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Single- versus Multi-Carrier Terahertz-BandCommunications: A Comparative Study
Simon Tarboush, Hadi Sarieddeen, Member, IEEE, Mohamed-Slim Alouini, Fellow, IEEEand Tareq Y. Al-Naffouri, Senior Member, IEEE
Abstract—The prospects of utilizing single-carrier (SC) andmulti-carrier (MC) waveforms in future terahertz (THz)-bandcommunication systems remain unresolved. On the one hand, thelimited multi-path (MP) components at high frequencies resultin frequency-flat channels that favor low-complexity widebandSC systems. On the other hand, frequency-dependent molecularabsorption and transceiver characteristics and the existence ofMP components in indoor sub-THz systems can still result infrequency-selective channels, favoring off-the-shelf MC schemessuch as orthogonal frequency-division multiplexing (OFDM).Variations of SC/MC designs result in different THz spectrumutilization, but spectral efficiency is not the primary concern withsubstantial available bandwidths; baseband complexity, power ef-ficiency, and hardware impairment constraints are predominant.This paper presents a comprehensive study of SC/MC modula-tions for THz communications, utilizing an accurate widebandTHz channel model and highlighting the various performanceand complexity trade-offs of the candidate schemes. Simulationsdemonstrate the robustness of discrete-Fourier-transform spreadOFDM (DFT-s-OFDM) to THz impairments and orthogonaltime-frequency space (OTFS) to THz Doppler spreads.
Index Terms—THz Communications, CP-OFDM, SC-FDE,DFT-s-OFDM, OQAM/FBMC, OTFS.
I. INTRODUCTION
THE successful deployment of millimeter-wave (mmWave)communications [1] has encouraged researchers to ex-
plore the last piece of available spectrum, the terahertz (THz)band over 0.3− 10 THz, which promises to be an essentialingredient of future ultra-broadband wireless communications[2], [3]. Moving towards beyond-fifth generation (B5G) andsixth-generation (6G) wireless networks [4], [5], a plethoraof services are expected to be supported, such as ultra-lowlatency communications, ubiquitous connectivity, and veryhigh data rates (up to several terabits-per-second (Tbps)). Suchfeatures can be leveraged in novel use cases in fixed radiolinks, wireless local area networks, nano cells, or inter-chipcommunications. Furthermore, accurate localization, sensing,and imaging applications are promised in the THz band[6], [7]. However, researchers should first overcome severalchallenges in THz materials and technologies (photonic andelectronic) and the corresponding system designs and hard-ware complexity [8], [9].
S. Tarboush is a researcher from Damascus, Syria (e-mail:[email protected]). The rest of the authors are with theDepartment of Computer, Electrical and Mathematical Sciences andEngineering (CEMSE), King Abdullah University of Science and Technology(KAUST), Thuwal, Makkah Province, Kingdom of Saudi Arabia, 23955-6900 (e-mail: [email protected]; [email protected];[email protected]).
The THz-band channel’s peculiarities (frequency/distance-dependency and sparsity) impose challenging constraints onthe physical layer of future wireless standards. THz signalssuffer from severe path loss, which limits the transmissiondistances to a few meters [10]. The frequency-selective molec-ular absorption further results in distance-dependent spectrumfragmentation and shrinking (variable-bandwidth (BW) trans-mission windows) [11]. Hence, ultra-massive multiple-inputmultiple-output (UM-MIMO) antenna arrays and intelligentreflecting surfaces (IRSs) are essential for extending the THzcommunication range [12], [13]. Furthermore, since the line-of-sight (LoS) path dominates THz-band signal propagation,THz channels tend to be flat-fading. However, a few multi-path (MP) components might persist, especially in indoorscenarios, resulting in frequency-selective channels (FSCs) ofcoherence BWs of hundreds of megahertz (MHz) over mediumcommunication distances [14]. Therefore, THz multi-carrier(MC) schemes are called for despite LoS dominance.
Designing efficient THz-specific waveforms (WFs) is cru-cial for unleashing the THz-band’s true capabilities. BecauseBW and spectral efficiency (SE) are not yet a THz bottle-neck; low complexity, robustness to hardware impairments andDoppler spreads, and high power efficiency are prioritized.The first sub-THz standard (IEEE 802.15.3d [15]) supportsswitched point-to-point connectivity with data rates exceeding100 Gbps, offering two modes: (1) single-carrier (SC) modu-lation (long-range; high-rate) and (2) on-off keying (OOK)(low-complexity; short-range). OOK utilizes femtosecond-long pulses that could span an ultra-wideband THz spec-trum [16]. However, temporal broadening [14] and the chal-lenging synchronization procedure question the feasibility ofpulse-based modulation. IEEE 802.15.3d-compliant WFs areproposed in [17], where novel pulse-shaping designs reduceout-of-band (OOB) emissions. Several other projects revisit thephysical layer for future B5G sub-THz systems. Most notably,the BRAVE project [18] advocates for modified SCs schemes,such as continuous phase modulated single-carrier frequency-division multiple-access (CPM SC-FDMA), constrained enve-lope CPM-SC, and differential modulation (like differentialphase-shift keying). Block-based SC WFs, such as discrete-Fourier-transform spread OFDM (DFT-s-OFDM) [19] can alsobe explored.
A variety of THz MC schemes can be explored. In thesimplest form, multiple (quasi)-orthogonal non-overlappingSC modulations can be combined with some form of car-rier aggregation [20]. Cyclic-prefix orthogonal frequency-division multiplexing (CP-OFDM) is well investigated, but it
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is discouraged at THz [21], [22] due to its strong spectralleakage (high OOB emissions), unfavorable peak-to-averagepower ratio (PAPR) properties (limitations in state-of-the-art THz power amplifiers (PAs) [23]), strict synchronizationprocedures, and high sensitivity to Doppler spread. Other MCschemes such as novel fifth-generation new-radio (5G-NR)filter-based candidates provide their prospects and challenges.Such filtering is on the whole band in filtered-OFDM (f-OFDM) [24], per-subband (a set of contiguous subcarriers) inuniversal filtered multi-carrier (UFMC) [25], or per-subcarrierin offset quadrature amplitude modulation-based filter-bankmulti-carrier (OQAM/FBMC) [26] and generalized frequency-division multiplexing (GFDM) [27]. Although filter-basedschemes overcome some CP-OFDM limitations, reducingOOB emissions and enhancing SE, their increased imple-mentation complexity can be prohibitive in Tbps basebandsystems. For example, the single-tap equalizer is no longersufficient with CP-free OQAM/FBMC, requiring more com-plex equalization. Other works propose windowed overlap-and-add OFDM (WOLA-OFDM) [28], or combinations suchas OQAM/GFDM [29].
THz-specific multiple-access techniques are also emerging,such distance-adaptive MCs [21], hierarchical-BW modula-tions [22], and distance-/frequency-dependent adaptive CP-OFDM [30], which optimize distance-dependent spectral win-dow utilization. Spatial-spread orthogonal frequency-divisionmultiple-access (SS-OFDMA) is another THz MC candidatethat realizes frequency-based beam spreading by allocatingsubcarriers for users in different directions [31]. Similarly,beam-division multiple-access (BDMA) [32] schedules mu-tually non-overlapping beam subsets for users, followed byrelaxed per-beam synchronization. Moreover, THz-band non-orthogonal multiple access (NOMA) techniques are argued tobe feasible, despite the narrow beams that make user clusteringdifficult [33]. Other conventional techniques that improve SEat a reduced power cost, PAPR, and transceiver complexity, arealso being studied for THz communications, including spatial[13] and index modulation [34], [35] paradigms.
Other novel WFs target specific THz use cases and con-straints. For instance, zero-crossing modulation [36] uses tem-poral oversampling and 1-bit quantization to relax hardwarerequirements, such as in the digital-to-analog converter (DAC)and analog-to-digital converter (ADC). Furthermore, orthogo-nal time-frequency space (OTFS) modulation [37] is tailoredfor time-variant (TV) channels and high Doppler spreads,which arise in high-speed THz communication scenarios suchas vehicle-to-everything (V2X), drone, and ultra-high-speedrail communications. OTFS is superior in block error rateperformance to CP-OFDM when assuming mmWave LoSV2X channels [38]; also when accounting for oscillator phasenoise (PHN) impairments [39].
Many performance metrics need to be considered whendesigning THz WFs, such as bit error rate (BER), PAPR,and baseband computational complexity. Furthermore, hard-ware imperfections and radio frequency (RF) impairmentscritically impact THz WF design, where candidate THz materi-als/hardware are still under development. Hardware imperfec-tions include PA non-linearity, wideband in/quadrature-phase
imbalance (IQI) [40], phase uncertainty in the phase-shifters(PSs) [41], and PHN (studied for SC schemes [42] and CP-OFDM [43] in sub-THz and THz [40] systems). THz channel-induced phenomena such as beam split and misalignment[44] are also critical, especially with UM-MIMO systems.Moreover, the synchronization becomes more challenging withcarrier frequency offset (CFO) and symbol-timing offset (STO)at THz frequencies. Subcarrier spacing (SCS), its impact onPHN, and the design of phase-tracking reference signals arestudied in [45], [46] to assess whether CP-OFDM and DFT-s-OFDM can support mmWave and sub-THz communications.Moreover, a THz SC frequency-domain equalization technique(SC-FDE) is developed in [40], and a pilot design strategybased on index modulation is proposed in [47]. SC systemsare found superior to CP-OFDM in mmWave systems [48].For indoor THz scenarios, SC-FDMA with linear equalizationis shown to be superior to CP-OFDM and SC with linear-/decision-feedback-equalization [49].
The literature lacks a holistic and fair comparative studyof THz-band SC/MC schemes, and this work attempts to fillthis gap. We aim to analyze a plethora of candidate schemesto be able to draw recommendations on the suitable WFs forspecific THz use cases. The main contributions of this paperare summarized as follows:
• We study the THz compatibility of multiple modulationschemes, namely, SC-FDE, CP-OFDM, DFT-s-OFDM,OTFS, and OQAM/FBMC, adopting our newly developedaccurate THz channel model/simulator (TeraMIMO [44]).
• We analyze SE, physical layer end-to-end (E2E) latency,OOB emissions (reference THz IEEE 802.15.3d spectralmask), PAPR (theoretical bounds), and computationalcomplexity.
• We provide a fair comparison of modulation schemesunder THz-specific scenarios such as oscillator PHN (westudy a Gaussian uncorrelated PHN model), mobility, andbeam split.
• We promote DFT-s-OFDM and OTFS as promisingschemes for future B5G/6G networks.
The remainder of this paper is organized as follows: We firstintroduce the system and channel models in Sec. II. Then, wepresent a general framework for analyzing the studied SC/MCschemes and key performance indicators (KPIs) in Sec. III.Afterward, extensive simulation results validate our analysesin Sec. IV, where we recommend suitable WFs for specificscenarios. We conclude in Sec. V. Regarding notation, non-bold lower case, bold lower case, and bold upper case letterscorrespond to scalars, vectors, and matrices: 𝑎[𝑛] denotes the𝑛th element of a and a[𝑚] and 𝑎[𝑛, 𝑚] denote the 𝑚th columnand the (𝑛, 𝑚)th element of A, respectively. I𝑁 is the identitymatrix of size N, 0𝑁×𝑀 is a zero matrix of size N×M, and a𝑁 isa vector of size N. The superscripts (·)T, (·)∗, (·)H, (·)−1, and(·)𝑛 stand for the transpose, conjugate, conjugate transpose, in-verse, and 𝑛th-power functions, respectively. |·| is the absolutevalue (or set cardinality), diag(𝑎0, 𝑎1, . . . , 𝑎𝑁−1) is an 𝑁 ×𝑁diagonal matrix of diagonal entries 𝑎0, 𝑎1, . . . , 𝑎𝑁−1, vec(A)is the vectorized matrix representation that stacks the columnsof A in a single column, E(·) is the expectation operator,
3
and Pr (·) is the probability density function. The notations ⊗,[·]𝑁 , 〈 , 〉, R(·), and 𝑗 =
√−1 denote the Kronecker product,
remainder modulo 𝑁 , inner product, real part, and imagi-nary unit, respectively. The superscripts (t) and (r) denotetransmitter (Tx) and receiver (Rx) parameters, respectively.N(𝜑, 𝜎2) is the distribution of a Gaussian random variableof mean 𝜑 and variance 𝜎2, CN(a,𝚺) is the distribution ofa complex Gaussian random vector of mean a and covariancematrix 𝚺. The normalized 𝑁-point DFT and IDFT matrices aredenoted by F𝑁 and FH
𝑁, respectively. The used abbreviations
are summarized in Tables VI and VII.
II. SYSTEM AND CHANNEL MODEL
We aim to evaluate the performance of candidate SC/MCschemes in realistic THz settings, including massive antennadimensions and ultra-wide BWs. We adopt the array-of-subarrays (AoSA) architecture of TeraMIMO [44], in whicheach subarray (SA) is composed of many antenna elements(AEs), as depicted in Fig. 1. AoSAs can mitigate THzhardware constraints and combat the limited communicationdistance problem using low-complexity beamforming [9]. Weassume 𝑄 = 𝑄𝑎 ×𝑄𝑏 SAs, and �� = ��𝑎 × ��𝑏 tightly-packeddirectional AEs per SA. Each AE is attached to a widebandTHz analog PS of acceptable phase error, return loss, and in-sertion loss [41] (such PSs can be implemented using graphenetransmission lines in plasmonic solutions [50]). We assumethe AoSAs to realize sub-connected hybrid beamforming, withanalog beamforming over the AEs of each SA. Each RF chainthus drives one disjoint SA, reducing power consumption andcomplexity; the SAs provide the spatial diversity gain.
For SCs, we consider SC-FDE and DFT-s-OFDM. ForMCs, we investigate CP-OFDM, OQAM/FBMC, and OTFS,assuming 𝑀-subcarriers. The 𝑚th-subcarrier Rx signal is
y[𝑚] = WHBB [𝑚]W
TRFH[𝑚]x[𝑚] + WH
BB [𝑚]WTRFn[𝑚], (1)
where assuming perfect time and frequency synchronization(no STO or CFO), the received signal is processed using an RFcombining matrix, WRF ∈C𝑄 (r) �� (r)×𝑄 (r)
, and a digital basebandcombining matrix, WBB [𝑚] ∈C𝑄 (r)×𝑁tot ; n[𝑚] ∈C𝑄 (r) �� (r)×1 isthe additive white Gaussian noise (AWGN) vector of indepen-dently distributed CN(0𝑄 (r) �� (r)×𝑄 (r) �� (r) , 𝜎2
𝑛I𝑄 (r) �� (r) ) elementsof noise power 𝜎2
𝑛 . Note that 𝑁tot = 𝑁st×𝑁 , where 𝑁st ≤𝑄 (t)
is the number of data streams (𝑄 (t) is also the number of TxRF chains), and 𝑁 is the number of MC symbols per frame.
The UM-MIMO channel matrix, H[𝑚] ∈ C𝑄 (r) �� (r)×𝑄 (t) �� (t),
represents the overall complex channel at the 𝑚th-subcarrier;assuming a time-invariant (TIV)-FSC, H can be expressed as
H[𝑚] =
H1,1 [𝑚] · · · H1,𝑄 (t) [𝑚]...
. . ....
H𝑄 (r) ,1 [𝑚] · · · H𝑄 (r) ,𝑄 (t) [𝑚]
, (2)
where H𝑞 (r) ,𝑞 (t) [𝑚] ∈C�� (r)×�� (t)denotes the channel response
between the 𝑞 (t) th Tx SA and the 𝑞 (r) th Rx SA. Further detailson the channel model can be found in [44] and equationstherein (Eqs. (15) and (16) define H𝑞 (r) ,𝑞 (t) [𝑚] ∈C�� (r)×�� (t)
in
…
𝑁
𝑁
𝑀
𝐬[0]
𝐬[M − 1]
……
……
……
……
𝐏RF
SC/MCWaveform
DACRF
ChainAnalog
Beamforming
𝑁st
SC/MCWaveform
DACRF
ChainAnalog
Beamforming
SC/MCWaveform
DACRF
ChainAnalog
Beamforming
𝐏BB[M− 1]
𝐏BB[1]
𝑄𝑏(𝑡)
𝑄𝑎(𝑡)
BasebandPrecoding𝐏BB[0]
…
……
Fig. 1: Block diagram of a THz-band UM-MIMO transmitter.
𝑫DDDelay
Doppler1
𝑁𝑇
1
MΔ𝑓
ISFFTΛ Λ⊥
𝑑DD 𝑙, 𝑘
Frequency
Time𝑇 𝑁 − 1
𝑑TF 0,0
01
𝑀 − 1
Δ𝑓
𝑫TF
𝑑TF 𝑚,𝑛
𝑑TF 𝑀−1,𝑁 − 1
𝑑TF 0,𝑁 − 1
SFFT
0
1
𝑀−1
0 1
𝑁 − 10 1𝑑DD 0,0
Fig. 2: Illustration of time-frequency delay-Doppler lattices.
the delay and frequency domains, respectively). The discrete-time Tx complex baseband signal at the 𝑚th-subcarrier is
x[𝑚] = PRFPBB [𝑚]s[𝑚], (3)
where PBB [𝑚] ∈ C𝑄 (t)×𝑁tot is the digital baseband precod-ing matrix per subcarrier, PRF ∈ C𝑄 (t) �� (t)×𝑄 (t)
is the analogRF beamforming matrix, and s[𝑚] =
[𝑠1, 𝑠2, . . . , 𝑠𝑁tot
]T ∈X𝑁tot×1 is the information-bearing symbol vector consisting ofdata symbols drawn from a quadrature amplitude modulation(QAM) constellation, X. We assume normalized symbols,E (s[𝑚]s∗ [𝑚]) = 𝑃t
𝑀𝑁totI𝑁tot , where 𝑃t is the average total Tx
power over 𝑀-subcarriers.
III. SC/MC WAVEFORMS: KEY PERFORMANCEINDICATORS AND ANALYSIS
Choosing a suitable WF is a challenging task that dependson several conflicting communication system performancerequirements and design criteria. For fairness of comparison,we consider the transmission of 𝑀×𝑁 complex symbols ofBW 𝐵 = 𝑀Δ 𝑓 , with SCS Δ 𝑓 and frame duration 𝑇 𝑓 = 𝑁𝑇 ,for both SC and MC schemes; the signal period (𝑇) differsbetween WFs. We discretize the time-frequency (TF) domaininto a lattice, Λ, by sampling time and frequency at integermultiples of 𝑇 and Δ 𝑓 , respectively.
Λ = {(𝑛𝑇, 𝑚Δ 𝑓 ) , 𝑛=0, . . . , 𝑁−1, 𝑚=0, . . . , 𝑀−1} . (4)
Similarly, we discretize the delay-Doppler (DD) plane into
Λ⊥ =
{(𝑘
𝑁𝑇,
𝑙
𝑀Δ 𝑓
), 𝑘 =0, . . . , 𝑁−1, 𝑙=0, . . . , 𝑀−1
}, (5)
4
where 1𝑁𝑇
, 1𝑀Δ 𝑓
define the Doppler and delay domain reso-lutions, respectively. The maximum supported Doppler anddelay spreads areamax=
𝜐𝑐𝑓𝑐 <1/𝑇 and𝜏max<1/Δ 𝑓, respectively,
where 𝜐 is the user velocity, 𝑐 is the speed of light, and 𝑓𝑐 isthe carrier frequency.
We denote by DTF and DDD ∈ C𝑀×𝑁 the data symbolmatrices (of elements 𝑑TF [𝑚, 𝑛] and 𝑑DD [𝑙, 𝑘]) in the TF andDD domains, respectively. In vector form, dTF=vec(DTF) anddDD=vec(DDD). Furthermore, dTF
𝑀∈C𝑀×1 is a column of DTF
(of elements 𝑑TF [𝑚]). In the case of DFT-s-OFDM, the datasymbol matrix is DTF ∈ C��×𝑁 , a sub-matrix of DTF, where�� represents the number of Tx symbols modulated over 𝑀subcarriers. We also denote by dTF
��∈C��×1 a column of DTF.
Note that S= [s[0], s[1], . . . , s[𝑀−1]] of (3), for a single datastream (𝑁st = 1; no digital baseband precoding), reduces toDT
TF. Both TF and DD lattices are shown in Fig. 2.The general form of a continuous-time MC modulator, 𝑥(𝑡),
can be expressed using the discrete Heisenberg transform [37],parameterized by a pulse-shaping prototype filter, 𝑔tx (𝑡), as
𝑥(𝑡) =𝑀−1∑𝑚=0
𝑁−1∑𝑛=0
𝑑TF [𝑚, 𝑛]𝑔𝑚,𝑛 (𝑡), (6)
𝑔𝑚,𝑛 (𝑡) = 𝑔tx (𝑡 − 𝑛𝑇)𝑒 𝑗2𝜋𝑚Δ 𝑓 (𝑡−𝑛𝑇 ) , (7)
where 𝑑TF [𝑚, 𝑛] represents the Tx symbol at subcarrier-index𝑚 and time-index 𝑛. The complex orthogonality condition forthe basis pulse 𝑔𝑚,𝑛 (𝑡) is expressed as 〈𝑔𝑚1 ,𝑛1 (𝑡), 𝑔𝑚2 ,𝑛2 (𝑡)〉=𝛿 (𝑚2−𝑚1) , (𝑛2−𝑛1) , with 𝛿 being the Kronecker delta function.The discrete-time representation of (6) (Nyquist sampling at𝐹s=
1𝑇s=𝐵; limited by ADC/DAC specifications) is
𝑥 [𝑢𝑇s] =𝑀−1∑𝑚=0
𝑁−1∑𝑛=0
𝑑TF [𝑚, 𝑛]𝑔𝑚,𝑛 [𝑢𝑇s], (8)
where 𝑢 = {0, 1, . . . , 𝑀𝑁−1}. The PAPR of a discrete-timesignal 𝑥 [𝑢] over a finite observation period 𝑁per is expressedas a random variable [51]
PAPR (𝑥 [𝑢]) = max𝑢∈[0,𝑁per−1]
(|𝑥 [𝑢] |2
)/E
(|𝑥 [𝑢] |2
), (9)
the statistical behavior of which can be estimated throughnumerical simulations; we characterize the complementarycumulative distribution function (CCDF) of PAPR. In theremainder of this section, we detail various KPIs and introduceseveral schemes, namely, CP-OFDM, DFT-s-OFDM, SC-FDE,OQAM/FBMC, and OTFS, as illustrated in Fig. 3.
A. Spectral Efficiency and End-to-End Latency
The SE (bits/sec/Hz) is an essential indicator of throughputand achievable rate for a given BW. Since the THz bandpromises huge available BWs, unlike below 6 GHz com-munications, SE is not a primary concern. However, SE isstill important for data demanding use cases, such as THz-enabled holographic video meeting, augmented reality (AR),and virtual reality (VR). Similarly, at the physical layer,the E2E latency, which is the delay between informationtransmission (bits at the output of the channel encoder) andrecovery (bits at the input of the channel decoder), is WF-
dependant (overlapping in OQAM/FBMC lengthens the frameduration, for example). Nevertheless, the ultra-broadband THzBW (𝐵) ensures a very small sampling period (𝑇s). Theprocessing time of the equalizer and channel encoder/decoder(not included in our E2E latency definition) are more criticalat THz frequencies and are also dependent on the used WF.
B. Power Spectral Density and Out-of-Band Emissions
The power spectral density (PSD) and OOB emissionsfollow strict standard regulations to meet spectrum mask re-quirements. For example, the international telecommunicationsunion (ITU) radio regulation 5.340 prohibits transmissions inten passive bands over 100−252 GHz to protect deep space ob-servatories and satellite sensors [52], resulting in a maximumavailable contiguous BW of 23 GHz. OOB emissions are alsocritical in integrated space-air-ground THz networks. It is thusimportant to study OOB-induced interference to neighboringsystems and among multiple users, highlighting the role ofcarrier-aggregation techniques. The severity of OOB emissionsis dictated by BW, required SE, and neighboring co-operatingsystems. The WF Tx spectrums in the IEEE 802.15.3d sub-THz standard are described for different BWs in [15].
C. Transceiver Complexity
The computational complexity of the studied SC/MCtransceivers is arguably the most important KPI to consider,given the limited processing capabilities at Tbps and theneed for low-cost and low-power solutions. Without loss ofgenerality, we only consider the number of real multiplicationsper unit of time in the modulation and demodulation processes.
The complexity of equalization, channel coding, and de-coding are important in their own right but not included inour study. The number of real multiplications in an 𝑀-pointfast Fourier transform (FFT)/inverse FFT (IFFT) (split-radixalgorithm) is [53]
CFFT (𝑀) = 𝑀 (log2 (𝑀) − 3) + 4. (10)
As illustrated in Fig. 3(a), IFFT/FFT is followed by rectan-gular pulse-shaping in CP-OFDM, resulting in a complexity(COFDM) and number of multiplications per unit time (COFDM):
C (t)/(r)OFDM = CFFT (𝑀) + 4(𝑀+𝑁CP),
C (t)/(r)OFDM =
𝑁 C (t)/(r)OFDM
𝑁 (𝑀+𝑁CP)𝑇s=
C (t)/(r)OFDM
𝑀+𝑁CP𝐹s.
(11)
Furthermore, in the case of DFT-s-OFDM (Fig. 3(b)), anadditional precoding FFT/IFFT block in Tx/Rx results in
C (t)/(r)DFTsOFDM= CFFT (𝑀) + CFFT (��) + 4(𝑀+𝑁CP),
C (t)/(r)DFTsOFDM=
C (t)/(r)DFTsOFDM𝑀+𝑁CP
𝐹s.(12)
SC-FDE enjoys relatively low Tx complexity as symbolsare directly transmitted after CP (Fig. 3(c)). However, withFFT/IFFT at Rx, the overall transceiver complexity is that ofCP-OFDM (complexity shift from Tx to Rx); COFDM= CSCFDE.For OQAM/FBMC, we consider the direct form polyphaseprototype filter realization, with a filter length of 𝐿p =𝑂×𝑀
5
(a): CP-OFDM
(b): DTF-s-OFDM
(e): OTFS
THz
channelCP-1 channel
equalizerDFTIDFT CP
𝐝𝑀TF
𝐅𝑀Η 𝐅𝑀𝐂CP
𝐱OFDM 𝐱OFDM 𝐝𝑀TF
𝐇s, 𝐧s
𝐲OFDM 𝐲OFDM
𝐄ZF/MMSE
THz
channelCP-1 channel
equalizerDFTCP
𝐝𝑀TF
𝐅𝑀𝐂CP
𝐱SCFDE 𝐝𝑀TF 𝐲SCFDE 𝐲SCFDE
𝐄ZF/MMSE
(c): SC-FDE
(d): OQAM/FBMC
THz
channel
analysis
filter bank
de-offset
QAM
sub-channel
equalizer
offset
QAM
synthesis
filter bank
𝐝TF
𝐆syn
𝐱FBMC 𝐲FBMC
THz
channelCP-1 ch.
equ.DFTDFT CP
𝐝 𝑀TF
𝐅𝑀𝐂CP
𝐱DFTsOFDM 𝐝 𝑀TF
𝐇s, 𝐧s 𝐄ZF/MMSE
IDFT
𝐅𝑀Η𝐅 𝑀
subcarrier
mapping
𝐌𝑀, 𝑀
IDFT
𝐅 𝑀Η
IDFT
𝐅𝑀Η
subcarrier
demapping
𝐆ana
THz TV
channel
Wigner
transform
ch.
equ.
Windowing
+ SFFT
SFFT-1 +
Windowing
Heisenberg
transform
𝐝DD
OTFS transform OTFS-1 transform
CP CP-1
𝐝TF
𝐝TF
𝐇DD, 𝐧OTFS𝐱OTFS 𝐲OTFS
𝐲DD𝐲TF
Time-Frequency Domain
𝐝DD
𝐇s, 𝐧s
Fig. 3: SC/MC transceiver block diagrams: (a) CP-OFDM, (b) DFT-s-OFDM, (c) SC-FDE, (d) OQAM/FBMC, and (e) OTFS.
(𝑂 is the pulse-shaping overlapping factor). Accounting forOQAM, phase offsets (for linear phase filters), IFFT, filtering,and 50% overlapping, CFBMC and CFBMC add up to [26]
C (t)/(r)FBMC =2CFFT (𝑀) + 4𝐿p + 4𝑀,
C (t)/(r)FBMC =
𝑁 C (t)/(r)FBMC
𝑀 (𝑁+𝑂−1/2) 𝐹s,(13)
where the first multiplication by a factor of 2 accountsfor complex-valued QAM symbols that are separated intotwo real-valued symbols. The OQAM/FBMC complexity isslightly dependant on (𝑂). OQAM/FBMC is clearly morecomplex than CP-OFDM. While for OTFS 1, based on (41),the complexity and number of multiplications per unit timeare expressed as
C (t)/(r)OTFS = CFFT (𝑁) + 4(𝑁+𝑁CP/𝑀),
C (t)/(r)OTFS =
𝑀 C (t)/(r)OTFS
𝑁𝑀+𝑁CP𝐹s.
(14)
Hence, COTFS/COTFS are functions of 𝑁 and not 𝑀 and areless than other WF complexities.
D. Peak to Average Power Ratio
PAPR is an important KPI for sub-THz/THz communica-tions as it dictates the Tx power efficiency, which affectsenergy efficiency, link budget, and coverage (processing ultra-wide BW sub-THz/THz signals is very power consuming).Large amplitude fluctuations in high PAPR lead to spectralregrowth and non-linear distortion; an output back-off isthus needed to retain the linear PA region, reducing powerefficiency. The saturated output power (𝑃sat) recordings in
1In this work, we use OTFS with rectangular Tx and Rx windowing andpulse-shaping, and consider one CP per frame (𝑀×𝑁 symbols), which resultsin a low-complexity implementation. This setting is different from the OTFSsetting in [37] with complexity C (t)/(r)
OTFS = 2CFFT (𝑀 ) + CFFT (𝑁 ) + 4(𝑁 +𝑁CP/𝑀 ) and the OFDM-based OTFS setting in [39] which adds one CPevery 𝑁 blocks (each block is of length 𝑀 ). See Sec. III-K for more details.
state-of-the-art THz PAs [23] reveal limited achievable outputpower that decreases drastically with operating frequency(the trend lines for different technologies follow a stepperincreasing-slope). For example, 𝑃sat≈20, 23 dBm and 28 dBmat 𝑓𝑐 = 100 GHz for CMOS, SiGe BiCMOS, and InP tech-nologies, respectively. Furthermore, high PAPR necessitateshigh dynamic-range THz ADCs of low signal-to-quantization-noise ratios, which are not cost- and power-efficient [54]. TheADC signal-to-noise and distortion ratio (SNDR) decreasesby increasing the Nyquist sampling rate. However, the energyper conversion step increases linearly with frequency beyond100 MHz [55]. For example, for an ADC of 𝐹s = 100 GHz,the power consumption and SNDR are approximately 0.3 Watt(very high) and 35 dB (very low), respectively. The PAPRCCDF of CP-OFDM is expressed as [51]
Pr (PAPR (𝑥OFDM) > 𝛾th) = 1 − (1 − 𝑒−𝛾th )𝑀 , (15)
for a PAPR threshold 𝛾th. Furthermore, the closed-form ap-proximation of the PAPR CCDF of OQAM/FBMC in [51]reveals higher PAPR values compared to CP-OFDM due toper-subcarrier filtering. In [56], the PAPR CCDF of OTFS isapproximated for high values of 𝑁 as
Pr (PAPR (𝑥OTFS) > 𝛾th) ≈ 1 − (1 − 𝑒−𝛾th )𝑀𝑁. (16)
OTFS has better PAPR characteristics when 𝑁 < 𝑀 , with anupper bound that grows linearly with 𝑁 [56], unlike TF MCmodulations where the PAPR depends on and grows linearlywith the number of subcarriers 𝑀 (15). This OTFS featurecan be leveraged in ultra-wideband THz communications withmany subcarriers. SCs inherently result in low PAPR, whetherin SC-FDE or in DFT-s-OFDM due to DFT-precoding.
E. Robustness to Hardware Impairments
THz-band transceivers are substantially more vulnerable toconventional RF impairments than microwave and mmWavetransceivers. Therefore, the following KPIs are crucial tomitigate the corresponding impairments in the design process.
6
1) Phase Noise: Due to time-domain instability, the localoscillator (LO) output can be a phase-modulated tone. PHNin THz devices (that are not yet mature) has more severeconsequences than in microwave or mmWave devices. Themotivation to use low-cost devices for THz communicationsis also limiting, where achieving low PHN requires advancedcomplex techniques such as phase-locked loops [57]. In par-ticular, if the THz LO signal is generated using a low-costlow-frequency oscillator followed by frequency multipliers,the required multiplication factor, b, is relatively high, whichfurther increases the PHN power by a factor of b2. Therefore,PHN increases by 6 dB for every doubling of the oscil-lation frequency [57]. Furthermore, PHN causes significantperformance degradation and reduces the effective signal-to-interference plus noise ratio (SINR) at the Rx, limiting bothdata rate and BER. Note that increasing the signal-to-noiseratio (SNR) does not mitigate the PHN effects. Therefore,optimized SC schemes and non-coherent modulations that areinherently robust to PHN are argued to be good candidates forsub-THz communications [18].
There are several approaches for modeling PHN, two ofwhich are most prominent. The first is a correlated model thatuses the superposition of Wiener (Gaussian random-walk) andGaussian processes; the second is an uncorrelated model thatconsiders only a Gaussian noise reflecting the white PHN floor.The appropriate choice of PHN models for sub-THz band isaddressed in [42], where it is argued that the uncorrelatedGaussian PHN model should be favored if the system BW (𝐵)is large enough compared to the oscillator corner frequency( 𝑓cor):
𝑁
(𝑓cor𝐵
)2≤ ln (2)
2𝜋. (17)
Therefore, the Rx signal, at instant 𝑢, is expressed as
𝑦[𝑢] =(ℎ[𝑢] ∗
(𝑥 [𝑢]𝑒 𝑗 𝜙 (t) [𝑢 ]
))𝑒 𝑗 𝜙
(r) [𝑢 ] + 𝑛[𝑢], (18)
where ∗ denotes linear convolution, and 𝜙 (t) [𝑢], 𝜙 (r) [𝑢] arediscrete stochastic processes representing Tx, Rx LO PHN,respectively. The correlated model is defined as
𝜙[𝑢] = 𝜙w [𝑢] + 𝜙g [𝑢], (19)
where the Wiener and Gaussian PHN models are expressed,respectively, as
𝜙w [𝑢] = 𝜙w [𝑢 − 1] + \w [𝑢], \w [𝑢]∼N (0, 𝜎2w), (20)
𝜙g [𝑢]∼N (0, 𝜎2g ). (21)
The uncorrelated PHN implies 𝜙[𝑢] = 𝜙g [𝑢]. The variancesare defined as 𝜎2
w=4𝜋2𝐾2𝑇 and 𝜎2g =𝐾0/𝑇 , where 𝐾0 and 𝐾2
are the PHN levels that can be evaluated from the measuredPHN PSD, the corner frequency is ( 𝑓cor =𝐾2/𝐾0), 𝑇 =1/𝐵 isthe modulated signal duration, and 𝐵 is the system BW [58](a strong dependence of system performance on BW).
2) Wideband IQI: The frequency-dependent wideband IQIis another dominant hardware impairment in THz transceiversoperating over ultra-wide BWs. Efficient signal processingtechniques have been extensively studied for narrowband IQIat both Tx (via digital pre-distortion) and Rx. However, only
a few works address the wideband IQI model in the THz-band, such as [40] for SC-FDE. Furthermore, wideband PAnon-linearity models still lack in the THz literature. Extensiveresearch to study such impairments is crucial.
F. Robustness to THz-specific ImpairmentsTHz-specific channel-induced impairments should also be
considered when studying candidate WFs. For example, THzpropagation suffers from misalignment between Tx and Rx,which is highly probable given the narrow THz beams [44].Another THz channel characteristic is the spherical wavepropagation model (SWM), which should be accounted forat relatively short communication distances [44]. More im-portantly, a beam split effect arises in wideband UM-MIMObeamforming. In particular, the difference between the carrierand center frequencies, 𝑓𝑚 and 𝑓𝑐 , results in THz path com-ponents squinting into different spatial directions at differentsubcarriers, causing severe array gain loss [44]. Such beamsplit is mainly caused by frequency-independent delays inanalog-beamforming PSs. Furthermore, large UM-MIMO THzarrays result in very narrow beamwidths that worsen thiseffect. Several beam-split mitigation methods are proposed inthe literature, such as delay-phase precoding in [59], whereCP-OFDM is assumed. However, the effect of beam split onother SC/MC schemes is not yet studied. This work onlystudies the impairment caused by beam split as it is morerelevant to WF design than misalignment and SWM.
G. CP-OFDMThe discrete-time Tx OFDM signal is derived from (8) (𝑁 =
1) using rectangular pulse-shaping:
𝑥OFDM [𝑢] =𝑀−1∑𝑚=0
𝑑TF [𝑚]𝑔tx [𝑢]𝑒 𝑗2𝜋𝑚𝑀
𝑢 ,
𝑔tx [𝑢] =
1√𝑀
𝑢 = 0, . . . , 𝑀−1
0 otherwise,
(22)
xOFDM = FH𝑀dTF
𝑀 . (23)
To combat inter-symbol interference (ISI) in a time-dispersivewireless channel of lengh 𝑁ch = 𝜏rms/𝑇s, where 𝜏rms is theroot mean square (RMS) delay spread, a guard interval of𝑁CP ≥𝑁ch samples is added to the Tx signal. The CP-OFDMsignal can thus be expressed as
xOFDM = CCPxOFDM; CCP =
[0𝑁CP ,𝑀−𝑁CP I𝑁CP
I𝑀
]. (24)
where xOFDM= [𝑥OFDM [[−𝑁CP]𝑀],· · ·, 𝑥OFDM [0],· · ·,𝑥OFDM [𝑀−1]]T, and CCP is the CP-insertion matrix of size 𝑀t ×𝑀 ,𝑀t = 𝑀 + 𝑁CP, defined as The total CP-OFDM symbolduration, 𝑇 = (𝑀+𝑁CP)𝑇s =𝑇u+𝑇CP, is that of the CP duration(𝑇CP) plus the useful symbol duration (𝑇u). Although CPreduces SE (Table IV), it emulates a cyclic convolution withthe channel, allowing simple FDE through FFT. The receivedsignal over a single-input single-output (SISO) channel ofimpulse response hs= [ℎ0, . . . , ℎ𝑁ch−1]T, after CP removal is
yOFDM=HsxOFDM+ns, (25)
7
where assuming perfect time and frequency synchronization,Hs is an 𝑀×𝑀 circular convolution matrix of band-diagonalstructure built upon hs, and ns∼CN(0, 𝜎2
𝑛I𝑀 ) is the AWGNvector. Note that the actual transmission is expressed asyOFDM = HsxOFDM + ns, where Hs ∈ C(𝑀t+𝑁ch−1)×𝑀t is de-rived from Hs, and ns ∼ CN(0, 𝜎2
𝑛I𝑀t+𝑁ch−1). The signal isthen processed by a DFT block F𝑀 . Equalization can beperformed using zero-forcing (ZF) or minimum mean-squarederror (MMSE), with corresponding equalization matrices
EZF = (HHs Hs)−1HH
s ,
EMMSE = (HHs Hs +
𝜎2𝑛
𝑃𝑥
I𝑀 )−1HHs ,
(26)
where 𝑃𝑥 is the signal power. The Tx symbol estimatesare retrieved as dTF
𝑀= 𝐽 (EF𝑀yOFDM), where 𝐽 (·) maps an
equalized symbol to the closest symbol in X. Note that (23)can be generalized to express the Tx CP-OFDM frame (𝑁symbols) as
XOFDM = CCPFH𝑀DTF,
xOFDM𝑀t𝑁
= vec(XOFDM) = (I𝑁 ⊗CCP)(I𝑁 ⊗FH
𝑀
)dTF.
(27)
Designing a CP-OFDM system requires tuning many pa-rameters such as the number of subcarriers (𝑀), the CPduration (𝑇CP), and the SCS (Δ 𝑓 ). Such parameters are chosensuch that
𝜏rms ≤ 𝑇CP ≤ 𝑇u � 𝑇coh; Δ 𝑓 =1𝑇u
=𝐵
𝑀, (28)
where 𝐵coh = 15𝜏rms
is the coherence BW and 𝑇coh =√9
16𝜋amax× 1
amaxis the coherence time. The SCS satisfies (28)
to ensure orthogonality and maximize SE.The SCS choice also affects E2E latency, PAPR, complexity,
and equalization performance. In particular, the SCS providesa trade-off between CP overhead, sensitivity to Doppler spread,and robustness to hardware imperfections. The CP length isalso a critical design parameter, where larger 𝑁CP relaxes timesynchronization constraints caused by STO, but also at theexpense of larger CP overhead (decreased SE). Furthermore,the number of subcarriers (𝑀) impacts the PAPR performance(15) and the FFT/IFFT complexity (Sec. III-C).
The transmission spectra and molecular absorption dictatethe large available BW in THz LoS scenarios [44]. However,in indoor THz scenarios, the channel can be LoS-dominant andnon-LoS (NLoS)-assisted, or only NLoS (MP). Based on thecoherence BW, for a given communication distance, we decideon the corresponding design parameters of a frequency-flatchannel or FSC per subcarrier. For example, for a communi-cation distance of 3 m, in the sub-THz band ( 𝑓𝑐 = 0.3 THz),𝐵coh = 1 GHz; in the THz band ( 𝑓𝑐 = 0.9 THz), 𝐵coh ≈ 5 GHz[14].
We list in Table I some of the expected CP-OFDM param-eters for sub-THz/THz band communications, alongside pa-rameters adopted in both 4G-LTE (below 6 GHz) and 5G-NR(below 6 GHz and mmWaves). In a nutshell, CP-OFDM enjoysa relatively low-complexity implementation (using FFT), isrobust to MP fading, and uses a simple FDE method (single-tap equalizer for a broadband FSC). However, the resultant
high PAPR is challenging for power-limited sub-THz/THzcommunications. Moreover, the CP-OFDM time tolerance forsymbol synchronization is very low (order of nanoseconds)due to the expected small 𝑇CP and 𝜏rms.
H. DFT-s-OFDM
DFT-s-OFDM, also known as precoded OFDM, is adoptedin 4G-LTE/5G-NR uplink and is a promising candidate forTHz communications. The use of a DFT-block at the Txreduces the PAPR and retains all SC benefits, albeit at amarginal complexity cost. DFT-s-OFDM thus aims at reducingpower consumption and PA costs at user terminal. When datasymbol blocks are assigned to different users, DFT-s-OFDMreduces to SC-FDMA in multi-user scenarios.
As illustrated in Fig. 3(b), data symbols are first spreadin DFT-precoding; the outputs are the complex symbols thatmodulate the OFDM subcarriers. For a selection of �� ≤ 𝑀
subcarriers to be modulated. The Tx signal is expressed as
xDFTsOFDM = CCPFH𝑀
(M𝑀,��F�� dTF
��
), (29)
where M𝑀,�� is a mapping matrix between data symbols andthe �� active subcarriers (zero insertion at 𝑀 − �� unusedsubcarriers). The mapping can be localized or distributed. Inthe localized mode, M𝑀,�� = [I�� , 0�� ,𝑀−�� ]T, and the DFToutputs are directly mapped to a subset of consecutive subcar-riers. In the distributed mode, the DFT outputs are assignedto non-continuous subcarriers over the entire available BW.The additional need for signaling, pilots, and guard bands (inmultiple access scenarios) in the distributed mode increases thesystem complexity, whereas the straightforward implementa-tion of equal SCS in the localized mode is favorable.
I. SC-FDE
A promising alternative to CP-OFDM is SC-FDE, whichcombines the benefits of CP and FDE, and has low PAPRdue to low envelope variations. Unlike in CP-OFDM, whereeach data symbol is allocated a small BW over a long symbolduration, in SC-FDE, data symbols are assigned to a singlelarge BW with short symbol durations. For the same CP-OFDM symbol duration, the SC-FDE Tx signal, containing𝑀 symbols, can be expressed as
xSCFDE = CCP
(FH𝑀F𝑀
)dTF𝑀 = CCPdTF
𝑀 . (30)
The remainder transmission, equalization, and demodulationstages are similar to those of CP-OFDM, as shown in Fig. 3(c).The SC-FDE synchronization algorithms are also very similarto those of CP-OFDM. Furthermore, the spectral shape of theSC-FDE WF is determined by the Tx pulse-shaping, usedDAC, and RF filtering stages. It is worth noting that thechoice of pulse-shaping affects the PAPR, OOB emissions,complexity, and immunity to hardware impairments. However,SC-FDE sub-THz/THz implementations still share some CP-OFDM problems in synchronization and robustness to impair-ments.
8
TABLE I: Key parameters for CP-OFDM at different bands
ParameterBand below 6 GHz mmWave sub-THz THz
SCS - Δ 𝑓 (KHz) 2` × 15, ` = 0 (4G-LTE)2` × 15, ` = 0, 1, 2 (5G-NR)
2` × 15,` = 2, 3, 4
2` × 5000,` = 0, 1
2` × 25000,` = 0, 1
System BW - 𝐵 (MHz) 1.25 up to 20 (4G-LTE)5 up to 100 (5G-NR)
50, 100,200, 400
[10−20]×103 [20−100]×103
Number of subcarriers - 𝑀 128 up to 2048 (4G-LTE)up to 4096 (5G-NR)
up to 4096 up to 2048 up to 2048
Number of used subcarriers 76 up to 1200 (4G-LTE)up to 3300 (5G-NR)
up to 3300 up to 1200 up to 1200
J. OQAM/FBMC
OQAM/FBMC is another promising WF candidate, es-pecially for cognitive radio (CR) and dynamic/intelligentspectrum sharing applications. OQAM/FBMC offers high SE(no need for CP), low OOB levels, and low sensitivity toCFO. Furthermore, by using a per-subcarrier well-localizedpulse-shaping filter in both time and frequency (such asPHYDYAS [60]), OQAM/FBMC supports enhanced synchro-nization procedures. However, such benefits come at the costof limited integration with MIMO systems (maintaining realorthogonality in OQAM complicates precoder design [61]),higher PAPR compared to OFDM (due to subcarrier filtering),and higher complexity (especially in the equalizer as thereis no CP). Given the importance of such KPIs at highfrequencies, OQAM/FBMC is not a good candidate for THzcommunications.
The direct form of an OQAM/FBMC system is illustratedin Fig. 3(d), consisting of OQAM pre-processing, a synthesisfilter bank (SFB), an analysis filter bank (AFB), and OQAMpost-processing. We assume a low-complexity implementationbased on a polyphase filter structure (PHYDYAS with over-lapping factor 𝑂) and FFT, as described in [60] (Figures (2-7) and (2-8)). OQAM/FBMC satisfies the real orthogonalitycondition, R
(〈𝑔FBMC
𝑚1 ,𝑛1 (𝑡), 𝑔FBMC𝑚2 ,𝑛2 (𝑡)〉
)= 𝛿 (𝑚2−𝑚1) , (𝑛2−𝑛1) , instead
of complex orthogonality. Thus, the useful symbol time stillsatisfies Δ 𝑓 = 1/𝑇u, but the symbol duration is 𝑇 =𝑇u/2. TheTx signal can be derived from (6) by adding to the Tx basispulse in (7) a phase shift, 𝛽𝑚,𝑛 = 𝜋
2 (𝑚+𝑛):
𝑔FBMC𝑚,𝑛 (𝑡) = 𝑔FBMC
tx (𝑡 − 𝑛𝑇)𝑒 𝑗2𝜋𝑚Δ 𝑓 (𝑡−𝑛𝑇 )𝑒 𝑗𝛽𝑚,𝑛 . (31)
Such a phase shift transfers the induced interference betweensymbols to the imaginary domain [62]. The resultant basispulse in (31) is a frequency- and time-shifted version of theprototype filter 𝑔FBMC
tx (𝑡). Furthermore, the prototype filter isdesigned using the frequency-sampling technique, with (2𝑂−1)non-zero frequency-domain samples for an overlapping factor𝑂. For filter of length 𝐿p and coefficients 𝜓 [𝑜]’s (defined in[60]), the impulse response is
𝑔FBMCtx [𝑖] = 1 + 2
𝑂−1∑𝑜=1
(−1)𝑜𝜓 [𝑜] cos(2𝜋𝑜𝐿p
(𝑖 + 1)). (32)
Then, the discrete-time Tx signal can be express as
xFBMC = GsyndTF, (33)
where the time interval is −𝑂𝑇u/2≤ 𝑡 <𝑂𝑇u/2+(𝑁−1)𝑇 andGsyn ∈C𝑁t×𝑀𝑁 is the Tx matrix that contains the basis pulse-
shaping vectors gFBMC𝑚,𝑛 ∈ C𝑁t×1 (𝑁t = (𝑂𝑇u+(𝑁−1)𝑇) 𝐹s),
defined as
Gsyn= [gFBMC0,0 · · · gFBMC
𝑀−1,0 gFBMC0,1 · · · gFBMC
𝑀−1,𝑁−1], (34)
gFBMC𝑚,𝑛 [𝑛t] = 𝑔FBMC
𝑚,𝑛 (𝑡) |𝑡=𝑛t𝑇s−𝑂𝑇u
2, 𝑛t = 0, 1, · · · , 𝑁t. (35)
At the Rx, the analysis filter, Gana=GHsyn, is used in matched-
filter decoding.
K. OTFSThe recently proposed OTFS WF [37] is tailored for high-
Doppler doubly-selective channels, typically arising in V2Xcommunications. Unlike the other WFs that modulate data inthe TF domain, OTFS modulates data in the DD domain,transforming the TV channel in TF into a 2D quasi-TIVchannel in DD. The corresponding transmission frame symbolsexperience a nearly constant channel gain [63], making OTFSa promising solution in high-Doppler MP channels, exploitingthe full diversity of TV-FSC and providing substantial delayand Doppler resilience [64]. OTFS is superior to CP-OFDMin this context.
OTFS modulation consists of two main blocks, OTFStransform and Heisenberg transform, as illustrated in Fig. 3(e).Furthermore, OTFS transform involves two stages, inversesymplectic finite Fourier transform (ISFFT) and windowing.ISFFT maps data symbols 𝑑DD [𝑙, 𝑘] in the DD domain tosamples 𝑑TF [𝑚, 𝑛] in the TF domain as follows [63]
𝑑TF [𝑚, 𝑛] =1
√𝑀𝑁
𝑀−1∑𝑙=0
𝑁−1∑𝑘=0
𝑑DD [𝑙, 𝑘]𝑒 𝑗2𝜋(𝑛𝑘𝑁
−𝑚𝑙𝑀 ) . (36)
A closer look into (36) reveals that the ISFFT of DDD isequivalent to an 𝑀-point DFT and an 𝑁-point IDFT of thecolumns and rows of DDD, respectively. Subsequently, (36)can be expressed in matrix and vectorized forms as
DTF = F𝑀DDDFH𝑁 ; dTF = (FH
𝑁 ⊗ F𝑀 )dDD. (37)
The OTFS transform applies a Tx window 𝑈tx [𝑚, 𝑛] to theTF signal in (36). Let Utx = diag(𝑈tx [𝑚, 𝑛]) ∈ C𝑀𝑁×𝑀𝑁
and assume rectangular windows for both Tx and Rx(Utx=Urx=I𝑀𝑁), the OTFS transform output is expressed as
dTF = UtxdTF. (38)
Heisenberg transform then forms the time-domain Tx signal;combining (6), (7), and (38)
𝑥OTFS (𝑡) =𝑀−1∑𝑚=0
𝑁−1∑𝑛=0
𝑑TF [𝑚, 𝑛]𝑔tx (𝑡−𝑛𝑇)𝑒 𝑗2𝜋𝑚Δ 𝑓 (𝑡−𝑛𝑇 ) . (39)
9
The Tx (𝑔tx (𝑡)) and Rx (𝑔rx (𝑡)) pulses ideally satisfy thebi-orthogonality condition [37], although not practical. LetGtx = diag(𝑔tx [0], 𝑔tx [𝑇/𝑀],. . ., 𝑔tx [(𝑚 − 1)𝑇/𝑀]) ∈ C𝑀×𝑀
be formed from samples of 𝑔tx (𝑡); Grx similarly defined(assuming rectangular pulse-shaping Gtx=Grx=I𝑀 [63], thenDTF =DTF). We can restructure (39) in matrix and vectorizedforms as
XOTFS = GtxFH𝑀
(F𝑀DDDFH
𝑁
)= GtxDDDFH
𝑁 , (40)
xOTFS =
(FH𝑁 ⊗ Gtx
)dDD. (41)
If 𝑔tx (𝑡) is a rectangle pulse-shape of duration 𝑇 , (39) reducesto IDFT, and for 𝑁 =1, the inner box of Fig. 3(e) is CP-OFDM.Therefore, one OTFS frame is effectively an ISFFT over 𝑁consecutive independent OFDM symbols with 𝑀 subcarriers.
As a spectral-efficient solution, we assume one CP for theentire OTFS frame, of the same duration 𝑇CP as in previousWFs. The Rx signal can be expressed as
𝑦OTFS (𝑡)=∫a
∫𝜏
ℎDD (𝜏, a)𝑥OTFS (𝑡−𝜏)𝑒 𝑗2𝜋a (𝑡−𝜏)d𝜏da+𝑛(𝑡), (42)
where 𝜏 and a are delay and Doppler variables, respectively,and ℎDD (𝜏, a) is the DD channel response that is typicallysparse [63] (a small number of reflectors with associateddelays and Doppler shifts; limited number of MPs) and canbe expressed as [63]
ℎDD (𝜏, a) =𝑁P∑𝑖=1
ℎ𝑖𝛿(𝜏 − 𝜏𝑖)𝛿(a − a𝑖), (43)
where 𝑁P is the number of paths, 𝛿(·) is the Dirac deltafunction, and ℎ𝑖 , 𝜏𝑖 , and a𝑖 are the 𝑖th-path gain, delay, andDoppler shift, respectively:
𝜏𝑖 =𝑙𝜏𝑖
𝑀Δ 𝑓, a𝑖 =
𝑘a𝑖
𝑁𝑇, (44)
for integers 𝑙𝜏𝑖 , 𝑘a𝑖 (indexes of the lattice in (5)). Note thatthe assumptions in (44) can be further extended to involvefractional Doppler shifts, which result in additional inter-Doppler interference. The resultant performance degradationcan be compensated in the equalizer, using the message-passing algorithm [63], for example. We can ignore fractionaldelays in a typical wideband THz system since the resolutionis sufficient to approximate the path delay to the nearest pointin the DD lattice [63]. The Rx signal, after discarding CP, issampled as
𝑦OTFS [𝑢]=𝑁P∑𝑖=1ℎ𝑖𝑒
𝑗2𝜋𝑘a𝑖 (𝑢−𝑙𝜏𝑖 )
𝑀𝑁 𝑥OTFS[[𝑢 − 𝑙𝜏𝑖 ]𝑀𝑁
]+𝑛OTFS [𝑢],
(45)yOTFS = HDDxOTFS + nOTFS, (46)
where yOTFS ∈ C𝑀𝑁×1, nOTFS ∼CN(0, 𝜎2𝑛I𝑀𝑁 ), and HDD ∈
C𝑀𝑁×𝑀𝑁 is the channel matrix
HDD =
𝑁P∑𝑖=1
ℎ𝑖𝚷𝑙𝜏𝑖𝚫𝑘a𝑖 , (47)
with 𝚷𝑙𝜏𝑖∈R𝑀𝑁×𝑀𝑁 being the delay matrix, a forward cyclic
shifted permutation of 𝚷 of delay 𝑙𝜏𝑖 (𝚷𝑙𝜏𝑖=1=𝚷 and 𝚷𝑙𝜏𝑖=0=I𝑀𝑁 ),
𝚷 =
0 · · · 0 1
1. . . 0 0
.... . .
. . ....
0. . . 1 0
, (48)
and 𝚫𝑘a𝑖 ∈ C𝑀𝑁×𝑀𝑁 is the Doppler shift matrix whichmodulates the Tx signal with a carrier at frequency 𝑘a𝑖 , where𝚫 = diag(𝜔0, 𝜔1, · · · , 𝜔𝑀𝑁−1) and 𝜔=𝑒
𝑗2𝜋𝑀𝑁 .
The Rx signal is then transformed into TF using Wignertransform, which match filters 𝑦OTFS (𝑡) with an Rx pulseshape, 𝑔rx (𝑡), and samples it at the lattice points defined in(4). The Wigner transform is given by
𝑦TF [𝑚, 𝑛]= 𝐴𝑔rx ,𝑦OTFS (𝑡, 𝑓 ) |𝑡=𝑛𝑇 , 𝑓 =𝑚Δ 𝑓 ,
𝐴𝑔rx ,𝑦 (𝑡, 𝑓 )=∫𝑔∗rx (𝑡 ′ − 𝑡)𝑦OTFS (𝑡)𝑒− 𝑗2𝜋 𝑓 (𝑡′−𝑡)𝑑𝑡 ′.
(49)
We can express (49) (similar to (40)) after building YOTFS ∈C𝑀×𝑁 from yOTFS’s in (46) as
YTF = F𝑀GrxYOTFS, (50)
where YTF ∈ C𝑀×𝑁 consists of elements 𝑦TF [𝑚, 𝑛]. The Rxwindowing operation is similar to (38). Thus, yTF=vec(YTF)=UrxyTF (in our case Urx = I𝑀𝑁 ). Then, the TF domain signal,��TF [𝑚, 𝑛]= 𝑦TF [𝑚, 𝑛], is mapped back to the DD domain usingthe symplectic finite Fourier transform (SFFT) as
𝑦DD [𝑙, 𝑘] =1
√𝑀𝑁
𝑀−1∑𝑚=0
𝑁−1∑𝑛=0
𝑦TF [𝑚, 𝑛]𝑒− 𝑗2𝜋( 𝑛𝑘𝑁
−𝑚𝑙𝑀 ) , (51)
YDD = FH𝑀YTFF𝑁 = FH
𝑀 (F𝑀GrxYOTFS) F𝑁 , (52)
where YDD ∈C𝑀×𝑁 is composed of 𝑦DD [𝑙, 𝑘]. We can write(52), after substituting (41) in (46), in a vectorized form
yDD = vec(YDD) = (F𝑁 ⊗ Grx) yOTFS
= (F𝑁 ⊗ Grx) HDD
(FH𝑁 ⊗ Gtx
)dDD+(F𝑁 ⊗ Grx) nOTFS
= HeffDDdDD + nOTFS,
(53)where Heff
DD denotes the effective channel matrix in the DDdomain, and nOTFS is the modified noise vector. OTFS equal-ization and detection can be applied directly on the vectorizedform in (53), where message passing is shown to be effi-cient [63]. However, we only consider the linear equalizersZF/MMSE [65] (not the low-complexity version in [65]) fora fair comparison with other candidate WFs.
As illustrated in Sec. III-D, the maximum PAPR CCDFof OTFS is limited by 𝑁; an enhanced PAPR performancecompared to CP-OFDM [56]. Moreover, due to the potentialfull TF channel diversity, OTFS outperforms reference MCschemes. The OTFS complexity is also shown to be less thanthat of traditional CP-OFDM (see (41) and Sec. III-C). Allthat emphasizes the prospects of OTFS in emerging V2X usecases in THz-enabled B5G/6G. However, the price to pay is inincreased Rx detection and DD channel estimation complexity,especially in the presence of fractional Doppler [66]. Note
10
that 𝑀 determines the delay resolution and the channel’smaximum supported Doppler spread (amax) for a given BW(𝐵); 𝑁 dictates Doppler resolution and latency (𝑇 𝑓 = 𝑁𝑇).OTFS system design parameters are thus related (we onlychoose three from Δ 𝑓 , 𝑇, 𝑀, 𝑁), where Δ 𝑓 = 𝐵/𝑀 = 1/𝑇 ischosen such that
amax < Δ 𝑓 < 1/𝜏max. (54)
Using more subcarriers (𝑀) results in smaller SCS (Δ 𝑓 ) fora fixed 𝐵, which in turn results in a longer slot duration (𝑇).The OTFS design should thus observe the maximum latencyconstraints of novel use cases. Furthermore, the OTFS PAPR,complexity, and decoding delay are proportional to 𝑁 , so lower𝑁 values are favored. However, increasing frame size (large𝑁) results in enhanced BER performance [64] (higer diver-sity). Therefore, a careful trade-off between latency, PAPR,complexity, and performance is crucial with OTFS.
Table II presents maximum Doppler spread (amax) valuesin different bands, from below 6 GHz to THz. The notedsevere changes in amax impose many challenges on both WFdesign and receiver components (automatic frequency controlrange and synchronization). Frequency synchronization in thepresence of large CFO (tens/hundreds of KHz) is challengingfor CP-OFDM, even for low user mobility, where complexcircuits are required at the Rx side. The resilience of OTFS toCFO and Doppler spreads reduces such complexity, which ismuch needed in THz communications.
IV. SIMULATION RESULTS AND DISCUSSION
This section presents the results of extensive simulationsinvestigating relevant modulation KPIs under realistic THzconditions. The default simulation settings are listed in Ta-ble III (modifications are declared subsequently and channelparameters are taken form [44]). We list in Table V a summaryof the WF comparisons under the studied KPIs.
A. SE and E2E Latency
Table IV compares the SE and E2E latency of the studiedschemes, assuming a fixed modulation order of log2 ( |X|);Fig. 4 further plots the SE values versus the number of sub-carriers (𝑀). Assume a target of 10 Gbps at a system BW of𝐵 = 10 GHz, with 𝑁CP = 48. OQAM/FBMC achieves high SEfor large 𝑁 (asymptotic SE of 1 as 𝑁 goes to infinity; absenceof CP) and low SE for short frames (per-subcarrier pulse-shaping extends frame duration by 𝑂−1/2). OTFS achievesthe best SE performance, outperforming both CP-OFDM andSC-FDE (mainly due to CP overhead). An additional SE lossis introduced in DFT-s-OFDM where only �� ≤ 𝑀 symbolsallocated over 𝑀 subcarriers (29). However, such loss isnegligible in multi-user scenarios as vacant subcarriers can beallocated to other users. The SE of CP-OFDM increases with𝑀 , where a CP-OFDM symbol transmits 𝑀 QAM symbolsover 𝑀 subcarriers of duration 𝑇 = 𝑇u + 𝑇CP, repeated for𝑁 symbols per frame. However, the SE of CP-OFDM doesnot reach 1 bit/sec/Hz and is independent of 𝑁 , which is animportant feature for controlling the E2E latency. Note that
200 400 600 800 1000 1200 1400 1600 1800 2000
Number of subcarriers (M)
0
0.2
0.4
0.6
0.8
1
SE
(bit/s
ec/H
z)
OTFS(N = 64)
OTFS(N = 32)
OTFS(N = 4)
OQAM/FBMC(N = 64)
OQAM/FBMC(N = 32)
CP-OFDM
OQAM/FBMC(N = 4)
DFT-s-OFDM(M = 64)
DFT-s-OFDM(M = 16)
Fig. 4: Comparison of SE.
500 1000 1500 2000
Number of subcarriers (M)
0
10
20
30
40
50
60
70
E2E
Latency
(µsec)
OQAM/FBMC
CP-OFDM
OTFS
N=64
N=32
N=4
Fig. 5: Comparison of E2E latency.
SC-FDE has the same SE as CP-OFDM; we thus exclude itsresults (one CP for every 𝑀 QAM symbols (30)).
Regarding E2E latency, Fig. 5 illustrates that OTFS haslower latency than CP-OFDM, the latter which has the samelatency as both SC-FDE and DFT-s-OFDM. OQAM/FBMChas lower latency for smaller 𝑁 and 𝑀 values, but higherlatency for long frame durations.
B. PSD and OOB
The OOB emissions and the impact of adjacent channelleakage are studied in Fig. 6 by comparing the PSD oftwo users utilizing various WFs under the IEEE Tx spectralmask specifications (Sec. 13.1.3 [15]). Each user occupies aBW of 𝐵 = 2.16 GHz, with 𝑓𝑐 = 305.64 GHz for the firstuser (ID = 25 [15]) and the second user is assigned thenext channel. The results confirm that OQAM/FBMC has thebest frequency localization, thanks to its pulse-shaping filteron each subcarrier; other WFs respect the specified mask.However, it is expected that a much lower spectral emissionmask will be specified for B5G/6G networks. Thus, all WFsother than OQAM/FBMC would show a high interferencelevel. Note that without additional pulse shaping, the OOB
11
TABLE II: Maximum Doppler spread (amax) at different terminal speeds and frequency bands
Frequency band below 6 GHz mmWave sub-THz THz
Speed 𝜐 (km/hr)cen. freq. 𝑓𝑐 (Hz) 3 GHz 28 GHz 60 GHz 150 GHz 300 GHz 0.8 THz 1.2 THz
5 14 130 278 695 1390 3706 555930 83 778 1668 4170 8339 22238 33356120 334 3113 6671 16678 33356 88950 133426300 834 7783 16678 41696 83391 222376 333564500 1390 12972 27797 69493 138985 370627 555940
TABLE III: Simulation parameters
Common parameters ValuesOperating frequency 𝑓𝑐 0.3 THz
System BW 𝐵 10 GHzNumber of subcarriers 𝑀 256
Modulation scheme 4-QAMOverlapping factor (PHYDYAS) 𝑂 4
DFT-s-OFDM mode LocalizedChannel parameters Values
Molecular absorption coefficient 0.0033 m−1
Cluster arrival rate 0.13 nsec−1
Ray arrival rate 0.37 nsec−1
Cluster decay factor 3.12 nsecRay decay factor 0.91 nsec
Tx and Rx Antenna gains 0 dBi
TABLE IV: SE and E2E latency of SC/MC schemes
Waveform SE E2E latencyCP-OFDM 𝑀
𝑀+𝑁CP𝑁 × 𝑀+𝑁CP
𝐹s
SC-FDE 𝑀𝑀+𝑁CP
𝑁 × 𝑀+𝑁CP𝐹s
DFT-s-OFDM ��𝑀+𝑁CP
𝑁 × 𝑀+𝑁CP𝐹s
OQAM/FBMC 𝑁𝑁+𝑂−1/2 𝑀× 𝑁+𝑂−1/2
𝐹s
OTFS 𝑀𝑀+(𝑁CP/𝑁 )
𝑁×𝑀+𝑁CP𝐹s
performances of OTFS, DFT-s-OFDM, and SC-FDE are thoseof CP-OFDM; thus excluded from Fig. 6. Although DFT-s-OFDM suffers from high OOB emissions, variants such aszero-tail DFT-s-OFDM [19] could overcome this limitation.Furthermore, the performance in the presence of a pulse-shaping filter depends on parameters such as the roll-off factor,oversampling ratio, and raised-cosine (RC) filter length (out ofthis work’s scope). Therefore, including a guard band is crucialto achieving the required OOB and interference levels; acareful trade-off between OOB and SE needs to be maintained.
C. Complexity Analysis
The computational complexities of different WFs(Sec. III-C) are compared in Fig. 7. For a maximum𝐹s = 1 GHz (due to hardware constraints), OTFS is shownto be the least complex (complexity that is a functionof 𝑁; not 𝑀). However, state-of-the-art OTFS equalizersare much more complex than the CP-OFDM ZF/MMSEequalizers. CP-OFDM has the same overall complexity asSC-FDE; CP-OFDM and SC-FDE enjoy lower complexitiesthan both DFT-s-OFDM (due to additional DFF/IDFTprecoding blocks in Tx/Rx) and OQAM/FBMC (due to pulse-shaping overlapping and OQAM processing that doubles thecomplexity). OQAM/FBMC is the most complex scheme.
-3 -2 -1 0 1 2 3 4 5frequency f − fc (GHz)
-50
-40
-30
-20
-10
0
PS
D (
dB
r)
IEEE 802.15.3d mask
CP-OFDM
OQAM/FBMC
Fig. 6: Comparison of OOB emissions between two users’PSD.
500 1000 1500 2000
Number of subcarriers (M)
5
10
15
20
25
30
Num
ber
of M
ultip
lications p
er
tim
e u
nit (
Gig
a)
OQAM/FBMC (N=32)
OQAM/FBMC (N=4)
DFT-s-OFDM (M = 64)
CP-OFDM
OTFS (N=32)
OTFS (N=4)
Fig. 7: Comparison of computational complexity.
D. CCDF of PAPR
The WF’s PAPR CCDFs are compared in Fig. 8 fordifferent scenarios assuming Nyquist sampling. Figure 8ashows that SC-FDE achieves the best performance (lowestPAPR), followed by DFT-s-OFDM and OTFS, which out-perform CP-OFDM and OQAM/FBMC. In particular, DFT-s-OFDM with �� = 64 and OTFS with 𝑁 = 4 show 2.1 dBand 1.9 dB PAPR gains compared to CP-OFDM, respectively.Furthermore, DFT-s-OFDM PAPR is dependent on �� (2.2 dBincrease between �� = 8 and �� = 64), whereas OTFS PAPRis dependent on 𝑁 (16) (2.7 dB increase between 𝑁 = 2 and
12
0 2 4 6 8 10 12
γth
(dB)
10-5
10-4
10-3
10-2
10-1
100
Pr(
PA
PR
>γ
th)
SC-FDE
OTFS(N=2)
DFT-s-OFDM M = 8
DFT-s-OFDM M = 64
OTFS(N=4)
CP-OFDM
CP-OFDM(theo)
OQAM/FBMC(O=4)
(a) Performance using 16-QAM and 𝑀 =128.
0 2 4 6 8 10
γth
(dB)
0
0.2
0.4
0.6
0.8
1
Pr(
PA
PR
>γ
th)
α = 1
α = 0.5
α = 0
SC-FDE
DFT-s-OFDM M=16
DFT-s-OFDM M=64
(b) Effect of pulse-shaping for SC WFs (𝑀 =128).
2 4 6 8 10 12γ
th(dB)
0
0.2
0.4
0.6
0.8
1
Pr(
PA
PR
>γ
th)
M=32
M=128
M=256
CP-OFDM
OTFS
OTFS
CP-OFDM
(c) Effect of changing 𝑁 and 𝑀 for CP-OFDM and OTFS (𝑁 =4).
7.5 8 8.5 9 9.5 10 10.5 11 11.5 12γ
th(dB)
0
0.2
0.4
0.6
0.8
1
Pr(
PA
PR
>γ
th)
M=128
M=256
OTFS
CP-OFDM
Theoretical Bound
(d) Effect of changing 𝑁 and 𝑀 for CP-OFDM and OTFS with theoreticalbounds (𝑁 =32).
Fig. 8: PAPR CCDF comparison for different scenarios.
𝑁 = 4). OQAM/FBMC is worst performing (0.5 dB worsethan CP-OFDM) due to per-subcarrier pulse-shaping. TheCP-OFDM simulations are in agreement with theory (15).However, we did not include OTFS theoretical bounds as theapproximation in (16) is only valid for high 𝑁 values (Fig. 8d).All schemes (except OQAM/FBMC) apply rectangular pulse-shaping; no pulse-shaping with SC-FDE.
The effect of pulse-shaping on PAPR performance inSC-FDE and DFT-s-OFDM is illustrated in Fig. 8b, vary-ing the roll-off factor (𝛼 = {0, 0.5, 1}) of the RC filter:cos (𝜋𝛼𝑡/𝑇 )sinc(𝑡/𝑇 )
(1−4𝛼2 (𝑡/𝑇 )2) (an RC filter of 6 symbols; oversamplingfactor of 4; normalized to unit energy). We notice that in-creasing 𝛼 significantly improves the PAPR performance inSC-FDE, but at the expense of excess BW; DFT-s-OFDM isnot as highly affected. Moreover, the PAPR variations with 𝛼are negligible for small values of �� .
Figures 8c 8d show the effect of changing 𝑁 and 𝑀 onPAPR with 𝑁 = 4,𝑀 = {32, 128, 256} and 𝑁 = 32,𝑀 =
{128, 256}, respectively; we only simulate CP-OFDM andOTFS (CP-OFDM findings also apply to OQAM/FBMC). Wenote that increasing either 𝑀 or 𝑁 increases PAPR ((15)and (16)). Nevertheless, the maximum PAPR in OTFS growslinearly with 𝑁 , and the CCDF is zero for 𝛾th values greaterthan a threshold related to 𝑁 (for example, the maximumPAPR for 𝑁 = 4 is 10 log (4) = 6.02. Note that for low 𝑁 inOTFS, increasing 𝑀 does not increase the PAPR significantly.Furthermore, we note good agreement between simulated andanalytical results of OTFS and CP-OFDM for large 𝑁 .
E. Phase Noise
We first verify our assumption of Gaussian PHN for THzcommunications. We incorporate the PHN measurement re-sults in [67] for a 300 GHz signal source, a PHN floor levelof 𝐾0 = −110 dBc/Hz, and 𝐾2 = 10 ( 𝑓cor = 1 MHz); weset 𝐵 = 10 GHz, which satisfies (17). We consider Tx PHNwithout loss of generality. For an AWGN channel plus Tx
13
0 1 2 3 4 5 6 7 8 9 10
SNR (dB)
10-6
10-5
10-4
10-3
10-2
10-1
BE
R
TX PHN: Wiener+Gaussian
TX PHN: Gaussian
TX PHN: Wiener
AWGN without PHN (theo)
AWGN without PHN (sim)
(a) CP-OFDM performance with three PHN models.
0 2 4 6 8 10 12
SNR (dB)
10-4
10-3
10-2
10-1
BE
R
SC-FDE
OTFS (N = 4)
OTFS (N = 12)
CP-OFDM
OQAM/FBMC, (O = 4)
DFT-s-OFDM, (M = 128)
DFT-s-OFDM, (M = 16)
AWGN without PHN (theo)
(b) Effect of Gaussian PHN model for different WFs.
10-3
10-2
10-1
σ2g
10-6
10-5
10-4
10-3
10-2
BE
R
SC-FDE
OTFS (N = 4)
CP-OFDM
OQAM/FBMC, (O = 4)
DFT-s-OFDM, (M = 128)
DFT-s-OFDM, (M = 16)
(c) Effect of changing Gaussian PHN variance at 10dB SNR.
2.510 40 160
SCS (∆f) (MHz)
10-5
10-4
10-3
10-2
BE
R
OTFS (N = 4)
OTFS (N = 16)
CP-OFDM
OQAM/FBMC, (O = 4)
DFT-s-OFDM, (M/M = 4)
DFT-s-OFDM, (M/M = 16)
(d) Effect of changing the SCS with Gaussian PHN.
Fig. 9: BER performance of various schemes with Tx PHN.
PHN (ℎ[𝑢] = 1, for all 𝑢 in (18)), the CP-OFDM resultsin Fig. 9a illustrate that the models in (19) and (21) areequivalent. Hence, for large BWs, the uncorrelated modelof (21) is sufficient. The reason behind this observation is thatthe Wiener model PSD decreases with frequency, resultingin PHN power levels lower than the white floor noise of theGaussian model at frequencies higher than ( 𝑓cor). Note thatthe PHN power of the Gaussian model is constant; we adda reference AWGN lower bound. PHN leads to inter-carrierinterference (ICI) and adjacent channel interference, whichexplains the resultant degradation. The comparison assumingGaussian PHN in Fig. 9b illustrates that DFT-s-OFDM is themost robust to PHN, and decreasing �� enhances the perfor-mance. OQAM/FBMC outperforms other schemes because ofits good time and frequency localization. Furthermore, CP-OFDM is more robust than SC-FDE and OTFS (increasing𝑁 in OTFS enhances performance). SC-FDE has the worstperformance. We also analyze the effect of changing the noisevariance (𝜎2
g ) assuming THz-band Gaussian PHN and an SNR
of 10 dB in Fig. 9c. We vary 𝜎2g between low (10−3), medium
(10−2), and strong (10−1) values (as indicted in Table I in [42]),retaining a system BW of 𝐵=10 GHz; this changes the spectraldensity (𝐾0) of the white PHN floor. Increasing 𝜎2
g increasesthe BER, where DFT-s-OFDM is the best performing.
In Fig. 9d, we study the effect of changing SCS by changing𝑀 = {4096, 2048, 1024, 256, 64} and fixing 𝐵 = 10.24 GHz.Surprisingly, the WF BERs are retained, which is an importantfeature that relaxes other design parameters. For example,we can use a small 𝑀 to ensure low PAPR and high PHNrobustness concurrently. Such results are not observed below6 GHz, where increasing SCS ensures high robustness to PHN(different low-frequency models).
F. Beam Split
We compare all WFs using a stochastic THz channelsimulator, TeraMIMO [44], in Fig. 10, for 𝐵 = 50 GHz and𝑓𝑐 = 0.325 THz. We consider an UM-MIMO system withbeamforming, where both Tx and Rx have uniform linear
14
arrays of 32 AEs; we set the communication distance to 1 m.The channel is LoS-dominant with a few MP components,which tends to be almost flat-fading. We first plot the BERsassuming the absence of beam split; all WFs achieve similarperformance except for OQAM/FBMC (we considered simpleMMSE equalization). When adding beam split, OQAM/FBMCis shown to be less affected compared to CP-OFDM. OTFS,SC-FDE, and DFT-s-OFDM (with large ��) have high robust-ness to THz-induced impairments, securing multiple-dB BERgains over both CP-OFDM and OQAM/FBMC.
G. Performance in Doubly-Selective ChannelsDoppler spreads in THz channels are orders-of-magnitude
larger than those in the conventional microwave and mmWavechannels (Table II). In Fig. 11, we compare the BERs ofthe studied schemes in a doubly-selective THz channel. Weconsider 𝑓𝑐 = 0.5 THz, 𝐵 = 0.5 GHz, 𝑀 = 128, 𝑁 = 32(for fairness between OTFS and CP-OFDM, we concatenate𝑁 = 32 CP-OFDM symbols per frame), and user velocity𝜐 = {120, 500 km/hr}. Note that the communication distanceis 2 m, and cluster/rays parameters are taken from Table III(WF parameters are derived following (54)). We considerMMSE detectors for OTFS and CP-OFDM (other WFs suchas OQAM/FBMC show the same performance as CP-OFDMand are thus omitted). OTFS is more robust than CP-OFDMand other WFs in TV-FSC, even for larger user velocity(𝜐), showing multiple-dB BER gains. Such advantages renderOTFS exceptionally suitable for high-mobility, high-carrierscenarios (THz V2X scenarios, for example).
V. CONCLUSION
In this paper, a comprehensive study of SC/MC modula-tions for THz communications is conducted. The analysesand simulation results demonstrate that the candidate 5GWFs (filtered-based OFDM, such as OQAM/FBMC) are notsuitable for future B5G/6G networks because of the increasedPAPR and complexity. Furthermore, CP-OFDM and SC-FDEshare similar characteristics: good SE, moderate E2E latency,high UM-MIMO compatibility, acceptable to high OOB emis-sions (without any additional pulse-shaping), and relativelylow implementation complexity (especially with a single-tapZF/MMSE equalizer). SC-FDE is shown to be less robust touncorrelated Gaussian PHN, but it results in low PAPR andhigh robustness to THz beam split. DFT-s-OFDM is furthershown to offer low PAPR and high robustness to both THzPHN and beam split at a slight increase in complexity. Finally,OTFS is illustrated to achieve high SE, low E2E latency,good PAPR characteristics, low implementation complexity,and high robustness to THz impairments, but it suffers fromhigh OOB emissions and low robustness to the THz GaussianPHN model. OTFS is proved to outperform all other WFsin doubly-selective channels. In a nutshell, the findings ofthis work recommend the use of DFT-s-OFDM and OTFS inB5G/6G sub-THz/THz communications; CP-OFDM can stillbe used in sub-THz indoor scenarios (TIV-FSC).
In future work, other relevant performance metrics can beconsidered. For instance, researchers should study the WF ro-bustness to asynchronous access, synchronization procedures
50 55 60 65 70 75
10-5
10-4
10-3
10-2
10-1
100
BE
R
DFT-s-OFDM (M = 128)DFT-s-OFDM (M = 256)DFT-s-OFDM (M = 512)SC-FDECP-OFDMOTFS (N=4)OTFS (N=8)OQAM/FBMCno beam splitwith beam split
Fig. 10: BER performance of various WFs in the presenceand the absence of beam split.
90 100 110 120 130 140SNR (dB)
10-4
10-3
10-2
10-1
100
BE
R
CP-OFDM υ = 500km/hr
CP-OFDM υ = 120km/hr
OTFS υ = 500km/hr
OTFS υ = 120km/hr
Fig. 11: BER performance of various schemes in a THz TV-FSC.
in the presence of both STO and CFO, wideband IQI, PA non-linear distortion, multi-user scheduling, and flexible resourceallocation. Joint sensing and communication (JSC) metricsshould also be designed. Novel studies illustrate the benefits ofusing DFT-s-OFDM, with some modifications, for JSC [68];OTFS-enabled JSC is yet to be studied.
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TABLE V: Performance evaluation metrics for different SC/MC WFs
MetricWaveform SC-FDE DFT-s-OFDM CP-OFDM OQAM/FBMC OTFS
Spectral efficiency medium low/medium medium low/high highE2E latency low medium low/medium low/high low/medium
UM-MIMO compatibility high high high low highPAPR low low/medium high high low/medium
OOB emissions medium/high high high low highGaussian PHN robustness low high low/medium medium low
Beam split robustness high high low low/medium highRobustness to doubly selec. chan. low low low low high
Complexity medium medium/high medium high low/medium
TABLE VI: Summary of Frequently-Used Abbreviations
Abbreviation Definition4G-LTE fourth-generation long term evolution5G-NR fifth-generation new-radio
6G sixth-generationADC analog-to-digital converterAE antenna element
AFB analysis filter bankAoSAs array-of-subarraysAWGN additive white Gaussian noise
B5G beyond-fifth generationBDMA beam-division multiple-access
BER bit error rateCCDF complementary cumulative distribution functionCFO carrier-frequency offset
CP-OFDM cyclic-prefix orthogonalfrequency-division multiplexing
CP cyclic-prefixCPM continuous phase modulationCR cognitive radio
DAC digital-to-analog converterDD delay-Doppler
DFT-s-OFDM discrete-Fourier-transformspread OFDM
DFT-s-OFDM discrete-Fourier-transform spread OFDME2E end-to-end
FBMC filter-bank multi-carrierFDE frequency domain equalization
FDMA frequency-division multiple-accessFSC frequency-selective channelFFT fast Fourier transform
f-OFDM filtered-OFDMGFDM generalized frequency-division multiplexingIFFT inverse FFT
ISFFT inverse SFFTICI inter-carrier-interferenceIQI in/quadrature-phase imbalanceISI inter-symbol-interferenceIRS intelligent reflecting surfaceITU international telecommunications unionKPI key performance indicatorLAN local area networkLoS line-of-sight
MMSE minimum mean-squared errormmWave millimeter-wave
MC multi-carrierMHz megahertzMP multi-path
NLoS non-LoSNOMA non-orthogonal multiple-access
TABLE VII: Summary of Frequently-Used Abbreviations
Abbreviation DefinitionOFDM orthogonal frequency-division multiplexingOOB out-of-bandOOK on-off keying
OQAM offset quadrature amplitudemodulation
OQAM offset quadrature amplitude modulationOTFS orthogonal time-frequency space
PA power amplifier
PAPR peak-to-averagepower ratio
PAPR peak-to-average power ratioPHN phase noisePS phase-shifterRF radio frequencySA subarray
SC-FDE single-carrier frequency-domainequalization
SC-FDE single-carrier frequency-domain equalizationSCS subcarrier spacingSE spectral efficiency
SFB synthesis filter bankSISO single-input single-output
SS-OFDMA spatial-spread orthogonalfrequency-division multiple-access
SFFT symplectic finite Fourier transformSTO symbol-timing offsetTbps terabits-per-secondTHz terahertzTIV time-invariantTV time-variant
UFMC universal filtered multi-carrierUM-MIMO ultra-massive multiple-input multiple-output
UM-MIMO ultra-massive multiple-inputmultiple-output
WOLA-OFDM windowed overlap-and-add OFDMZF zero-forcing
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