Performance of CDRT-Based Underlay Downlink NOMA Network With Combining at the Users

The performance of an underlay coordinated direct and relay transmission (CDRT) based non-orthogonal multiple access (NOMA) network is limited by the random nature of the transmit powers due to the interference temperature limit (ITL) imposed by the primary network. In this paper, we show that combining the signals at both near and far users can significantly enhance throughput. Considering the practical scenario of imperfect successive interference cancellation, the performance of near and far user nodes is analyzed in terms of throughput and outage probability. The traditional case of static ITL is considered first. Then, analysis is presented for the case when the ITL depends on the primary channel state information, and it is shown that the gain in performance can be very large over static ITL. Closed-form expressions for NOMA power apportioning are presented for a special case. Further, it is shown that how optimum parameters can be chosen to maximize the performance. Simulations validate the derived analytical expressions.


I. INTRODUCTION
T HE RAPID increase in the number of data services in recent years has resulted in an acute spectrum scarcity [1]. A tremendous amount of research effort has therefore focused on approaches to improve spectrum utilization efficiency (SUE). While several different frequency reuse approaches have been proposed, cognitive radio (CR) technology has received the most research attention [2]. Among its various basic modes of operation -underlay, overlay 1  The authors are with the Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India (e-mail: anandjee7@ ee.iitd.ac.in; kamal.agrawal@ee.iitd.ac.in; deepali28johari@gmail.com; shankar@ee.iitd.ac.in).
Digital Object Identifier 10.1109/TCCN.2023.3234975 1 In overlay type, the secondary nodes use sophisticated signal processing and coding to maintain the communication of the primary nodes while also obtaining some additional bandwidth or time-slots for their own communication. A high degree of cooperation between primary and secondary networks is implied. In interweave type, the secondary nodes periodically monitor the radio spectrum and opportunistically communicate without disrupting the active primary users when a spectrum hole is detected. In addition, it avoids simultaneous transmissions of primary and secondary users which limits secondary performance.
interweave-it is the underlay type that has shown the greatest promise in improving SUE [3], [4]. In underlay CR, the secondary nodes transmit concurrently with the primary transmitter (PT) but with carefully controlled transmit power to ensure that the interference temperature limit (ITL) imposed by the primary user (PU) is not exceeded [5], [6]. The ITL is a measure of the peak interference that the PU can tolerate [7]. This ITL can be a statistically fixed quantity designed to satisfy the outage quality of service (QoS) constraint of the primary network [8].
With the rapidly growing demands of wireless data services and Internet of Things (IoT) devices, efficient utilization of spectrum has become a critical challenge [9]. The conventional orthogonal multiple access (OMA) used in earlier generation radio networks is not sufficient to ensure services to such a large number of devices, and the use of non-orthogonal multiple access (NOMA 2 ) has been suggested [10], [11]. In downlink NOMA, the symbols of several users are superposed (multiplexed in the power domain) with more power allocated to the symbol intended for the user with poorer channel gain [12], and successive interference cancellation (SIC) is used at the receivers [13]. NOMA is known to outperform OMA in several scenarios. In underlay networks, the use of power domain NOMA appears to be contra-indicated. However, it is shown in [14], that by carefully switching between OMA and NOMA and intelligent user selection, NOMA can vastly outperform OMA even in the underlay context and can result in more efficient spectrum utilization.

A. State-of-Art
Integration of NOMA with underlay CR network further enhances the spectral efficiency (SE) and also helps to support a large number of devices while maintaining the desired QoS at PU [15], [16]. Since the transmit power of secondary nodes is restricted in the underlay NOMA due to the ITL constraints of the primary network [17], it is intuitive that the presence of a dedicated relay station or a user for relaying in the secondary network can ensure better outage performance at the secondary receivers [18], [19]. In view of this, various types of NOMA techniques like relayed-NOMA, cooperative-NOMA, and coordinated direct and relay transmission (CDRT) based NOMA has been studied. In cooperative-NOMA, a near user (NU), which has a good channel gain relays the information to far user (FU) [16], whereas in relayed-NOMA, a dedicated relay assists in communication to the FU [18] 3 [19].
The CDRT framework allows the source to communicate directly with the NU while the FU receives its signal via a dedicated relay (thus ensuring higher SE [20]). The authors in [21], [22], [23] have shown that NOMA in CDRT results in significant performance gain (sum-capacity) as compared to NOMA in the non-CDRT cooperative framework. However, they did not consider the underlay scenario, where due to the random nature of the transmit powers and consequent large variation in the link SNRs, an extension of these ideas from cooperative literature is far from obvious. NOMA may result in poor performance in underlay networks since the source (S) transmit power is constrained. Also, in the existing CDRT-NOMA scheme, the absence of an S-FU direct link limits the performance of both NU as well as FU. Furthermore, the consideration of perfect SIC [20], [21], [22] leads to a simpler but impractical scenario. Existing underlay CDRT-NOMA literature has the following limitations: • In many applications like multimedia, low outage is a basic requirement. Existing CDRT schemes fail to utilize the relay transmissions to improve outage performance at the NU. In underlay CR, where the link signal-tonoise ratios (SNRs) show large variation due to transmit power control, this causes degradation in the outage performance. • The relay node has to decode the FU's symbols successfully to assist in information transmission to the FU in the second phase. Due to power allocated to FU, it is evident that NU loses on performance due to participation in NOMA signalling. • Due to the absence of direct link from the source, there is no assistance to the users from the first phase of signalling, which causes a loss in users' performance. • The performance of the users completely depends on the decoding status of the symbols at the relay in the first phase of signalling. If the source to relay link goes into a deep fade in the first phase, then the relay will not be able to assist the second phase transmission and hence none of the users will be able to decode the desired symbols and will remain in outage. Therefore, the decoding status at the relay in the first phase is a bottleneck for the users' performance. • NU first decodes the FU symbols and then performs SIC to decode its own symbol. If the NU fails to decode the FU symbol, then the NU will not be able to decode its own symbol and loses its performance due to participating in NOMA signalling. Therefore, decoding of the FU symbol at NU is a bottleneck for the NU's performance. 3 Different from [18], we consider direct links to both U 1 and U 2 and analyse the performance for the case of SI as well as CI while taking SIC error into account. Due to the combining at both the users (along with SIC at U 1 ) the analysis of the considered system is quite challenging and very much different from [18]. Further, we derive the optimal value of the power apportioning parameter to maximize throughput at NU while guaranteeing a fixed desired throughput at the FU. Moreover, we demonstrate that the combining of the first and the second phase signals at both users provide a substantial performance gain in comparison to [18].

B. Motivations and Contributions
From the above-mentioned limitations of CDRT-NOMA, it is evident that NU loses on performance due to participation in NOMA signalling (cooperating with FU). Also, the absence of direct links from the source to users limits performance. Moreover, in underlay CR, the transmit powers of the source are random due to the interference constraints imposed by the primary network. Therefore, exploiting the direct links is especially important in such scenarios. To overcome the limitations of conventional CDRT-NOMA, in this paper, the direct and the relayed signals at both the users are combined to ensure better performance. At the NU, this combining of direct and relayed link signals along with SIC presents several critical challenges. In addition, it is important to appreciate that we consider imperfect SIC at the relay as well as NU due to the possibility of severe degradation in the NU's performance. Further, consideration of channel-dependent ITL can be analytically complex in comparison to statistical ITL. Therefore, considering the impact of SIC errors as well as the channeldependent ITL in the underlay CDRT-NOMA framework is of utmost importance. The availability of the direct link to both the users can significantly enhance the performance of the secondary network due to the combining of the relayed and direct link signals.
Motivated by the above, we suggest a scheme in which we combine signals from the S and the relay (R) at both NU and FU in this CDRT based framework using a half-duplex (HD) relay. Considering the effect of imperfect SIC, the analytical expressions for the outage probability and throughput of the NU and FU are derived in closed-form, assuming that the ITL to be a fixed quantity. It is shown that the suggested scheme ensures much better performance at the NU (due to the combining of symbols) compared to conventional OMA and relayed-NOMA schemes. Next, we exploit the fact that to satisfy the outage QoS of the PU, the SINR at PU is larger than the required threshold value with a very high probability, which makes ITL a CSI-dependent random quantity in any coherence interval. We thus analyze the performance of secondary receivers when the value of ITL in each signalling interval is broadcast by the PU and show that the suggested CDRT scheme ensures high throughput at both NU and FU.
The major contributions of this paper are as follows: • We consider a CDRT based downlink NOMA network specifically suited for underlay scenario in which both NU and FU combine the signals from source and relay. Considering the imperfect SIC at both R and NU, the closed-form expressions for the outage probability and throughput are derived for both NU and FU assuming fixed-rate transmission when the ITL is a static quantity. • We also investigate the case when ITL is a CSI-dependent quantity and derive closed-form expressions for the outage probability and throughput at both NU and FU while considering the effect of imperfect SIC at both R and NU. • The conventional relayed-NOMA is a special case of CDRT-NOMA framework considered in this paper (with direct links to both NU and FU in deep fade), and its performance analysis follows as a special case.  I  COMPARISON OF LITERATURE ON UNDERLAY/NON-UNDERLAY DIRECT/CONVENTIONAL NOMA,  COOPERATIVE-NOMA, RELAYED-NOMA, CDRT-NOMA, AND OUR WORK • From the derived expressions, we also investigate the influence of target information rates and power apportioning parameters at source and relay on the performance of NU as well as FU. • We also derive asymptotic expressions (floor analysis) for the outage probability of both the NU and the FU. Using the asymptotic expressions, we demonstrate that the proposed scheme (for the cooperative scenario when ITL goes to infinity) allows both NU and FU to harness a diversity of 2. • Finally, expressions are derived in both static ITL and CSI-dependent ITL cases for the optimal value of the power apportioning parameter at the source that maximizes throughput at NU while maintaining a fixed desired throughput at the FU. • In addition, the performance of the proposed scheme is compared with the relayed-NOMA scheme as well as its OMA counterpart, and it is shown that the proposed scheme ensures huge throughput gain.

C. Structure of the Paper
The remainder of this paper is organized as follows: The system model of the considered CDRT based underlay NOMA system is described in Section II. Performance analysis of the proposed system model in terms of outage probability for both statistical ITL and CSI-dependent ITL is presented in Section III. Optimization of the power apportioning factor to maximize the throughput is discussed in Section IV. Numerical results based on mathematical analysis and computer simulations are discussed in Section V. Finally, Section VI concludes this paper.

D. Notations Used in the Paper
Throughout this paper, f X (x ), F X (x ) andF X (x ) denote the probability density function (PDF), cumulative distribution function (CDF) and complementary CDF of a random variable (RV) X taking value x, and F X | Y (x ) denotes the CDF of X given Y. CN (0, σ 2 ) represents the distribution of a zero-mean Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. complex Gaussian RV of variance σ 2 . The generalized hypergeometric function and the exponential integral of type 1 are denoted by p F q (.) and E 1 [.], respectively.

II. SYSTEM MODEL
As depicted in Fig. 1, we consider an underlay CDRT downlink NOMA network consisting of a secondary source S, a primary user PU, an NU U 1 , an FU U 2 and an HD decodeand-forward (DF) relay R. All the nodes are equipped with a single antenna. The communication to U 1 and U 2 in the secondary network takes place in two phases. In the first phase, S communicates to U 1 and R over a direct link, whereas in the second phase, communication to U 1 and U 2 is assisted by R. The primary network consists of a PT and a PU. Considering that PT is located far away from the secondary receiving nodes as compared to the secondary transmitters, the interference from PT at these nodes is ignored. This assumption has been justified on information-theoretical grounds and has been made in most literature on CR [32], [33], [34], [35], [36].
All the channel coefficients are independent and are of the quasi-static Rayleigh fading type. Denote by h AB ∼ AB with d AB denoting the normalized distance between A and B, and ς is TABLE II DECODING STATUS-BASED TRANSMISSION WITH "1" AND "0" DENOTING THE SUCCESS AND FAILURE, RESPECTIVELY the path-loss exponent. Superscripts "I" and "II" represent the first and the second phase quantities, respectively. As illustrated in Table II, in the first phase, S transmits a superposition of unit-energy symbols x 1 and x 2 intended for U 1 and U 2 with respective target rates R 1 and R 2 . The fraction of power allocated to U 1 and U 2 at S is α S and (1 − α S ), respectively. The superimposed symbol is expressed as where I P denotes the peak interference power tolerated at PU 4 (the ITL). The received signals at U 1 , U 2 and R are For ease of exposition, we assume |h , and then performs SIC to decode x 1 . The signalto-interference-plus-noise ratio (SINR) Γ I 12 and the SNR Γ I 11 at U 1 to decode x 2 and x 1 are expressed as 4 Note that the peak power constraints are not introduced. However, this is not a major limitation since outage performance in underlay networks exhibits a floor (outage with finite peak power is the same as that with infinite peak power), and it is in this low outage region that the system is operated. It is already demonstrated that "imposing a constraint on the peak power does not yield a significant impact on the performance as long as the ITL is constrained" [33]. Such an assumption is reasonable, and commonly made in the existing literature on underlay CR [8], [14], [28], [37], [38].
respectively, where γ 2 = 2 R 2 −1. The SINR Γ I 12 and the SNR Γ I 11 at U 2 to decode x 2 is given by Similarly, the SINR Γ I R2 and SNR Γ I R1 at R to decode x 2 and x 1 are expressed as After successfully decoding both symbols, R forwards a re-encoded superimposed symbol |h RP | 2 x 2 in the second phase with α R and (1 − α R ) denoting the fraction of power allocated to U 1 and U 2 , respectively. The signals received at U 1 and U 2 in this second phase are In this paper, we start with the motivation that the second phase signal does have information about x 1 and x 2 transmitted in the first phase, and should ideally be used by U 1 to improve its decoding. Unlike all other papers to date, we make use of the second phase transmissions at U 1 . The SINR Γ II 12−S and SNR Γ II 11−S at U 1 to decode x 2 and x 1 are given by where (12) holds provided Γ II 12−S ≥ γ 2 . In the above, the subscript −s is used to emphasize that R transmits a superimposed symbol. The SINR Γ II 2−S to decode x 2 is expressed as If R decodes x 2 successfully but fails to decode x 1 then it transmits only x 2 . The signals received at U 1 and U 2 are respectively, where the subscript −f is used to emphasize that R transmits only the FU symbol x 2 . As in the earlier case, U 1 uses the received signal y II 1−f with information on x 2 to enhance its performance. Before transmission in the second phase, R sends pilot bits to both U 1 and U 2 to inform them of its decoding state, allowing U 1 and U 2 to decode accordingly. The respective SINRs Γ II 12−f and SNR Γ II 2−f to decode x 2 at U 1 and U 2 are expressed as

III. OUTAGE PROBABILITY ANALYSIS
In this Section, we derive the outage probability of U 2 and U 1 . Two scenarios are possible. In the first scenario for which performance is analyzed in Section III-A, I P is determined based solely on statistical parameters. In the second scenario for which performance is analyzed in Section III-B, PU computes the I P based on CSI and broadcasts the same in every coherence interval. For ease of exposition, we define the following quantities: , .

A. Statistical Properties Dependent ITL (SI Scheme)
The PT transmits a unit-energy symbol x P with power P P to PU with a fixed information rate R P . The sampled matched filter output at PU is where h PP ∼ CN (0, 1/λ P ), λ P = d ς PP with d PP denoting the normalized distance between PT and PU and w P ∼ CN (0, σ 2 ) is the additive noise at PU. The SINR Γ P at PU to decode x P is given by The outage probability at PU can be expressed as where γ P = 2 R P − 1 is the threshold SNR at PU. Now p o P is equated to a desired outage of p o,t P at PU. It can be shown that I P is then given by The PU broadcast this quantity very infrequently (not in every coherence interval) and is stored by all secondary nodes. In this Subsection, we derive the performance of the proposed system model with this value of I P . 1) U 2 Outage Probability: U 2 is in non-outage when: a) R decodes both x 2 and x 1 (Γ I R2 ≥ γ 2 , Γ I R1 ≥ γ 1 ), and U 2 decodes x 2 successfully after combining the direct and relayed signal ( , and U 2 decodes x 2 successfully after combining (Γ II 2−f + Γ I 2 ≥ γ 2 ), and c) R fails to decode both x 1 and x 2 (Γ I R2 < γ 2 , Γ I R1 < γ 1 ) but U 2 decodes x 2 only from the first phase signal (Γ I 2 ≥ γ 2 ). Clearly, the outage probability p o 2−SI 5 of U 2 is given by Theorem 1: The closed-form expression for the outage probability of U 2 with imperfect SIC is given by where the respective expressions for A 1−SI , A 2−SI and A 3−SI are as in (23), (24) and (25), respectively, shown at the bottom of the page. Proof: To obtain p o 2−SI , we solve for term A 1−SI , A 2−SI and A 3−SI given in (21). Please Refer to Appendix A for the detailed proof.
Corollary 1: The outage probability of U 2 for CDRT-NOMA without S − U 2 direct link can be readily derived by substituting T = 0 in (6) (which implies Γ I 2 = 0 in (21)) and then using a procedure similar to that used to derive (22). The closed-form expression for p o 2−SI with imperfect SIC is obtained as follows: Corollary 2: The outage probability of U 2 for conventional NOMA in which S communicates to U 1 and U 2 only 5 The subscript −SI is used to represent the case of static ITL.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. over the direct links (which is a special case of proposed CDRT-NOMA, when R is absent) can be readily derived by substituting X = 0 in (7) and (8) and Z = 0 in (13) and (16), which implies that Γ I R2 = 0, Γ I R1 = 0, Γ II 2−S = 0 and Γ II 2−f = 0 in (21). Thereafter, using a procedure similar to that used to derive (22), the closed-form expression for p o 2−SI is obtained as follows: The expressions obtained in (22) involve the generalized hypergeometric function due to which the diversity cannot readily be established from them. We, therefore, present a high-SNR approximate expression to p o 2−SI in the following corollary.
Corollary 3: The asymptotic expression of the U 2 outage probability p o 2−SI for SI case can be obtained by substituting I P → ∞ into (22) followed by linear approximations to exponential terms. After neglecting higher order terms of 1 I P , the outage probability of U 2 can be expressed as Remark 1: The diversity order D 2 achieved at U 2 in the cooperative case (I P → ∞) is 2, which can be easily obtained using D 2 = − lim 2−SI is given by (28).
2) U 1 Outage Probability: In the CDRT scheme of this paper, we ensure that U 1 enhances its performance at each stage of SIC using the signal transmitted by R. Clearly, U 1 will be in non-outage when a) R decodes both symbols (Γ I R2 ≥ γ 2 , Γ I R1 ≥ γ 1 ) correctly, and after combining the direct and relayed signals, both the symbols are decoded correctly at U 1 , and c) R fails to decode both symbols (Γ I R2 < γ 2 ), but U 1 decodes successfully based only on the signal from S (Γ I 12 ≥ γ 2 , Γ I 11 ≥ γ 1 ). The outage probability of U 1 is defined as: Theorem 2: The closed-form expression for the outage probability of U 1 with imperfect SIC 6 is given by where the closed-form expressions for B 1−SI , B 2−SI and B 3−SI are one by one given below where and I 2 , I 3 , I 4 and I 5 are given in (33), (34), (35) and (36), respectively, shown at the bottom of the next page, and Proof: To obtain the closed-form expression of p o 2−SI , we solve for the terms B 1−SI , B 2−SI and B 3−SI in (29). For the detailed derivation of B 1−SI in terms of I 1 , I 2 , I 3 , I 4 and I 5 , please refer to Appendix B, and for the derivation of B 2−SI and B 3−SI , please refer to Appendix C.
Corollary 4: Relayed-NOMA is a special case of the proposed CDRT-NOMA in which the direct S-U 1 and S-U 2 links are both shadowed. The outage probability of U 1 can be readily derived by substituting W = 0 in (4) and (5), which results in Γ I 12 = 0 and Γ I 11 = 0 (no combining at U 1 ). It can be seen that p o 1−SI in (29) becomes Further, using a procedure similar to that used to derive (30), a closed-form expression for p o 1−SI with imperfect SIC is obtained as follows: 6 The expressions for the outage probability of U 1 and U 2 for the case of SI with perfect SIC can be readily obtained by substituting I 1 = II 1 = I R = 0 into (22) and (30), respectively.
Note that the outage probability of U 2 for the case of relayed-NOMA is given by (26).
Corollary 5: The outage probability of U 2 for conventional NOMA can be readily derived by substituting X = 0 in (7) and (8) and Y = 0 in (11) and (12), which implies that Γ I R2 = 0, Γ I R1 = 0, Γ II 12−S = 0 and Γ II 11−S = 0 (there will be no combining at U 1 ) in (29). With this, p o 1−SI becomes Further, using a procedure similar to that used to derive (30), a closed-form expression for p o 1−SI with imperfect SIC is obtained as follows: In the following corollary, we present an asymptotic expression for p o 1−SI to obtain the diversity order at U 1 . Corollary 6: The asymptotic expression of the U 1 outage probability p o 1−SI for SI case can be obtained by substituting I P → ∞ into (30) followed by linear approximations to exponential terms. After neglecting higher order terms of 1 I P , the outage probability of U 1 can be expressed as Remark 2: The diversity orders D 1 attained by U 1 in the cooperative case (I P → ∞) can be expressed as which can be readily obtained using D 1 = − lim 1−SI is given by (44). From the above, it is clear that judicious choice of α S is essential to attain the maximum diversity of order 2.

B. Channel State Information Dependent ITL (CI Scheme)
In Section III-A, the ITL I P was a fixed quantity. For any desired outage of the primary network, the SINR at PU is larger than the threshold with a very high probability. If PU can compute and broadcast the ITL in every coherence interval, much better secondary performance can be achieved, especially because the NOMA scheme suggested here is specifically designed to work with low source transmit powers and is suited for underlay CR. Equating the primary SINR expression in (18) to the threshold SNR γ P , the expression of I P in the CI case can be expressed as Note that, Γ p < γ P is same as I P < 0, the outage probability of PU becomes The ITL parameter I P is now a random variable, which can be clearly seen from (46). The PDF of I P is obtained as (max( ψ4, ψ4) − min(ψ1, ψ1, β1)) (λv + λw min(ψ1, ψ1, β1) + λx max(ψ1, ψ2))(λv + λw max( ψ4, ψ4) + λx max(ψ1, ψ2)) Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Using the expression of I P given in (46) and its PDF given in (48), we derive the outage probability p o 1−CI of U 1 and p o 2−CI of U 2 for the CI case 7 in this Subsection. Theorem 3: The closed-form expression for the outage probability of U 2 with imperfect SIC in the CI case is given by where A 1−CI , A 2−CI and A 3−CI can be obtained after averaging A 1−SI in (23), A 2−SI in (24) and A 3−SI in (25), respectively, over the PDF of I P which is given by (48). Proof: To find the outage probability of U 2 for CSIdependent I P case, we first average A 3−SI in (25) over the PDF of I P in (48) to get where Using partial fraction, the above can be rewritten as Using relation [39, 3. 352. 1], A 3−CI is obtained as Similarly to evaluate A 1−CI and A 2−CI , we average A 1−SI in (23) and A 2−SI in (24) over the PDF of I P followed by some mathematical rearrangements along with using the relation [39, 3. 352. 4]. Next, we use Gauss-Laguerre quadrature approximation to obtain the expression of A 1−CI and A 2−CI as follows The subscript −CI is used to denote the case of CSI-dependent ITL. where and C 5 = γ 2 λz λu , and I A 1−CI and I A 2−CI are given by (55) and (56), respectively, shown at the bottom of the next page.
Proof: To find the outage probability of U 1 for CSIdependent I P case, we first average B 3−CI in (40) over the PDF of I P (after substituting for ψ 4 and ψ 1 ) as where ξ 3 = λw λv max( γ 2 χ 1 , γ 1 χ 3 ) and ξ 4 = ξ 3 + λx γ 2 λv χ 1 . Solving the above integrals using relation [39, 3.352.1], B 3−CI is obtained as 8 The expressions for the outage probability of U 1 and U 2 for the case of CI with perfect SIC can be readily obtained by substituting I 1 = II 1 = I R = 0 into (49) and (59), respectively.
Corollary 9: For relayed-NOMA, the outage probability of U 1 for the CI scheme can be derived by averaging (41) over the PDF of I P . Further, using the relation [39, 3.352.4] followed by mathematical simplifications the closed-form expression for p o 1−CI with imperfect SIC is obtained as follows: Corollary 10: For conventional NOMA, the outage probability of U 1 with the CI scheme is derived by averaging (43) over the PDF of I P . Using the relation [39, 3.352.4] followed by mathematical simplifications, the closed-form expression for p o 1−CI with imperfect SIC is obtained as follows: where ζ 5 = max( γ 2 χ 1 , γ 1 χ 3 ).

IV. U 1 THROUGHPUT MAXIMIZATION
Increasing α S from 0 initially improves the performance of U 1 and as well as decodability at the relay, but a larger α S causes failure of SIC at both R and U 1 (and thereby poorer performance at U 1 ). In this CDRT network, where the signal from R is combined at U 1 , a careful choice of the power allocation parameter α S at S is important to ensure good secondary performance. In comparison, the choice of power allocation α R is less crucial. Performance of U 1 continuously increases with an increase in α R , while that of U 2 decreases. For this reason, in this paper, we choose α R to ensure desired performance at U 2 and then choose α S to maximize the performance of U 1 .

A. Statistical Properties Dependent ITL (SI)
Throughput at U 1 (τ 1−SI ) and U 2 (τ 2−SI ) for the case of SI are given by where p o 2−SI and p o 1−SI are defined in (22) (in (26) when there is no direct link to U 2 ) and (30), respectively. Since the expression of p o 2−SI in (22) is quite mathematically involved, it is extremely difficult to obtain the optimum value of α S to maximize U 1 throughput while ensuring a desired target U 2 throughput. However, in the absence of S-U 2 direct link, using (65) and fixing τ 2−SI = τ , the closed-form expression for the optimum value of power allocation factor α o S in terms of τ is obtained as in (66) and (67), where κ 1 = λu I P χ 2 +λz γ 2 . The derivation is tedious but relatively straightforward, and is therefore omitted.

B. Channel State Information Dependent ITL (CI)
Similar to the SI case, throughput at U 1 and U 2 for the CSI-dependent I P is defined as where p o 2−CI and p o 1−CI are defined in (49) (in (57) when there is no direct link to U 2 ) and (59), respectively. Since the expression of p o 2−SI in (49) is quite mathematically involved, it is extremely difficult to obtain the optimum value of α S to maximize U 1 throughput while ensuring a desired target U 2 throughput. However, in the absence of S-U 2 direct link, we determine an approximate expression for the outage probability of U 2 to determine the optimum value of α S . Using relation [40, 5.1.19] (57), the approximate outage probability of U 2 for the case of CI is The above is valid for α S ≤ Ω 3 . It has been verified that this approximation matches closely with the exact analysis for all range of parameter values. Substituting (69) in (68) and fixing τ 2−CI = τ , the closed-form expression for the optimum value of power allocation factor α o S is obtained for CSI-dependent I P as given in (70) and (71), as shown at the bottom of the next page, where κ = R 2 2 , η 5 = ζ 2 2 1+ζ 2 δ and η 6 = λx λv . Assuming a certain desired FU throughput, we obtain α S in (66)-(67) and (70)-(71) respectively for SI and CI schemes when there is no direct link to U 2 . Using these values of α S in the expression of NU throughput, it can be inferred that NU throughput completely depends on the choice of target rates R 1 and R 2 . Therefore, the optimization problem can be formulated as follows: Solving the above joint optimization problem analytically is difficult due to the complex nature of the resulting mathematical expressions. However, numerical techniques can readily be used to pick the optimal R N and R F . It can be clearly seen from (65) and (68) that the choices of target symbol rates R 1 and R 2 along with power allocation coefficient α S is crucial in attaining superior throughput at both U 1 and U 2 . We first note that τ 1−SI and τ 1−CI are both concave functions of α S . Intuitively, this is because when α S is small, the U 2 symbol cannot be decoded by U 1 (and throughput should therefore be optimized hence U 1 symbols SI Scheme Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. detection fails), and when α S is close to 1, the U 1 symbol cannot be decoded. For any specified τ that is smaller than the maximum U 2 throughput, α S can take values only between α min S and α max S . Denote this range of values by B. The FU throughput should therefore be optimized for α S ∈ B. Therefore, the optimization problem is formulated as Solving the above joint optimization problem analytically is difficult due to the complex nature of the resulting mathematical expressions. However, numerical techniques can readily be used to pick the optimal R N and R F and α S . The sum throughput with SI and CI schemes can be expressed as where k ∈ {SI , CI }, and the expression for τ 1−SI , τ 2−SI , τ 1−CI and τ 2−SI are as in (65) and (68), respectively.

V. SIMULATION RESULTS
In this Section, computer simulation results are presented to verify the accuracy of derived analytical expressions. We assume the following system parameters: [36]. The noise variance is normalized to unity [14]. Unless mentioned otherwise we consider R 1 = R 2 = R [24]. Fig. 2 plots the outage probability p o 2−SI vs. I P for various values of α S and α R with R = 0.5, 1 bits per channel use (bpcu). We present a comparison of the performance of three different schemes, namely CDRT-NOMA with S-U 2 link, CDRT-NOMA without S-U 2 link, and conventional NOMA. Clearly, the analytical expressions for the outage probability of U 2 for CDRT-NOMA with S-U 2 link in (22), CDRT-NOMA without S-U 2 link in (26), and conventional NOMA (27) are validated by numerical simulations. In all three cases, with increasing I P , p o 2−SI decreases. Further, it can be clearly seen that for high target rates requirements, the outage probability degrades. Also, the choice of power allocation factor at S as well as R majorly affects p o 2−SI . In CDRT-NOMA with S-U 2 link, p o 2−SI is mainly affected by the power allocation at S (this is because of the optimal combining of incoming symbols from the direct and the relayed links at U 1 and U 2 ). In contrast, when the S-U 2 link is absent, the performance of U 2 depends solely on the relayed link, and the outage probability depends on power allocation at R. Also, CDRT-NOMA with S-U 2 link performs better, followed by CDRT-NOMA without S-U 2 link and conventional NOMA. Further, it can be clearly seen from Fig. 2 that the SIC error doesn't affect the performance of U 2 event after considering 30% of error. Fig. 3 depicts the variation p o 1−SI vs. I P for different combinations of α S and α R with R = 0.5, 1 bpcu. Clearly, the analytical expressions for the outage probability of U 1 for CDRT-NOMA with S-U 2 link in (30), relayed-NOMA in (41), and conventional NOMA (43) are validated by numerical simulations. As expected, the outage decreases with the increasing I P . As in Fig. 2, the outage probability with CDRT-NOMA with S-U 2 link is lower in comparison to the other two cases. However, conventional NOMA performs better than that of relayed-NOMA here (this is because U 1 is nearer to the BS in comparison to the relay (d S 1 < d SR )). The effect of SIC error on the outage performance of NU and FU is also shown in Fig. 3. It is clear that the SIC error at the relay ( I R ) as well CI scheme α o S = −4δγ 2 1 η 2 6 τ (−δe −δσ 2 κη 5 + e −δσ 2 κζ 2 − τζ 2 ) + (−e −δσ 2 γ 1 κη 6 − δ 2 e −δσ 2 γ 1 κη 5 η 6 + γ 1 η 6 τ + δe −δσ 2 γ 1 κη 6 ζ 2 − δγ 1 η 6 τζ 2 ) 2  as at U 1 ( I 1 & II 1 ) affect the U 1 outage performance. However, the effect of SIC error is not significant (even at 10% of SIC error), and we have therefore considered I 1 = II 1 = I R = 0 in all the remaining figures.
In Fig. 4, we plot the sum throughput vs. I P for different combinations of the power allocation factors α S and α R with R = 1, 1.5. We present a comparison of the proposed CDRT-NOMA with S-U 2 direct link with relayed-NOMA [18] where only the relay node assists in information transmission to both the users. It can be clearly seen that for high values of I P , the throughput saturates at the maximum target information rate. We also observed that CDRT-NOMA always outperforms relayed-NOMA with a huge margin. This is due to the presence of a direct link to both the users. Fig. 5 shows the variation of U 1 throughput vs. α S for both SI and CI schemes at different target information rates. It can be clearly seen from Fig. 5 that increasing the target information rate increases the throughput but narrows down the valid range of the power allocation coefficient. It is also clear that the U 1 throughput first increases with α S and then decreases. We observe that U 1 attains a maximum value at α S = γ 1 γ 1 (1 + γ 2 ) + γ 2 (α S = 0.33 for R = 1, α S = 0.26 for R = 1.5, and α S = 0.2 for R = 2). Beyond this value of α S , both NU and FU symbols cannot be decoded at U 1 and R. The sharp decrease beyond this value is a natural consequence of the change in relay transmission strategy and combining technique at U 1 when both symbols cannot be decoded. Note that throughput is 0 for α S ≥ 1 1 + γ 2 (α S = 0.5 for R = 1, α S = 0.3535 for R = 1.5, and α S = 0.25 for R = 2). This is due to the fact that when α S ≥ 1 1 + γ 2 , SIC fails at U 1 and hence U 1 fails to decode its own symbol. While the range of α S remains the same, notice that the performance with the CI scheme does not change much in the range of permissible α S values (which implies that a larger power allocation to U 2 is possible, and therefore larger throughput at U 2 ). Fig. 6 depicts U 2 throughput vs. α S for both SI and CI schemes for different target information rates. U 2 attains the maximum throughput at α S = 0 for both SI and CI schemes. For increasing α S , U 2 throughput decreases and becomes zero for α S ≥ 1 − γ 1 1+γ 2 (as shown analytically). 9 Clearly, the CI scheme provides a huge throughput gain in comparison to the SI scheme. Also, for the CI scheme, U 2 throughput is 9 Since α S depends on choice of target rates and share a relation α S ≥ 1 − . Clearly, a decrease in target information rates results in a wider range of valid power allocation which can also be inferred from Fig. 5 and Fig. 6.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  insensitive to α S in the range α S < 1 − γ 1 1+γ 2 whereas with the SI scheme, it decreases with an increase in α S . Therefore, the CI scheme requires lower transmit power in comparison to the SI scheme. In Fig. 7, we plot the throughput of both U 1 and U 2 vs. α S for higher target rates (R = 3, 4). Clearly, the CI scheme provides a huge throughput gain over the SI scheme. Unlike in the case of low target rates (in Fig. 5 and Fig. 6), significant variation in U 1 and U 2 throughput w.r.t α S with the CI scheme can be seen, and U 1 achieves maximum throughput at α S = Ω 1 and becomes 0 for α S ≥ Ω 2 . Fig. 8 illustrates the variation of both U 1 and U 2 throughput vs. α R for both SI and CI schemes. Clearly, for the CI scheme, the choice of α R has an almost negligible effect on the throughput performance of U 1 as well as U 2 . In contrast, for the SI scheme, the U 2 throughput depends on α R -it is maximum when α R is 0, decreases with the increase in α R , and becomes almost negligible beyond α R > Ω 2 (this is due to unsuccessful decoding of symbol x 2 at U 2 ). Fig. 9 plots the sum throughput vs. target rate of both CI and SI schemes. We also compare the same with relayed-NOMA, 10 OMA and its relayed-OMA counterpart. 11 Clearly, our proposed scheme outperforms both relayed-NOMA and its OMA counterpart by a huge margin for both SI and CI schemes. Also, relayed-NOMA outperforms relayed-OMA. The CI scheme with the proposed CDRT-NOMA performs best among all the schemes. We also observe that there exists an optimum target rate at which the sum throughput is maximum. Fig. 10 depicts the variation of U 1 throughput vs. R 1 for different choices of fixed target throughput at U 2 (τ ). We also show the variation for different arbitrary target rates R 2 and compare the same with the optimal choice of R 2 , i.e., R * 2 . We observe that optimal values of R 1 and R 2 (R * 1 and R * 2 ) exist at which U 1 throughput is maximum while ensuring the desired target throughput at U 2 . It is also observed that with increasing desired throughput requirement at U 2 , U 1 throughput decreases. 10 For the case of relayed-NOMA, the sum throughput is evaluated by using p o 2−SI and p o 1−SI in (26) and (41) into (74). 11 In the considered relayed-OMA system with direct link to both U 1 and U 2 , S transmits x 2 to U 1 , U 2 and R in the first signalling phase. In the second signalling phase, R forwards x 2 , while S transmits x 1 to U 1 . Using the decoded x 2 in the first phase, U 1 cancels the interference caused by transmission by R before decoding, and x 1 transmitted from S acts as interference at U 2 . However, if U 1 fails to decode x 2 in the first phase, then the signal received from the R in the second phase acts as interference at U 1 .   Fig. 11 depict user fairness 12 versus the target rate of users. User fairness depends on the choice of target rate and NOMA power allocation factor. It is clear from the figure that the proposed scheme is very much fair for the lower values of the target rate. It is also clear that the fairness decreases with increasing α S . On the other hand, the proposed CI scheme 12 In NOMA signalling, due to apportioning of power between users, the user performance differs and, a user may have better performance in comparison to others. To maintain user fairness, the performance gap among users should not be large. Similar to [41], we also define the proportional fairness in terms of the ratio of FU to NU throughput as results in better fairness in comparison to the SI scheme for every value of α S .

VI. CONCLUSION
In underlay networks, the random nature of the transmit power limits the performance that can be attained. Despite this, it was shown in this paper that good performance can be attained when signalling to two users through a relay (the CDRT-NOMA scheme) provided the combining of the direct and relayed signals is employed at both the users, and not just at the distant user. Analysis of performance was presented for both users assuming imperfect successive interference cancellation (SIC). The case of static interference temperature limit (ITL) was considered, and the gain in throughput compared to conventional CDRT was shown to be large. Then, it was shown that when primary channel variations are also exploited, performance gain can be substantial. It was shown how such a network can be optimized by NOMA power allocation and careful selection of the target rates.
Extension of the analysis to the scenario of multiple users in the secondary network and multiple transmit antennas is an open problem for future work.

APPENDIX A
To evaluate the outage probability of U 2 , we derive A 1−si , A 2−si and A 3−si one by one as follows: Evaluation of A 1−si : We evaluate the term A 1−SI of (21) using (6), (8), (7) and (13) as follows: After some mathematical manipulations, conditioned on V, A 1−SI | V is given by The above is due to independence of RV X, U, and Z. A 1−SI | V can be rewritten as where ψ 3 = max(ψ 1 , ψ 2 ) and ψ 2 = γ 1 χ 4 I P . After averaging the above equation over the PDF of RV V, we obtain On evaluating the above integrals, A 1−SI can be rewritten as After some mathematical manipulations, we obtain ). Using relation [39, 3.352.1], we obtain After some mathematical rearrangements, the above equation can be rewritten as Solving the above using relation [42, 4.2.11], we obtain the closed-form expression of A 1−SI as in (23). Evaluation of A 2−si : Similar to A 1−SI , using (6), (8), (7) and (16), the expression for A 2−SI in (21) is given by After some mathematical rearrangements and using the independence of RVs X, U, and Z, A 2−SI | V is expressed as The above can be written in the integral form as follows: where ψ 2 = γ 1 χ 4 I P . The above equation is averaged over the PDF of RV V to obtain The above is valid if ψ 2 > ψ 1 (α S < γ 1 (1+γ 2 I R ) γ 1 +γ 2 (1+γ 1 +γ 1 I R ) ), otherwise, A 2−SI will be 0. Solving the above integrals, we obtain Using partial fraction expansions, we obtain where On solving, the above can be rewritten as Further, using [42, 4.2.21], finally A 2−SI is obtained as in (24). Evaluation of A 3−SI : Using (6) and (8), the expression for A 3−SI in (21) can be expressed as Further, we average over RV V to get A 3−SI as APPENDIX B Based on the conditions in (29), the obtained ranges of RV W, X, and Y for different α S are given in Table III. Using (4), (5), (7), (8), (11) and (12), B 1−SI in (29) can be expressed as From the above equation, the conditions obtained over RVs X, Y, W, U, and V for different ranges of α S are listed in Table III, where Further, we use conditions given in Table III and performing mathematical rearrangements to obtain B 1−SI as where terms I 1 , I 2 and I 3 correspond to first second and third row of B 1−SI of Table III, and I 4 , I 5 correspond to the fourth row. To obtain the closed-form expression of B 1−SI , we solve for I 1 , I 2 , I 3 , I 4 and I 5 one by one as follows: Evaluation of I 1 : From Table III, it can be clearly seen that On evaluating the integrals, I 1 is obtained as Evaluation of I 2 : Using Table III, I 2 in (91) can be expressed as Substituting t = (w − ψ 1 v ) and using the standard integral [39, 3. 352. 1], we obtain Next, we use relation [42, 4.2.11] and perform mathematical manipulation to obtain the closed-form expression of I 2 as in (33).
Evaluation of I 3 : Similar to I 2 , I 3 can be expressed as On solving the above integral using relation [42, 4.2.11], a final closed-form expression for I 3 is obtained as in (34).