Time_and_Frequency_Synchronization_for_OTFS.pdf

1
Time and Frequency Synchronization for OTFS
Mohsen Bayat, Arman Farhang
Abstract—In this letter, we propose timing offset (TO) and
carrier frequency offset (CFO) estimators for orthogonal time
frequency space modulation (OTFS). The proposed estimators
do not require any additional training overheads as they deploy
the same pilot signal that is used for channel estimation. Hence,
no additional training overhead is required for synchronization,
thanks to the periodic properties of the pilot signal in the delay-
time domain. Our proposed TO estimator finds the start of each
OTFS block by searching for a periodic sequence in delay and
time dimensions. The CFO is then estimated by finding the mean
of the angles of a two-dimensional correlation function at the
best timing instant. As a novel aspect of these estimators, the
multipath diversity of the channel is exploited to achieve a high
estimation accuracy. Finally, we show the efficacy of our proposed
synchronization techniques through extensive simulations.
Index Terms—OTFS, synchronization, timing offset estimation,
carrier frequency offset estimation.
I. INTRODUCTION
Due to its attractive properties, orthogonal time frequency
space modulation (OTFS), [1], has become a strong candidate
waveform for the sixth generation wireless networks (6G).
These properties include resilience to the time-varying channel
effects, backward compatibility with previous generation wire-
less systems and joint radar and communication capabilities,
[2]. Synchronization is a challenge in the design of any prac-
tical communication system, [3], specially when the channel
is time-varying, [4]. Due to their close relationship, OTFS
inherits sensitivity to synchronization errors from orthogonal
frequency division multiplexing (OFDM). While there exists a
large body of work on the synchronization aspects of OFDM,
[5]–[8], OTFS literature on this topic is very limited [9]–[11].
In particular, there is no work on joint timing offset (TO) and
carrier frequency offset (CFO) estimation for OTFS.
As the initial work, the authors in [9] designed a random
access preamble and developed a TO estimation method for
the uplink of OTFS. In this method, the TO estimates need to
be fed back to the users as the timing reference for the next
uplink transmission. However, these TO reports may become
outdated. Thus, the authors in [10], proposed a preamble-based
TO and cell identity estimation method in downlink. In this
method, a linear frequency modulated (LFM) signal is utilized
as a preamble. Both solutions in [9] and [10] are limited
to time synchronization and the only work on the frequency
synchronization of OTFS can be found in [11]. In this work,
the CFO effect is considered as a part of the channel and a joint
CFO and channel estimation method using an isolated pilot in
the delay-Doppler domain is proposed. The drawback of this
technique is that CFO cannot be separated from the channel
Mohsen Bayat and Arman Farhang are with the Department of Electronic
and Electrical Engineering, Trinity College Dublin, Dublin 2, Ireland (Email:
{bayatm,arman.farhang}@tcd.ie). This publication has emanated from re-
search conducted with the financial support of Science Foundation Ireland
under Grant number 19/FFP/7005(T). For the purpose of Open Access, the
authors have applied a CC BY public copyright licence to any Author
Accepted Manuscript version arising from this submission.
estimate. Consequently, the users cannot pre-compensate the
CFO before the uplink transmission, [8]. Based on the above,
the existing synchronization techniques for OTFS are limited
to only TO or CFO estimation while they may require the
extra preamble overhead.
Hence, in this letter, we propose TO and CFO estimation
techniques for OTFS. Our proposed techniques do not require
any additional training overhead as the same pilot that is often
utilized for channel estimation in OTFS literature, [12], is
also deployed for synchronization. Additionally, for a further
reduced overhead, similar to [13], we only use one cyclic
prefix (CP) at the beginning of each OTFS block instead
of using multiple CPs within each block. The isolated pilot
in the delay-Doppler domain that is utilized for channel
estimation has periodic properties in the delay-time domain.
Being inspired by OFDM literature on synchronization, [5]–
[8], we exploit the periodic structure of the pilot signal and
develop TO and CFO estimation techniques for OTFS.
The delay-time domain pilot signal has a periodic structure
on a given delay bin, i.e., a row on the delay-time grid where
the pilot is located. Hence, the start of each OTFS block can be
found by searching for this periodic pilot signal on the delay-
time grid. To this end, in our proposed TO estimation tech-
nique, we form a two dimensional (2D) correlation function to
find the TO in delay and time dimensions. The proposed TO
estimation technique is attractive to completely asynchronous
users as it does not have any acquisition range limitations.
Finally, we obtain the CFO by finding the angle of the 2D
correlation function at the estimated timing instant. As a novel
contribution of this letter, we exploit the multipath diversity
of the channel that significantly improves the accuracy of both
TO and CFO estimates. To corroborate our claims, we analyze
the performance of our proposed TO and CFO estimation
techniques through simulations. In our simulations, we study
the mean and variance of the TO estimation error and the
mean square error (MSE) of the CFO estimates. We also
analyze the bit error rate (BER) performance of our proposed
synchronization techniques. In this analysis, we show that our
proposed TO and CFO estimation techniques lead to the same
performance as a fully synchronous system even when the
channel is highly time-varying.
II. OTFS PRINCIPLES
OTFS signal at baseband is generated by translating the
delay-Doppler domain data symbols to the delay-time domain.
This process can be implemented by taking inverse discrete
Fourier transform (IDFT) across the Doppler dimension at
different delay bins, [14], [15], i.e., the rows on the delay-
Doppler grid, see Fig. 1. Considering M delay and N Doppler
bins, and the corresponding delay-Doppler domain quadra-
ture amplitude modulated (QAM) data symbols, D[m, n] for
m = 0, . . . , M − 1 and n = 0, . . . , N − 1 , the signal
samples on the delay-time grid can be obtained as X[m, l] =
This article has been accepted for publication in IEEE Wireless Communications Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LWC.2022.3214002
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

2
Fig. 1. OTFS signal as a combination of interleaved OFDM signals.
1
√
N
PN−1
n=0 D[m, n]e
j2πln
N where l = 0, . . . , N − 1 is the time
and m is the delay index. To form the OTFS transmit signal,
these samples need to be converted to a serial stream. This
is done by concatenating the samples on the columns of the
delay-time grid through a parallel-to-serial converter block,
i.e., P/S in Fig. 1. Thus, the resulting signal, x[k], takes the
values X[m, l] for k = Ml + m. From Fig. 1, the OTFS
signal before appending the CP can be interpreted as the
combination of M up-sampled and delayed OFDM signals,
each with N Doppler domain subcarriers. Finally, a CP with
the length LCP is appended at the beginning of each OTFS
block with MN samples to form the transmit signal s[k].
LCP ≥ L − 1 is chosen to guarantee inter-block interference-
free communication where L is the channel length.
The received signal in presence of TO and CFO after
transmission over the linear time-varying (LTV) channel can
be expressed as
r[k] = e
j2πεk
MN
B−1
X
i=0
L−1
X
ℓ=0
h[ℓ, k]s[k − ℓ − θ − iNT] + η[k], (1)
where η[k] ∼ CN(0, σ2
η) is the complex additive white Gaus-
sian noise (AWGN) with the variance σ2
η, NT =MN +LCP is
the OTFS block length, and B is the number of OTFS blocks in
each data frame. The variables θ and ε are the normalized TO
and CFO values to the delay and Doppler spacings ∆τ = Ts
and ∆ν = 1
MNTs
, respectively, for a given sampling period Ts.
h[ℓ, k] =
PΓ−1
i=0 αiej2π κmax
MN (k−ℓ) cos ψi
δ[ℓ − ℓi] is the channel
response at the delay-tap ℓ and sample k where Γ, κmax, ψi αi,
and ℓi are the total number of paths, maximum Doppler shift
normalized to the Doppler spacing, the arrival angle, gains and
delays corresponding to a given path i, respectively.
III. PROPOSED TO ESTIMATION TECHNIQUE
In this section, we propose a TO estimation technique for
OTFS that exploits the periodic properties of the commonly
deployed pilot for channel estimation, [12]. Thus, no additional
signaling overhead is required for synchronization. Further-
more, our proposed technique does not have any estimation
range limitations. Based on the 2D structure of the OTFS block
in delay-time domain, we decompose the TO as θ = θd +Mθt
where θd and θt represent the TO in delay and time, re-
spectively. Then, in the following, we develop a two-stage
technique to estimate these offsets.
A. TO estimation in delay dimension
According to the explanations in Section II and Fig. 1, the
rows of the delay-time grid are composed of OFDM signals,
each with N Doppler domain subcarriers. On this basis, one
may think of adapting OFDM synchronization techniques to
OTFS. In pilot patterns (a) and (b), shown in Fig. 2, an
impulsive pilot at the delay-Doppler bin (mp, np) with the
power ρp is surrounded by zero guard symbols. These pilot
patterns are the same as the ones that are used for channel
estimation in OTFS literature, [12]. The guard symbols of
2L − 1 and 4(κmax + κ̂) + 1 along the delay and Doppler
dimensions, respectively, are required to avoid interference
between the pilot and data symbols caused by the channel,
where κ̂ is a design parameter, [12]. While our proposed
synchronization techniques in this paper work with both pilot
patterns, in the following, we consider the pilot pattern (a) for
the ease of explanations.
Taking the isolated pilot from delay-Doppler to delay-time
domain leads to a constant amplitude sequence on the row mp
of the delay-time grid with the linear phase e
j2πnpl
N for l =
0, . . . , N − 1, i.e.,
p
ρp/N[1, e
j2πnp
N , . . . , e
j2πnp(N−1)
N ]. This
sequence can be split into two halves with the same amplitude
and a constant phase difference of πnp. Thus, the Schmidl and
Cox (S&C) method, [5], can be adapted to OTFS. The main
assumption of the S&C method is that the two halves of pilot
signal remain identical at the channel output. However, this
assumption does not hold in LTV channels, as the channel
deteriorates the similarity of the two pilot halves. To tackle
this issue, the identical parts of the pilot can be brought closer
to one another where the time-selective channel only slightly
affects the similarity of the adjacent parts. This is inline with
the idea of increasing the number of the repetitive parts of the
pilot that was proposed to address the limitations of the S&C
method in OFDM, [6]. The extreme case for this is when all
the pilot samples are the same. Since our pilot has a constant
amplitude and a constant phase difference of 2πnp/N between
the adjacent samples, this extreme case applies to it. Hence,
we have the opportunity to develop a TO estimation technique
robust to the time variations of the channel for OTFS.
To estimate the TO, we need to search for the pilot sequence
on a row of the delay-time grid. Thus, we convert the received
signal from serial to parallel, with blocks of M samples in each
parallel stream that represent the samples on the columns of
the grid. Consequently, we rearrange the received signal as
r[m, l] = r[Ml + m] with the delay and time indices m and
l, respectively. For a given row m on the delay-time grid, we
consider a sliding window with length N that searches for the
pilot sequence with N identical samples. This process can be

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Fig. 2. Pilot patterns in the delay-Doppler domain [12].
implemented by forming the correlation function
P[m, l] =
N−2
X
q=0
r∗
[m, l + q]r[m, l + q + 1], (2)
whose samples can be iteratively calculated as P[m, l + 1] =
P[m, l]−r∗
[m, l]r[m, l+1]+r∗
[m, l+N−1]r[m, l+N], where
m=0, . . . , M −1 and l=0, . . . , N −1. Hence, considering the
CP and pilot position in delay, θd can be estimated by finding
the peak of the timing metric Pd[m]=
PN−1
l=0 P[m, l] as
θ̂d = arg max
m
{|Pd[m]|} − mp − LCP. (3)
If θ < M is guaranteed, the TO estimate in (3) is accurate.
However, when θ ≥ M an additional TO estimation step in
time dimension is also required.
B. TO estimation in time dimension
As a single estimation error in time leads to the large error
of M samples, accurate estimation of θt is of a paramount
importance. The peak of the correlation function in (2) on
the row m′
p = θ̂d + mp + LCP of the delay-time grid can
provide an estimate of θt. However, as it will be shown in
Section V, this estimate may not be accurate. This is because
the time-selective channel may distort the pilot samples on
the edges of the row m′
p on the delay-time grid and result in
estimation error. Therefore, as a novel aspect of our proposed
TO estimation technique, we exploit the multipath diversity of
the channel to improve the estimation accuracy. In particular,
multipath effect creates independent copies of the pilot signal
at the delay bins m = m′
p + ℓ for ℓ = 0, . . . , L − 1, that can
provide diversity gains in estimating θt. Hence, assuming that
θ̂d is accurate, θt can be estimated by finding the peak of the
timing metric Pt[l] =
Pm′
p+L−1
m=m′
p
P[m, l]1
as
θ̂t = arg max
l
{|Pt[l]|}. (4)
To gain a deeper insight into the mechanism of exploiting
multipath diversity in (4), we substitute the received pilot
samples, rp[m′
p+ℓ, l]=e
j2πε(Ml+m′
p+ℓ)
MN h[ℓ, Ml+m′
p]X[mp, l−
θt] + η[m′
p + ℓ, l] for ℓ = 0, . . . , L − 1, into the correlation
function (2). Thus, at the exact timing estimate, P[m′
p+ℓ, θt]=
PN+θt−2
q=θt
r∗
p[m′
p + ℓ, q]rp[m′
p + ℓ, q + 1] can be expanded as
P[m′
p +ℓ, θt]=Apejϕp
N+θt−2
X
q=θt
Γ−1
X
i=0
|αi|2
ejφq
i,i δ[ℓ−ℓi] +
Γ−1
X
i,i′
=0,i̸=i′
α∗
i αi′ e
jφq
i,i′
δℓi,ℓi′ + η′
[m′
p + ℓ, q]
!
, (5)
1If m ≥ M due to the pilot position mp, the delay and time indices in
the timing metric will change to (m mod M) and (l + ⌊ m
M
⌋), respectively.
where η′
[m′
p + ℓ, q] = η∗
[m′
p + ℓ, q]η[m′
p + ℓ, q + 1], φq
i,i′ =
2π κmax
N ((cos ψi′ −cos ψi)(q +
m′
p−ℓ
M )+cos ψi′ ), δℓi,ℓi′ =δ[ℓ−
ℓi]δ[ℓ−ℓi′ ], and Ap =
|D[mp,np]|2
N and ϕp =
2π(ε+np)
N represent
the amplitude squared and the CFO affected phase of the pilot
in delay-time, respectively. Since the noise samples and the
path gains are independent and identically distributed random
variables, taking expected value from (5), we have
E{P[m′
p +ℓ, θt]} = (N − 1)Apejϕp
ρ[ℓ]J0(2π
κmax
N
), (6)
where E{
PΓ−1
i=0
|αi|2
δ[ℓ−ℓi]} = ρ[ℓ] is the total power that
is received at delay tap ℓ, Ei̸=i′ {α∗
i αi′ } = 0, and E{η′
[m′
p +
ℓ, q]} = 0. According to the Jakes’ channel model [16], ψ is
uniformly distributed in [−π, π) and thus E{ej2π κmax
N cos ψ
} =
1
2π
R π
−π
ej2π κmax
N cos ψ
dψ = J0(2π κmax
N ), where J0(.) de-
notes zeroth-order Bessel function of the first kind. Finally,
from (6) and (4), one may realize that E{|Pt[θt]|} =
(N−1
N )|D[mp, np]|2
|J0(2π κmax
N )| as
PL−1
ℓ=0 ρ[ℓ] = 1. Hence,
by combining the received pilot signals at the delay bins m′
p+ℓ
for ℓ = 0, . . . , L − 1, using (4), we can collect the maximum
received pilot energy and achieve multipath diversity gain.
Additionally, from (3) and the above discussion, it is obvious
that neither the CFO nor the pilot position in delay-Doppler
plane have any effect on the TO estimates.
Fig. 3 depicts a snapshot of the timing metrics Pd[m] and
Pt[l] at the SNR of 10 dB for an OTFS system with M = 128
and N = 32 delay and Doppler bins, respectively, for both LTI
and LTV channels. After timing recovery of the received signal
with the estimated TO, θ̂ = θ̂d + Mθ̂t, a residual timing error
∆θ = θ − θ̂ may still remain in r̂[k] = r[k + θ̂d + Mθ̂t
]. In
Section V, we numerically evaluate the performance of our
proposed TO estimation technique and show that the mean
and variance of ∆θ are very small that can be easily absorbed
into the CP. The next step after timing acquisition is frequency
synchronization. Thus, in the following Section, we develop a
CFO estimation technique for OTFS.
IV. PROPOSED CFO ESTIMATION TECHNIQUE
As it was mentioned earlier, the phase difference between
the pilot samples in the delay-time domain is 2πnp/N. When
the channel is LTI over each OTFS block, the delay-time
pilot samples are affected by the same channel. Thus, the
angle of P[mp, l]/e
j2πnp
N at the best timing instant l = θ̂t
in the absence of the noise is equal to b
ϕ = 2πb
ε
N . This
leads to a reliable CFO estimate. However, when the channel
is LTV, the adjacent pilot samples are affected by different
channel coefficients that result in an inaccurate CFO estimate.
To tackle this issue, similar to our proposed TO estimation
technique in Section III, we exploit the multipath diversity of
the channel. Based on our simulation results in Section V, this
brings over an order of magnitude higher estimation accuracy
than the case where multipath diversity is not considered.
Hence, we can find the mean of the angles of the phase-
corrected correlation functions P[m, θ̂t]/e
j2πnp
N at the delay
bins m = m′
p, . . . , m′
p + L − 1 and estimate the CFO as
b
ε =
N
2πL
m′
p+L−1
X
m=m′
p
∠P[m, θ̂t]
!
− np, (7)

4
-64 -32 0 30 64
Delay
0
500
1000
1500
max
T=0
max
T=1.46
-16 -8 -2 8 16
Time
1000
1800
2600
3400
max
T=0
max
T=1.46
Fig. 3. A snapshot of the timing metrics Pd[m]
and Pt[l] at SNR=10 dB for M=128 and N=32.
0 4 8 12 16 20 24 28
SNR (dB)
10
0
10
1
10
2
Mean
(Sample)
w/o multipath diversity
w/ multipath diversity
Pilot pattern (a)
Pilot pattern (b)
0 4 8 12 16 20 24 28
SNR (dB)
10
0
10
1
10
2
Variance
(Sample
2
)
Pilot pattern (a)
Pilot pattern (b)
3
1.1
Fig. 4. Mean and variance of the proposed TO esti-
mator for M=128 and N=32 and νmaxT ≈1.46.
0.35 0.874 1.165 1.457
max
T
1
2
3
4
5
6
Variance
(Sample
2
)
/
Mean
(Sample)
M=256 & N=16
M=128 & N=32
M=64 & N=64
Variance
Mean
Fig. 5. Mean and variance of the proposed TO
estimator at SNR=20 dB for pilot pattern (b).
where ∠A represents the angle of the complex argument A.
Therefore, the TO and CFO corrected signal can be obtained
as y[k] = e− j2πε̂k
MN r[k + Mθ̂t + θ̂d].
Based on the results of [7], in LTI channels, for a pilot
with I identical parts, the CFO acquisition range is [−I
2 , I
2 ).
Consequently, as the pilot for OTFS has N identical samples,
the acquisition range for our proposed technique is [−N
2 , N
2 )
when the channel is LTI. However, Doppler spread in LTV
channels imposes an ambiguity at the edges of the range.
Hence, considering the Doppler effect, the acquisition range
for reliable CFO estimation becomes [−N−κmax
2 , N−κmax
2 ).
The estimated CFO in (7) can be considered as the normal-
ized CFO that is contaminated by the phase variations of the
LTV channel. Considering the guard symbols around the pilot,
ignoring the noise and using the equations (1), (2), and (7),
the CFO with estimation errors can be represented as
b
ε = ε +
N
2πL
m′
p+L−1
X
m=m′
p
∠C[m], (8)
where C[m] =
PN−2
q=0 h∗
[m−m′
p, Mq+m′
p]h[m−m′
p, M(q+
1) + m′
p]. In Section V, we numerically evaluate the perfor-
mance of our proposed CFO estimation technique. We show
that both the CFO and TO estimation errors can be absorbed
into the channel and compensated at the equalization stage.
Even though in our developments, we only considered pilot
pattern (a), in the following section, we show that our proposed
synchronization techniques provide a satisfactory performance
when any of the pilot patterns shown in Fig. 3 are deployed.
V. SIMULATION RESULTS
In this section, we numerically analyze the performance of
our proposed TO and CFO estimation techniques. We consider
an OTFS system with M = 128 and N = 32 delay and
Doppler bins, respectively, unless otherwise is stated. One
CP with the duration longer than the channel is appended
at the beginning of each OTFS block. We use the extended
vehicular A (EVA) channel model, [17], the bandwidth of
7.68 MHz and the delay-Doppler resolution (∆τ, ∆ν) =
(130.21 nsec, 1.875 kHz). When pilot pattern (a) is deployed,
the power of the surrounding zero guard symbols are allocated
to the pilot. This results in an increased pilot power by the
factor of (2L − 1)N compared to the data symbols. As it is
shown in [12], for pilot pattern (b), allocating the power of the
guard symbols to the pilot does not lead to accurate channel
estimates. Thus, similar to [12], we set the pilot power 60 dB
higher than data symbols and we choose κ̂ = 2. Throughout
our simulations, the normalized TO and CFO values are
randomly generated from a uniform distribution in the range
[−MN
2 , MN
2 ) and [−N−κmax
2 , N−κmax
2 ), respectively.
In Fig. 4, we analyze the performance of our proposed
TO estimation technique. We study the estimation error mean
and variance as a function of signal to noise ratio (SNR),
for the normalized Doppler spread of νmaxT ≈ 1.46. The
results in Fig. 4 show that our proposed technique leads to
very small estimation error at SNRs above 8 dB for both pilot
patterns. This estimation error originates from the fact that the
correlation function in (6) at the correct timing depends on the
PDP, ρ[ℓ]. Hence, the delay tap ℓ from which the maximum
power is received determines θ̂d which is not necessarily the
first tap. This error can be absorbed into the CP given that
LCP ≥ L − 1 + ∆θ which will appear as a phase factor in the
channel estimate. The higher estimation accuracy at low SNRs
when pilot (b) is deployed is due to its higher power than pilot
(a). Fig. 5 shows the behaviour of our proposed TO estimator
as the normalized Doppler spread increases for different values
of M and N and a fixed OTFS block length. An interesting
observation here is the improved estimation accuracy as the
Doppler spread of the channel increases. This is due to the
diversity that is provided by the time-selective channel.
To solely analyze the performance of our proposed CFO
estimator, we assume perfect knowledge of TO for the results
provided in Figs. 6 and 7. In Fig. 6, we evaluate the MSE
performance of our proposed CFO estimation technique as a
function of SNR for two different values of Doppler spread
and the two pilot patterns. As discussed in Section IV,
CFO estimation accuracy can be substantially improved by
exploiting the multipath diversity. Hence, in Fig 6, we observe
that the accuracy of the CFO estimates can be improved
by over an order of magnitude by exploiting the multipath
diversity compared with the single-path case. The error floor
that is observed in Fig. 6 is due to the Doppler spread of
the LTV channel. As it is shown in Fig. 8, this error can
be estimated as a part of the channel and compensated at
the equalization stage. Similar to the TO estimation case, the
superior performance of the pilot pattern (b) to pilot pattern
(a) in SNRs below 20 dB is due to its higher power. In Fig. 7,
we study the performance of our proposed CFO estimator
as a function of Doppler spread for different pairs of M

5
0 4 8 12 16 20 24 28
SNR (dB)
10-2
10
-1
10
0
101
max
T=1.46
max
T=0.87
0 4 8 12 16 20 24 28
SNR (dB)
10
-2
10
-1
10
0
max
T=1.46
max
T=0.87
Pilot Pattern (a) Pilot Pattern (b)
Fig. 6. MSE performance of the proposed CFO
estimation technique for M=128 and N=32.
0.35 0.874 1.165 1.457
max
T
10
-3
10
-2
M=256 & N=16
M=128 & N=32
M=64 & N=64
Fig. 7. MSE vs. Doppler spread of the proposed
CFO estimator at SNR=20 dB for pilot pattern (b).
0 4 8 12 16 20 24 28
SNR (dB)
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
BER
Channel estimation with our proposed synchronization tech-
nique
Channel estimation with perfect synchronization
Perfect knowledge of the channel with perfect synchronizat-
ion
Fig. 8. BER of the proposed technique for M=
128, N=32 and νmaxT≈1.46.
and N, and a fixed OTFS block length. Our results show
that the MSE performance degrades as the Doppler spread
increases. To tackle this issue, one may choose to reduce M
and thus alleviate the time-varying effects on the pilot samples.
However, while reducing M leads to lower values of ∠C[m]
in equation (8), it results in a larger value of N that amplifies
the MSE. Hence, we observe about the same performance for
different pairs of M and N in Fig. 7.
Finally, in Fig. 8, we assess the overall BER performance
of our proposed TO and CFO estimation techniques. We
deploy the channel estimation method in [12] using the
pilot pattern (b) that is more bandwidth efficient than pilot
pattern (a). For channel equalization, we utilize the least
squares minimum residual based technique with interference
cancellation (LSMR-IC) in [18]. We set the case where the
channel is estimated and we have perfect synchronization as
a benchmark. As shown in Fig. 8, the BER performance of
our proposed synchronization technique perfectly matches that
of the benchmark. This proves the efficacy of our proposed
technique. Fig. 8 shows that the small gap between the
BER performance of our proposed technique, and a fully
synchronous system with the perfect knowledge of the channel
originates from the channel estimation error.
VI. CONCLUSION
In this letter, we proposed TO and CFO estimation tech-
niques for OTFS. We showed that the periodic properties of
the pilot signal that is used for channel estimation can be
used for synchronization. Our proposed TO estimator forms
a 2D correlation function and finds the start of each OTFS
block by searching for a periodic sequence in delay and time
dimensions. While a small TO estimation error in delay can
be tolerated by using a slightly longer CP, a single estimation
error in time leads to total misalignment of the received
OTFS block. We addressed this issue and improved the TO
estimation accuracy by exploiting the multipath diversity of
the channel. We estimated the CFO by finding the angle of
the aforementioned 2D correlation function at the best timing
instant while considering the channel multipath diversity.
Finally, we analyzed our proposed estimators in terms of
the mean and variance of the TO estimation error and the
MSE of the CFO estimates through simulations. We assessed
the BER performance of our synchronization techniques and
showed that our proposed estimators can provide the same
BER performance as that of a fully synchronous system.
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