Exact BER Performance Analysis for Downlink NOMA Systems Over Nakagami-m Fading Channels

In this paper, the performance of a promising technology for the next generation wireless communications, non-orthogonal multiple access (NOMA), is investigated. In particular, the bit error rate (BER) performance of downlink NOMA systems over Nakagami-m flat fading channels, is presented. Under various conditions and scenarios, the exact BER of downlink NOMA systems considering successive interference cancellation (SIC) is derived. The transmitted signals are randomly generated from quadrature phase shift keying (QPSK) and two NOMA systems are considered; two users' and three users' systems. The obtained BER expressions are then used to evaluate the optimal power allocation for two different objectives, achieving fairness and minimizing average BER. The two objectives can be used in a variety of applications such as satellite applications with constrained transmitted power. Numerical results and Monte Carlo simulations perfectly match with the derived BER analytical results and provide valuable insight into the advantages of optimal power allocation which show the full potential of downlink NOMA systems.


I. INTRODUCTION
The expeditious development of the mobile Internet and Internet of Things (IoT) has obtruded several challenges on the Fifth Generation (5G) and Beyond 5G (B5G) wireless communications systems. Such challenges include capacity increase by a factor of 1000, data rates exceeding 10 Gb/s, 10 years battery life for machine-to-machine (M2M) communications, and network latency less than 1 ms [1]. Consequently, extensive research has been focused in the last few years to develop the enabling technologies that can satisfy such requirements, which include dense heterogeneous networks, full-duplex communication, energy-aware communication and energy harvesting [2], [3], cloud-based radio access networks [4], wireless network virtualization [5], advanced antenna systems [6], and efficient error correction coding [7]. Moreover, great advancements have been achieved in terms of multiple access technologies such as the non-orthogonal multiple access (NOMA), which may improve the spectral efficiency and latency with respect to orthogonal multiple access (OMA) [8]. Several NOMA schemes have been actively investigated, which can be categorized into two main categories, powerdomain NOMA, [9]- [10] and code-domain NOMA [11]- [12]. The focus of this paper is on the power-domain NOMA.
The performance of NOMA systems has attracted extensive literature, which is mainly focus outage probability and capacity. For example, the outage achievable rate region is studied in [13], where the results imply that NOMA outperforms OMA under similar conditions. In [14], the authors investigate a two-users NOMA system in terms of their power allocation, through the maximization of the ergodic sum capacity with the constraint of minimum sum rate requirement, fixed total transmit power, and partial channel state information (CSI) availability. In [15], a novel NOMA clustering scheme using a power allocation mechanism is presented to reduce the computational complexity at the expense of user fairness compared to the full search method. In [16], with the objective of providing proportional fairness, optimal power allocation and user pairing problems in multiuser downlink NOMA are investigated. A sum rate comparison between multipleinput multiple-output (MIMO) NOMA and OMA clusters is conducted in [17], where each cluster consists of two users. In [18], various user scheduling strategies are presented to accomplish flexible and efficient trade-offs between capacity and user fairness in NOMA systems. In [19] and [20], user clustering and power allocation for downlink NOMA system have been studied thoroughly.
More recently, noticeable efforts have focused on evaluating the bit error rate (BER) performance of NOMA systems with imperfect successive interference cancellation (SIC). For example, the authors of [21] derived closed-form expressions for the union bound on the BER of downlink NOMA with imperfect SIC over Nakagami-m fading channels. Although the derived bounds are useful to estimate the BER, the results presented in [21] show that the bounds may deviate significantly from the simulation results in certain scenarios. The average BER performance of a NOMA system using spaceshift keying (SSK) in Rayleigh fading channels is investigated in [22] where the exact BER is expressed in closed-form only for users two and three in a three users scenario. The exact symbol error rate (SER) for a downlink NOMA with imperfect SIC is presented in [23]. Nevertheless, using the BER is more informative when comparing different systems with different modulation orders, and the results are limited only to the two users scenario and Rayleigh fading channels. The exact SER for the two users scenario in Rayleigh fading channels is arXiv:1908.01357v1 [cs.IT] 4 Aug 2019 also investigated in [24]. The BER of uplink NOMA for the two users scenario is considered in [25] under imperfect SIC scenarios. The main limitation of this work is that the channel fading is considered constant over the transmission block, and hence, the channel becomes effectively an additive white Gaussian noise (AWGN) channel. An asynchronous uplink NOMA system based on triangle-SIC error is presented in [26]. Similar to [25], the channel coefficients are assumed to be fixed, hence the derived closed-form expressions can not be used in random fading scenarios. In [27], the BER is derived for a two users downlink and uplink NOMA systems over Rayleigh fading channels. However, the assumption that the links in downlink NOMA follows independent and identically distributed (i.i.d.) Rayleigh fading overlooks the large scale fading factor, which limits the contribution of this work. Moreover, the paper lacks in-depth insights into the analysis of the obtained BER results.

A. Motivation and Main contributions
As can be noted from the aforementioned literature survey, to the best of the authors' knowledge, there is no work reported that considers the exact BER analysis of downlink NOMA in Nakagami-m fading channels. Moreover, the works considered the special case where m = 1, i.e. Rayleigh, focus on the twousers scenario. Therefore, the main contributions of this paper are summarized as follows: 1) The exact BER performance analysis of a downlink NOMA with imperfect SIC over Nakagami-m flat fading channel is considered, where exact analytical BER expressions are derived for each user individually for the cases of two and three users' scenarios. The derived BER expressions are verified by Monte Carlo simulations. 2) The exact BER is derived in terms of a closed-form expressions for the special case of m = 1, Rayleigh fading, for two and three users scenarios.
3) The optimal power allocation for all users is investigated based on the derived BER expressions for two different criteria. In the first, the power allocation for each user is allocated optimally to guarantee fairness among all users, which is expressed in terms of equal BER for all users. In the second, the power allocation coefficients are selected optimally to minimize the average BER for all users.

C. Paper organization
The rest of the paper is organized as follows. In Sec. II, the system and channel models are presented. This is followed by exact BER analysis for the two-users and threeusers downlink NOMA systems are presented in Sec. III and Sec. IV, respectively. The optimal power allocation problem is formulated in Sec. V, while numerical and simulation results are shown in Sec. VI. Finally, the work is concluded in Sec. VII.

II. SYSTEM AND CHANNEL MODELS
This work considers a power-domain downlink NOMA system with N users, U 1 , U 2 , . . ., U N . The users' equipment (UEs) and the base station (BS) are equipped with single antennas [21]. Therefore, the transmitted signal from the BS can be expressed as where s n is the information signal of the nth user selected uniformly from a particular symbol constellation, P T is the BS transmit power, and β n is the allocated power coefficient for the nth user such that N n=1 β n = 1. For the rest of the paper, the transmit power P T is normalized to unity.
In flat fading channels, the received signal at the nth UE can be written as where h n represents the link between the BS and the nth user whose probability density function (PDF) is described in Appendix I and w n is the AWGN, w n ∼ CN (0, N 0 ) . Given that the channel phase θ n is estimated and compensated perfectly at the receiver, then the received signal after phase compensatioň r n = r n e −jθn = α n x +w n , wherew n = w n e −jθn and α n = |h n | is the channel gain. Assuming that the AWGN is circularly symmetric, thenw n and w n have identical PDFs, consequentlyw n and w n can be used interchangeably. Without loss of generality, it is assumed that the first user has the lowest channel gain, and the second user has the second lowest channel gain, and so forth, i.e., α 1 < α 2 < · · · < α N . To enable reliable detection of all users, it is necessary to cancel the inter-user interference (IUI), which is typically performed using SIC. Therefore, the power should be allocated in the opposite order of the channel gains, i.e., β 1 > β 2 > · · · > β N .
To detect the signal of the nth user, the signals of U 1 , U 2 , . . ., U n−1 should be detected and scaled, then subtracted from r n , the IUI for users U n+1 , U n+2 , . . ., U N is considered as unknown additive noise. For the first user, the IUI from all users will be treated as noise, and thus, the maximum likelihood detector (MLD) given that the channel gain h 1 is known perfectly at the receiver can be expressed as [28], whereŝ 1 is the estimated data symbol, S is the set of all possible constellation points for U 1 , ands 1 are the trial values of s 1 . For the nth user, the detector can be described bŷ In the following sections, the exact BER is derived for a power-domain NOMA system with two and three users. Although the presented approach can be applied to any phase shift keying (PSK) or quadrature amplitude modulation (QAM), the derivation becomes intractable for a modulation order M > 4, particularly for N > 2. Therefore, the analysis presented in this work considers As can be noted from (1), the transmitted symbol x for N = 2 is the superposition of two QPSK symbols, and hence, it should correspond to one of the 16 constellation points shown in Fig. 1. The bit representation for each constellation point is given in the form of b 11 b 12 b 21 b 22 , for each bit b ni , {n, i} ∈ {1, 2}, where n denotes the user index while i denotes the bit index.
For the first user, the detection is performed using (3), and thus, no SIC is required. Based on the specific value of s 2 , the IUI caused by U 2 , the symbol x may become one of the four constellation points in the neighborhood of s 1 . The shaded blocks in Fig. 1 show the four possible values that a particular symbol s 1 may take. For example, given that s 1 = 10 then x| s1 ∈ {1000, 1001, 1010, 1011}. The amplitudes of the inphase x I (x) and quadrature x Q (x) for each constellation point are defined as A u1u2u3 = u 1 β 1 +u 2 β 2 +u 3 β 3 , u i ∈ 0, 1,1, 2 (5) where1 −1. For example, given that s 2 , then x = 1010 and It is worth noting that for N = 2, u 3 = 0 regardless the values of s 1 or s 2 , however, it is used to unify the notation throughout the paper. The probability of error for each bit actually depends on the values of s 1 and s 2 . For example, given that s (2) 1 , the first bit b 11 might be detected incorrectly ifŝ 1 = a 0 (00) or a 1 (01), as shown in Fig. 1. However, P(ŝ 1 = a 0 or a 1 ), denoted as P(ŝ 1 = a 0 ; a 1 ) , depends on s 2 as well. Therefore, the average BER should consider all possible combinations of s 1 and s 2 , By noting that s 1 and s 2 are independent, then (6) can be written as, Case 1: s 1 , s 2 : For this case, x| a2,a0 = −A 110 +jA 110 , to simplify th notations x| a l ,a k is written as x l,k . Consequently, the error probability of b 11 is given by, As can be noted from Fig. 1, P(ŝ 1 = a 0 ; a 1 ) depends only on the inphase component ofř 1 , i.e., (ř 1 ) r 1 and the specific value of x. Thus, where r 1 = −α 1 A 110 + n 1 , (w 1 ) n 1 . Therefore, where γ 1,1 = α 2 1 A 2 110 /σ 2 n1 and Q (.) denotes the Gaussian Q function.
Case 2: s 2 : This case is similar to the case of s (0) 2 , hence, the error probability is given by (9) as well.

:
The probability of error in this case is similar to the case of s 2 . The remaining cases, Case 5 to Case 16 are similar to Case 1 to Case 4 except that the value of s 1 is replaced by a 0 , a 1 , a 2 and a 3 . Substituting the results of the 16 cases in (7) gives It is also straightforward to show that P 12 = P 11 . Therefore, the conditional BER of the first user is given: To detect its own symbol s 2 , the second user should initially detect s 1 as described in (3), and then compute, where r 2,sic = r 2 − h 2 β 1ŝ1 and x sic = x − β 1ŝ1 . Therefore, given thatŝ 1 = s 1 , then x sic = β 2 s 2 and r 2,sic corresponds to IUI-free QPSK signal. The constellation diagram of x sic |ŝ 1=s1 is shown in Fig. 2. On the other hand, ifŝ 1 = s 1 , then x sic = β 1 s 1 + β 2 s 2 − β 1ŝ1 and its constellation diagram depends onŝ 1 . For example, given that 1 , the constellation diagram of x sic becomes as shown in Fig. 2.
The BER of U 2 depends on s 1 , s 2 andŝ 1 . Therefore, the probability of error should be averaged over all possible combinations, Using the chain rule P s and noting that s 1 and s 2 are independent, then (14) can be written as, It should be noted that the probability of correct or incorrect detection of b 12 does not affect the error probability of the second user bits P 2i . Therefore, it is assumed that P b12 = 1 for all cases. The case where v = l corresponds to the event thatŝ 1 = s 1 , and the corresponding probabilities can be computed as follows: Case 1: s can be obtained by considering the error probability of U 1 given that b 2 . In this case, the conditional error probability of b 21 can be computed by noting that P b21 |A 221 = . The transmitted signal amplitude for this case is x = A 110 +jA 110 , thus, x sic = A 010 + jA 010 and r 2,sic = α 2 x sic + w 2 . As can be noted from Fig. 2, P(ŝ 2 = a 0 ; a 1 | A221 ) depends only on the inphase component ofř 2,sic , i.e., (ř 2,sic ) r 2,sic . Thus where r 2,sic = α 2 A 010 + n 2 , (w 2 ) n 2 and −A 010 = A 010 . Using Bayes' theorem and considering the results in (15) and (16), we obtain By noting that the right hand side of (RHS) of (17) is equal to P(n 2 ≥ α 2 A 010 , n 2 ≥ α 2 A1 10 ), then where γ 2,1 = α 2 2 A 2 010 /σ 2 n2 . For b 22 , the error probability is obtained using the approach used with b 21 As the RHS of (19) can be simplified to is computed for x sic = A 110 + jA 110 and r 2,sic = α 2 x sic + w 2 , Since the RHS of (21) can be simplified to where γ 2,3 = α 2 2 A 2 110 /σ 2 n2 . For b 22 , the error probability is given by The RHS of (23) can be simplified to P(q 2 ≤ α 2 A 010 ) P(n 2 ≥ α 2 A11 0 ), and thus (24) The remaining cases, Case 3 to Case 9 where the constellation points are 0110, 0011, 0111, 1000, 1001, 1100, 1101 are similar to Case 1, and Case 10 to Case 16 where the constellation points are 0001, 0100, 0101, 1010, 1011, 1110, 1111 are similar to Case 2. By using (15), the BER of the second user given thatb 11 = b 11 can be expressed as The constellation diagram of x sic , after subtractingŝ 1 |b 11 =b11 is shown in Fig. 2 (solid diamonds). The total error probability of the second user whenb 11 = b 11 can be derived by considering all the cases in (15).
2 : The transmitted signal amplitude of this point is x = A 110 + jA 110 , thus, x I,sic = A 210 . The error probability for this case is 11 , s (2) 2 . The RHS of (26) can be simplified to P(α 2 A2 10 ≤ n 2 ≤ α 2 A1 10 ). Thus, By noting that the RHS of (28) can be simplified to Case 2: s 2 : The error probability of this case can be computed as The RHS of (30) can be expressed as P(n 2 ≤ α 2 A21 0 , n 2 ≤ α 2 A11 0 ), and hence, where For b 22 , the error probability is evaluated as Since the RHS of (32) can be simplified to For the remaining cases, Case 3 to Case 9 where the associated symbols are 0011, 0110, 0111, 1000, 1001, 1100, and 1101 are identical to Case 1. Case 10 to Case 16 where the constellation points are 0001, 0100, 0101, 1010, 1011, 1110, and 1111 are similar to Case 2. Therefore, the total error probability whenb 11 = b 11 can be expressed as Finally, the exact total BER for the second user is given as the sum of the results of the two scenarios whereb 11 = b 11 andb 11 = b 11 , C. Average BER, N = 2 The average BER can be evaluated by averaging over the PDFs of all γ n,c values, which are given in Appendix I. Therefore, by substituting N = 2 and n = [1,2] in the ordered PDF in (93), the exact average BER of the first and second users can be simplified to It is interesting to note that for the special case of Rayleigh fading channel where m = 1, the BER for both users can be expressed in closed-form as,

IV. NOMA BIT ERROR RATE (BER) ANALYSIS: THREE USERS (N = 3)
This section presents the derivation of the BER for a three-users NOMA system, N = 3, which generally follows the derived N = 2 case. The transmitted signal constellation is given in Fig. 3. The first, second, and third users' signals are shown in the form of [s 1 , s 2 , s 3 ]. The binary bit representation for the three users are represented The error probability of the first user can be obtained directly from Fig. 3. The probability of error for each bit depends on the values of s 1 , s 2 and s 3 . For example, given that s 3 , the first bit b 11 might be detected incorrectly if s 1 = a 0 (00) or a 1 (01), as shown in Fig. 3. Therefore, the average BER should consider all possible combinations of s 1 , s 2 and s 3 , Because s 1 , s 2 and s 3 are mutually independent, (40) can be written as The error probability of the first user is derived by considering all possible combinations, which can be derived as follows: 2 , s 3 : For this case, x = A 111 + jA 111 , consequently, the error probability of b 11 is given by, As can be noted from Fig. 3, P(ŝ 1 = a 2 ; a 3 ) depends only on the inphase component ofř 1 , i.e., r 1 = α 1 A 111 + n 1 , and the specific value of x. Thus, 2 ,s  TO Following the same approach, Table I shows the summary of Case 2 to Case 4. The remaining 60 cases, Case 5 to 64 in Fig. 3, can be obtained following the same approach, and hence, the total BER of the first user can be expressed as The BER of the second user depends on the detection result of the first user and the SIC process as illustrated in Fig. 4. The first case is whenŝ 1 = s 1 , which is represented in Fig.  4.a while the other case is whenŝ 1 = s 1 , which is depicted in Fig. 4.b. To detect its own symbol s 2 , the second user should follow the SIC process described in (13). It should be noted that the error probability of the second user is not affected by the detection result of the second bit of the first user b 12 .
The BER of U 2 depends on s 1 ,ŝ 1 , s 2 and s 3 . Therefore, the average BER for b 2i is the average of all possible combinations, With the aid of Fig. 4.a, the total BER for the scenario wherê s 1 = s 1 , or more specifically, whereb 11 = b 11 is obtained as follows: 2 , s 3 : 2 , s can be evaluated by considering the error probability of U 1 given A 321 . In this case P b21 can be computed by noting that 2 , s . The transmitted signal amplitude of this case is x = A 111 +jA 111 , thus, x sic = A 011 + jA 011 and r 2,sic = α 2 x sic + w 2 . Therefore, where r 2,sic = α 2 A 011 + n 2 . Therefore, 2 ,s where γ 3,5 = α 2 2 A 2 011 /σ 2 n2 . For b 22 bit, the error probability is obtained in a similar approach as b 21 2 ,s 2 , s 3 : Based on Fig. 4.a, the error probability of b 21 is given by 2 ,s By noting that the RHS of (49) can be simplified to P(n 2 ≥ α 2 A 011 ), and consequently it is straightforward to show that 2 ,s where 2 , s (0) 3 .
For b 22 bit, the error probability is obtained as 2 ,s Case 3: s 3 : The error probability of this case can be evaluated as 2 ,s where 11 , s 2 , s . For b 22 bit, the error probability is obtained as 2 ,s The error probability of b 21 can be evaluated as where 11 , s 2 , s (0) 3 . For b 22 , the error probability can be expressed as By taking into account the remaining 60 cases from Case 5 to 64, the total BER of the second user whenb 11 = b 11 can be expressed as The same approach is adopted for the scenario whereb 11 = b 11 . The transmitted signal constellation after subtractingŝ 1 whenb 11 = b 11 is shown in Fig. 4 (b).
2 , s 3 : The transmitted signal in this case is x = A 111 + jA 111 , which after subtractingŝ 1 = − β 1 + j β 1 becomes x sic = A 211 + jA 011 . The error probability can be computed as 2 ,s 3 = P(α 2 A2 11 ≤ n 2 ≤ α 2 A1 11 ) 11 , s 2 , s 2 ,s 2 , s 3 : The error probability of b 21 for this case can be derived as, 2 ,s where 11 , s 2 , s (0) 3 . The second bit b 22 error probability can be represented as 2 ,s 3 : Similar to the previous cases, the error probability for this case can be derived as follows.
where γ 3,9 = α 2 2 A 2 211 /σ 2 n2 and B 323 → b 11 , s 2 , s 2 ,s 3 : The probability of error for b 21 can be computed as: where By considering the other 60 cases, Case 5 to Case 64, the total BER of the second user whenb 11 = b 11 can be represented as The total BER for the second user is evaluated by combining (56) and (66) (67)

C. BER of Third User
The error probability of the third user is calculated based on Fig. 5. The BER of U 3 depends on s 1 , s 2 , s 3 ,ŝ 1 andŝ 2 and . Therefore, the average BER for b 3i is the average of all possible combinations. It should be noted that both b 12 and b 22 bits do not affect the error probability of the third user, hence it is assumed that P b12 = P b22 = 1. As for the first scenario whereb 11 = b 11 andb 21 = b 21 , the error probability of the third user is derived according to Fig. 5 (solid circles) and (68).
3 : The transmitted signal x = A 111 + jA 111 is subtracted bŷ s 1 andŝ 2 , hence, x sic = A 001 + jA 001 . The error probability for this case can be obtained as follows.
The error probability of this case is expressed as (71) . For b 32 , the error probability can be obtained as (72) 3 : The error probability of this case is derived as . The error probability for b 32 is (74) Case 4: s The error probability of this case is obtained as (75) . For b 32 is as follows By taking into consideration the other 60 cases, Case 5 to Case 64, which are obtained in a similar manner, the total BER for the first scenario whereb 11 = b 11 andb 21 = b 21 can be represented as As for the other scenarios, a similar procedure is followed as for the caseb 11 = b 11 andb 21 = b 21 . The analysis for b 11 Fig. 5 and (68). The final total BER results for this scenario is respectively given by, and P (4) where . By combining the results in (77), (78), (79) and (80), the total BER for the third user can be computed as 2,4,11,13,14,15,16,17,18] , D. Average BER, N = 3 Similar to the N = 2 NOMA system, the average BER of N = 3 NOMA system can be evaluated by averaging over the PDFs of all γ n,c values, which are given in Appendix I. Therefore, by substituting N = 3 and n = [1,2,3] in the ordered PDF in (93), the exact average BER of the first and second users can be simplified tō The closed-form average BER for the first, second, and third users over Rayleigh fading channel (m = 1) are shown in (85), (86), and (87), respectively. The optimal power allocation problem is formulated for two different objective functions using the derived exact BER expressions for N = 2 and N = 3 NOMA systems. The first objective is to obtain the optimum power coefficients β n that minimize the overall average BER of all users. The second objective function is to evaluate the optimum power coefficients which guarantee fairness among all users. The fairness in this work indicates equal BER for all the users.
The optimal power allocation for minimizing the average BER is formulated as follows: where the first constraint limits the maximum transmit power, which is normalized to unity. The second constraint is used assure that the power allocated to each user is inversely proportional to its channel gain, i.e., β 1 > β 2 > · · · > β N are assigned for the users with the channel gains α 1 < α 2 < · · · < α N , respectively. The problem in (88b) is a constrained nonlinear optimization problem which is solved using the Interior-Point Optimization (IPO) algorithm [29]. The optimal power allocation for achieving fairness among the users in the NOMA system is formulated as follows: The constraints for the second optimization problem are similar to those in the first optimization problem and the solution can be obtained using the same approach.

VI. NUMERICAL AND SIMULATION RESULTS
This section presents numerical and Monte Carlo simulation results for N = 2 and N = 3 downlink NOMA systems. All users are assumed to be equipped with a single antenna, and the channel between the BS and each user is modeled as an ordered Nakagami-m flat fading channel. The randomly generated channels are ordered based on their strength, where the weakest channel is assigned to the first user and the strongest channel is assigned to N th user. The transmitted symbols for all users are selected uniformly from a Gray coded QPSK constellation. The total transmit power from the BS is unified for all cases, P T = 1.   7 is generally similar to Fig. 6, except that it considers the three users scenario, i.e., N = 3. The power allocation coefficients are β 1 = 0.8, β 2 = 0.15 and β 3 = 0.05. The figure clearly shows the perfect match between the analytical results obtained using (82)-(87) and simulation results for all m values and over the entire E b /N 0 range. As can be seen from Figures 6 and 7, the performance of the first user is more sensitive to the variations of m as compared to the second and third users, which is due to the fact that the fading effect becomes less significant for the near users. Fig. 8 compares the exact BER and union bound [21] for N = 2, m = 1, β 1 = 0.7 and β 2 = 0.3. It can be noted that the union bound is generally tight in the high E b /N 0 range, particularly for the second user. For example, the gap between the exact BER and union bound at E b /N 0 = 20 dB is about 2 dB for the first user and 1 dB for the second user. At low E b /N 0 , the gap may increase to 3 dB. Therefore, using the exact BER expression is critical when accurate BER estimates are desired. Fig. 9 shows the BER for N = 3 under perfect and imperfect SIC for different values of m. Although the perfect SIC assumption may tremendously reduce the BER analysis, the results presented in Fig. 9 show that such assumption is too optimistic, particularly for U 3 . As expected, the results for U 1 with/without SIC are identical because U 1 detection does not involve a SIC process. For U 2 , the impact of the perfect SIC assumption is apparent at low SNRs, because at high E b /N 0 , U 1 signal is mostly detected correctly, and thus, the results with/without SIC converge. The third user U 3 is the one who will experience the maximum difference because the probability that the two SIC operations are performed successfully is relatively small. Therefore, the BER with and without SIC for U 3 will exhibit substantial difference. Moreover, as m increases, the BER with perfect and imperfect BERs become closer, particularly at high SNRs, which is due to that fact that for high values of m the fading is less severe, which implies that the probability of having successful SIC operations is higher as compared to the low m values.
Tables III and IV present the optimal power coefficients that provide equal BER for all users. The results are obtained for different values of E b /N 0 , N = 2, 3, and m = 1, 3. As can be noted from the results in Table III, most of the power is actually allocated for U 1 , particularly at high SNRs. For N = 2, the first user is allocated more than 98% of the total power at E b /N 0 = 30 dB, and it is about 89% for N = 3. Moreover, the range of values that the power coefficients might be allocated depends drastically on N . The same trends can be noted for the m = 3 case in Table IV. Nevertheless, the power given to the first user generally decreases by increasing m as the impact of the AWGN becomes more noticeable. Table V shows the optimal power coefficients which minimize the average BER for N = 2, 3, and m = 1. As can be noted from the table, the power allocation should be performed meticulously to achieve the desire results. Although       Fig. 10 presents the BER using the optimum power coefficients that guarantees minimizing the average BER, in addition, two other curves obtained using fixed power values are used where β 1 = 0.7, and 0.9 for N = 2, and {β 1 = 0.84, β 2 = 0.12}, and {β 1 = 0.9, β 2 = 0.07} for N = 3. As can be noted from the figure, allocating the power coefficients appropriately might save the need to use adaptive power values. On the other hand, large deviations from the optimum power values might result in severe BER degradation. Figures 11 and 12 show the effect of the fading factor on the performance of each user for NOMA systems with N = 2, 3 where E b /N 0 = 10, and 18 dB. Power allocation coefficients at E b /N 0 = 10 dB are assigned as follows, β 1 = 0.84, β 2 = 0.16 and β 1 = 0.67, β 2 = 0.27, and β 3 = 0.06 for the two and three users' systems, respectively. For the case where E b /N 0 = 18 dB, the power allocation coefficients are β 1 = 0.88, β 2 = 0.12 and β 1 = 0.84, β 2 = 0.13, and β 3 = 0.03 for the two and three users' systems, respectively. It should be noted that the power allocation coefficients are the optimal values that minimize the average BER. As can be seen from Figures 11 and 12, the BER performance of all users highly depends on the fading parameter m. Additionally, it is shown that m affects the performance of the higher order users more than the lower order users, which is due to the ordering of users based on the channel conditions that resulted in an enhanced performance for higher order users. Moreover, when E b /N 0 increases, the effect of m on the performance increases because the BER will be mostly determined by the fading.

VII. CONCLUSION
This work presented the performance of a downlink NOMA system in terms of BER where exact BER expressions were derived for different users over Nakagami-m fading channels for two and three users' scenarios, where imperfect SIC is considered. The BER can be evaluated numerically for general m values, as one of the integrals does not have an analytical solution. For the special case of Rayleigh fading, m = 1, closed-form expressions are derived for several cases of interest. Moreover, constrained nonlinear optimization problems which aim to find the optimum power coefficients that minimize the average BER and achieve fairness among the users were formulated. Each objective can be used for different purposes, for example, if the main goal of NOMA system is to fairly acquire equal average BER for all users such as satellite systems, then the fairness optimal power allocation should be evaluated. On the other hand, when the purpose of a NOMA system is to minimize the overall average BER for applications such as , then the latter problem formulation can be used.

APPENDIX I: AVERAGE BER OVER NAKAGAMI-m FADING CHANNEL
The average BER for a NOMA system over Nakagamim fading channel follows the order statistics of Nakagamim distribution. Based on order statistics theory, the general   ordered PDF of the channel gain of the nth user can be expressed as [30], where K n = N ! (n−1)!(N −n)! , f (α n ) and F (α n ) are respectively the PDF and CDF of Nakagami-m distribution with parameters m and Ω, where Γ(m) is the upper incomplete Gamma function, Ω = E α 2 n , m = Ω 2 V ar(α 2 n ) , and Φ (a, z) = z 0 t a−1 e −t dt is the lower incomplete Gamma function [31]. Therefore, the ordered PDF of the nth channel gain over Nakagami-m channel is Because the lower incomplete Gamma function is raised to a power, rendering the integral analytically for m > 1 is intractable. Therefore, the infinite series representation of the lower incomplete Gamma function can be used [32], In addition, the term respectively, where γ n,c = A 2 u2u2u3 Ω/σ 2 n and c is the index parameter.
The ordered PDF of γ n,c of the nth channel can be expressed using (90), (94), and (97), as follows In order to evaluate the average BER of the nth user over Nakagami-m channel, (99) and the alternative representation of the Q function defined by [34] are utilized. Moreover, the following integral is used [33], ∞ 0 x t e −bx dx = t! b t+1 , t ∈ {0, 1, . . . , b} > 0. (100) The general average BER for user n is given by, Although the integral in (101)  Hatem Zeineldin (M'06-SM'13) received the B.Sc. and M.Sc. degrees in electrical engineering from Cairo University, Giza, Egypt, in 1999 and 2002, respectively, and the Ph.D. degree in electrical and computer engineering from the Univer-sity of Waterloo, Waterloo, ON, Canada, in 2006. He was with Smith and Andersen Electrical Engineering, Inc., North York, ON, USA, where he was involved in projects involving distribution system designs, protection, and distributed generation. He was a Visiting Professor with the Massachusetts Institute of Technology, Cambridge, MA, USA. He is currently a Professor with the Khalifa University of Science and Technology, Abu Dhabi, UAE, and is currently on leave from the Faculty of Engineering, Cairo University, Giza. His current research interests include distribution system protection, distributed generation, and micro grids. He is currently an Editor for the IEEE Transactions on Energy Conversion and the IEEE Transactions on Smart Grid.