Ashwitha NAIKOTI, Ananthanarayanan CHOCKALINGAM

Abstract：Orthogonaltimefrequencyspace（OTFS）modulationisarecentlyproposedmodulation scheme that exhibits robust performance in high-Doppler environments. It is atwo-dimensional modulation scheme where information symbols are multiplexed in the de ?lay-Doppler （DD） domain. Also， the channel is viewed in the DD domain where the chan ?nel response is sparse and time-invariant for a long time. This simplifies channel estima?tion in the DD domain. This paper presents an overview of the state-of-the-art approachesin OTFS signal detection and DD channel estimation. We classify the signal detection ap ?proaches into three categories， namely， low-complexity linear detection， approximate max?imum a posteriori （MAP） detection， and deep neural network （DNN） based detection. Simi?larly， weclassify theDDchannelestimationapproachesinto threecategories ， namely，separate pilot approach， embedded pilot approach， and superimposed pilot approach. Wecompile and present an overview of some of the key algorithms under these categories andillustrate their performance and complexity attributes.

Keywords： OTFS modulation; delay-Doppler domain; high-Doppler channels; signal detection; channel estimation

Citation （IEEE Format）： A. Naikoti and A.Chockalingam， “Signal detection and channel estimation in OTFS ， ”ZTE Communications， vol. 19， no. 4， pp. 16 –33， Dec. 2021. doi： 10. 12142/ZTECOM.202104003.

1 Introduction

Next-generation wireless systems are expected to sup ? port a variety of use cases with a wide range of per? formancerequirements.Interestinhigh-mobility use cases involving high-speed trains， unmanned ve? hicles/cars， drones， aeroplanes， etc. ， is on the rise.Also， in order to meet the growing bandwidth requirement， a wireless spectrum in the mmWave frequency band is preferred. Com? munication in such high-mobility and/or high carrier frequen ? cy scenarios has to deal with high Doppler shifts which are commoninsuchenvironments. Orthogonal frequencydivi ? sion multiplexing（OFDM） isa widely usedcommunication waveform in the current generation of wireless systems. De ?spite its popularity and adoption in current standards， OFDM suffers from severe performance degradation in high-Doppler scenarios.This is because of the increased loss of orthogo ? nality among subcarriers and the resulting inter-carrier inter? ference （ICI）.

Orthogonal time frequency space（OTFS） modulation is a recently introduced 2-dimensional （2D） modulation[ 1]. There hasbeengrowinginterestinthismodulationrecently ，be? cause of its superior performance compared withOFDM in high-Doppler environments[2 –6]. InOTFSmodulation， infor? mationsymbolsaremultiplexedin thedelay-Doppler（DD） domain. The symbols in the DD domain are converted to the time domain and transmitted. At the receiver， the receivedsignal in the time domain is converted back to the DD do ? main where the information symbols are recovered. The DD domain to time domain conversion and vice versa can be done using two approaches. In the first approach， symbols from the DD domain are mapped to the time domain in two steps： DD to time-frequency （TF） domain conversion using inverse symplectic finite Fourier transform （ISFFT）， followed by the TF domain to time domain conversion using Heisen ? berg transform[1]. The corresponding inverse transforms map the received time-domain signal to the TF domain and then to the DD domain （Wigner transform followed by SFFT）. The second approach is a direct one-step approach， where DD do? main to time domain mapping is done using inverse Zak transform[7]. At the receiver， the Zak transform maps the sig? nal from the time domain directly to the DD domain. While the first approach has been adopted in most of the studies re ? ported in the literature so far， the second approach is also gaining popularity. While the first approach can be imple ? mented as an overlay on existing TF modulation schemes （such as OFDM）， the second approach has the benefit of re ? duced implementation complexity.

Since the introduction of OTFS in 2017， there has been a spurt of research activities in OTFS leading to an increasing volume of publications on OTFS[5 –51]. Some of the key areas of focus in these works include DD signal representation in OTFS， input-output relation in the DD domain in the form of a linear vector channel model， framework for DD signal pro? cessing， signal detection algorithms， techniques for DD chan? nel estimation， characterization of the peak-to-average power ratio （PAPR）， the effect of practical pulse shapes， diversity analysis of OTFS， the effect of oscillator phase noise and IQ imbalance， multi-antenna OTFS， space-time coding and pre? coding in OTFS， multiuser OTFS on the uplink and down? link， etc. Recognizing that efficient signal detection and channel estimation techniques are crucial for the successful realization of OTFS systems in practice， we focus on these two key receiver functions in this paper.

We classify the OTFS signal detection approaches into three broad categories. The first is the linear detection ap ? proach， where the focus is on exploiting the structure inher? ent in the effective channel matrix for reducing complexity. The second approach is based on approximations to maxi ? mum a posteriori （MAP） detection， which aim near-optimal performance at reduced complexity. The last one is a recent approach involving deep neural networks （DNN）. We high? light some of the algorithms in these categories reported in the literature. In highlighting various detection algorithms， perfect DD channel knowledge will be assumed at the re ? ceiver.

Similarly， we classify the DD channel estimation approach ? es into three categories. In the first approach， separate pilot frames are employed for DD channel estimation. The channel estimates obtained during the pilot frames are used for detection during data frames. The second approach involves em ? bedding both pilot and data symbols in a frame. This further improves the throughput but the interference between pilot and data symbols needs to be taken into account by way of providing guard symbols around pilot symbols and/or inter? ference cancellation. The last approach is the superimposed pilot approach， where pilot symbols are superimposed on da? ta symbols. This further increases the throughput while de ? manding sophisticated signal processing （e. g.， interference cancellation） to perform joint channel estimation and detec ? tion. We present algorithms reported in the literature under these categories.

The rest of this paper is organized as follows. Section 2 in? troduces the OTFS system model. Section 3 presents the state-of-the-art approaches and algorithms for OTFS signal detection. Section 4 presents the approaches and algorithms for DD channel estimation. Section 5 provides the conclu? sions.

2 OTFS Modulation and System Model

2.1 OTFS Modulation

In OTFS modulation， MN information symbols are multi? plexed onto an N × M DD grid， where N is the number of Doppler bins and M is the number of delay bins， as shown in Fig. 1. The information symbols， denoted by x k，l，k = 0，...，N - 1，l = 0，...，M - 1， come from a modulation alphabet （e.g.， QAM/PSK）. The NM symbols are transmitted over a time duration of NT， occupying a bandwidth of MΔf， whereΔf = 1/T. The Doppler resolution isand the delay resolution is1

The symbols in the DD grid are mapped to a time-domain signal x tfor transmission. This can be done in two ways as shown in Fig. 2. In a two-step approach， the DD signal is first mapped to a time-frequency （TF） signal which is then mapped to a time-domain signal. The DD-to-TF domain map? ping is done using ISFFT and the TF-to-time domain map? ping is done using Heisenberg transform. In a one-step ap? proach， the DD signal is directly mapped to a time-domain signal using inverse Zak transform. The corresponding in? verse transforms are used at the receiver to demap the re ? ceived time-domain signal to the DD domain. In this paper， we adopt the two-step approach which has been widely fol? lowed in the literature so far.

2.2 OTFS System Model

In this subsection， we present the OTFS system model forthe two-step approach of DD-to-time domain conversion， as shown in Fig. 2（a）. The symbols x k，lin the DD domain are mapped to the TF domain using ISFFT， as

X n，m= x k，lej2π-.（1）

This TF signal is transformed into a time-domain signal us? ing Heisenberg transform， as

x t=X n，mgtxt - nTej2πmΔft - nT，

where gtxtdefines the transmit pulse shape. The transmit? ted signal is passed through the channel whose response inthe DD domain is given by

h τ，ν=hi δ τ - τ iδ ν - ν i，

where hi， τ i， and ν iare the channel gain， delay， and Doppler shift associated with the i-th path， respectively， and P is thenumber of resolvable paths in the DD domain.

The received time domain signal y tat the receiver is giv? en by

y t= ∫ν∫τ h τ，νx t - τej2πνt - τdτdν + v t，（4）

where v tis the additive white Gaussian noise. Wigner transform is applied to y tto transform it into a TF domain signal， as

Y n，m= Agrx ，y t，ft = nT，f = mΔf ，

Agrx ，y t，f=rx（*） t′ - ty te-j2πf t′ - tdt′ ， （5）

where grxtdefines the receive pulse shape. If grxtand gtxtsatisfy the biorthogonality condition， the input-output relation in the TF domain is given by Ref. [11]：

Y n，m= H n，mX n，m+ V n，m，

where V n，mis noise in TF domain and H n，mis

H n，m= ∫τ∫ν h τ， νej2πνnT e-j2π ν + mΔfτdνdτ.

The TF signal Y n，mis transformed to the DD domain signal y k，lusing SFFT， as

y k，l=Y n，me-j2π-.

The above DD domain signal at the output of the SFFT can be derived to be of the form1

y k，l=hi′xk - β iN ，l - α iM+ v k，l，（9）

where hi′ = hi e-j2πνiτi ， his are i. i. d and are distributed as0， 1/Pwith uniform scattering profile， α iand β iare integers2corresponding to indices of delay and Doppler， respectively， for the i-th path， i.e.， τ i?and ν i?， andv k， l is the additive white Gaussian noise. By vectorizing the input-output relation in Eq. （9）， we can write[11]

y = H% + v，（10）

where %，y，v∈MN × 1， the （k + Nl） -th entry of %，xk + Nl=x [ k，l ]，k = 0，...，N - 1，l = 0，...，M - 1 and x [ k，l ] ∈. Simi? larly， yk + Nl= y [ k，l ] and vk + Nl= v [ k，l ]， k = 0，...，N - 1，l =0，...，M - 1， and H ∈MN × MNis the effective channel matrix，whose j-th row （j = k + Nl）， denoted by H [j ]， is given by H [j ] = ?（「）（（k - 0）N ，（l - 0）M ）（（k - 1）N ，（l - 0）M ）...（（k - N - 1）N ，（l - M - 1）M ）」（ù） ， where（k，l） denotes the （k，l）-th element of the N × M DD channel matrix， given by（k，l） = {hi′， if k = β，0（i ），lr（i） i，wi（）se（∈） { 1， 2，...，P }.（11）

It can be seen from the above that the effective channelmatrix H has only P non-zero entries in each row and col? umn， i.e.， there are only MNP non-zero elements in H. The linear vector channel model in Eq. （10） is used for signal de? tection/equalization and channel estimation in OTFS.

3 OTFS Signal Detection

In this section， we present some of the signal detection al? gorithms proposed in the literature for OTFS modulation. Based on the approaches used， these algorithms are catego ? rized into three groups， namely， 1） low-complexity linear de? tection， 2） approximate MAP detection， and 3） neural net? works based detection， as shown in Fig. 3. We present algo? rithms under these categories in the following subsections， assuming perfect knowledge of the DD channel matrix. Later， in Section 4， we will present techniques/algorithms to esti ? mate the channel matrix.

3.1 Low-Complexity Linear Detection

Linear equalizers detect the transmitted symbols by apply ? ing a linear transformation to the received vector y followed by mapping to a symbol in the modulation alphabet which is closest in terms of euclidean distance. The linear transfor? mation matrix is represented by G and the mapping function is represented byf （.）. Therefore， the estimate of the transmit vector x is given by

=f （Gy）.???????? ?（12）

The transformation matrix for linear minimum meansquared error （LMMSE） equalization is given by

Glmmse= （HH H + σ2I）- 1HH ，

where σ2 is the noise variance and the transformation matrix for zero-forcing （ZF） equalization is given by

Gzf= （HH H）- 1HH.

These equalizers in the case of OTFS have a computationcomplexity of （M3 N3）. However， the structure of the channel matrix in the DD domain can be exploited to reduce the complexity of these operations[22 –23].

3.1.1 Low-Complexity LMMSE Equalization

In the low-complexity linear minimum mean square error （LMMSE） equalization[22]， the channel matrix H in Eq. （10） has a block circulant structure with M circulant blocks denot? ed by Ai（i = 0，1，...，M - 1） of size N × N. Using this property of the channel matrix， a low-complexity algorithm for imple ? menting LMMSE equalization has been proposed in Ref. [22]. Let M，Ndenote the class of such block circulant matri? ces. These matrices have the following exploitable properties.

· Any matrix H ∈M，N can be unitarily diagonalizable as

H = （FM? FN ）H Λ （FM? FN ），（15）

where Λ = diag{ λ1，λ 2， … ，λMN} such that λ iis the i-th eigen? value of H， FM is the DFT matrix of size M， and ? is the Kro? necker product operator.

· The matrix Λ can be written as

where ΩM=diag{ 1， ω ，...， ωM - 1 } and ω = ej2π/M. Λ iis N × N di? agonal matrix with eigenvalues of N × N circulant block Ai on the diagonal.

· For A，B ∈，， the matrices AT， AH， AB = BA ， c1 A + c2B，crAr（crare all scalars） and A- 1 （if exists） are also block circulant and belong to M，N.

As H ∈M，N， using the above properties， we have the LMMSE transformation matrix Glmmse∈M，N. Thus， by substi? tuting Eq. （15） in Eq. （13）， we get

Glmmse= FM? FN H Ψ FM? FN ，（17）

where Ψ is a diagonal matrix containing the eigenvalues of Glmmse， given by

Ψ = Λ* Λ + σ2I- 1 Λ* ，（18）

where Ψi=，i = 1， 2，...，MN. To reduce the complexity of computing FM? FN y， write y as an N ×M ma? trix Y such that vecY= y. This gives

z = FM? FN y = vecFN YFM .（19）

Now， compute q = Ψz and write q as a N × M matrix Q such that vecQ= q. Finally， compute the estimated x as

= Glmmse y = vecFN（H）QFM ，（20）

which gives the exact LMMSE solution at a much less com? putational complexity. The complexity of computing z in? volving N-point DFT and M-point IDFT operations isMNlogMNand the complexity of computing qis MN. Again， the computation ofinvolves N-pointIDFTandM-pointDFToperationswithcomplexityMNlogMN.Therefore，theoverallcomplexityis 2MNlogMN + MNwhich is much small compared to the M3 N3complexity of conventional LMMSE detec? tion using matrix inversion.

3.1.2 Low-Complexity LMMSE Equalization

As shown in Fig. 4， a low-complexity LMMSE equalization method that takes advantage of the sparse and quasi-banded nature of the OTFS demodulation matrices has been proposed in Ref. [23]. Here， a different representation of the system is used. The transmit vector x in the DD domain is writ? ten as an M × N matrix X such that vecX= x. Assume that E x k，lx* k′，l′= σx（2）δ k - k′，l - l′. Using this represen? tation， we obtain vector s as

s = vecXFN（H）.（21）

This vector s can also be written in the form s = Ax， where A = FN（H）? IMis a unitary matrix. The received vector r at the receiver is given by

r = H?s + n = H?Ax + n，（22）

where H? =hi Π αi Δ βi ， Π = circ010...0M（T）N × 1is a circunal Doppler matrix， and n is i.i.d Gaussian noise vector with variance σ n（2）. The detected symbol vector for this system mod? el in Eq. （22） using LMMSE equalization is given by

= H?AHH?AH?AH+I- 1 r.（23）

Due to the unitary nature ofA， the above equation reduces to

This detected vectoris obtained in two steps. The first step involves a calculation of= Heqr and the second step in? volves the matched filter operation AHto get. Most com? plexity is in the first step as it involves an inverse of Ψ =H?H? H+Ito obtain Heq. This complexity is reduced byusing a low-complexity LU decomposition of Ψ[23]. With LU decomposition of Ψ， we can write Eq. （24） as

The computational complexity can be further reduced byusing the quasi-banded nature of lower triangular matrix L and upper triangular matrix U. In Eq. （25）， r1 is computed using the forward substitution method for quasi-banded lower triangular matrix and r2 is computed using backward substitution method for quasi-banded upper triangular ma? trix. The final computation of= AH H? H r2 is done by first obtaining= H? H r2. This vectoris arranged in an M ×N matrix as

Using the matrix Y? ， x? is obtained using DFT operation as

The main reduction in the complexity is in the computa?tion of r?. All the steps involved in obtaining r2 together have a complexity of O （MN） and the final computation in Eq. （27） to obtain x? has a complexity of O （ MN 2 log2N）. On the whole， this LMMSE equalization method has a complexity of just O （ MN 2 log2N + MN） compared to O （M3 N3 ） of the conven?tional LMMSE equalization.

3.2 Approximate MAP Detection

Signal Detection and Channel Estimation in OTFSSpecial Topic Ashwitha NAIKOTI， Ananthanarayanan CHOCKALINGAM

From the vectorized representation y = Hx + v in Eq. （10）， the MAP decision rule for detection of x is given by

When the transmit symbol vectors are equally likely， the decision rule for maximum-likelihood （ML） detection is

Assuming the noise vector v to be i.i.d Gaussian， the opti? mum decision rule is given by

The complexity of the optimum detector grows exponential?ly in MN because of the exhaustive enumeration/search in? volved. Therefore， several suboptimum detection algorithms have been proposed that are efficient with low complexity. In the following， some of the popular low-complexity approximate MAP algorithms for OTFS signal detection are presented.

3.2.1 Message Passing Algorithm

One of the popular detectors reported in the early OTFS literature is an approximate MAP detector based on low-com?plexity message passing[11] . The key advantages of this detec? tor are its linear complexity in MN and very good perfor?mance. The message passing algorithm involves the computa?tion of approximate a posteriori probability of the modulation symbols by passing messages on a factor graph. The transmit? ted vector x is represented by MN variable nodes and the received vector y is represented by MN check nodes in the graph. As noted earlier in Section 2， H is sparse with only L non-zero elements in any row or column （generally L ?MNand L = P for non-fractional delays and Dopplers）.

Let rand cdenote the indices corresponding to non-zero elements in the r-th row and c-th column， respectively，such that r= c= L for all rows and columns. In thefactor graph， each variable node x c has connections with Lcheck nodes y r ， r ∈cand each check node y r hasconnections with L variable nodes x c ，c ∈ras shown inFig. 5. The symbol-by-symbol decision rule is given by

This approximation is obtained assuming that the compo ? nents of vector y are independent for a given x c because of the sparsity of H matrix. From the system model， we can write

In the i-th iteration， the complete interference plus noiseterm in Eq. （32） is modelled as a single Gaussian randomvariable ζ ri，c with mean μri，c and variance σri，c2.

1） Message from check node y r to variable node x c ： The check nodes pass mean and variance information to the variable nodes， where

and σ2 is the variance of v r .

2） Message from variable node x c to check node y r ：The variable nodescompute the probability mass functionand pass it to the check nodes. Each component of the proba? bility mass function pis given by

3） Compute convergence parameter ηifor some γ > 0， as

and

4） If ηi<ηi - 1for c = 1， 2，...， MN， then update

5） Stop the iterations if any one of the following holds：

· The maximum limit set for the number of iterations has reached;

·ηi= 1;

·ηi<ηs*- ? for some small ? ， where ηs*= msi（x）ηs.

Modified variants of this message passing algorithm have also been proposed. For example， a low complexity variant that exploits channel hardening through match filtering oper? ation on y and message passing on the resulting system mod ? el is presented in Ref. [15]. Another variant in Ref. [ 17] is presented below.

3.2.2 Gaussian Approximate Message Passing Algorithm

In this variant of message passing， the a posteriori proba? bility of each transmitted symbol is assumed to be Gaussian distributed instead of assuming the inter-symbol interference to be Gaussian as done earlier[ 17]. We have

As the channel matrix is sparse， we get

where %rcontains the elements x c ， c ∈r. The Gauss? ian assumption is that

where Hris a row vector containing the non-zero elements in r-th row of H. This approach has been proposed in Ref. [ 17] and the results presented show that this approach has su ? perior bit error rate （BER） performance with the same com ? plexity order.

3.2.3 Variational Bayes Detection

An iterative algorithm that approximates the optimal MAPdetection and has a faster convergence compared to the messagepassingalgorithmhasbeenproposedinRef.[20].Anapproximationof theaposterioriprobability p %|yisobtainedbyusingKullback-Leibler（KL）divergence（q||p）and the corresponding evidence lower bound （ELBO） is maxi? mized iteratively using the variational Bayes approach. The convergence is guaranteed because the ELBO maximization problem is convexly resulting in a globally optimum solution. In this way， the marginal distribution for each symbol is ob ? tained which is used for symbol-by-symbol MAP detection. The approximate distribution q %is obtained by searching over a family of distributions such that

The ELBO is given by which is the expectation over x having distribution q %. In particular， when a family with mutually independent variables is considered，

qisobtainediterativelyi = 1， 2，...， MN，andthe symbols are detected as

In addition to having complexity lower than that of MAP detection， this detection method has performance significant? ly close to the performance of MAP detection.

3.2.4 Hybrid MAP and PIC Detection

Another approximation to the MAP detection using a parti ?tioning method based on path gains has been proposed inRef. [16]. The hybrid MAP-PIC algorithm is a combination ofboth symbol-by-symbol MAP detection and message passingalgorithm. The received symbols are partitioned into two sub?sets based on the path gains of the channel. On one part withgood path gains MAP detection is used， and on the remainingpart parallel interference cancellation （PIC） method is used.Definethefollowingsets：i? { hj | 1 ≤ j ≤P，j≠i } ，k，l? { y [（k + βj ）N ，（l + αj ）M ] | 1 ≤ j ≤ P } ， l（）? { y [（k +βi- βj ）N ，（l + αi- αj ）M ] | 1 ≤ j ≤P，j≠i } ， where Pis thenumber of channel paths and i is the path index. The P re?ceived symbols that are associated with the transmitted sym?bol x k， l are in set k，l. Similarly， the P - 1 transmittedsymbols， other than x k， l ， corresponding to the receivedsymbol k，liare in setl（）. The path gains in iare ar?ranged in decreasing order such that |hm |2 > |hn |2whenm < n. The setl（）is partitioned into two subsets by enumer?ating different possible combinations of S （where S is the sizeof the first subset with good path gains） as follows：

It is proposed to perform MAP detection on X?（i） k，l and PIC on Xˉ （i） k，l . The message passing algorithm that we discussed ear?lier is a special case of Hybrid-MAP-PIC detection when S = 0. Results have shown that choosing S = P/2 gives good error performance. A trade-off can be established between BER performance and computation complexity by choosing a suit? able value for S.

3.2.5 MCMC Sampling Based Detection

This detection algorithm proposed in Ref. [8] uses the Mar?kov chain Monte Carlo （MCMC） sampling method to obtain an approximate solution to Eq. （28）. The joint probability dis?tribution is given by

The algorithm starts by initializing a random initial vector x（t = 0） ， where t denotes the iteration number. The MN coordi? nates of the x vector are updated in each iteration based on the coordinates of the previous iteration as follows：

After updating over a certain number of iterations， the dis? tribution obtained approximately converges to the distribution in Eq. （49）. For a received vector， the symbol vectorwhich has minimum ML cost y - Hx2 in all the iterationsis chosen as the detected symbol vector. A modification to this has also been proposed to reduce the number of itera? tionsand also to overcome the phenomenon of stalling seen in the Gibbs sampling method at high SNRs which limits the BER performance. The modified joint distribution involves a temperature parameter α chosen based on the operating SNR and is given by

Alternately， the sampling can be randomized by updating the parameters in each iteration using the conventionalGibbs sampling method with probability q （e.g.， q =） andobtaining samples from a uniform distribution with probabili ? ty 1 - q. This randomized sampling has been shown to avoid stalling problems and achieve good BER performance.

3.3 Neural Networks Based Detection

Apart from the detection methods based on conventional approaches， detectors based on DNN have been proposed re? cently. Two DNN approaches have been presented in Ref. [18]. One approach is to use a single fully-connected DNN to detect the signal vector. The detection problem is formulated as a multi-class classification problem where each class cor? responds to each vector in the transmitted signal set， en? abling joint detection of the transmitted symbol vector. The number of input neurons in the network is decided by the size of the received vector （MN） and the number of output neurons is decided by the size of the multi-dimensional mod? ulation alphabet （MN）. This approach requires a large num? ber of parameters to be learned and is computationally com? plex because of the exponential growth in the number of out? put neurons with the size of transmit symbol vector. The ar? chitecture of this fully connected DNN is shown in Fig. 6. The real and imaginary parts of the received vector y are giv? en as input to the DNN. The activation function used in the output layer is Softmax activation so that the output of each output neuron gives the probability of the corresponding transmitted signal vector and all these probabilities sum to one. The detected symbol vector is the one that has maxi? mum probability.

Another architecture that uses multiple DNNs is shown in Fig. 7. In this approach， each symbol in the transmitted vec? tor is detected by an individual DNN. In this way， each DNNhas the number of output neurons growing linearly in the size ofmodulationalphabet（）.Also，thenumberofDNNs growslinearlywiththesizeof thetransmitsymbolvector （MN）. This symbol-DNN architecture does symbol-by-symbol detectionatlowercomplexityandachievesBERperfor? mance almost the same as that of full-DNN. Each DNN uses Softmax activation in the output layer and obtains probabili ? ties corresponding to each symbol in .

The training of the DNNs is done by pseudo-randomly gen? erating a set of training examples xT which are known both at the transmitter and the receiver. These training examples are sent to the receiver through the channel. The transmitted sig? nal vector xTand the corresponding received signal vector y form the training pair at the receiver. The real and imaginary parts of y are given as input to the DNNs. The DNN trained in this manner learns the mapping from the received vector to the corresponding symbol in the transmitted vector.Anoth? er approach of using neural networks for signal detection has been reported in Ref. [ 19]， where the two-dimensional struc ? ture of theOTFS frame with data augmentation based pre- processing is given as input to two-dimensional convolution? al neural networks （CNN） for signal detection.

Thebenefitsof usingDNNsforsignaldetectioncanbe predominantlyseen when therearedeviationsin thenoise model from the standard i. i. d. Gaussian model. In situationswhere there are deviations from the Gaussian as well as inde ? pendenceassumptionsinthestandardnoisemodel，DNN baseddetectioncouldoutperformconventionalMLdetec ? tion. This is because ML detection is optimum only for the standard i. i. d. Gaussian noise model and the DNNs have the ability to learn the underlying deviations in the model.

3.4 Performance Results

Inthissubsection， wepresenttheBERperformanceof some of the detectors presented in the previous subsections ， assumingperfectDDchannelknowledgeatthereceiver. Thesystemparametersconsideredareaccordingtothe IEEE. 802. 11p standard for wireless access in vehicular en ? vironments （WAVE）[52]. A carrier frequency of 5.9 GHz with asubcarrierspacingof 0. 156MHz，amaximumspeedof 220km/h，andamultipathchannelwithP = 5pathsare considered. BPSK modulation alphabet is used with a frame size of M = 32 and N = 12.

Fig. 8 shows the BER performance of the linear detectors includingconventionalMMSE/ZFdetectorsandlow-com? plexity MMSE/ZF detectors. It can be observed from Fig. 8 that the performance of the conventional MMSE detector and thelow-complexityMMSEdetectorsinRefs. [22]and[23] are the same. However， the detectors in Refs. [22] and [23] achieve this with significantly lower complexities compared to the conventional MMSE detector complexity. This is illus ? trated in Fig. 9 where the computational complexities （in the number of real operations） for these detectors are plotted.

The BER performance of the message passing detector in Ref.[ 11]and thesymbol-DNNbaseddetector inRef.[ 18] are shown in Fig.10. MMSE detection performance is also shown for comparison. In this figure， a system with a carrier frequencyof 4GHz，subcarrierspacingof15kHz，OTFS frame size of M = N = 16， and a uniform power delay profilechannel with P = 8 are considered. The delay-Doppler pro? file considered is shown in Table 1. For the message passing algorithm， a damping factor of Δ = 0.6 and maximum itera? tions of 30 are used. For the symbol-DNN based detection， the parameters of the neural network are shown in Table 2. It can be seen from Fig. 10 that the symbol-DNN performs bet? ter than the MMSE detector， and the message passing detec? tor gives the best performance among them.

Next， the performance superiority of DNN-based detection compared to ML detection in correlated noise is illustrated in Fig. 11. This figure shows the BER performance of the MLdetector and symbol-DNN detector for a MIMO-OTFS system when the noise is correlated. The correlated noise vector is taken to be vc= Nc v， where v is the i.i.d Gaussian noise vectorandis the correlation coefficient （0 ≤ρ≤ 1）， ntis the number of transmit antennas， and nris the number of receive antennas. The following system parameters are considered in Fig. 11： carrier frequency of 4 GHz， subcarrier spacing of 3.75 kHz， frame size of M = N = 2， uniform power delay profile chan? nel with P = 4， MIMO configuration with nt= nr= 2， and a correlation coefficient ρ = 0.4. The symbol-DNN architecture has an input layer with 16 nodes， one hidden layer with 32 nodes， and an output layer with 2 nodes. The hidden layer has ReLU activation and the output layer has Softmax activation. The DNN is trained at an SNR of 8 dB for 60 epochs with 50 000 training examples. It can be seen from Fig. 11 that the symbol-DNN based detector outperforms the ML de ? tector by almost 1 dB at BER of 10-4. This is because the ML detector is optimal only when the noise is i.i.d Gaussian and is suboptimal in correlated noise. On the other hand， the sym? bol-DNN based detector performs well as it effectively learns the noise correlation leading to superior BER performance.

4 DD Channel Estimation

The task of channel estimation at the receiver is crucialas signal detection operation requires the knowledge of the channel state information. In OTFS， signal detection is car? ried out in the DD domain. In the system model in Eq. （10）， knowledge of the DD channel matrix H is needed for detection. In order to estimate H， % vector consisting of known pilot symbols is sent. Given the knowledge of the pi? lot symbol（s） in % and the observation vector y， channel es? timation algorithms estimate H. For the purpose of exposi? tion， we classify the channel estimation approaches into three broad categories based on the OTFS frame pattern used to transmit the pilot and data symbols. Fig. 12 shows this classification consisting of 1） separate pilot approach， 2） embedded pilot approach， and 3） superimposed pilot ap? proach. In the first approach， pilot frames consisting of on? ly pilot symbols are used for channel estimation. The esti? mated channel matrix obtained during the pilot frame is used for detection during data frames. The second ap? proach involves embedding both pilot and data symbols in a frame. In the third approach， pilot symbols are superim ? posed on data symbols. Some of the channel estimation techniques/algorithms employing these approaches are presented in the following subsections.

4.1 Separate Pilot Approach

As mentioned earlier， in this approach， separate framesare used for sending pilot symbols and data symbols. A pilot frame consists of only pilot symbol（s）. One pilot frame per spatial coherence interval of the DD channel is sent. The channel estimated during the pilot frame is used for the de ? tection of symbols in the data frames in that coherence inter? val. In the following， we present three-channel estimation methods using this approach.

4.1.1 Impulse Based Channel Estimation

In this method， impulses in the DD domain are sent as pi? lots[13]， i.e.， the pilot symbol is given by

For this transmitted pilot， the received signal at the receiv? er is

As the receiver knows the pilot locations kpand lpa priori，mated channel matrix H? can be obtained.

4.1.2 PN Pilot Based Estimation

Instead of impulses as pilots， this method uses PN se? quence as a pilot[8]. The estimation is done in the discrete do ? main and the parameters to be estimated are delay tap τi， Doppler shift ν i， and channel fade coefficient hi′. The input- output relation for a P path channel in the time domain can be obtained as

Let denote the vector space of complex-valued func? tions on the set of finite integers Np= { 0，1，...，Np- 1} with an inner product defined as

A signal is transmitted which is given by

where S ∈，M = Np+ Wmaxτi≥ Np. For some S ∈that is transmitted， the received sequence R n is

where e t= ejt， δ i ，ω i∈Np ， α i∈ C， and v n ∈. Eq.（56） can be simplified as

such that δ 0， ω 0∈Np× Np. The δ 0， ω 0pairs are estimated using the time-frequency shift problem. A matched filter matrix for R and S is defined as

4.1.3 Compressed Sensing Based Estimation

The channel estimation problem can be formulated as a sparse signal recovery problem using compressed sensing based methods like orthogonal matching pursuit （OMP） and modified sub-space pursuit （MSP）[28]. The channel is estimat? ed by sending a pilot matrix Xpin the DD domain with i.i.d Gaussian random sequences as pilots. The system model is rewritten as

such that Xp∈MN × MNand h ∈MN × 1 have P non-zero ele? ments. The channel estimation problem as a sparse signal re? covery problem is given by

OMP algorithm is used when the knowledge of the number of paths P is available. Initialize h0 = 0， S0 = ? ， and r0 = yp. The following operations are performed in the ith iteration.The indices of the highest correlated columns are obtainedSi - 1 ∪Ti. The non-zero values corresponding to the supportare hSi= Xp（S）i? yp ， where .? is the pseudo-inverse operator.Finally， the residue is updated as ri= yp- Xp（S）ihSi. Stop the iterations when ri2 is less than a threshold ? and obtain the estimate asSi= Xp（S）i? yp andSˉi= 0.

When the knowledge of the number of channel paths P is not known， the subspace pursuit algorithm is modified to esti? mate the channel and the corresponding support using Algo? rithm 1. yp ，Xpand ? are given as input andis obtained as output.

Algorithm 1. MSP based channel estimation[28]

Inputs： y，X，?

Initialize： i = 1，r1 = y

4.2 Embedded Pilot Approach

In this approach， instead of allocating an entire OTFS frame for pilot transmission， pilot symbols are transmitted in the same frame as data symbols with guard symbols around to prevent interference between pilot and data symbols. If no guard symbols are provided， then more sophisticated algo? rithms may be needed to handle the interference. In the fol? lowing， we present two estimation algorithms for the embed? ded pilot approach. The first algorithm is applicable when guard symbols are provided. The second algorithm， based on sparse Bayesian learning， is applicable for embedded frames without guard symbols.

4.2.1 Embedded Pilot Based Estimation

the pilot location such that 0 ≤kp≤ N - 1 and 0 ≤lp≤ M - 1. Define α = max { αi } corresponding to the largest delay and β = max { βi } corresponding to the largest Doppler. Moreprecisely， it is better to choose kp ，lpsuch that 0 ≤lp- α ≤lp≤lp+ α ≤ M - 1， and 0 ≤kp- 2β ≤kp≤kp+ 2β ≤ N -1.

The pilot， guard， and data symbols in an OTFS frame are arranged as follows （see Fig. 13 for an example）：

At the receiver， the subgrid in y k，lused for channel esti? mation is given by kp- β ≤ k ≤kp+ β，lp≤ l ≤lp+ α . Within this subgrid， if y k，l≥for a detection threshold > 0， thenk - kp ，l - lp= y k，l/xpandk - kp ，l - lp= 0 otherwise. If a path exists， then it must be seen in thereceived frame as a scaled version of the pilot plus Gaussian noise. It has been shown that choosing = 3σ gives good es? timation where σ2 is the noise variance. An extension to this method has also been proposed in Ref. [27] considering the fractional Doppler scenario. A method to select a threshold value based on the receiver operating characteristics （ROC） curve has been demonstrated in Ref. [32].

4.2.2 Sparse Bayesian Learning Based Estimation

In this method， the problem of channel estimation is con ? verted to a problem of sparse signal recovery by exploiting the sparsity of the channel in the DD domain[29]. This method does not require guard symbols and uses pilot SNR to be the same as data SNR. This approach considers the case of frac ? tional Dopplers as well. The structure of the OTFS frame isgiven by （see Fig. 14 for an example）

where kpand lpare chosen to be N 2 and M 2， respectively， and Q is a parameter that approximates the effect of Doppler. Results in Ref. [27] show that the channel approximation isgood when Q = 5. Further， lris a parameter that obtains atrade-off between the error performance and pilot overhead.are used for channel estimation. The system model for L = 2β + 2Q + 1× α + 1pilot symbols （xpk，l） is modified as

where X ∈MN × L， h ∈P × 1， and v ∈MN × 1. B is the phase compensation matrix which is a block diagonal matrix with the conjugate of the phase terms on the diagonal， and ⊙ is Hadamard product operator. The noise in Eq. （63） is assumed to be Gaussian with zero mean and variance 1/ν0，sumed to be Gamma distributed with parameters a and b. With this assumption， the distribution of the received vector is given as Pry|Φh，ν 0（-）1I. By modelling the conjugate prior as per the Bayesian estimation framework， we get

where Λ = diagνis the covariance matrix and ν iis the variance of hi. All the hi 's are identical with a Laplace sparse prior distribution

The joint probability distribution of this model is written as

and

where μ = ν 0 ΣΦT y and Σ = ν 0 Φ T Φ + Λ- 1- 1. The valuesof ν and ν 0 are obtained by solving expectation maximization （EM） algorithm as

which gives

After obtaining μ， the first P 2Q + 1largest values in μ are selected as h.

4.3 Superimposed Pilot Approach

In this approach， pilot symbols are superimposed on data symbols in an OTFS frame. For example， each bin in the DD grid has a data symbol and a pilot symbol superimposed on it as shown in Fig. 15[35]. Fig. 16 shows another example where all bins have data symbols and only one among them has a superimposed pilot symbol[36].

4.3.1 Superimposed Pilots Based Estimation in Ref. [35]

In this method of estimation， low-powered pilot symbols are superimposed on the data symbols in the DD grid （Fig. 15）. The mutual interference between the pilot and the data sym ? bols is handled by optimum selection of pilot SNR and byadopting an iterative approach that iterates between channelestimation and data detection. The system model considered is

where X ∈MN × P， h ∈P × 1， and v ∈MN × 1. When the pilot symbols are superimposed on the data symbols， the system model is given by

where Xpcorresponds to pilot symbols， Xdcorresponds to da? ta symbols， and vdis the noise plus interference term having mean μ vd= 0MN × 1 and covariance

where σ d（2）= Exdk，l2， σ h（2）iis the variance of ith channelcoefficient， and σ v（2）is the noise variance. Under these as? sumptions， the MMSE estimate of the channel using superim ? posed pilots is given by

where Ch= diagσ h（2）1 ，σ h（2）2 ，...，σ h（2）P . Message passing algorithmsalong with the MMSE estimated channel are used to detectthe data symbols X?0. This is used as initialization of Xdforan iterative channel estimation algorithm which is more ro ? bust to the interference than the MMSE estimate. With this， the system model in Eq. （73） can be rewritten as

Here， the data-aided pilot is given by Xand the inter? ference plus noise term is given by ξwhere

For the system model in Eq. （76）， the MMSE channel esti? mate in the n-th iterationnis obtained using Eq. （75） as

Thus， the expression in Eq. （78） gives the channel estimation after n iterations.

4.3.2 Superimposed Pilot Based Estimation in Ref. [36]A data-aided channel estimation that uses the whole OTFSframe for data transmission scheme with one pilot symbol sugrid is reported in Ref. [36]. The allocation of symbols in the DD grid is given by

The energy of the pilot symbol is denoted by Ep= xp2and the average energy of the data symbol is denoted by Ed=Ex k，l2. With this frame structure， the received signalin the DD domain is given by

where k，l is the interference due to data symbols， given by

The channel is initially estimated using a modified threshold which incorporates the effect due to k，l. Using this estimated channel， the data symbols are detected by a sum-product algorithm. The interference term can be sim ? plified as

where k，lis the set of indices of the data symbols that con the interference part is obtained as

WhenE { hi 2 } = 1， we get E { k，l 2 } = Es.Using theinterference energy， the threshold is obtained as

If y k，l≥ γ ， the estimates of the channel coefficientscan be obtained as

The data symbols are detected using the estimatedwby the decision rule：

The marginal PDF is obtained by using a sum-product al? gorithm. After detecting the data symbols， the interference caused by them is cancelled and the resultant symbols are given by

These symbols would contain only the pilot information if the interference was completely cancelled. However， due to imperfect estimates， this method of channel estimation fol? lowed by data detection and interference cancellation is per? formed iteratively to obtain better estimates.

4.4 Performance Results

In this subsection， we present the performance of some of the channel estimation methods presented in the previous subsections. The OTFS system considered has a carrier fre ? quency of 4 GHz， a subcarrier spacing of 15 kHz， and a DD grid with M = N = 32. A multipath channel with P = 5 paths， exponential power delay profile， and delay-Doppler profile shown in Table 3 are considered. Fig. 17 shows the mean squared error （MSE） performance as a function of pilot SNR for 1） OMP， 2） impulse based estimation， and 3） embed? ded pilot based estimation. A pilot frame with i.i.d Gaussian random sequences occupying the entire DD grid is used forOMPbasedchannelestimation. Apilot frame withanimpulse at location （kp， lp） = （15， 15） and zeros elsewhere is considered for impulse based channel estimation. An embed?ded frame is used for embedded pilot based channel estima?tion， with impulse as a pilot at location （kp， lp） = （15， 15）， guard symbols in the locations 7 ≤ k ≤ 23， 10 ≤ l ≤ 20， and data symbols elsewhere

It can be seen from Fig. 17 that the OMP algorithm gives the channel estimate with small MSE， which is in the order of 10- 3for a pilot SNR of 15 dB. Impulse based channel esti ? mation scheme is simpler but its MSE is high. The MSE of theembeddedpilotbasedchannelestimationisalsohigh. Fig. 18 shows the BER performance of these estimation meth ? ods using MMSE detection as a function of data SNR for a pi ? lot SNR of 20 dB. BPSK modulation is used. The BER perfor? mance of OMP based channel estimation is close to the per? formanceusingperfectchannelknowledge. Impulsebased estimation performance is inferior compared to OMP perfor? mancebutissuperiorcomparedto thatof embeddedpilot basedestimation. Embedded pilot basedestimationcan be used with high pilot SNRs to achieve good BER performance and higher throughput.

5 Conclusions

OTFS modulation is regarded as an attractive physical layer waveform for future wireless systems. It has demonstrated robust performance in high-Doppler scenarios which are expect? ed in emerging standards. In this paper， we presented an over? view of the state-of-the-art approaches in OTFS signal detec ? tion and DD channel estimation. We classified the detection approachesaslow-complexitylinearapproach，approximate MAP approach， and DNN approach. Low complexities possi ? ble in the linear approach due to the structure of the channel matrix make it attractive for practical implementations. The it? erative MAP approach （e.g.， message passing） is known for its goodperformanceatlowcomplexities.DNNapproachis emerging with good promise particularly when there are model deviations that are typical in practice. In the DD channel esti? mation space， we highlighted approaches based on exclusive pilot frames， embedded pilot frames， and superimposed pilot frames. More research in OTFS transceiver designs using the DNN approach can be pursued as future work.

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Biographies

Ashwitha NAIKOTI （ashwithan@iisc. ac. in） received the B. Tech. degree in electronicsandcommunicationengineeringfromtheNationalInstituteof Technology， Warangal， India in 2017. She is currently pursuing M. Tech （Re? search） degree with the Department of Electrical Communication Engineer? ing，IndianInstituteof Science（IISc），Bengaluru，India.Shewaswiththe Center forDevelopmentof Telematics ，Bengaluru，asaResearchEngineer from 2017 to 2019. Her current research interests include orthogonal time fre ? quency space modulation and transceiver design using neural networks.

AnanthanarayananCHOCKALINGAMreceivedthePh. D.degreein electrical communication engineering （ECE） from the Indian Institute of Sci? ence （IISc）， Bangalore， India. He was a post-doctoral fellow and an assistant project scientist with the Department of Electrical and Computer Engineer? ing， University of California， San Diego， USA. He was with Qualcomm， Inc.， SanDiego，USAasaStaff Engineer/Manager. Currently，heisaprofessor with the Department of ECE， IISc， Bangalore. He served as an associate edi? tor for the IEEE Transactions on Vehicular Technology， an editor for the IEEE TransactionsonWirelessCommunications，andaguesteditor for the IEEE Journal on Selected Areas in Communications and the IEEE Journal of Select? ed Topics in Signal Processing. He is an author of the book Large MIMO Sys? tems published by Cambridge University Press.