Estimation and Equalization of residual synchronization errors in CP-OTFS

In this work, we analyze the effect of residual frame timing offset and carrier frequency offset along with the presence of fractional Doppler on the bit error rate performance of OFTS system. We show in this work that channel equalization with appropriate channel estimates can compensate for the combined effect of the time-varying multipath channel (TVMC) as well as residual synchronization errors. We also present a unique time-domain channel estimation method for OTFS, which can be used along with the above-mentioned channel equalization. It is seen from the extensive Monte Carlo simulation results of the LDPC coded OTFS system that the estimation and compensation methods presented here provide necessary resilience properties to OTFS against the effects of TVMC channel and residual synchronization errors. A uniﬁed signal processing ﬂow for OTFS and OFDM is also described in this work to motivate studies on coexistence between the two as well as to encourage investigations on a seamless transition from OFDM based system to OTFS based systems for future air interface design.


I. INTRODUCTION
IMT-2020 [1], [2], aka 5G, aims to provide high spectral efficiency and reliability in high mobility scenarios, where the wireless link becomes a highly time varying multipath channel Vivek Rangamgari, Shashank Tiwari,and Suvra Sekhar Das are with G. S. Sanyal School of Telecommunications, Indian Institute of Technology, Kharagpur, India. Subhas Chandra Mondal is with Wipro Limited, Bangalore, India e-mail: rkvivek97@gmail.com, shashankpbh@gmail.com, suvra@gssst.iitkgp.ernet.in, and subhas.mondal@wipro.com. June 12, 2020 DRAFT ii (TVMC) [3], [4]. The modified avatar of orthogonal frequency division multiplexing (OFDM), namely OFDM numerology [5] is the fundamental physical layer signalling technique, which is used in 5G-New radio (5G-NR) to address this operating requirement. OFDM numerology is essentially variable subcarrier bandwidth [6] along with variable guard interval [7] as described in [8]. Such modifications not only improve the immunity of OFDM to inter-carrier interference (ICI) caused by high Doppler in high mobility conditions and phase noise at high carrier frequencies but also allow a smooth transition from 4G (IMT-Advanced [9]) to 5G with minimalistic changes in the system configuration. In a recent work by the authors [10], it is shown that orthogonal time frequency space (OTFS) [11]- [13] outperforms adaptive OFDM used in 5G-NR in such high ICI conditions.
In OTFS, delay-Doppler domain data is first converted to frequency-time domain through a unitary transform, which is then converted to time domain by OFDM modulation. A cyclic prefix (CP) can be added before a block of OFDM symbols resulting in reduced CP-OTFS, namely RCP-OTFS [14], whereas in CP-OTFS a CP is added before each OFDM symbol [15], which is considered in this work.
Although OTFS is an orthogonal modulation scheme, yet when the signal passes through a TVMC the received signal in delay-Doppler domain, encounters inter symbol interference.
To demodulate the interference affected signal, the works [16] and [17] describes a belief propagation receiver for OTFS. In [18] Markov chain Monte Carlo sampling based on the lowcomplexity OTFS signal detection scheme is presented. The work [19] describes localized search based non-linear receiver for rectangular pulse shaped OTFS. These receivers have a non-linear signal decoding structure and result in very high complexity. The work [19] also presents a linear receiver architecture however it is limited because it considers ideal pulse shape and hence is not practical. In [20] a low complexity linear receiver for rectangular orthogonal pulse shaped OTFS system is presented. In this work, we restrict ourselves to linear receiver owing to its lower complexity.
schemes such as OTFS and OFDM, which helps in paving the path for a flexible reconfigurable air interface for future air interface design.

II. NOTATIONS
We use the following notations throughout the paper. We consider x, X and x to be vectors, matrices and scalars respectively. Complex conjugate value of x is given byx whereas j = √ −1. We

III. SYSTEM MODEL
We consider a CP-OTFS system with M sub-carriers, each of ∆f Hz bandwidth, N symbols of duration T u = 1 ∆f sec. each with T CP sec. long CP. The system has bandwidth B = M ∆f Hz. and total frame duration T f = N T seconds, where T = T u + T CP .

A. Transmitter
X(m, n) is converted to a time domain signal s (t) through a Heisenberg transform as, After appending CP to the baseband signal s(t) we get, where, g(t) is transmitter pulse of duration T . In this work, we use rectangular pulse i.e. g(t) = 1 if 0 ≤ t ≤ T , and g(t) = 0, otherwise. The baseband signal s(t) is up-converted to the RF carrier frequency f c to obtain the RF signal s RF (t) = s(t)e j2πfct .
We consider a baseband time varying channel with P paths having h p complex attenuation, τ p delay and ν p Doppler value for the p th path where p ∈ Z[1 P ]. The delay-Doppler channel spreading function is written as, The delay and Doppler values for p th path is given as τ p = lp M ∆f and ν p = kp N T where l p and k p are delay and Doppler bin numbers on Doppler-delay lattice Λ for p th path. Let τ max and ν max be the maximum delay and Doppler spread respectively. Channel delay length l τ = τ max M ∆f and channel Doppler length, k ν = ν max N T . The RF equivalent channel can be given as, The received signal can be written as, The received signal after downconversion to base band is where f c = f c − δf c is the receiver carrier frequency with offset δf c . The signal r(t) is, The signal r(t) sampled at F s = B = 1/T s = (M + L)/T , where L = T CP B is the length of sampled CP and T = T u + T CP = (M + L)T s , becomes, The n th OTFS symbol with CP can be collected from the samples of received signal as r(n (M + L) + l), ∀ l = 0, 1, · · · , (M + L) − 1 and can be written as, Ts) e j2πm∆f ([l− τp Ts ]Ts) )e j2π(δfc+νp)Ts(n (M +L)+l− τp Ts ) .
We assume that, after initial coarse synchronization, a residual synchronization error of l o  Fig.1 exists. We also assume (l 0 T s + τ max ) ≤ T CP so that no interference is experienced from the neighbouring OTFS symbols. The first step towards decoding the signal is to perform discrete Fourier Transform DRAFT June 12, 2020 vii (DFT) on the CP removed samples of n th OTFS symbol to obtain the time-frequency data and can be given as, Using (7) Since, g(t) is a rectangular pulse, Then, (9) can be written as, which can be simplified to (as shown in appendix A) y(k , l ) =

IV. EFFECTS OF RESIDUAL SYNCHRONIZATION ERRORS
In this section, we describe the effects of residual FTO and CFO errors in the receiver. From (13), it may be noted that y(k , l ) experiences ISI in both delay and Doppler dimension.

A. Integer delay and integer Doppler values
Whenl p andk p are integers, then (13) simplifies as, If we consider an AWGN scenario, then integer time and frequency errors result in a cyclic shift in delay and Doppler direction respectively thus cyclically shifting the origin of delay-Doppler grid to (l o , δfc ∆ν ). It can also be observed that, (15) resembles the received delay-Doppler signal as described in equation (24) of [3]. Thus the effect of synchronization error can be considered as a part of the channel itself, with modified channel tapsh p = h p e j2πδfcτp ,τ p = τ p + l 0 T s & ν p = ν p + δf c . Thus, it may be conjectured that channel equalization with appropriate channel coefficients may be able to equalize the effects of TVMC and residual synchronization errors.

B. Integer delay and fractional Doppler values
By observing the the summation terms on the running variables l and k in (14) we can infer the following. If any Doppler or delay value of the modified channel is fractional, i.e,k p orl p / ∈ Z, then every symbol experiences interference in Doppler or delay dimension accordingly. The DRAFT June 12, 2020 ix interference in the Doppler axis is observed frequently because the Doppler values of channel are not usually resolved to integer values. For example, the imperfect compensation of CFO during initial synchronization, which results in δf c , can lead to a modified fractional Doppler value even though the actual channel Doppler values (k p ) are integers. This is becausek p = k p + δfc ∆ν .
Fractional delay values are not observed since the sampling of the received signal in time domain approximates the effect of fractional channel delay to the nearest integer time bin [3]. Thus Therefore (14) becomes, In (16), each received symbol experiences interference from all other symbols in Doppler dimension. In the sections that follow, we describe the construction of the equivalent channel matrix, its estimation and compensation of the effects of synchronization errors described here.

V. EQUIVALENT CHANNEL MATRIX FOR OTFS INCLUDING SYNCHRONIZATION ERRORS
In this section, we derive the expression of the equivalent channel matrix in time domain, which includes the effect of residual synchronization errors and time varying channel. For this, we establish the equivalent system model in matrix-vector form for CP-OTFS. Symbols d(k, l) are arranged in M × N matrix as, Delay-Doppler to frequency-time domain conversion (after the ISFFT) is done following X = is along m and time is along n. Frequency-time domain to time domain signal is obtained using The pulse shaped samples of the signal is written as, The CP appended signal is given as,  is the operator for appending CP. Thus, the transmit signal can be given as, We also introduce the transmit signal vector s = vec{S} without CP added and can be given as s = Ad by putting (18). At the receiver, the noiseless received signal in discrete form [3] can be written as, where l ∈ Z[0 ((M +L)N −1)]. We collect the samples to obtain qth OTFS symbol vector r q CP with CP. Then the CP removed vector r q can be given as, Therefore, DRAFT June 12, 2020 xi With the introduction of CFO at the receiver, the samples of qth received OTFS symbol r f q (l) is where k 0 = δfc ∆ν . With residual time synchronization error of l o samples, From (19), (21) and (22), We assume l o + l τ < L. Therefore, Then,r where The above can be modified as, wherel τ = l τ + l 0 is maximum excess delay bin value of channel. We let k ν l be the set of Therefore, the concatenation of CP removed vectors can be given as, which can be written as,r = Hs Gaussian noise vector. Equation (26) suggests that the effect of synchronization errors can be considered as part of time domain channel matrix. Equation (27) shows that the structure of the channel matrix is invariant to the introduction of synchronization errors which is an added advantage as the number of elements in the matrix does not change with residual synchronization errors and hence the sparsity of the matrix is unaltered. Next, we propose an estimation algorithm to estimate this equivalent channel matrix, however we first give a short justification for choosing

A. Pilot structure in delay-Doppler domain
We extend on the pilot structure described for OTFS in [14]. The pilot is a 2-dimensional (2D) where P P LT = N p = 2N L is the pilot power. At the receiver, the received June 12, 2020 DRAFT xiv delay-Doppler signal corresponding to pilot signal, with synchronization error is, From the above, one may infer that the transmitted 2D impulse pilot, after going through the channel, spreads over the entire Doppler axis while the spread in delay is limited to (L p +l τ ) starting from L p . We collect received signal y(k , l ), ∀k ∈ [0 N − 1], l ∈ [L p L p + L − 1] for channel estimation. This part of the received signal contains the response of the channel to the 2D-delay-Doppler impulse pilot signal, which is not-interfered by data symbols. This is because we have assumed that L ≥ l τ + l o − 1.

B. Channel Estimation
Here When IFFT of (M+L)N point is applied and it can be shown that, From (32), where spline interpolate is implemented using 'interpl1', a inbuilt function in Matlab R . Algorithm 1 describes how channel estimates for the entire time duration are generated from the received signal as well as how to estimate qth channel matricesĤ q,i &Ĥ q , which are described in (31) & (30) respectively. for i ∈ l do 18:

VII. LMMSE EQUALIZATION
In this section, we explain a low complexity LMMSE receiver for CP-OTFS based on [20].
The LMMSE equalization of r in (29) results in estimated data vectord given as, When g(t) is rectangular, A becomes unitary. Thus (37) can be written as, Thus LMMSE equalization can be performed as a two stage equalizer. In the first stage, LMMSE channel equalization is performed to obtain r ce = H eq r. Second stage is a OTFS matched filter receiver to obtaind = A † r ce . The direct implementation of r ce = H eq r requires inversion of Thus we can write, . From this, it can be concluded that the maximum shift of diagonal elements in ∆ can be ±(l τ − 1). Additionally, due to the cyclic nature of the shift, Ψ q is quasi-banded with bandwidth of 2l τ − 1. Asl τ << M , Ψ q is also sparse for typical wireless channel. Structure of Ψ q is similar to the channel matrix of RCP OTFS as described in equation (13) specified in [20]. Thus Ψ −1 q can be computed using LU factorization of Ψ q in a similar way as described in Sec. III B of [20], i.e. Ψ q = L q U q .

B. Computation ofd
After LU decomposition of Ψ q , r ce,q is simplified to, r ce, As L q is a quasibanded lower triangular matrix, r (1) q = L −1 q r q can be computed using low complexity forward substitution as explained in Algorithm 2 in [20]. Algorithm 3 of [20] can be used to r Using the definition of H q , r ce,q =H † q r (2) q can be written as, r ce,q = P p=1h p ∆ −kp Π −lp r (2) q circular shift . To compute r ce,q , r (2) q is first circularly shifted by '−l p ' and then multiplied byh p diag{∆ −kp } using point-to-point multiplication for each path p. All vectors obtained above are summed to obtain r ce,q . Then, {r ce,q } N −1 q=0 are concatenated to obtain r ce . Finally,d = A † r ce can be implemented using M number of N -point FFTs (Sec. III-C, [20]).

C. Computation complexity
With some effort it can be shown that the number of CMs required to implement our proposed The order of complexity achieved through our receiver is M N log(M N ), which is significantly lower than the direct implementation, which is of the order of M 3 N 3 . Our proposed receiver requires around 10 7 x lower CMs than the direct implementation following (37), if we consider a typical OTFS system with ∆f = 15 KHz, f c = 4 GHz, N = 128, M = 512, speed of 500 kmph and the extended vehicular A (EVA) 3GPP channel model [26] with P = 9 and τ max = 2.51 µ sec.

VIII. UNIFIED FRAMEWORK FOR ORTHOGONAL MULTICARRIER SYSTEMS
In this section, we describe a generalized framework for orthogonal waveforms. Modulation techniques following this framework will be able to take advantage of the channel estimation and equalization algorithms proposed earlier. Let D be the data matrix of size (M ) × N . Then the transmit signal can be of the form where B and C are modulation specific matrices. A in (37) can be computed using A = (C ⊗ I)(I ⊗ B). We define S p as the constant pilot matrix of size 2L × N as The combined time domain matrix with N symbols each containing M samples can be given

IX. RESULTS
In this section, we present low density parity check (LDPC) coded performance of CP-OTFS system with residual FTO and CFO errors. Since we present a unified model for constructing OTFS signal as well as OFDM signal, therefore we also present the performance of an equivalent OFDM system. The simulation parameters are mentioned in Table I. For each channel delay tap value, Doppler is generated using Jake's formula, ν p = ν max cos(θ p ), where θ p is uniformly distributed over [−π π]. CP length is chosen longer than τ max of the TVMC.

A. BLER Performance
We begin with the block error rate (BLER) performance of CP-OTFS system through Let us first consider the 'Ideal' performance . It is observed that as dpt increases, the performance of the system improves. At BLER of 10 −1 , when dpt increases from 1 to 2 the performance improved by 4.5 dB. The improvement is 2 dB when dpt changes from 2 to 3. At BLER of 10 −2 , the SNR gain is approximately 6 dB and 2.5 dB. This gain in performance with increasing value of dpt can be attributed to Doppler diversity. With increasing dpt, the number of independent channel paths increase. Thus resulting in increased channel diversity, which is extracted by the LMMSE receiver.
Next, we consider the performance of CP-OTFS with the proposed channel estimation algo-DRAFT June 12, 2020 xxiii rithm but without residual synchronization error, i.e. legends 'Ch est but without l 0 and k o . The degradation in performance from 'Ideal' is limited to approximately 1 dB for all considered dpt. Now we turn our attention to the performance of OTFS system with synchronization errors while using the estimation and compensation techniques described above. The values of l o is 2 samples. Normalized cfo error k 0 = δfc ∆ν = 20 results in δf c = k 0 ∆ν = 2.33 KHz. We find that at BLER of 10 −1 the performance degradation is less than 1 dB when compared to the situation when such residual synchronization errors are not present. However, when compared with ideal performance at the BLER threshold of 10 −2 , the maximum SNR degradation is about 1.5, which is observed for dpt=3. Therefore, it may be stated that the performance of CP-OTFS while using the proposed estimation and compensation techniques for channel and residual synchronization errors is within acceptable limits.
Now we present the performance of OFDM system, which is described in Section VIII, in Fig.   4. We find that almost similar degradation in the performance of coded OFDM system is observed under identical test conditions. However, when we compare the ideal performance for dpt=1 of CP-OTFS and OFDM from figures 3 & 4 respectively, we find that CP-OTFS outperforms OFDM by approximately 3 dB. When dpt=2, the gap is about 4 dB. At dpt=3, the gap increases further.
It can be noted from the figures that OFDM's performance at BLER of 10 −2 would only be significantly worse. The results with channel estimation and synchronization errors also bring out the fact that OTFS is significantly more reliable than OFDM in a TVMC. From the above discussions, it can be said that the proposed algorithms can provide sufficient resilience to OTFS against synchronization errors while compensating for TVMC, thus making OTFS a potential transmission technology candidate for use, especially in high mobility scenarios.
Having exposed the most important performance metrics of OTFS under TVMC and residual synchronization errors, we now take a look at the mean square error (MSE) of the channel estimates for the strongest tap against varying SNR for different dpt shown in Fig. 5. It can be observed that the MSE increases with dpt. Thus, one can infer that the channel coefficients June 12, 2020 DRAFT xxiv obtained from the interpolation based channel estimation deviate from actual channel coefficients as dpt increases. This is reflected in the SNR degradation observed in Fig. 3. It can also be observed that MSE saturates even when SNR increases. This indicates that the proposed estimation and compensation methods are effective only in the mid-SNR region. One may wonder that the saturation in MSE should be observed in the BLER curves as well. This is indeed true, however, the saturation in BLER occurs below the level of 10 −3 (100 times less compared to BLER 10 −1 ), which is not captured in the figure.

B. Limit of CFO tolerance
Since we are concerned with the effect of residual synchronization errors, of which the CFO has more significant effect on the received signal, we present the BLER performance against normalized CFO ( δfc ∆ν ) for different dpt at SNR of 15 dB in Fig. 6 In this work, we have described the system model of a rectangular pulse shaped CP-OTFS system with residual synchronization errors. We exposed that integer time and frequency errors DRAFT June 12, 2020 xxvii result in a cyclic shift in delay and Doppler dimensions. We also show that fractional Doppler or delay causes interference in Doppler or delay dimension respectively. We have brought out that the effect of synchronization errors can be considered as a part of the channel itself, however with modified channel taps. Initially we began with the conjecture that channel equalization with appropriate channel coefficients may compensate for the effects of synchronization errors along and TVMC. Accordingly we presented methods to estimate and equalize this effective channel matrix and the conjecture was finally shown to be valid. Using the estimates of the effective channel matrix, we derive a low complexity LMMSE receiver to cancel the interference, which grows log-linearly and has about 10 7 times lower complexity than the direct implementation.
With the proposed method, the maximum SNR loss at BLER 10 −2 is around 1 dB in the absence of residual synchronization errors. It is also shown that the proposed compensation method shows promise to compensate for residual CFO of up to 15% of subcarrier bandwidth or 0.4 ppm of carrier frequency with a loss in SNR up to 1.5 dB at BLER 10 −2 . We have shown that OTFS with practical channel estimation and residual synchronization errors significantly outperforms OFDM (3 dB or more) especially when such errors are high as encountered in high ICI conditions. Considering that OFDM with frequency-domain adaptive modulation and coding is known for providing high spectral efficiency especially in low mobility conditions and that OTFS provides much superior reliability in high mobility conditions, the unified framework for representing both modulations presented in this work can help pave the path for a flexible and reconfigurable future air interface.