This article has been accepted for publication in IEEE Transactions on Wireless Communications. 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/TWC.2022.3215991
1
2D Off-grid Decomposition and SBL Combination
for OTFS Channel Estimation
Qianli Wang, Yu Liang, Zhengquan Zhang, Pingzhi Fan Fellow, IEEE,
Abstract—Orthogonal time-frequency space (OTFS) is a
promising technique for high mobility wireless communications,
as it approaches a sparse representation of doubly-selective
fading channel and can obtain delay-Doppler (DD) diversity
gain. However, in practical OTFS systems, fractional channel
parameter is a great challenge and will seriously deteriorates
the sparsity and the performance of channel estimation. In this
paper, we propose a novel channel estimation scheme for doubly
selective channel in the presence of both fractional Doppler and
fractional delay. First, we analyze the input-output relationship
of single-input single-output (SISO) OTFS system based on the
integer sampling DD grid, then extend it to the fractional DD
grid. Compared with the integer DD domain, the model in the
fractional DD domain is sparser and more accurate. Based on
our analysis, two kinds of off-grid models, i.e., doubly fractional
model and mixed one- and two-dimensional (1&2D) fractional
models are proposed and compared. The mixed 1&2D fractional
model has an advantage of low complexity, but encounters a
tandem off-grid distortion. To tackle this distortion problem, as
well as to balance the accuracy and the computational workload,
a novel two dimensional off-grid decomposition and combination
scheme is proposed based on the doubly and mixed 1&2D
fractional model. Simulation results demonstrate the effectiveness
and superiority of the proposed channel estimation schemes,
compared with the existing work for OTFS systems under doubly
selective channels.
I. I NTRODUCTION
New generation of wireless communication is expected to
support reliable communications in high-mobility scenarios up
to 500 km/h, such as on high speed trains [1], [2], vehicle-toeverything (V2X) [3], [4], as well as unmanned aerial vehicle
(UAV) [5], [6]. To satisfy the requirements for high mobility, orthogonal frequency division multiplexing (OFDM)based 5G cellular systems have made some enhancements
such as additional demodulation reference signal (DMRS)
design. However, 5G still regards high mobility as a negative
factor and suffers from serious performance loss in high
mobility scenario. However, high mobility results in serious
Doppler shift, which renders the wideband wireless channel
time as well as frequency-selective, i.e. doubly-selective [7].
The conventional time-frequency (TF) domain representation
of doubly-selective channel is time-varying, which is related
to the velocity. For a certain carrier’s frequency, the higher the
velocity, the faster the time-varying. As a result, features of
the wireless channel becomes difficult to extract owing to the
Authors’ addresses: All of the authors are with the School of Information
Science and Technology, Southwest Jiaotong University, Chengdu, China.
Email: qlwang@swjtu.edu.cn. This work was funded by Fundamental Research Funds for the Central Universities(2682021CX048), Natural Science
Foundation of Sichuan Province (2022NSFSC0957), and was supported in
part by NSFC (62020106001, 61871334).
parameters are non-stationary. Therefore, it becomes important
to explore techniques and methods to overcome the challenge
of the doubly-selective channels.
Some pioneers, Liu et al. [8] and Bajwa et al. [9], realize
that the doubly-selective channels for high mobility wireless
communications systems tend to exhibit a sparse multi-path
structure in delay-Doppler (DD) domain. Recently, a novel
two dimensional (2D) modulation scheme in the DD domain,
called Orthogonal time-frequency space (OTFS), is proposed
by Hadani et al [10], [11]. The OTFS modulator spreads
each information symbol over a 2D orthogonal basis function.
The information symbols are spanned across the entire TF
frame to go through same fading from the perspective of
the DD domain, leading to performance gain [12]. In the
DD domain, the Doppler shifts caused by high-mobility are
regarded as statistical stable, thus the time-varying channels in
the time-frequency domain could be converted into the timeinvariant channels in the delay-Doppler domain. This change
of representation domain brings not only the statistic stability,
but also compact representation of channel parameters and
sparsity. A few principal components in the DD domain
could draw the main feature of the channel, which leads to
a significant reduction in the channel’s dimensionality [13].
Furthermore, the compact representation of parameter and
sparsity also provide extra possibility to reduce pilot overhead
[14] or even to cancel the pilot guard [15], [16]. However,
these advantages require accurate channel state information
(CSI). Hence, accurate CSI estimation plays a central role in
realizing the potential gain promised by the OTFS system [17].
Early contributions proposed impulse-based CSI estimation
scheme for single-input single-output (SISO) OTFS system
[10]. A single impulse pilot is used to test the channel impulse
response (CIR). The channel parameters are directly calculated
by symplectic finite Fourier transform (SFFT) and shifted
by a threshold. This scheme is then extended to multipleinput multiple-output (MIMO) OTFS systems [18]. Though
this scheme is very low complexity, it requires an entire OTFS
frame for pilot transmission, which significantly reduces the
spectral efficiency (SE). To this end, an embedded pilot pattern
design is proposed in [14]. The pilot, as the terminology
implies, are embedded in a frame with data and separated from
data with protect guard. Owing to the compact representation
of parameter, an appropriately designed guard can reduce
interference between the data and the pilot to an acceptable
level. Besides, sequences and matrices of pilots are designed,
including pseudo-random noise (PN)-based pilot sequences
[19] and deterministic pilot matrix [20].
Channel estimation algorithm is one key component in CSI
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2
acquisition beyond the scheme. Though the SFFT is with
low complexity, it encounters with Rayleigh limit resulting in
high side-lobe. Alternatively, sparse signal representation and
recovery methods are introduced by innovative contributions
to explore the sparsity of wireless channel [21]–[24]. Shen
et al. [21] explore the 3D sparse representation for massive
MIMO OTFS system in the delay-Doppler-angle domain. And
orthogonal matching pursuit [25] is extended here to its 3D
version to solve the sparse signal reconstruction problem.
Rasheed et al. [22] present OMP and modified subspace
pursuit (MSP) based algorithms improving the performance
of normalized mean square error (NMSE) and bit error rate
(BER). Another algorithm strategy gained extensive attention
here is the sparse Bayesian learning (SBL) [26], [27]. Zhao et
al. [23] adopted the expectation maximum (EM) based SBL to
solve the sparse recovery problem. Liu et al. [24] proposed a
channel estimation method based on the expectation maximum
variational Bayesian (EM-VB) SBL algorithm. And by using
the fast Bayesian inference, one low complexity approach is
further constructed.
Good progress has been achieved by the noted contributions,
but a few shortcomings remain. Most of the works assume
that the DD frame is large enough to form a compact and
sparse channel parameter representation. In practice, the timefrequency resource is limited, i.e., the bandwidth and duration
corresponding to one DD frame is limited. Therefore, the
resolution in the DD domain is restricted by the Rayleigh resolution limit. The exact Doppler frequency shift or delay shift
may straddle a pair of finite-resolution bins, rather than falling
exactly into a single bin [28]. Thus the diversity gain promised
by the OTFS in DD domain is hard to reach. This phenomenon
is known as off-grid [29], [30] or fractional Doppler/delay
[12], [31], [32]. Cause the fractional Doppler is much serious
compared with fractional delay in current wideband system,
several works have explored methods to estimate the fractional
Doppler. N. Hashimoto et al. [33] analyze the impact of
fractional Dopplers and a local maximum search method is
proposed to locate the fractional Doppler. S. Srivastava et al.
[7], [17] explore the row and group (RG)-sparsity and relax
the off-grid effect by using a dense grid. D. Shi [20] represent
the fractional Doppler corresponding to each integer delay,
and the fractional Doppler is estimated by interpolation of
Doppler corresponding to the maximal complex path gain and
the Doppler corresponding to the second largest complex path
gain. F.Liu et al. [34] represent the fractional Doppler based
on the first order Taylor expansion, then message passing
framework is used to estimate the fractional Doppler. The
Cramer Rao Lower Bound (CRLB) is also deduced.
Though these works have achieved performance gain by
estimating the fractional Doppler, fractional delay is not
considered. Actually for a fixed data rate per frame, it is
impossible to obtain high precision Doppler without sacrificing
resolution of delay [12]. To cope with this 2D fractional
parameter estimation problem, Z. Wei et al. [28] firstly estimate the delay, Doppler and complex gain in the original
DD domain, instead of in the integer sampling based effective
DD domain. A 1D and a 2D off-grid channel models as well
as their corresponding estimation methods are proposed. The
fractional delay and the fractional Doppler are estimated by
the off-grid sparse Bayesian inference (OGSBI) framework
[30]. By adopting the 2D off-grid representation, significant
performance gain has been achieved by this work. Besides,
some other works, though not based on the OTFS, studied the
off-grid channel estimation or tracking in high mobility scenes
[6], [35], [36].
However, less thought was given to the problem that the estimation of Doppler and delay will affect each other, especially
referring to their fractional parameters . For example, the 2D
off-grid model and the two-step estimation algorithm proposed
in [28] ignore the mutual influence between fractional delay
and Doppler, leading to performance degradation. We name
this phenomenon as tandem off-grid distortion. Additionally,
the actual sparsity influenced by the off-grid effect is rarely
studied. This problem becomes even more important in MIMO
OTFS system as large pilot overhead may occupy the limited
OTFS frame and interfere each other because of their degraded sparsity. Furthermore, the CSI acquisition problem may
become a high complexity four-dimensional (4D) parameter
estimation problem because of the off-grid representation in
the delay and Doppler dimension separately. How to relax the
computational complexity is also a key point needed to be
considered. These challenges motivate us to develop a novel
2D fractional delay and Doppler estimation scheme that is
able to overcome the shortcomings of the existing schemes.
The main contributions are as following:
•
•
•
The actual sparsity of the channel parameters affected by
both fractional delay and fractional Doppler is analyzed,
especially when the OTFS frame is small. Two main
effects, i.e., the spread and the effective signal-to-noise
(SNR) loss, are analyzed. Our results show that the
channel estimation performance would be deteriorated
significantly due to the fractional DD parameters, even
the bandwidth and duration become large.
By considering the fractional delay and the fractional Doppler simultaneously, an end-to-end input-output
model in vector and matrix form is presented, together
with the corresponding transformation based on the stateof-the-art assumptions. Based on this model, the state-ofthe-art off-grid assumptions, models and their transformation are formulated. Two mixed one- and two-dimensional
(1&2D) fractional models are proposed, which decompose the received OTFS signal into two related mesh grid.
Each of the mesh prominently indicates partial channel
characteristics in the DD domain.
The tandem off-grid distortion caused by the off-grid
representation in the 2D off-grid model [28] is analyzed.
To tackle the tandem off-grid distortion, a novel scheme
based on 2D off-grid decomposition and sparse Bayesian
learning (SBL) combination is proposed. This scheme
firstly decomposes the received signal into two related
mesh grid for channel feature extraction, which are then
combined under the SBL framework. It has been shown
that the proposed scheme is more accurate, and has lower
computational cost compared with the doubly fractional
model based method in [28].
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3
The remainder of this paper is organized as follows. Sec.
II recalls the OTFS principles, and analyses the deterioration
caused by fractional delay and Doppler. Sec. III expands
the on grid model to the off-grid model. Several off-grid
models are reformulated and compared from the perspective
of their assumptions. To solve the tandem off-grid distortion
problem, Sec. IV proposes the 2D off-grid decomposition and
combination scheme. And the OGSBI framework is briefly
introduced, which is used as estimator in several steps in the
proposed scheme. Sec. V gives the simulation results. And
Sec. VI is the conclusion.
A matched filter grx (t) is used here to transform the signal
back to the time-frequency domain,
Z
′
∗
Y (t, f ) = grx
(t′ − t)r(t′ )e−j2πf (t −t) dt′ .
(5)
Sampling at t = nT and f = m∆f , the signal in the discrete
time-frequency domain [12],
Y [n, m] = Y (t, f )|t=nT,f =m∆f =
n′ =0
·
OTFS is a 2D modulation/de-modulation which transforms
the information symbol between the TF domain and the DD
domain. Consider the OTFS system having the frame duration
of Tf = N T and bandwidth of B = M ∆f , where T (seconds)
and ∆f (Hz) is the slot duration and subcarrier spacing,
respectively. N and M represent the number of symbols along
the time and frequency axes in the corresponding TF-grid,
respectively.
Let x[k, l] denote the information symbols in the DD
domain. The OTFS transmitter firstly map the information
symbols to the time-frequency domain by the inverse symplectic finite Fourier transform (ISFFT), that is [12],
x[k, l]e
ml
j2π ( nk
N −M )
,
(1)
k=0 l=0
where Agrx ,gtx ((n − n′ )T − τi , (m − m′ )∆f − νi ) is the
ambiguity function. Ideal waveforms are assumed here, thus
Agrx ,gtx ((n − n′ )T − τi , (m − m′ )∆f − νi ) = 1, for n =
n′ , m = m′ and zero otherwise. Though the ideal pulse cannot
be realized in practice, it will help us to focus on the fractional
parameter analysis. The rationality of this assumption will be
discussed in the Sec. II-B.
Finally, by applying the symplectic finite Fourier transform
(SFFT), the received signal y[k, l] in the DD domain could be
represented as that in [28],
y[k, l] =
s(t) =
X[n, m]gtx (t − nT )e
,
(2)
where gtx is a transmit waveform, which could be defined as
that in [20].
The signal s(t) is transmitted over a time-varying channel.
Regardless of noise, the channel could be represented by
channel impulse response h(ν, τ ) [9],
h(ν, τ ) =
P
X
hi δ (ν − νi ) δ (τ − τi ) ,
(3)
i=1
where P is the number of paths in the channel, hi is the
channel coefficient of the i-th path, and δ(·) is the Dirac delta
function. τi ∈ (0, τmax ) and νi ∈ (−νmax , νmax ) denote the
delay and Doppler shifts of the i-th path, respectively. τmax
and νmax are the maximum time delay and Doppler shift,
respectively. Then, the signal at the receiver is
Z Z
r(t) =
h(ν, τ )s(t − τ )ej2πν(t−τ ) dτ dν
P
X
i=1
x [k ′ , l′ ] hw [k − k ′ − kνi , l − l′ − lτi ] ,
(7)
where x [k ′ , l′ ] is the information symbol arranged in the DD
domain with replaced subscript.
hw [k − k ′ − kνi , l − l′ − lτi ]
=
P
X
h̃i w (k − k ′ − kνi , l − l′ − lτi )
(8)
i=1
j2πm∆f (t−nT )
n=0 m=0
=
N
−1 M
−1
X
X
k′ =0 l′ =0
where n = 0, . . . , N −1, m = 0, . . . , M −1. Then Heisenberg
transform is used to converts the X[n, m] into a continues time
waveform,
−1
N
−1 M
X
X
(6)
′
hi ej2πm ∆f (−τi ) ej2πνi (−τi ) ej2πνi (nT )
· Agrx ,gtx ((n − n′ )T − τi , (m − m′ )∆f − νi ),
A. OTFS channel estimation model
1
X[n, m] = √
NM
P
X
X[n′ , m′ ]
m′ =0
i=1
II. BASIC P RINCIPLES OF F RACTIONAL OTFS
N
−1 M
−1
X
X
N
−1 M
−1
X
X
(4)
hi s(t − τi )ej2πνi (t−τi )
′
where k ∈ {0, . . . , N − 1}, l′ ∈ {0, . . . , M − 1}. kνi =
νi N T and lτi = τi M ∆f are the corresponding normalized
νi and τi in sampling grid, respectively. To simplify the
parameters, the phase e−j2πνi τi is absorbed into the channel
coefficient, or so called complex gain, h̃i = hi e−j2πνi τi .
w (k − k ′ − kνi , l − l′ − lτi ) can be decomposed into the delay and the Doppler dimension,
w (k − k ′ − kν , l − l′ − lτ ) = wν (k, k ′ , kν ) wτH (l, l′ , lτ ) ,
(9)
where,


′
k′ +kν −k sin (π (k − k − kν ))
1
ej(N −1)π N
,
wν (k, k ′ , kν ) =
′
N
sin π(k−k −kν )
N
(10)


wτ (l, l′ , lτ ) =
1  j(M −1)π l′ +lτ −l sin (π (l − l′ − lτ )) 
M
e
.
′
M
sin π(l−l −lτ )
M
(11)
If
a
single
impulse
pilot
is
used
as
that
in
[28],
i.e.,
x kp′ , lp′ ̸= 0, and others are zero. The received signal is
y[k, l] = x kp′ , lp′ hw k − kp′ , l − lp′ .
(12)
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And its matrix form could be written as
YDD
B. Sparsity over Fractional Parameter
P
X
= x kp′ , lp′
h̃i wν (kνi )wτH (lτi )
i=1
H
= x kp′ , lp′ UDD HDD VDD
,
(13)
where YDD ∈ CN ×M is the matrix form of y[k, l]. UDD ∈
CN ×P and VDD ∈ CM ×P are the measurement matrix
representing the Doppler effect and time delay, respectively.
Define the true Doppler kν = [kν1 , · · · , kνP ]T , and the true
time delay lτ = [lτ1 , · · · , lτP ]T , then
UDD (kν ) = wν (kν1 ) wν (kν2 ) · · · wν (kνP ) , (14)
VDD (lτ ) = wτ (lτ1 ) wτ (lτ2 ) · · ·
wτ (lτP ) ,
(15)
A significant property of OTFS is that the channel
parameter h̃i is compact and sparse in the DD domain. The sparsity relies on the property of the weight
w (k − k ′ − kνi , l − l′ − lτi ). Based on Eqs. (9)-(11), for integers lτi and kνi , |w (k − k ′ − kνi , l − l′ − lτi ) | = 1, if and
only if k and l satisfy
k = kp′ + kνi and l = lp′ + lτi .
The weight in Doppler dimension and delay dimension is given
by wν (k, k ′ , kν ) = 0, wτ (l, l′ , lτ ) = 0, for other [k, l]. Since
kνi and lτi are integers, kν ⊆ k̃ν and lτ ⊆ l̃τ . Based on Eq.
(13) and Eq. (19), it could be found that only P elements of
YDD , as well as of H̃DD , are nonzero, which correspond to
the P paths. For k̃νn0 = k̃νi and ˜lτm0 = ˜lτi ,
h̃n0 ,m0 = h̃i .
where
wν (kν ) = [wν 0, kp′ , kν , · · · , wν N − 1, kp′ , kν ]T , (16)
wτ (lτ ) = [wτ 0, lp′ , lτ , · · · , wτ M − 1, lp′ , lτ ]T ,
(17)
are the vectors corresponding to kν and lτ , respectively. The
channel coefficient matrix is in the form of
HDD = diag([h̃1 , . . . , h̃P ]) ∈ CP ×P ,
(18)
where diag(a) denotes the diagonal matrix with elements of
a being the diagonal elements.
In practice, the true Doppler kν and delay lτ are unknown.
A mesh grid is usually initialized for representation as k̃ν =
{k̃ν1 , · · · , k̃νN0 } and l̃τ = {˜lτ1 , · · · , ˜lτM0 }, respectively. N0
and M0 are the number of grid in the Doppler and delay dimension, respectively. k̃νn = − N2 + NN0 (n−1) and the Doppler
∆
∆
frequency corresponding to k̃νn is ν̃n = − 2f + Nf0 (n − 1).
˜lτ = M (m − 1) and the corresponding time delay is
m
M0
τ̃m = MT0 (m − 1).
N0 X
M0
X
YDD = x kp′ , lp′
h̃n0 ,m0 wν (k̃νn0 )wτH (˜lτm0 )
n0 =1 m0 =1
H
= x kp′ , lp′ ŨDD H̃DD ṼDD
,
(19)
where, ŨDD = UDD (k̃ν ) ∈ CN ×N0 , ṼDD = VDD (l̃τ ) ∈
CM ×M0 , by using the definition of Eq. (14) and Eq. (15),
respectively. H̃DD is the effective complex gain in the discrete
DD domain with its element h̃n0 ,m0 corresponding to the
Doppler k̃νn0 and delay ˜lτm0 . N0 = N and M0 = M are
usually used, thus leading to an integer sampling grid based
model.
The CSI acquisition task is finally transformed to estimate
the parameters h̃n0 ,m0 from Eq. (19). Estimation of h̃n0 ,m0 is
equal to estimate h̃i , lτi and kνi , i = 1, . . . , P , if fractional
delay and Doppler is not considered. However, sparsity degradation as well as performance deterioration will be introduced
in practice because of the off-grid effect.
(20)
(21)
Let h̃ = vec(H̃DD ), where vec(A) is the vectorization of
matrix A. ∥h̃∥0 = P , thus we could use P prominent values to represent the time-varying channel which significantly
decrease the complexity of detection.
However, the sparsity cannot reach P in practice, because
the assumption that lτi = τi M ∆f and kνi = νi N T are
integers is not always true. Actually, the delay τi and Doppler
νi are usually non-integer multiplying of 1/N T and 1/M ∆f .
Let ˜lτi and k̃νi denote the nearest integer sampling grid
index in the delay dimension and in the Doppler dimension,
respectively.
τi =
˜lτ + βl
k̃νi + βkνi
kν
lτi
i
τi
(22)
=
and νi = i =
M ∆f
M ∆f
NT
NT
where βlτi , βkνi ∈ (−1/2, 1/2) represent the fractional delay
and Doppler, respectively.
Note that, usually two approximations are used in previous
studies, i.e., the ideal bi-orthogonal waveform assumption and
the integer assumption. Both of them will affect the sparse
representation of signals in the DD domain. We have assumed
the bandwidth B = M ∆f , thus the sampling frequency for a
complex signal should at least be fs = B and the sampling
interval be Ts = 1/fs = 1/M ∆f . Therefore, the duration of
one slot is T = M Ts = 1/∆f , which is also the maximum
supportable delay. While the total time-width is Tf = N T .
The sampling interval in frequency is 1/N T . With N slots,
the maximum supportable Doppler is N/Tf = ∆f . For the
ambiguous function Agrx ,gtx ((n−n′ )T −τi , (m−m′ )∆f −νi ),
if we want it to be sharp in the time dimension, then for a fixed
n − n′ , larger T is helpful. Unfortunately, this widening the
sidelobe in the frequency dimension cause ∆f gets smaller,
or vice versa. We cannot expect both of T and ∆f are
large enough, referred as the uncertainty property. However,
the fractional delay and fractional Doppler are caused by
discretization of truncated signal with limited bandwidth and
duration. With the bandwidth of M ∆f , the resolution in the
delay dimension is the time sampling interval 1/M ∆f . The
sampling grid may not be right on the true delays, resulting
in the off-grid gap βlτi . While the resolution in the Doppler
dimension is the frequency sampling interval 1/N T with the
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5
time-width N T . Similarly, the sampling grid may not be
right on the true Dopplers, resulting in the off-grid gap βkνi .
The two off-grid gaps are related to the bandwidth B and
time-width Tf , which is irrelevant to the uncertainty property
(choice of T and ∆f ). Therefore, we could assume an ideal
waveform and analyze the fractional delay and Doppler based
on Eq. (7).
There are two phenomena caused by the off-grid gaps, i.e.,
spread and effective SNR loss, as shown in Fig. 3b.
1) Spread: Considering kνi = k̃νi +βkνi and lτi = ˜lτi +βlτi
are not integers in Eq. (8), i.e., βlτi βkνi ̸= 0. Then, for any
k and l that related to the paths, y[k, l] ̸= 0. This could be
easily verified by considering the absolute value of Eq. (10),
sin π k − k ′ − k̃νi − βkνi
. (23)
|wν (k, k ′ , kν ) | =
π (k−k′ −k̃νi −βkν )
i
N sin
N
Let η = k − k ′ − k̃νi ∈ {−N + 1, . . . , N − 1} be integer, then
the numerator
| sin π η − βkνi |
= | sin(πη) cos(πβkνi ) − cos(πη) sin(πβkνi )|
(24)
= | sin(πβkνi )| > 0
Thus |wν (k, k ′ , kν ) | > 0. Similarly, |wτ (l, l′ , lτ ) | > 0. And
according to Eq. (12), y[k, l] ̸= 0. This leads to a phenomenon
that the elements of YDD which have similar index to the
paths, either in delay dimension or Doppler dimension, are
nonzero. So according to Eq. (19), the corresponding sampling
points in model are nonzero. Thus, in the discrete DD domain,
∥h̃∥0 ≫ P . This is a significant defect that we cannot just
use only P prominent elements to describe the channel as that
by taking the integer assumption. Fortunately, the information
is still basically compact in general.
Firstly, a peak corresponding to the ith path in the discrete
Doppler dimension will be at k = k ′ + k̃νn , where k̃νn = k̃νi
is the nearest grid point to kνi . Recall that, k̃νn , n = 1, . . . , N0
is the discrete sampling point in Doppler dimension and k =
k ′ + k̃νn in practice. From Eq. (23) and Eq. (24),
sin πβkνi
sin πβkνi
≤
πβ .
|wν (k, k , kν ) | =
kν
π (η−βkν )
i
i
N sin
N sin
N
N
′
(25)
This can be easily verified by
!
π η − βkνi
sin
≥ sin
N
π βkνi
N
!
(26)
because sin(x) is monotone increasing for x ∈ [−π, π],
sin(0) = 0 and |η − βkνi | ≥ |βkνi |.
Secondly, we could use the beam width BW0 and half
power beam width BW0.5 to describe the compactness. From
Eq. (10), the first zero point corresponding to the ith path
in the Doppler dimension is at k = k ′ + kν ± 1. Therefore,
the width in the discrete Doppler dimension is 2 and in the
Doppler dimension
BW0Doppler = 2/N T
(27)
Let |wν (k, k ′ , kν ) |2 = 1/2, for large N , we will have
Doppler
≈ 0.886/N T.
BW0.5
(28)
Similarly, the peak corresponding to the ith path in the
discrete delay dimension will be at l = l′ + ˜lτm , where
˜lτ = ˜lτ . And the beam width
m
i
delay
BW0delay = 2/M ∆f and BW0.5
≈ 0.886/M ∆f .
(29)
Thirdly, the spread is monotonically decreasing. Based on
Eq. (26), it’s easily obtained,
!
!
π η2 − βkνi
π η1 − βkνi
sin
> sin
, (30)
N
N
if |η1 | > |η2 |. Thus, the weight |wν (k, k ′ , kν ) | and
|wτ (l, l′ , lτ ) | will get smaller as k moving away from k ′ + k̃νi
and l moving from l′ + ˜lτi .
These results imply that, in the discrete DD domain the
information is still gathering around the true Dopplers and
delays, though the sparsity ∥h̃∥0 ≫ P . Thus we could use
the spreading information in the algorithm to enhance the
performance.
2) Effective SNR loss: Another influence is that the actual
SNR is decreased. At the peaks, as shown in Eq. (25), the
weight
|w (k − k ′ − kνi , l − l′ − lτi ) |
= |wν (k, k ′ , kν ) wτ (l, l′ , lτ ) |
sin πβkνi
sin πβlτi
πβ πβ ≤
kν
l
i
N sin
M sin Mτi
N
(31)
Therefore, the peak power corresponding to the ith path in the
DD domain is
2
sin πβkνi
sin πβlτi
2
2
πβ πβ , (32)
|h̃n0 ,m0 | = |h̃i |
kν
lτ
i
i
N sin
M
sin
N
M
for k − k ′ − k̃νi = 0 and l − l′ − ˜lτi = 0. Compared with Eq.
(21), the loss of SNR is LSNR (dB),
sin πβkνi sin πβlτi
πβ πβ LSNR = 20 log
kν
l
i
M N sin
sin Mτi
N
(33)
sin πβkνi sin πβlτi
≈ 20 log
,
π 2 βkνi βlτi
where sin(x) = x if x is very small, is used for approximation.
Figure 1 shows the SNR loss vs. normalized fractional delay
and Doppler. It could be found that the SNR loss could reach
to 7.8 dB in the worst case. And an important fact is that,
according to Eq. (33), increasing M and N is no helpful to
decrease the SNR loss. The loss may also reach to 7.8 dB even
if we use a very large bandwidth and duration. We could also
find the similar phenomena from Fig. 2. Giving the normalized
fractional delay and Doppler βkνi = βlτi = 0.25, it could be
found that, increasing M and N has no helpful to improve
the LSNR for a certain fractional parameter.
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Fig. 1. SNR loss vs. normalized fractional delay and Doppler
Fig. 2. SNR loss vs. M and N
This conclusion doesn’t sound like common sense. It’s
because we use the normalized fractional parameter in the
discrete DD domain instead of parameter in the real and continues DD domain. Actually, for a real delay τi and Doppler νi ,
increasing M and N decrease their possibility corresponding
to a large normalized fractional parameter by shrinking 1/N T
and 1/M ∆f . Though, the fractional part cannot be eliminated.
As shown in Fig. 3b, the spread could interfere the detection
in the embedded frame based methods [14], and the SNR loss
will deteriorate the estimation performance.
So could it be possible to retrace the P-sparsity and the full
SNR gain? We will introduce the off-grid models and estimate
the parameters in the fractional DD domain in the next section.
Based on our analysis in the Sec. II-B, the spread and the
effective SNR loss is caused by the off-grid gap. There are
some classical methods which could decrease the off-grid gap,
such as the chirp-z transform [37], in which a denser grid is
applied to represent the signal. Similar to this idea, larger M0
and N0 could be used in Eq. (19) for a finer representation,
as that in [17]. However, it’s still an on-grid representation,
which assume that the true delays and Dopplers are on the
predefined or refined sampling grid. Though the SNR loss is
relieved, the sparsity cannot be improved, as shown in Fig. 4.
The spread would still have impact on the whole dimension.
III. O FF - GRID MODELS FOR FRACTIONAL OTFS
Another way to cope with the off-grid gap is to approximate
the fractional delays and Dopplers. We have derived the ideal
input-output channel response as Eq. (13), where the channel
could be represented by the complex gain h̃i , delay τi and
Doppler νi , i = 1, . . . , P . No discretization is made to these
parameters, thus we call it is the representation in the original
DD domain as that of [28]. In practice, the received signal
YDD (Eq. (19)) is discretized by sampling and deteriorated by
the off-grid effect. We call it is the representation in the integer
DD domain. Fig. 3 shows the representation in the original
and the integer DD domain, respectively. In this section, we
will try to approximate the original DD domain based on
the measurements in the integer DD domain by using two
dimensional off-grid expansion in the fractional DD domain.
For any of i ∈ {1, · · · , P }, the true normalized Doppler kνi
can be represented by its nearest grid k̃kνi ∈ {k̃ν } adds an
off-grid parameter βkνi . Then, wν (kνi ) in Eq. (14) could be
approximated by Taylor expansion,
In this section, we will deduce the off-grid model from the
uniformly sampling representation model shown in Eq. (19).
The CSI acquisition task will be finished in the fractional
DD domain instead of the integer DD domain (also called
the effective DD domain in [28]).
A. fractional Doppler on-grid model
From Eq. (7), the CSI is in the function
hw [k − k ′ − kνi , l − l′ − lτi ], which depends on six
independent variables. If the integer delay and Doppler
assumption is applied, then it has to be k − kνi = k ′ and
l − lτi = l′ for the nonzero elements. Thus, the CSI could
be represented by hw [k ′ , l′ ] because the other parameters are
absorbed. This is right the input-output model in [21].
y[k, l] =
N
−1 M
−1
X
X
k′ =0
x [k ′ , l′ ] hw [k ′ , l′ ]
(34)
B. From the on-grid to the off-grid
wν (kνi ) = wν (k̃νi + βkνi )
l′ =0
= wν (k̃νi ) + βkνi wν′ (k̃νi ) + o(βkνi )
Based on the model (34), [20] considers fractional Doppler
corresponding to the integer delays.
y[k, l] =
N
−1 M
−1
X
X
′
′
′
′
′
x [k , l ] hw [k , l ] ϕ(l, l ),
(35)
k′ =0 l′ =0
where ϕ(l, l′ ) is compensation related to each lτ . It is right the
basic model (Eq. (7)) with the assumption βlτi = 0. Though
the performance is improved by compensating the fractional
Doppler corresponding to all the mesh grid, this model still
adopts a fixed predefined mesh grid which introduces the offgrid gaps.
(36)
where o(βkνi ) is the infinitesimal of higher order. Similarly,
wτ (kτi ) = wτ (˜lτi ) + βlτi wτ′ (˜lτi ) + o(βlτi ). Therefore, we
could expand Eq. (19) to
N0 X
M0
X
YDD = x kp′ , lp′
h̃n0 ,m0 w̃ν (k̃νn0 )w̃τH (˜lτm0 ),
n0 =1 m0 =1
(37)
where,
w̃ν (k̃νn0 ) ≜ wν (k̃νn0 ) + βkνn wν′ (k̃νn0 ),
0
(38)
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(a) Representation in the original DD domain.
(b) Representation in the integer DD domain.
Fig. 3. The signal representation in the original and integer DD domain. M = 64, N = 16 and ∆f = 15 KHz.
(a) M0 = M = 64, N0 = N = 16
(b) M0 = 2M = 128, N0 = 2N = 32
Fig. 4. The representation of channel impulse response with different M0 and N0 in Eq. (19).
w̃τ (˜lτm0 ) ≜ wτ (˜lτm0 ) + βlτm0 wτ′ (˜lτm0 ),
(39)
where wν′ (k̃νn ) and wτ′ (˜lτm ) are partial derivative with respect
to k̃νn and ˜lτm , respectively.
wν′ (k̃νn0 )
N
1 X j2πn −j2πn k−kp −k̃vn0
N
e
=
N n=1 N
M
1 X j2πm j2πm l−lp −l̃τm0
M
wτ′ (˜lτm0 ) = −
e
M m=1 M
(40)
(41)
βkνn and βlτm are off-grid parameters denoting the underlying
fractional Doppler or delay. If k̃νn and ˜lτm are the nearest grid
to a true Doppler kνi and delay lτi , then the off-grid parameters
are βkνn = kνi − k̃νn and βlτm = lτi − ˜lτm . Otherwise,
βkνn = βlτm = 0. Denote the off-grid parameters as the vector
form βkν = [βkν1 , · · · , βkνN ] and βlτ = [βlτ1 , · · · , βlτM ],
0
where the measurement matrix in Doppler dimension is
′
= UDD (k̃ν ) + diag(βkν )UDD
(k̃ν ),
Ṽ (βlτ ) = [w̃τ (˜lτ1 ), · · · , w̃τ (˜lτN0 )]
′
= VDD (l̃τ ) + diag(βlτ )VDD
(l̃τ ),
(44)
′
(l̃τ ) is the partial derivative of VDD (l̃τ ) with
where VDD
respect to l̃τ . Note that, Ũ (βkν ) ∈ CN ×N0 , Ṽ (βlτ ) ∈
CM ×M0 . The unknown parameters are H̃DD , βkν and βlτ .
H̃DD ∈ CN0 ×M0 , βkν ∈ RN0 ×1 and βlτ ∈ RM0 ×1 .
With these definitions, Eq. (13) can be approximated by Eq.
(42). The off-grid parameters are continuous in the Doppler
dimension and delay dimension, respectively. Thus, the true
delays and Dopplers could be estimated by additionally estimating the off-grid parameters βkν and βlτ . The signal is
represented in the fractional DD domain instead of the integer
DD domain.
0
corresponding to the grid k̃ν and l̃τ , respectively, the estimation model could be written as
YDD = x kp′ , lp′ Ũ (βkν )H̃DD (Ṽ (βlτ ))H ,
(42)
Ũ (βkν ) = [w̃ν (k̃ν1 ), · · · , w̃ν (k̃νN0 )]
′
(k̃ν ) is the partial derivative of UDD (k̃ν ) with
where UDD
respect to k̃ν . The measurement matrix in delay dimension
can be represented as
(43)
C. doubly off-grid expansion
Though Eq. (37) approaches to Eq. (12) from the delay
dimension and the Doppler dimension respectively, discretization is still here in the DD domain. In Eq. (12), HDD is
a diagonal matrix because of infinitesimal resolution. But in
practice, M0 and N0 cannot be infinite, thus multiple Dopplers
may correspond to a same interval of delay grid, or vice
versa, resulting in that H̃DD is no longer a diagonal matrix.
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Therefore, we have to consider the mesh grid k̃ν × l̃τ instead
of k̃ν and l̃τ separately. In some cases, such as [20], only
the fractional Doppler is considered, and others considers part
of fractional DD grid [28]. These assumptions lead to several
different off-grid models, which are shown in Fig. 5.
1) doubly fractional model: Considering for a same delay
interval, multiple Dopplers correspond to the delay and their
fractional part are nonzero. Thus, we need extra delay index to
distinguish the off-grid parameters in the Doppler dimension,
or vice versa. Therefore, Eq. (37) is extended to
N0 X
M0
X
YDD = x kp′ , lp′
h̃n0 ,m0 w̃ν (k̃νn0 ,m0 )w̃τH (˜lτm0 ,n0 ),
(a) 1D-expansion off-grid model (Eq. (37) and Eq. (42))
n0 =1 m0 =1
(45)
where k̃νn0 ,m0 and ˜lτm0 ,n0 represent the grid corresponding
to the m0 th delay and n0 th Doppler, respectively. The offgrid parameters corresponding to the grid are βkνn0 ,m0 and
βlτm ,n , respectively. Denote
0
βk2D
ν
0
= [βkν1,1 , . . . , βkνN
0 ,1
, βkν1,2 , . . . , βkνN
0 ,2
, . . . , βkνN
βl2D
= [βlτ1,1 , . . . , βlτ1,N , βlτ2,1 , . . . , βlτ2,N , . . . , βlτM
τ
0
0
The model Eq. (45) could be vectorized as,
y = x kp′ , lp′ Φ̃ βk2D
, βl2D
h̃ + n,
ν
τ
0 ,M0
0 ,N0
]T
]T
(46)
where y = vec(YDD ), h̃ = vec(H̃DD ). The matrix Φ̃ ∈
CM N ×M0 N0 ,
Φ̃ βk2D
, βl2D
= Φ + Φν diag βk2D
+ Φτ diag βl2D
, (47)
ν
τ
ν
τ
The ((m0 − 1)N0 + n0 )-th column of Φ, Φν and Φτ
are vec(wν (k̃νn0 )wτH (˜lτm0 )), vec(wν′ (k̃νn0 )wτH (˜lτm0 )) and
vec(wν (k̃νn0 )(wτ′ (˜lτm0 ))H ), respectively. n is a noise vector
and absorbing the model error of infinitesimal of higher
order. It could be found from Fig. 5b, the off-grid expansion
is used corresponding to each grid of k̃ν × l̃τ . Thus, the
fractional Dopplers of the path denoted by the triangle could
be represented by this model. As a comparison, the fractional
Dopplers of the two paths in Fig. 5a cannot be distinguished
in the model Eq. (37).
Applying the doubly fractional model, the unknown parameters are h̃, βk2D
and βl2D
. h̃ ∈ CM0 N0 ×1 . The off-grid
ν
τ
M0 N0 ×1
parameters belongs to R
.
2) mixed 1&2D fractional models: The complexity of
solving the doubly fractional model is very high because we
need to solve three unknown parameters with the dimension
of M0 N0 . In many cases, the bandwidth is large thus we could
assume that the delays of the paths are in different mesh grid
while their Dopplers are in the same interval as shown in
Fig. 5c. The fractional delay is represented by βlτ , while its
fractional Doppler is represented by βk2D
.
ν
Based on Eq. (45), the model could be rewritten as
!
M0
N0
X
′ ′ X
w̃ν (k̃ν
)h̃n ,m w̃H (˜lτ ).
YDD = x k , l
p
p
n0 ,m0
m0 =1
0
0
τ
(b) Doubly fractional model (Eq. (45) and Eq. (46))
(c) mixed 1&2D fractional model with 2D extending in the Doppler
dimension (Eq. (48))
m0
n0 =1
(48)
Recall w̃τ (˜lτm0 ) = wτ (˜lτm0 ) + βlτm wτ′ (˜lτm0 ) as Eq. (39),
0
and in the Doppler dimension,
w̃ν (k̃νn0 ,m0 ) = wν (k̃νn0 ,m0 ) +
βkνn0 ,m0 wν′ (k̃νn0 ,m0 ),
(49)
(d) mixed 1&2D fractional model with 2D extending in the delay
dimension (Eq. (52))
Fig. 5. The off-grid models with different off-grid assumption. The dot and
the triangle represent two independent paths in the DD domain, respectively.
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Therefore, the input-output relationship could be decomposed
as,
M0
′ ′ X
YDD = x kp , lp
dVm0 w̃τH (˜lτm0 )+nH
(50)
τ ,
m0 =1
where
dVm0 =
N0
X
w̃ν (k̃νn0 ,m0 )h̃n0 ,m0 +nτ ν .
(51)
n0 =1
nτ and nτ ν denote the uncertainty in the delay dimension and delay-Doppler domain, respectively. By applying
this model, it could be found the unknown fractional parameters βlτ ∈ RM0 ×1 and βk2D
∈ RM0 N0 ×1 . Though
ν
2D
M0 N0 ×1
βkν ∈ R
, it could be calculated from Eq. (51) by
estimating βk2D
|
= [βkν1,m , · · · , βkνN ,m ]T ∈ RN0 ×1 ,
m=m
0
ν
0
0
0
m0 = 1, . . . , M0 , separately. Thus the actual dimension is
N0 , which is significantly decreased. Because this model uses
1D off-grid parameter in the delay dimension and 2D off-grid
parameter in the Doppler dimension, we call this model mixed
one- and two-dimensional (1&2D) fractional model with 2D
extending in the Doppler dimension.
Symmetrically, we could obtain the mixed 1&2D fractional
model with 2D extending in the delay dimension, as shown in
Fig. 5d,
!
N0
M0
X
′ ′ X
H ˜
YDD = x k , l
w̃ν (k̃ν )
h̃n ,m w̃ (lτ
) .
p
p
n0
n0 =1
0
0
τ
m0 ,n0
m0 =1
(52)
And the input-output relationship can be written as
YDD
N0
′ ′ X
w̃ν (k̃νn0 )dH
= x kp , lp
Un0 +nν ,
(53)
Fig. 6. The scheme of 2D off-grid decomposition and SBL combination.
deteriorated in the situation shown in Fig. 5d owing to the
fact that βkν2 cannot represent the fractional Dopplers of the
two paths simultaneously. We call this error as tandem offgrid distortion because it only happens in the mixed 1&2D
fractional models for its tandem steps.
IV. 2D OFF - GRID DECOMPOSITION AND SBL
COMBINATION BASED CHANNEL ESTIMATION
The number of the dimension of the doubly fractional model
is too high, while the mixed 1&2D fractional models encounter
the tandem off-grid distortion. In this section, a dimensional
decomposition and combination scheme is proposed to mitigate the tandem off-grid distortion and to reduce the dimension
of the unknown parameters. We will firstly decompose the
received OTFS symbols in two different structures to obtain
the channel responses and the mesh grid, respectively. Then,
these estimated channel responses and the mesh grid are
combined in a SBL framework to reconstruct the final channel
responses. The scheme is illustrated as Fig. 6.
n0 =1
A. 2D Off-grid Decomposition
where
dH
Un0 =
M0
X
h̃n0 ,m0 w̃τH (˜lτm0 ,n0 )+nH
ντ .
(54)
m0 =1
nν and nντ denote the uncertainty in the Doppler dimension and Doppler-delay domain, respectively. The off-grid
parameter in Doppler dimension is βkν ∈ RN0 ×1 , while the
off-grid parameter in the delay dimension is βl2D
∈ RN0 M0 ×1 .
τ
2D
Similarly, βlτ |n=n0 = [βlτ1,n , · · · , βlτM ,n ] ∈ RM0 ×1
0
0 0
could be estimated corresponding to different n0 separately.
Thus the actual dimension is M0 . From the Eqs. (46), (53)
and (54), it could be found that the doubly fractional model
and the mixed 1&2D fractional model with 2D extending in
the delay dimension are right the 1D off-grid model and the
2D off-grid model in [28], respectively. To avoid confusion,
we don’t use the term 1D off-grid model, because the off-grid
parameters are 2D in the 1D off-grid model referring to our
discussion here.
The mixed 1&2D fractional model assumes that for different
delay, the paths may share the same Doppler basis w̃ν (k̃νn0 ),
or for different Doppler, the paths may share the same delay
basis w̃τ (˜lτm0 ). This assumption results in a significant reduction in parameter dimensions. However, it also introduces
extra representation error to the estimation procedure in the
second step. For example, the performance of [28] will be
2D off-grid decomposition consists of two branches, i.e.,
the V-branch and the U-branch. The two branches decompose
the received symbols YDD based on the model of Eq. (48)
and Eq. (52), respectively.
1) V-branch: In the V-branch, we firstly decompose the
received symbols in the delay dimension based on the Eq.
(50). Then the result is further decomposed into the Doppler
dimension corresponding to each delays as the Eq. (51). Thus,
Solving the original model (Eq. (48)) could be transformed to
two steps.
Step 1:
H
H
YDD
= Ṽ (βlτ )DV
+nτ ,
(55)
where Ṽ (βlτ ) is defined in Eq. (44). The unknown paramH
eters are βlτ and DV
. The estimator used here could be
OGSBI [30], which is introduced in Sec. IV-C Let D̂V =
[dV1 , · · · , dVM0 ] be the estimated DV in the first step. Each
column of D̂V corresponds to a slice with different delay
˜lτ + βl .
m0
τm0
Step 2:
dVm0 = Ũ (βk2D
|
)hVm0 +nτ ν (m0 ),
ν m=m0
(56)
where βk2D
|
is the fractional Doppler corresponding to
ν m=m0
the delay ˜lτm0 + βlτm0 . hVm0 and nτ ν (m0 ) are the m0 th
column of the channel matrix H̃DD and the uncertainty nτ ν ,
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respectively. Similarly, the OGSBI could be used to estimate
the unknown parameters as β̂k2D
|
and ĥVm0 .
ν m=m0
Finally, through these two steps, we could get the estimation
of βlτ , H̃DD and βk2D
, denoted as β̂lτ ,
ν
h
i
ĤV = ĥV1 , . . . , ĥVM0 ,
h
i
(57)
2D
2D
β̂k2D
=
vec
β̂
|
,
.
.
.
,
β̂
|
.
kν m=1
kν m=M0
ν
2) U-branch: Symmetrically, in the U-branch, we firstly
decompose the received symbols in the Doppler dimension,
and then in the delay dimension based on the Eqs. (53) and
(54). Solving the model of Eq. (52) could be transformed to
two steps,
H
(58)
Step 1: YDD = x kp′ , lp′ Ũ (βkν )DU
+nν ,
where Ũ (βkν ) is defined in Eq. (43). The unknown parameters
H
are βkν and DU
. Let D̂U = [dU1 , · · · , dUN0 ] be the estimated
DU in the first step. Each column of D̂U corresponds to a
slice in different Doppler k̃νn0 + βkνn .
0
Step 2: dUn0 = Ṽ (βl2D
|n=n0 )hUn0 +nντ (n0 ),
τ
(59)
where βl2D
|n=n0 is the fractional delay corresponding to
τ
the Doppler k̃νn0 + βkνn0 . hH
Un0 and nντ (n0 ) are the n0 th
column of the channel matrix H̃DD and the uncertainty nντ ,
respectively.
Finally, through these two steps, we could get the estimation
of βkν , H̃DD and βl2D
, denoted as β̂kν ,
τ
h
iH
ĤU = ĥU1 , . . . , ĥUN0
,
h
(60)
iT 2D
2D
2D
β̂lτ = vec β̂lτ |n=1 , . . . , β̂lτ |n=N0
.
B. SBL Combination
The previous steps have decomposed the received YDD in
two related mesh grid. Then we will combine these information to reconstruct as the model Eq. (45). Expanding Eq.
(38) and Eq. (39), the basis corresponding to the h̃nm can be
approximately represented as
w̃ν (k̃νn,m ) ≈ wν (k̃νn + β̂kνn,m )
(61)
w̃τ (˜lτm,n ) ≈ wτ (˜lτm + β̂lτm,n )
(62)
Therefore, the Eq. (46) can be written as
y = x kp′ , lp′ Φh̃ + n.
(63)
The ((m0 − 1)N0 + n0 )-th column of the Φ is
vec(w̃ν (k̃νn,m )w̃τ (˜lτm,n )) as Eqs. (61) and (62). So the offgrid problem becomes to an on-grid problem.
Sparse Bayesian learning (SBL) framework is used here to
estimate the channel matrix h̃ in Eq. (63), denoted as ĥ. The
SBL framework could be simply implemented by considering
β ≡ 0 for the OGSBI in Sec.IV-C. To fully make use of
the previous results as prior in SBL combination, ĤU and
ĤV which are estimated in the U-branch and the V-branch
respectively, are used as initial prior of h̃. That is, vec( 21 ∗
(|ĤU | + |ĤV |)). The prior is close to the convergence.
C. Off-grid Sparse Bayesian Inference
Off-grid sparse Bayesian inference (OGSBI) [30] is used
here to estimate the unknown parameters in Eqs. (55), (56),
(58), (59) and (63). For simplicity, we take Eq. (58) as an
example.
Assume x kp′ , lp′ = 1, n is a complex Gaussian noise with
α0 = 1/σ 2 , σ 2 being the noise variance. Then the YDD could
be assumed with the probability distribution function of
H
H
p(YDD |DU
, βkν , α0 ) = CN (YDD |Ũ (βkν )DU
, α0−1 I),
(64)
where α0 adopts a Gamma hyper-prior, i.e.
p(α0 ; c, d) = Γ(α0 |c, d),
(65)
[Γ(c)]−1 dc α0c−1 e−dα0
where Γ(α0 |c, d) =
with Γ(·) being the
H
Gamma function. And c, d → 0. Assume that DU
∈ CN0 ×M
follows a two-stage hierarchical prior, i.e.,
M
Y
H
p(DU
; ∆) =
H
CN (DU
(m)|0, ∆),
(66)
m=1
H
H
where DU
(m) is the mth column of DU
. ∆ = diag(δ)
denotes the covariance matrix. δ = [δ1 , . . . , δN0 ]T ,
p(δ; ρ) =
N0
Y
Γ(δn0 |1, ρ),
(67)
n0 =1
where ρ > 0 is a small positive constraint (e.g., ρ = 0.01
[27]). Assume that βkνn , which is the n0 th element of βkν ,
0
follows a uniform prior,

 U([− 12 r(n0 − 1), 12 r(n0 )]) if 1 < n0 < N0 ;
if n0 = 1;
U([0, 12 r(1)])
βkνn ∼
0

U([− 12 r(n0 − 1), 0])
if n0 = N0 ,
(68)
where r(n0 ) = k̃νn0 +1 − k̃νn0 .
With the assumptions above, Bayesian framework can be
H
used to obtain E(DU
|YDD ) = µ.
µ = α0 ΣŨ H YDD ,
Σ = (α0 Ũ H Ũ + ∆−1 )−1 = ∆ − ∆Ũ H Σ−1
x Ũ ∆
(69)
(70)
where Σx = σ 2 I + Ũ ∆Ũ H . And denote Ũ = Ũ (βkν ) for
simplicity, which is defined in Eq. (43). µ and Σ are the
H
estimated expectation and variance of DU
, respectively. And
these hyper-parameters, i.e., α0 , ∆ and βkν can be obtained
H
by maximizing p(α0 , δ, βkν , DU
, YDD ). According to [30],
they can be calculated iteratively.
r
hP
i
M
H (m)| + M Σ
|µ(m)µ
−M
M 2 + 4ρ
m=1
n0 n0
it +1
,
δ n0 =
2ρ
(71)
MN + c − 1
α0it +1 =
,
(72)
d + ∥YDD − Ũ µ∥2F + M tr(Ũ ΣŨ H )
βkitν+1 = arg min βkTν P βkν − 2v T βkν ,
(73)
βkν
where it is the number of iterations, µ(m) is
the mth column of µ. δnit0+1 , α0it +1 and βkitν+1
are updates of δ, α0 and βkν , respectively. Let
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n
o
′
′
Ξ ≜ µ(µ)H + Σ, P = Re UDD
(k̃ν )H UDD
(k̃ν ) ◦ Ξ ,
where the symbol n◦ is the Hadamard product.
o
PM
1
′
H
v
=
t=1 Re diag (µ (m)) UDD (k̃ν ) ∆Y (m) ,
M
where ∆Y (m) = YDD (m) − UDD (k̃ν )µ (m). Note that Eq.
(73) can be solved as that of [30].
it
it −1
∥2
The stopping criterion is attained if ∥δ ∥δ−δ
< τ or
it −1 ∥
2
the number of iteration reaches the maximum number itmax ,
where τ is a settled tolerance.
TABLE I
T HE 2D OFF - GRID DECOMPOSITION AND SBL COMBINATION SCHEME
Input: received YDD , predefined grids k̃ν and l̃τ
V-branch
= 0. Prior δ =
1: Initialization: Off-grid parameters βlτ = 0, βk2D
ν
1 PN
H Y H (n) and noise estimate α .
Ṽ
(0)
0
n=1
DD
N
Solve Eq. (55) based on the OGSBI and obtain D̂V , βlτ .
for all m0 = 1, . . . , M0 do
Solve Eq. (56) based on the OGSBI and obtain β̂k2D
|m=m0 , ĥVm0 .
ν
end for
Reconstruct ĤV and β̂k2D
according to Eq. (57)
ν
U-branch
7: Initialization: Off-grid parameters βkν = 0, βl2D
= 0. Prior δ =
τ
1 PM
HY
and
noise
estimate
α
.
Ũ
(0)
(m)
0
DD
m=1
M
2:
3:
4:
5:
6:
Solve Eq. (58) based on the OGSBI and obtain D̂U , βkν .
for all n0 = 1, . . . , N0 do
Solve Eq. (59) based on the OGSBI and obtain β̂l2D
|n=n0 , ĥUn0 .
τ
end for
2D
Reconstruct ĤU and β̂lτ according to Eq. (60)
SBL combination
13: Initialization: construct Φ according to the 2D grid extending Eqs. (61)
and (62), the prior δ = vec( 12 ∗ (|ĤU | + |ĤV |)) and the noise estimate
α0 .
14: Solve Eq. (63) based on the SBL and obtain ĥ, which is the estimation
of h̃
Output: ĥ, β̂k2D
and β̂l2D
ν
τ
8:
9:
10:
11:
12:
D. Computational cost analysis
To make the discussion easier, assume M0 ≥ M , N0 ≥ N
for over-complete dictionary in the DD domain and M0 ≈ N0 .
We also take Eq. (58) as an example. In each iteration,
we firstly calculate Eqs. (69) and (70) with the complexity O(N02 N ) + O(N0 N M ) = O(N02 N ) and O(N02 N ) +
O(N 2 N0 ) + O(N0 N 2 ) + O(N N02 ) = O(N02 N ). The inverse
of Σx has the complexity of O(N 3 ). Updating the parameters
δ and α0 costs O(N02 M ) and O(N N0 M ) + O(N 2 M ) +
O(N02 N ) + O(N02 N ) = O(N02 N ). Calculate the off-grid
parameter by Eq. (73) costs O(N03 ), but we can reduce the
dimension of βkν in the computation by discarding negligible
components as that of [30]. Therefore, the complexity of
the step of off-grid update is about O(P 3 ), recall P ≤ N
is the number of paths. It could be found that with this
procedure, the complexity of this step is O(N02 N ). Similarly,
we can deduce the complexity of solving Eq. (56) is O(M02 M )
for n0 = 1, . . . , N0 , respectively. Thus, the complexity of
obtaining ĤU is O(M02 M N0 ). And the complexity of the
U-branch is O(M02 M N0 ) + O(N02 N ) = O(M02 M N0 ). The
complexity of these off-grid methods are summarized in
Table. II. Note that, though the complexity of the proposed
method is O(M02 M N0 ) + O(N02 N M0 ) + O(N02 M02 N M ) =
O(N02 M02 N M ), the major computational complexity is determined in the combination step, in which a prior close to the
solution is used for initialization. Thus the actual computation
cost is far less than that of the doubly fractional model based
method.
TABLE II
C OMPLEXITY AND COMPUTATIONAL WORKLOAD COMPARISON OF
OFF - GRID METHODS
Method
U-branch (2D off-grid method in [28])
V-branch
doubly fractional (1D off-grid method in [28])
2D off-grid decomposition and SBL combination
Complexity
O(M02 M N0 )
O(N02 N M0 )
O(N02 M02 N M )
O(N02 M02 N M )
V. S IMULATION
Several simulations are performed to demonstrate the tandem off-grid distortion phenomenon and the effectiveness of
the proposed method. The OTFS frame size is M = N = 64,
the carrier frequency is 3 GHz, the subcarrier spacing is 30
KHz. 5 taps time-delay and Doppler-velocity are uniformly
and randomly generated in [0, τmax ] and [−vmax , vmax ], respectively. τmax = 2.6 ∗ 10−6 s and vmax = 70 m/s. SNR is 10
dB. A single pilot is used and the channel coefficients hi are
generated as that in [28]. To show the off-grid effect as well
as the tandem off-grid distortion phenomenon, we exclude the
data in this experiment and the grid resolution is 0.2. For the
parameters in Bayesian framework, c = d = 10−4 , ρ = 10−2 ,
the tolerance τ = 10−3 except for the SBL combination step
where τ = 10−1 . itmax = 200.
Figs. 7 and 8 show the absolute value of the estimated
channel responses reconstructed by the proposed U-branch
(i.e., the 2D off-grid method in [28]), the proposed V-branch,
the proposed 2D off-grid decomposition and SBL combination
method and the traditional OTFS method (as that of [10]),
respectively. The red lines with the mark ”*” denote the true
randomly generated channel coefficients in the DD domain.
It could be found in Fig. 8 that the traditional OTFS cannot
obtain a sparse CSI in the DD domain even a denser grid is
used. While based on the proposed off-grid model, we can
obtain a much more sparse estimation results compared with
that of traditional method.
However, it can be also found in Fig. 7a and Fig. 7b that
the estimated channel response in the rectangle is interfered
by other channel responses with similar Doppler and different
delay. The similar phenomenon could be found in Fig. 7c
that the result in the rectangle is interfered by other channel
responses with similar Delay and different Doppler. Both of
the U-branch and the V-branch show partial information of the
true channel responses. By combining appropriately the results
of the two branches, we could get more accurate channel
estimation result as shown in Fig. 7e. Furthermore, it could
be found that the Doppler corresponding to different delay in
Fig. 7b, and the delay corresponding to different Doppler in
Fig. 7d are the same in same grid interval. While, all the grid
points in the Fig. 7e are relatively independent with their own
fractional delays and fractional Doppler, resulting in closer
grid point to the true Dopplers and delays.
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(a) Off-grid decomposition result of the U-branch.
(b) result of the U-branch (top view).
(c) Off-grid decomposition result of the V-branch.
(d) result of the V-branch (top view).
(e) SBL combination estimation result.
(f) SBL combination estimation result (top view).
Fig. 7. The tandem off-grid effect and the estimation result of the channel matrix in the fractional DD domain.
TABLE III
C LASSIFICATION OF W IRELESS C HANNELS I N S IMULATION
Channel Classification
Frequency Selective
Doubly Selective
Time Selective
M ∆f τmax
≥1
≥1
≪1
N T vmax fc /c
≪1
≥1
≥1
To further demonstrate the effectiveness of the proposed
method, normalized mean square error (NMSE) [28] is used
for comparison. τmax = 8.3 ∗ 10−6 s corresponding to the
maximum propagation distance about 2.5 km and highest
velocity vmax = 500 km/h. N0 = M0 = 2N = 2M = 64
as the virtual sampling rate 0.5 in [28]. Data is added to the
OTFS grid and the received YDD is also truncated as that in
[28], where 7 sampling points are used in Doppler dimension
( −3 ∼ 3 for the normalized Doppler shift) while 5 sampling
points are used in delay dimension (0 ∼ 4 for the normalized
exp(−0.1lτi )
delay shift). hi ∼ CN (0, q lτi ) where q lτi = P exp(−0.1l
τi )
i
and other parameters are chosen as that in [28]. In order to
show the influence of the tandem off-grid distortion, three
types of channel, i.e, the frequency selective (FS) channel,
the time selective (TS) channel and the doubly selective (DS)
channel, are studied here. Based on the classification in [38],
the frequency, time and doubly selective channel are defined in
the Table. III. In the simulation, the FS channel is assumed to
have no Doppler spread, that is the Doppler velocity is 0, while
delay is still generated uniformly and randomly in [0, τmax ].
The TS channel is assumed to have no delay spread, that is
the delay is assumed to be 0, while Doppler velocity is still
generated uniformly and randomly in [−vmax , vmax ]. For the
DS channel, the delay and Doppler are generated uniformly
and randomly as that of [28].
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Fig. 8. Channel estimation in the integer DD domain (oversampling).
It can be found from Fig. 9 that the methods based on the
off-grid model are more accurate than the traditional OTFS
estimator because the sparsity of the channel response as well
as fractional DD grid are considered. Compared with the result
of the 2D off-grid method (or equivalently the U-branch only)
in [28]), a lower NMSE can be achieved by the proposed
method in all types of channels. Note that, 7 measurements are
used in the Doppler dimension and 5 measurements are used
in the delay dimension and the number of path is 5 as that of
[28]. In the case of the FS channel, Doppler spread is assumed
to be zero, so fractional delay dominates the performance with
5 measurements being used. While in the case of the TS
channel, time spread is assumed to be zero thus fractional
Doppler dominates the performance with 7 measurements
being used. Therefore, the performances of these methods
in the TS channel are better than that in the FS channel
because of number of measurements. For the case of the DS
channel, 5 paths are randomly generated in the whole 7 × 5
delay-Doppler area, which leads to smaller mutual interference
between paths but introduces the tandem off-grid distortion. So
the performance of the 2D off-grid is deteriorated as shown in
Fig. 7a in the DS channel, which is even worse compared with
its performance in the TS channel. For the proposed method,
the tandem off-grid distortion is mitigated as shown in Fig.
7e. Its performance in the DS channel will not be bounded
by the line of the FS or TS channel. Owing to smaller close
interference between paths, its performance in the DS channel
is better than that in the TS channel. In addition, the accuracy
of the proposed method is surprisingly even better than that
of the doubly fractional model based method (or equivalently
the 1D off-grid method in [28]) in cases that SNR is relatively
high or the number of measurements is relatively large. This
implies that the decomposition step may obtain extra gain
which might owes to its ability to suppresses the mutual effect
between delay and Doppler. And the 2D structure information
is reserved.
Though the SBL combination step has the same complexity
as that of the doubly fractional model based method, the
average number of iterations of this step is 2.70 to get
convergence, i.e., E[it ] = 2.70. Usually, hundreds iterations
are needed for the SBL to get convergence for the doubly
Fig. 9. Comparison of the NMSE.
fractional model. Thus the computational workload of the
proposed method is much smaller than the doubly fractional
model based method in [28].
VI. C ONCLUSIONS
In this paper, the impact of fractional Doppler and fractional
delay on the OTFS system is studied. It is found that the
fractional delay and Doppler breaks the sparsity and deteriorate the effective SNR. An important result is that the severe
performance loss caused by fractional parameters cannot be
eliminated even a large bandwidth and frame duration is used.
To relieve the degradation caused by fractional parameters, we
expand the on-grid model to an off-grid model. Based on this
model, the state-of-the-art off-grid assumptions, models and
their transformations are formulated. Furthermore, we found
the existence of tandem off-grid distortion in estimating 2D
off-grid parameters. In order to mitigate the tandem off-grid
distortion, a 2D off-grid decomposition and SBL combination
method is proposed. This tandem off-grid distortion is eliminated by firstly decomposing the received signal in two related
mesh grid for channel feature extraction, then a combination
algorithm based on SBL is designed to merge the results.
Our simulation results show that the proposed scheme is more
accurate, and has lower computational cost compared with the
existing doubly fractional model based method.
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