Cancel-Decode-Encode Processing on Two-Way Cooperative NOMA Schemes in Realistic Conditions

This paper considers the effects of perfect/imperfect successive interference cancellation (SIC) and perfect/imperfect ` information (CSI) in a multiple-relay two-way cooperative network using nonorthogonal multiple access (NOMA) and digital network coding (DNC). In this model, a relay is selected by maximizing estimated channel gains to enhance the decoding capacity of the nearer source and minimize the collection time of imperfect CSI. Spectrum utilization efficiency is enhanced two times by a mixture of the SIC and DNC techniques at the selected relay (called as the SIC-2TS protocol). The system performance is considered through analysis of the exact and asymptotic expressions of the system outage probabilities and throughput. The major thing is exposed as the proposed SIC-2TS protocol can reach the best performance at optimal positions of the selected relay. Besides, the system throughput of the proposed protocol outperforms a SIC-utilized two-way relaying scheme without the DNC (called as the SIC-3TS protocol) and a conventional two-way scheme (called as the CONV-4TS protocol) for all signal-to-noise ratio regions. Lastly, the validity of the analytical expressions is verified by the Monte Carlo simulation results.


Introduction
Recently, wireless networks have rising challenges in enhancing system throughput and spectrum efficiency owing to the increasing user devices and increasing various Internet of Things applications. A key technology for the fifthgeneration wireless network to solve these challenges is NOMA technology because of its attainments to help grow spectral efficiency, enlarge connections, decrease access latency, and increase the users' fairness [1][2][3]. Power domain NOMA uses the superposition coding to allocate different power levels for transmitted signals to the multiusers at the same time, frequency, and code domains. At receivers, the successive interference cancellation method is applied to decode the received signals [2,3]. However, unexpected errors in decoding when using SIC still occur due to the complexity scale and error propagation, leading to the near user enduring a residual interference signal and the NOMA system performance impacted by this imperfect SIC (ipSIC) [3][4][5][6]. In [7], the authors investigated the reliability and security of the ambient backscatter NOMA systems, where the source was aimed at communicating with two NOMA users in the presence of an eavesdropper. The authors in [7] considered a more practical case that nodes and backscatter devices suffer from in-phase and quadrature-phase imbalance.
Besides, cooperative communication has also been widely studied because its spatial diversity advantage helps to reduce fading, widen coverage, and increase communication preciseness [6,[8][9][10]. In conventional cooperative communications, relaying nodes apply the decode-and-forward (DF) method or the amplify-and-forward (AF) method to process their received and transmitted signals [6,11]. The DF method is better because it decodes received signals at the relay, then reencodes them for forwarding to the destination so it does not amplify noises in received signals like the AF.
Cooperative models show that the selection of the bestrelaying devices, including partial relay selection and opportunistic relay selection, is necessary to improve system performance [5,[11][12][13][14][15][16][17][18][19]. These methods are based on the collection of channel state information to select the optimal relay to support communication. The partial relay selection does not offer the full spatial diversity, but it is not as complicated as the full relay selection, and it is useful for applications in industrial IoT and wireless sensor networks. In most practical applications, CSIs cannot be perfectly measured and there are some mismatch, known as imperfect CSIs (ipCSIs) [12,14,16,[19][20][21][22]. The mismatch can happen due to the feedback delays of the CSIs [12,14,16,19,22] or the faults in the CSI estimating process [20,21]. One-way NOMA and cooperative NOMA (CNOMA) networks with the SIC have been widely discussed to increase spectral efficiency gain, improve secure performance, enlarge system energy efficiency, and enhance significantly sum throughput in several studies [1,4,[23][24][25]. Besides, two-way cooperative networks have advantages in using frequency spectral efficiency over one-way networks because two sources are able to both transmit and receive signals. Moreover, network coding has the advantages of compressing data and high spectral efficiency, and it plays a crucial role in two-way relay networks [5].
In this paper, we propose a two-way cooperative network with a cluster of DF relays and two sources in which the relay will be chosen in the setup phase. The relay selection method in our work helps to enhance the decoding capacity of the nearer source and decrease the collection time of the CSIs than the opportunistic relay selection in [5]. To achieve higher spectral efficiency, we use the NOMA protocol for uplink and the DNC technique for downlink. At the selected relay, the SIC is used to decode sequentially the received signals; next, the DNC is applied to create a new encoded signal and then, this signal is transmitted back to sources, called as the SIC-2TS protocol. Moreover, this paper also investigates the effect of realistic conditions as the ipCSIs and the ipSIC on the system performance. Afterward, the system performance of the SIC-2TS protocol is evaluated based on the analysis expressions of the outage probabilities and the system throughput. Lastly, we compare the proposed protocol with the conventional two-way DF protocol, denoted as CONV-4TS protocol, and the SIC-utilized two-way relaying without the DNC, denoted as the SIC-3TS protocol.
The contributions in this paper are summarized as follows. Firstly, the exact and asymptotic expressions of the outage probabilities and throughput of two sources in the proposed scheme are analyzed and verified under the overall effects of pSIC/ipSIC and pCSIs/ipCSI conditions. Secondly, the exact and asymptotic closed-form expressions are proved valid by the simulation results. Next, the simulation results show that the ipSIC and the ipCSI conditions significantly affect the system performance. Fourthly, the performance of the SIC-2TS protocol is improved by the increased number of relays as well as the perfect operations of the SIC process and the CSI estimations. Moreover, the performance of SIC-2TS can attain the best level at optimal locations of the selected relay and the power suitable coefficients of two sources. Last but not the least, the system throughput of the proposed method outperforms the conventional CONV-4TS and SIC-3TS protocols in both cases of pCSIs and ipCSIs for all SNR regions.
The rest of our paper is organized as follows. Section Section 2 shows some related works. Section 3 describes the system model. Section 4 analyzes the system performance of the SIC-2TS, SIC-3TS, and CONV-4TS protocols. Section 5 shows the results and respective discussions. Finally, section 6 summarizes contributions in this paper.
Notations used in this paper are listed in Table 1.

Related Works
In recent researches, specific two-way CNOMA (TWR CNOMA) networks have been investigated to take benefit on system performance. The performance of the NOMA-based two-way relaying network for uplink and downlink of two users or two groups in the perfect SIC (pSIC) and ipSIC conditions and Rayleigh fading was analyzed with a half-duplex DF relay in [26,27] and a full-duplex DF relay in [28]. The works in [26][27][28] showed that two-way NOMA is superior to two-way orthogonal multiple access (OMA) in terms of outage probability in low signal-to-noise ratio (SNR) regimes. In [29], the joint effects of in-phase and quadrature-phase imbalance and ipSIC on the performance of TWR CNOMA networks over the Rician fading channels were studied. Besides, the realistic assumptions of the residual hardware impairments or ipCSIs of two-way or multiway CNOMA networks have also been considered in the articles [21,22,30]. Moreover, DNC and NOMA techniques can be combined to decreasing transmission time between devices and improve the performance system [5,31,32]. In [31,32], the authors combined NOMA and DNC techniques in a twoway DF relay cooperative scheme confirming that performance in this proposed asymmetric scheme had better spectrum utilization efficiency than the traditional two-way DF OMA scheme, the two-way DF with only using the CNOMA, and the two-way relaying system with OMA in the uplink and DNC in the downlink. The authors in [31,32] only used  [31] was considered in the case of the ideal conditions. In [5], the system performance was investigated in perfect CSI (pCSI) conditions with the opportunistic relay selection.

System Model
A cooperative two-way network has two sources S 1 and S 2 and a closed group of N half-duplex DF relays R i with i ∈ f 1, 2, ⋯Ng, as depicted in Figure 1. This system model can be applied for data transmission in heterogeneous cellular networks. The sources and the relays in the SIC-2TS are single antenna and HD devices. We assume that the direct link between the two sources does not exist due to severe fading and path loss, and the information exchange can be performed only via relays [32][33][34]; the relays are close together as a cluster [17] so the link distances between each source and the relays are identical; hence, we denote d 1 and d 2 as the normalized distances between S 1 and R i and between S 2 and R i , respectively. We also assume that the flat and block Rayleigh fading channels with the fading coefficients for links S k ⟶ R i and R i ⟶ S k are denoted as h S k R i and h R i S k , respectively, where k ∈ f1, 2g. In addition, perfect knowledge of all links is assumed at the receivers by channel estimators that are error-free [35], and the ipCSIs are only caused by the feedback delay with a time-variant channel representation and is described by [12,14,16,20,22,32,35] where ν ∈ fS k R i , R i S k g. Here, ε ν stand for the estimation errors and h ∧ ν describes the estimated CSIs. ε ν and h ∧ ν are independent complex Gaussian random variables (RVs) with zero means and variances λ ν , ðε ν ,ĥ ν~C Nð0, λ ν ÞÞ [14,16].
Prior to transmitting data, the setup phase is performed firstly by request and feedback messages through the cooperative medium access control (MAC) protocol [8]. The nearer source, denoted as S n , n ∈ f1, 2g, receives all channel coefficients from it to the relays under effect of feedback delay and performs the cooperative relay selection. The SIC-2TS protocol uses two time slots for signal communication. In the first slot, two sources S 1 and S 2 send concurrently their signals x 1 and x 2 , respectively, to all relays, and at the selected relay, the SIC technique is applied to decode the received signals. In the second slot, the DNC technique is used to create a new signal x = x 1 ⊕ x 2 [5,15,32] and the selected relay sends it back to two sources S 1 and S 2 with transmit power P R .
The signal which the relays receive in the first time slot is a weighted sum of x 1 and x 2 as follows: where α 1 P S and α 2 P S are transmit powers to carry x 1 and x 2 , respectively; α 1 and α 2 are power allocation coefficients to fairness between two sources (a lower power should be given to the source which is nearer relays) [28], ð0 < α 1 , α 2 < 1Þ; and n R i is the AWGNs with the variance N 0 at the nodes R i .
First time slot Second time slot Figure 1: Two-way cooperative model of the SIC-2TS protocol.

Wireless Communications and Mobile Computing
Substituting (1) into (2), the received signal is given by Due to the symmetry of the proposed system model in Figure 1, without loss of generality, assume that S n and S f are near and far sources from the relays, respectively, where n, f ∈ f1, 2g and n ≠ f . Applying the SIC technique [5,6,24,27,32], firstly, the relays decode the signal x n of the nearby source which has better average channel quality while the signal x f is considered as interference. The received signal-to-interference-plus-noise ratio (SINR) for detecting x n is given by where γ is the transmit SNR, γ = P S /N 0 . In this paper, the relay selection method is used by maximizing estimated channel gains to enhance the decoding capacity of the nearer source. This method has an outstanding advantage in minimizing the collection time of ipCSIs. The relay selection criterion based on the estimated channel gains has been used in [36][37][38] to achieve the better performance. From (4), the selected relay R b is expressed as follows: After decoding x n successfully, the relay R b deletes the component containing the x n signal in (3); then, it decodes the x f signal and the received SINR for detecting x f is given by where h R b is a remaining interference signal with zero mean and variance Ω at the relay R b [27], [27,28]. ε = 0 and ε = 1 correspond to pSIC and ipSIC at the relay R b , respectively [5,28]. In the second time slot, at the relay R b , the signal x = x 1 ⊕ x 2 is synthesized and transmitted to two sources. And the received signal at the source S k , k ∈ f1, 2g, is described as follows: where n S k is the AWGNs at the sources S k with the variance N 0 . Next, the x signal is detected at the two sources with the SINR as follows: where η = P R /P S and η > 0.

Remark 1.
If the proposed SIC-2TS protocol operates without the DNC at the selected relay, the signals x 1 and x 2 are sent sequences by the R b to the sources S 2 and S 1 in two different time slots (the second and third time slots). The received signals and the corresponding SINRs are expressed identically as formulas (7) and (8) in which the symbol x is changed to x 1 (to send S 2 ) and x 2 (to send S 1 ). We denoted the SIC-2TS protocol in this case (without the DNC) as the SIC-3TS protocol to distinguish in the rest of this paper.
We also investigate a conventional two-way CONV-4TS protocol using four time slots with relay selection. This protocol's transmission process is as follows: Wireless Communications and Mobile Computing Firstly, the data signal x 1 is transmitted from the source S 1 to all the relays; the received signal and the SINR at the relay R i are described by Secondly, the relay R b 1 is selected as by formula R b 1 = arg max i=1⋯N g S 1 R i and then R b 1 decodes and transmits the signal The received signal and the SINR at the source S 2 are described by In the same way, in the third and fourth time slots, the source S 2 transmits the signal x 2 to the source S 1 via the best relay R b 2 . We have the SINRs to decode the signal x 2 at the relay R i and the source S 1 as follows:

Performance Analysis
Section IV presents expressions of the outage probability and throughput for the protocols. We assume that the outage occurs at the nodes R i and S k if their SINRs are less than a predefined target γ t . Conversely, these nodes decode signals successfully.

Outage Probability Analysis
The outage of the system occurs in this link when the relay R b fails to decode the signal x n or it decodes successfully the signal x n but the source S f fails to decode the signal x. Besides, the outage probability can also be calculated by the complementary event of the success transmission probability. The successful transmission is the signal x n , and the signal x is received and decoded successfully at the R b and the source S f , respectively [32]. At the source S f , the outage probability of the signal x n can be described as A point to remark is that γ S n R b ⟶x n |d n ≤d f ≥ γ t and γ R b S f ⟶x ≥ γ t are separate events. Thus, the OP S f j d n ≤d f can be given by Lemma 2. The probability Pr ½γ S n R b ⟶x n |d n ≤d f ≥ γ t is calculated by where Proof. See the proof in "Appendix A." The probability Pr ½γ R b S f ⟶x ≥ γ t is solved as Substituting (17) and (18) into (16), the outage probability at the source S f is obtained as (2) The Outage Probability at the Source S n for the S f ! x f R b ! x S n Link. The outage of the system occurs in this link when the signal x n is not decoded successfully at relay R b ; or it is decoded successfully but the signal x f is not decoded successfully at relay R b ; or both the signals x n and x f are decoded successfully at the relay R b but the source S n decodes unsuccessfully the signal x. Conversely, the success transmission of the signal x f occurs when the relay R b and the source S n decode successfully the signals ðx n , x f Þ and the signal x, respectively. At the source S n , the outage probability of the signal x f can be described as Wireless Communications and Mobile Computing Lemma 3. The probability Pr ½γ S n R b ⟶x n |d n ≤d f ≥ γ t , where ϕ 4 = γ t ερ 2 /α f and ϕ 5 = γ t ϕ 1 /α f .
Proof. See the proof in "Appendix B." The final probability Pr ½γ R b S n ⟶x ≥ γ t in (20) is answered as By substituting (21) and (22) into (20), the outage probability at the source S n is solved as Wireless Communications and Mobile Computing CONV-4TS protocol is described as follows: Substituting γ S 1 R b1 and γ R b1 S 2 in (10) and (12) into ((26), we obtain Similarly, the outage probability at the source S 1 for the S 2 !
S 1 link is expressed as By substituting γ S 2 R b2 and γ R b2 S 1 in (13) and (14), respectively, into (28) and after some manipulations as finding the outage probability OP C S 2 , we get a final result as

Numerical Results and Discussion
In this section, the outage probabilities and system throughput of three protocols SIC-2TS, SIC-3TS, and CONV-4TS are analyzed and evaluated. The exactness of the asymptotic and exact theory extractions is validated by Monte Carlo simulations (simulated results are shown by the marker point in all figures). We default the threshold SINR as γ t = 1 and the path-loss exponent as β = 3 in all the analyses and evaluations. From Figures 2-5, the distance d 1 has smaller value and d 2 = 1 − d 1 .
In Figure 2, we examine the outage probabilities of the two sources S 1 and S 2 in the proposed SIC-2TS protocol as a function of the P S /N 0 (dB) with assuming perfect CSIs ðρ = 1Þ when Ω = −10 (dB), d 1 = 0:4, d 2 = 1 − d 1 , N ∈ f2, 6g, and α 1 = α 2 = η = 1 in both pSIC case (ε = 0) and ipSIC case (ε = 1). Figure 2 shows that the outage probabilities of the source S 2 are equal in the pSIC case and the ipSIC case as formula (19). The outage probabilities of the source S 1 in the pSIC case are higher than those of the source S 2 at the low P S /N 0 (dB) regions, and they move to the same saturation values at the high P S /N 0 (dB) regions. The outage 9 Wireless Communications and Mobile Computing probabilities of the source S 1 in the ipSIC case are higher than those in the pSIC case with every P S /N 0 (dB) due to adding the residual interference signals to the SINR of the signal x 2 at the relay as in formulas (6). Furthermore, the system diversity capacity increases because of using the relay selection methods as in (5) and (17) so the system performance of the proposed SIC-2TS protocol is better when the number of relays increases. Finally, the asymptotic and exact theory analysis lines of the outage probabilities also coincide well with their Monte Carlo simulation lines. Figure 3 illustrates the outage probabilities of the sources S 1 and S 2 in the proposed SIC-2TS protocol as a function of P S /N 0 (dB) in both ideal (pSIC-pCSIs) and practical (ipSIC-ipCSIs) conditions when Ω = −10 (dB), d 1 = 0:4, d 2 = 1 − d 1 , N = 6, and the power allocation coefficients α 1 = α 2 = η = 1. Figure 3 shows that the outage probabilities of the two sources with the pSIC-pCSI condition are better than with the ipSIC-ipCSI condition. In the pSIC-pCSI condition, the outage probabilities of the two source nodes have a small difference. But in the ipSIC-ipCSI case, the system outage probability for the source nodes S 1 is a lot higher. These results happen because the SIC technique is used to decode the signal at the relay to make the signal of the farther source more influenced in imperfect cases. In order to have fairness, meaning the two sources can have the nearly same system outage probability in the ipSIC-ipCSI condition, we can provide the higher transmit power for the farther source by changing the transmit power coefficients (α 1 ,α 2 ) in formula (2). Finally, the asymptotic and exact theory analysis lines of the outage probabilities also coincide well with their Monte Carlo simulation lines.
In Figure 4, we consider the outage probabilities of the two sources S 1 and S 2 in the proposed SIC-2TS and CONV-4TS protocols as a function of P S /N 0 (dB) with assuming perfect SIC ðε = 0Þ when Ω = −10 (dB), d 1 = 0:4, d 2 = 1 − d 1 , N = 6, and α 1 = α 2 = η = 1 [5,13] in both pCSI case (ρ = 1) and ipCSI case (ρ = 0:92). Considering the SIC-2TS protocol in Figure 4, firstly, the outage probabilities of two sources in the pCSI case are smaller than those in the ipCSI case and all of them have the floor values when P S / N 0 (dB) is large. Secondarily, the outage probabilities of the source S 1 has higher than the source S 2 . Thirdly, if the P S / N 0 (dB) has enough large value the outage probability of the two sources will be equal in the pCSI condition, but the source S 1 outage probabilities are always bigger than the outage probabilities of the source S 2 at all P S /N 0 (dB) values in the ipCSI condition. Those SIC-2TS protocol results occur because the negative effects of imperfect CSIs lead to channel gain coefficients decrease as formula (1); and in case of d 1 ≤ d 2 , decoding the signal x 2 is decided by the SIC technique     (4) and (6) so the SINR of the x 2 is affected by the ipCSIs of both links S 1 ⟶ R i and S 2 ⟶ R i . Moreover, we also see that the CONV-4TS protocol has a smaller outage probabilities than the SIC-2TS protocol in both the pCSI and the ipCSI conditions, but this conventional protocol will take a lot of time and energy to transmit the signals. Lastly, the asymptotic and exact theory analysis lines of the outage probabilities coincide well with their Monte Carlo simulation lines. Figure 5 plots the system throughput for the SIC-2TS, SIC-3TS, and CONV-4TS protocols as a function of P S /N 0 (dB) with pCSIs/ipCSIs ρ ∈ f0:92, 1g when Ω = −10 (dB), d 1 = 0:4 , d 2 = 1 − d 1 , N = 6, and the power allocation coefficients α 1 = α 2 = η = 1 for case d 1 ≤ d 2 in formulas (31), (32), and (33), respectively. We can see that the proposed SIC-2TS protocol has the ability to achieve higher throughput than the CONV-4TS and SIC-3TS protocols in all P S /N 0 (dB) for both pCSI and ipCSI cases because it combines the NOMA, SIC, and DNC techniques to help degrade the number of the time slot of the transmission between two sources. In addition, the interference parts on the received SINRs are skipped in the case of pCSIs so the throughput of protocols in this condition is always better than that in the ipCSI condition. Furthermore, the SIC-2TS protocol throughput converges at the same value in the high P S /N 0 (dB) regions ðP S /N 0 > 15 dBÞ. Finally, the exact theory values of the system throughput of three protocols fix well the Monte Carlo simulations. Figure 6 demonstrates the system throughput of the SIC-2TS and CONV-4TS protocols versus d 1 in cases of pCSIs/ipC-SIs ρ ∈ f0:92, 1g when P S /N 0 = 5ðdBÞ, Ω = −10 (dB), N = 6, and the power allocation coefficients α 1 = α 2 = η = 1. Figure 6 shows that the SIC-2TS protocol has the system throughput higher than the CONV-4TS protocol in the pCSI case. But in the ipCSI case, its throughput is only better when the distances d 1 are about from 0.3 to 0.7. Moreover, the throughput of the SIC-2TS protocol reaches the highest values at optimal locations of the selected relay as d 1 = 0:4 (in the pCSIs) and d 1 = 0:45 (in the ipCSIs). Besides, the CONV-4TS protocol has the highest system throughput when the relay is at an equidistant point of the two sources (d 1 = 0:5). Lastly, in the perfect CSI ðρ = 1Þ, the throughput of the two protocols is always better than in the imperfect CSI ðρ = 0:92Þ case. Figure 7 observes the throughput of the proposed SIC-2TS protocol versus α 1 and d 1 . The scopes of α 1 and d 1 are from 0.05 to 0.95. The throughput of the proposed SIC-2TS protocol

12
Wireless Communications and Mobile Computing is the highest at about 0.8853 when joint pairs fα 1 , d 1 g = f 0:4,0:5g and fα 1 , d 1 g = f0:6,0:5g. Table 2 shows the detail of the maximum throughput value corresponding to the distance d 1 and the power coefficient α 1 of the S 1 . The coefficients α 1 and α 2 help to adjust the transmit powers of the source nodes; a smaller power is set to the source node nearer to the relay cluster and higher power for the farther source node. The transmit power allocation can achieve the best throughput performance for the proposed SIC-2TS protocol. Figure 8 presents the throughput of the proposed SIC-2TS protocol as functions of ρ and d 1 . The range of ρ is from 0.8 to 1, and the range of d 1 is set from 0.05 to 0.95. The power coefficients α 1 and α 2 also vary according to distance d 1 and d 2 , respectively, to achieve the best throughput performance as mentioned in Figure 7. It is seen that a small range decrease in ρ will result in a large range reduction in throughput so obviously, it is necessary to consider the effect of feedback delay when examining a real system. In other words, channel error estimation in cooperation networks becomes principally important and any ineffective estimation can have detrimental consequences for system performance and it should be not omitted when surveying a cooperation network model. Furthermore, the relative distance between the two sources and the relay cluster also affects different throughput decreases as ρ decreases. When the distance d 1 is in the range ½0:3 : 0:4 and ½0:6 : 0:7, the throughput performance of the system is affected by reduction less than the rest. At last, the larger the number of the relay is, the larger the throughput; therefore, this shows the advantage of using multiple relays.

Conclusion
In this article, a two-way cooperative NOMA model with two sources and multiple relaying nodes under the reality conditions as the ip/pCSIs and the ip/pSIC is studied. In the proposed protocol, a relay was selected in the setup phase by the MAC layer protocol to enhance the decoding capacity of the nearer source and minimize the collection time of imperfect CSIs. Spectrum utilization efficiency was improved by using the SIC and DNC techniques at the selected relay. In order to analyze and evaluate the system performance, exact and asymptotic closed-form outage probabilities and throughput expressions were considered and demonstrated by the Monte Carlo simulations. Our results showed that the performance of the proposed SIC-2TS protocol is significantly improved by the increased number of relays as well as the perfect operations of the SIC process and the CSI estimations. Besides, the system performance is decreased in the ipSIC and the ipCSI conditions. The noteworthy thing is found as the proposed SIC-2TS protocol can reach the best performance at optimal locations of the relay cluster and suitable values of power coefficients. In the pCSI condition, the proposed SIC-2TS protocol always has the system performance much better than the CONV-4TS and SIC-3TS protocols. However, in the ipCSI condition, the SIC-2TS protocol only performs better if the distances from two sources to the relay cluster are not very different. Finally, the analysis expressions of the outage probabilities and system throughput are validated by the Monte Carlo simulations. A. Proof of Lemma 2 We have an equivalent expression of Pr ðγ S n R b ⟶x n |d n ≤d f ≥ γ t Þ as Pr γ S n R b ⟶x n |d n ≤d f ≥ γ t = 1 − Pr γ S n R b ⟶x n |d n ≤d f < γ t : ðA:1Þ Substituting in (4) into (A.1), we have Pr γ S n R b ⟶x n |d n ≤d f ≥ γ t = 1 − Pr α n g S n R b ðA:2Þ In (A.2), F g S n R b ðxÞ is the CDF of g S n R b and can find the following:  ðB:2Þ Therefore, Lemma 3 is proven completely.

Data Availability
The data used to support the findings of this study are included in the article.