Performance Analysis of Non-Orthogonal Multiple Access under I/Q Imbalance

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1 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 1 Performance Anaysis of Non-Orthogona Mutipe Access under I/Q Imbaance Bassant Seim, Student Member, IEEE, Sami Muhaidat, Senior Member, IEEE, Paschais C. Sofotasios, Senior Member, IEEE, Bayan S. Sharif, Senior Member, IEEE, Thanos Stouraitis, Feow, IEEE, George K. Karagiannidis, Feow, IEEE, and Naofa A-Dhahir, Feow, IEEE Abstract Non-orthogona mutipe access NOMA has been recenty proposed as a viabe technoogy that can potentiay improve the spectra efficiency of fifth generation 5G wireess networks and beyond. However, in practica communication scenarios, transceiver architectures inevitaby suffer from radiofrequency RF front-end reated impairments that can ead to degradation of the overa system performance with inphase/quadrature-phase imbaance IQI constituting a major impairment in direct-conversion transceivers. In this artice, we quantify the effects of IQI on the performance of NOMA based singe-carrier SC and muti-carrier MC systems under mutipath fading conditions. This is reaized by first deriving anaytic expressions for the signa-to-interference-pus-noise ratio and the outage probabiity of both SC and MC NOMA systems subject to IQI at the transmitter and/or receiver RX side. Furthermore, we derive asymptotic diversity orders for a considered impairment scenarios. Capitaizing on these resuts, we demonstrate that the effects of IQI differ consideraby among NOMA users and depend on the underying systems parameters. For exampe, for a target data rate and power aocation ratio satisfying a given condition, IQI does not affect the asymptotic diversity of SC NOMA systems whereas the asymptotic diversity of MC NOMA systems, suffering from RX IQI, is aways zero. Moreover, it is shown that for both SC and MC NOMA systems, the first sorted user appears more robust to IQI, which indicates that higher order users are more sensitive to the considered impairment. Index Terms Non-orthogona mutipe access, hardware impairments, I/Q imbaance, outage probabiity. B. Seim is with the Department of Eectrica and Computer Engineering, Khaifa University of Science and Technoogy, PO Box , Abu Dhabi, UAE emai: bassant.seim@kustar.ac.ae. S. Muhaidat is with the Department of Eectrica and Computer Engineering, Khaifa University of Science and Technoogy, PO Box , Abu Dhabi, UAE and with the Institute for Communication Systems, University of Surrey, GU2 7XH, Guidford, UK emai: muhaidat@ieee.org. P. C. Sofotasios is with the Department of Eectrica and Computer Engineering, Khaifa University of Science and Technoogy, PO Box , Abu Dhabi, UAE, and with the Department of Eectronics and Communications Engineering, Tampere University of Technoogy, Tampere, Finand emai: p.sofotasios@ieee.org. B. Sharif is with the Department of Eectrica and Computer Engineering, Khaifa University of Science and Technoogy, PO Box , Abu Dhabi, UAE and with the Schoo of Eectrica and Eectronic Engineering, Newcaste University, NE1 7RU, Newcaste upon Tyne, UK, emai: bayan@ieee.org. T. Stouraitis is with with the Department of Eectrica and Computer Engineering, Khaifa University of Science and Technoogy, PO Box , Abu Dhabi, UAE and with the Department of Eectrica and Computer Engineering, University of Patras, , Patras, Greece, emai: thanos.stouraitis@kustar.ac.ae. G. K. Karagiannidis is with the Department of Eectrica and Computer Engineering, Aristote University of Thessaoniki, Thessaoniki, Greece e-mai: geokarag@auth.gr. N. A-Dhahir is with the Department of Eectrica Engineering, University of Texas at Daas, TX Daas, USA e-mai: adhahir@utdaas.edu. I. INTRODUCTION The emergence of the Internet of Things IoT together with the ever-increasing demands of mobie Internet impose high spectra efficiency and massive connectivity requirements on fifth generation 5G wireess networks and beyond. Furthermore, future communication systems are expected to support heterogeneous devices with various service types, essentia high throughput and ow atency requirements. Based on this, non-orthogona mutipe access NOMA was recenty introduced as an effective approach that is capabe of overcoming these chaenges, and thus rendering it a promising candidate for 5G systems. The distinct characteristic of NOMA is that it can be reaized by aowing mutipe users to share the same frequency bands and time sots through power-domain or code-domain mutipexing, whie successive interference canceation SIC is utiized to enabe eimination of the resuting muti-user interference. The key concept underying NOMA is to make use of nonorthogona resources, such as power or code-domain for mutipe access MA, instead of the time or frequency domain, as in orthogona mutipe access OMA schemes. Aso, contrary to OMA schemes, NOMA does not aocate orthogona resources to the different users and it instead performs SIC. Hence, NOMA can provide better spectra efficiency, higher ceedge throughput and reaxed channe feedback as ony the received signa strength is practicay required. In addition, it is capabe of providing ow transmission atency since scheduing requests from users to the base station are not necessary [1]. Moreover, the number of supported users in OMA is stricty restricted to the amount of avaiabe resources, whereas the non-orthogona resource aocation in NOMA can potentiay ead to a significant increase of the number of connected devices in the network [2], [3]. A critica component of the massive number of interconnected devices is the radio frequency RF transceiver, which faciitates communication between the individua devices and/or their respective base stations. However, the continuousy increasing demands in appications of RF transceivers has ed to harsh design targets incuding increased performance and efficiency, ow cost, ow power dissipation, and sma form factor. In this context, direct-conversion transceivers represent an effective RF front-end soution, as they demand neither externntermediate frequency fiters nor image rejection fiters. Hence, these transceiver architectures can be integrated on chip fairy easiy whie their cost is c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

2 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 2 rather ow. However, in practica communication scenarios, these monoithic architectures suffer from inevitabe RF frontend reated imperfections due to components mismatch and manufacturing nonideaities, which imit the overa system performance. A critica exampe of these impairments is the inphase I/quadrature-phase Q imbaance IQI, which refers to the ampitude and phase mismatch between the I and Q branches of the transceiver, eading to imperfect image rejection, and utimatey resuting to performance degradation of the overa communication system [4], [5]. In idea scenarios, the I and Q branches of a mixer have equa ampitude and a phase shift of 90, providing an infinite attenuation of the image band; however, in practice, direct-conversion transceivers are sensitive to certain anaog front-end reated impairments that introduce errors in the phase shift as we as mismatches between the ampitudes of the I and Q branches which corrupt the down-converted signa consteation, thereby increasing the corresponding error rate [4]. I/Q signa processing is commony reaized in modern communication transceivers, which gives rise to the probem of matching the ampitudes and phases of the branches, and thus resuting in interference from the image signa [6]. Motivated by this practica concern, severa recent works have attempted to mode, mitigate or even expoit IQI, see [7] [9] and the references therein. Specificay, the authors in [10] derive the signa-to-interference-pus-noise-ratio SINR taking into account the channe correation between the subcarriers, in the context of orthogona frequency division mutipexing OFDM systems. Assuming IQI at the receiver RX ony, the SINR probabiity distribution function PDF of generaized frequency division mutipexing under Weibu fading channes was derived in [11] and was then used in the formuation of the corresponding average symbo error rate SER for the case of M-ary quadrature ampitude moduation. In the same context, the ergodic capacity of OFDM systems with RX IQI and singe-carrier frequency-division-mutipe-access systems with joint transmitter TX/RX IQI under Rayeigh fading conditions was investigated in [12] and [13], respectivey. Likewise, the bit error rate BER of differentia quadrature phase shift keying DQPSK was recenty derived in [14] for singe-carrier SC and muti-carrier MC systems in the presence of IQI. Moreover, the authors in [15] derived the SER of OFDM with M QAM consteation, over frequency seective channes with RX IQI, whereas the authors in [16] quantified the effects of IQI on the outage probabiity of both SC and MC systems over cascaded Nakagami-m fading channes. Likewise, the error rate of subcarrier intensity moduated based QPSK over Gamma-Gamma fading channes with RX IQI was investigated in [17], whie the impact of IQI on differentia space time bock coding STBC-based OFDM systems was recenty anayzed in [18] and [19] by deriving an error foor and approximations for the corresponding BER. Finay, IQI has been aso studied in haf-dupex and fu dupex ampify and forward AF and decode and forward cooperative systems [20] [23], as we as in two-way reay systems and muti-antenna systems [24] [28]. Despite the impact of RF front-end impairments on the system performance, their detrimenta effect is often negected. Therefore, it is necessary to investigate this topic, which even given its paramount importance for the actua reaization of NOMA systems, it has not yet, to the best of the authors knowedge, been addressed in the open technicterature [29]. Motivated by this, the present investigation is devoted to the quantification and anaysis of the effects of IQI on NOMA based systems over mutipath fading channes. Specificay, the main objective of this artice is to deveop a genera framework for the comprehensive anaysis of NOMA based systems under different IQI scenarios. To this end, a downink NOMA system consisting of a base station BS and M users is considered. The cases of TX IQI ony, RX IQI ony and joint TX/RX IQI are formuated for both SC and MC transmission. Specificay, the SINR of both SC and MC NOMA systems in the presence of TX and/or RX IQI is derived. In order to quantify the effects of IQI on the considered NOMA based systems, the outage probabiity OP is used to evauate the performance of the aforementioned impairments scenarios. In more detais, the main contributions of this work are summarized as foows: We derive nove anaytic expressions for the SINR of SC and MC NOMA based systems over Rayeigh fading channes with TX and/or RX IQI aong with nove cosed form expressions for the corresponding SINR cummuative distribution function CDF. We derive a unified cosed form expression for the OP of SC NOMA systems under different impairment scenarios as we as OP expressions for MC NOMA systems subject to TX and/or RX IQI. For both SC NOMA and MC NOMA systems, under the different impairment scenarios considered, we derive the asymptotic diversity. We demonstrate that for SC NOMA systems and MC NOMA with TX IQI ony, IQI does not affect the asymptotic diversity of the system as ong as the target data rate satisfies a given condition. Extensive Monte-Caro simuation resuts are presented in order to corroborate the derived exact and asymptotic expressions. The remainder of the artice is organized as foows: Section II presents the system mode of the considered NOMA scenario and derives the SINR for both SC and MC systems subject to IQI. The OP and the asymptotic diversity orders of both SC and MC NOMA systems experiencing TX and/or RX IQI are derived in Section III. The corresponding numerica resuts for each considered scenario aong with insights and discussions are provided in Section IV. Finay, cosing remarks are given in Section V. II. SYSTEM MODEL It is recaed that the basic idea behind NOMA is to aow a certain eve of interference from adjacent users by aocating non-orthogona resources to the different users in the network. In the present investigation, we consider the case of downink NOMA where a users are served by a BS at the same time and frequency, but with different power eves. It is worth mentioning that power domain mutipexing NOMA can be efficienty reaized by appying superposition coding at the TX and SIC at the RX. Hence, in order to serve severa users c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

3 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 3 Fig. 1: Typica M user NOMA communication scenario. simutaneousy, the BS divides its transmission power between a users, whie at the receiver end, muti-user detection is performed using SIC [2]. One of the key chaenges in NOMA is how to aocate the power amongst the invoved users. A common and fairy simpe power aocation strategy is the fixed power aocation, where the power ratios are fixed and ordered according to the users channe gains. In this scenario, we assume that more power is aocated to users with poorer channe conditions. In this context, we assume a singe-ce NOMA downink system consisting of a BS and M users, depicted in Fig. 1, with transceivers equipped with a singe antenna. Based on this, we et h i represent the sma scae fading coefficient between the i th user and the BS which foows a Rayeigh distribution. Therefore, for h 1 2 h h M 2, and assuming an idea RF front end, the baseband equivaent transmitted signa is given by [30] x = Pi s i 1 where P i = E t and s i are the power and information symbo of the i th sorted user, respectivey. E t is the transmit power of the BS, M = 1 and a 1 > a 2 >... > a M. At the receiver RF front end, the received RF signa undergoes the necessary processing stages incuding fitering, ampification, anaog I/Q demoduation down-conversion to baseband and samping. Hence, assuming an idea RF front end, the baseband equivaent received signa at the j th sorted user is given by r j = h j M Pi s i + n j 2 where the subscript j refers to the j th sorted user whie h and n denote the channe coefficient and circuary symmetric compex additive white Gaussian noise AWGN signa, respectivey. The j th sorted user wi then perform SIC in order to cance the resuted interference for a users i, where i < j, whereas the signas intended for a other users with i > j wi be treated as noise. Hence, assuming perfect channe state information CSI and perfect canceation, the instantaneous SINR per symbo of the m th user s message at the j th sorted user, m < j, is given by [31] j m = a m + 1 ρ j 3 where ρ j is the j th user s instantaneous signa-to-noise ratio SNR given by ρ j = E t N 0 h j 2 4 where N 0 is the singe-sided AWGN power spectra density. It is noted here that the exact modeing and simuation of the RF front-end is typicay compex and time consuming. Thus, appying baseband equivaent impairment modes, instead of modeing the actua RF front-end, represents a pragmatic approach to the anaysis of RF reated impairments as this modeing reates the baseband representation to the bandpass signa with the RF impairments [32]. Based on this, we assume that the RF subcarriers in the present anaysis are up/down converted to the baseband by direct conversion architectures and consider frequency independent IQI caused by the gain and phase mismatches of the I and Q mixers. Thus, the timedomain baseband representation of the IQI impaired signs c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

4 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 4 given by [33] g IQI = µ t/r g id + ν t/r g id 5 where denotes conjugation, g id is the baseband IQIfree signa, gid is due to IQI and the subscripts t/r denote the up/down-conversion process at the TX/RX, respectivey. Furthermore, the IQI coefficients µ t/r and ν t/r are given by [32] and µ t = ϵ t exp jϕ t, 6 ν t = ϵ t exp jϕ t, 7 µ r = ϵ r exp jϕ r 8 ν r = ϵ r exp jϕ r 9 where j = 1, whereas ϵ t/r and ϕ t/r are the TX/RX ampitude and phase mismatch eves, respectivey. It is noted that for idea RF front-ends, ϕ t/r = 0 and ϵ t/r = 1, which impies that µ t/r = 1 and ν t/r = 0. Moreover, the TX/RX image rejection ratio IRR is given by µt/r 2 IRR t/r = ν t/r 2 10 where denotes absoute vaue operation. It is noted that in SC systems, IQI causes distortion to the signa by its own compex conjugate whie in MC systems, IQI causes distortion to the transmitted signa at subcarrier k by its image signa at subcarrier k. In the foowing we present the signa mode of both SC NOMA and MC NOMA based systems in the presence of IQI at the TX and/or RX. A. Singe-Carrier NOMA Systems Impaired by IQI It is recaed that SC moduation is receiving considerabe attention due to its robustness towards RF impairments compared to MC moduation [34]. As a resut, it is increasingy rendered more suitabe for ow compexity and ow power appications. In this context, this section presents the signa mode and a comprehensive OP anaysis for downink SC NOMA based systems in the presence of TX and/or RX IQI effects. 1 TX Impaired by IQI: This case assumes that the RF front-end of the RX is idea, whie the TX experiences IQI. Based on this, for the case of an M user downink NOMA system, the baseband equivaent transmitted signs represented as x IQI = µ t Pi s i + ν t Pi s i. 11 where µ t and ν t correspond to the BS s IQI parameters and are given in 6 and 7, respectivey. Hence, at the j th sorted user, the baseband equivaent received signs given by r jiqi = µ t h j Pi s i + ν t h j M Pi s i + n j. 12 By aso considering perfect CSI at the receiver, user j wi perform SIC in order to cance the interference from the j 1 users aocated more power than it, which under the assumption of perfect canceation yieds r jiqi j 1 = µ t 1 h j Pi s i + µ t h j + ν t h j Pi s i + n j. M i=j Pi s i 13 Hence, for 1 m j M, the instantaneous SINR per symbo of the m th user s message at the j th user is given by [35] ] E [µ t h j Pm s m µt h j Pm s m E [ ] 14 Υ j Υ j where E [ ] denote statistica expectation and j 1 Υ j = µ t 1 h j Pi s i + µ t h j + ν t h j Pi s i + n j. M i=j+1 Pi s i 15 Finay, since M i = 1 and for E [s i ] = 0, E [s i s k ] = 0 and E [ ] s 2 i = 0, which is vaid for most commony used moduation schemes such as M-PSK and M-QAM, the instantaneous SINR per symbo of the m th sorted user s message at the j th sorted user s receiver is derived as j m = µ t 2 a m µ t 1 2 m 1 + µ t 2 + ν t ρ j RX Impaired by IQI: This case assumes that the RF front-end of the TX is idea, whie the RX is subject to IQI. Thus, for a M user downink NOMA system, the baseband equivaent received signa at the j th sorted user is expressed as r jiqi = µ rj h j Pi s i + ν rj h j Pi s i + µ rj n j + ν rj n j. 17 where µ rj and ν rj correspond to the j th user s IQI parameters and are given in 8 and 9, respectivey. Simiary, assuming perfect CSI and perfect interference canceation, foowing the SIC, one obtains r jiqi = µ rj 1 j 1 h j Pi s i + µ rj h j + ν rj h j M i=j Pi s i + µ rj n j + ν rj n j. Pi s i 18 Based on this, for E [s i s k ] = 0, the instantaneous SINR per c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

5 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 5 symbo of the m th user s message at the j th user is deduced, namey j m = µ rj 2 a m µ rj 1 2 m 1 + µ rj 2 + ν rj 2 + Ξr j ρ j 19 where Ξ rj = µ rj 2 + ν rj 2. 3 Joint TX/RX impaired by IQI: This case assumes that both TX and RX are impaired by IQI. For a M user downink NOMA system, the baseband equivaent received signa at the j th sorted user is given by r jiqi = ξ 11j h j + ξ 22j h j Pi s i + ξ 12j h j + ξ 21j h j Pi s i + µ rj n j + ν rj n j. 20 where ξ 11j = µ rj µ t, ξ 22j = ν rj ν t, ξ 12j = µ rj ν t and ξ 21j = ν rj µ t. Hence, assuming perfect CSI and perfect canceation, foowing the SIC, one obtains r jiqi = ξ 11j 1 h j + ξ 22j h j Pi s i j 1 + ξ 11j h j + ξ 22j h j Pi s i i=j + ξ 12j h j + ξ 21j h j Pi s i + µ rj n j + ν rj n j. 21 Finay, foowing a simiar approach as in II-A1, for E [s i ] = 0, E [s i s k ] = 0 and E [ ] s 2 i = 0, and after some mathematica manipuations, we obtain the instantaneous SINR per symbo of the m th user s message at the j th user receiver in 22, at the top of the next page. Given that for direct conversion transceivers, the IRR is typicay in the range of 20 40dB [16], it can be safey assumed that [36] ξ 11j h 2 + ξ 22j h [ ] 2 2R ξ 11j hξ22 j h 23 and ξ12j h 2 + ξ21j h [ ] 2 2R ξ 12j hξ21 j h 24 Hence, the SINR can be approximated as 25 at the top of the next page. wireess services and appications [37]. It is recaed that MC systems are based on the division of the avaiabe signa bandwidth among K subcarriers, which has been shown to have severa advantages such as enhanced robustness against mutipath fading. Based on this, Long-Term Evoution LTE empoys orthogona frequency division mutipexing OFDM, which is a MC moduation with orthogona subcarriers, in the downink. In what foows, we derive the SINR CDF and the OP of MC NOMA systems in the presence of IQI assuming that the RF subcarriers are down converted to the baseband by wideband direct conversion. We aso denote the set of subcarriers as { S = K 2,..., 1, 1,..., K } 26 2 and assume that there is an information signa present at the image subcarrier and that the channe responses at the k th subcarrier and its image are uncorreated. 1 TX Impaired by IQI: This case assumes that the RF front-end of the RX is idea, whie the TX experiences IQI. For the case of a M user downink NOMA system, the baseband equivaent transmitted signa at the k th subcarrier is given by x IQI k = µ t Pi s i k + ν t Pi s i k 27 where the subcarrier k is the image of the subcarrier k. Based on this and assuming perfect CSI at the receiver, user j wi perform SIC in order to cance the interference from the users aocated a higher power ratio. Hence, for the case of perfect canceation, it foows that j 1 r jiqi k = µ t 1 h j k Pi s i k + µ t h j k + ν t h j k Pi s i k i=j Pi s i k + n j k. 28 Based on this, foowing the SIC operation at the k th subcarrier, the instantaneous SINR per symbo of the m th user s message at the j th user s receiver is represented as B. Muti-Carrier NOMA Systems Impaired by IQI Athough NOMA has demonstrated considerabe capacity improvement and reduced atency, it suffers from severa drawbacks, particuary, in sma ces, where users coud possiby experience simiar channe conditions. In this respect, NOMA is envisioned to coexist with other OMA schemes, such as orthogona frequency division mutipe access OFDMA, which wi resut in a significanty better overa performance and, potentiay, fufi the diverse requirements of different j m k = µ t 2 a m µ t 1 2 m 1 + µ t 2. + ν t ρ jk 29 2 RX Impaired by IQI: This case assumes that the RF front-end of the TX is idea, whie the RX is subject to IQI. Foowing a simiar approach, at the k th subcarrier and the j th sorted user, the baseband equivaent received signs given c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

6 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 6 j m = ξ 11j h j + ξ 22j h j 2 a m ξ 11j 1 h j + ξ 22j h m 1 j 2 + ξ 11j h j + ξ 22j h j 2 + ξ 12j h j + ξ 21j h j 2 + Ξ r j N 0 E t 22 j m ξ11j 2 + ξ 22j 2 a m ξ11j ξ 22j 2 m 1 + ξ 11j 2 + ξ 22j 2 + ξ 12j 2 + ξ 21j 2 + Ξr j ρ j. 25 by r jiqi k =µ rj h j k Pi s i k + ν rj h j k Pi s i k + µ rj n j k + ν rj n j k. 30 Therefore, assuming perfect CSI at the receiver and perfect canceation, foowing the SIC, one obtains r jiqi = µ rj 1 j 1 h j k Pi s i k + µ rj h j k + ν rj h j k Pi s i k i=j Pi s i k + µ rj n j k + ν rj n j k. 31 Finay, since E [ h 2] = 0 and assuming E [s i s k ] = 0, at the k th subcarrier, the instantaneous SINR per symbo for the m th user s message at the j th user is given by 32 at the top of the next page, where ρ j k = E t N 0 h j k Joint TX/RX Impaired by IQI: Here, we consider the case of MC NOMA where both the TX and RX suffer from IQI. To this effect, at the j th user s receiver, the baseband equivaent received signs given by r jiqi = ξ 11j h j k + ξ 22j h j k K Pi s i k + ξ 12j h j k + ξ 21j h j k K Pi s i k + µ rj n j k + ν rj n j k. 34 Hence, by aso assuming perfect CSI and canceation, foowing the SIC we obtain r jiqi = ξ 11j 1 h j k + ξ 22j h j k j 1 Pi s i k + ξ 11j h j k + ξ 22j h j k Pi s i k i=j + ξ 12j h j k + ξ 21j h j k Pi s i k + µ rj n j k + ν rj n j k. 35 Simiary, since E [ h 2] = 0 and assuming E [s i ] = 0, E [s i s k ] = 0, foowing a simiar approach as in II-A3, at the k th subcarrier, the instantaneous SINR per symbo of the m th user s message at the j th user s receiver is approximated 36, at the top of the next page. III. OP OF NOMA WITH IQI The OP can be defined as the probabiity that the symbo error rate is greater than a certain quaity of service requirement and it can be computed as the probabiity that the SNR or SINR fas beow a corresponding threshod which depends on the detection technique as we as on the moduation order [38]. In this section, assuming Rayeigh fading, we derive an anaytica framework for the OP of SC NOMA and MC NOMA systems subject to the aforementioned IQI scenarios. A. Singe-carrier Systems With the aid of 3, 16, 19 and 25, the SINR per symbo of the m th user s message at the j th sorted user s receiver can be expressed as IQI,j m = α m β m + A 37 ρ j where the parameters α m, β m, and A depend on the considered impairment scenario and are depicted in Tabe I, at the top of the next page. Proposition 1. Assuming downink SC NOMA systems with TX and/or RX IQI, the OP of the j th sorted user is given by P out,j = =j M [ 1 exp ψ m ] [ exp ψ m ] M c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

7 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 7 j m k = m 1 µ rj 1 2 µ rj 2 a mρ j k. 32 ρ j k + µ rj 2 ρ j k + ν rj 2 ρ j k + µ rj 2 + ν rj 2 ξ11j 2 ρ j k + ξ 22j 2 ρ j k a m j m k ρ j k ξ 11j 1 2 m 1 + ξ 12j 2 + ξ 11j 2 + ρ j k. 36 ξ 22j 2 1 a m + ξ 21j 2 + Ξ rj TABLE I: SC systems impaired by IQI parameters α m β m A m 1 Idea a m 1 TX IQI µ t 2 a m µ t 1 2 m 1 + µ t 2 RX IQI µ rj 2 a m µ rj 1 2 m 1 + µ rj 2 Joint IQI ξ11j 2 + ξ 22j 2 a m ξ11j ξ 22j 2 m 1 + ξ 11j 2 + ξ 22j 2 + ν t ν rj 2 + ξ 12j 2 + ξ 21j 2 Ξ rj Ξ rj where = E t /N 0 is the average transmit SNR and ϕ m A ψ m = max m j α m ϕ m β m where ϕ m = 2 R m 1 and R m is the targeted data rate of the m th user [31]. Moreover, 38 is vaid for otherwise P out,j = 1. 0 ϕ m < α m β m, 40 Proof. The proof is provided in Appendix A. Capitaizing on Proposition 1, the asymptotic diversity order of SC NOMA with IQI is evauated as [39] og P out d a = im og Since exp x x 0 1 x, substituting 38 in 41, yieds d e = im og =j M ψ m 1 ψ m M og Moreover, since M ψm im 1 ψ M j m M ψm = im, j =j 43 the asymptotic diversity can be approximated by M og j + j og og ψ m d e = im 44 og = j 45 It is noted here that the above expression is ony vaid for ϕ m satisfying the condition stated in 40, otherwise P out,j = 1 and the asymptotic diversity is nu. In addition, it is interestingy noticed that, for SC NOMA systems, the impairment scenario does not affect the asymptotic diversity, which is equa to the user s order. B. Muti-Carrier Systems 1 TX Impaired by IQI: Here we assume MC transmission in the aforementioned downink NOMA scenario where the TX ony is impaired by IQI. Proposition 2. Assuming downink MC NOMA systems with TX IQI ony, the OP of the j th sorted user is given by [ M P out,j = 1 exp ψ ] i [ m exp ψ ] M i m i i=j 46 where ψ m is given in 47, at the top the next page, whie 46 is vaid for 0 ϕ m < µ t 1 2 m 1 otherwise P out,j = 1. µ t 2 a m + µ t 2 + ν t c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

8 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 8 ψ m = max 1 m j ϕ m m 1 µ t 2 a m ϕ m µ t 1 2 µ t 2 + ν t 2 47 Proof. Since 29 is simiar to 16, it foows that the OP of MC NOMA system with TX IQI ony is given by 46. Moreover, the asymptotic diversity of MC NOMA systems with TX IQI ony is deduced from 45 for ϕ m incuded in the range in 47 otherwise it is nu. 2 RX Impaired by IQI: This case assumes that the RF front-end of the TX is idea, whie the RX is subject to IQI. Proposition 3. Assuming the aforementioned downink MC NOMA system with RX IQI ony, the OP of the j th sorted user is expressed as P out,j = where ψ m = max 1 m j =j p=0 M µ rj 2 a m ϕ m whie 49 is vaid for p 1 p exp ψ mξ rj M +p 1 + ψm ν rj 2 M + p 1 µ rj 1 2 m 1 + µ rj 2 µ rj 2 a m 0 ϕ m < µ rj 1 2 m 1 + µ rj 2 otherwise, P out,j = 1. Proof. The proof is provided in Appendix B Substituting 49 in 41, the asymptotic diversity of MC NOMA systems with RX IQI ony is obtained as d a = im og M =1 p=0 M 1 p 1 ψ mξr M +p j p og 1+ψ m ν rj 2 M +p 52 = which impies that for MC systems with RX IQI, an error foor is reached. This error foor can be obtained by taking = in 49, yieding P out,j = =j p=0 M p 1 p 1 + ψm ν rj 2 M + p 54 3 Joint TX/RX impaired by IQI: Here, we consider the case of MC NOMA where both the TX and RX experience IQI effects. Proposition 4. Assuming downink MC NOMA systems with joint TX/RX IQI, the OP of the j th sorted user is given by 55, at the top of the next page, where ψ m is given in 56. moreover 55 is vaid for 0 ϕ m < ξ 11j 1 2 m 1 ξ 11j 2 a m + ξ 11j 2 If the above condition is not satisfied, the OP is 1. Proof. The proof is provided in Appendix C. + ξ 12j 2 57 Substituting 55 in 41, the asymptotic diversity is obtained in 58. Importanty, this reveas that MC NOMA systems with joint TX/RX IQI asymptoticay reach an error foor, regardess of the eve of IQI considered. This error foor is obtained by taking = in 55 yieding 59. It is noted that in the above anaysis we considered an idea system where the users have perfect knowedge of the CSI. This is not usuay the case in practice, and hence NOMA systems performance is expected to further degrade under this condition. Hence, the present paper provides a ower bound to actua NOMA systems OP performance. The anaysis of imperfect CSI is out of the scope of this paper and can be found in [30]. IV. NUMERICAL AND SIMULATION RESULTS Considering the aforementioned NOMA approach, this section investigates the effect of TX and/or RX IQI on the performance of NOMA based communication systems. To this end and assuming Rayeigh fading conditions, extensive Monte Caro simuations have been executed in order to investigate the OP performance of NOMA under IQI effects. It is noted that, uness otherwise stated, the number of users considered is M = 2 and that the power aocation coefficients are a 1 = 4 5 and a 2 = 1 5 for M = 2, and a 1 = 1 2, a 2 = 1 3 and a 3 = 1 6 for M = 3 [31]. Moreover, for a fair comparison, we assume that the transmit power eve is aways fixed. This impies that the transmitted signs normaized by µ t 2 + ν t 2 for TX IQI, by µ r 2 + ν r 2 for RX IQI and by µ t 2 + ν t 2 µ r 2 + ν r 2 for joint TX/RX IQI. Figs. 2 5 and 6 8 present the OP of SC and MC NOMA based systems in the presence of IQI, respectivey. It is noted that the numerica resuts are shown with soid ines, whereas markers are used to iustrate the respective computer simuation resuts. For both SC and MC systems, it is observed that the derived outage probabiity for TX and RX IQI ony perfecty matches the simuations whereas for the case of joint TX/RX IQI, the proposed approximation accuratey characterizes the system s outage probabiity c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

9 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access P out,j = =j p=0 M 1 p exp ϕmξr j M +pψm p 1 + ψm M + p ϕ m ξ22j 2 1 a m + ξ 21j ξ 22j 2 a m 9 ψ m = max 1 m j 1 m 1 ξ 11j 2 a m ϕ m ξ 11j ξ 11j 2 + ξ 12j d a = im = 0. og M =1 p=0 M 1 p p exp ϕmξr j M +pψm 1+ψ m M +pϕ m ξ 22j 2 1 a m + ξ 21j 2 ξ 22j 2 a m og 58 P out,j = =j p=0 M p 1 p 1 + ψm M + p ϕ m ξ22j 2 1 a m + ξ 21j ξ 22j 2 a m Average Outage Probabiity 10-3 Idea TX/RX TX IQI ony RXIQI ony TDMA NOMA Eb/No db Fig. 2: Comparison between NOMA and TDMA in terms of the average OP for 3 user, R = 0.9 bits/s/hz IRR t = IRR r = 20dB and ϕ = 3. A. SC NOMA Systems The effects of TX IQI ony, RX IQI ony and joint TX/RX IQI on the OP performance of a 3 user NOMA based system compared to a 3 user time division mutipe access TDMA based system are shown in Fig. 2. The considered IRR, which determines the amount of attenuation of the image frequency band, is 20dB. It is noted here that for a fair comparison, we fix the transmit power E t. This impies that for NOMA, E t is divided between the users during one time sot whie for TDMA the transmit power is E t /N users for N users time sots. The target data rate is R = 0.9 bits/s/hz. It is first observed that RX IQI has more detrimentmpact on the system performance than TX IQI. This resut is expected since RX IQI affects both the signa and the noise whie TX IQI impairs the information signa ony. Second, the detrimenta effects of IQI are more pronounced for NOMA based systems compared to TDMA systems. In fact, whie the average OP of TDMA seams uninfuenced by TX IQI and quite robust to RX IQI, NOMA observes some performance degradation, speciay from RX IQI. This is expained by the fact that in NOMA, the power domain mutipexing causes interference not ony from U i s signa conjugate but from a the other NOMA users signa conjugate as we. Moreover, the interference canceation performed by NOMA users does not take into account the IQI, eaving a portion of the signa uncanceed and since a 1 > a 2 >... > a M, this interference becomes significant as the number of users increase. This highights the importance of modeing the effects of the we known RF impairments in the context of NOMA systems, where the power domain mutipexing as we as the SIC are utimatey expected to render the system more sensitive to RF impairments than OMA schemes. Fig. 3 shows the OP as a function of the target data rate. We consider that the transmit SNR is = 20dB and IRR t = IRR r = 20dB. Here, it is observed that the eve of performance degradation caused by IQI depends on the target data rate, on the order of the considered sorted user and the incurred impairments. In fact, it is noticed that IQI affects U 2 more severey than U 1. This resuts from the SIC operation performed by U 2 without taking into account the IQI effects. Hence, foowing the SIC operation, U 2 is eft c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

10 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 10 Outage Probabiity Rate, bits/s/hz U 1 U 2 Idea TX/RX TX IQI ony RXIQI ony In Fig. 4 the OP of a 3 user NOMA system is depicted as a function of the IRR for IRR t = IRR r, = 25dB and R = 0.95 bits/s/hz. As mentioned earier for a 2 user NOMA system, it is once more observed that IQI does not affect a the users equay. In fact, it is quite obvious that the performance degradation is directy proportiona to the order of the user, i.e. to the number of SIC operations performed. Moreover, for the assumed scenario, the effects of IQI can be considered negigibe for reativey high vaues of TX/RX IRR. Hence, this highights the importance of the correct modeing of the effects of RF impairments, such as IQI on the performance of NOMA based systems. Furthermore, depending on the severity of the impairment, compensation techniques shoud be studied and impemented, if required, as in some cases the compexity engendered by compensation schemes is not justified with regards to the eve of performance degradation caused. Fig. 3: SC system OP as a function of the target data rate for = 20dB, IRR t = IRR r = 20dB and ϕ = 3. with uncanceed interference from U 1. For instance, assuming R = 2 bits/s/hz, joint TX/RX IQI increases the OP of U 2 of more than 200% whereas U 1 s OP is increased of around 33% ony. Moreover, it is noticed that the target data rate aso affects the eve of performance degradation caused since, for U 2, the increase in the OP goes from 200% to amost 500% when increasing the rate from 2 bits/s/hz to 4 bits/s/hz. Average Outage Probabiity Idea TX/RX TX IQI ony RXIQI ony a 1 U 1 Fig. 5: SC system OP as a function of a 1 for = 20dB, R = 1.5 bits/s/hz, IRR t = IRR r = 20dB and ϕ = 3. Outage Probabiity U 2 Idea TX/RX TX IQI ony RXIQI ony U 3 In addition, the effects of IQI on the average OP for different power spitting ratios is depicted in Fig. 5. It is observed that, assuming = 20dB and R = 1.5 bits/s/hz, the eve of performance degradation caused by the impairment is aso dependent upon the power spitting ratio. In fact, it is noticed that IQI can ater the optimum power spitting ratio that minimizes the average OP. This further highights the importance of investigating the optimum power aocation taking into account RF impairments and/or the impementation of efficient dynamic compensation schemes IRR, db Fig. 4: SC system OP as a function of the normaized IRR for R = 0.9 bits/s/hz, = 20dB and ϕ = 1. B. MC NOMA Systems Fig. 6 depicts the OP of MC NOMA systems as a function of the target rate for = 20dB and IRR t = IRR r = 20dB. It is quite obvious that the effect of RX IQI is far more significant than TX IQI ony. This is because TX causes interference from c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

11 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 11 Outage Probabiity Rate, bits/s/hz U 1 U 2 Idea TX/RX TX IQI ony RXIQI ony Average Outage Probabiity Idea TX/RX TX IQI ony RXIQI ony a 1 Fig. 6: MC system OP as a function of the target rate with IRR t = IRR r = 20dB, = 20dB and ϕ = 3. the transmit signa ony and not the noise or the channe gain as in RX IQI. This observation was made earier for SC NOMA systems as we; however, it is observed that the gap between TX IQI ony and RX IQI is even more significant for MC NOMA. For instance, for R = 4 bits/s/hz TX IQI increases the OP of U 2 by more than 110%, whereas RX IQI increases it by more than 980%. Fig. 7 shows the average OP of MC NOMA systems for R = 1.25 bits/s/hz. Once more, we observe the significant impact of RX IQI on MC NOMA based systems. In fact, we notice that both TX and RX IQI cause a reativey constant shift of the average OP where the performance penaty caused by TX IQI ony is fairy sma and coud be negected, whereas the effects of RX IQI are quite significant and require compensation in order to achieve a reiabe communication ink. Furthermore, once more, it is highighted that this impairment can ater the optimum power aocation amongst the users. Finay, Fig. 8 dispays the OP of a 3 user MC NOMA system vs the IRR, assuming joint TX/RX IQI with IRR t = IRR r. For R = 0.9 bits/s/hz and = 20dB, it is observed that, even though the eve of OP increase depends on the user s index, IQI causes a quite significant degradation of the OP of a the NOMA users. In fact, for a reativey high IRR of 30dB, joint TX/RX IQI increases the OP of U 3 by neary 68%, whie U 1 s OP is increased by approximatey 13%. C. Comparison and Discussion Fig. 9 shows the effects of RX IQI on a SC NOMA and a MC NOMA based system. It is observed that IQI causes a performance penaty in both systems; however, its impact is more significant in MC NOMA. More precisey, we observe that IQI causes a reativey constant shift of the OP in SC Fig. 7: MC system average OP as a function of a 1 for = 20dB, R = 1.25 bits/s/hz, IRR t = IRR r = 20dB and ϕ = 3. Outage Probabiity Idea TX/RX TX IQI ony RXIQI ony IRR, db Fig. 8: MC system OP as a function of the IRR for 3 user NOMA system with R = 0.9 bits/s/hz, = 20dB, = 25 and ϕ = 1. systems, whereas in MC NOMA, an OP foor is observed for a the users suffering from RX and consequenty joint TX/RX IQI. This is in ine with the derived asymptotic diversity order of 0, which confirms the anaytica resuts derived in Section III-B. Moreover, for SC NOMA, with joint TX/RX IQI, Fig. 3 shows that the OP of U 2 is increased by 200% for R = 2 bits/s/hz whereas from Fig. 6, for the same rate U 2 U 1 U c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

12 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 12 Outage Probabiity U 1 Ergodic sum rate NOMA TDMA Idea TX/RX TX IQI ony RXIQI ony U 2 U 3 Idea TX/RX MC IQI SC IQI Eb/N, db 0 Fig. 9: Comparison between the effects of joint TX/RX IQI on SC and MC NOMA systems for 3 user with R = 0.95 bits/s/hz IRR t = IRR r = 20dB and ϕ = Eb/N 0, db Fig. 10: Comparison between SC NOMA and TDMA in terms of the ergodic sum rate for 2 user, IRR t = IRR r = 20dB and ϕ = 3. and transmit SNR, MC NOMA systems OP increases from to when subject to joint TX/RX IQI, which corresponds to an increase of more than 1350%. Simiary, in Fig. 4, it is shown that joint TX/RX IQI effects can increase U 3 s OP by neary 35% in SC NOMA systems whereas MC NOMA systems OP witnesses an increase by more than 1600%. The reason underying this is that in MC systems, IQI causes interference from the image subcarrier which affects the orthogonaity of the system. Moreover, IQI in MC systems causes interference from the image subcarrier, which coud benefit from better fading conditions than the desired signa, whie SC IQI causes interference from the signa s own compex conjugate. Finay, in both SC and MC NOMA systems, it was shown in Figs. 5 and 7 that IQI can affect the optimum power aocation between the different NOMA users. Hence, the effective impementation of NOMA requires modeing of this impairment prior to the optimization of the system. Moreover, dynamic compensation, i.e. impemented by the users that are affected by the impairment ony, can be effectivey considered in order to avoid added compexity. Finay, we compare the ergodic sum rate [31] of idea and I/Q impaired SC and MC NOMA and TDMA systems in Figs. 10 and 11, respectivey. The considered IRR is 20dB. Interestingy, in both cases, it is observed that the effect of IQI is more pronounced in the ergodic sum rate of NOMA than TDMA. Moreover, for the considered IQI eves, for SNR greater than 20dB, under joint TX/RX IQI, the ergodic sum rate of MC NOMA is ower than MC TDMA. Ergodic sum rate NOMA TDMA Idea TX/RX TX IQI ony RXIQI ony Eb/N 0, db Fig. 11: Comparison between MC NOMA and TDMA in terms of the ergodic sum rate for 2 user, IRR t = IRR r = 20dB and ϕ = 3. V. CONCLUSION We investigated the effects of IQI on the OP of both SC and MC NOMA based systems. The reaistic cases of TX IQI ony, RX IQI ony and joint TX/RX IQI were considered and the corresponding OP over Rayeigh fading c 2018 IEEE. Transations and content mining are permitted for academic research ony. Persona use is aso permitted, but repubication/redistribution requires IEEE permission. See for more information.

13 This artice has been accepted for pubication in a future issue of this journa, but has not been fuy edited. Content may change prior to fina pubication. Citation information: DOI /ACCESS , IEEE Access 13 channes was derived. In addition, the asymptotic diversity of both SC and MC NOMA systems was derived assuming a the different considered impairment scenarios. The derived anaytic resuts were corroborated with respective resuts from computer simuations. It was shown that the eve of performance degradation caused by IQI depends on numerous factors incuding the power aocation ratio, the impairment scenario, the target rate and the order of the considered user. This is due to the uncanceed interference from IQI that affects the performance of the empoyed SIC. Hence IQI affects SC NOMA based systems by affecting the SIC s performance whie in MC NOMA systems both the SIC and the orthogonaity of the subcarriers are compromised eading to more significant performance degradation. Moreover, it was shown that IQI can deviate the optimum power spitting ratio and hence compromise the efficiency of both SC and MC NOMA systems. To this end, this highights the importance of effective modeing, optimization and dynamic compensation for the efficient impementation of the NOMA paradigm in future communication systems, such as 5G. APPENDIX A SC NOMA SYSTEMS IMPAIRED BY TX AND/OR RX IQI OP DERIVATION It is highighted that the j th user is required to detect the messages intended for a users aocated a higher power ratio than itsef before detecting its own message. Without oss of generaity, et the channe gains of a users be sorted in an ascending order as h 1 2 h h M 2. Then, the power aocation coefficients are chosen such that a 1 > a 2 >... > a M. Assuming 1 < m < j, the outage probabiity is defined as the probabiity that user j cannot detect its own signa or the signntended for any user in the SIC j > m, which is obtained as { } P out,j = 1 Pr Ej 1 c E c j j 60 where E j m = {R j m < R m } is the event that the j th user cannot detect the m th user s message, R j m denotes the rate for the j th user to detect the m th user s message, R m is the targeted data rate of the m th user and Ej m c the compementary set of E j m which can be written as { } α m Ej m c = β m + A > ϕ m ρ { j } ϕ m A = ρ j > α m ϕ m β m which is vaid for 0 ϕ m < α m βm and ϕ m = 2 R m 1. An outage event occurs at the j th user if it is not abe to decode its own message or the message of any of the users m in the SIC i.e m < j. Hence, from 60, the OP of the j th sorted user is obtained as as P out,j = 1 Pr {ρ j > ψ m } 63 = F ρj ψ m 64 where F X x denote the CDF of X and ψ m is given in 39. Making use of order statistics [40], the cumuative distribution function CDF of the j th user s instantaneous SINR ρ j is given by =j F ρj x = M [Fρ x] [1 F ρ x] M, 0 ϕ m < αm β m 1, ϕ m α m βm65 where F ρ is the CDF of the unordered SNR which assuming Rayeigh fading, foows an exponentia distribution. Hence, assuming TX and/or RX IQI, the OP of SC systems is obtained by 38. APPENDIX B MC NOMA SYSTEMS IMPAIRED BY RX IQI OP DERIVATION Since the instantaneous SINR per symbo of the m th user s message at the j th user is given by 32, and foowing the same approach as in Appendix A, the conditiona SINR CDF for a given ρ j k is given by 66, at the top of the next page, which is vaid for the range in 51. Meanwhie, for ϕ m not satisfying 51, the OP is 1. Based on this, the unconditiona CDF is obtained by integrating 66 over the distribution of ρ j k, namey the Rayeigh PDF in the considered scenario. Hence, expanding the binomia and after some mathematica manipuations, equation 67 is deduced. Finay, by evauating the invoved integra and recaing that an outage event occurs at the j th user if it is not abe to decode its own message or the message of any of the users m in the SIC i.e m < j, the OP in 49 is deduced, which competes the proof. APPENDIX C MC NOMA SYSTEMS IMPAIRED BY JOINT TX/RX IQI OP DERIVATION Considering the instantaneous SINR per symbo of the m th user s message at the j th user given in 36, for a given ρ j k, foowing the same approach as in Appendix B, the conditiona SINR CDF is given by 68 and is vaid for the range presented in 57, whie for ϕ m not satisfying 57, the OP is 1. Based on this, the unconditiona CDF is obtained foowing the same approach as in Appendix C. Hence, expanding the binomia, performing some mathematica manipuations and soving the integra, yieds 55, which competes the proof. REFERENCES [1] S. M. R. Isam, N. Avazov, O. A. Dobre, and K. S. 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