request, data packets can be successfully transmitted without collision under ideal channel conditions. However, GB scheme does not suit these scenarios due to high latency and heavy signaling overhead [2, 3]. Moreover, the initial request transmission is still subject to collision and could require multiple transmissions depending on traffic load and the available resources at the BS.

To tackle these issues, grant-free (GF) transmission schemes were introduced in [4, 5], in which multiple users may occupy the same resource without the initial access request procedure. Unlike the GB principle, no dedicated request transmission for granting access and allocating resource blocks is required for GF communications before starting a data transmission. Although the GF scheme makes it possible to allow users to choose resource blocks independently and transmit data directly to reduce signaling overhead and latency effectively, collisions will become severe when multiple users select the same resource block to transmit simultaneously [6]. The collision issue can be resolved using massive multiple-input multiple-output (MIMO) or non-orthogonal multiple access (NOMA) technologies. The former solution utilizes spatial degrees of freedom to mitigate multi-user collisions, while the latter focuses on spectrum sharing among multiple users with successive interference cancellation (SIC) [7, 8, 9, 10].

Even though the massive connectivity can be supported through GF schemes, GB schemes are still desired, especially when strict quality of service (QoS) requirements exist [10]. The GB and GF transmission scheme must coexist in scenarios where URLLC applications are served by the GB scheme and mMTC applications in the same system are served by the GF scheme. For example, a new hybrid access scheme was proposed in [11] to meet the various requirements of IoT networks wherein machine-type users with small data packets and delay-tolerant traffic utilized the GF scheme, and some users with large data packets and delay-sensitive traffic used the GB scheme. NOMA-aided Semi-GF (SGF) transmission scheme was first explicitly introduced in [12] to alleviate the collisions and obtain massive connectivity. A single GB user with multiple GF users to perform NOMA and two contention control mechanisms were proposed to suppress the interference on the GB user from the GF users. Closed-form expressions for the outage probability (OP) of GF users were derived and the impact of different SIC decoding orders was investigated. Their results demonstrated the superior performance of NOMA-aided SGF schemes. Based on the relationship between the GB user’s targeted rate and channel conditions, an adaptive power allocation strategy was proposed to control the transmit power of GB users to ensure that the GB user’s signals are always decoded in the second stage of SIC [13]. In [14], the authors investigated the performance of an uplink SGF system with multiple uniformly distributed GF and GB users considered, in which the GF user whose received power is lower at the BS than that of the GB user was selected to pair with the connected GB user. Closed-form expressions for GB and GF users’ exact and approximate ergodic rates were derived. Further, the authors in [15] studied the effect of random locations of GF users on the performance of NOMA-assisted SGF systems by utilizing stochastic geometry. A dynamic threshold protocol was proposed to reduce the interference to GB users, and the outage performance was analyzed and compared with the open-loop protocol.

Relative to the SGF schemes proposed in [12], a new QoS-guarantee scheme for NOMA-aided SGF systems was proposed in [16] to ensure that the QoS of the GB user is the same as that when it solely occupies the channel. Closed-form expressions were derived for the exact and asymptotic OP with the best-user scheduling (BUS) scheme and a hybrid SIC scheme. The results demonstrated that the proposed scheme could significantly improve the reliability of the GF users’ transmissions. Based on [16], a new adaptive power control strategy was proposed to solve OP error floors entirely by adjusting the GF user’s transmit power to change the decoding order of SIC in [17]. In [18], the authors analyzed the outage performance of the NOMA-aided SGF systems with multiple randomly distributed GF users with fixed power and dynamic power schemes. As discussed in [18], the BUS scheme may lead to a fairness problem because the users closer to the base station may be scheduled more frequently due to weak path loss. To solve the fairness problem, a cumulative distribution function (CDF)-based user scheduling (CUS) scheme was proposed where the GF user with the maximal CDF value will be admitted to the channel. The analytical expressions for the OP with the CUS and BUS schemes were derived and the impacts of small-scale fading, path loss, and random user locations were jointly investigated.

Recently, physical layer security for NOMA systems has attracted considerable attention [19] - [26]. In [19], the authors investigated the secrecy performance of NOMA systems. Stochastic geometry was utilized to model the locations of legitimate and illegitimate receivers and the analytical expressions for the exact and asymptotic secrecy outage probability (SOP) for both single-antenna and multi-antenna scenarios were derived. In [20], the authors investigated the optimal decoding order, transmission rates, and power allocation in the design of NOMA systems. Two optimization problems were proposed and solved: the transmission power was minimized subject to the secrecy outage and QoS constraints and the minimum secrecy rate was maximized subject to the secrecy outage and transmit power constraints, respectively. Their results indicated that the optimal decoding order would not vary with the secrecy outage constraint in the considered problems and the power allocation ratio to the user must be increased as the secrecy constraint becomes more stringent. In [21], Lv et al. proposed a new NOMA-inspired jamming and forwarding scheme to improve the security of cooperative communication systems and derived the analytical expressions for the lower bound of the ergodic secrecy sum rate (ESSR) and the asymptotic ESSR. Three relay selection schemes were proposed to enhance the secrecy performance of the multi-relay cooperative NOMA systems and the analytical expressions for the exact and asymptotic SOP were derived in [22]. In [23], the authors proposed a novel downlink multi-user transmission scheme to meet the heterogeneous service requirements for the airborne NOMA systems consisting of security-sensitive users and QoS-sensitive users. The scenario where the QoS-sensitive users act as potential internal eavesdroppers were considered. The achievable secrecy rate was maximized through the joint optimization of user scheduling, power allocation, and trajectory design. In [24], two new schemes were proposed to enhance the security of airborne NOMA systems by the single user requiring and multiple users requiring security, respectively, and the effectiveness of the proposed schemes in ensuring secure transmissions were analyzed. In [25], the relationship between the reliability and security of a two-user NOMA system was investigated. Considering different decoding capabilities at eavesdroppers and imperfect SIC, the analytical expressions of the SOP under the reliability outage probability constraint were derived. In [26], the authors investigated the secrecy performance of a NOMA-based MEC system using the hybrid SIC decoding scheme. The latency was minimized by jointly optimizing the power allocation, task allocation, and computational resource allocation. A reinforcement learning-based and a matching-based algorithm were proposed to solve the optimization problems for the single-user and multi-user scenarios.

Based on the authors’ knowledge, there are two main differences between traditional NOMA and SGF schemes: 1) In traditional NOMA systems, all the NOMA users can utilize the resource blocks, such as time slots or subcarriers. In NOMA-based SGF systems, only the selected GF users based on scheduling schemes are allowed to opportunistically gain access to those resource blocks that GB users would exclusively occupy. 2) For the conventional NOMA systems, the static SIC technology, either channel state information (CSI)-based SIC or QoS-based SIC, is utilized to cancel inter-user interference. The method in SGF systems to enhance spectral efficiency is through the hybrid (dynamic) SIC scheme. For these reasons, although the secrecy performance of NOMA systems has been investigated in many works, the results are not applicable to NOMA-based SGF systems. This is the motivation for this work. Technically speaking, it is much more challenging to investigate the secrecy performance with a hybrid (dynamic) SIC scheme than that with a static SIC scheme.

We investigate a NOMA-aided SGF system with a single GF user, and then the results have been extended to SGF systems with multiple GF users. The main contributions of this paper are summarized as follows.

1. 1.

   We analyze the secrecy performance of an uplink NOMA-aided SGF system with a single GF user as a benchmark. The analytical expression for the exact SOP of the GF user is derived. To obtain more insights, we derive asymptotic expressions for the SOP of the GF user in the high transmit signal-to-noise ratio (SNR) regime.

2. 2.

   We further investigate the secrecy performance of NOMA-aided SGF systems with multiple GF users. The analytical expression for the exact and asymptotic SOP with the BUS scheme is developed based on order statistics to facilitate the performance analysis. Monte Carlo simulation results are provided and compared with two different scheduling schemes. The effects of system parameters on the SOP of the considered system are demonstrated and the accuracy of the developed analytical results is verified.

3. 3.

   In contrast to the metrics, such as OP and ergodic rate, derived in [12]-[18], the secrecy performance of SGF systems is investigated in this work. Note that it is much more challenging to obtain the analytical expressions of the SOP relative to that of the OP for SGF systems, especially in the presence of multiple GF users.

The rest of this paper is organized as follows. Section II describes the considered system model. The SOP of the SGF systems with a single GF user and multiple GF users are analyzed in Sections III and IV, respectively. Section V presents the numerical and simulation results to demonstrate the analysis and the paper is concluded in Section VI. The notations utilized in this paper are summarized in Table I, which is shown at the top of this page.

Consider an uplink SGF system illustrated in Fig. 1, a GB user ($U_{B}$) transmits signals to the BS ($S$), and the channel is re-used by $K$ GF user ($U_{k},k=1,\cdots,K$) in SGF mode. In other words, $U_{k}$ is allowed to utilize the resource block that would be solely occupied by $U_{B}$ employing NOMA technology while $U_{B}$’s QoS experience is the same as when it occupies the channel alone. All the GF users are assumed to transmit signals with the same power ${\rho_{F}}$ and the channel gains are ordered as ${\left|{{h_{1}}}\right|^{2}}\leq\cdots\leq{\left|{{h_{K}}}\right|^{2}}$, where ${\left|{{h_{1}}}\right|^{2}}=\mathop{\min}\limits_{1\leq k\leq K}\left({\frac{{{{\left|{{g_{k}}}\right|}^{2}}}}{{r_{k}^{\alpha}}}}\right)$ and ${\left|{{h_{K}}}\right|^{2}}=\mathop{\max}\limits_{1\leq k\leq K}\left({\frac{{{{\left|{{g_{k}}}\right|}^{2}}}}{{r_{k}^{\alpha}}}}\right)$ where $g_{k}$ denotes $U_{k}$’s channel coefficient, $r_{k}$ denotes the distance between $U_{k}$ and $S$, and $\alpha$ signifies the path loss exponent. All the channels are assumed to undergo an independent identically and quasi-static Rayleigh fading model. To facilitate performance analysis, it is assumed that all the GF users are located in a small size cluster, such that the distances between $U_{k}$ and $S$ are same $\left({{r_{k}}={r_{F}}}\right)$.

The received signal at $S$ is expressed as ${y_{B}}=\sqrt{{P_{B}}}{h_{B}}{x_{B}}+\sqrt{{P_{F}}}{h_{k}}{x_{F}}+n$, where $P_{i}$ ($i\in\left\{{B,F}\right\}$) denotes the transmit power, ${{\left|{{h_{B}}}\right|^{2}}}=\frac{{{{\left|{{g_{B}}}\right|}^{2}}}}{{r_{B}^{\alpha}}}$, $g_{B}$ denotes $U_{B}$’s channel coefficient, $r_{B}$ denotes the distance between $U_{B}$ and $S$, $x_{i}$ is the signals from $U_{i}$ with unit power, i.e., $\mathbb{E}\left[{{{\left|{{x_{i}}}\right|}^{2}}}\right]=1$, and $n$ is the additive white Gaussian noise (AWGN) with zero mean and variance ${\sigma^{2}}$.

In this work, the BUS scheme is considered, which means the GF user achieving the maximum data rate is scheduled to transmit signals [16, 18]. The admission procedure consists of the following steps [18]: 1) The $S$ sends pilot signals, 2) Each user estimates its own channel state information (CSI), 3) $U_{B}$ feedbacks its transmit SNR, target rate, and CSI to $S$, 4) The $S$ calculates $U_{B}$’s decoding threshold and broadcasts $U_{B}$’s effective received SNR and decoding threshold to all GF users, 5) Each GF user calculates its transmit data rate, and 6) Each GF user sets its back-off time, which is a strictly decreasing function of the user’s data rate. Then the GF user with the maximal data rate will be admitted to transmitting through distributed contention control protocol [12].

To ensure the $U_{B}$’s QoS, there must have ${\log}_{2}\left({1+\frac{{{\rho_{B}}{{\left|{{h_{B}}}\right|}^{2}}}}{{1+\tau\left({{{\left|{{h_{B}}}\right|}^{2}}}\right)}}}\right)\geq{R_{B}}$, where ${\rho_{B}}=\frac{{{P_{B}}}}{{{\sigma^{2}}}}$, ${R_{B}}$ denotes the target data of $U_{B}$ and $\tau\left({{{\left|{{h_{B}}}\right|}^{2}}}\right)=\max\left\{{0,{\tau_{B}}}\right\}$ denotes the maximum interference power tolerated when $U_{B}$’s signal is decoded during the first stage of SIC [16] ¹¹1 The availability of perfect CSI is crucial in deciding the decoding order and the implementation of hybrid SIC. The imperfect power gain caused by imperfect CSI could lead to an inappropriate SIC decoding order being selected and SIC decoding failures occurring [27]. , ${\tau_{B}}=\frac{{{{\left|{{h_{B}}}\right|}^{2}}}}{{{\alpha_{B}}}}-1$, ${\alpha_{B}}=\frac{{\varepsilon_{B}}}{{{\rho_{B}}}}$, ${\varepsilon_{B}}={\theta_{B}}-1$, and ${\theta_{B}}={2^{{R_{B}}}}$.

$S$ first broadcasts $\tau\left({{{\left|{{h_{B}}}\right|}^{2}}}\right)$ before scheduling. By comparing their received power of GF’s signals on $S$ to $\tau\left({{{\left|{{h_{B}}}\right|}^{2}}}\right)$, all the GF users are divided into two groups (${{\cal S}_{\mathrm{I}}}$ and ${{\cal S}_{\mathrm{II}}}$).

• •

  For ${U_{k}}\in{{\cal S}_{\mathrm{I}}}$ $\left({1\leq k\leq\left|{{{\cal S}_{\mathrm{I}}}}\right|\leq K}\right)$, they experience ${\rho_{F}}{\left|{{h_{k}}}\right|^{2}}>\tau\left({{{\left|{{h_{B}}}\right|}^{2}}}\right)$ with ${\rho_{F}}=\frac{{{P_{F}}}}{{{\sigma^{2}}}}$, which will lead to ${\log}_{2}\left({1+\frac{{{\rho_{B}}{{\left|{{h_{B}}}\right|}^{2}}}}{{1+\tau\left({{{\left|{{h_{B}}}\right|}^{2}}}\right)}}}\right)<{R_{B}}$. This signifies that ${U_{k}}$’s signals must be decoded before decoding $U_{B}$’s signals to guarantee that $U_{B}$’s QoS experience is the same as when it occupies the channel alone ²²2 Since the signals from GF users were decoded before decoding those from the GB user in this case, additional latency for GB users will occur. Thus, the GF scheme in the NOMA-aided SGF systems are suitable for such applications with more stringent QoS than latency requirements. In other words, the NOMA-aided SGF systems aim to make a channel reserved by a GB user that can be shared by GF users, improving connectivity and spectral efficiency through collaboration between GF transmission and conventional GB schemes.. Then, the achievable rate of $U_{B}$ and $U_{k}$ are expressed as ${R_{B}^{\mathrm{I}}}={\log_{2}}\left({1+{\rho_{B}}{{\left|{{h_{B}}}\right|}^{2}}}\right)$ and ${R_{k}^{\mathrm{I}}}={\log}_{2}\left({1+\frac{{{\rho_{F}}{{\left|{{h_{k}}}\right|}^{2}}}}{{1+{\rho_{B}}{{\left|{{h_{B}}}\right|}^{2}}}}}\right)$, respectively.

• •

  For those GF users in ${U_{k}}\in{{\cal S}_{\mathrm{II}}}$ $\left({1\leq k\leq\left|{{{\cal S}_{\mathrm{II}}}}\right|\leq K}\right)$, they experience ${\rho_{F}}{\left|{{h_{k}}}\right|^{2}}<\tau\left({{{\left|{{h_{B}}}\right|}^{2}}}\right)$, which will lead to ${\log}_{2}\left({1+\frac{{{\rho_{B}}{{\left|{{h_{B}}}\right|}^{2}}}}{{1+\tau\left({{{\left|{{h_{B}}}\right|}^{2}}}\right)}}}\right)>{R_{B}}$. This signifies that the GF user’s signal in this group will be decoded at either the first or the second stage of SIC. Accordingly, ${U_{k}}$ will achieve a data of ${R_{k}^{\mathrm{I}}}={\log}_{2}\left({1+\frac{{{\rho_{F}}{{\left|{{h_{k}}}\right|}^{2}}}}{{1+{\rho_{B}}{{\left|{{h_{B}}}\right|}^{2}}}}}\right)$ or ${R_{k}^{{\mathrm{II}}}}={\log}_{2}\left({1+{\rho_{F}}{{\left|{{h_{k}}}\right|}^{2}}}\right)$. Due to ${R_{k}^{\mathrm{II}}}>{R_{k}^{\mathrm{I}}}$, to achieve the maximum data rate at the GF user, $U_{B}$’s signal must be decoded during the first stage of SIC [16, 18]. Thus, the achievable rate of $U_{B}$ and $U_{k}$ are expressed as ${R_{B}^{\mathrm{II}}}={\log_{2}}\left({1+\frac{{{\rho_{B}}{{\left|{{h_{B}}}\right|}^{2}}}}{{1+{\rho_{F}}{{\left|{{h_{k}}}\right|}^{2}}}}}\right)$ and $R_{k}^{{\mathrm{II}}}={\log_{2}}\left({1+{\rho_{F}}{{\left|{{h_{k}}}\right|}^{2}}}\right)$, respectively.

Then, the achievable rate of ${U_{k}}$ ($1\leq k\leq K-1$) is expressed as

It must be noted that only one GF user is selected to access the channel. The grouping stated before is logically grouped for analysis of the achievable rate of the selected GF user. Specifically, the signals from the users in different groups have different decode orders at the base station.

In this work, we consider the worst-case security scenario wherein $E$ is equipped with $N$ antennas using maximal ratio combining (MRC) scheme to fully decode the users’ information ³³3 In this case, it is assumed that the eavesdropper has a powerful multi-user detection capability (e.g. parallel interference cancellation) so that the received data stream can be distinguished and the interference generated by the superimposed signals can be subtracted [37]. As stated in [19, 25], this case is the worst-case scenario where the decoding capability of the eavesdropper has been overestimated, which makes the analysis and design robust for the practical scenario and is sensible and desirable from a security perspective.. Then, the eavesdropping rate is expressed as ${R_{E}}={\log_{2}}\left({1+{\rho_{F}}{{\left|{{H_{E}}}\right|}^{2}}}\right)$, where ${{{\left|{{H_{E}}}\right|}^{2}}}\buildrel\Delta\over{=}\sum\limits_{i=1}^{N}{{{\left|{{h_{{k_{i}}}}}\right|}^{2}}}$, ${\left|{{h_{{k_{i}}}}}\right|^{2}}=\frac{{{{\left|{{g_{{k_{i}}}}}\right|}^{2}}}}{{r_{E}^{\alpha}}}$, ${{\left|{{g_{{k_{i}}}}}\right|}^{2}}$ denotes channel coefficient between $k$-th GF user and $i$-th receive antenna at $E$, and $r_{E}$ denotes the distance between the GF users and $E$.

This subsection provides the statistical law of channel power gains, laying the performance analysis foundation. The probability density function (PDF) of ${{{\left|{{H_{E}}}\right|}^{2}}}$ is expressed as ${f_{{H_{E}}}}(x)=\frac{{r_{E}^{N\alpha}}}{{\Gamma\left(N\right)}}{x^{N-1}}{e^{-r_{E}^{N\alpha}x}}$, where $\Gamma(z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt$ is the Gamma function as defined by [30, (8.310.1)]. The CDF of ${\left|{{h_{K}}}\right|^{2}}$ is expressed as ${F_{{{\left|{{h_{K}}}\right|}^{2}}}}\left(x\right)=\sum\limits_{i=0}^{K}{{\varphi_{i}}}{e^{-ir_{F}^{\alpha}x}}$, where ${\varphi_{i}}=\left({{}_{\,\,i}^{K}}\right){\left({-1}\right)^{i}}$, and $\left({{}_{\,\,i}^{K}}\right)=\frac{{K!}}{{i!\left({K-i}\right)!}}$.

The joint PDF of ${\left|{{h_{i}}}\right|^{2}}$ and ${\left|{{h_{j}}}\right|^{2}}$$\left({1\leqslant i<j\leqslant K}\right)$ is expressed as [16]

where ${\phi_{1}}=\frac{{K!{{\left({-1}\right)}^{m+n}}\left({{}_{\quad n}^{j-i-1}}\right)\left({{}_{\,m}^{i-1}}\right)r_{F}^{2\alpha}}}{{\left({i-1}\right)!\left({K-j}\right)!\left({j-i-1}\right)!}}$, ${\phi_{2}}=r_{F}^{\alpha}\left({m+j-i-n}\right)$, and ${\phi_{3}}=r_{F}^{\alpha}\left({K-j+n+1}\right)$. Then, the joint CDF of ${\left|{{h_{i}}}\right|^{2}}$ and ${\left|{{h_{j}}}\right|^{2}}$ $\left({1\leqslant i<j\leqslant K}\right)$ is obtained as

When $i=1,j=K$, we obtain

and

respectively, where ${\mu_{0}}={\rm{}}\frac{{K!{{\left({-1}\right)}^{n}}\left({{}_{\;n}^{K-2}}\right)r_{F}^{2\alpha}}}{{\left({K-2}\right)!}}$, ${\mu_{1}}=\frac{{{\mu_{0}}}}{{r_{F}^{2\alpha}K\left({n+1}\right)}}$, ${\mu_{2}}=\frac{{{\mu_{0}}}}{{r_{F}^{2\alpha}K\left({K-n-1}\right)}}$, ${\mu_{3}}=\frac{{{\mu_{0}}}}{{r_{F}^{2\alpha}\left({n+1}\right)\left({K-n-1}\right)}}$.

The joint PDF and CDF of ${{{\left|{{h_{k}}}\right|}^{2}}}$, ${{{\left|{{h_{k+1}}}\right|}^{2}}}$ $\left({1\leq k\leq K-2}\right)$, and ${{{\left|{{h_{K}}}\right|}^{2}}}$ is given as [16]

and

respectively, where ${\varsigma_{0}}=\frac{{K!{{\left({-1}\right)}^{m+n}}\left({{}_{n}^{K-k-2}}\right)\left({{}_{m}^{k-1}}\right)r_{F}^{3\alpha}}}{{\left({K-k-2}\right)!\left({k-1}\right)!}}$, ${A_{0}}=r_{F}^{\alpha}\left({m+1}\right)$, ${B_{0}}=r_{F}^{\alpha}\left({K-k-n-1}\right)$, ${C_{0}}=r_{F}^{\alpha}\left({n+1}\right)$, ${W_{0}}={B_{0}}+{C_{0}}$, ${\varsigma_{1}}=-{\varsigma_{2}}=\frac{{\varsigma_{0}}}{{{A_{0}}{B_{0}}{C_{0}}}}$, ${\varsigma_{3}}=-{\varsigma_{4}}=-\frac{{\varsigma_{0}}}{{{A_{0}}{B_{0}}{W_{0}}}}$, ${\varsigma_{5}}=-{\varsigma_{6}}=-\frac{{\varsigma_{0}}}{{{A_{0}}{C_{0}}{W_{0}}}}$, ${A_{1}}={A_{3}}={A_{5}}=0$, ${A_{2}}={A_{4}}={A_{6}}={A_{0}}$, ${B_{1}}={B_{3}}={B_{5}}={A_{0}}$, ${B_{2}}={B_{4}}={B_{6}}=0$, ${C_{1}}={C_{2}}={B_{0}}$, ${C_{3}}={C_{4}}=0$, ${C_{5}}={C_{6}}={W_{0}}$, ${W_{1}}={W_{2}}={C_{0}}$, ${W_{3}}={W_{4}}={W_{0}}$, and ${W_{5}}={W_{6}}=0$.

For $k=K-1$, we have ${\left|{{h_{k+1}}}\right|^{2}}={\left|{{h_{K}}}\right|^{2}}$, the joint PDF and CDF of ${\left|{{h_{K-1}}}\right|^{2}}$ and ${\left|{{h_{K}}}\right|^{2}}$ are expressed as

and

respectively, where ${a_{1}}={a_{4}}=0,{a_{2}}={a_{3}}={C_{0}},{b_{1}}={b_{4}}={C_{0}},{b_{2}}={b_{3}}=0,{c_{1}}={c_{2}}=0$, ${{c_{3}}={c_{4}}=r_{F}^{\alpha}}$, ${q_{1}}={q_{2}}=r_{F}^{\alpha}$, and ${q_{3}}={q_{4}}=0$.