Impact of Hardware Impairment on the Uplink SIMO Cooperative NOMA With Selection Relay Under Imperfect CSI

Non-orthogonal multiple access (NOMA) has emerged as a promising solution for enabling massive connectivity in future wireless networks. A great deal of research has extensively considered the implementation of the NOMA with other technologies such as multi-antenna and cooperative communications. Most of the previous studies focused on investigating the downlink NOMA networks, while the uplink papers received relatively less attention. Besides, the majority of the current studies on uplink NOMA schemes have neglected practical limitations. Motivated by this, we investigate the uplink single-input multiple-output cooperative NOMA (SIMO-CNOMA) performance in the presence of hardware impairment (HWI), imperfect channel state information (ipCSI), and imperfect successive interference cancellation (ipSIC). We expand the scope of the proposed system to encompass multiple relay schemes, within which the selection relay technique is executed. We derive the outage probability (OP) and the system throughput of the considered system over the Rayleigh fading channels. In this respect, we provide performance analysis for different combinations of relays and antennas. The OP and system throughput analytical expressions are validated through computer simulations. In addition, the effects of HWI, ipCSI, and ipSIC on the performance of the uplink SIMO-CNOMA are discussed. The results show that the system’s performance can be improved with an increase in the relays and antenna numbers. Also, there is an error floor at the high signal-to-noise ratio (SNR) in ideal conditions (i.e., in the absence of HWI, ipCSI, and ipSIC), and the error floor is increased in non-ideal conditions (i.e., in the presence of HWI, ipCSI, and ipSIC). The arbitrary number of users in the presence of non-ideal conditions increases the error and reduces the system’s performance. Finally, although the performance gains obtained through the use of multiple antennas and relays, the adverse effects of HWI, ipCSI, and ipSIC on system performance cannot be avoided.


I. INTRODUCTION
Meeting the demands of the 6th-generation (6G) and beyond wireless communication networks, which support a vast The associate editor coordinating the review of this manuscript and approving it for publication was Stefan Schwarz .number of connected devices with low communication latencies and improved data rates, poses a significant challenge.To address this, researchers are exploring new communication technologies and system designs.Among these technologies, non-orthogonal multiple access (NOMA) has captured the attention of the research community as a promising solution to meet the requirements of 6G networks [1], [2].The NOMA is based on the concept of utilizing the power domain for multiple access, which enables multiple users to be multiplexed at the same time, frequency, or code resources but at different power levels.This technique employs successive interference cancellation (SIC) to separate overlapping messages at the receiver, resulting in significant improvements in both spectrum efficiency and user fairness compared to the conventional orthogonal multiple access (OMA) [3], [4], [5].The evaluation of performance in uplink NOMA systems is extensively discussed in [6], [7], [8], and [9], where metrics such as outage probability (OP) and ergodic capacity (EC) are commonly utilized, taking into account both perfect and imperfect channel state information (CSI).
On the other hand, cooperative communication is a successful strategy for addressing issues related to multipath propagation, extending coverage, and improving the reliability of communication systems [10].A combination of cooperative communication and NOMA has resulted in the development of cooperative NOMA (CNOMA) systems, which have garnered significant attention in recent papers.The OP and throughput performance of the downlink CNOMA over the Nakagami-m fading channels are evaluated in [11].The authors in [12] evaluate the OP performance of CNOMA over the Rayleigh fading.The authors in [13] examine the effectiveness of hybrid downlink-uplink NOMA cooperative relay networks by analyzing the OP, diversity order, and throughput.In [14], underlay cognitive hybrid satellite-terrestrial for CNOMA has been analyzed for OP over Rician fading and Nakagami-m fading.Furthermore, multi-relay in NOMA systems can improve wireless network coverage, capacity, reliability, and interference reduction.These systems also enhance network resilience through redundancy and diversity.The effect of the relay selection on the performance of CNOMA is investigated in [15].The authors of [16] present the performance of the NOMA system with the aid of multiple relays where two-stage relay selection with decode-and-forward (DF) and amplifyand-forward (AF) relaying protocols.The performance of the multiple relays assisted NOMA over the Nakagami-m fading channels is investigated in [17].In addition, advanced approaches related to NOMA are being considered through the adoption of emerging transmission techniques, such as full-duplex communications and multi-antenna systems.The authors in [18] explore the use of NOMA with a multiple-antenna AF relaying network as a means of enhancing system performance.The performance of a twouser with simultaneous wireless information and power transfer (SWIPT) of the cooperative multiple-input singleoutput (MISO) NOMA system is examined in [19].In [20], the authors study the performance of the uplink singleinput multiple-output (SIMO) NOMA with joint maximum likelihood detector in terms of bit error rate.
The aforementioned studies assume ideal hardware, perfect CSI, and SIC.In realistic communication systems, the hardware transceivers suffer from several imperfections due to in-phase/quadrature (I/Q) imbalance in modulators, phase noise in local oscillators, and non-linearity power amplifiers [21], [22], [23].The effect of the hardware impairment (HWI) on the NOMA-based relay systems over the Nakagami-m channels in terms of OP and ergodic sum rate performance is evaluated in [24] and [25].The BER and OP performance of the CNOMA under practical assumptions where the imperfect SIC (ipSIC), imperfect channel state information (ipCSI), and hardware impairments (HWI) are investigated in [26].In [27] and [28], downlink NOMA with cognitive radioassisted satellite-terrestrial relay networks under joint effects of channel estimation errors and HWI has been investigated in terms of secrecy OP.In [29], the OP and intercept probability of cognitive ambient backscatter downlink NOMA internet-of-vehicle enabled maritime transportation systems communication with HWI has been examined.The ambient backscatter-downlink NOMA communication with both IQI and residual HWI has been analyzed in terms of OP and ergodic rate [30].The authors of [22] and [31] evaluate the performance of OP and system throughput of downlink and uplink CNOMA with IpCSI and IQI.
As previously mentioned, considerable attention has been given to the downlink CNOMA system with multiple relays or/and antennas in previous literature, as in [10], [11], [12], [13], [14], [15], [16], [17], [18], and [19].However, the uplink NOMA and CNOMA system has not garnered an equivalent level of focus.The evaluation of the uplink NOMA with and without ipCSI has been analyzed in [6], [7], [8], and [9].Moreover, multi-antenna technology plays a crucial role in enhancing the performance of NOMA and CNOMA schemes.These schemes have been predominantly investigated for downlink improvements [19], [20], while their potential in the uplink remains absent.Also, most papers consider the effect of HWI for downlink NOMA and CNOMA schemes [27], [28], [29], [30], while it is missing in the uplink NOMA and CNOMA schemes.To the best of the authors' knowledge, the investigation of uplink SIMO-CNOMA, both with and without multiple relays has not been investigated.Besides, the practical constraints have been neglected in all previous studies of the uplink of NOMA and CNOMA.Therefore, it is important to consider the practical constraints in the uplink CNOMA for realistic evaluations.Motivated by this, the performance of the uplink SIMO-CNOMA with the aid of multiple relays and an arbitrary number of users is studied in this paper in the presence of HWI, ipCSI, and ipSIC.We analyze the OP and system throughput of our considered system.
Accordingly, the main contributions of this work can be summarized as follows: • We examine a practical multi-user uplink NOMA scheme affected by HWI, ipCSI, and ipSIC.Our analysis includes the multi-relay scenario with the selection relay (SR) technique and explores the use of SIMO to enhance performance.The transmitter has a single antenna, while the receiver uses multiple antennas during the cooperative scheme's two transmission phases.
• We derive the OP and system throughput of our proposed system with multiple relays and SIMO.We take into account practical constraints such as HWI, ipCSI, and ipSIC.The analytical expressions are validated by computer simulations.An asymptotic analysis of the system's performance is examined in high SNR regions.The objective of this analysis is to uncover valuable insights regarding the effects of HWI, ipCSI, and ipSIC parameters on the system's performance.The results reveal an impact of HWI, ipCSI, and ipSIC parameters on the system's performance in high SNR regions.
• The impact of the system parameters, such as HWI, ipCSI, relay number, and the number of antennas on the OP and system throughput performance of the considered schemes are extensively investigated.Numerical results show that the systems are limited by the effects of HWI and ipCSI which have a degradation on the system performance despite increases in the number of antennas and relays.The rest of the paper is organized as follows.The proposed system is introduced in Section II.In Section III, we analyze the OP and system throughput expressions.The simulation results are presented in Section IV.Finally, Section V concludes the paper.

II. SYSTEM MODEL
We consider an uplink CNOMA system consisting of an access point (AP), J users, and L relays denoted as {U j , j ∈ [1, J ]} and {R l ∈ [1, R L ]}, respectively, as presented in Fig. 1.It is assumed that the AP and relays are equipped with N r receive antennas given by {m ∈ [1, N r ]}.The SR is supposed to be used in our system as in [32], where the selected relay denotes as R l .The direct links between the users and the AP are neglected due to the obstacles, so the AP receives the users' signals with the help of relays.The complex Rayleigh channel coefficients between the users-relays and relays-AP are presented as h R l are distances between related nodes and µ is the path loss exponent.We assume that CSI between all nodes is imperfect and relay nodes work in half-duplex (HD) mode.The estimated channel coefficients are given as h{m,R l } j = h {m,R l } j − e and g{m,R l } = g {m,R l } − e, where e is the channel estimation error as in [33].and e ∼ CN (0, σ 2 ϵ ).Based on the uplink CNOMA scheme, in the first phase, the users transmit their signals simultaneously to the selected relay R l .Hence, the received signal at relay l in the presence of HWI and ipCSI can be expressed as where P is the total transmit power, α j is power allocation (PA) coefficients in which α j < . . .< α 3 < α 2 < α 1 and j α j = 1, n is the additive white Gaussian noise (AWGN) n ∼ CN (0, σ 2 n ), x j is the users signal with E[|x j | 2 ] = 1, where E[.] is the expectation operator.
Since R l node has N r antennas at the receiver, we suppose that the selection combining (SC) technique 1 is used to select the best-received signal.Thus, it is assumed the SC is implemented based on the received signal-to-interference plus noise ratios (SINR) for each signal i.e., the SIC for each antenna is performed then the SC is used to select a better received SINR.In general, the R l node decodes the received signals using the SIC process, where the signal with a higher power (i.e., the signal of U 1 ) is decoded first then the signal with lower power (i.e., the signal of U 2 ), and so on until the last signal (i.e., the signal of U j ).The SIC is supposed to be imperfect in this paper.The received SINRs for the signal transmitted by the U j to the R l in the first phase can be defined as in (2), as shown at the bottom of the next page.In which R l re-combines the decoded signals of the first phase in new superimposed coding signals with different power allocations according to the channel gain as in the first phase (we assume the distribution of power in the second phase will be the same as the first phase, i.e., α j < . . .< α 3 < α 2 < α 1 , where j α j = 1) and forward it to AP in the second phase.The received signal in the presence of HWI and ipCSI at the 1 As it is known in the diversity combining techniques the maximal ratio combining (MRC) is better than SC.In the uplink CNOMA, the received signal consists of different signals coming from various sources and channels.In this case, the implementation of SC at the receiver requires less computational complexity compared to MRC to evaluate the system's performance.In the SC, the receiver chooses the signal with the highest signal-to-noise ratio (SNR) and discards the others, while in MRC, all signals are combined.Both of them are part of the diversity combining technique, which is utilized to increase the reliability of wireless communication systems.Thus, to evaluate the system performance of this paper and for less complexity, we use the SC techniques.
106708 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

AP can be given as
where are the distortion term of the HWI at the first and the second phase which are defined respectively as in [26] with AP decodes the signals using the SIC process, where a higher strength signal is decoded first then a lower strength signal, and so on until the last signal (i.e., the U 1 is decoded first then U 2 and so on until U j ).In SR schemes, only one relay will be allowed to forward the signal.If relay l is selected, it will transmit the signal to the AP in the second phase.The relay whose path has the maximum SINR is selected.Since the receiver at AP has multiple antennas, it is assumed that the SC is used to select the best-received signal based on the highest SINR.Hence, the SINR at AP in the second phase can be given in ( 4), as shown at the bottom of the next page.In which At the receiver of R l and AP, the SC technique is used to select the best-received signal according to the highest received SINR.It is supposed also the SR technology is adopted, so only one path U j -R l and R l -AP will be allowed to select, i.e., only one relay will be allowed to forward the signal to the AP.Hence, the SINR of the optimal SR for U j based on max-min relay selection criteria as in [34] can be written as with

III. PERFORMANCE ANALYSIS
In this section, we derive the OP and system throughput expressions of the considered system under HWI, ipCSI, and ipSIC over the Rayleigh fading channel.Thus, OP and system throughput of the considered system are derived in the below subsections, respectively.

A. OUTAGE PROBABILITY
In this subsection, we derive the OP expressions of the uplink SIMO-CNOMA assisted multiple relays with an arbitrary number of users in the presence of HWI, ipCSI, and ipSIC.To detect multiple signals in a specific order, R l uses SIC.The SIC starts with the signal that has the highest power and ends with the signal that has the lowest power.In the first step, R l uses maximum likelihood detection (MLD) to detect the U 1 signal.Then, the SIC is performed to detect the subsequent signals.Once the U 1 signal is detected, it is removed from the total received signal, and the same process is repeated to detect the next signal.This process is repeated for all the signals until the U j signal is detected.Thereafter, the R l re-encodes the detected signals in phase one and combines them in a new superimposed coding signal, and forwards it to AP.In this latter, the detection will be according to the power allocation of each signal, in which the signal of the highest power (signal of U 1 ) is detected first.Then, using the SIC detects the signals that have lower powers.Assuming that the receivers are equipped with multiple antennas while the transmitter has a single antenna, in the two phases as presented in Fig. 1.It is assumed that the SC technique is used at R l and AP to select the best-received signals according to their SINRs.Also, SR technology is used to choose the best signal transmission path according to the SINRs of each path as described in (5).The OP of each user is calculated in the subsubsections below.

1) CASE OF TWO USERS
• Outage probability of U 1 : The OP of the U 1 in the phase one and two are defined as the probability of the received SINR of the U 1 is less than the threshold γ th,1 , where γ th,1 = 2 2r 1 −1, r 1 is the target rate of x 1 .The end-2-end (e2e) OP of U 1 signal, when SR and SC are implemented under HWI and ipCSI is expressed as Each term of (8) can be computed as given in ( 9) and (10), as shown at the bottom of the next page, respectively, shown at the top of the next page.In which Thus, by substituting ( 9) and ( 10) into (8), we can obtain the OP of the U 1 as given in (11), as shown at the bottom of the next page.
• Outage probability of U 2 : The outage of the U 2 signal at R l and AP occurs if they cannot successfully decode the U 1 and U 2 signals.Thus, the e2e OP of the U 2 signal under HWI, ipCSI, and ipSIC can be written as in (12), as shown at the bottom of the next page.
Where γ th,2 = 2 2r 2 − 1, r 2 is the target rate of x 2 .Hence, we can re-write (12) as where P 2,I and P 2,II are the OP of U 2 in the first phases and the second phase, respectively, and Hence, each term of ( 14) can be computed as in ( 16) and (17), as shown at the bottom of the next page, as where By substituting ( 16) and ( 17) into ( 14), we get the OP of U 2 in the first phase as (18), as shown at the bottom of the next page.Likewise, each term in (15) can be calculated as in (19) and (20), as shown at the bottom of the next page.By substituting (19) and ( 20) into (15), we get the OP of the U 2 in the second phase as in (21), as shown at the bottom of the next page.To find the e2e OP of the U 2 , we substitute ( 18) and ( 21) into (13).

2) CASE OF ARBITRARY NUMBER OF USERS
In order to obtain the OP for U j (where 1 < j < J ), it is required to determine the probability that user U j will successfully detect both its own signal and the signals of users who are further away in the two phases (i.e., at R l and AP).In other words, the corresponding e2e OP of U j underHWI, 106710 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
ipCSI, and ipSIC can be computed as where γ th,j = 2 2r j − 1, r j is the target rate of x j .Hence, ( 22) can be re-written as where Pr Pr where P j,I (out) and P j,II (out) are the OP of U j in the first and the second phases, respectively.The term of ( 24) can be calculated as in (26), as shown at the bottom of the next page.
Proof: Please see Appendix A. The term of (25) can be calculated as given in (27), as shown at the bottom of the next page.By using the probability density functions (PDF) and cumulative distribution function (CDF) of Rayleigh fading, which are defined in [10], ( 27) can be obtained as in (28), as shown at the bottom of the next page.Thus, to find the OP of the first and the second phase of U j , we substitute ( 26) and ( 28) into ( 24) and ( 25), respectively.Finally, to obtain the e2e OP of U j we substitute (24) and ( 25) into (23).

B. ASYMPTOTIC OUTAGE PROBABILITY
In this subsection, we investigate the behavior of the asymptotic OP in high SNR regimes which can be defined as exp(−x) ≈ (1 − x) as in [35], and [26] to gain insights into our considered scenarios.

1) CASE OF TWO USERS
The asymptotic OP of the U 1 is given in (29), as shown at the bottom of the next page.In order to obtain the asymptotic OP of the U 2 , we need to find the OP of the first and second phases as given in (30), as shown at the bottom of the next page, and (31), as shown at the bottom of page 9. Replacing (30) and ( 31) into (13), we find the asymptotic OP of the U 2 .

2) CASE OF ARBITRARY NUMBER OF USERS
To obtain the asymptotic OP of U j , we need to find the asymptotic OP of the first and second phases as given in (32) and (33), as shown at the bottom of page 9. Substituting (32) and ( 33) into ( 24) and ( 25) to find the asymptotic OP of the first and second phases of the U j , respectively.Thus, the asymptotic OP of the U j can be expressed by replacing the asymptotic OP of the first and second phases into (23).

C. SYSTEM THROUGHPUT
In this subsection, we provide the analytical expression of the system throughput which is expressed as [22] where P j (out) is the e2e OP of U j .

IV. NUMERICAL RESULTS
In this section, the analytical OP and system throughput expressions are validated by computer simulations.Particularly, we investigate the impact of the HWI, ipCSI, and ipSIC on the uplink SIMO-CNOMA with SR and an arbitrary number of users.The simulation results in all figures match 106712 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Nr m=1
Pr 2 perfectly with the numerical results.To validate our analysis, the parameters are set as in Tab.1.
In Fig. 2, we present the OP performance of the uplink SIMO-CNOMA system with SR for two users under HWI, ipCSI, and ipSIC versus SNR with different numbers of relays and users when l = 1, 2, 3 and j = 1, 2, 3.The asymptotic OP curves given for the two users at the high SNR are limited over the theoretical curves.We observe that the OP performance of the users improves as the number of relays and antennas increases.There is an error floor at all curves of the users despite the increase in the number of relays and antennas.NOMA supports more users, so we evaluate the OP performance with more than two users.In Fig. 3, we present an evaluation of the OP performance of the uplink SIMO-CNOMA system with three users under HWI, ipCSI, and 106714 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

4.
OP versus SNR for uplink SIMO-CNOMA comparison with the ideal case.
ipSIC versus SNR with different numbers of relays and users when l = 1, 2, 3 and j = 1, 2, 3.The asymptotic OP curves for the three users are limited within the theoretical curves, specifically in high SNR scenarios.It is observed that the users' performance improves as the antenna and relay number increase.However, there is an error floor in the high SNR regime as in Fig. 2 (case of two users), which can be attributed to two primary factors.The first factor is the interference between the users' channels.The OP in the uplink CNOMA depends on the first and second phases.As we know, in the uplink CNOMA schemes in the first phase, all users transmit their signals with different power simultaneously to the R l node, so different signals are received at R l node from different users with different channel quality, and this causes interference between users channel which lead to errors in the signals detecting during the SIC process, especially at high SNR.Secondly, the imperfections in the system from HWI and ipCSI also contribute to the error floor.Through the comparison between Fig. 2 and 3, we observe that the OP with two users under the HWI, ipCSI, and ipSIC is better than the case of three users.In the case of three users, it can be easily seen that the performance of the U 3 was greatly affected than the other users, as it records a very poor performance in the presence of practical imperfections (HWI, ipCSI, and ipSIC).In these imperfections, regardless of increasing the number of relays and antennas, the performance of U 3 does not improve well.In the case of two users, the two users have good performance, and they become better when the number of relays and antennas increases in the presence of practical imperfections.The increasing number of users affects the distribution of power to the rest of the users, which affects the SIC process.The presence of practical imperfections with more users reduces the system's performance.Fig. 4 presents the OP of the uplink SIMO-CNOMA with two and three users under ideal and non-ideal conditions (HWI, ipCSI, and ipSIC) when l = 3 with two and three users.The performance of the users improves as the antenna number increases, and there is an error floor at the high SNR.In the non-ideal case, we observe that the presence of the HWI and ipCSI increases the error floor of the users' signals.However, the effect of HWI and ipCSI was bigger in the case of the three users than in the case of the two users, especially at the U 3 .As we know, in the uplink CNOMA scheme, the OP depends on the first and second phases.Thus, the error floor comes from the first or second phases or both.Through previous analysis in the literature, when there are no interference channels as in the downlink NOMA schemes [26], there is no error floor in the ideal case.Based on that the error floor comes from the first phase in the uplink SIMO-CNOMA, where multiple users with different channels are sent their signals simultaneously to R l node.The interference of channels occurs at the R l node which affects the SIC and causes an error floor on the users' signals with growth SNR.In the practical imperfections (HWI, ipCSI, and ipSIC) case, it can be seen that the OP performance of the system gets worse, and the U 3 has a poor OP performance that is close to 1 despite increasing antennas number.It proves the previous discussions in Fig. 2 and 3, that the imperfections of HWI and ipCSI contribute to the error floor in the system performance, and the more number of users affects the system performance of the users.
The impact of the channel estimation error and HWI factors on OP performance with SNR = 20 dB are evaluated in Figs.5(a) and 5(b), respectively when l = 3 and j = 3.We can observe that increasing the channel estimation error and HWI factors decreases the OP performance of all users.It can be easily seen that the HWI has less effect on the outage performance compared to the ipCSI effects with all antenna numbers.The U 3 diminishes very quickly with an increase ipCSI factor compared to the HWI factor although the increase in antenna number.This implies that the performance of the system depends more significantly on channel estimation error than on HWI.However, the presence of HWI and ipCSI affect the SIC and the detection signals.It is clear that the large number of users in the presence of practical imperfections will affect the performance of users more.Because it reduces the power allocation of users and causes complexity in the detection.Under these circumstances, with the effects of practical imperfections, the error will be greater, so it is recommended to use fewer number users to obtain better performance.This proves the validity of previous analyzes of NOMA schemes as [36] and [37] (downlink NOMA schemes), that a large number of users reduces performance and increases error.
In Fig. 6, we illustrate the system throughput of the uplink SIMO-CNOMA with different numbers of relays and antenna in the presence of HWI, ipcSI, and ipSIC w.r.t SNR when l = 1, 2, 3 and j = 1, 2. We can observe that by increasing the number of relays and antennas the system throughput improves.In another scenario, when there are more than two users, in Fig. 7, the system throughput of the uplink SIMO-CNOMA with SR in the presence of HWI, ipCSI, and ipSIC with a different number of relays and users when l = 1, 2, 3 and j = 1, 2, 3.It is observed that the throughput performance increases as the relay and antenna numbers increase.By comparing Fig. 6 and 7, it can be easily seen that increasing the number of users increases the system throughput performance.
To examine the impact of imperfections (HWI and ipCSI) on the system throughput performance, Fig. 8 presents the uplink SIMO-CNOMA when l = 1, 2, 3 in the case of two and three users.We can observe that the system throughput of the ideal HWI and ipCSI (K = 0 and σ 2 e = 0) achieves a higher performance gain compared to the nonideal condition (K = 0.05 and σ 2 e = 0.001).Also, of HWI and ipCSI was higher with three users compared to the two users.Increasing the number of users reduces the power allocation of the other users which effect the SIC and the signal detection.It recommends that the use of a lower number of users is better to achieve higher performance gain.
In order to examine the impact of HWI and ipCSI, in Fig. 9, we evaluate the impact of the HWI and ipCSI on the considered system in the presence of ipSIC when SNR = 20 dB, l = 3 and j = 3.We observe that both HWI and ipCSI have a negative impact on the system throughput performance.In Fig. 9 (a), we can easily see that increasing the ipCSI factor decreases system throughput faster, while as be seen in Fig. 9 (b), an increase in the HWI factor decreases system performance to a lesser extent.Accordingly, system performance is affected by channel estimation error much more than HWI.Also, the HWI and ipCSI weaken the SIC to detect users' signals.

V. CONCLUSION
In this paper, we examine the performance of the OP and system throughput of the uplink SIMO-NOMA system with multiple relays, taking into account practical considerations such as HWI, ipCSI, and ipSIC.By deriving mathematical expressions for the OP and system throughput, we demonstrate that the analytical results match the simulation results.In addition, we discuss the effect of the HWI, ipCSI, and ipSIC on the performance of the uplink SIMO-CNOMA with multiple relays and its crucial role in the system.The findings reveal that irrespective of the number of relays and antennas employed under HWI, ipCSI and ipSIC conditions, the presence of a high number of users reduces the performance of the system.A lower number of users is better for high-performance gain with fewer errors.The ipCSI has a higher impact on the performance compared to HWI on the considered system.However, the system performance improves as the number of antennas and relays increases.This paper can give valuable insights into the practical implementation of the uplink SIMO-CNOMA system.For future work, it is important to take into account the investigation of multiple input multiple output (MIMO) technology during both phases in the uplink CNOMA to enhance the system's performance.

APPENDIX A
The OP of the U j in the first phase is presented in (24).Hence, we can calculate the term Nr m=1 Pr γ {m,R l } j,I < γ th,j of (24) as given in (35), as shown at the bottom of the page.By using the PDF and CDF, we determine (35) as given in (36), as shown at the bottom of the page.
The integral of (36) can be calculated as in [38].Hence, we can obtain (36) as given in (37), as shown at the top of the page.The proof is completed.

FIGURE 2 .
FIGURE 2. OP w.r.t.SNR for uplink SIMO-CNOMA with SR of two users.

FIGURE 3 .
FIGURE 3. OP w.r.t.SNR for uplink SIMO-CNOMA with SR of three users.

FIGURE 6 .
FIGURE 6. Throughput w.r.t.SNR for uplink SIMO-CNOMA with SR of two users.

FIGURE 8 .
FIGURE 8. Throughput versus SNR for uplink SIMO-CNOMA comparison with the ideal case.