Non Orthogonal Multiple Access with Orthogonal Time Frequency Space Signal Transmission

arXiv:2003.06387v2 [cs.IT] 31 May 2020 LR pcrlefficiency. spectral BLER, infiatybte hnNM-FMi em fbokerror block throughout. of show and the terms goodput results at in (BLER), link-level NOMA-OFDM rate improved performs Whereas than receiver is better NOMA-OTFS SE. based NOMA-OTFS significantly mean CWIC developed of in the SE that decrease NOMA-OFDM, outage of to that cost compared When find OMA-OTFS. we syst than higher SE provides w NOMA- NOMA-OTFS sum evaluation, that and performance interestingly system-level observe (OMA-OFDM) From system schemes. multiplexing OFDM NOMA-OTFS and division proposed level OMA-orthogonal frequency OMA-OTFS, the system benchmark against of The compared are performance system. performan level NOMA-OTFS level along link practical link codes evaluating such LDPC for using of equalization realizab receiver MMSE a (CWIC) with design SIC th further level in We SE NOMA. codeword sum of the version maximize to high-mobility (SE) schemes efficiency spectral allocation post-proces sum power symbol-wise system develop evaluate for to order expression in derive SINR NOMA-O we successive for which (MMSE)- receiver for error based (OMA square (SIC) whi access cancellation mean multiple interference conditions, minimum mobilit orthogonal low a mobility in present to efficiency We high comparison spectral to high in yield scenarios resilience to known its is T NOMA for tech its perspective. transmission OTFS known level evaluate while that link is and fact the and in system lies level challenge OTFS ‘NOMA-OTFS’ system transmission based called from work, wireless performance (NOMA) be this In suitable may access scenarios. which multiple a mobility nonorthogonal high as in propose times use for recent technology in pursued ihG .Sna colo eeomnctos ninInst Indian Telecommunications, aritrachatterjee@ of e-mail: suvra@gssst School India. [email protected]; [email protected]; Sanyal Kharagpur, S. Technology, G. with 1] lhuhi h poig5 e ai N) the (NR), adapt radio to [13] new flexible 5G made is upcoming OFDM of the bandwidth subcarrier in scena such inter Although in from Doppler suffers high [12]. to OFDM due that (ICI) known interference 4G-LTE well carrier [10], is WiFi it systems However [9], [11]. [8], communication DVB-S wireless DVB-A, DVB-T, broadband namely popular waveforms successful is most used the of divi one frequency is Orthogonal (OFDM) transmissi multiplexing [7]. quality challenge existing a high using is ve- providing technologies service aerial where communication unmanned etc., (V2V) wireless [6] [3], vehicle-to-vehicle [5], [2], [4], communications [1], communications (UAV) railways hicle speed high as Motivation and Background A. cetdfrpbiaini EESsesJunl uhr a Authors Journal. Systems IEEE in publication for Accepted ne Terms Index Abstract eaeeprecn e ihmblt cnro such scenarios mobility high new experiencing are We o rhgnlMlil ceswt Orthogonal with Access Multiple Orthogonal Non Otooa iefeunysae(TS sbeing is (OTFS) space frequency time —Orthogonal ieFeunySaeSga Transmission Signal Space Frequency Time rtaCatre,VvkRnagr,Sahn iaiadS and Tiwari Shashank Rangamgari, Vivek Chatterjee, Aritra NM,OF,SC DC oe allocation, power LDPC, SIC, OTFS, —NOMA, .I I. NTRODUCTION ikpa.n rk- iitkgp.ac.in; .iitkgp.ernet.in. tt of itute nology TFS, We . sion sing rios em we on he ce le le re ). y e e akt rnmte eoe udtdvr atwihlimits which fed fast CSI very The outdated work. becomes high this transmitter in in to considered rapidly back as instan- change user’s scenarios which the mobility coefficients, of aware channel made taneous be must transmitter [30]. NOMA [29], in available is systems, analysi transmit performance NOMA at and power-domain (CSI) allocation of resource information on state overview eit channel An on partial based and r done or and downlink is [28] full allocation both [27], resource [26], Such for [25], therein. [24], ences length [23], allocation at in resource directions addressed T-F uplink are and issues BS grouping at user Such power (ii) transmit available users, total among of division (i) are NOMA aim its OMA-OTFS. We compare and now. and NOMA-OFDM NOMA-OTFS against till the performance attention investigate and limited develop only to attracted has [21], which performance (SE) efficiency spectral [20] sum multiple [22]. schemes of NOMA orthogonal code-division terms as outperform in well as significantly (OMA) to access the schemes achieve found PD-NOMA known channel. are is broadcast receiver Gaussian interf the of Power-domain successive at capacity with unit. (SIC) along cancellation superposition transmitter resource ence the using one at realized (SC) in coding schemes user (PD-NOMA) one NOMA than more D-D different allocated are D [19]: users (ii) OTFS resources. where and MA [14]: resources two T-F T-F MA-OTFS OTFS, different Do (i) to allocated reported are reference users With are where user. OMA-OTFS u one of resource only types one to i.e. allocated resource maintained is OMA, is (ii) In orthogonality and (NOMA). (OMA) allocation access access multiple multiple orthogonal non orthogonal on (i) namely focus nique we work technology. transmission this OTFS In for access access. multiple multiple and technology 1] 1] 1] 1] si rvdsgetipoeet in improvements scenarios great mobility high provides globe new it such the being in as across especially is [18], performance researchers plane, [17], by (T-F) [16], [15], enthusiasm time-frequency with to (De-Do explored opposed delay-Doppler On in as [14] sections. constellation (OTFS) plane later space signal frequency in places time sever discussed which orthogonal to be hand due other will limited the is as it constraints yet other conditions, channel various to sidctdaoe nodrt civ pia gains, optimal achieve to order in above, indicated As of implementation the to pertaining aspects Important work, this in NOMA-OTFS investigate to aim we Therefore allocate schemes NOMA methodology, OMA to contrast In tech- (MA) access multiple of class broad two are There transmission of comprises (RAT) technology access radio A vaSka Das, Sekhar uvra ebr IEEE Member, efer- her ter. er- nit e- al . s 1 ) . 2 the achievable gain of using T-F domain NOMA. performance of the proposed NOMA-OTFS in practical Further, because of OFDM’s limited capability to handle ITU De-Do channel model [33]. The results provide high Doppler restricts the choice of NOMA-OFDM as a RAT better estimate of such NOMA-OTFS in future realistic in high mobility scenarios. Since OTFS is resilient to high scenarios for 5G and beyond. Doppler in comparison to OFDM, we aim to investigate the • The framework developed in this work is made flexible use of NOMA with OTFS so that multi user extension of so as to handle OTFS and OFDM in an unified matrix OTFS can be achieved in such high-mobility conditions. Such representation. It is also worth noting that the modified investigation is expected to pave the path for future research OFDM framework we adopt in this work use block cyclic on methods for multi-user spectral efficiency enhancement prefix (CP) along with MMSE equalizer. Accordingly we techniques in high mobility scenarios. analyze the performance of OFDM with block processing and ICI canceling receiver for comparison against OTFS. B. Related Works and Contribution The above discussions are for system level performance evaluation. While SE performance analysis gives us one perspective, The interplay of two futuristic technologies namely OTFS it is also vital to evaluate the link level performance for such and NOMA has attracted the attention of researchers as re- NOMA-OTFS system in order to have a comprehensive view ported in [31], [32]. In [31] use of NOMA with OTFS has been of the performance of such newly proposed system namely presented in order to serve users with heterogeneous mobility NOMA-OTFS. Accordingly, the following are included: profiles for uplink and downlink. In [32], beamforming aspect of ‘OTFS assisted NOMA’ networks has been explored (in • In order to get a link level performance estimation, we presence of multi-antenna base station) to maximize the low- need to develop a receiver for NOMA-OTFS system. mobility NOMA users’ data rate while maintaining high- Accordingly we have developed a codeword level low mobility OTFS user’s target data rate. density parity check (LDPC)-SIC receiver which uses In light of the limited state-of-the-art available as indicated symbol-level log-likelihood ratio (LLR) values of the above, the major contributions of this work are outlined as MMSE based ICI canceling receiver (see Sec. V), which follows: to the best of the authors’ knowledge first such attempt. • The performance of such realistic LDPC enabled MMSE- • The system model in [31], [32] considers only the user SIC receiver is further compared with NOMA-OFDM with highest velocity is served in De-Do plane (using and OMA-OTFS in terms of block error rate (BLER), OTFS scheme). Whereas, the rest of low-mobility users throughput (in bits/sec/Hz) and goodput (in bits/sec/Hz). are served in same T-F plane (using OFDM modulation) • A comprehensive performance analysis taking into ac- which are multiplexed in PD-NOMA. Therefore, the user count the system level and link level performance has with high mobility (served with OTFS) does not partic- been presented in this work (see Sec. VI). ipate in De-Do NOMA transmission. In this work we propose and develop a holistic framework to obtain the Notations: We use the following notations throughout the De-Do PD-NOMA-OTFS where multiple high-mobility paper. We let x, x and X represent scalars, vectors and users are served by OTFS in the same De-Do resource matrices respectively. The superscripts (.)Tand (.)† indicate block, which is the first such proposal to the best of our transpose and conjugate transpose operations, respectively. IN knowledge. and WL represents identity matrix with order N and L-order • The pulse shape used in [31], [32] is considered to normalized IDFT matrix respectively. Kronecker product be ideal in nature, which is not realizable in practice operator is represented by . The Frobenius norm of any ⊗ due to time-frequency uncertainty principle [17]. Such matrix X is denoted by X F . diag[.] denotes a diagonal || || ideal assumption simplifies the system equations which matrix whose diagonal elements are formed by the elements yields to block circulant system matrices. In this work we of the vector inside. circ . denotes a circulant matrix whose { } consider realizable time domain rectangular pulse which first column is given by the vector inside. The expectation does not offer such simplification. parameter is denoted by E[.]. Column-wise vectorization of • Furthermore, in [31, Sec. VII], NOMA users are allocated matrix (.) is represented by vec . . The ceiling operator is { } fixed power without taking into account their channel denoted as . . N[a b] represents the set of natural numbers ⌈ ⌉ condition. Such elementary power allocation restricts ranging from a to b. j = √ 1. − NOMA gain. We evaluate the performance of NOMA- OTFS with different dynamic power allocation strategies suitably designed for high-mobility environments. II. OTFS SIGNAL MODEL • In [31], SE results are obtained using Shannon’s expression using ideal SIC at the receiver. In this work we We consider a multi-carrier and multi time-slot system with compute post-processing symbol-level SINR for practical total Tf sec. duration and B Hz. bandwidth. We have total ICI canceling MMSE with SIC NOMA receiver, which M number of sub-carriers having ∆f sub-carrier bandwidth renders the results more close to reality (in Sec. III-A2 and N number of symbols having T symbol duration, thus and III-B2). B = M∆f and Tf = NT . • In [31], [32] arbitrary De-Do channel is considered For a user (termed as i-th user henceforth), the QAM for performance analysis, whereas we have evaluated modulated Delay-Doppler data symbols, di(k,l) C, k ∈ ∈ 3

N[0 N 1], l N[0 M 1], are arranged over Doppler- We consider linear time varying (LTV) channels for all the − ∈ k − l delay lattice Λ = ( , ) . Data symbols di(k,l) is users. Let, the i-th user’s channel consists of Pi paths with h i { NT M∆f } p mapped to time-frequency domain data Xi(n,m) on lattice complex attenuations, τpi delays and νpi Doppler values for ⊥ i i Λ = (nT, m∆f) , n N[0 N 1] and m N[0 M 1] p th path where p N[1 Pi]. Thus, Delay-Doppler channel by using{ inverse symplectic} ∈ fast Fourier− transform∈ (ISFFT)− . spreading function for∈ the i-th user can be given as,

Thus Xi(n,m) can be given as [16], Pi N−1 M−1 hi(τ,ν)= hpi δ(τ τpi )δ(ν νpi ), i =1, ,K. (6) 1 j2π[ nk − ml ] − − ··· Xi(n,m)= di(k,l)e N M . (1) pXi=1 √NM Xk=0 mX=0 i The delay and Doppler values for p th path is given as τpi = i i Next, a time-frequency modulator modulates Xi(n,m) to time lp kp i i and ν i = , where l N[0 M 1] and k domain using Heisenberg transform as, M∆f p NT p ∈ − p ∈ N[0 N 1] are the number of delay and Doppler bins on the Doppler-delay− lattice for pith path. We assume that N and M N−1 M−1 j2πm∆f(t−nT ) are sufficiently large so that there is no effect of fractional si(t)= (√ Xi(n,m))g(t nT )e , P − delay and Doppler on the performance. We also assume the nX=0 mX=0 i perfect knowledge of (h i , l i , k i ), p N[0 Pi 1], at the (2) p p p ∈ − where, g(t) is transmitter pulse of duration T and transmit receiver of i-th user, as previously considered in [16], [18]. i power is denoted by . Further, si(t) is sampled at the One work on such estimation is given in [34]. Let τmax and PT i sampling interval of . We collect samples of si(t) in νmax be the maximum delay and Doppler spread for users. M i i si = [si(0) si(1) si(MN 1)]. The QAM symbols di(k,l) Channel delay length α = τmaxM∆f and channel Doppler ··· − i i ⌈ ⌉ i are arranged in M N matrix as, length, β = νmaxNT . LCP = max (α ). × ⌈ ⌉ i=1,··· ,K di(0, 0) di(1, 0) · · · di(N − 1, 0) At the i-th user’s receiver, after removal of CP, the received  di(0, 1) di(1, 1) · · · di(N − 1, 1)  signal can be written as [17], Di = . . . .  . . . .  r H s n  . . . .  i = i + i, i =1, ,K. (7) di(M − 1, 0) di(M − 1, 1) · · · di(N − 1, M − 1) ··· (3) where, ni is white Gaussian noise vector of length MN with 2 The transmitted signal can be written as matrix-vector multi- elemental variance σn and Hi is a MN MN channel matrix plication as: for ith user which is given by, × s A√ d i = i, (4) Pi P i li ki d D A W I Hi = h Π p ∆ p , i =1, ,K, (8) where, i = vec i . Finally, MN×MN = N M p ··· denotes the OTFS{ modulation} matrix. A cyclic prefix⊗ (CP) pXi=1 ′ of length L LCP 1 is appended at the starting of the Π T CP with MN×MN = circ [0 1 0 0]MN×1 is a circulant ≥ − − s, where L is the channel’s maximum excess delay length. { ···j2π 1 }j2π MN 1 CP delay matrix and ∆ = diag[1 e MN e MN ] is a In order to implement OFDM in the same framework, the diagonal Doppler matrix. Using the above··· mentioned signal modulation matrix is modified as A = IN WM . ⊗ model for the De-Do PD-NOMA-OTFS in downlink, we proceed to evaluate the corresponding SINR and SE experienced III. DELAY-DOPPLER POWER-DOMAIN NOMA-OTFS by the non-orthogonally multiplexed users. In this section, we further extend the OTFS signal model presented in Sec. II in order to develop the multi-user De-Do PD-NOMA-OTFS signal models and derive the SINR and SE expressions of the same for both downlink and uplink. We 2) Receiver Processing, SINR and SE Analysis: In OTFS, consider K users with high velocity are multiplexed in power when the signal passes through the LTV channel, it encounters domain all of which are served by OTFS (unlike [31], [32]) both ICI and inter-symbol interference (ISI), thereby degrading in both downlink and uplink transmission. its performance. In order to negate such induced ICI and ISI, we equalize the received signal through LMMSE equalizer, as A. De-Do PD-NOMA-OTFS Downlink done in [35]. Furthermore, in the later stage, SIC has been performed in order to mitigate the NOMA interference at the 1) Transmit Signal Model: Among the K high mobility receiver, which has been discussed subsequently. users multiplexed in power domain, we assume The i-th user K The total effective noise at the i-th receiver amounts to: is allocated βi fraction of total power . Clearly, βi =1. P i=1 K Choice of βi is described in detail in Sec. IV. FollowingP the n˜i = βi′ HiAdi′ + ni. (9) principle of superposition, the composite transmitted signal DL P i′=1X,i′=6 i p from the transmitter intended for all users can be written by modifying (4) as: Assuming the total effective noise following Gaussian distribution, LMMSE equalization on the received signal r in (7) K i results in estimated data vector for i-th user as given in (10), s = A βi di. (5) P where Γi denotes the average SNR of i-th user. Rewriting (10) Xi=1 p 4

K −1 † † † 1 † † 1 −1 dˆi = βi(HiA) βi(HiA)(HiA) + βi′ (HiA)(HiA) + I ri = βi(HiA) [(HiA)(HiA) + I] ri, (10) Γi Γi p i′=1X,i′6=i p

by using (7) and (5), we obtain: assumed that the receiver BS has perfect knowledge about

K the channels from transmitting users. The OTFS modulated transmitted vector from i-th user is given by: dˆi = Bi (βi )di + Bi (βi′ )di′ (11) P ′ ′ P u u desiredp signal i =1X,i =6 i p s = A ud , (15) i Pi i | {z } NOMA interference u du p where i and i denote the transmit power and vectorized + | Cini {z, i =1, } ,K, P ··· transmit data of the i-th user respectively. The uplink average noise component u u 2 SNR of the i-th user is given by Γi = i /σn. The aggregate | {z } P where, for notational simplicity, we assign Ci = received signal at the base station after removal of CP is given † † 1 −1 √βi(HiA) [(HiA)(HiA) + I] and Bi = CiHiA. by: Γi At this point, without loss of generality, we consider that K r Husu n from the transmitting BS the distance of i-th user is higher u = i i + , (16) than the (i + 1)-th user for i = 1, , (K 1), thus Xi=1 in terms of received average SNR, it··· can be− written as: where Hu denotes the MN MN delay-Doppler uplink i × Γ1 < Γ2 < < Γi−1 < Γi < < ΓK . Thus, we assume channel matrix from i-th user to the BS. Similar to the that following··· the principle of··· NOMA, the i-th user will downlink scenario presented before, we further proceed to not face any interference due to the signals intended for 1st, evaluate the SINR and SE experienced by the PD-NOMA- 2nd, , (i 1)-th users through perfect SIC1. Using these OTFS users in uplink direction in the following section. assumptions··· − and expanding (11), the symbol-wise pre- and post SIC received SINR at any user can be formulated. For the i-th user, the downlink pre- and post-SIC SINR for j-th symbol (denoted as ΥPre-D and ΥPost-D respectively) can be ij ij 2) Receiver Processing, SINR and SE Analysis: During given by (12) and (13) respectively at the top of the next uplink transmission in NOMA, the signal from the users page, with i = 1, ,K and j = 1, ,MN. bi and ci ··· ··· pq pq with higher SNR are sequentially decoded and successively denote the (p, q)th elements of B and C respectively. In i i canceled from the aggregate signal. For the same user ordering OTFS, the SINR achieved in all symbols are nearly same for as considered in downlink transmission (Γu < < Γu ), large M and N values [38] and thus, henceforth we drop 1 K while decoding the i-th user’s signal, the BS will consider··· the subscript j and represent the pre- and post-SIC SINR of i-th 1st, 2nd, , (i 1)-th users’ signals as noise. Thus, for i-th user as ΥPre-D and ΥPost-D respectively. Thus, the downlink i i user, the effective··· − noise can be denoted as: sum rate of the system in bps/Hz is given by: i−1 K u u u n˜i = ′ H ′ Ad ′ + n. (17) RDL = log (1+ΥPost-D). (14) Pi i i sum 2 i iX′=1 p Xi=1 Subsequently, the noise variance for the i-th user is given by: It is noteworthy that the SE performance presented here for downlink (and subsequently for uplink in Sec III-B2) are done i−1 2 H u H 2 σ˜ = E[n˜ n˜ ]= ′ H H + σ I. (18) for such realizable MMSE-SIC receiver only. SE calculation ni i i i i i n ′ P using log-determinant method2 of the delay-Doppler channel iX=1 r Hi is beyond the scope of the current work. After processing the received signal ( u) through LMMSE equalizer (similar to (10) for downlink), the estimated data vector for the i-th user at the BS B. De-Do PD-NOMA-OTFS Uplink can be expressed by (19). For notational simplicity, 1) Transmit Signal Model: For uplink OTFS-NOMA, all we denote Cu = (HuA)†[(HuA)(HuA)† + u i i i i the users transmit data simultaneously to the base station i−1 Γi′ u u † 1 −1 u u u K ′ u (H A)(H A) + u I] , B = C H A i =1 Γ i i Γ ii i i in delay-Doppler plane, thus making it a multiple-access Bu i CuHu A i andP ii′ = i i′ . Thus combining (15) and (16), (19) channel (MAC). For the sake of simplicity, perfect carrier and can be rewritten as: clock synchronization among the transmitting users has been i−1 assumed. It has also been assumed that both the users are u u u u u u dˆ = B d + B ′ ′ di′ (20) implementing same OTFS grid size (M,N). It has also been i ii Pi i ii Pi i′=1 desiredp signal X p 1Consideration of imperfect SIC and subsequent error propagation can be | {z } NOMA interference Cun seen as a potential future work [36], [37]. + | i {zi , i =1} , ,K, 2as usually done in conventional point-to-point multiple-input multiple- ··· output (MIMO) systems [39] noise component | {z } 5

i 2 i 2 βiP|bjj | βiP|bjj | desired power desired power ΥPre-D = ,ΥPost-D = , ij MN MN| {z }MN MN ij MN MN| {z MN} MN i 2 i 2 i 2 2 i 2 i 2 i 2 2 βiP |b | + β ′ P( |b | ) + |c | σ βiP |b | + β ′ P( |b | ) + |c | σ jl i jl jl n jl i jl jl n l=1X,l6=j i′=1X,i′6=i Xl=1 Xl=1 l=1X,l6=j i′=Xi+1 Xl=1 Xl=1

inter-symbol interference NOMA interference noise power inter-symbol interference NOMA interference noise power | {z } | {z } | {z (12)} | {z } | {z } | {z (13)} i−1 u 2 i−1 u u u † u u † Pi′ u u † σn −1 u † u u† Γi′ u u † 1 −1 dˆ = (H A) [(H A)(H A) + (H A)(H A) + I] ri = (H A) [H H + H ′ H ′ + I] ri, (19) i i i i Pu i i Pu i i i Γu i i Γu iX′=1 i i iX′=1 i i

Expanding (20), the uplink SINR for j-th symbol of the i-th strategies [23], [40]. The power fractions are independent of user can be formulated as: user channel conditions and system performance. In order u uii 2 to maintain fairness among users, it is a general practice to i bjj P | | allocate more power to the users with lower received average desired power ΥU = , (21) SNR. Thus, for the SNR order mentioned in III-A2, the ﬁxed ij MN i−1 MN | {z } ′ u uii 2 u uii 2 transmit power fractions will be ordered as: β1 >β2 >βK , i bjl + i′ ( bjl ) + K ··· P | | P | | with the constraint i=1 βi =1. l=1X,l=6 j iX′=1 Xl=1 P NOMA interference inter-symbol interference B. Fractional Transmit Power Allocation (FTPA) | {z } MN| {z } cui 2σ2 In this dynamic power allocation scheme, the fraction of | jl | n Xl=1 power allocated to any user is proportional to the inverse of its channel gain so that the users with lower channel gain gets noise power greater transmit power in order to maintain system fairness ′ | {z } uii uii ui with i = 1, ,K and j = 1, ,MN. bpq , bpq and cpq [24, Sec. II.B]. Depending on the nature of information about ··· th B···u Bu Cu denote the (p, q) elements of ii, ii′ and i respectively. the channel available at the BS, following two FTPA schemes Using similar assumptions made for downlink direction, the have been investigated in this work: sum rate (in bps/Hz) in uplink direction is given by: 1) Average SNR based FTPA: In this scheme, it is assumed

K that the BS has the access to only slowly time-varying average UL U received SNR values (Γi) of the users (which can be fed back Rsum = log2(1+Υi ). (22) Xi=1 to the BS from user through feedback channel or measured in reverse channel). Therefore, we propose to use the average IV. POWER ALLOCATION SCHEMES AMONG DOWNLINK received SNR values to allocate the users’ power fractions NOMA-OTFS USERS using FTPA. The fraction of power allocated to the i-th user is given by: In this section, we briefly outline the various power allo- Γ−1 β = i . (23) cation schemes used for performance evaluation of NOMA- i K −1 OTFS in downlink direction. i′=1 Γi′ It is noteworthy that in high-mobility scenarios, where 2) Channel Norm based FTPA:P The base station has access OTFS offers distinguishable advantages over OFDM, feeding to the partial CSI of all users in terms of the instantaneous back full instantaneous CSI to the BS becomes increasingly channel norms, it is assumed that we use those values to difficult due to coherence time constraint of the channel. evaluate the users’ power fractions. Therefore, the fraction of However, we also describe the full CSI based power allocation power allocated to the i-th user can be expressed as: algorithm, which is used as a benchmark in comparative study. H −1 It is also worth noting that in this work, while consid- β = i F , (24) i K|| || −1 ′ Hi′ ering power allocation for weighted sum rate maximization i =1 || ||F (discussed in Sec. IV-C), we have considered only two users where Hi is defined in (8).P multiplexed together following the principle of NOMA for analytical simplicity. Extension to generalized K-user scenario C. Power Allocation for Weighted Sum Rate Maximization can be seen as a potential future investigation, which is beyond (WSRM) the scope of the current work. Similar to the case for FTPA, we present two weighted sum rate maximization framework based on average SNR A. Fixed Power Allocation (FPA) information and instantaneous channel information at the base This power allocation scheme is simplistic in nature where station. the fractions of transmit power for different users are deter- 1) Average SNR based WSRM: In case the base station mined a priori. Such conventional scheme has been used in has access to the average SNR information of the users, the NOMA performance analysis for simplicity and in order to optimization problem can be formulated based on the AWGN have a benchmark for other sophisticated power allocation rates as described below: 6

It is also important to note that in order to implement the power allocation schemes described in Sec. IV-B2 and IV-C2, AWGN β1Γ1 Maximize Rsum = w1log(1 + ) (25) channel information (like channel norm for the first scheme 1+ β2Γ2 and B and C matrices for the second scheme) have to be + w log(1 + β Γ ) 2 2 2 either fed back to the BS by an error-free feedback channel, or subject to β + β =1, 0 β ,β 1; 1 2 ≤ 1 2 ≤ measured at BS itself by exploiting uplink-downlink duality. where w1 and w2 are the weights assigned to the two users in Such channel measurement and feedback has to be done at order to maintain fairness in power allocation. This being an least once in every delay-Doppler coherence time of the OTFS early work, we obtain a suboptimal solution of the maximiza- channel. It has been reported in literature that the delay- tion problem by differentiating the cost function, as done in Doppler coherence time of OTFS channel is significantly [28, Sec. III.A], although concavity of such cost function in larger than the coherence time in time-frequency domain for (25) is not straightforward to be shown. Reducing the problem OFDM [15]. Thus, instantaneous De-Do CSI based NOMA AWGN power allocation schemes in high Doppler scenarios are easily in terms of only β2, differentiating Rsum w.r.t. β2 and equating it to zero finally yields: realizable in OTFS systems compared to OFDM systems.

w1Γ2 w2Γ1 =0. (26) V. LINK LEVEL PERFORMANCE ANALYSIS OF 1+ β2Γ2 − 1+ β2Γ1 NOMA-OTFS SYSTEMS Solving the linear equation, the optimal value of β2 can be obtained as: It has been reported in literature that scheduling high Opt w2Γ1 w1Γ2 number of users (more than 2-3) in power domain NOMA β2 = − . (27) (w1 w2)Γ1Γ2 in same resource block does not offer much gain despite − prohibitively increasing the complexity of transmit signal pro- In order to impose the associated constraints stated in (25), Opt Opt Opt cessing, signaling overhead as well as realizing the successive we assign β2 = max(0, min(1,β2 )). Clearly, β1 =1 Opt − interference canceler receiver in polynomial or exponential β . 2 order [42]. Therefore, we limit this early investigating work 2) Instantaneous Channel Information based WSRM: If to 2 user multiplexed system as in [43], [44], [45]. the base station has access to partial information about the instantaneous channel of each user (in terms of Bi and Ci matrices defined after (10)), the 2-user optimization problem A. Downlink MMSE SIC Receiver with LDPC coding can be formulated using the exact post-SIC SINR expression This section illustrates the practical realization of a 2-user (for j-th symbol) derived in (13) as follows: LDPC enabled codeword level SIC OTFS-NOMA receiver Inst Post-D Post-D for downlink transmission as highlighted in Fig. 1. The base Maximize Rsum = w1log(1+Υ1j )+ w2log(1+Υ2j ) station generates the data for both the users (denoted as b and subject to β + β =1, 0 β ,β 1. 1 1 2 1 2 b respectively), encode using the LDPC encoder and then ≤ ≤(28) 2 modulate the data using modulation supported by the user. MN 1 2 1 2 The encoded signals for both the users are denoted as d1 and Using the notations bjj =Γ1d, l=1,l=6 j bjl =Γ1ISI, MN 1 2 P| | 2 MN P1 2 | | 2 2 d2 respectively in Fig. 1. The modulated symbols are further ( b )=Γ1N, σ c P = P1n, b = P l=1 | jl| n l=1 | jl| P| jj | superimposed with allocated power (β ). The superimposed Γ P, MN b2 2 = Γ P, and σ2 MN c2 2 = P , i 1d l=1,l=6 j jl 2ISI n l=1 jl 2n time-domain signal is further modulated for OTFS using SFFT the instantaneousP | weighted| sum rate in terms| of | can be P P β2 and the Heisenberg transform. The resulting signal s (refer to expressed as: (5) for its mathematical expression) is broadcast through the Inst (1 β2)(Γ1d +Γ1ISI)+ β2Γ1N + P1n delay-Doppler channel to both the users. Rsum = w1log[ − ]+ (1 β )Γ ISI + β Γ N + P n Since in this work, we consider 2 users, we let K = 2 in − 2 1 2 1 1 (29) (7). Both the users first perform LMMSE equalization in order to mitigate the ISI and ICI. Additionally, as it is assumed that β2(Γ2d +Γ2ISI)+ P2n w2log[ ]. SNR of second user is higher than the first user, thus second β Γ + P 2 2ISI 2n user performs the SIC. As done in previous section, we use the differentiation 1) Processing at First User: The equalized data using the method in order to obtain a suboptimal solution of β2 [28, MMSE equalizer can be described using (10) with i = 1. In Inst Sec. III.A]. Differentiating Rsum w.r.t. β2 and equating it order to decode the equalized data using LDPC decoder, the to zero results in (30). By numerically solving (30) using Opt channel LLR values are calculated from the equalized symbols available software tools, optimal value of β2 (β2 ) can as, be obtained. Similar to the earlier case, we finally assign Opt Opt Opt Opt dˆ 2 dˆ 2 β = max(0, min(1,β )) and β =1 β . j ˆ 1(η) s 1(η) s 2 2 1 2 (b d1(η)) (min || 2 − || ) (min || 2 − || ) − 1η 0 σ 1 σ It is to be noted that judicious assignment of weights for the L | ≈ sǫSj 1(η) − sǫSj 1(η) users has been addressed in literature considering proportional (31) th fairness [41], [28]. However, this being an early investigation, where di(η) is the η element of di mapped from the bits 0 1 Ki−1 for simplicity we consider assignment of fixed weights as: biη biη biη ,Ki is the number of bits per symbol for user ···2 2 1 2 † w1 =0.6, w2 =0.4. i and σ1(η) is the ηth element of σ1 = diag(σ C1C + β1 n 1 7

w1(Γ1 − Γ1 − Γ1 ) w1(Γ1 − Γ1 ) w2(Γ2 +Γ2 ) w2Γ2 N ISI d − N ISI + d ISI − ISI = 0 (30) (1 − β2)(Γ1d +Γ1ISI)+ β2Γ1N + P1n (1 − β2)Γ1ISI + β2Γ1N + P1n β2(Γ2d +Γ2ISI)+ P2n β2Γ2ISI + P2n

The algorithm terminates when the termination conditions of LDPC termination are satisﬁed or iteration count

reaches the maximum number of iterations(Nimax ) and tot the decoded codeword bit cµ = 1 if Zµ >= 0 and tot cµ =0 if Zµ < 0. 2) Processing at Second User: Since second user experi- ences higher SNR, it performs the SIC in which it decodes first user’s data and then uses it to cancel the interference to decode its own data. The detected first user’s data at the second user is given as, 1 dˆ = β (H A)†[(H A)(H A)† + I]−1r (35) 12 1 2 2 2 Γ 2 p 2 Corresponding LLR of the equalized data of fist user is calculated as, dˆ (η) s 2 dˆ (η) s 2 j ˆ 12 12 (b d1 (η)) (min || 2 − || ) (min || 2 − || ), 12η 2 0 σ 1 σ L | ≈ sǫSj 12 (η) − sǫSj 12 (η) (36) Fig. 1: Representative block diagram of 2-user NOMA-OTFS system 2 th 2 1 2 † where σ1 (η) is the η element of σ1 = diag(σ C2C + in downlink. 2 2 β1 n 2 B B† β2 2 2). the residual received signal at second user after canceling the interference due to first user is given by, B B† β2 1 1). The aggregate interference and noise is assumed r r H Ad˜ ˜2 = 2 β1 2 12 , (37) to follow Gaussian distribution, as previously stated in Sec. − p P k d˜ III-A2. Sj denotes the set of constellation symbols in which where 12 is generated at second user after passing the LDPC j dˆ the bit b = k. See the paragraph following (11) for the defini- decoded codeword obtained from 12 through symbol mapper. tion of B1 and C1 matrices. These LLRs are then fed into the After doing MMSE equalization on the residual signal given LDPC decoder to decode first user’s data. Let L1 denotes a in (37), the detected second user’s data at the second user itself L1 j dˆ is given by, matrix where (η, j)= (b1η 1(η)) for η =1, 2, ,MN L L|1 ··· and j = 0, 1, ,Ki 1. is reshaped to Lcl Ncw 1 ··· − × dˆ = β (H A)†[β (H A)(H A)† + I]−1˜r (38) matrix where Lcl and Ncw denote the LDPC codeword length 2 2 2 2 2 2 2 Γ2 and number of codewords respectively. Each column of L1 p As done for first user, the bit level LLRs for second user from subsequently regenerates codeword c1 for ι = 1, 2 ,N ι cw the symbols are calculated as, using the Min-Sum Algorithm [46] employed by the··· LDPC decoder. This algorithm iteratively updates the variable node dˆ 2 dˆ 2 j ˆ 2(η) s 2(η) s (b d2(η)) (min || 2 − || ) (min || 2 − || ) and check node equation as discussed below. 2η 0 σ 1 σ L | ≈ sǫSj 2(η) − sǫSj 2(η) • Variable Node Update (39) 2 th 2 1 2 † where σ2(η) is the η element of σ2 = diag(σ C2C ) (l−1) β2 n 2 (l) ′ mµ,ν = Zµ +Σν =6 ν mµ′,ν (32) (see the paragraph following (11) for the definition of C2 matrix). The LLRs are updated and the data of user 2 is where the Z is the channel LLR calculated from (31) µ generated by the LDPC decoder to generate the data using for the µth bit in the codeword and m(l) is the message ν,µ Min-Sum algorithm as described in detail for user 1 in Sec. received from the νth check node to the µth variable node V-A1. in the iteration l. • Check Node Update (l) (l) (l) ′ mν,µ = Πµ =6 µsign(mµ′,ν ) min( mµ′,ν ) (33) µ′=6 µ | | B. Uplink MMSE SIC Receiver with LDPC coding where the product and the minimum operator is taken This section focuses on a realization of an OTFS-NOMA over the set of neighboring variable nodes except the link coupled with LDPC codes for a two user scenario in message recipient itself. uplink transmission as highlighted in Fig. 2. For two user case, • Decoding Decision the received signal can be expressed using (16) by putting tot (l) u u Zµ = Zµ +Σν mν,µ (34) K = 2. At the BS, since it is assumed Γ1 < Γ2 , we first 8

TABLE I: Key system parameters Parameter Value LTV Delay-Doppler channel ‘Extended Vehicular model A (EVA)’[33] Doppler slots (N) 16 Delay slots (M) 256 Number of NOMA users 2 User speed 500 kmph Carrier frequency 5.9 GHz Subcarrier Bandwidth ∆f 15 KHz Total Bandwidth B 3.84 MHz Frame Duration Tf 1.08 ms LDPC codes. Code Error Correction Codes length = 648,code rate (R) = 2/3 [10] Γ = 15 dB, Γ = Downlink average SNR 1 2 Fig. 2: Representative block diagram of 2-user NOMA-OTFS system 25 dB in uplink. Γu = 10 dB, Γu = decode second user’s data as, Uplink average SNR 1 2 u 30 dB dû HuA † HuA HuA † Γ1 HuA HuA † 1 I −1r 2 = ( 2 ) [( 2 )( 2 ) + u ( 1 )( 1 ) + u ] u. Γ2 Γ2 (40) LLR values of second user can be computed as, NOMA-OTFS schemes in high speed scenario through exten- sive Monte-Carlo simulation for both downlink and uplink. dû 2 dû 2 uj dû 2 (η) s 2 (η) s The important simulation parameters are listed in Table I. (b η (η)) (min || 2 − || ) (min|| 2 − || ), 2 2 0 σ 1 σ L | ≈ sǫSj 2(η) − sǫSj 2(η) Doppler is generated using Jake’s formula, νp = νmaxcos(θp), (41) where θp is uniformly distributed over [ π π]. The CP is where ηth element of du, du(η) is mapped from bits − − i i chosen long enough to accommodate the maximum excess u0 u1 u(Ki 1) biη biη biη ,Ki is the number of bits per sym- delay of the channel. We present results of an equivalent ··· 2 th 2 bol for user i and σ2(η) is the η element of σ2 = OFDM system with synchronous CP length and block based 1 diag(σ2 CuCu† + Bu Bu†). The matrices Cu and Bu signal structure as described in Sec. II. The MMSE equalizer P2 n 2 2 1 21 21 2 21 are defined after (19). TheP calculated LLR values are processed implemented in this work can efficiently cancel the ICI at by LDPC decoder in order to produces the message word. The the receiver, unlike the single tap equalizer used in traditional obtained message is again encoded and modulated to generate systems [47]. d˜u the recovered data 2 for second user, which is used to cancel While 5G-NR has provision for variable subcarrier band- the interference from aggregate received signal to decode the width [12], [48], the EVA channel model restricts us to first user’s data as, a subcarrier bandwidth of upto 60 KHz corresponding to

u u u numerology 2 contrary to the maximum of 240 KHz with nu- ˜ru = ru H Ad˜ . (42) 1 − P2 2 2 merology 4 due to the coherence bandwidth of about 56 KHz. p After doing MMSE equalization of residual signal at the BS However, it is worth mentioning that from ICI perspective, given by (42), the detected first user’s data is given by, numerology 4 with maximum of 240 KHz is more desirable. It is important to note that the receiver used here is MMSE with dû H A † H A H A † 1 I −1r 1 = ( 1 ) [( 1 )( 1 ) + ] ˜u1 . (43) ideal channel estimates thus serving the purpose to cancel inter Γ1 sub-carrier interference of LTV channel which appear due to dû The equalized data is 1 is used to calculate the LLR as Doppler spread. The system performance is then accordingly follows: evaluated using a subcarrier bandwidth of 15 KHz which is dû 2 dû 2 also valid for system design of 4G systems [49], [11]. j dû 1 (η) s 1 (η) s (b1η 1 (η)) (min|| 2 − || ) (min || 2 − || ), In Sec. VI-A, we present the system level performance L | ≈ sǫS0 σ (η) − sǫS1 σ (η) j 1 j 1 of downlink and uplink in terms of sum SE results for (44) the MMSE-SIC receiver developed in III-A2 and III-B2 where assuming perfect SIC, dû(η) and σ2(η) are the ηth 1 2 respectively. In Sec. VI-A1, we evaluate the performance dˆ σ2 1 2 CuCu† Cu element of and 1 = u diag(σn 1 1 ). The matrix 1 P1 results of NOMA-OTFS and OMA-OTFS in order to find is also defined after (19). The calculated LLR values are then the feasibility of NOMA-based multi-user multiplexing in fed to LDPC decoder to reproduce the data of user 1. high mobility scenarios. We then conduct a comparative performance analysis between OTFS based and OFDM based VI. SIMULATION RESULTS AND DISCUSSION NOMA implementation with an aim to explore relative gains In this section we present detailed performance analysis in that can be obtained by such system design in high Doppler terms of system level and link level evaluation of the presented scenarios (in Sec. VI-A2). Since we evaluate different power 9

SNR based FTPA power allocation with respect to OMA- 1 OTFS. The gain is even higher for power allocation schemes like channel-norm based FTPA and instantaneous CSI based 0.9 WSRM NOMA-OTFS schemes. 0.8 2) Comparison between OTFS and OFDM performances: From the CDF trends in Fig. 3 and tabulated values in Table 0.7 II, it can be observed that the 5% outage sum SE shows signiﬁcant improvement for OTFS with respect to OFDM. For 0.6 OTFS

Abscissa) example, in case of average SNR based FTPA and channel 0.5 OFDM norm based FTPA schemes, an improvement of around 19.7%

0.4 and 27.5% respectively is observed. The gain is even higher for Fixed ( = 0.7, 1 weighted sum rate maximization schemes, reaching to nearly = 0.3) 0.3 2 26% and 55% respectively. Similar improvement (around FTPA Av SNR P(Sum Rate WSRM with Avg. SNR 10%) has been observed for OMA scheme as well, high- 0.2 5% Outage FTPA Chan Norm lighting the utility of OTFS over OFDM even for orthogonal Level WSRM full CSI 0.1 OMA multi-user scenario. Exact values of gains are tabulated in last 0.05 column of Table II. The outage improvement in OTFS over 0 0 2 4 6 8 10 12 OFDM reflects the diversity gain of OTFS with respect to Downlink Sum Rate (bps/Hz) OFDM. However, we note that OFDM based NOMA provides nominally higher mean SE (in the order of 5-14%) than Fig. 3: CDF of downlink sum rate for various NOMA power OTFS based NOMA. Detailed analytical treatment in order allocation schemes under OTFS/OFDM for Γ1 = 15 dB, Γ2 = 25 dB to investigate this issue is a potential future work. for user velocity = 500 kmph. Solid and dashed lines represent OTFS Similar improvement in 5% outage sum rate ( 12%) and and OFDM results respectively. Markers denote results corresponding ∼ to various power allocation schemes described in Sec. IV reduction in mean sum rate ( 17%) in OTFS compared to OFDM has also been reflected∼ for uplink NOMA in Fig. 4 for the MMSE-SIC receiver designed in Sec. III-B2. allocation for NOMA-OTFS in Sec. IV, we analyze their relative performance in Sec. VI-A3. After system-level performance evaluation, we delve into link level performance 1 verification of LDPC coded NOMA-OTFS CWIC MMSE-SIC Mean Sum SE receiver in Sec. VI-B. We also compare the same against the 0.9 7.1716 equivalent NOMA-OFDM as well as OMA-OTFS systems as NOMA-OTFS bps/Hz 0.8 described earlier in this work. The performance is evaluated 8.6769 NOMA-OFDM bps/Hz in terms of metrics like BLER (Pe), throughput and goodput. 0.7 The throughput of a link is defined as the number of bits NOMA-OTFS 0.6 transmitted per unit time and is given by (45). NOMA-OFDM Abscissa) 2 0.5 Throughput = Σi=1RKi bits/s/Hz, (45) 0.4 where R and Ki denote code rate and bits per QAM symbol respectively. Whereas, the goodput of a link is defined as the 0.3 5% Outage sum

number of bits that are successfully received and expressed in P(Sum Rate SE(NOMA-OFDM) 5% Outage sum 0.2 = 5.532 bps/Hz 5% Outage (46). SE(NOMA-OTFS) Level 2 = 6.199 bps/Hz Goodput = Σ RKi(1 Pe ) bits/s/Hz, (46) i=1 − i 0.1 0.05 where Pe denotes BLER for i-th user respectively. i 0 2 4 6 8 10 12 14 Uplink Sum Rate (bps/Hz) A. System Level Spectral Efﬁciency Results Fig. 4: CDF of uplink sum rate for NOMA users under OTFS/OFDM u u 1) Comparison between NOMA/OMA-OTFS: In Fig. 3, the for Γ1 = 10 dB, Γ2 = 30 dB for user velocity = 500 kmph. cumulative distribution functions (CDF) of downlink sum rates achieved under various OMA and various NOMA power 3) Comparison of various NOMA power allocation allocation schemes for OTFS and OFDM are shown. The exact schemes: The CDF curves in Fig. 3 and tabulated values in values of mean and 5% sum rate are given in Table II. From Table. II of downlink sum rates for various NOMA power the CDF curves and tabulated values we observe a signiﬁcant allocation schemes reveal important assessment of the schemes increase in mean and outage sum rate in the NOMA-OTFS in the considered system model. The average SNR based scheme compared to the OMA-OTFS. We notice that there is weighted sum rate maximization scheme’s (described in Sec. more than 16% improvement in both mean and 5% outage IV-C1) performance is seen to be the worst compared to the sum SE respectively in case of NOMA-OTFS with average other schemes. This is mainly due to the fact that strong 10

user’s average received SNR (Γ2) is significantly higher than system is able to support required BLER with these βis. This the weak user’s average received SNR (Γ1), thus resulting observation suggests that the Gaussian assumption considered in allocation of full power to the weak user (see (27)), for evaluating SINR may not hold true. User 1(weak user) effectively turning the scheme to OMA with only weak user. decodes the signal using MMSE equalization as outlined in Judicious choice of weights (wi) incorporating proportional Sec V-A1. On the other hand, as detailed in Sec. V-A2, fairness can be used to alleviate the issue. The average SNR user 2(strong user) experience higher SNR and thus perform based FTPA scheme (described in Sec. IV-B1) also gets SIC at codeword level. Same SINR thresholds are used for partially affected due to the same issue and thus the scheme uplink modulation scheme selection as in downlink. BLERs marginally outperforms the OMA scheme. The channel norm are evaluated using Monte-Carlo link level simulation for each based FTPA (described in Sec. IV-B2) and instantaneous CSI user for downlink as well as uplink. based weighted sum rate maximization (described in Sec. Based on the obtained BLER results, we compute through- IV-C2) schemes have all achieved better performance in terms put (45) and goodput (46) for each user. Table III is generated of mean and outage spectral efficiency which is significantly keeping user 1’s modulation as QPSK, as Γ1 = 15 dB. Γ2 higher than the OMA scheme, which advocates for use of is varied such that the higher modulation schemes can be De-Do PD-NOMA for better system performance. It is worth supported by user 2. Γ2 = 22, 30, 35 dB are considered. noting that the NOMA power allocation methods discussed Corresponding to these Γ2 values, Υ˜ 2 obtained from (48) here are suboptimal. The optimistic results obtained here support modulation schemes QPSK,16-QAM and 64-QAM promotes further investigation of such schemes which can be respectively following the discussion made earlier about the implemented in high Doppler scenario. SNR thresholds corresponding to the modulation schemes. Though the SNR range for both users can be between -3 dB B. Link Level Performance of NOMA-OTFS and 40 dB, only these combinations of SNRs are selected as representative values in order to demonstrate system perfor- In this section, we discuss about the link-level performance mance. An important aspect of NOMA name user selection is of LDPC coded codeword level SIC NOMA-OTFS system and dependent on achievable NOMA gain, which in turn depends compare it against OMA-OTFS and NOMA-OFDM system. on supportable data rate. Here, such consideration are made We first discuss about the downlink NOMA performance from link level perspective. In operational system, EESM [50], and then the uplink. Tables III and IV show the goodput [51] based mapping of user’s experienced SINR can be done performance along with BLER values of the users for down- to choose appropriate rate while satisfying required BLER. link and uplink respectively. While generating such results, For the SNR pair, Γ1 = 15dB and Γ2 = 35dB in the the average SNR values are considered for choosing the table III, user 1 and user 2 are assigned QPSK (K=2) and modulation scheme in order to guarantee that the experienced 64-QAM (K=6) respectively resulting in a throughput of 5.33 −1 BLER remains below the threshold 10 [43]. bits/sec/Hz (which is evaluated from (45)) for OTFS while 1) Performance of NOMA-OTFS in Downlink: In the down- the goodput achieved is 5.10 bits/sec/Hz, which is evaluated link direction, at base station, each user’s data is encoded using −2 −3 by taking K1 =2,K2 =6, Pe1 =5.6 10 , Pe2 =5 10 LDPC with code rate R = 2/3. The encoded bit stream is and R = 2/3 in (46). For the same× scenario in OMA× case, modulated using QPSK, 16QAM or 64QAM (Ki = 2, 4 and user 1 can support upto 16QAM while user 2 can support 6 respectively). In order to achieve BLER less than 0.1 with upto 64QAM resulting in throughput of (4*2/3 + 6*2/3)/2 = LDPC code rate and length shown in Table I, SNR thresholds 3.33 bits/sec/Hz. Here the percentage gain in throughput with to support QPSK,16QAM and 64QAM for OTFS are 9.5 dB, NOMA-OTFS over OMA-OTFS is 37.52%. 15 dB and 23.5 dB respectively. For OFDM, the thresholds When NOMA-OFDM is employed for the same conditions, are 10.8 dB, 18 dB and 26 dB respectively. The modulation user 1 is assigned QPSK and user 2 is assigned 16QAM in schemes for both the users are chosen based on the average order to satisfy BLER threshold resulting in a goodput of 3.99 ˜ SINR(Υ) experienced by the users, which are functions of bits/s/Hz. Thus, NOMA-OTFS offers 21.76% gain in goodput Γ1, Γ2,β1 and β2. In downlink, the average SINR of user 1 over NOMA-OFDM. are obtained assuming interference as Gaussian noise is given 2) Performance of NOMA-OTFS in Uplink: In uplink di- by u rection, the table IV is generated by keeping Γ1 = 10 dB and ˜ β1Γ1 u Υ1(in dB)=10log10( ). (47) varying the user 2’s SNR, Γ2 = 25, 30, 40 dB thus varying the β2Γ1 +1 user 2’s modulation scheme as QPSK,16QAM and 64QAM The post SIC average SINR for user 2 assuming perfect SIC respectively. It can be observed that NOMA using OFDM is can be expressed as: unable to support user 1 as Pe1 is above threshold. For the SNR pair, Γu = 10dB and Γu = 40dB, P = 3.6 10−2, Υ˜ (in dB)=10log (β Γ ). (48) 1 2 e2 2 10 2 2 thus user 2 can be supported with modulation scheme× 64QAM The modulated data of the users is transmitted using superpo- but user 1 is unable to transmit even using QPSK due to sition coding with β =0.9 and β =0.1 as described in Sec. the resulting BLER of about 4.1 10−1, as a result of 1 2 × III-A1. This choice of βis results in Υ˜ 1 =8.35 dB for Γ1 = 15 error propagation. The resulting NOMA-OFDM goodput is dB which puts the system close to minimum operational range. 4.64 bits/s/Hz compared to NOMA-OTFS goodput of 5.19 Though the SINR of 8.35 dB is insufficient to satisfy the BLER bits/s/Hz, thus a gain of 10.60% is shown for NOMA with threshold as per previous discussion but it is observed that the OTFS over OFDM in uplink. If OMA is employed for the 11

TABLE II: Mean and 5% outage SE (in bps/Hz) for downlink NOMA in OTFS and OFDM for Γ1 = 15 dB, Γ2 = 25 dB for user velocity = 500 kmph. NOMA PowerAllocation Schemes Mean SE 5% Outage SE OTFS OFDM % gain OTFS OFDM % gain OMA 4.7618 5.5852 -14.74% 3.931 3.544 10.92% Fixed-I (β1 =0.7, β2 =0.3) 5.9499 6.2898 -5.40% 4.925 3.9 26.28% Fixed-II (β1 =0.9, β2 =0.1) 5.546 6.5398 -15.19% 4.5 3.8 18.42% FTPA (Avg SNR) 5.5487 6.1500 -9.77% 4.574 3.821 19.70% WSRM (Avg SNR) 3.496 4.0838 -14.39% 2.574 1.658 55.24% FTPA (Channel Norm) 5.9977 6.3075 -4.91% 4.874 3.823 27.46% WSRM (Full CSI) 6.0254 6.2922 -4.24% 4.617 3.654 26.35%

TABLE III: Userwise BLER results for downlink NOMA in OTFS and OFDM. (β1 = 0.9 and β2 = 0.1, code rate= 2/3,UE1 using QPSK Modulation with SNR 15 dB resulting SINR 8.35 dB) SNR(dB) SINR(dB) UE2 Modulation BLER UE2 BLER UE1 Goodput(bits/s/Hz) UE2 UE2 OTFS OFDM OTFS OFDM OTFS OFDM OTFS OFDM % gain 22 12 QPSK(2) QPSK(2) 4.7x10−2 1.2x10−1 3x10−3 1x10−3 2.6 2.51 3.46 25 15 QPSK(2) QPSK(2) 0 1.3x10−3 1x10−3 3x10−3 2.67 2.66 0.34 30 20 16QAM(4) 16QAM(4) 2x10−3 2.2x10−2 0 6.5x10−3 3.99 3.93 1.50 35 25 64QAM(6) 16QAM(4) 5.6x10−2 3x10−4 5x10−3 5x10−3 5.10 3.99 21.76 TABLE IV: Userwise BLER results for uplink NOMA in OTFS and OFDM. (Code rate= 2/3,UE1 using QPSK Modulation with SNR 10 dB resulting SINR 10 dB) SNR(dB) SINR(dB) UE2 Modulation BLER UE2 BLER UE1 Goodput(bits/s/Hz) UE2 UE2 OTFS OFDM OTFS OFDM OTFS OFDM OTFS OFDM % gain 25 15 QPSK(2) QPSK(2) 0 1.3x10−3 7x10−2 1.8x10−1 2.57 2.42 5.84 30 20 16QAM(4) 16QAM(4) 1x10−3 2.8x10−2 6.7x10−2 3x10−1 3.91 3.52 9.97 40 30 64QAM(6) 64QAM(6) 3.4x10−3 3.6x10−2 9.7x10−2 4.1x10−1 5.19 4.64 10.60 same scenario, then user 1 and user 2 can support upto QPSK reduced by using NOMA. However we note that mean sum and 64QAM respectively resulting in a throughput of 2.67 SE of appropriately modified NOMA-OFDM is better than bits/s/Hz, while NOMA throughput is 5.33 bits/s/Hz. Thus a NOMA-OTFS. Thus we find that there is a tradeoff between gain of 50% in throughput can be obtained in NOMA with mean and outage SE. The improved outage sum SE indicates respect to OMA. a more resilient system in high-mobility scenario, which is highly desirable. VII. CONCLUSION The link level performance obtained from the developed codeword level SIC receiver shows that the NOMA-OTFS In this paper, we have presented the performance analysis system has upto 21.76% and 10.60% improved goodput in of a superposition coding based De-Do domain PD-NOMA- downlink and uplink respectively compared to NOMA-OFDM. OTFS system in high mobility scenarios. In order to real- It also shows 37.52% and 50% better throughput for NOMA- ize NOMA-OTFS, we have presented a linear MMSE-SIC OTFS over OMA-OTFS system in downlink and uplink re- receiver. Symbol-wise post-processing SINR is derived for spectively. both downlink and uplink for subsequent SE analysis of such Therefore, based on the system developed and presented a system. We have realized a few partial CSI-based power performance analysis we find that NOMA-OTFS has the allocation techniques among downlink NOMA users. In order potential to improve the performance of regular OMA-OTFS to investigate the practical applicability of such a system, and NOMA-OFDM in high mobility conditions. we have also develop a CWIC receiver with LDPC error- correcting codes along with MMSE equalization for 2-user NOMA case. REFERENCES Results show that the De-Do domain two-user PD-NOMA- [1] B. Ai et al., “Challenges toward wireless communications for high-speed OTFS, as presented in this work, is better than traditional railway,” IEEE Transactions on Intelligent Transportation Systems, OMA-OTFS by upto 16% in terms of both mean and outage vol. 15, no. 5, pp. 2143–2158, Oct 2014. sum SE performance. [2] R. He et al., “High-speed railway communications: From GSM-R to LTE-R,” IEEE Vehicular Technology Magazine, vol. 11, no. 3, pp. 49– We have also observed that NOMA-OTFS has upto 50% 58, Sep. 2016. better outage sum SE when compared to NOMA-OFDM [3] F. Hasegawa et al., “High-speed train communications standardization for partial-CSI based power allocation schemes. For full-CSI in 3GPP 5G NR,” IEEE Communications Standards Magazine, vol. 2, no. 1, pp. 44–52, MARCH 2018. based power allocation schemes, the gain is in the order of [4] S. Hayat, E. Yanmaz, and R. Muzaffar, “Survey on unmanned aerial 27%. This also indicates that the OTFS gain over OFDM is not vehicle networks for civil applications: A communications viewpoint,” 12

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