Progress In Electromagnetics Research M, Vol.

99, 191–200, 2021

AE-STAP Algorithm for Space-Time Anti-Jamming

Ruiyan Du1, 2, * , Fulai Liu1, 2 , Xiaodan Chen1, * , and Jiaqi Yang1

Abstract—Space-time adaptive processing (STAP) algorithms can provide effective interference

suppression potential in global navigation satellite system (GNSS). However, the performance of these
algorithms is limited by the training samples support in practical applications. This paper presents
an effective STAP based on atoms extension (named as AE-STAP) algorithm to provide better anti-
jamming performance even if within a very small number of snapshots. In the proposed algorithm, a
spatial-temporal plane is constructed firstly by the sparsity of received signals in the spatial domain.
In the plane, each grid point corresponds to a space-time steering vector, named as an atom. Then,
the optimal atoms are selected by searching atoms that best match with the received signals in the
spatial-temporal plane. These space-time steering vectors corresponding to the optimal atoms are used
to construct the interference subspace iteratively. Finally, in order to improve the estimation accuracy
of interference subspace, an atoms extension (AE) method is given by extending the optimal atoms in
a diagonal manner. The STAP weight vector is obtained by projecting the snapshots on the subspace
orthogonal to the interference subspace. Simulation results demonstrate that the proposed method can
provide better interference suppression performance and higher output signal-to-interference-plus-noise
ratios (SINRs) than the previous works.

1. INTRODUCTION

Space-time adaptive processing (STAP) is a critical technique for global navigation satellite system
(GNSS) to suppress the influence of multipath signals, radio frequency interference (RFI), etc. [1,2]. It is
known that the full-dimension (FD) STAP algorithm requires at least two times of the degrees of freedom
(DoFs) in independent and identically distributed (IID) samples to achieve a signal-to-interference-
plus-noise ratio (SINR) loss within 3 dB, which is usually impractical for heterogenous environments,
especially with large arrays [3]. Furthermore, high computational complexity and storage space are
needed to compute the FD STAP filter. Therefore, STAP algorithms with attractive performance at
low sample support and low computational complexity are of great importance in practical applications.
To deal with such issues, numerous STAP algorithms have been proposed in the last decades. Reduced-
dimension (RD) STAP algorithms, such as the multiple Doppler channels joint processing scheme
(mDT) [4], joint-domain localized algorithm (JDL) [5], space-time multiple beam (STMB) algorithm [6],
and robust two-stage RD SA-STAP considering inaccurate prior knowledge (RTSKA-RD-SA-STAP) [7]
are proposed by employing a low dimension for reducing the computational complexity and sample
support requirements. These algorithms have limited steady-state performance due to reduced system
DoFs. In this context, reduced-rank STAP algorithms, such as the principle components (PC) [8], cross-
spectral metric (CSM) [9], and multistage Wiener filter (MSWF) [10], can provide high steady-state
performance by using two times of the clutter rank of IID samples. However, the advantage of these
algorithms comes at the expense of reduced system DoFs. Recently, knowledge-aided STAP algorithms
Received 14 September 2020, Accepted 15 October 2020, Scheduled 15 December 2020
* Corresponding author: Ruiyan Du (ruiyandu@126.com), Xiaodan Chen (cxddxc@163.com).
1 Engineer Optimization & Smart Antenna Institute, Northeastern University at Qinhuangdao, Qinhuangdao, China. 2 School of
Computer Science and Engineering, Northeastern University, Shenyang, China.
192 Du et al.

have shown improved performance with a small number of training samples by exploiting prior knowledge
of environments or radar systems [11]. However, these algorithms suffer from performance degradation
in the presence of prior knowledge errors.
Owing to the successful application of compressive sensing (CS) in the parameter estimation, the
attention has been focused on STAP based on sparse recovery (SR-STAP) techniques by exploiting the
sparsity of the jamming signals [12,13]. Although SR-STAP algorithms show tremendous advantages in
the case of minimal training samples as compared to conventional STAP ones, their disadvantages should
not be ignored. It should be noted that the conventional SR-STAP is implemented under the condition
that the jamming signal is matched with the spatial-temporal grids, i.e., the spatial-temporal plane is
divided into a large number of grids where the jamming patches are assumed to exactly lie on some of
the grids. However, when the actual jamming signal is mismatched with the spatial-temporal grids, the
performance of conventional SR-STAP will decrease significantly. This phenomenon is known as the off-
grid problem in SR-STAP. The off-grid in the SR-STAP is a special form of the usual off-grid in CS. It is
easy to design a compact over-complete STAP dictionary to mitigate the off-grid issue. However, a dense
grid set not only increases the correlation between two adjacent spatial-temporal steering vectors but also
gives rise to the computational load [14], which restricts its practical application. Some researchers have
focused on the off-grid cases in the direction of arrival (DOA) estimation, where the DOAs of interest
may not exactly lie on the grids [15, 16]. An efficient root sparse Bayesian learning (SBL) method is
proposed to speed up the off-grid SBL method [15]. To solve the nonuniformity of the noise variance
and off-grid error, SBL based method of off-grid DOA estimation under nonuniform noise is proposed
in [16]. However, it is difficult to apply the first-order approximation method to off-grid SR-STAP.
A parameter-searched orthogonal matching pursuit (PSOMP) algorithm is proposed to eliminate the
basis mismatch in SR-STAP, which has a better performance and lower computational complexity than
orthogonal matching pursuit (OMP) algorithm [17]. However, the optimization problem in PSOMP is
not easy to solve.
In this paper, an STAP based on atoms extension (AE-STAP) algorithm is proposed to solve the
off-grid problem of SR-STAP, which can effectively select more appropriate atoms that are matched with
jamming signals. The proposed algorithm can mitigate the influence of off-grid in SR-STAP and also
can efficiently improve the performance of SR-STAP in limited-training-sample scenarios. The paper is
organized as follows. Section 2 outlines the signal model of STAP. Section 3 introduces the proposed
AE-STAP algorithm. Simulations are carried out to demonstrate the performance of our algorithm in
Section 4. Finally, the conclusion is summarized in Section 5.
N otation: (·)T , (·)H , and (·)−1 denote the transpose operation, Hermitian transpose operation,
and matrix inverse, respectively. ⊗ represents the Kronecker product. C expresses the complex field.
Moreover,  · 0 ,  · 2 , and  · F stand for l0 norm, l2 norm, and Frobenius norm, respectively. ∅ stands
for empty set. E(·) indicates the statistical average operation.

2. DATE MODEL

Consider an STAP model with a uniform linear array (ULA) consisting of M elements equispaced by d,
as shown in Fig. 1. Each element in the ULA is equally spaced with N taps. The M N × 1 space-time
observation vector x(t) = [x11 (t), x12 (t), . . . , x1N (t), x21 (t), . . . , x2N (t), . . . , xM N (t)]T at time t can be
expressed as
 L
x(t) = a(θs , fs )s1 (t) + a(θl , fl )jl(t) + n(t) = As(t) + n(t) (1)
l=1

where θs and θl (l = 1, 2, . . . , L) denote the DOAs of the desired signal and L interference signals,
respectively. fs and fl (l = 1, 2, . . . , L) are normalized frequencies of the desired signal and L
interference signals, respectively. s(t) = [s1 (t), j1 (t), . . . , jl (t)]T with s1 (t) denoting the desired
signal and jl (t) (l = 1, 2, . . . , L) the interference signals, respectively. The noise vector n(t) =
[n11 (t), n12 (t), . . . , n1N (t), . . . n21 (t), . . . , nM N (t)]T with nmn (t) (m = 1, 2, . . . , M, n = 1, 2, . . . , N )
standing for the additive noise. The M N × 1 spatial-temporal steering vector a(θ, f ) stands for
the array response with regard to the signal where DOA θ and f denote the DOA and normalized
Progress In Electromagnetics Research M, Vol. 99, 2021 193

1 Time-domain filter

x11 x12 x13 x

... 1(N-1) x1N
Z −1 Z −1 Z −1

w11 ... w1N

w12 w13
...

2
x21 x22 x23 x
... 2(N-1)
x2N
Z −1 Z −1 Z −1
Satellite
w21 w22 w23 ... w2N ∑
.. navigation
. ...
receiver
M
xM1 xM2 xM3 xM(N-1) xMN
Z −1 Z −1 ... Z −1
wM1 wM2 wM3 ... wMN
...
Spatial filtering

Figure 1. STAP model.

frequency, respectively. The array manifold matrix A = [a(θs , fs ), a(θ1 , f1 ), . . . , a(θL , fL )]. The space-
time steering vector a(θ, f ) is given by a(θ, f ) = as (θ) ⊗ at (f ). as and at (f ) can be written as
 T
as (θ) = 1, e−j2πd sin(θ)/λ , . . . , e−j2πd(M −1) sin(θ)/λ (2)
 T
at (f ) = 1, e−j2πf Ts , . . . , e−j2π(N −1)f Ts (3)

where as and at (f ) are the spatial steering vector and temporal steering vector, respectively. Ts
represents the delay time of each tap.
Further, the STAP filter output y(t) at time t can be written as
y(t) = w H x(t) (4)
where the M N ×1 complex-valued weight vector w = [w11 , w12 , . . . , w1N , w21 , . . . , w2N , . . . , wM N ]T with
wmn (m = 1, 2, . . . , M, n = 1, 2, . . . , N ) being the spatial-temporal weight for the nth tap on the mth
antenna.

3. ALGORITHM FORMULATION

In this section, the proposed algorithm will be introduced in detail. Firstly, the received signals are
represented sparsely in Subsection 3.1. In Subsection 3.2, the received signals are sparse restored to
obtain the optimal atoms in the space-time dictionary, and then the optimal atoms are extended.
Finally, the space-time optimal weight vector is solved in Subsection 3.3.

3.1. Sparse Representation of the Received Signal

The desired signals are submerged below the noise, and the power of narrow-band compression
interference signal is above 20 dB of the noise. The number of interference signals is much smaller
194 Du et al.

than the DoF of space-time two-dimensional processing. Therefore, the received signals are sparse in
space-time spectrum. In this subsection, the sparse representation of the received signals is given.
The entire normalized spatial-Doppler plane is uniformly discretized into Ns = ρs × M and
Nf = ρf × N grid points along the space and time axes, where ρs , ρf  1 denote the resolution
scales of the discretized plane. Each grid point corresponds to a space-time steering vector. The
M N × Ns Nf matrix Φ is the overcomplete STAP dictionary, as given by
 
Φ = a(θ11 , f11 ), . . . , a(θ1Nf , f1Nf ), . . . , a(θNs Nf , fNs Nf ) (5)
where θij and fij (i = 1, 2, . . . , Ns , j = 1, 2, . . . , Nf ) denote the quantised elevation angles and quantised
Doppler frequencies, respectively. The spatial-temporal steering vectors a(θij , fij ) (i = 1, 2, . . . , Ns , j =
1, 2, . . . , Nf ) are called atoms in the overcomplete STAP dictionary.
Assume that all the interference signals lie exactly on some of the grids, and then the model in
Eq. (1) can be rewritten as
x(t) = Φγ + n(t) (6)
where γ ∈ CNs Nf ×1 is the sparse signal with the non-zero elements representing the locations of
interference signals.
The problem in Eq. (6) is the canonical signal for the SR problem, which can be interpreted as
estimating a sparse vector γ from the received signal x(t). Specifically, γ could be formulated as the
following minimal optimisation problem
min γ0 s.t. x − Φγ22 ≤ ε (7)
γ

where the l0 norm measures the number of non-zero elements in a vector; .2 stands for the l2 norm;
and ε is the noise error allowance.

3.2. Extension of the Optimal Atoms

Although the problem in Eq. (7) has been shown to be NP-hard problem, many low complexity
algorithms have been proposed, and one of them, known as orthogonal matching pursuit (OMP)
algorithm [18], is selected in this paper. In each iteration, the optimal atoms are calculated by the
maximum correlation between the received signals and atoms in the STAP dictionary Φ. The indices
of atoms in Φ are recorded as λi (i = 1, 2, . . . , L). After sparse recovery processing by OMP algorithm,
the angles and frequencies of interference signals can be estimated by the optimal atomic matrix Φ. 
In practice, there is always a bias between the interference signals and the discrete grids. No
matter how dense we divide the spatial-temporal plane, the bias always exists. In order to reduce the
estimation bias of the optimal atomic matrix Φ,  the angles and frequencies of the estimated interference
signals need to be extended. Therefore, two atoms extension (AE) methods are considered. The first
AE method extends the optimal atoms along the angle and frequency axis. The second one extends the
optimal atoms in a diagonal manner. Taking the kth optimal atom a(θλk , fλk ) in Φ  as an example, two
forms of AE methods are considered, as shown in Fig. 2 and Fig. 3, respectively.
As shown in Fig. 2 and Fig. 3, the atomic cluster u k1 and u k2 with the center of atom a(θλk , fλk )
can be written as
k1 = [
u a(θλk − Δθ, fλk + Δf ), a  (θλk − Δθ, fλk ), a  (θλk − Δθ, fλk − Δf ),
a  (θλk , fλk − Δf ),
 (θλk , fλk + Δf ), a(θλk , fλk ), a (8)
 (θλk + Δθ, fλk + Δf ), a
a  (θλk + Δθ, fλk ), a  (θλk + Δθ, fλk − Δf )]

 k2 = [
u  (θλk − Δθ, fλk − Δf ), a(θλk fλk ),
a(θλk − Δθ, fλk + Δf ), a
(9)
 (θλk + Δθ, fλk + Δf ), a
a  (θλk + Δθ, fλk − Δf )]
where a denotes the space-time steering vector of the estimated interference signals. Δθ and Δf
 respectively. It should be noted that the values of Δθ
represent angular and frequency intervals in Φ,
and Δf are less than the error allowance; otherwise, they may suppress the desired signals.
Progress In Electromagnetics Research M, Vol. 99, 2021 195

f f

a (θ λk + Δθ , f λk + Δf )
a(

a (θ λk + Δθ , f λk ) a (θ λk + Δθ , f λk )

θ θ
a (θ λk + Δθ , f λk )
the optimal atom the optimal atom
the extend atoms the extend atoms

a((θ λk ++ Δθ , f λk − Δf )
a
a (θ λk , f λk − Δf )

Figure 2. Rectangular AE method of the optimal Figure 3. X-shaped AE method of the optimal
atom. atom.

Compared with Fig. 2, X-shaped AE method has a negative influence on a smaller number of
adjacent atoms. Then, the atomic cluster u  k . A new
k2 can be regarded as the optimal cluster u
 ∗
optimal atoms matrix U consisting of the optimal clusters can be expressed as
 ∗ = [ũ1 , ũ2 , . . . , ũL ]
U (10)
 k (k = 1, 2, . . . , L) is the kth optimal cluster.
where u

3.3. Solution of the Space-Time Optimal Weight Vector

The optimal weight vector wopt with minimum variance distortionless response (MVDR) algorithm can
be written as
min w H Rx w s.t. w H a(θs , fs ) = 1 (11)
w

where Rx = E(x(t)x(t)H ) stands for the convariance matrix of the received signals.
The optimization problem in Eq. (11) can be solved by Lagrange multiplier method. Then the
optimal weight vector wopt can be given by
wopt = (a(θs , fs )H R−1 −1 −1 −1
x a(θs , fs )) Rx a(θs , fs ) = μRx a(θs , fs ) (12)
where μ = (a(θs , fs )H R−1
x a(θs , fs ))
−1 is a constant. Thus, the optimal weight vector w
opt is related to
the covariance matrix Rx of received signals. Rx can be formulated as Rx = Rs +Rj +Rn with Rs , Rj ,
and Rn denoting the covariance matrix of desired signal, interference signals, and noise, respectively.
Generally, the power of narrow-band compression interference signals is much greater than the
noise and the desired signal. Hence, the covariance matrix Rx can be formulated by a block diagonal
matrix as
Rx ≈ Rj + Rn = Rj+n = (σ 2 I + Aj Rj AH j ) (13)
where σ 2 is the power of white Gaussian noise, and I is an M N × M N identity matrix.
Rj = diag(δ1 , δ2 , . . . , δL ) with δi denoting the power of the ith interference signal. Aj =
[a(θ1 , f1 ), a(θ2 , f2 ), . . . , a(θL , fL )] represents the interference signals.
According to matrix inversion lemma, R−1 j+n can be written as
1 1  
R−1
j+n = (I + σ −2
A R A
j j j
H −1
) = I − A j (σ 2 −1
R j + AH
j Aj )−1 H
Aj (14)
σ2 σ2
The power of the narrow-band interference signals is much larger than that of noise. Then Eq. (14) can
be rewritten as
1   1 ⊥
R−1 −1 H
j+n ≈ 2 I − Aj (Aj Aj ) Aj = 2 PAj
H
(15)
σ σ
196 Du et al.

where PA ⊥ = I −A (AH A )−1 AH stands for the orthogonal projection matrix of the interference signals
j j j j j
matrix Aj .
Accordingly, the optimal weight vector in Eq. (12) can be rewritten as
−1 μ ⊥
wopt = μR−1 x a(θs , fs ) ≈ μRj+n a(θs , fs ) ≈ 2 PAj a(θs , fs ) (16)
σ
It is evident that the space-time weight vector wopt depends on the orthogonal projection matrix of
the interference signals matrix Aj when the angle and frequency of the desired signal are confirmatory.
Let the interference signals matrix Aj be expressed by the optimal atoms matrix U  ∗ , that is Aj = U
 ∗.
On this basis, the space-time optimal weight vector in Eq. (16) can be solved with the optimal atoms
matrix U  ∗ . Then, the space-time optimal weight vector wopt is given by
μ ⊥ μ  −1
wopt = 2 PA a(θ , f ) = {I −  ∗ (U
U  ∗ )H U
∗  ∗ )H }a(θs , fs ) = μ P ⊥ a(θs , fs )
(U (17)
σ 2 U ∗
j s s 2
σ σ
 −1
where P ⊥∗ = I − U  ∗ (U  ∗ )H U ∗  ∗ )H is the orthogonal projection matrix of interference signals
(U
U
matrix.
The procedure of the proposed algorithm can be summarized as in Table 1.

Table 1. AE-STAP algorithm.

Input : The received signals x(t) and the number of interference signals L
Output : Space-time weight vector wopt
Procedure
Step 1. Construct space-time dictionary Φ and formulate optimisation problem according to (7).
Step 2. Obtain the optimal atoms matrix Φ  and support set Λ by OMP algorithm.
Step 3. Estimate the interference subspace with AE methods according to (8)–(10).
Step 4. Calculate the orthogonal projection matrix P ⊥∗ of the interference signals matrix:
 −1 U
 ∗ (U
P ⊥∗ = I − U ∗
 ∗ )H U  ∗ )H .
(U
U
Step 5. Calculate the optimal weight vector: wopt = μ ⊥
σ 2 PU
 ∗ a(θs , fs ).

4. SIMULATION RESULTS

In this section, several simulations are constructed to evaluate the performance of the proposed method.
Meanwhile, the proposed algorithm is compared with MVDR algorithm [19], PI algorothm [21], CS-SF
algorithm [20], and OMP-ISRP algorithm [22].
Assume that the ULA is composed of 8 antennas with equally spaced d = 0.098m. Each antenna
element is equally spaced with 7 delay taps. Assume that the desired signal whose DOA θ = 0◦ and
frequency f = 1.575 GHz (the normalized frequency is 1 GHZ). There are three narrow-band interference
signals whose DOAs and normalized frequencies are (−60◦ , 0.9), (20◦ , 1.1), and (50◦ , 1.2), respectively.
The input signal-to-noise ratio (SNR) is set to −20 dB. The noise is a zero-mean white Gaussian noise
with power σ 2 = 1. For the interval unit of dictionary, the angular interval is fixed as 1◦ , and the
normalized frequency interval is set as 0.01.
Figure 4 illustrates the matching degree between the presupposed interference signals and the
optimal atoms in the space-time dictionary. The matching degree is determined by the vertical
projection length in the direction of each optimal atom. As shown in Fig. 4, there are three peaks located
at (−60◦ , 0.9), (19◦ , 1.09), and (51◦ , 1.22), respectively. The DOAs and normalized frequencies of peaks
do not accord with the presupposed interference signals. Two main factors are considered. Firstly,
matching bias may be caused by strong correlation of adjacent atoms in the space-time dictionary.
Secondly, the noises affect the degree of matching between the interference signals and the atoms for
Progress In Electromagnetics Research M, Vol. 99, 2021 197

X: 19 X: 20
X: 51 X: 51
2500 Y: 1.09 Y: 1.09
Y: 1.22 Y: 1.21
Length of vertical projection

Z: 2286 2500 Z: 2284

Length of vertical projection

Z: 1745 Z: 1746
2000
X: -60 2000 X: -60
1500 Y: 0.9 Y: 0.9
Z: 2002 1500
Z: 2004
1000
1000
500
500
0
0
1.4 1.4
1.2 1.2
100
1 100 1 50
50 0
0.8 0 0.8
-50 -50
0.6 -100 0.6 -100 DOA(°)
DOA(°)
Normalized frequency(fs) Normalized frequency(fs)

Figure 4. Matching degree for the received Figure 5. Matching degree for the received
signals versus the dictionary atoms(with the signals (without the noises) versus the dictionary
noises). atoms.

the space-time dictionary. Fig. 5 illustrates the matching degree between the presupposed interference
signals and the space-time optimal atoms in the case without the noises. As shown in Fig. 5, the
simulation results are similar to the results in Fig. 4. The above situations occur because the noises
may have a negative influence on the matching bias, but it is not the main factor. Matching bias is
mainly caused by strong correlation of adjacent atoms in the space-time dictionary. Therefore, OMP
algorithm can just identify the optimal atoms approximately.
Figures 6 and 7 show the space-frequency response and the contour map of AE-STAP algorithm
when the optimal atoms are extended with the rectangular AE method. As shown in Fig. 6, three
nullings are formed at the DOAs of the interference signals, and a distortionless response is maintained
for the desired signal. In order to show the range of the formed nullings distinctly, the contours of
AE-STAP algorithm with the rectangular extension method is plotted in Fig. 7. It is evident that
there are three interference signals located at around (−60◦ , 0.9), (20◦ , 1.1), and (50◦ , 1.2), respectively.
In addition, the nullings are wide, which have negative influences on the gain of the signals near
interferences. When the DOAs and frequencies of the interference signals are close to the desired

1.4

40
1.3
20

0 1.2
Normalized frequency(fs)

-20
interference 1
Gain (dB)

-40 1.1

-60
1 interference 2
-80

-100 0.9
-120 interference 1
1.5 interference 3 interference 2
0.8
1 100 interference 3
50
0
0.5 -50
-100
DOA(°) 0.7
-80 -60 -40 -20 0 20 40 60 80
Normalized frequency(fs) DOA (¡ã)

Figure 6. Space-frequency response of AE-STAP Figure 7. Contour map of AE-STAP algorithm

algorithm (rectangle). space-frequency response (rectangle).
198 Du et al.

1.4

1.3

1.2

Normalized frequency(fs)
1.1 interference 1

1
interference 2

0.9

interference 3
0.8

0.7
-80 -60 -40 -20 0 20 40 60 80
DOA (¡ã)

Figure 8. Space-frequency response of AE-STAP Figure 9. Contour map of AE-STAP algorithm

algorithm (X-shaped). space-frequency response (X-shaped).

signal, it will cause distortion of the desired signal.

Figures 8 and 9 show the space-frequency response and the contour map of AE-STAP algorithm
with the X-shaped AE method. It can be seen that the proposed algorithm can also form nullings in the
corresponding DOAs and frequencies of the interference signals. Compared with the rectangular AE
method, AE-STAP algorithm with the X-shaped AE method to form deeper nullings and the points near
nullings are less negative affected. The anti-jamming performance is better than the rectangular AE
method. Therefore, in the subsequent simulation experiments, the optimal AE mode is the X-shaped
AE method.
To further verify the effectiveness of AE-STAP algorithm, the output SINRs under different DOAs
of an interference signal are simulated, and the simulation results are shown in Fig. 10. In this
simulation, the angles of the first interference signal and second interference signal are set to −60◦
and 50◦ , respectively. The DOA of the third interference signal is traversed from −90◦ to 90◦ . The
number of snapshots is set to 64. It is evident that the proposed method achieves higher output SINRs
than other methods. This is due to the estimation errors of signal covariance matrix for traditional

0 0

-10 -5

-20 -10
output SINR(dB)

output SINR(dB)

-30 -15

-40 -20

AE-STAP AE-STAP
-50 -25
MVDR MVDR
PI CS-SF
CS-SF OMP-ISRP
-60 -30
-100 -80 -60 -40 -20 0 20 40 60 80 100 50 100 150 200 250
Angle of interference 3( o ) snapshots

Figure 10. Output SINR versus the angle of Figure 11. Output SINR versus the number of
interference 3. snapshots.
Progress In Electromagnetics Research M, Vol. 99, 2021 199

MVDR and PI anti-jamming algorithms when there are fewer snapshots. CS-SF method can achieve
output SINRs the same level as the proposed algorithm in certain angles. However, CS-SF method
has poor performance in other angles, which is caused by the optimal atomic estimation error in the
sparse recovery process. The proposed algorithm extends the optimal atoms, which can estimate the
interference subspace more accurately. Therefore, the proposed method achieves significantly higher
SINRs than CS-SF and MVDR methods under a small number of snapshots. Besides, AE-STAP
method is an interference nulling algorithm so that it can form deep nullings, which does not need the
power of the interference signals and only uses the DOAs and frequencies of the interference signals.
As shown in Fig. 11, the anti-jamming performances of each method are compared in terms of the
output SINRs under different numbers of snapshots. The proposed method still achieves stable output
SINRs even when the number of snapshots is fewer than 55. With the number of snapshots growing,
it can be seen that the output SINRs of the proposed algorithm are similar to MVDR and OMP-ISRP
algorithms. And CS-SF algorithm has lower output SINRs than other methods. It means that the
proposed algorithm has better anti-jamming performance than the other methods even under a small
number of snapshots.

5. CONCLUSION

In this paper, an effective AE-STAP algorithm is proposed to solve the off-grid SR-STAP problem even
in the case of low sample support. Based on the sparsity of received signals, OMP algorithm is used
to effectively select more appropriate atoms that are matched with interference signals. Then, an AE
method is proposed to reduce the estimation bias. On this basis, the interference subspace is constructed.
Finally, the space-time optimal steering vector is calculated with the orthogonal projection matrix of
interference subspace. Simulation results demonstrate that the proposed algorithm can achieve superior
interference suppression performance.

ACKNOWLEDGMENT

This work was supported by the National Natural Science Foundation of China (Grant No. 61971117), by
the Natural Scienence Foundation of Hebei province (Grant No. F2020501007) and by the Fundamental
Research Funds for the Central Universities (Grant No. N172302002).

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