Received 19 July 2023, accepted 2 August 2023, date of publication 9 August 2023, date of current version 7 September 2023.
Digital Object Identifier 10.1109/ACCESS.2023.3303814
3D-IPRDSOMP Algorithm for Channel Estimation
in Massive MIMO With OTFS Modulation
GUO-RUI WANG
AND ZHONG-XUN WANG
School of Physics and Electronic Information, Yantai University, Yantai 264005, China
Corresponding author: Zhong-Xun Wang (ytdxwzx@163.com)
This work was supported in part by the Graduate Innovation Foundation, Yantai University, under Grant KGIFYTU2306.
ABSTRACT In this paper, we propose a channel estimation method for massive multi-input multioutput (MIMO) with orthogonal time frequency space (OTFS) modulation. Our method aims to reduce
inter-symbol interference (ISI) when the subtraction of the two delay paths is smaller than the resolution
of the system. We propose a 3D inner product proportion reduce difference structured orthogonal matching
pursuit (3D-IPRDSOMP) algorithm that combines the characteristics of 3D structured sparse channels. With
knowledge of the number of dominant paths, we use the number of local optimums to match the number of
distinguishable dominant paths with backtracking. The algorithm approximates similar and indistinguishable
dominant paths as identical ones. The estimated number of distinguishable paths as the iteration’s termination
condition increases the channel estimation’s accuracy. Simulation results show better performance than the
traditional algorithm under different conditions.
INDEX TERMS Massive MIMO, OTFS, OMP, compressed sensing, channel estimation.
I. INTRODUCTION
In the future, more wireless communication scenarios
will be considered, particularly wireless communication in
high-mobility situations become more important [1], [2].
Motivated by the demand, the amount of related research
is increasing. However, orthogonal frequency division multiplexing (OFDM) modulation is sensitive to Doppler shift,
significantly affecting communication efficiency [3]. The two
modulations are related in the mathematical construction, and
OTFS modulation is considered a precoding scheme based
on OFDM [4]. To solve the problem, an OTFS modulation
scheme was proposed in 2D modulation, which provides
both time and frequency diversity gain, transforming the
time-varying channel into a quasi-time-invariant channel [5].
The efforts have addressed the formulation, equalization,
estimation, and detection of the input-output relations in vectorization. OTFS massive MIMO allows for higher spectrum
efficiency and robustness in rapidly changing channels [6],
[7], [8].
A framework for channel estimation schemes in OTFS
massive MIMO systems was proposed in [7]. There is sparThe associate editor coordinating the review of this manuscript and
approving it for publication was Li Zhang.
VOLUME 11, 2023
sity in the delay and Doppler dimension. This property
is exploited to reduce the pilot overhead and complexity
channel estimation. Massive MIMO systems offer substantial benefits in enhancing spectrum efficiency. However, the
downlink channel ought to be estimated to obtain channel
state information (CSI). The uplink and downlink channel state information for time division duplexing (TDD) is
complex conjugate and interconverted without transmitting
CSI information [9]. In [10], the authors proposed methods
for the semi-blind estimation of sparse channels in MIMO
systems. This approach reduces the computational complexity by converting the semi-blind estimation problem into a
reduced-rank filtering problem through a novel formulation
of the sparse matrix. In [11], the authors estimated channels
in MIMO systems with discrete priors. Channel estimation
is an essential component of signal demodulation, and the
channel estimation algorithms in the delay-Doppler domain
are described in [12] and [13]. Similarly, obtaining CSI for
OTFS massive MIMO systems is essential, and accurately
estimating the CSI of the channel is a challenging topic.
In [14] and [15], the authors proposed a sparse reconstruction
algorithm for block sparse signals, using the prior information
of block sparsity. This approach enables the recovery of the
estimated sparse block signals at low sampling rates.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.
For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/
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TABLE 1. Summary of literature survey.
In OFDM massive MIMO systems, [6] proposed a sparse
reconstruction channel estimation scheme. With the idea,
we apply it to OTFS massive MIMO channel estimation
problem, where we reconstruct the channel matrix using
fewer pilots because of the prior information that the channel is known to be sparse. In [20], the authors proposed a
compressed sensing channel estimation model for traditional
multipath channels. The application of compressive sensing
in the channel and a novel method for estimating sparse
multipath channels are proposed in [21].
The input-output models of OTFS single-input singleoutput (SISO) systems use a method that transmits the
received signal as a two-dimensional (2D) convolution in the
delay-Doppler domain and a pulse-based channel estimation
technique. A pseudo-random noise (PN) sequence was proposed by the pilots in the delay-Doppler domain, and they
proposed that a long PN sequence achieved better performance [22]. In [7], the authors proposed an estimation channel model with compressed sensing for OTFS massive MIMO
systems. They compared the traditional orthogonal matching pursuit (OMP) algorithm with the proposed 3D structured orthogonal matching pursuit (3D-SOMP) algorithm.
The 3D-SOMP channel estimation algorithm increases the
performance with the number of dominant paths as the
prior information. However, this is not a practical sparse
channel recovery algorithm. There may be similar delay
paths in the channel model, for example, the spatial channel
model (SCM) channel [23]. If the interval of the dominant paths is below the delay resolution of OTFS systems, it is considered a single path, and the estimation
based on the number of dominant paths necessarily introduces error paths, leading to a degradation of the estimation
performance.
To solve this problem, we analyze the reasons for the poor
performance of the traditional OMP algorithm. We optimize
the 3D-SOMP algorithm by backtracking. We work backward
from the number of dominant paths to match the local optimum near the number of dominant paths. We approximate
that the solution is the number of distinguishable dominant paths. The proposed algorithm effectively reduces the
selection of extra atoms because of coherent inter-symbol
interference and increases channel estimation accuracy.
The specific contributions are summarized as follows.
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1) We formulate the channel estimation problem as a sparse
signal recovery problem based on the previously demonstrated characteristics of the sparsity of the OTFS massive
MIMO channel structure. Then, we analyze the reason for the
poor performance of the traditional OMP algorithm for sparse
channel reconstruction. The traditional iterative approach
gets stuck in the local optimum because of a lack of error correction mechanisms, resulting in poor channel reconstruction.
Although this deficiency is compensated by setting different
steps, they are required to be different in different cases,
which does not fundamentally solve the problem.
2) The 3D-SOMP algorithm is designed to exploit sparsity
in the time domain, block sparsity in the Doppler domain,
and burst sparsity in the angular domain. It combines different methods for different dimensions. We propose the
3D-IPRDSOMP algorithm by exploiting the property that the
inner product descent ratio difference of the OMP algorithm
has a maximum value near sparsity. We match the index with
a maximum value smaller than the number of dominant paths
and use it as the number of distinguishable paths and one of
the conditions for iteration termination.
The rest of the paper is organized as follows. In Section II,
we introduce the structural framework for OTFS SISO systems, OTFS massive MIMO systems, and discrete inputoutput relations. In Section III, the reason why the number
of distinguishable dominant paths is smaller than the number
of dominant paths is presented, as is the core principle of
the proposed algorithm. We introduce the 3D sparsity of
the OTFS massive MIMO downlink, the traditional OMP
algorithm, and illustrate the principle of the 3D-IPRDSOMP
algorithm. In Section IV, we present the simulation results.
In Section V, we present the conclusions of the paper.
II. SYSTEM MODEL
In Fig. 1, the OTFS modulation is located before OFDM
modulation, and the demodulation part is located after
OFDM modulation, essentially the precoding operation of
OFDM systems. The time-varying channel is converted into
a time-invariant channel by performing a fast Fourier transform (FFT) and an inverse fast Fourier transform (IFFT)
on the two dimensions of OFDM, respectively, thus reducing the performance degradation caused by the time-varying
channel.
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FIGURE 1. OTFS SISO architecture. OTFS modulation is seen as a combination of OFDM modulation and processing.
A. OTFS SISO MODULATION
The SISO model of the OTFS systems consists of two components: modulation and demodulation. OTFS modulation is
a 2D data manipulation of the time and frequency dimensions
of the OFDM channel, transforming the symbols from the
time-frequency dimension to the delay-Doppler dimension.
The modulation takes the M × N 2D constellation symbols (after QAM, QPSK, etc. mapping) in the delay-Doppler
domain XDD [l, k] and performs an inverse symplectic finite
Fourier transform (ISFFT) to transform the symbols from
the delay-Doppler domain to the frequency-time domain,
which are represented as XFT [f , t]. The ISFFT is divided
into two parts. First, the IFFT modulation is performed on the
time-delay dimension symbols in the delay-Doppler domain,
transforming the symbols from the delay dimension to the
frequency dimension. Then, the FFT is performed on the
Doppler dimension symbols, transforming the symbols from
the Doppler dimension to the time dimension. Where the
mathematical expression of ISFFT is
XISFFT [m, n] = √
1
N
−1 M
−1
X
X
MN
k=0 l=0
j2π
XDD [l, k]e
nk ml
N −M
,
(1)
where XDD [l, k] is the transmit symbol in the delay-Doppler
domain, and XISFFT [m, n] is the transmit symbol in the
frequency-time domain, and l = 0, 1, · · ·, M − 1, k = 0, 1,
· · ·, N − 1, m = 0, 1, · · ·, M − 1, n = 0, 1, · · ·, N − 1 [19].
We convert the mathematical expression (1) into a matrix
expression [7] as
XISFFT = FM XDD FH
N,
(2)
where FM ∈ CM ×M is the discrete Fourier transform (DFT)
N ×N is the inverse discrete Fourier
matrix and FH
N ∈ C
transform (IDFT) matrix, and XDD is a matrix consisting
of XDD [l, k]. The symbol XISFFT needs to be added to the
transmit windowing Wtx ∈ CM ×N after ISFFT, and the transmit windowing symbol obtains the frequency-time domain
symbol XFT by performing Hadamard product operation with
XISFFT , which XFT can be expressed as
XFT = XISFFT ⊙ Wtx .
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(3)
In this paper, we use the ideal transmit windowing and the
ideal receive windowing. A transmit windowing Wtx is an allone matrix, so the time-frequency symbols in the transmitter
are approximated by the symbols after ISFFT. Therefore, XFT
is similar to XISFFT as
XFT [f , t] = XISFFT [m, n] ,
(4)
where f = 0, 1, · · ·, M − 1, t = 0, 1, · · ·, N − 1. The signal
is converted from the frequency-time domain XFT [f , t] to
the delay-time domain XIFFT [τ, t] by performing an IFFT on
the frequency dimension in the 2D frequency domain symbol
XFT [f , t] as
M −1
1 X
j2π fMτ
,
XFT [f , t]e
XIFFT [τ, t] = √
M m=0
(5)
where XFT [f , t] is the transmit symbol in the frequency-time
domain, XIFFT [τ, t] is the transmit symbol in the frequency
domain, and τ = 0, 1, · · ·, M − 1, t = 0, 1, · · ·, N − 1 [19].
The corresponding matrix representation as
FT
XIFFT = FH
MX ,
(6)
M ×M denotes the IDFT matrix, XIFFT ∈
where FH
M ∈ C
M
×N
C
is a matrix representation of XIFFT [τ, t]. For convenience of representation, the 2D matrix S ∈ CM ×N is used to
denote the time-delay domain matrix XIFFT . The IFFT modulation is similar to the traditional OFDM modulation in that
both convert the frequency dimension symbols into the time
dimension, enabling channel transmission. The integration of
the equations (2)(6) is obtained as
DD H
S = FH
FN = XDD FH
M FM X
N,
(7)
where S = [s1 , s2 , · · · , sN ] is a M × N matrix, each vector
si is an OFDM symbol frame, the bandwidth occupied by
each OFDM symbol frame is M 1f , the duration is T , and
the subcarrier interval is 1f .
Then we transform the matrix S into a one-dimensional
time vector by vectorization. To avoid inter-symbol interference, we add a cyclic prefix (CP) to the modulation by
multiplying the CP addition matrix ACP ∈ C(M +NCP )×M with
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G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO
the matrix S. The specific formula is
s = vec {ACP S} ,
(8)
where vec {} denotes the vectorizing of a matrix [7]. The
resulting vector s ∈ C(M +NCP )N ×1 is the vector that is transformed into a signal that is transmitted in the time dimension
after adding cyclic prefixes and vectorizing.
B. OTFS SISO DEMODULATIO
In this section, we describe the demodulation of OTFS SISO
systems. OTFS demodulation is the modulation inverse at the
receiver. First, the received time series is the vector r, which
results from the vector through the channel response. The
received vector r needs to be matrixed as
R = invec {r} ,
(9)
where invec {} denotes the invectorizing of a vector and R is
the result of matrixing the receiver vector r, which contains
the CP. Correspondingly, we use the CP deletion matrix at the
receiver. In contrast to the modulation, the signal is restored
to the delay-Doppler domain after FFT and SFFT.
Then, the matrix R after removing CP, which is abbreviated
as RDCP ∈ CM ×N , needs to be FFT in the time dimension
to transform the signal from the delay-time domain to the
frequency-time domain, thus obtaining the frequency-time
domain receive matrix. The corresponding matrix-form transformation equation is
YFT = FM RDCP ,
(10)
where YFT ∈ CM ×N is the frequency-time domain receive
matrix. The corresponding mathematical expression for the
FFT is
YFFT
M −1
1 X
j2π τMf
,
RDCP [τ, t]e
[f , t] = √
M m=0
(12)
(13)
where the mathematical expression of YDD is denoted by
YFT ,W [m, n], where m = 0, 1, · · ·, M − 1, n = 0, 1, · · ·, N −
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YDD [l, k] = √
MN
N
−1 M
−1
X
X
j2π
YFT ,W [m, n]e
kn lm
N −M
,
k=0 l=0
(14)
where YDD [l, k] is the received symbol in the matrix
YDD . Combining equation (14), the total matrix equation is
obtained as
YDD = FH
M FM RDCP FN ,
(15)
YDD
where
is the matrix in the delay-Doppler domain, corresponding to the delay-Doppler domain matrix XDD at the
transmitter. The relationship between the receive matrix YDD
and the transmit matrix XDD is a 2D convolution concerning
the delay-Doppler channel HDD ∈ CM ×N .
C. OTFS MASSIVE MIMO
Previously, we introduced the discrete input-output relation
for the OTFS SISO systems. To improve spectrum efficiency,
we are increasingly adopting massive MIMO [24]. The next
step is to extend the discrete input-output relationship from
the OTFS SISO system to the OTFS massive MIMO system.
The symbol Nt is the number of antennas on the transmitter
and Nr is the number of antennas on the receiver. Each
antenna uses OTFS modulation. In general, the rectangular
windowing Wtx [n, m], Wrx [n, m] is used to match the filters
at the transmitter and receiver.
For every single transmitting and receiving antenna,
we consider that there are P paths (taps). The index i represents the i-th delay tap τi , the Doppler shift tap νi and the
channel gain hi . The impulse response in the delay-Doppler
domain is
h(τ, ν) =
The received matrix YFT,W is subjected to an M -point IFFT
that converts the symbols from the frequency dimension to the
delay dimension and an N -point FFT that converts the symbols from the time dimension to the Doppler dimension. The
matrix of the delay-Doppler domain signal transformation is
obtained as
FT,W
YDD = FH
FN ,
MY
1
(11)
where RDCP [τ, t] is receive matrix after removing CP. After
obtaining the frequency-time domain signal YFT (consisting
of YFFT [f , t]), it needs to be transformed into the delayDoppler domain. The matrix YFT,W ∈ CM ×N is obtained by
Hadamard product with the corresponding receive windowing matrix Wrx ∈ CM ×N in the previous section, where the
corresponding receive windowing matrix is also the all-one
matrix. The matrix form of the operation is
YFT,W = YFT ⊙ Wrx .
1 and YFT ,W [m, n] is the mathematical expression consists of
YFT,W . The corresponding mathematical expression is
P
X
hi δ (τ − τi ) δ (ν − νi ) ,
(16)
i=1
where δ (τ − τi ) and δ (ν − νi ) are the Dirac delta function
with τi and νi shift, separately. Considering the number of taps
present between the p-th transmitter and the q-th receiver, the
impulse response in the delay-Doppler domain is
hqp (τ, ν) =
P
X
hqpi δ (τ − τi ) δ (ν − νi ) ,
(17)
i=1
where p = 1, 2, · · ·, Nt , q = 1, 2, · · ·, Nr . In the following,
the input-output relation between the transmitter and receiver
antennas will be described.
βi
i
, νi = NT
where ai
For SISO systems, we define τi = Mα1f
and βi denote the delay taps and Doppler taps, respectively.
Among them, ai and βi are considered integer parts, although
fractional delay and fractional Doppler effects will be present
in practical situations. The authors generalized this from
SISO to the MIMO modulation in [16]. From the MIMO
channel input-output relationship, we deduce that
y = Hx + v,
(18)
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G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO
where the vectors x, y, v ∈ CNM ×1 , H ∈ CNM ×NM [24].
Then, we derive the discrete input-output relation in OTFS
massive MIMO. With the equations of the SISO modulation,
we deduce the input-output relation model in OTFS massive
MIMO is
y1 = H11 x1 +H12 x2 + · · · + H1Nt xNr + v1
y2 = H21 x1 +H22 x2 + · · · + H2Nt xNr + v2
(19)
..
..
..
..
..
.
.
.
.
.
yNr = HNr 1 x1 +HNr 2 x2 + · · · + HNr Nt xNr + vNr
where Hqp denotes the equivalent channel matrix between the
p-th transmit antenna and the q-th receive antenna and (19)
represents a linear model of the input-output relation, which
is transformed into a matrix


H11
H12
...
H1Nr
 H21
H22
...
H2Nr 


HMIMO =  .
(20)
.
..  .
.
..
..
 ..
. 
HNt 1 HNt 2 . . . HNt Nr
After further simplifying the matrix HMIMO , we obtain
yMIMO = HMIMO xMIMO + vMIMO , wherevMIMO ∈ Cnr NM ×1
xMIMO ∈ Cnt NM ×1 , HMIMO ∈ Cnt NM ×na NM , which denotes
the OTFS massive MIMO discrete input-output matrix.
III. 3D-IPRDSOMP BASED CHANNEL ESTIMATION IN
OTFS MASSIVE MIMO SYSTEMS
In this section, we analyze different channel dimensions to
transform a channel estimation problem into a sparse signal
recovery problem. We performed an analysis of estimation
errors in the OMP algorithm with different steps. Using the
observation that the inner product in sparse signal recovery continuously decreases with the number of iterations
and that the difference in decrease ratio is the largest value
after all paths are selected. We proposed the 3D-IPRDSOMP
algorithm to achieve better performance.
A. DELAY PATHS AND SYSTEM RESOLVABLE DELAY PATHS
Simulated channels typically include the common Rayleigh
and Rice fading channels. We use the 3GPP SCM channel
model based on geometric random variables to characterize the communication channel more accurately. This model
includes three types: urban macro-cell, urban micro-cell, and
suburban macro-cell. In the case of the urban macro as an
example, the number of delay paths is six, and the delay
correlation coefficient is determined by
τn′ = −rDS σDS ln zn ,
n = 1, . . . , N ,
(21)
where N represents the number of delayed paths,
zn (n = 1, 2, · · ·, N ) obeys the uniform distribution within
(0, 1), rDS is the scale factor of the delay distribution, σDS
is the delay spread. The delay values of the N paths satisfy
′ > τ′
′
the descending order τ(N
)
(N −1) > τ(1) .
The specific value of the delay tap τn is determined by
( ′
)
′
τ(n) − τ(1)
Tc
floor
, n = 1, 2, · · · , N , (22)
τn =
16
Tc /16
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where the function floor is obtained by rounding the nearest
′ − τ ′ = 0,
integer to negative infinity [25]. Because of τ(1)
(1)
τ1 = 0. It represents the main path. The minimum spacing
between different path delays is 1τ = Tc /16. For OTFS
or OFDM systems, the matrix estimated by the channel is
discrete and does not distinguish all the delay paths, which
is related to the sampling rate of the system. We consider
the minimum delay 1τ that is distinguished by the OTFS
systems. When ∃τi − τi−1 < 1τ, i = 2, 3, · · ·, N , if the
minimum interval in N delay paths is less than the resolution
of the OTFS systems, the number of distinguishable delay
paths of the system K < N . For the whole OTFS system,
the number of delayed paths of the channel is K , not N .
It is found that the relationship between inner product
summation and residuals is more significant with the number
of iterations in the compressed sensing channel estimation.
When the dominant path N is six, the number of distinguishable delay paths of the system K ≤ 6. The most likely to
occur are K = 4, 5, 6.
(n+1)
uτ
,
(23)
(n)
uτ
where uτ represent the inner product after selecting dominant
paths. It represents the correlation between the received signal
and the sparse basis vector. It represents the proportional
decrease of the inner product. The ∇u denotes the division
between inner products. To equalize the inner product decline
ratio using the residual, the formula as
n
∇u
n
ω =1
,
(24)
γn
∇un =
where γ represents the residual in the reconstruction process,
and the confidence of the reconstruction vector increases
gradually as γ decreases. After all distinguishable delay paths
are selected, the decrease in the inner product significantly
reduces, and the residuals become smaller and stabilize.
In this case, the difference between the inner product uτ and
the residual vector γ is obtained by taking the difference
between the inner product and the residual vector. There is
a local maximum ωNp around the number of distinguishable
paths.
As shown in Fig. 2, the number of iterations at the local
maximum value approximates the number of distinguishable
dominant paths in OTFS systems. Therefore, the number
of distinguishable dominant paths is estimated in channel
reconstruction with compressive sensing, reducing selected
wrong dominant paths.
B. 3D STRUCTURED SPARSITY CHANNEL AND TRAINING
PILOTS
Multiple antennas in massive MIMO systems allow the signal
to be studied in spatial dimensions, increasing the information
dimension. We consider a base station with Nt antennas at the
base station and K single-antenna users at the receiver. The
users are assumed to have some degree of concentration, but
the direction between them and the base station is random.
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G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO
FIGURE 2. Variation of differential values with the number of iterations in
the channel reconstruction process in different distinguishable delay
paths.
A uniform linear array (ULA) of antennas is used, where the
spatial angle ψsi is defined as
d
sin θsi ,
(25)
λ
where d is the antenna spacing and λ is the wavelength of the
carrier frequency [26]. The model assumes that the system
has multiple dominant paths, each with multiple sub-paths,
and ignores the difference in delay in the sub-paths.
The channel response in the delay-Doppler-space domain
is obtained by the discrete Fourier transform (DFT) of the
equation
ψsi =
DDS
Hℓ,k,p
=
N
X
recovery problem while reducing the pilot overhead [7].
However, the sparse characteristics of the three dimensions
are different, and with prior information, we adopted different schemes to recover the channel information in different
dimensions.
The figure above shows the location of the pilots in the
delay-Doppler domain. The guard intervals are left around
the pilots to prevent mutual interference between the pilot
response and the data symbol response. This interference
affects subsequent channel estimation and symbol detection.
To reduce the correlation between pilots and increase channel
recovery performance, the PN sequence is used for the pilots.
Different pilots need to be adopted for different antennas, and
the orthogonalized pilots distinguish users at the receiver end
and reduce inter-user interference. The pilots are placed in
the p-th antenna as follows. The delay domain is placed as
ℓ = 0, 1, · · · , Mτ − 1, the Doppler dimension is placed as
k = −Nν /2, · · · , 0, · · · , Nν /2−1, and the spatial dimension
is placed on the antenna p = 0, 1, · · · , Nt − 1 for the pilots.
The desired signals and the pilots on the Nt antennas of the
transmitter transmit simultaneously, and the received signals
are separated from the desired signals and the pilots. The
received pilots in the delay-Doppler domain at the user side
can be expressed as
yℓ,k
=
g −1
NX
t −1 M
X
Ng
2 −1
e
p=0 ℓ′ =0 k ′ =− Ng
2
(M +NCP ) H DDS
ℓ′ ,k ′ ,p xℓ−ℓ′ ,k−k ′ ,p
+ vℓ,k .
(29)
k
h(n−1)(M +NCP )+1,(ℓ)M ,p e−j2π(n−1) N ,
(26)
n=1
DDS channel is
where the delay-Doppler-space-domain Hℓ,k,p
DDA as
transformed to Hℓ,k,r
DDA
Hℓ,k,r
NX
t −1
To simplify the equation (29), denote ϕℓ−ℓ′
(ℓ−ℓ′ )k ′
j2π
e N (M +NCP ) . By combining (27) and (29),
yℓ,k =
rp
DDS j2π Nt
,
Hℓ,k,p
e
(27)
and
−1 Mg −1 2 −1
X
X X
′
r=− N2t ℓ =0
×
DDS
Hℓ,k,p
NtX
/2−1
r=−Nt /2
DDA
Hℓ,k,r
e
pr
−j2π N
t
=
Ng
Nt
2
p=0
NX
t −1
k ′ =−
−j2π Nrp
e
t
ϕℓ−ℓ′ HℓDDA
′ ,k ′ ,r
Ng
2
xℓ−ℓ′ ,k−k ′ ,p + vℓ,k ,
(30)
p=0
,
(28)
DDS , H DDA ∈ CM ×N ×Nt represent the channel
where Hℓ,k,p
ℓ,k,r
response in the delay-Doppler-space and delay-Dopplerangle dimension separately [7].
In OTFS massive MIMO systems, considering the case of
finite channel dominant paths, the locations of the delay taps
are related to the dominant channel and are relatively random.
Here, the delay-time domain in the channel is considered to
be absolutely sparse. Ideally, the value of the delay index is
zero except for the delay tap location.
In summary, the sparsity properties of channel models
in three dimensions have been demonstrated, and we transformed the channel estimation problem into a sparse signal
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(ℓ−ℓ′ )k ′
j2π N
X
where k
= −Nν /2, · · · , 0, · · · , Nν /2 − 1, ℓ =
0, 1, · · · , Mτ − 1, k ′ = −Ng /2, · · · , 0, · · · , Ng /2 − 1, ℓ′ =
0, 1, · · · , Mg − 1 [7]. The authors proposed the relationship
between the response of the pilots after passing the system and the input signals, thus establishing a model of the
DDA provide theoretical support for
input-output relation to Hℓ,k,r
subsequent channel estimation.
The derivation process is omitted, and the final expression
that conforms to the standard form of compressive sensing is
obtained as
y = Zc,W h + v,
(31)
where Zc,W = [W ⊙ Zc,−Nt /2 , . . . , W ⊙ Zc,0 , . . . , W ⊙
Zc,Nt /2−1 ] ∈ CMτ Nν ×Mg Ng Nt , Zc,r ∈ CMτ Nν ×Mg Ng , where r =
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G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO
−Nt /2, · · · , 0, · · · , Nt /2−1, the (ℓNν +k +1+Nν /2, ℓ′ Ng +
P t −1 −j2π rp
Nt x
k ′ + 1+ Ng /2)-th element equal to N
ℓ−ℓ′ ,k−k ′ ,p
p=0 e
in Zc,r , where k = −Nν /2, · · · , 0, · · · , Nν /2 − 1, ℓ =
0, 1, · · · , Mτ − 1, k ′ = −Ng /2, · · · , 0, · · · , Ng /2 − 1, and
ℓ′ = 0, 1, · · · , Mg − 1. Where W ∈ CMτ Nν ×Mg Ng , the
(ℓNν +k +1+Nν /2, ℓ′ Ng + k ′ +1+Ng /2)-th element equal to
iT
h
ϕℓ−ℓ′ in W, and h = hT−Nt /2 , · · · , hT0 , · · · , hTNt /2−1 − 1 ∈
CMg Ng Nt ×1 , where hr ∈ CMg Ng ×1 from HℓDDA
′ ,k ′ ,r , it correspondence the (ℓNg + k + 1 + Ng /2)-th element in hr .
By simplifying the complex formula to the above form,
the sensing matrix in compressive sensing is represented
as Zc,W , and the channel vector is transformed into a 3D
delay-Doppler-angle domain tensor. The received vector is
known, and the sensing matrix (obtained from the receiver)
is used to estimate the channel vectors through the sparse
reconstruction principle. In this paper, the OMP and 3DSOMP algorithms are used for sparse channel reconstruction,
and we analyze the reasons for the poor performance of the
OMP algorithm and propose a 3D-IPRDSOMP algorithm.
FIGURE 3. An OTFS frame in the delay-Doppler domain with pilots and
guard intervals.
C. OMP ALGORITHM ANALYSIS AND 3D-IPRDSOMP
ALGORITHM PRINCIPLE
The traditional OMP algorithm, as well as the 3D-SOMP
algorithm, rely on sparsity as prior knowledge. However,
sparsity is unknown in wireless channels, making these
algorithms impractical. To solve the problem, we proposed
an algorithm to estimate the number of dominant paths in
the channel. It is based on the decreasing differential ratio
between inner products and is applied to sparse channel
reconstruction. We analyze the poor performance of the traditional OMP algorithm and propose a solution to the problem.
The traditional OMP algorithm [27] recovers signals
from random measurements via orthogonal matching pursuit.
As one of the earliest and most widely used iterative algorithms, the matching pursuit (MP) algorithm [28] is known
as the base algorithm of the greedy algorithm. For channel
estimation in massive MIMO systems, the compressive sensing formulation is
y = 9h + v,
(32)
where 9 ∈ CMτ Nν ×Mg Ng Nt is the sensing matrix, the noise
vector v ∈ CMg Ng Nt ×1 [21]. The channel vector h ∈
CMg Ng Nt ×1 of massive MIMO systems is converted from the
channel matrix in the delay-Doppler-space domain. The core
of the OMP algorithm is to select the column with the highest
correlation with the current residual vector, perform Schmitt
orthogonalization and normalization on the selected atom,
and update the residuals. This process ensures that the residuals are orthogonal to the previously selected atoms. In each
iteration, the algorithm updates the observation matrix by
removing the selected columns, which ensures that the atoms
are not repeatedly selected and the residuals are rapidly
reduced. By iteratively selecting the atoms with the highest
correlation with the residuals and updating the residuals and
VOLUME 11, 2023
FIGURE 4. The normalized mean square error (NMSE) performance
curves of OMP algorithm with different step sizes.
observation matrix, the OMP algorithm effectively estimates
the sparse channel vector.
The main steps of the OMP algorithm are as follows.
First, it only selects one atom at each iteration, and once an
atom is selected, it cannot be changed in the next iteration.
Therefore, the larger the length of the vector h, the greater
the number of atoms to be selected, and the larger the number
of iterations, the greater the probability of selecting the error
atom in the same case, with a certain sparsity. When only one
atom is selected at each iteration, it is easy to introduce the
wrong atoms and get stuck in the local optimum that does not
reconstruct the channel well in subsequent iterations.
Fig. 4 shows the simulation results of the OMP algorithm
for different step sizes. The channel reconstruction performance increases as the step size increases and more atoms
are selected at each iteration. The improvement results from
the reduced probability of getting stuck in the local optimum.
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G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO
However, excessive steps also lead to the wrong selection of
atoms, significantly degrading the performance of channel
reconstruction. Therefore, selecting only one atom at each
iteration with the traditional OMP algorithm results in poor
channel reconstruction.
To solve this problem, the 3D-SOMP algorithm was proposed to increase the atom selection efficiency based on the
structured characteristics of the channel and the correlation
between the indexes of the vector [7]. For the delay-Dopplerangle domain, with a fixed number of dominant paths, ideally,
there are non-zero values only at the delay index of the
channel and zero values for the other delay indexes.
The decreased ratio of the inner product of the dominant
path changes drastically when the orthogonal matching pursuit algorithm selects the wrong atoms. To mitigate the effect
of these extreme values in the early stage of reconstruction,
we proposed using each iteration’s residual as the confidence level of the channel reconstruction effect. Specifically,
we increase the estimation performance of distinguishable
dominant paths by dividing the differential between indicators of the inner product decrease ratio by the residual. Then,
we backtrack to find the local optimum index. The index
is used as the number of distinguishable dominant paths,
and the corresponding reconstructed channel vector is used
as the estimated channel vector. The approach increases the
reconstruction accuracy, especially for scenarios with many
dominant paths and significant channel sparsity.
Algorithm 1 Proposed 3D-IPRDSOMP Algorithm
Input:
Vector y, Sensing matrix 9
Output:
Recovered channel vector b
h  = h(Pos) 
1 Initialization:
2 i ← 0, j ← 0, max_ω = 0
(i) ← 0, r(i) = y − 9h(i) , γ i = r(i)
3  ← ∅, h
2
4 for i ≤ Np do
5
i = i + 1;
6
e = 9 H r;
7
E = invec{e};
8
eτ (m) = E(1) (m, :) ;
9
uτ (i) = max(eτ (m));
(i)
10
mτ = arg max
m eτ (m);
11
12
13
14
15
16
17
18
19
D. 3D-IPRDSOMP ALGORITHM
The proposed 3D-IPRDSOMP algorithm is presented in
Algorithm 1. The proposed algorithm exploits the structured
characteristics of the channel and incorporates the fundamental concept of the OMP algorithm to increase channel
estimation accuracy. Furthermore, it exploits the differential
between mean inner products during the iterative process to
reach the local maximum when searching from the known
number of dominant paths to estimate the number of distinguishable dominant paths. The proposed method enables
more efficient and accurate channel estimation.
The traditional OMP algorithm considers that the vector h
is sparse. However, it is known from [7] that the channel has
different sparse characteristics in the delay-Doppler-angle
domain, which is exploited by the 3D-SOMP algorithm.
It has different characteristics for different algorithms, but
the number of dominant paths is assumed to be known. In a
wireless channel, if the delay intervals between dominant
paths are smaller than the resolution of the system, it causes
selected wrong paths and overestimates the channel. Therefore, the number of distinguishable dominant paths is used as
an approximate number of dominant paths.
The inner product vector between the residuals and the
observation matrix is solved by
e = 9 H r,
94686
(33)
20
21
(i)
eν (n) = E mτ , n, : ;
N
N
(i)
nν = arg minn eν 2g − n : 2g + n − 1 , s.t.
eν N2t − n : N2t + n − 1 ≥ ϵ ∥eν ∥;
n
o
N
N
N
(i)
(i)
(i)
3ν = 2g − nν , · · · , 2g , · · · , 2g + nν − 1 ;
(i)
(i)
eθ (r) =?E mτ , 3ν , r ;
dθ = LH eθ ;
gθ (r) = ∥Dθ (r, :)∥;
ps = arg maxr gθ (r);
(i)
3θ = {ps, ps + 1, · · · , p
s + D − 1};
(i)
(i)
(i)
 =  ∪ mτ , 3ν , 3θ ;
†
h(i)  = 9 y, h(i)
r(i) = y − 9h(i) ;
†

= 9c y;
22
23
24
25
26
27
28
29
end
for j = 0 to Np − 1 do
∇uNp −j =
(Np −j+1)
uτ
ωNp −j = 1
(Np −j)
uτ
;
∇uNp −j
;
γ Np −j
N
−j
ω p then
if max_ω <
max_ω = ωNp −j ;
else
max_index = Np − j + 1;
break;
30
end
31
32
end
Pos = max_index;
where r ∈ CMτ Nν ×1 is the residual vector and e ∈ CMg Ng Nt ×1
is the correlation of the observation matrix with the channel
residual vector.
The structural characteristics of the channel are known to
provide prior information. When one of the dominant path
indexes is determined, the indexes of the other Doppler-angle
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G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO
where the equation is found in the maximum gθ index, it is
regarded as the starting location of the burst sparsity. Then we
find the angle-dimension support with ps and D as
(i)
3θ = {ps , ps + 1, · · · , ps + D − 1},
(39)
(i)
where 3θ selects the angle-dimension indexes with starting
location ps and totaling D, representing the total number of
indexes in the angle-dimension. Then we select all the indexes
of the delay-Doppler-angle domain by
(i)
(i)
 =  ∪ m(i)
,
(40)
τ , Av , Aθ
FIGURE 5. Extracting the same delay indexes in the channel vector.
(i)
domains corresponding are also non-zero, as well as the delay
indexes that are not selected. This will greatly reduce the
number of atom selections and thus avoid getting stuck in the
local optimum, which greatly increases the accuracy of channel reconstruction and reduces the probability of selecting the
wrong atoms. Then, we use the structural characteristics of
the channel to find the indexes in the same dominant path,
as shown in Fig. 5. The same color indexes in the figure
indicate that they have the same delay index, representing that
they have the dominant delay indexes. The specific equation
is
(i)
(i)
where mτ , Av , Aθ representing delay, Doppler, and angle
domains, respectively.
After the above algorithm steps, the maximum inner prod(i)
uct uτ corresponding to Np delayed dominant paths and the
L2-norm γ i of the residuals are obtained by
ri = y − 9h(i) ,
(41)
where the residual vector ri after the i-th iteration is obtained
from (41). The residual vector represents the similarity
between the estimated and wireless channels. Smaller residuals represent more accurate estimates. The proposed equation
(Np −j+1)
E = invec{e}.
(34)
To simplify the summation, the (34) is used to organize
indexes e into a tensor E ∈ CMg ×Ng ×Nt . The tensor E is
rearranged to obtain E(1) ∈ CMg ×Ng Nt , and calculate the
corresponding residual
eτ (m) = E(1) (m, :) ,
(35)
where E(1) (m, :) is the inner products of all indexes of the
same dominant path, we obtain the L2-norm eτ (m) of the total
inner product of each delay domain, which represents the total
inner product of the τ -th delay, and find the index with the
largest total inner product among them by
m(i)
τ = arg max eτ (m).
m
(36)
With the 3D-IPRDSOMP algorithm, the index of the inner
product and the dominant path corresponding to the maximum inner product after summation is obtained by (36).
The arg max denotes the index corresponding to finding the
(i)
maximum value. Where mτ denotes the index of the dominant delay of the sum of the maximum inner product. The
(i)
corresponding inner product eτ (mτ ) = max eτ (m) is also
recorded. In [7], step 11-step 15 has been explained.
gθ (r) = ||Dθ (r, :)||,
(37)
CNt ×D
where matrix Dθ ∈
obtained from, then, obtain the
L2-norm gθ (r) of Dθ (r, :). To record the starting location of
the burst sparse, we find the maximum value of gθ as
ps = arg maxr gθ (r),
VOLUME 11, 2023
(38)
∇uNp −j =
uτ
(Np −j)
,
(42)
uτ
represents the proportion of the maximum inner product uτ
for each iteration The division between ∇u and the residual
γ as
Np −j+1 ∇u
,
(43)
ωNp −j = 1
γ Np −j
where the symbol denotes the difference between them.
We obtain the value ω of variation in the scaled decrease
of the maximum inner product. The corresponding values ω
change from fast to slow when the overestimating channel and
selection of wrong atoms result in a maximum inner product
and slow variations after max_ω < ωNp −j .
Then, starting from the Np -th value, the local maximum
max_ω and the corresponding index max_index are obtained
by calculating ωNp −j and comparing them with the maximum
with backtracking. The corresponding index Pos is marked as
the corresponding index vector h| = h(Pos)  .
IV. SIMULATION RESULTS
In this section, we compared the performance of four different
algorithms: the traditional least squares (LS) algorithm, the
traditional OMP algorithm, the 3D-SOMP algorithm, and
the proposed 3D-IPRDSOMP algorithm. We use 3GPP Standardized Channel Model (SCM)to create the channel. We use
the normalized mean square error (NMSE), which represents
the error of channel estimation. The NMSE of the proposed
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G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO
TABLE 2. Simulation parameters.
FIGURE 7. The NMSE performance comparison against the pilot overhead
ratio η.
FIGURE 6. The NMSE performance comparison against the SNR.
3D-IPRDSOMP based channel estimation technique is computed as
k= N2 ℓ=M −1
P
P
NMSE =
k=− N2
ℓ=0
DDA − H DDA
Ĥℓ,k,r
ℓ,k,r
k= N2 ℓ=M −1
P
P
k=− N2
ℓ=0
2
,
DDA
Hℓ,k,r
(44)
2
DDA ∈ CMg ×Ng ×Nt , H DDA ∈ CMg ×Ng ×Nt are
where Ĥℓ,k,r
ℓ,k,r
obtained by matrixing the channel vectors h and h. The effectiveness of channel reconstruction is proven by evaluating the
channel estimation effect.
We construct the 3GPP SCM link channel. The channel
model is configured with six dominant paths and twenty
sub-paths for each dominant path. However, the path delay
does not have a one-to-one correspondence with the OTFS
modulation because of the resolution. The simulation parameters are summarized in Table 2.
94688
Fig. 6 shows the NMSE performance under different
signal-to-noise ratios (SNR)s. The number of antennas is 32,
the user velocity is 100 m/s, and the pilot overhead ratio is
0.5. Notably, the proposed 3D-IPRDSOMP algorithm outperforms the traditional 3D-SOMP algorithm by approximately
2 dB. We observe that the NMSE performance of channel
estimation increases as the SNR increases. The performance
in the channel is compromised because of the low accuracy of
the OTFS massive MIMO delay dimension. Specifically, only
one dominant path exists on the integer delay index, despite
the presence of multiple paths in the channel. Consequently,
the number of distinguishable dominant paths is smaller
than the number of dominant channel paths. The proposed
algorithm matches the number of approximate primary paths
to achieve better performance. In this case, the traditional
algorithm estimates the atom selection based on the number
of dominant paths, resulting in wrong atom selection and the
consequent degradation of the NMSE performance.
Fig. 7 shows the NMSE performance under different pilot
overhead ratios. The number of antennas is 16, the user
velocity is 100 m/s, and the SNR is 5 dB. The proposed 3DIPRDSOMP algorithm outperforms the traditional algorithm
under the same pilot overhead ratio. We observe that the
NMSE performance of channel estimation increases as the
pilot overhead ratio increases. It is because the solvability
of the matrix is enhanced as the number of pilot overheads increases, leading to an approximate solution closer
to the optimal solution. Furthermore, the 3D-IPRDSOMP
algorithm exploits the structural characteristics of the channel, where the indexes between different dimensions are
correlated. It reduces the number of atom choices without
increasing the step, resulting in better channel reconstruction
performance.
Fig. 8 shows the performance of NMSE with different
numbers of BS antennas. The pilot overhead ratio is 0.5,
the user velocity is 100 m/s, and the SNR is 5dB. The
proposed 3D-IPRDSOMP algorithm outperforms traditional
algorithms with the same number of antennas. We observe
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G.-R. Wang, Z.-X. Wang: 3D-IPRDSOMP Algorithm for Channel Estimation in Massive MIMO
dimension increase. The larger Doppler shift for the same
number of samples, the smaller the resolution of the Doppler
dimension, and hence the accuracy of the channel estimation
decreases. Notably, the proposed 3D-IPRDSOMP algorithm
outperforms the traditional 3D-SOMP algorithm, especially
when the user velocity is low.
V. CONCLUSION
FIGURE 8. The NMSE performance comparison against the number of BS
antennas.
FIGURE 9. The NMSE performance comparison against the number of
user velocities.
In this paper, we studied the number of distinguishable dominant path problems in channel estimation in OTFS massive
MIMO systems. First, we analyze the reasons for the poor
performance of the traditional OMP algorithm. We adjust the
step size of the OMP algorithm to escape the local optimum
and be closer to the global optimum. Then, using residuals and the inner product to characterize the completion of
reconstruction, we proposed a method that approximates the
number of distinguishable dominant paths. It is used as an
iterative termination condition to select atoms, thus avoiding
performance degradation because of the introduction of erroneous atoms. The 3D-IPRDSOMP algorithm increases the
accuracy of channel estimation by exploiting the 3D structured characteristics of the channel and reducing inter-symbol
interference. The 3D-IPRDSOMP algorithm has better performance than traditional algorithms. In the future, we aim
to enhance the accuracy of the dominant path estimation
and eliminate inter-path interference. In addition, we plan
to increase the performance of estimating the number of
dominant paths in a channel without any prior information.
To solve the problem of low resolution in the number of distinguishable paths, we hope to find a solution that enhances
the resolution of the system from the design of the system and
improves the performance of distinguishing between similar
channel paths.
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Dec. 1993.
GUO-RUI WANG was born in Dongying,
Shandong, in 1999. He received the B.S. degree
in electronic information engineering from Yantai
University, in 2021, where he is currently pursuing the M.S. degree in electronic science and
technology.
ZHONG-XUN WANG was born in Yantai,
Shandong, in 1964. He received the B.S. degree
from the Physics Department, Shandong Normal
University, in 1986, the M.S. degree from the
Department of Hydroacoustic, Harbin Engineering University, in 2008, and the Ph.D. degree
from the Naval Aeronautical Engineering Institute, in 2009. His current research interest includes
source channel coding in wireless communication.
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