Blind Signal Classification Analysis and Impact on User Pairing and Power Allocation in Nonorthogonal Multiple Access

This paper studies blind signal classification (SC) in the nonorthogonal multiple access (NOMA) system, which especially determines whether or not the received NOMA signal requires successive interference cancellation (SIC) without a priori signal information. In this paper, two types of blind SC errors are analyzed: 1) the signal that has to cancel the superposed interference component is classified as one that does not require SIC and 2) the signal that has no need of performing SIC is classified as one that requires SIC. Here, the interesting observation is that the classification error of the first type decreases with the SNR, but that of the second type increases with the SNR. In this regard, this paper proposes the joint user pairing and power allocation policy for the NOMA system based on the above observation that the blind SC performance of the NOMA user who does not perform SIC decreases with the SNR. The joint optimization problem maximizes the sum-rate gain of NOMA over orthogonal multiple access (OMA) with constraints on the maximum classification error probability and the minimum data rate. Since the problem is nonconvex, we propose the numerical algorithm that iteratively finds the appropriate user scheduling and power allocation. Simulation results show that the proposed scheme outperforms existing user scheduling methods.


I. INTRODUCTION
In next-generation communications, it is necessary to provide high data rates to a massive number of devices with limited resources [1]. To achieve this goal, nonorthogonal multiple access (NOMA) has emerged as a promising technology to improve both the efficiency of resource utilization and the system performance in 5G networks [2], [3]. Recently, the 3rd Generation Partnership Project (3GPP) has also studied deployment scenarios and receiver designs for NOMA systems in Release 14 in the context of a work item called ''multiple user superposition transmission'' (MuST) [4]. Basically, NOMA superposes multiuser signals with the different power weights within the same frequency, time, and spatial domains [5]. Thus, the NOMA receiver should handle the interference in the superpositioned signal by using The associate editor coordinating the review of this manuscript and approving it for publication was Donatella Darsena . successive interference cancellation (SIC) and can significantly improve the sum-rate compared to orthogonal multiple access (OMA) with the assumption of ideal SIC [6].
NOMA has been widely developed with other technologies in various environments, such as energy harvesting [7], visible light communications with NOMA [8], and wireless caching networks [9]. The application of multiple-input multiple-output (MIMO) to NOMA has also been investigated in [11]. In addition, the data rate of a cell-edge user can be increased by using NOMA in a cooperative system [12] or a distributed antenna system [13]. The cooperative NOMA system has been proposed in an environment where cooperation among users is possible [14], [15].
Blind modulation classification (MC) for OMA has been originally developed for military applications, such as electronic warfare [16], [17]; therefore, the existing MC techniques were employed at the receiver when no high-layer signaling for signal information is transmitted to prohibit the electronic surveillance from enemy. Since the multi-user signals are superposed in the NOMA system, the additional information of the superposed signal is required. According to [18], blind MC at the NOMA receiver has to identify the following signal information in advance to handle the interference: 1) signal identification, i.e., whether the signal is modulated by OMA or NOMA; 2) modulation classification; and 3) whether or not SIC is required for the received signal. We call these steps of classifying the signal information as blind signal classification (SC). As much signal information as the receiver requires for signal detection in the NOMA system, the risk of electronic surveillance of the transmitted signal is reduced. Even when the modulation scheme of the target signal is known at the receiver, anyone cannot identify the received signal because the transmitter superposes the multiple-user signals. The 3GPP has also discussed blind SC in NOMA system [19], [20].
In this regard, this paper explores user scheduling and power allocation in the downlink NOMA system when blind SC is required at the receiver. On the basis of an ML-based classifier, which has been researched for OMA for a long time [21]- [23], the performances of blind signal identification and blind MC are improved as the SNR increases. Therefore, it is sufficient for scheduling users whose SNRs are larger than a certain threshold for guaranteeing reliable signal identification and blind MC. As the reader will see in detail later, however, the performance of blind SC for the presence of interference decreases with the SNR especially for the user who does not perform SIC. That is because the user who does not perform SIC has to determine by itself that there is no interference in the received signal, even though the signals of multiple users are superposed at the transmitter.
In a traditional two-user NOMA system, the user with high SNR performs SIC and another one with low SNR does not require SIC in general. Here, as the SNR of the weak user decreases, the strong user performs SIC better and the sum-rate increases; however, in practical scenarios, the minimum data rate should be guaranteed for the weak user for its reliability and user fairness. The minimum data rate constraint makes the weak user have the sufficiently large SNR. On the other hand, because the performance of blind SC for the presence of interference at the weak user becomes worse as its SNR grows, the SNR of the weak user should be limited to guarantee the minimum performance of blind SC. Accordingly, this paper proposes the advanced user scheduling and power allocation scheme in the NOMA system where users perform blind SC for the presence of interference. The scenario is that the base station (BS) or the command center transmits the signal to army units without conveying signaling information in the downlink NOMA system where receivers cannot exchange their information for military purposes not to be intercepted. The receivers are assumed to perform blind SC, and the transmitter applies appropriate user scheduling and power allocation for NOMA signaling to guarantee the reliable blind SC as well as the sufficiently large data rate. Here, we suppose that the full channel state information (CSI) is known at the transmitter.
User scheduling and resource allocation for NOMA have been extensively researched. The impact of user pairing on NOMA transmissions in a hybrid multiple access system has been researched in [24], and the optimal user pairing for downlink NOMA was proposed in [25]. The power allocation problem has also been extensively researched with fixed user pairing for NOMA [26]- [28]. The authors of [26]- [28] focus on joint optimization of subchannel assignment and power allocation but consider the situation where user scheduling for NOMA signaling is already given. The joint optimization of power allocation and user scheduling for NOMA systems has been researched in [29]- [31]. The distributed matching algorithm was used for the optimal two-user schedulings and power allocation in [29], and the authors of [30] proposed the globally optimal two-user pairings and power allocation. In [31], the dynamic algorithm for user scheduling and power allocation is presented for reducing the time-average queuing delay. However, all of the above studies do not consider blind SC in an NOMA system.
The main contributions of this paper are as follows: • The two types of blind SC errors for the presence of interference in two-user NOMA are analyzed. In the first type, the signal arrived at the SIC user is classified as one that does not require SIC. In the second type, the non-SIC user classifies the signal that should perform SIC. In addition, mathematical forms of these two error probabilities are derived.
• The impacts of blind SC for the presence of interference on user scheduling and power allocation in the NOMA system are investigated. We show that appropriate user scheduling and power allocation balance the trade-off between the data rate and the performance of blind SC.
• The joint optimization problem of user scheduling and power allocation in the NOMA system is formulated. We then propose the iterative algorithm for finding the optimal user scheduling and power allocation.
• Our extensive simulation results show that the proposed scheme outperforms other existing user scheduling methods. We also numerically investigate how many data samples are required for reliable blind SC for the presence of interference. The rest of this paper is organized as follows. In Section II, the system model is described. Section III derives the mathematical forms of the classification error probabilities for the presence of interference. The joint optimization problem of user scheduling and power allocation is formulated, and an algorithm is proposed to solve this problem in Section IV. Simulation results are presented in Section V. Finally, the conclusion follows in Section VI.
P{.} and p(.) represent the probability of event occurrence and the probability density function of a random variable, respectively. VOLUME 8, 2020

II. SYSTEM MODEL A. CELLULAR MODEL AND DOWNLINK NOMA TRANSMISSION
Consider downlink communications in a cellular model where a BS transmits signals to K users simultaneously on the basis of hybrid multiple access, where NOMA and OMA coexist. We focus on the downlink scenario without any high-layer signaling that conveys the information required for signal detection, and users should perform blind SC. The multiple two-user pairings could be globally made at the BS; however, we suppose that the BS pairs two users for NOMA signaling among K users. In practical scenarios, data requests from users are asynchronous in general; therefore, global multiple user pairings at the same time are rarely performed. Further, multiple user pairings for NOMA signaling can be obtained by employing the optimal two-user pairing with appropriate power allocations.
Suppose that users k and n are scheduled for NOMA at the BS, k, n ∈ {1, · · · , K }. The BS intentionally superposes the signals for both users with different power weights; thus, the signal received by user i is given by where y, s, h, and w correspond to the received signal, transmitted symbol, channel, and noise, respectively, and the subscript i ∈ {k, n} indicates the user's index. Denote γ k and γ n as the power ratios for users k and n, respectively, and assume the normalized power; then, E[|s i | 2 ] = γ i for i ∈ {k, n} and γ k + γ n = 1. Let χ k and χ n be the power weighted constellation sets of users k and n, respectively; then, s k ∈ χ k and s n ∈ χ n . Let χ = χ k ⊕ χ n be the composite constellation set of the superpositioned NOMA signal, which means that the set χ consists of the sums of all possible combinations of two elements in χ k and χ n , respectively. The examples of χ and χ k are shown in Fig. 2. For simplicity, we call the user who requires SIC and the user who does not perform SIC as the SIC user and non-SIC user, respectively. Then, suppose that user k is the non-SIC user and user n is the SIC user throughout the paper. The Rayleigh fading channel from the BS to user i is defined as controls the path loss; d i is the distance between the BS and user i; and g i represents the fast fading component having a complex Gaussian distribution, g i ∼ CN (0, 1). Without loss of generality, assume that |h 1 | 2 ≥ |h 2 | 2 ≥ · · · ≥ |h K | 2 and |h k | 2 ≤ |h n | 2 for two NOMA users. In addition, suppose that the BS knows the instantaneous channel gains, and w i ∼ CN (0, σ 2 ), where σ 2 is the normalized noise variance.
Larger power is usually allocated to the user with the weak channel condition in the NOMA system, i.e., γ k ≥ γ n . With large power allocation, user k does not perform SIC and just decodes s k without cancellation of s n . Therefore, the data rate of user k is given by (2) Meanwhile, SIC is necessary for user n to cancel user k's signal component of s k , and its SINR is |h n | 2 γ k |h n | 2 γ n +σ 2 . However, R k remains the same as (2) because |h k | 2 ≤ |h n | 2 . After performing SIC, the data rate of user n becomes OMA is the general baseline multiple access scheme; therefore, the fair allocation of frequency resources is reasonable in a hybrid multiple access system. Therefore, the data rate of user i for i = k, n served by OMA is given bỹ Since (2) and (3) depend on γ k and γ n , finding the optimal power allocation ratios is as important as appropriate user scheduling. In a hybrid multiple access system, several user groups consisting of more than two users can be created for NOMA signaling. However, this paper considers two-user pairings for NOMA, owing to the excessive complexity of blind SC, as we will see later.
We define the two feasible SNR regions: 1) the operating SNR region and 2) the high SNR region. Simply speaking, the operating SNR region guarantees reliable data detection within the constellation set which the received signal is based on, and respectable performances of data detection as well as blind SC are achieved in the high SNR region.
Definition 1: From (1) with the assumption of γ k > γ n , the operating SNR region is defined as the region in which the receiver can decode the signal of the non-SIC user (i.e., s k ); however, the correct decoding of the SIC user's signal (i.e., s n ) is not ensured.
Definition 2: Suppose that there are some competing constellation sets when blind SC is required at the receiver side. The high SNR region is defined as the region in which the receiver can correctly decode the signals of both non-SIC user and SIC user.

B. ML-BASED BLIND SIGNAL CLASSIFICATION
The concept of ML-based MC [21] is applied to blind SC in the NOMA system, and this paper especially focuses on blind SC for the existence of interference [18]. Let two hypotheses, H S and H N , represent the cases in which the received signal requires SIC and the receiver should not perform SIC for the signal, respectively. In other words, H S means that the user who received the signal is the SIC user, and H N means that the signal is for the non-SIC user. Note that the transmitted superposed signal is identical and delivers data for both users; however, each user has to determine whether to perform SIC or not after receiving the signal, i.e., which hypothesis is correct. Here, we can define the probability density function of y that requires SIC as p(y|H S ). Similarly, p(y|H N ) is the probability of y that should not perform SIC. Denote the transmitted symbol by s 0 (l 0 ) = s n (l n ) + s k (l k ), then the received signal is y 0 = h 0 (s n (l n ) + s k (l k )), where l 0 , l n and l k are the indices of symbols in χ, χ n and χ k , respectively. Here, if the receiver is the non-SIC user, it just needs to directly detect s k (l k ) from y 0 so that detection is based on χ k only. On the other hand, if the receiver is the SIC user, then χ k is used to cancel s k (l k ) from y 0 , and after SIC, s n (l n ) is detected based on χ n ; therefore, the signal detection step of the SIC user depends on χ. Then, the likelihood probabilities of the SIC user and non-SIC user are given by respectively, where the symbol s is assumed to be equally probable, and the cardinalities of χ and χ k are denoted by |χ| and |χ k |, respectively. When p(y 0 |H S ) > p(y 0 |H N ), the user performs SIC; otherwise, the user directly detects the symbol based on χ k . Therefore, comparison of (5) and (6) represents ML-based blind SC for the presence of interference.
The following lemma gives the unique property of blind SC for the presence of interference.
Lemma 1: p(y 0 |H S ) increases with the SNR, but p(y 0 |H N ) decreases with SNR in the operating SNR region (i.e., the extremely low or high SNR is not considered).
Proof: Again, let the transmitted symbol be s 0 (l 0 ) = s n (l n ) + s k (l k ). Since the received signal is originally generated from χ, p(y 0 |H S ) obviously increases with the SNR. According to Definition 1, in the operation SNR region, the SNR is not excessively low so that at least detection of s k (l k ) is guaranteed. In χ k , the most confusing symbol point with s 0 (l 0 ) is s k (l k ). In this situation, as the SNR grows and approaches to the high SNR, the received symbol becomes closer to s 0 (l 0 ) and farther away from s k (l k ). Therefore, as the SNR grows in the operating SNR region, |y 0 −h 0 s k (l k )| 2 σ 2 is likely to increase; therefore, p(y 0 |H N ) decreases.
According to Lemma 1, it is beneficial to schedule the user with a high SNR and the user with a low SNR as the SIC user and the non-SIC user, respectively, for reliable blind SC for the presence of interference. In Fig. 2, received signals with high and low SNRs are shown with constellation sets. Since the signal is originally generated from χ, the received symbols are very close to the points of χ with the high SNR and p(y 0 |H S ) > p(y 0 |H N ) for most of received symbols. On the other hand, as the SNR decreases, some of the received symbols become to be distanced from the points of χ and satisfy p(y 0 |H S ) < p(y 0 |H N ). In this case, however, there is a risk that the data rate of the non-SIC user would not be sufficiently large. In this regard, more elaborate user scheduling and power allocation are necessary in the NOMA system where users perform blind SC for the presence of interference. Thus, this paper considers two conflicting constraints for the non-SIC user, the minimum data rate and the maximum error probability of blind SC for the presence of interference.
Remark 1: The equations (5) and (6) appear to be similar to the blind signal identification (i.e., OMA/NOMA classification) in [18]; however, they are different. If we consider that χ k is the constellation set of the OMA signal, then (6) becomes the likelihood probability of the OMA user, but the hypothesis H N is not the same as that of the OMA user. In the OMA/NOMA classification step, the likelihood probability becomes p(y|H O ), where H O is the hypothesis for the case in which y is not superposed which means the OMA signal. In this case, p(y|H O ) increases with the SNR because y is originally generated from χ k exactly. On the other hand, p(y|H N ) decreases with the SNR because y is not generated from χ k only but from χ .
Remark 2: In practice, blind signal identification and blind MC should be also considered as in [18]. The performances of the ML-based classifiers for these steps increase with the SNR on every NOMA user side [21]; therefore, the BS only needs to schedule the users with sufficiently large SNRs to guarantee the reliability of blind signal identification and blind MC. Accordingly, this paper only considers blind SC for the presence of interference while optimizing user scheduling and power allocation. From now on, the term ''blind SC'' means determination of whether to perform SIC for decoding the received signal on the user side or not.

C. EXTENSION TO M-USER GROUPING FOR NOMA
Although this paper mainly focuses on a two-user NOMA system, blind SC can be performed in the general M -user NOMA model. Again, |h 1 | 2 ≥ |h 2 | 2 ≥ · · · ≥ |h M | 2 , and consider M users are grouped for NOMA signaling. Then, in general, user m decodes its data after canceling the interference components of user M , user M − 1, · · · , user m + 1 in order. Denote χ n−m as the composite constellation set of χ n , χ n+1 , · · · , and χ m , i.e., χ n−m = χ n ⊕ χ n+1 ⊕ · · · ⊕ χ m . Let H m be the hypothesis indicating that the target user is user m; i.e., M − m − 1 SIC steps are required. The likelihood probability of user m is given by The receiver then determines itself to be user m 0 , where As the number of hypotheses increases, i.e., M increases, the classification performance is expected to be significantly degraded. In addition, massive computations are required to obtain the likelihood probabilities of all users; therefore, blind SC is not preferred when a large number of users is served by NOMA. Thus, it is reasonable to focus on a two-user NOMA system where blind SC for the presence of interference is performed at the user side.

III. ERROR TYPES OF BLIND SIGNAL CLASSIFICATION FOR THE PRESENCE OF INTERFERENCE
In this section, we analyze two types of incorrect blind SC for the presence of interference. The first is that the SIC user determines the received signal not to require SIC, and in the second type, the non-SIC user classifies itself as the SIC user. The mathematical forms of two classification error probabilities are derived.

A. CLASSIFICATION ERROR AT THE SIC USER
Denote the probability that user n (i.e., the SIC user) incorrectly determines that it should not perform SIC by P n {Ĥ N |H S }, whereĤ N indicates that the receiver classifies itself as the non-SIC user. Suppose that s k (i 0 ) ∈ χ k and s n (l 0 ) ∈ χ n are the transmitted signal components for users k and n, respectively, for i 0 ∈ N k = {1, · · · , |χ k |} and l 0 ∈ N n = {1, · · · , |χ n |}. The superpositioned signal becomes s 0 (m 0 ) = s k (i 0 ) + s n (l 0 ) ∈ χ for m 0 ∈ N = {1, · · · , |χ|}, and the received signal is y 0 = h(s k (i 0 ) + s n (l 0 )) + w. For simplicity, we define two hypotheses as follows: The classification error probability at user n as a SIC user is then given by where the expectation is over the random generations of the signal and noise components but the channel gain is static. According to (5) and (6), p(y 0 |H N ) and p(y 0 |H S ) consist of terms corresponding to symbols in χ and χ k , respectively. If the received symbol y 0 is much closer to one of symbols in χ and χ k than others, then the corresponding term in (5) and (6) dominates the likelihood value, respectively. Since the user with a high SNR is chosen as the SIC user, according to Definition 2, p(y 0 |s 0 (m 0 ))p(s 0 (m 0 )) and p(y 0 |s k (i 0 ))p(s k (i 0 )) dominate p(y 0 |H S ) and p(y 0 |H N ), respectively. Then, P{Ĥ N |H S } can be approximated into where P{G k (i 0 )} = 1 and P{G(i 0 , l 0 )} = 1 because arg max s k p(y 0 |s k ) = s k (i 0 ) and arg max s p(y 0 |s 0 ) = s k (i 0 ) + s n (l 0 ) in the high SNR region. In addition, the input symbol is equally probable, i.e., p(s k (i 0 )) = 1 |χ k | and p(s 0 (m 0 )) = 1 |χ| . For simplicity, let the following functions represent normal distributions given only s k (i) and s k (i) + s n (l), respectively; In addition, we assume that M -QAM is used for both NOMA users as in MuST adopted by 3GPP [4]. Then, the equation (11) can be expressed as

B. CLASSIFICATION ERROR AT THE NON-SIC USER
Denote the classification error probability at user k as a non-SIC user by P k {Ĥ S |H N }. When the non-SIC user classifies itself as the SIC user, the receiver would cancel the target signal by performing SIC; therefore, the incorrect SC for the presence of interference is directly linked with inaccurate signal detection. In a similar manner as that in Section III-A, P k {Ĥ S |H N } can be obtained by where the expectation is over the random generation of the signal and noise components. As mentioned earlier, the non-SIC user experiences the low SNR compared to the SIC user; therefore, the high-SNR approximation cannot be applied. Suppose that s k (i 1 ) and s n (l 1 ) are detected based on χ k and χ, respectively. Then, P k {Ĥ S |H N } can be expressed as where P{f k (i 1 ) < f (i 1 , l 1 )|G k (i 1 ), G(i 1 , l 1 )} is given by (16), as shown at the bottom of this page, when the signal is modulated by M -QAM.

C. NUMBER OF DATA SAMPLES REQUIRED FOR BLIND SIGNAL CLASSIFICATION
In practical scenarios, it is important to find how many data symbols are required for the reliable blind SC for the presence of interference. The prior sections assume that only one symbol is used for blind SC. However, the more symbols for blind SC, the better performance; on the other hand, the use of many data samples results in heavy signal processing tasks and large computational complexity. Suppose that L symbols are used for blind SC and that all symbols experience the same channel gain h. The likelihood probabilities of the SIC user and non-SIC user are then given by where y 0 = [y 1 , · · · , y L ] and s 0 = [s 1 , · · · , s L ] are the vectors of the received signals and transmitted symbols, respectively. The numbers of summations in (27) and (28) exponentially increase with L; thus, massive computations are required to compute (27) and (28), when L is large. Therefore, we independently compute the likelihood probability of each y l , denoted by p(y l |Ĥ i ) for l = 1, · · · , L. The probability that all of L data samples are likely to predict H i for i ∈ {N , S} becomes L l=1 p(y l |Ĥ i ). Since all data samples are based on the same hypothesis, it would be reasonable thatĤ i is true if more than half of the data samples predict H i . Let P n {Ĥ S |H N , y l } and P k {Ĥ N |H S , y l } be the classification error probabilities computed from y l . Assuming that all elements y l of y 0 are independent, P n {Ĥ S |H N , y 1 } = · · · = P n {Ĥ S |H N , y L } = P n0 and P k {Ĥ N |H S , y 1 } = · · · = P k {Ĥ N |H S , y L } = P k0 , because L samples experience the same channel gain. The classification error probabilities can then be approximately computed by with the odd number L. Since the receiver does not have to know any information about the data samples required for blind SC, the information bits can be used as samples for blind SC. Thus, the system does not need additional signaling overhead for blind SC.

IV. JOINT OPTIMIZATION PROBLEM OF USER SCHEDULING AND POWER ALLOCATION
In this section, the effects of blind SC on user scheduling and power allocation in NOMA systems are presented. For the joint optimization problem of user scheduling and power allocation, the sum-rate gain of NOMA over OMA is considered an optimization metric. The gains at users n and k are respectively given by Again, users k and n are scheduled for NOMA as the non-SIC user and SIC user, respectively, and |h k | 2 < |h n | 2 . The total sum-rate gain of NOMA over OMA is (k,n) = k + n . Then, the max-sum-rate problem for user pairing and power allocation for NOMA signaling can be formulated by {n * , k * , γ * n * ,(k * ,n * ) , γ * k * ,(k * ,n * ) } = arg max n,k,γ n ,γ k (k,n) (33) (34) γ n,(k,n) + γ k,(k,n) = 1, , where n * and k * are the indices of the optimally scheduled users for NOMA, . γ * n * ,(k * ,n * ) and γ * k * ,(k * ,n * ) are the optimal power coefficients for users n * and k * , respectively, when users n * and k * are scheduled for NOMA as the SIC user and non-SIC user, respectively. The constraint in (34) is for the minimum data rate of NOMA users. and the normalized power is given by (35).
However, since P n {Ĥ S |H N } and P k {Ĥ N |H S } are too complicated to deal with, we approximate the maximization problem by maximizing the lower bound of (k,n) . Let P t be the threshold of the error probability for reliable blind SC such that max{P n {Ĥ N |H S }, P k {Ĥ S |H N }} ≤ P t . The lower bound of (k,n) is then obtained by Therefore, the approximated joint optimization problem of user scheduling and power allocation which maximizes the lower bound of the sum-rate gain of NOMA over OMA is formulated as follows: {n * , k * , γ * n * ,(k * ,n * ) , γ * k * ,(k * ,n * ) } = arg max n,k,γ n ,γ k (k,n) (37) where the constraint in (39) guarantees the reliability of SC for the presence of interference. The tightness of (k,n) with respect to (k,n) can be controlled by P t . If the system takes a small P t , the problem of (37)-(40) becomes almost identical to the original problem of (33)-(35). A closed-form solution of (37)-(40) is difficult to obtain; therefore, we decouple (37) into separate problems of user scheduling and power allocation. First, start with the arbitrary scheduling of users k and n as the non-SIC and SIC users, respectively, and find their optimal power allocation ratios. Next, a different user pairing is tested depending on the tightness of the two constraints in (38) and (39). In this way, we will solve the problem in (37)-(40) through an iterative algorithm that obtains the optimal power allocation for fixed user scheduling and gradually replaces the scheduled users for NOMA.

A. POWER ALLOCATION PROBLEM
Assuming that users k and n are already scheduled for NOMA, the problem for finding the optimal power allocation is given by Note that R k and R n are decreasing and increasing functions of γ n , respectively, which is easily proved by differentiating (2) and (3). Similarly, (k,n) is an increasing function of γ n . On the other hand, the impacts of power allocation γ n on the SC error probabilities are described in the following lemma.
Lemma 2: P n {Ĥ N |H S } and P k {Ĥ S |H N } decrease and increase with γ n , respectively.
Proof: Since E[|s n (l 0 )| 2 ] = γ n and Q function is a decreasing function, according to (12), P n {Ĥ N |H S } decreases with γ n . On the other hand, (25) and (26) increase with γ n ; thus, P k {Ĥ S |H N } increases as γ n grows, according to (23).
Suppose that there exists a certainγ n ∈ [0, 1] that satisfies R k = R n = R 0 . In other words, the graphs of R k and R n for γ n ∈ [0, 1] intersect each other atγ n as shown in Figs. 3 and 4. Since R n increases and R k decreases with γ n ,  R 0 ≥ R t is necessary for the constraint in (42). Similarly, considerγ n ∈ [0, 1] such that P k {H S |H N } = P n {H N |H S } = P 0 . Then, P 0 should then be smaller than P t to satisfy the constraint in (43). Assuming that R 0 ≥ R t and P 0 ≤ P t , the following theorem gives the optimal solution of (41)-(44).
Theorem 1: Suppose that users k and n are already scheduled with |h n | 2 > |h k | 2 . When R 0 ≥ R t and P 0 ≤ P t are satisfied, the optimal solution of (41)-(44) is given as where γ R n,(k,n) and γ P n,(k,n) are the power allocation ratios of user n when R k = R t and P k {H S |H N } = P t , respectively. I R and I P are the intervals of γ n satisfying the constraints in (42) and (43), respectively.
Proof: (k,n) can be shown to be a nondecreasing function of γ n , as given by which satisfies ∂γ n > 0 because |h n | 2 > |h k | 2 . First, it is obvious there is no solution when I R ∩ I P = φ because any power allocation rule cannot satisfy both of the constraints in (42) and (43). Consider the case I R ∩ I P = φ. VOLUME 8, 2020 Without the classification error constraint in (43), because (k,n) monotonically increases and R k decreases with γ n , the upper boundary of I R , i.e., γ R n, (k,n) , is the optimal solution satisfying R k = R t , as shown in Figs. 3 and 4. On the other hand, without the minimum data rate constraint in (42), the optimal power allocation becomes γ P n,(k,n) when P k {Ĥ S |H N } = P t . Therefore, (45) could be achieved to satisfy both constraints for the minimum data rate and maximum classification error probability.
Note that Theorem 1 can be interpreted as the fact that γ * n,(k,n) is the upper boundary of I R ∩ I P . For example, in Fig. 3, γ P  n,(k,n) is the optimal solution for the constraint of the classification error probability only, but it does not satisfy the minimum data rate constraint. In this case, it can be stated that the minimum data rate constraint is tighter than the maximum classification error probability constraint. Therefore, γ P n,(k,n) > γ R n,(k,n) and γ * n,(k,n) = γ R n,(k,n) . On the other hand, Fig. 4 shows the case in which the maximum classification error constraint is tighter than the minimum data rate constraint, where γ P n,(k,n) < γ R n,(k,n) and γ * n,(k,n) = γ P n,(k,n) . It remains to find γ R n,(k,n) and γ P n,(k,n) . γ R n,(k,n) can be directly obtained from (2) with knowledge of the instantaneous channel gain, as given by However, a closed-form expression for γ P n,(k,n) is difficult to derive because P k {Ĥ S |H N } in (24) is not simply expressed with γ P n,(k,n) . Since P k {Ĥ S |H N } is monotonically increasing with γ n , the bisection method can be used to compute γ P n,(k,n) . The details are in Algorithm 1.

B. USER SCHEDULING PROBLEM
In Section IV-A, the optimal power allocation ratios are derived for fixed scheduling of user k and user n. We then aim at finding better user scheduling, and the following lemma gives an insight for user scheduling.
Lemma 3: (k,n) is an increasing function of the received SNR of the SIC user (user n) in the high SNR region. Moreover, if 1−2γ n −2P t −2γ n P t γ n > |h k | 2 σ 2 , (k,n) is an increasing function of the received SNR of the non-SIC user (user k); otherwise, (k,n) is a decreasing function of the received SNR of the non-SIC user. Proof: Since γ n , γ k , and σ 2 are constants, the received SNRs of both users depend only on the channel gains. Differentiating (36) with respect to |h n | 2 , Because the nonextreme value of γ n > 0, as |h n | 2 → ∞, is always satisfied, and ∂ (k,n) ∂|h n | 2 converges to zero. Therefore, (k,n) is a nondecreasing function of |h n | 2 in the high-SNR region. Similarly, However, the high-SNR approximation cannot be applied to user k because |h k | 2 is quite small for blind SC. Recall that P k {Ĥ S |H N } increases with the received SNR. According to Lemma 3, the user with the strongest channel conditions is the best choice for the SIC user (user n) because (k,n) increases and P n {Ĥ N |H S } decreases with the received SNR of the SIC user. Recall that |h 1 | 2 ≥ |h 2 | 2 ≥ · · · ≥ |h K | 2 ; thus, the optimal choice of the SIC user becomes n * = 1. On the other hand, scheduling of the non-SIC user (user k) is not clear, because (k,n 0 ) could be increasing or decreasing depending on the received SNR of the non-SIC user. Instead, we can see that the global maximum of * (k,n) is obtained when user k satisfies |h k | 2 σ 2 = 1−2γ n −2P t +2γ n P t γ n . However, the number of users is finite, and it is almost impossible that a user exists whose SNR is exactly 1−2γ n −2P t +2γ n P t γ n ; therefore, we will gradually find the better user to be the non-SIC user instead of user k. The better non-SIC user can be chosen in two ways: scheduling of the user with the larger or smaller received SNR than the previously scheduled user k. The next step for scheduling of the non-SIC user follows one of the following four cases: The minimum data rate constraint is tighter than the classification error constraint, and (k,n) increases with |h k | 2 σ 2 by Lemma 3. To relieve the constraint of the data rate, the choice of user k whose SNR is larger than that of user k, i.e., |h k | 2 ≥ |h k | 2 , would be beneficial. Then, R k ≤ R k and P k {Ĥ S |H N } ≤ P k {Ĥ S |H N }. In Fig. 5, some arrows and dashed lines indicate which values are changed with the newly chosen user k . (k ,n) , R k , and P k {Ĥ S |H N } increase; therefore, γ P n,(k n) < γ P n,(k,n) and γ R n,(k ,n) > γ R n,(k,n) . I P becomes narrower but I R is wider than before, and this means that the minimum data rate constraint is relieved and the classification error probability constraint becomes tighter.  Note that (k,n) increases with the received SNR of the non-SIC user and γ n . Accordingly, if γ * n,(k ,n) > γ * n,(k,n) , (k,n) ≤ (k ,n) always as shown in Fig. 5, and user k is better to be scheduled as the non-SIC user compared to user k. However, there is no guarantee that γ * n,(k ,n) > γ * n,(k,n) always; therefore, if γ * n,(k ,n) < γ * n,(k,n) , then the comparison step of * (k ,n) and * (k,n) is necessary. If * (k ,n) > * (k,n) , user k is scheduled as the non-SIC user; otherwise, user k remains as the non-SIC user.
The minimum data rate constraint is tighter than the constraint of blind SC; therefore, user k with a larger SNR than user k, i.e., |h k | 2 ≥ |h k | 2 , is chosen as the non-SIC user, similar to Case 1. The changes in R k , P k {Ĥ S |H N }, γ R n,(k ,n) , and γ P n,(k ,n) are all the same as those in Case 1. However, (k,n) decreases with |h k | 2 in this case; therefore, even though γ * n,(k ,n) > γ * n,(k,n) , there is no guarantee that * (k,n) < * (k ,n) . Accordingly, a comparison of the newly updated * (k ,n) with the previously obtained * (k,n) is necessary. If * (n,k) < * (n,k ) , user k is preferred as the non-SIC user rather than user k. Otherwise, user k is determined as the non-SIC user.   In this case, the classification error constraint is tighter than the minimum data rate constraint. Therefore, the BS takes user k with a smaller SNR rather than user k as the non-SIC user; i.e., |h k | 2 ≤ |h k | 2 . P k {Ĥ S |H N } and R k then decrease, and therefore I P is enlarged but I R becomes narrower. Note that (k,n) decreases increases as the received SNR of the non-SIC user decreases and increases with γ n . If |h k | 2 ≤ |h k | 2 and γ * n,(k,n) ≤ γ * n,(k ,n) as shown in Fig. 7, * (k,n) ≤ * (k ,n) obviously. Then, user k replaces user k as the non-SIC user. However, if γ * n,(k,n) > γ * n,(k,n) , user k is paired for NOMA only when * (k,n) < * (k ,n) , similar to Case 1.
The maximum classification error constraint is tighter than the minimum data rate constraint; therefore, user k with the smaller SNR is chosen as the non-SIC user rather than user k.
The changes in the parameters are the same as those in Case 3, except that (k,n) increases with the received SNR of the non-SIC user. In this case, if * (k,n) < * (k ,n) , user k is scheduled; otherwise, user k remains as the non-SIC user.
By iteratively examining users in increasing or decreasing order of the SNR, depending on each case, the BS can find the most appropriate one as the non-SIC user giving the maximum value of * (k 0 ,n 0 ) . In addition, given the user scheduling, the optimal power allocation rule is obtained by Theorem 1. The details of this iterative approach are presented in Algorithm 2. Since the appropriate user scheduling is determined by a greedy search-based method and the mathematically optimal power allocation is given, it can be said that the optimality of Algorithm 2 is guaranteed.
The computational complexity of Algorithm 2 depends on the iteration number and computations of γ R n * ,(k * ,n * ) and γ P n * ,(k * ,n * ) . First, the appropriate initial decision of the non-SIC user can largely reduce the iteration number, i.e., computational complexity of the algorithm. As mentioned before, we already know that |h k | 2 is the best choice of the non-SIC user; however, it is almost impossible to find the user who satisfies this condition exactly. However, we can run Algorithm 2 with the user whose SNR is the closest to 1−2γ n −2P t +2γ n P t γ n as the initial non-SIC user, and the worst case requires K /2 iterations. In addition, in order to find the approximated γ R n * ,(k * ,n * ) and γ P n * ,(k * ,n * ) in (12) and (24), |χ n | and |χ k,1 ||χ n,1 ||χ n | 2 computations are required. In the practical scenario, QPSK or 16-QAM is available for both NOMA users, so at most 2 12 computations are required for finding γ R n * ,(k * ,n * ) and γ P n * ,(k * ,n * ) in the worst case. Also, the number of users (i.e., K ) is finite and we focus on the scenario where there are not very many users; therefore, the computational complexity of Algorithm 2 is expected not to be excessively large.

V. NUMERICAL RESULTS
In this section, we show that the analytically obtained error probability of blind SC for the presence of interference is well matched to numerical results. In addition, the proposed user scheduling and power allocation policy provides better sum-rate performances when blind SC for the presence of interference is required at the receiver side. The impacts of the number of data symbols for SC and the threshold of classification error probability are also shown by simulation results.

A. BLIND CLASSIFICATION ERROR PROBABILITY
In Section III, the approximated form of the blind classification error probabilities were mathematically derived. In Fig. 9, we compare the classification error probabilities obtained by the analytical method and Monte Carlo simulation. Assume that the non-SIC user and SIC user are modulated by QPSK and 16-QAM, respectively, and γ n = 0.24. In addition, the L = 1 data sample is used for blind SC. Fig. 9 shows the plots of two classification error types, i.e., P n {Ĥ N |H S } and P k {Ĥ S |H N }, versus the received SNR. As we explored in Section IV, P n {Ĥ N |H S } decreases as the SNR increases, but P k {Ĥ S |H N } increases with the received SNR. If P t = 0.1, the SNR of the non-SIC user has to be smaller than 5 dB, but that of the SIC user should be larger than approximately 23 dB, as shown in Fig. 9. In addition, the approximated P n {Ĥ N |H S } based on (12) is Algorithm 2 Iterative Algorithm for Jointly Optimizing User Scheduling and Power Allocation Ratios Precondition: 1: • |h 1 | 2 ≥ · · · ≥ |h K | 2 : Channel gains of K users • R t : minimum threshold of the data rate • P t : maximum threshold of the classification error probability • σ 2 : normalized noise variance 2: Choose n * = 1 3: Arbitrarily choose k * = k, for k ∈ {2, · · · , K } 4: while True do 5: Compute γ R n * ,(k * ,n * ) , γ P n * ,(k * ,n * ) , and (k * ,n * ) 6: if γ R n,(k * ,n * ) < γ P n,(k * ,n * ) then 7: k = k * + 1 8: if k == K + 1 then 9: * (k * ,n * ) = (k * ,n * ) , γ * (k * ,n * ) = γ R (k * ,n * ) 10: break; 11: end if 12: Compute γ * n,(k * ,n) and γ * n,(k ,n) by Theorem 1 13: end if 36: end while very similar to the numerical result; thus, we can conclude that the approximation used in (12) is reliable. The graphs of P k {Ĥ S |H N } show a small difference, but they are still similar. As mentioned in Section III, the high-SNR approximation cannot be applied to compute P k {Ĥ S |H N }, but we further approximate it by (23). This approximation explains the differences between the analytical and simulation results. To verify that Algorithm 2 works well for the joint optimization problem of user scheduling and power allocation for NOMA transmissions, the sum-rate gain of NOMA over OMA is compared with those of other scheduling methods.
Consider the cellular downlink model as shown in Fig. 1. K = 40 users are randomly located in the circular region of radius r = 50 centered at the BS. The BS chooses two users among K users for NOMA transmissions to provide an increased data rate compared to OMA while guaranteeing the minimum data rate constraint and reliable blind SC. For the simulation, R t = 0.8, P t = 10 −2 , a transmit SNR of 10 dB, and L = 5 are used unless otherwise noted.
According to [24], a pair of users having the strongest and weakest channel gains is the best choice for NOMA with fixed power allocation (F-NOMA). On the other hand, the performance gain of NOMA over OMA could be maximized when the users with the strongest and second-strongest channel gains are paired for cognitive-radio-inspired NOMA (CR-NOMA). In CR-NOMA, it is important to guarantee the weak user's quality of services (QoS); thus, it is consistent with the minimum data rate constraint in (38). We compare those two methods with the proposed algorithm. Along with the proposed algorithm, the user with the strongest channel condition, i.e., user n * = 1, is always chosen as the SIC user for those comparison schemes; therefore, what remains is to find the appropriate non-SIC user. Since [24] does not consider blind SC, we assume that the comparison schemes find the non-SIC user in the user set U P , whose members satisfy both constraints of the minimum data rate and maximum classification error probability. In other words, when I R ∩ I P = φ, user n * and any user k ∈ U P are scheduled as the SIC user and non-SIC user, respectively, and there exists a γ n ∈ [0, 1] satisfying both constraints of the data rate and classification performance. In summary, the comparison methods are explained as follows:

1) STRONGEST-STRONGEST
The BS schedules the user having the second-strongest channel gain as the non-SIC user among the user set U P . If U P includes all K users, n * = 1, and k * = 2.

2) STRONGEST-WEAKEST
The BS schedules the user having the weakest channel gain as the non-SIC user among the user set U P . If U P includes all K users, n * = 1, and k * = K . The above comparison methods also find the optimal power allocation ratio by Theorem 1 after user scheduling. However, the main difference between the proposed algorithm and the above two methods is the dependency between user scheduling and power allocation. The steps of user scheduling and finding the power allocation for NOMA users are independent in the above comparison methods; however, the proposed algorithm iteratively finds the appropriate non-SIC user with consideration of the optimal power allocation corresponding to the iterative decision of user scheduling. Fig. 10 shows plots of the sum-rate gain of NOMA over OMA versus the transmit SNR. We can easily see that the proposed algorithm gives the best performance compared to ''Strongest-Strongest'' and ''Strongest-Weakest.'' The sum-rate gains of the proposed algorithm and ''Strongest-Weakest'' increase with the transmit SNR. However, the performance of ''Strongest-Strongest'' maintains almost the same value for all transmit SNRs. Since ''Strongest-Strongest'' chooses the user with the second-strongest channel gain as the non-SIC user, its SNR is generally too large to satisfy the constraint of blind SC due to its large SNR. Note that the error probability of blind SC at the non-SIC user decreases as the SNR grows.
Meanwhile, the performances of the proposed algorithm and that of ''Strongest-Weakest'' are shown to converge as the SNR increases. The high transmit SNR is beneficial for the data rate but not for classification of the non-SIC user, i.e., P k {Ĥ N |H S }. This means that the classification error constraint is tighter than the minimum data rate constraint in the high-SNR region; thus, the optimal solution is heavily dominated by blind SC. This situation corresponds to Case 3 in Section IV-B because the tight constraint of SC makes γ * n,(k,n) = γ P n, (k,n) , and the high SNR generally satisfies 1−2γ n −2P t −2γ n P t γ n < |h k | 2 σ 2 . In Case 3, scheduling of the user VOLUME 8, 2020 with a lower SNR as the non-SIC user is preferred; therefore, the performance gap between the proposed method one and ''Strongest-Weakest'' decreases as the SNR grows. On the other hand, the performance gap between the proposed scheme and ''Strongest-Weakest'' increases in the low-SNR region. The low SNR is likely to satisfy 1−2γ n −2P t −2γ n P t γ n > |h k | 2 σ 2 . Moreover, since both R k and P k {Ĥ N |H S } decrease as the SNR decreases, the optimal solution is usually dominated by the minimum data rate constraint, unlike the situation in the high-SNR region. Accordingly, most situations in the low SNR region correspond to Case 1. In this case, scheduling of the user with the stronger channel condition as the non-SIC user makes * larger. Thus, a larger gain of the proposed scheme over ''Strongest-Weakest'' is obtained in the low-SNR region.
We also observe the effect of modulation in Fig. 10, where the solid and dashed lines indicate the results obtained when the SIC user is modulated by 16-QAM and QPSK, respectively, and QPSK is used for the non-SIC user. The performance trends of the comparison technologies are generally similar; however, the improvement in the data rate with increasing SNR is not much large when the SIC user is modulated by QPSK, compared to the case of 16-QAM. According to (5) and (6), the classification of the SIC user becomes easier and the classification of the non-SIC user becomes more difficult, when the composite constellation χ and non-SIC user's constellation χ k are more clearly distinguishable. A large minimum distance between χ and χ k can be obtained with the small modulation order of the SIC user; thus, P n {Ĥ N |H S } is lower, but P k {Ĥ S |H N } is higher with QPSK for the SIC user. The higher P k {Ĥ S |H N } makes the classification error constraint tighter, and hence the advantage of increasing the SNR is not shown clearly when QPSK is used for the SIC user compared to the case of 16-QAM.
C. SYSTEM PARAMETERS L AND P T Figs. 11 and 12 show plots of * versus the number of data samples of L and P t , respectively. We can easily see that * of ''Strongest-Strongest'' decreases with L, which means that blind SC does not work well in ''Strongest-Strongest'' owing to the high SNR of the non-SIC user; i.e., P k {Ĥ S |H N } > 0.5. On the other hand, the performances of the proposed scheme and ''Strongest-Weakest'' increase as L grows; therefore, blind SC is conducted properly in those techniques. The sum-rate gains of the proposed method and ''Strongest-Weakest'' are converging, and Fig. 11 can be used to determine how many data samples are sufficient for blind SC. When QPSK and 16-QAM are used for the non-SIC user and SIC user, respectively, L = 3 is sufficient to boost the sum-rate gain as shown in Fig. 11. Thus, not much excessive complexity is expected for blind SC.
In addition, the system designer could choose the appropriate value of P t by observing the trade-off between the data rate and the performance of blind SC. As mentioned before, if P t is too small, the user with a very low SNR is scheduled as the non-SIC user to satisfy P t ; therefore, the data rate  decreases. On the other hand, if the classification constraint becomes loose, i.e., P t is large, incorrect SC occurs more frequently, and the data rate slowly decreases. From Fig. 12, the appropriate value of P t can be found; e.g., P t = 0.01 when QPSK and 16-QAM are used for the non-SIC user and SIC user, respectively. Note that even though the sum-rate gain of NOMA over OMA remains constant for a certain interval of P t , a small P t is preferred to avoid the classification failure as much as possible.

VI. CONCLUSION
This paper mainly focuses on blind SC of NOMA signals for the presence of interference that determines whether to perform SIC or not, without any useful high-layer signaling. The error probabilities of blind SC are mathematically derived for the two different cases in which the SIC user is classified as the non-SIC user and the non-SIC user classifies itself as the SIC user, respectively. On the basis of analytical results, we formulate the joint optimization problem of user scheduling and power allocation to maximize the sum-rate gain of NOMA over OMA subject to the constraints of the maximum classification error probability and minimum data rate. To solve this problem, we investigate the effects of blind SC on user scheduling and power allocation for NOMA users. An iterative algorithm is then proposed for user scheduling for NOMA signaling and finding the power allocation ratios of the scheduled users. Finally, numerical results confirm that the proposed scheme provides better sum-rate gains over OMA compared to conventional user scheduling methods.