TELK OMNIKA T elecommunication, Computing, Electr onics and Contr ol V ol. 23, No. 4, August 2025, pp. 918 ∼ 931 ISSN: 1693-6930, DOI: 10.12928/TELK OMNIKA.v23i4.26822 ❒ 918 Ener gy sca v enging-aided NOMA uplink communications: perf ormance analysis Huu Q . T ran 1 , Khuong Ho-V an 2 1 F aculty of Electronics T echnology , Industrial Uni v ersity of Ho Chi Minh City , Ho Chi Minh City , V ietnam 2 F aculty of Electrical and Electronics Engineering, Ho Chi Minh City Uni v ersity of T echnology (HCMUT), VNU-HCM, Ho Chi Minh City , V ietnam Article Inf o Article history: Recei v ed Dec 5, 2024 Re vised Apr 12, 2025 Accepted May 27, 2025 K eyw ords: Nonorthogonal multiple access Outage probability Shado wed f ading Throughput Uplink communications ABSTRA CT Ener gy sca v enging-aided nonorthogonal multiple a ccess (NOMA) netw orks sig- nicantly ameliorate ener gy-and-spectral ef ciencies thanks to superimposing a multitude of user signals for concurrent transmission and harv esting radio fre- quenc y ener gy . Practically , ener gy harv esters possess non-linear characteristic and their ef cienc y is enhanced considerably with deplo yment of multiple anten- nas. Moreo v er , communication reliability and harv ested ener gy are directly in- uenced by wireless propag ation w hich induces simultaneous ef fects of shado w- ing, path loss, and f ading. Accordingly , the current paper assesse s analytically outage probability and throughput of ener gy sca v enging (ES)-aided NOMA up- link communications (eNOMAu) taking int o account the abo v e-addressed re- alistic f actors ( κ − µ shado wed f ading, multi-antenna deplo yment, ES non- linearity). The results re v eal considerable performance de gradation caused by ES non-linearity and wireless propag ation. Additionally , desired system perfor - mance can be reached e xibly with appropriate specication selection. In addi- tion, accreting a quantity of antennas drastically mitig ates the outage probability of eNOMAu, which can be minimized with optimal ES time select ion. Further - more, the proposed e NOMAu is considerably superior to its eOMAu counter - part. This is an open access article under the CC BY -SA license . Corresponding A uthor: Huu Q. T ran F aculty of Electronics T echnology , Industrial Uni v ersity of Ho Chi Minh City Go V ap District, Ho Chi Minh City ,V ietnam Email: tranquyhuu@iuh.edu.vn 1. INTR ODUCTION 5G/6G systems grant a multitude of emer ging wireless applications for a ma ssi v e number of de vices b ut enforce serious challenges on telecommunication infrastructure, particularly in current scenarios of spec- trum and ener gy shortages, to supply suf ciently spectrum and ener gy for these de vices [1], [2]. Therefore, solutions for impro ving ener gy and spectral ef ciencies are essential and ur gent. Nonorthogonal multiple access (NOMA), which w as recommended for 5G/6G netw orks, i s a feasible solution to ameliorate spectral ef cienc y [3]. By distrib uting distinct po wer le v els to di v erse users, NOMA e xploits ef ciently successi v e decoding in combination with interference cancellation to further mitig ate system outage. Furthermore, harv esting radio frequenc y ener gy present in wireless signals by NOMA users can be considered as a helpful solution in meliorating ener gy ef cienc y . Notably , an impl ementable ES circuit is deplo yed in 5G/6G transcei v ers [4]. Notwithstanding, a majority of performance analyse s pertinent to ES ha v e J ournal homepage: http://journal.uad.ac.id/inde x.php/TELK OMNIKA Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 919 characterized ES to be linear for tractability [5]. Realistically , ES circuit is implem ented by nonlinear elements, namely capacitors, inductors, diodes. Thereby , characterizing ener gy harv esting ought to tak e the non-linearity of circuit elements int o account. So f ar , v arious non-linear ener gy harv esting models ha v e model led such non-linearity [3], [6], [7]. Ener gy sca v enging-aided NOMA uplink communications (eNOMAu), viz. Figure 1, enables tw o NOMA transmitters (U1 and U2) to concurrently send their data to the same recei v er (NR) and to sca v enge ener gy from a po wer source (PS) for their transmissi on wherein high-and-stable po wer tele vision/radio broad- casting transmitters can play a role as PS. T o further ameliorate ener gy harv esting ef cienc y , PS should emplo y multiple antennas, which is the case in this paper . W ireless propag ation causes a multitude of impairments lik e path loss, f ading, shado wing, which in- uence drastically reliability of wireless transmission. F or eNOMAu under consideration in this w ork, wireless channels impact an amount of sca v enged ener gy and communication reliability . F or performance assessment realistically , wireless propag ation must be characterized properly to t real-w orld data. κ − µ shado wed f ading paradigm is w i d e l y a c k n o w ledged to feature appropriately concurrent ef fects of f ading, path loss, shado wing [8]. In summary , eNOMAu e xposes adv antages of high spectrum and ener gy ef ciencies. Notwit hstand- ing, the outage probability of eNOMAu is impacted by numerous real-w orld impairments such as wireless channel (path loss, shado wing, f ading), ES non-linearity , and multi-antenna deplo yment. Thence, performance assessment of eNOMAu is essential to conrm whether eNOMAu possesses such superiority as w orking in such impairments. This paper proposes outage probability and throughput assessment under consideration of all these f actors. NR PS v P ow e r  s our c e N O MA  r e c e i ve r N O MA  us e r  2 N O M A  us e r  1 Energy Scavengi ng NOMA Uplink S t a ge  1:   α T  S t a ge  2:   (1 - α ) T  V 1 U2 U1 Figure 1. Ener gy sca v enging-aided NOMA uplink communications in this paper W ang and Men [3], eNOMAu w as studied whereby multiple NOMA terminals that broadcast infor - mation to a common recei v er NR need tw o stages (Figure 1). In stage 1, NOMA terminals utilize nonlinear ener gy harv esters (NLEHs) to harv est radio frequenc y ener gy from a s table PS while in stage 2, the y use har - v ested ener gy to send data to NR. W ang and Men [3] sho wed each stage is optimized for the best performance. Nonetheless, W ang and Men [3] o v erlook ed the e xplicit analysis of secrec y outage probability . The partic- ular e xample of [3] with tw o NOMA t erminals w as demonstrated in [9] that presented approximated outage probability analysis. Researchers [10]-[12] studied ener gy sca v enging-aided NOMA do wnlink communica- tions (eNOMAd) in which NR sends information using NOMA to tw o recei v ers (U1 and U2). Afterw ards, Le et al. [13] performed an e xtension of [10]-[12] to multiple recei v ers. Researchers [10]-[13] deri v ed the approximated formulas of outage probability (OP) and throughput (TP). In addition, Le et al. [13] maximized sum-rate. Ho we v er , researchers in [10], [11], the single-antenna NOMA de vice pro vides ener gy for NR. In contrast, Le et al. [13] the multi-antenna NOMA de vice supplies ener gy for NR. Ne v ertheless, researchers [10], [11], [13] utilizes unrealistic linear ener gy harv ester (LEH). Furthermore, U1 harv ests radio frequenc y ener gy from the single-antenna NR by NLEH in [12]. eNOMAd with tw o recipients (U1 and U2) w as in v estig ated whereby transmission to U1 is aided by a relay in [14]-[16] and by U2 in [17]-[21]. The relay and U2 in [14]-[21] sca v enge po wer from the NOMA transmitter . Researchers [16], [17] studi ed NLEH at the relay in t heir approximated OP analysis. In the meantime, researchers [14], [19], [20] solv ed the aggre g ate capacity optimization problem. Ne v ertheless, Liu Ener gy scavenging-aided NOMA uplink communications: performance analysis (Huu Q. T r an) Evaluation Warning : The document was created with Spire.PDF for Python.
920 ❒ ISSN: 1693-6930 et al. [14] considered LEH wherea s Si et al. [19] and Garcia et al. [20] implemented NLEH. Researchers in [22], [23] performed an e xtension of [16] to multiple relays as well as selected the best relay to aid NOMA transmission from NR to U1 and U2. Furthermore, Zhai et al. [24] e xtended Agra w al et al. [16] to assist U2 by utilizing tw o relaying de vices. Ho we v er , Zhai et al. [24] o v erlook ed the closed-form TP analysis. Instead of emplo ying a multitude of relays in [22], [23], [25] e xploited a multitude of near NOMA terminals and performed selecting solely one near NOMA terminal to help the f ar NOMA terminal. Another e xtension of Agra w al et al. [16] is to in v estig ate se v eral recei v ers in [26], [27]. Notwithstanding, researchers [15], [18], [21]-[27] conducted performance analysis for LEH. Moreo v er , researchers [28]-[30] e xploited recongurable intelligent surf ace to relay messages from NR to U1 and U2 in order to maximize aggre g ate capacity . Ho we v er , L yu et al. [28] considered NLEH whereas [29], [30] in v estig ated LEH. Additionally , system performance w as not analyzed in [14], [19], [20], [28]-[30]. T o sum up, the aforementioned publicat ions pertinent to performance analysis in [3], [9]-[13], [15]- [26], [29], [30] in v estig ated a tri vial system model with the single-antenna PS, thence barely meliorating ES ef cienc y . Merely Aldababsa and Basar [27] studied se v eral antennas emplo yed at all users. Additionally , only tri vial f ading models without shado wing such as Rayleigh ([3], [12], [13], [16]-[21], [24], [25], [29]), Rician ([14], [15], [28], [30]), W eib ull [26], Nakag ami- m ([10], [11], [22], [23], [27]) ha v e been considered in the pre vious publications. Thence, the updated publications o v erlook ed the OP/TP anal yses for the conguration in Figure 1 when considering path loss, shado wing, generalized κ − µ f ading, and se v eral antennas at PS such that U1 and U2 harv est more ener gy , ultimately ameliorating system performance at NR. This paper pioneers in carrying out such an analysis that is useful in e v aluating swiftly and maximizing system performance prior to practical deplo yment. The contrib utions are itemized to be: (i) we recommend eNOMAu in Figure 1 in which the arbitrary number of antennas can be deplo yed at PS for ameliorating ener gy harv esting ef cienc y , e v entually impro ving system perf o r mance; (ii) to e v aluate communication reliability promptly , we perform the OP/TP analyses for the proposed eNOMAu considering the non-linearity of ener gy harv esting, the multiple antenna consideration, and d i f ferent channel impairments lik e path loss, f ading, shado wing; and (iii) we rate and maximize syst em per - formance in distinct realistic scenarios. Multiple results demonstrate a considerable performance de gradation o wing to non-linearity of ener gy sca v engers and impairments of wireless propag ation. Moreo v er , communica- tions reliability of eNOMAu can be controlled and maximized by adopting appropriately and e xibly se v eral specications. Additionally , accreting a quantity of antennas impro v es signicantly syst em performance. Fur - ther , the proposed eNOMAu is considerably superior to its eOMAu counterpart. Section 2 e xplains the recommended eNOMAu. Subsequently , section 3 carries out the OP and TP analyses. Then, section 4 presents the asymptotic performance analysis in the re gime of high transmit po wer . Ne xt, section 5 analyzes t he benchmark eOMAu scheme for con v enience in comparing with the proposed eNOMAu. Then, section 6 discusses simulated/analytical results in dif ferent practical settings. Ev entually , section 7 concludes the paper . Notable symbols are presented in T able 1. T able 1. Notable symbols Symbol Interpretation N (0 , b ) Zero-mean and b -v ariance comple x Gaussian random v ariable Γ( · ) Complete g amma function F Q ( · ) Cumulati v e distrib ution functon (CDF) of Q Pr {·} Probability operator ¯ F Q ( · ) Complementary cumulati v e distrib ution function (CCDF) of Q E {·} Expectation operator f Q ( · ) Probability density function (PDF) of Q Θ Q ( · ) Moment generating function (MGF) of Q Γ( · , · ) Incomplete upper g amma function C l k = k ! l !( k − l )! Binomial coef cient 2. ENERGY SCA VENGING-AIDED NOMA UPLINK COMMUNICA TIONS 2.1. System model Figure 1 depicts eNOMAu comprising U2, U1, NR, PS. eNOMAu e x emplies uplink communications in mobile wireless systems. U2 and U1 ha v e po wer limitation. Thereby , the y ought to sca v enge po wer from PS. In eNOMAu, PS pro vides po wer for operations of U2 and U1 in α T time unit of Stage 1 wherein T TELK OMNIKA T elecommun Comput El Control, V ol. 23, No. 4, August 2025: 918–931 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 921 mentions the frame duration and α ∈ (0 , 1) means the time splitting parameter , whereas U2 and U1 conduct uplink communications using NOMA to NR in the remainder of T in stage 2. T o ameliorate ener gy sca v enging ef cienc y , ultimately enhancing system performance, V antennas are supposed to be deplo yed at PS, which is possible for high-ener gy PS. More particular , multi-antenna PS enables U2 and U1 to sca v enge ener gy ef fecti v ely . Nonetheless, U1, U2 and NR may be mobile terminals and thence, assumption of single antenna deplo yed at them is more appropriate. 2.2. Channel model Our w ork represents g s v i and g i correspondingly as channel g ains from the v th antenna of PS to U i and from U i to NR where i = { 1 , 2 } . It also supposes block frequenc y non-selecti v e κ − µ shado wed f ading links. In particular , a cluster of specications ( µ, κ, β , ϑ xy ) determines denitely g xy with xy = { s v i, i } . In other w ords, by changing a parameter set ( κ, µ, β , ϑ xy ) of such κ − µ shado wed f ading model, one can control e xibly dif ferent impairment le v els of shado wing, path loss, f ading. The parameter ϑ xy = E { g xy } represents channel po wer , which comprises path loss, µ notates the quantity of multi-path groups, β indicates shado wing ef fect, κ indicates the Rician- K parameter . Accordingly , this model characterizes a majority of practical wireless channels. As per [8], the PDF and CDF of g xy are gi v en by f g xy ( w ) = G X l =0 H l Λ β l xy Γ ( β l ) w β l − 1 e − w Λ xy and F g xy ( w ) = 1 − G X l =0 β l − 1 X c =0 H l Λ c xy c ! w c e − w Λ xy , (1) whereby Λ xy = ϑ xy ( β + κµ ) µβ ( κ +1) , H l = κµ β + κµ G − l β β + κµ l C l G , β l = β − l , G = β − µ with µ ≤ β . T o represent path loss, we model ϑ xy as ϱd − σ xy with ϱ being f ading po wer at 1 meter (m), σ being path loss decay , d xy being transcei v er distance [7]. Independent and identically distrib uted (i.i.d) f ading channels between the NOMA user i and PS’ s antennas are supposed and er go, the subscripts s v i related to channel specications ( g s v i , Λ s v i , ϑ s v i ) can be written shortly as si in ( g si , Λ si , ϑ si ) if not inducing an y confusion, viz. ϵ s v i = ϵ si , ∀ v , ϵ = { g , Λ , ϑ } . 2.3. Signal r epr esentation PS grants ener gy for U2 and U1 o v er multiple-input si ng l e-output links in Stage 1, considerably increasing a mount of collect ed po wer at U2 and U1. Accordingly , the ener gy a v ailable at the NOMA user i is E i = ρα T P V P v =1 g s v i where P is per -antenna po wer of PS, g s v i = | h s v i | 2 with h s v i being channel g ain between the NOMA user i and the v th antenna of PS, and ρ ∈ (0 , 1) means ener gy con v erting ef cienc y . Po wer a v ailable for St age 2 transferred from E i is E i (1 − α ) T due to Stage 2 prolonging (1 − α ) T time unit. Thereby , the NOMA user i conducts transmission in Stage 2 with the follo wing po wer aligned with NLEH in [6]: P i = ραP 1 − α V P v =1 g s v i , α P V P v =1 g s v i ≤ ψ ραψ 1 − α , α P V P v =1 g s v i > ψ = AQ i , Q i ≤ B D , Q i > B (2) wherein ψ is the po wer saturation threshold, A = ραP 1 − α , D = ραψ 1 − α , B = ψ αP , and Q i = V P v =1 g s v i . Stage 2 is for uplink communications using NOMA where U2 and U1 conduct concurrent transm is- sions of x 1 and x 2 with po wers of P 1 and P 2 to NR where E n | x 1 | 2 o = E n | x 2 | 2 o = 1 . Accordingly , NR recei v es a signal as y = h 1 p P 1 x 1 + h 2 p P 2 x 2 + ξ , (3) wherein NR suf fers the noise ξ ∼ N (0 , ζ ) , h 1 and h 2 are channel g ains correspondingly pertinent to channel po wer g ains as g 1 = | h 1 | 2 and g 2 = | h 2 | 2 . Relied on (3), NR performs decoding x 1 and x 2 using NOMA principle. T w o cases of the order to reco v er x 1 and x 2 are possible. If the si g na l from U1 is stronger than U2 ( g 1 P 1 > g 2 P 2 ), then NR rsly Ener gy scavenging-aided NOMA uplink communications: performance analysis (Huu Q. T r an) Evaluation Warning : The document was created with Spire.PDF for Python.
922 ❒ ISSN: 1693-6930 reco v ers x 1 by beha ving x 2 as noise. Thereby , NR restores x 1 with signal-to-interference plus noise ratio (SINR) deri v ed from (3) as ˜ γ x 1 = g 1 P 1 g 2 P 2 + ζ . (4) After suppressing noise which x 1 generates, NR k eeps reco v ering x 2 from ˜ y = y − h 1 √ P 1 x 1 = h 2 √ P 2 x 2 + ξ . Accordingly , in line with ˜ y , NR reco v ers x 2 with signal-to-noise ratio (SNR) to be ˜ γ x 2 = g 2 P 2 ζ . (5) Similarly , if the signal from U2 is stronger than U1 ( g 2 P 2 > g 1 P 1 ), then NR rstly reco v ers x 2 under consideration of x 1 with the role of noise. Afterw ards, NR decodes x 2 with SINR deri v ed by (3) as ˆ γ x 2 = g 2 P 2 g 1 P 1 + ζ . (6) Subsequently after suppressing noise that x 2 induces, NR k eeps reco v ering x 1 from ˆ y = y − h 2 √ P 2 x 2 = h 1 √ P 1 x 1 + ξ . Thereby , in line with ˆ y , NR decodes x 1 with SNR to be ˆ γ x 1 = g 1 P 1 ζ . (7) 3. PERFORMANCE AN AL YSIS OF ENOMA U This part initially mak es the OP analysis of eNOM Au. The OP stands for the probability which the channel capacity lo wers the predetermined transmission speed C 0 . Thereafter , OP analysis is reused to attain TP analysis. Such analyses of fer prompt TP/OP assessment irrespecti v e of e xhausti v e simulations. 3.1. OP of U1 There are three scenarios that cause U1 to be in outage: - The 1st scenario is when the signal at NR from U1 is stronger than that from U2 ( g 1 P 1 > g 2 P 2 ) and NR f ails to decode x 1 ( ˜ γ x 1 < γ 0 ) where γ 0 = 2 C 0 / (1 − α ) − 1 . - The 2nd scenario is when the signal at NR from U2 is stronger than that from U1 ( g 2 P 2 > g 1 P 1 ) and NR f ails to decode x 2 ( ˆ γ x 2 < γ 0 ). - The 3rd scenario is when the signal at NR from U2 is stronger than that from U1 ( g 2 P 2 > g 1 P 1 ) and NR decodes x 2 successfully ( ˆ γ x 2 ≥ γ 0 ) b ut f ails to decode x 1 ( ˆ γ x 1 < γ 0 ). The OP of U1 is thereby e xpressed as: Υ 1 = Pr { g 1 P 1 > g 2 P 2 , ˜ γ x 1 < γ 0 } | {z } Υ 11 + Pr { g 1 P 1 < g 2 P 2 , ˆ γ x 2 < γ 0 } | {z } Υ 12 + Pr { g 1 P 1 < g 2 P 2 , ˆ γ x 2 ≥ γ 0 , ˆ γ x 1 < γ 0 } | {z } Υ 13 . (8) No w , we deri v e all terms ( Υ 11 , Υ 12 , Υ 13 ) to complete the deri v ation of (8). First, inserting (4) into (8) yields. Υ 11 = Pr g 1 P 1 > g 2 P 2 , g 1 P 1 g 2 P 2 + ζ < γ 0 = Pr { z 1 > z 2 , z 1 < γ 0 z 2 + γ 0 ζ } = Pr { z 2 < z 1 < γ 0 z 2 + γ 0 ζ , z 2 < γ 0 z 2 + γ 0 ζ } = ˜ Υ 11 z }| { Pr { z 2 < z 1 < γ 0 z 2 + γ 0 ζ } , γ 0 ≥ 1 Pr z 2 < z 1 < γ 0 z 2 + γ 0 ζ , z 2 < γ 0 ζ 1 − γ 0 | {z } ˜ Υ 12 , γ 0 < 1 (9) where z 1 = g 1 P 1 and z 2 = g 2 P 2 . TELK OMNIKA T elecommun Comput El Control, V ol. 23, No. 4, August 2025: 918–931 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 923 Comment 1 : One sees from (9) that since γ 0 = 2 C 0 / (1 − α ) − 1 , selecting the tar get spectral ef c ienc y C 0 and the time portion α leads to γ 0 < 1 or γ 0 ≥ 1 , inducing Υ 11 to accept dif ferent v alues and e v entually causing dif ferent outage le v els for U1. Also, since ˜ Υ 11 > ˜ Υ 12 and γ 0 represents the SNR/SINR threshold, Υ 11 when selecting C 0 and α in order for γ 0 to be lar ge (i.e. γ 0 ≥ 1 ) is lar ger than that when selecting C 0 and α in order for γ 0 to be small (i.e. γ 0 < 1 ). ˜ Υ 11 and ˜ Υ 12 are represented through f z 2 ( · ) and F z 1 ( · ) as follo ws: ˜ Υ 11 = ∞ Z 0 [ F z 1 ( γ 0 x + γ 0 ζ ) − F z 1 ( x )] f z 2 ( x ) dx, ˜ Υ 12 = γ 0 ζ 1 − γ 0 Z 0 [ F z 1 ( γ 0 x + γ 0 ζ ) − F z 1 ( x )] f z 2 ( x ) dx, (10) where the CDF of z i , i = { 1 , 2 } , is pro v ed in [31] as: F z i ( x ) = B 2 U X u =1 π U p 1 − φ 2 u f Q i ( ϖ u ) F g i x Aϖ u + ¯ F Q i ( B ) F g i x D , (11) wherein U is a parameter of Gaussian-Chebyshe v quadrature [32], ϖ u = B 2 ( φ u + 1) , and φ u = cos 2 u − 1 2 U π . Moreo v er , [31] sho wed f Q i ( x ) and ¯ F Q i ( x ) as: f Q i ( x ) = X G P v =0 a v = V V ! ( G Y t =0 ( H t ) a t a t ! ) x ϕ − 1 Λ ϕ si Γ( ϕ ) e − x Λ si , ¯ F Q i ( x ) = X G P v =0 a v = V Γ ϕ, x Λ si V ! Γ ( ϕ ) G Y t =0 ( H t ) a t a t ! , (12) where ϕ = G P t =0 a t β t . The deri v ati v e of F z i ( x ) yeilds the PDF of z i to be: f z i ( x ) = d F z i ( x ) dx = B 2 U X u =1 p 1 − φ 2 u π f Q i ( ϖ u ) U Aϖ u f g i x Aϖ u + ¯ F Q i ( B ) D f g i x D . (13) Inserting (11) and (13) into (10) and after some careful simplications, one attains the e xplicit e xpres - sion of ˜ Υ 11 to be: ˜ Υ 11 = ¯ F Q 1 ( B ) ¯ F Q 2 ( B ) D Ψ ( D , D ) + π B 2 U U X u =1 p 1 − φ 2 u f Q 1 ( ϖ u ) ¯ F Q 2 ( B ) D Ψ ( Aϖ u , D ) + f Q 2 ( ϖ u ) ¯ F Q 1 ( B ) Aϖ u Ψ ( D , Aϖ u ) + π B 2 U U X v =1 p 1 − φ 2 v f Q 1 ( ϖ u ) f Q 2 ( ϖ v ) ϖ v A Ψ ( Aϖ u , Aϖ v ) ) , (14) where φ v and ϖ v are dened in the same manner as φ u and ϖ u whilst the function Ψ ( · , · ) has a closed form as (17). Ψ ( · , · ) is a function of tw o ar guments ( I , J ) as: Ψ ( I , J ) = ∞ Z 0 F g 1 γ 0 x + γ 0 ζ I − F g 1 x I f g 2 x J dx. (15) Ener gy scavenging-aided NOMA uplink communications: performance analysis (Huu Q. T r an) Evaluation Warning : The document was created with Spire.PDF for Python.
924 ❒ ISSN: 1693-6930 In v oking f g i ( · ) and F g i ( · ) in (1) and using the binomial e xpansion, one simplies Ψ ( I , J ) as: Ψ ( I , J ) = ∞ Z 0 " G X k =0 β k − 1 X u =0 H k (Λ 1 I ) u u ! x u e − x Λ 1 I − e − γ 0 ζ Λ 1 I γ u 0 ( x + ζ ) u e − γ 0 x Λ 1 I # × " G X m =0 H m J 1 − β m Λ β m 2 Γ ( β m ) x β m − 1 e − x Λ 2 J # dx = G X k =0 β k − 1 X u =0 G X m =0 H k H m J 1 − β m (Λ 1 I ) u u !Λ β m 2 Γ ( β m ) ∞ Z 0 x u + β m − 1 e − 1 Λ 1 I + 1 Λ 2 J x − e − γ 0 ζ Λ 1 I γ u 0 e − γ 0 Λ 1 I + 1 Λ 2 J x x β m − 1 u X l =0 C l u x l ζ u − l # dx = G X k =0 β k − 1 X u =0 G X m =0 H k H m J 1 − β m (Λ 1 I ) u u !Λ β m 2 Γ ( β m ) ∞ Z 0 x u + β m − 1 e − 1 Λ 1 I + 1 Λ 2 J x dx − e − γ 0 ζ Λ 1 I γ u 0 u X l =0 C l u ζ u − l ∞ Z 0 x l + β m − 1 e − γ 0 Λ 1 I + 1 Λ 2 J x dx . (16) The last inte grals in (16) are solv ed with the help of [33] (3.351.3) Grad, e v entually leading to the closed form of Ψ ( I , J ) as: Ψ ( I , J ) = G X m =0 G X k =0 β k − 1 X u =0 H k H m J 1 − β m (Λ 1 I ) u u !Λ β m 2 Γ ( β m ) " Γ ( u + β m ) 1 Λ 1 I + 1 Λ 2 J − ( u + β m ) − e − γ 0 ζ Λ 1 I γ u 0 u X l =0 C l u ζ u − l Γ ( l + β m ) γ 0 Λ 1 I + 1 Λ 2 J − ( l + β m ) # . (17) Subsequently , by e xploiting the Gaussian-Chebyshe v quadrature, one tightly approximates ˜ Υ 12 as: ˜ Υ 12 = γ 0 ζ 2 (1 − γ 0 ) U X u =1 π U p 1 − φ 2 u [ F z 1 ( γ 0 χ u + γ 0 ζ ) − F z 1 ( χ u )] f z 2 ( χ u ) , (18) where χ u = γ 0 ζ 2(1 − γ 0 ) ( φ u + 1) . Ne xt, inserting ˆ γ x 2 in (6) into Υ 12 yields. Υ 12 = Pr g 1 P 1 < g 2 P 2 , g 2 P 2 g 1 P 1 + ζ < γ 0 . (19) By comparing Υ 12 in (19) with Υ 11 in (9), one recognizes that the e xplicit formula of Υ 12 is obtained from that of Υ 11 by interchanging (Λ 1 , Λ s 1 ) and (Λ 2 , Λ s 2 ) . Accordingly , the deri v ation of Υ 12 is omitted for compactness. Finally , inserting ˆ γ x 2 in (6) and ˆ γ x 1 in (7) into Υ 13 yields. Υ 13 = Pr g 1 P 1 < g 2 P 2 , g 2 P 2 g 1 P 1 + ζ ≥ γ 0 , g 1 P 1 ζ < γ 0 = Pr { g 1 P 1 < g 2 P 2 , g 2 P 2 ≥ γ 0 g 1 P 1 + γ 0 ζ , g 1 P 1 < γ 0 ζ } = Pr { g 2 P 2 ≥ γ 0 g 1 P 1 + γ 0 ζ , g 1 P 1 < γ 0 ζ } = γ 0 ζ Z 0 [1 − F z 2 ( γ 0 x + γ 0 ζ )] f z 1 ( x ) dx = F z 1 ( γ 0 ζ ) − γ 0 ζ Z 0 F z 2 ( γ 0 x + γ 0 ζ ) f z 1 ( x ) dx. (20) TELK OMNIKA T elecommun Comput El Control, V ol. 23, No. 4, August 2025: 918–931 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 925 γ 0 ζ R 0 F z 2 ( γ 0 x + γ 0 ζ ) f z 1 ( x ) dx has a tight approximation by emplo ying Gaussian-Chebyshe v quadra- ture, yielding. Υ 13 = F z 1 ( γ 0 ζ ) − γ 0 ζ 2 U X u =1 π U p 1 − φ 2 u F z 2 ( γ 0 δ u + γ 0 ζ ) f z 1 ( δ u ) , (21) where δ u = γ 0 ζ 2 ( φ u + 1) . 3.2. OP of U2 There are three scenarios that cause U2 to be in outage: - The 1st scenario is when the signal at NR from U2 is stronger than that from U1 ( g 2 P 2 > g 1 P 1 ) and NR f ails to decode x 2 ( ˆ γ x 2 < γ 0 ). - The 2nd scenario is when the signal at NR from U1 is stronger than that from U2 ( g 1 P 1 > g 2 P 2 ) and NR f ails to decode x 1 ( ˜ γ x 1 < γ 0 ). - The 3rd scenario is when the signal at NR from U1 is stronger than that from U2 ( g 1 P 1 > g 2 P 2 ) and NR decodes x 1 successfully ( ˜ γ x 1 ≥ γ 0 ) yet f ails to reco v er x 2 ( ˜ γ x 2 < γ 0 ). The OP of U2 is thereby e xpressed as: Υ 2 = Pr { g 1 P 1 < g 2 P 2 , ˆ γ x 2 < γ 0 } | {z } Υ 21 + Pr { g 1 P 1 > g 2 P 2 , ˜ γ x 1 < γ 0 } | {z } Υ 22 + Pr { g 1 P 1 > g 2 P 2 , ˜ γ x 1 ≥ γ 0 , ˜ γ x 2 < γ 0 } | {z } Υ 23 . (22) By comparing (22) with (8), one infers that Υ 21 = Υ 12 and Υ 22 = Υ 11 . Moreo v er , inserting ˜ γ x 1 in (4) and ˜ γ x 2 in (5) into (22) yields. Υ 23 = Pr g 1 P 1 > g 2 P 2 , g 1 P 1 g 2 P 2 + ζ ≥ γ 0 , g 2 P 2 ζ < γ 0 . (23) By comparing Υ 23 with Υ 13 in (20), one recognizes that the e xplicit formula of Υ 23 is obtained from that of Υ 13 by interchanging (Λ 1 , Λ s 1 ) and (Λ 2 , Λ s 2 ) . Accordingly , the deri v ations of Υ 21 , Υ 22 , and Υ 23 are omitted for com pactness. Moreo v er , similar to Υ 1 , one sees that the outage le v el of U2, Υ 2 , can be adjusted by selecting properly C 0 and α . Comment 2 : Both Υ 1 and Υ 2 are functions of specications ( C 0 , α , P , V , ψ , ρ ) . This impl ies U1 and U2 is able to attain tar get performances by properly choosing these specications. 3.3. Thr oughput In delay-constrained communication, the throughput of U i for eNOMAu is computed as: ∆ i = (1 − α ) C 0 (1 − Υ i ) . (24) One sees from (24) that the TP of U i is jointly specied by parameters ( C 0 , α , P , V , ψ , ρ ) since this cluster impacts Υ i . Accordingly , the e xpected TP is obtained by appropriatel y choosing such parameters based on their preset v alue ranges. 4. ASYMPT O TIC AN AL YSIS OF ENOMA U This secti on studies the performance upper -bound of eNOM Au corresponding t o high trans mission po wer ( P → ∞ ). In such a re gime, ener gy harv ester denitely saturates and er go, P i → D with i = { 1 , 2 } . Accordingly , the CDF and PDF of z i = g i D reduce to F z i ( x ) = F g i x D and f z i ( x ) = 1 D f g i x D , corre- spondingly , when P → ∞ . Therefore, the OP of U i is summarized to be: Υ ∞ i = Υ ∞ i 1 + Υ ∞ i 2 + Υ ∞ i 3 , (25) where Υ ∞ 11 = Υ ∞ 22 , Υ ∞ 12 = Υ ∞ 21 , the e xplicit e xpressions of Υ ∞ 12 and Υ ∞ 23 are obtained from those of Υ ∞ 11 and Υ ∞ 13 by interchanging (Λ 1 , Λ s 1 ) and (Λ 2 , Λ s 2 ) , and Υ ∞ 11 = ˜ Υ ∞ 11 , γ 0 ≥ 1 ˜ Υ ∞ 12 , γ 0 < 1 (26) Ener gy scavenging-aided NOMA uplink communications: performance analysis (Huu Q. T r an) Evaluation Warning : The document was created with Spire.PDF for Python.
926 ❒ ISSN: 1693-6930 with ˜ Υ ∞ 11 = G X k =0 β k − 1 X u =0 G X m =0 H k H m D − u − β m Λ u 1 u !Λ β m 2 Γ ( β m ) " Γ ( u + β m )  1 Λ 1 + 1 Λ 2 1 D − u − β m − e − γ 0 ζ Λ 1 D γ u 0 u X l =0 C l u ζ u − l Γ ( l + β m )  γ 0 Λ 1 + 1 Λ 2 1 D − l − β m # (27) ˜ Υ ∞ 12 = G X k =0 β k − 1 X u =0 G X m =0 H k H m D − u − β m Λ u 1 u !Λ β m 2 Γ ( β m ) Φ γ 0 ζ 1 − γ 0 , u + β m − 1 , 1 Λ 1 + 1 Λ 2 1 D − e − γ 0 ζ Λ 1 D γ u 0 u X l =0 C l u ζ u − l Φ γ 0 ζ 1 − γ 0 , l + β m − 1 , γ 0 Λ 1 + 1 Λ 2 1 D # (28) ˜ ∆ ∞ 13 = G X k =0 β k − 1 X u =0 G X m =0 u X l =0 C l u Λ − β m 1 ζ u − l H m H k Γ ( β m ) u ! D u + β m +1 γ 0 Λ 2 u e − γ 0 ζ Λ 2 D Φ γ 0 ζ , l + β m − 1 , γ 0 Λ 2 + 1 Λ 1 1 D (29) Φ ( a, b, c ) = b ! c b +1 − e − ac b X v =0 b ! v ! a v c b − v +1 . (30) 5. ENERGY SCA VENGI NG-AIDED OR THOGON AL MUL TIPLE A CCESS UPLINK COMMUNI- CA TIONS (EOMA U) In ener gy sca v enging-aided orthogonal multiple access uplink communications (eOMAu), Stage 2 is separated equally into tw o sub-stages during which U1 and U2 transmit their information sequentially and directly to NR. Therefore, NR recei v es the signal from U i as y i = h i √ P i + ξ i where ξ i is the noise at NR and i = { 1 , 2 } . Accordingly , the channel capacity that U i achie v es is R i = 1 − α 2 log 2 1 + g i P i ζ where the f actor 1 − α 2 before the log arithm is since U i transmits only in 1 − α 2 T . Consequently , the OP of U i is: Υ O M A i = Pr { R i < C 0 } = Pr g i P i ζ < ˜ γ 0 = F z i ( ˜ γ 0 ζ ) , (31) where ˜ γ 0 = 2 2 C 0 / (1 − α ) − 1 . One notes that eOMAu is deemed as a baseline transmission scheme as compared to eNOMAu. Gi v en the closed-form formula of Υ O M A i , it is con v enient in quickly comparing the performances between eOMAu and eNOMAu from which the adv antages of NOMA are e xposed promptly . 6. EV ALU A TIONS This section discusses multiple analytical/simulated ndings to rate outage probabilities of U1 and U2 in eNOMAu and its eOMAu counterpart. The analytical formulas deri v ed in sections 3-5 yield analytical ndings (Ana.). Also, Monte-Carlo simul ation is run, yielding simulated ndings (Sim.). Comparing ‘Sim. ’ with ‘ Ana. ’ v alidates theoretical deri v ations. Since OP and TP are linearly proportional to each other , TP is an one-by-one mapping of OP . Accordingly , this section presents merely outage probabilities of U1 and U2. T erminals’ s positions are illustrated as NR (50 , 0) m, U1 (0 , 0) m, U2 ( − 10 , 15) m, PS ( − 15 , 0) m. Unless otherwise addressed, parameters are adopted as ( κ, β , µ ) = (3 , 4 , 2) , ϱ = 10 − 2 , ψ = − 10 dB, α = 0 . 4 , σ = 3 , P = 15 dB, C 0 = 0 . 5 bps/Hz, ρ = 0 . 7 , V = 4 , and ζ = − 90 dBm. Findings in subsequent gures demonstrate that i ) analysis matches simulation, conrming preciseness of e xpressions obtained in sections 3-5; ii ) U1 outperforms U2, which mak es sense since U1 is nearer to PS and NR than U2. Figure 2(a) demonstrates OPs of U1 and U2 v ersus P , which un v eils performance enhancement with accreting P for both U1 and U2. This mak es sense o wing to increasing sca v enged ener gy . Further , eNOM Au signicantly outperforms its eOMAu counterpart o v er the whole range of P , which e xposes the adv antage of NOMA in comparison with OMA. Also, Figure 2(b) re v eals outage performances of U1 and U2 v ersus ψ . One recognizes that the performances of both U1 and U2 are ameliorated with accreting ψ , as anticipated, for both TELK OMNIKA T elecommun Comput El Control, V ol. 23, No. 4, August 2025: 918–931 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA T elecommun Comput El Control ❒ 927 eNOMAu and eOMAu. Moreo v er , U1 and U2 saturate at high ψ because NLEH coincides LEH. Further , U1 and U2 are in a complete outage for lo w ψ , as anticipated. Furthermore, similar to Figure 2(a), ndings in Figure 2(b) illustrate that eNOMAu considerably outperforms eOMAu for an y ψ . Figure 3 e xposes OPs of U1 and U2 ag ainst V (Figure 3(a)) and ρ (Figure 3(b)). One e xpect s that accreting V and ρ f acilitates U1 and U2 in harv esting more ener gy and as a result, mitig ating OPs of both U1 and U2. Figure 3 un v eils e xactly such an e xpectation wherein accreting V and ρ dramatically impro v es performances of both U1 and U2. Further , similar to Figure 2, the results in Figure 3 il lustrate that eNOMAu is superior to its eOMAu counterpart o v er the whole range of V and ρ , which ag ain illustrates the adv antage of NOMA as compared to OMA. Figure 4 re v eals outage performances of U1 and U2 v ersus α (Figure 4(a)) and C 0 (Figure 4(b)). One observ es from Figure 4(a) that one can optimize α to reach minimum OPs for U1 and U2, and for eNOMAu and eOMAu. The optimal α is for poising durations for ES and transmission. Additionally , Figure 4(b) illustrates outage increase with increasing C 0 for both U1 and U2 as well as both eNOMAu and eOMAu, as e xpected. As analyzed in section 3., the SNR/SINR threshold γ 0 is controlled jointly by C 0 and α . Moreo v er , the analysis in section 3 e xposes that γ 0 ≥ 1 and γ 0 < 1 cause di f ferent outage le v els for eNOM Au. As such, the results in Figure 4(a) and Figure 4(b) mak e sense in that eNOMAu is not al w ays better than eOMAu. More specically , Figure 4(a) un v eils that the performance of U1 (or U2) in eNOMAu outperforms that in eOMAu when α < 0 . 47 (or α < 0 . 53 ) whilst Figure 4(b) illustrates that the performance of U2 in eNOMAu outperforms that in eOMAu for an y C 0 b ut the performance of U1 in eNOMAu outperforms that in eOMAu only for C 0 < 0 . 57 bps/Hz or C 0 > 1 . 29 bps/Hz. P (dB) 0 2 4 6 8 10 12 14 16 18 20 Outage probability 10 -4 10 -3 10 -2 10 -1 10 0 OMA: U1 - Ana. OMA: U2 - Ana. NOMA: U1 - Ana. NOMA: U2 - Ana. OMA: U1 - Sim. OMA: U2 - Sim. NOMA: U1 - Sim. NOMA: U2 - Sim. ψ  (dB) -55 -50 -45 -40 -35 -30 Outage probability 10 -2 10 -1 10 0 OMA: U1 - Ana. OMA: U2 - Ana. NOMA: U1 - Ana. NOMA: U2 - Ana. OMA: U1 - Sim. OMA: U2 - Sim. NOMA: U1 - Sim. NOMA: U2 - Sim. OMA: U1 OMA: U2 NOMA: U1 NOMA: U2 LEH NLEH (a) (b) Figure 2. Inuence of (a) OP v ersus P and (b) OP v ersus ψ V 1 2 3 4 5 6 7 8 9 10 Outage probability 10 -3 10 -2 10 -1 10 0 OMA: U1 - Ana. OMA: U2 - Ana. NOMA: U1 - Ana. NOMA: U2 - Ana. OMA: U1 - Sim. OMA: U2 - Sim. NOMA: U1 - Sim. NOMA: U2 - Sim. ρ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Outage probability 10 -2 10 -1 10 0 (a) (b) Figure 3. Inuence of (a) OP v ersus V and (b) OP v ersus ρ Ener gy scavenging-aided NOMA uplink communications: performance analysis (Huu Q. T r an) Evaluation Warning : The document was created with Spire.PDF for Python.