Coded Orthogonal Time Frequency Space Modulation

LIU Mengmeng, LI Shuangyang, ZHANG Chunqiong, WANG Boyu, BAI Baoming

Abstract： Orthogonal time frequency space （OTFS） modulation is a novel two-dimensionalmodulation scheme for high-Doppler fading scenarios， which is implemented in the delay-Doppler （DD） domain. In time and frequency selective channels， OTFS modulation is morerobustthanthepopularorthogonalfrequencydivisionmultiplexing（OFDM）modulationtechnique. To further improve transmission reliability， some channel coding schemes areused in the OTFS modulation system. In this paper， the coded OTFS modulation system isconsidered and introduced in detail. Furthermore， the performance of the uncoded/codedOTFSsystemandOFDMsystemisanalyzedwithdifferentrelativespeeds ，modulationschemes， and iterations. Simulation results show that the OTFS system has the potential offull diversity gain and better robustness under high mobility scenarios.

Keywords： OTFS modulation; OFDM; channel coding; fading channel

Citation （IEEE Format）： M. M. Liu， S. Y. Li， C. Q. Zhang， et al.， “Coded orthogonal time frequency space modulation， ”ZTECommunica ? tions， vol. 19， no. 4， pp. 54 –62， Dec. 2021. doi： 10. 12142/ZTECOM.202104006.

1 Introduction

The 5G network has achieved the peak rate of 10 – 20 Gbit/s， which is more than ten times that of 4G Long TermEvolution（LTE）cellularnetworks.Somenew scenarios with high mobility have emerged in 5G/B5G， suchasV2X（vehicle-to-vehicle—V2Vandvehicle-to-infra? structure—V2I） with the terminal speed up to 300 km/h， high speed train （HST） with the maximum speed up to 500 km/h and unmanned aerial vehicle （UAV）.In these cases， the high? er Doppler spread will be induced.In addition， a higher data rate is required， which is considered to be solved by using a higherfrequencyband，suchasamillimeterwavebandor even a terahertz band.Both high mobility and high frequencywill lead to large Doppler shifts， yielding the large frequency dispersion.Although orthogonal frequency division multiplex? ing （OFDM） modulation is used in 4G and 5G， it has good ro? bustness only in time-invariant channels and is very sensitive tocarrierfrequencyoffsets.However，thechannelistime- varying in high Doppler scenarios.The orthogonality of sub- carriers in an OFDM symbol is seriously damaged so that the channel estimation is no longer accurate， which will lead to se ? vere inter-carrier interference （ICI） and the disappearance of the near-capacity advantage.

Todeal withcommunicationscenarios withhighDoppler shifts，anoveltwo-dimensional（2D）modulationscheme calledorthogonaltimefrequencyspace（OTFS）modulation was proposed by R. HADANI et al. in 2017， whose pioneering works[ 1 –4]introducedtheprincipleof OTFSmodulationand demonstrateditssignificantperformanceonOFDMmodula? tioninchannelswithhighDopplerorathighfrequencies.Compared with the OFDM modulation， OTFS modulation has the potential of full diversity gain and better robustness， which can effectively deal with the impact of high Doppler shifts. One more advantage of OTFS is that it can be implemented as pre- and post-processing blocks applied to a time-frequency signal? ing scheme， such as OFDM[5]. Furthermore， an implementation scheme of OTFS modulation based on the OFDM has been pro ? posed in Ref. [6]， which greatly reduces the complexity of im ? plementation. In Ref. [7]， the vector form of concise and elegant input-output relationship of the OFDM-based OTFS system has also been derived by utilizing the properties of the Kronecker product in matrices and vectors， which is also suitable for gen? eral time-varying channels with arbitrary Doppler and window? ing functions. It is worth mentioning that this representation is very popular in subsequent research work.

As for the significant advantage of achieving the full diversi ? ty， the detailed formal analysis on the diversity order of OTFS in doubly-dispersive channels has been presented in Ref. [8]， whichpointsoutthatthefulldiversityinthedelay-Doppler （DD） domain can be extracted by using the phase rotation meth ? od. In addition， when the OTFS frame is long enough， even the uncoded OTFS modulation system can obtain almost full diver? sity in the case of path number P = 2[9]. In order to make full use of full diversity， effective equalization is needed， which de? pends on the accurate channel estimation. A well-known chan? nel estimation scheme for OTFS has been proposed in Ref. [ 10]， in which pilots， protection symbols， and data symbols are clev? erlyarrangedonthedelayDopplergridplanetoeffectively avoid the interference between pilots and data symbols at the re ? ceiver and enable the channel estimation and data detection to be performed in the same OTFS frame with the minimum over? head. However， the performance of such algorithms[ 10 – 11]is very sensitive to the availability of protection space. In fact， more ad? vanced channel estimation methods based on compressed sens ? ing[ 12]， orthogonal matching pursuit （OMP）[ 13 – 14]or sparse Bayes ? ian learning[15]algorithms have been proposed， which take ad? vantage of the channel sparsity in the DD domain. However， the channel in the DD domain may not always be sparse， especially in the case of fractional Doppler[5]. An effective solution is to en? hance channel sparsity by applying time-frequency （TF） domain windows，suchasDolph-Chebyshev（DC） window[ 16]. Inaddi? tion， advanced detection algorithms are also an important part of OTFS to achieve potential full diversity gain[ 17]. A message passingalgorithm（MPA） based on the maximuma posteriori probability（MAP）detectioncriterionhasbeenintroducedin Ref. [5]， which processes the interference from other informa? tion symbols as Gaussian variables to reduce the detection com ? plexity. However， due to theshort period of the probabilistic graphical model， the proposed MPA may not converge， result? ing in performance degradation. In order to solve this problem， a convergence protection receiver based on variable Bayes （VB） framework has been presented in Ref. [ 18]， which utilizes the relative entropy to approximate the optimal detection of the corresponding a posteriori distribution to realize the MPA on a sim ? ple graphical model. In addition， a hybrid detection scheme has been demonstrated in Ref. [ 19]， which takes the MAP and the parallelinterferencecancellation（PIC）intoaccountand achieves a good trade-off between the error performance and de ? tection complexity.

Channel coding is also one of the effective methods to ensure diversity gain and achieve reliable communications. However， most of the research work analyzes the performance of the un ? coded OTFS modulation system. In Ref. [20]， the BER perfor? mance of the coded OTFS system has been analyzed in detail. The derivation of pairwise-error probability （PEP） and its upper bounds have demonstrated a very interesting trade-off between the coding gain and diversity gain of the coded OTFS system. Moreover， a channel coding design criterion is derived， that is， maximizing the minimum Euclidean distance between all code ? word pairs. This criterion is very similar to the channel coding design in the additive white Gaussian noise （AWGN） channel. However，theOTFSmodulationexperiencesatime-varying channelwithbothtimedispersionandfrequencydispersion. Thus， both the inter-symbol interference （ISI） and ICI will be generated， which depend on the delay τ ， Doppler ν of the chan? nel and the cross-ambiguity function of pulses at the transmitter andreceiver. Theassumedidealpulse-shaping waveforms[ 1， 3] that satisfy the bi-orthogonality condition in both time and fre ? quency do not exist in practical applications. Therefore， ISI and ICI are inevitable in the actual OTFS system. Obviously， it is necessary to consider the properties of OTFS and characteris ? tics of the DD domain channel to design the channel code suit? able for the OTFS system.

In this paper， we consider the coded OTFS modulation and describe it in detail. Then， recent work about the coded OT? FS system analysis is summarized. On this basis， the upper bound on the unconditional PEP of the coded OTFS system is further supplemented. In addition， the joint iterative strate ? gy of detection and decoding is also considered to improve the system performance. According to the considered itera? tive system， the coding design scheme of the OTFS system is discussed. Finally， we analyze the performance of the coded/ uncodedOTFSandOFDMsystemswithdifferentrelative speeds，modulationschemesanditerations. Simulationre? sultsshowthatOTFSsystemshavesignificantrobustness compared with OFDM.

2 Principle and System Model

2.1 RelationshipBetweenTime-FrequencyDomainand Delay-Doppler Domain

In Ref. [4]， the authors proposed that the OTFS modulation can be viewed as a time-frequency spreading scheme， which was based on the Fourier duality relation between the time-fre? quency plane and the delay-Doppler plane， resulting in a simple pre-processing step over an arbitrary multicarrier modula? tion （such as OFDM）. In view of the importance of the trans? form between the DD domain and the TF domain， we will re? view this relationship in this subsection.

The grid in the TF domain and the corresponding reciprocal grid in the DD domain are shown in Fig. 1， with the size of M × N. According to Fig. 1， the TF grid can be represented as：

where T is the sampling interval along the time axis， Δf is the sampling interval along the frequency axis， and N and M are the corresponding numbers of sampling points on the TF plane. According to the principle of the time-frequency modu? lation explicated in Ref. [1]， the transmitted packet can be re? garded as a burst one with a total duration of NT seconds anda total bandwidth of MΔf Hz. Then， the modulated sym ? bols XTF m，n， m = 0，1，...，M - 1， n = 0，1，...，N - 1 are transmitted over the burst packet in the TF domain.

The reciprocal delay-Doppler grid is represented as：

whereand represent the sampling intervals along theDoppler axis and the delay axis， respectively.

The mapping between signals in the TF domain and DD do ? main depends on two-dimensional symplectic finite Fourier transform （2D SFFT） pairs， which can be exemplified as：

where h τ ，νand H t，fare the responses of linear time-vary? ing （LTV） wireless channels in the DD domain and TF domain，respectively. Without loss of generality， the DD domain repre? sentation of an LTV wireless channel can be expressed as：

where δ ?denotes the Dirac delta function， P is the number of resolvable paths， and hi， τiand ν iare the channel coeffi ? cient， delay and Doppler shift of the i-th path respectively. Here， τi and ν i are defined as：

In Eq. （6）， lirepresents the index of the delay with integer val? ues， kirepresents the index of the Doppler shift with integer values， and κ i∈ （-0.5，0.5] is the real number， indicating the fractional shift from the nearest Doppler index ki， which is al? so called fractional Doppler[5].

Specifically， the 2D SFFT pairs can be realized by simple discrete Fourier transform （DFT） pairs or fast Fourier trans? form （FFT） pairs. For example， in view of the DFT， the inverse symplectic finite Fourier transform （ISFFT） can be regarded as the M-point DFT along the delay axis and the N-point in? verse DFT （IDFT） along the Doppler axis for the two-dimen? sional signal with the size of M × N in the DD domain， result? ing in the corresponding TF domain signal.

2.2 Coded OTFS System Model

Fig. 2 shows the proposed coded OTFS system model in thispaper. Suppose that the information bit sequence u of length Kis encoded using a forward error correction （FEC） code， result?ing in the codeword c of length Nc. After interleaving， the in?terleaved sequence v is then mapped to an M-ary signal con?stellation ， such as M-ary phase shift keying （MPSK） or M-ary quadrature amplitude modulation （MQAM）， and the modu?lated symbol vector xDDof length MN is arranged as a two-di?mensional signal matrix xDD∈M × N in the DD domain， whereM is the number of the sub-carriers， and N is the number oftime slots for each OTFS symbol. The element x l，kof thexDDrepresents the modulated signals in the l-th Delay and kth Doppler grid， for l ∈0，1，...，M - 1and k ∈0，1，...，N - 1. Then the symbol X m，nin the m-th frequency and n-th time grid is obtained by the ISFFT， which is given as：

for m = 0，1，...，M - 1，n = 0，1，...，N - 1. The two-dimensional signal matrix in the TF domain is denoted by XTF∈M × N. The early literature[1 –3] explained that the composition of the ISFFT and the windowing function in the TF domain are re ? ferred to as the OTFS transform. The window operations of the transmitter and receiver affect the cross-symbol interference of the effective impulse response[1]. The window design has the potential to increase the effective channel sparsity in the DD domain， which is conducive to the channel estimation， as de? scribed in Refs. [6] and [7]. Furthermore， the influence of the design of the TF domain window on improving the perfor? mance of the channel estimation and data detection is dis ? cussed in Ref. [16]. Here， the rectangular window is consid ? ered.

The transmitted signal in the time domain is obtained fromthe TF domain symbols X m，nusing the Heisenberg trans? form parameterized by the pulse shaping filter gtxt， which can be written as：

This can be regarded as a general form of the OFDM modula? tion[3]. Moreover， OTFS modulation can be implemented as a cascade of a pre-coder （ISFFT） and a traditional OFDM modu? lator[6]， as shown in Fig. 2.

Assume that the channel is the LTV channel described in Eq. （5）， the received signal can be expressed as：

where w tis the additive white Gaussian noise with zeromean and one-sided power spectral density of N0.

At the receiver， r tis subject to the Wigner transform to ob ? tain the received symbols Y m，nin the TF domain， given by

where grxtis the pulse shaping filter at the receiver. Eq. （10） can be further written as[5]：

wherem，nis the noise sample in the TF domain， and Hm，nm ′，n ′is the channel impulse response in the TF do ? main， i.e.，

In Eq. （12）， Agrx，gtxτ ，ν is referred to as the cross-ambiguityfunction， which represents the interference between symbols in the DD domain caused by the channel dispersion[20]， and can be expressed as：

The received symbols in the DD domain are given as：

for l = 0，1，...，M - 1，k = 0，1，...，N - 1， where w l，kis thenoise sample in the DD domain. Upon the received symbolsy l，k， a signal detection algorithm is then performed. In ad? dition， a joint iterative strategy between detection and decod ? ing can also be considered.

It is generally known that the MAP detection is optimum for OTFS systems. However， the complexity of the MAP detection increases exponentially with the block size of each OTFS frame. As a compromise of the MAP detection， a lot of litera? ture has studied the massage passing detection algorithm based on the factor graph， which can effectively reduce the de ? tection complexity， such as Ref. [5]. For the coded system， the iterative signal processing of the detector and the decoder is usually considered at the receiver， as shown in Fig. 2. Corre? spondingly， soft decision detection algorithms should be adopt? ed， such as MP， unitary approximate MP （UAMP）， vector AMP （VAMP）， sum-product algorithm （SPA） and other message passing algorithms. With Log-Likelihood Ratios （LLRs）， the message Le， det passed from the detector to the decoder is calcu ? lated as：

where Lapp， det and La， det represent the a posteriori LLRs and the priori LLRs of the detector， respectively; P ?denotes the priori symbol probabilities， aj∈，j = 1，...，， and the binary vector （signal label） corresponding to ajcan be ex? pressed as vj. Similarly， the extrinsic LLRs of the decoder Le， dec are also obtained by subtracting priori LLRs La， dec from the a posteriori LLRs Lapp， dec， and La， dec is updated by the extrinsic LLRs of the detector. The iterative process between the detector and the decoder can be described as follows.

2.3 Vectorization Representation of the System

With respect to the vectorization， the following definitions are given， xDD = vec （xDD） ∈ AMN × 1 ， xTF = vec （XTF） ∈ CMN × 1 ， yTF = vec （YTF） ∈ AMN × 1 ， and yDD = vec （YDD） ∈ CMN × 1 ， where vec（ ? ） denotes the vectorized version of the 2D matrix formed by stacking the columns of the one into a single column vector. Besides， the N-point DFT matrix and its inverse are rep? resented by FN and F H N respectively and are assumed to be nor?malized so that FNF H N = IN. According to the introduction of the coded OTFS system in the above subsection， the relation? ship between the symbol matrix XTF ∈ CM × N in the TF domain and xDD∈ CM × N in the DD domain can be described as：

The vectorized form can be expressed as：

Considering that the pulse shaping filter gtxtis the rectan? gular form， the output of the Heisenberg transform is given by

where S ∈M × Nrepresents the transmitted signal matrix in the time domain. Vectorize S by stacking each column of S in? to a vector， we have

At the receiver， the received signal expressed by Eq. （9） in discrete form is

where ?MNindicates mod MN operation， r n， s nand w nare the corresponding discrete forms of r t， s tand w t， respectively. Thus， the received signal can be written in the vector form as

In the above formula， HTis an MN × MN matrix， given by Ref. [10]，

where Π is the permutation matrix （forward cyclic shift）， andΔkiis the diagonal matrix， as shown below.

The received signal vector r is devectorized into an M × N matrix R. Then， the Wigner transform and SFFT can be suc ? cessive to obtain the received signal matrix YDD with the size M × N in the DD domain as follows：

The vector form can be obtained by

Substituting Eqs. （19） and （21） with Eq. （26）， we can get the vector form of the input-output relation in the DD domain as follows：

The vectorized forms of each operation are simple and more vivid， which contribute to understanding the OTFS modulation more clearly and they are widely used in the research of the OTFS modulation.

3 Error Performance of the Coded OTFS System

3.1 Error Performance Analysis

PEP is commonly used in communication systems for analyz? ing the error performance of the system. In Ref. [8]， the achiev? able diversity order of the OTFS system is analyzed based on the PEP under the maximum likelihood （ML） detection. Similar? ly， the PEP under the ML detection is also used to analyze the error performance of OTFS modulation in Ref. [21]. On this ba? sis， the effective diversity （ED） is introduced from the perspec? tive of PEP[9]. In Ref. [20]， the conditional PEP and the uncondi? tional PEP are utilized to analyze the error performance of cod? ed OTFS systems. And an approximate upper bound on the un? conditional PEP for small P is derived by：

where e ?xDD-DDis the codeword difference vector， d E（2） e= eH e is the squared Euclidean distance between xDDandDD， and r is the rank of the positive semidefinite Hermite matrix Ω egiven by Eq. （18）[20]. In Eq. （28）， the exponent r and the term d E（2） e P are regarded as the diversity gain and the coding gain， respectively. According to the early works， e.g. Refs. [1] and [8]， OTFS can achieve full diversity， whose order is the number of the separable multipath P. When the channel code is given， the term d E（2） eis also fixed. Thus， as described in Corol? lary 1 in Ref. [20]， the diversity gain increases and the coding gain decreases with the increase of P， which reveals an interest? ing trade-off between them. In addition， an approximate upper bound on the unconditional PEP for large P is also given by

which only depends on the signal to noise ratio （SNR） and d E（2） e， and demonstrates that channels with a large number of resolvable paths approach an AWGN model.

More detailed derivation and illustration can be found inRef. [20]. It should be noted that the upper bound on the un? conditional PEP shown in Eqs. （28） and （29） are approximate. Referring to the appendix A in Ref. [22]， the upper bound onthe unconditional PEP has a more accurate display.

Note that Ω （e） is also a Gram matrix[23] corresponding to vectors {u1，u2，...，uP}， where ui?Ξie， and Ξi? （FN ? IM） Πli Δ（ki） （F H N ? IM）. According to the appen? dix A in Ref. [22] ， the determinant of Gram matrix Ω （e） can be calculated by

where u? j is the orthogonal projection of uj onto the orthogonal complement of span （u1，u2，...，uj - 1）. Besides， the maximum value of the rank of the matrix Ω （e） is the number of resolv? able paths P. In particular， when matrix Ω （e） is full-rank， we have r = P. Then the upper bound on the unconditional PEP of the coded OTFS system can be rewritten as

Furthermore， the equality holds if Ω eis a diagonal matrix.

3.2 Design Issues of Channel Codes for OTFS Systems

In Ref. [20]， the code design criterion for the coded OTFS is given based on the PEP analysis， which is to maximize the minimum squared Euclidean distance of all possible codeword pairs. Simulation results show the performance of the coded OTFS system under convolutional codes with different mini ? mum squared Euclidean distances and verify the proposed code design criterion. At present， most channel codes are de? signed for AWGN channels. In Ref. [20]， authors also reveal that the channel with a large number of diversity paths ap? proaches an AWGN channel when the number of resolvable paths Pis large enough. In this case， some good channel codes can be used in the OTFS system. However， the increase of P will bring about large ISI， making signal detection more complicated. Therefore， it is necessary to design channel codes according to the characteristics of the OTFS modula? tion. In particular， the joint iteration between decoding and de? tection is needed， when the channel conditions are poor. In a word， the design of the channel coding scheme is still an interesting challenge.

A simple and direct method to analyze coded OTFS systems istousetheextrinsicinformationtransfer（EXIT）chart[24] ， which is commonly used to aid the construction of good itera? tively-decoded error-correcting codes. EXIT charts are espe ? cially popular in the analysis of low-density parity-check （LD ? PC）codesandTurbocodes.Inthemostworksofcoded OFDMsystems， the tool， EXIT chart， is also commonly uti ? lized to optimize the performance of iterativedecoding，and parameters of the corresponding channel coding scheme and detection， such as Refs. [25 –26].

Another possible method is to learn from the code construc ? tion method under ISI channels. In general， ISI channels can be conveniently represented by a trellis[27]or a factor graph[28]. Codes such as Turbo codes and LDPC codes can also be repre ? sented by a trellis or a factor graph. Note this， the channel fac ? tor graph and the code factor graph are considered together to obtain the joint channel/code graph in Ref. [29]. The limits of the performance of LDPC codes over binary linear ISI chan ? nels are also studied in Ref. [29]. With the use of density evo? lution， the noise tolerance threshold is calculated. This may provide some reference for the design of coded OTFS system ， because the received signals with ISI can also be represented by a trellis or a factor graph.

4 Numerical Results

Numerical results of the considered coded OTFS system areprovidedinthissection. The5GLDPCcodeisused， whose code rate and length of information sequence are R = 1 2 and K = 1 024， respectively. Without loss of generality， quadrature phase shift keying （QPSK） and16QAM are cho? senasthetraditionalmodulationschemes ，whosecorre? spondingOTFS framesizes are M × N = 64 × 16 and M × N = 32 × 16， respectively. In all simulations， the LTV chan? nel with path number P = 4 is used， where the path gain fol? lows the Rayleigh distribution with respect to the exponen ? tial power delay profile. For the DD domain channel， the in? dices of delay and Doppler shifts are integers. Moreover， ac? cording to 4GLTEand5GNR， the carrier frequency and subcarrier interval are selected as 4GHz and15 kHz， re? spectively. Thus，weconsiderthemaximumdelayindex lmax= 5 and the maximum Doppler shift index kmax= 1， 2， 3， correspondingtothecasesinwhichrelativespeedsare around 275 km/h， 500 km/h， and 750 km/h， respectively. It should be noted that the delay and Doppler shift indices are generated uniformly at random. At the receiver， the near-op? timal symbol-by-symbol MAP detection algorithm[ 19]is used， unless otherwise specified. In order to accelerate the itera? tive convergence between the detector and the decoder， the offset min-sum algorithm （MSA） with an offset factor of 0.5 is adopted by the decoder. The maximum iteration number of the detection is 10， while that of the decoding is 50.

Fig. 3 shows the frame error rate （FER） performances of the uncoded OTFS and OFDM systems with 16QAM and different relative speeds， such as 275 km/h， 500 km/h and 750 km/h. For a fair comparison， we also apply the near-optimal symbol- by-symbol MAP detection[ 19]for OFDM systems， which is de? signed to exploit all the interference （including both ISI and ICI）. Unless otherwise specified， this detection algorithm will be used in subsequent simulations of uncoded/coded OFDM systems. As shown in Fig. 3， we first observe that both uncod ? ed OTFS and OFDM systems have good robustness at different relative speeds. This is because the channel coherence time （Tc= 1 fd= 0.981， 0.539， 0.360 ms corresponding to relative speeds of 275 km/h， 500 km/h and 750 km/h， respectively） is longer than the OFDM symbol time （Ts= 1 Δf = 0.067 ms）.

The channel variation isslowat considered relativespeeds， andtheinterferencebetweenadjacentsubcarriersdemon ? stratessimilar property. It is also assumed that the channel state information is perfectly known to the receiver. Thus， with the use of the near-optimal symbol-by-symbol MAP detection， all the ISI and ICI can be effectively cancelled. Furthermore， theerror performances forOFDMtransmission withconsid ? ered relative speeds are similar. On the other hand， for OTFS transmission， different Doppler shifts caused by different rela? tive speeds do not change the 2D convolution nature of the sig? nal-channel interaction in the DD domain. Therefore， OTFS is insensitive to Doppler effects. In addition， we notice that the OTFS system has a better error performance than the corre ? sponding OFDM system. Moreover， the slope of the FER curve for the OTFS system is greatly higher than that for the OFDM system， which indicates that OTFS enjoys a larger diversity advantage. Those observations align with the findings inRefs. [5] and [20].

The FER performances of the coded OTFS and OFDM sys ? tems without the joint iteration are also shown in Fig. 4， where the relative speeds are 275 km/h， 500 km/h， and 750 km/h， re? spectively. The 16QAM modulated symbols are considered in the simulation. Similar to Fig. 3， we observe that the FER per? formancesof bothcodedOTFSandOFDMsystemsdonot change much with different relative speeds， thanks to the near- optimal MAP detection. Furthermore， compared with uncoded cases，bothcodedOTFSandOFDMsystemsenjoyanim ? proved error performance. In addition， we also notice that the error performance of the coded OTFS is much better than that of the coded OFDM. However， it can be noticed that the cod? ingimprovementfortheOFDMsystemismoresignificant compared with that of the OTFS system. Moreover， the FER curveofthecodedOFDMsystemsharesalmostthesame slope as that of the coded OTFS system. This is because OTFS has the potential to achieve the full channel diversity and con ? sequently， channel coding cannot improve the diversity perfor? mance very much for OTFS systems. In contrast， OFDM sys? tems rely deeply on the channel coding to achieve the larger diversity gain. Those observations are also consistent with the analysis in Ref. [20].

The FER performance of the coded OTFS system with dif? ferent joint detection and decoding iterations is compared in Fig. 5， as well as the uncoded OTFS system. The modulation type is QPSK and the relative speed is 500 km/h in the simula? tion. We can obviously observe that the channel coding signifi ? cantly improves the error performance. In addition， we notice that the iterations between the detector and decoder do not improve the error performance very much. This is because the near-optimalsymbol-by-symbolMAPdetectionalgorithm used in the coded OTFS system exploits all possible interfer? encepatterns，andthereforetheaprioriinformationfrom channeldecodingcannotimprovetheextrinsicinformation fromthenear-optimaldetection fordecoding.Consequently， theiterationsbetweenthedetectoranddecodercannotim ? prove the error performance very much.

5 Conclusions

In this paper， the coded OTFS system is introduced and the existingresearchworkofthecodedOTFSissummarized. Based on this， the upper bound on unconditional PEP for the coded OTFS system is supplemented， and the design issues of thechannelcodingschemearediscussed.Furthermore，the performance of the uncoded OTFS and OFDM systems is ana? lyzed， as well as that of 5G LDPC coded systems. Simulation results show that the performance of the OTFS system signifi ? cantlyoutperformsthatofOFDMsystemunderdifferent speeds， modulation schemes and iterations， whether coded or uncoded.

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Biographies

LIU Mengmeng received her B. S. degree in communication engineering from Xidian University， China in 2017. She is currently pursuing her Ph. D. degree with XidianUniversity.Her researchinterestsincludesignal processingand channel coding for wireless communications.

LI Shuangyang received his B. S. and M. S. degrees from Xidian University， China in 2013 and 2016， respectively. He is currently pursuing his Ph. D. degree in Xidian University and University of New South Wales， Australia. His research interests includesignal processing， channel coding and their applications to communication systems.

ZHANG Chunqiong received her B. S. degree in communication engineering from Xidian University， China in 2019. She is currently pursuing the M. S. de gree with Xidian University. Her research interests include signal processing and channel coding for wireless communications.

WANG Boyu received his B. S. degree in mathematics from China University of Mining and Technology-Beijing， China in 2017， and M. S. degree in mathematics from University of Sheffield， UK in 2019. He is currently studying at Xidian University， China. His research interests include coding theory， and al gorithm design and analysis via convex optimization in LDPC decoding.

BAI Baoming （bmbai@mail. xidian. edu. cn） received his B. S. degree from the Northwest Telecommunications Engineering Institute， China in1987， and the M. S. and Ph. D. degrees in communication engineering from Xidian University， China in 1990 and 2000， respectively. From 2000 to 2003， he was a senior re? search assistant in the Department of Electronic Engineering ， City University of Hong Kong， China. Since April 2003， he has been with the State Key Laborato ? ry of Integrated Services Networks（ISN）， School of Telecommunication Engi? neering， Xidian University， where he is currently a professor. In 2005， he was with the University of California， USA as a visiting scholar. His research inter? ests include information theory and channel coding， wireless communication， and quantum communication.