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Radar signals modulation recognition based on
bispectrum feature processing
To cite this article: Xinping Mi et al 2021 J. Phys.: Conf. Ser. 1971 012099
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EEI 2021
Journal of Physics: Conference Series
1971 (2021) 012099
IOP Publishing
doi:10.1088/1742-6596/1971/1/012099
Radar signals modulation recognition based on bispectrum
feature processing
Xinping Mi1, Xihong Chen1, Qiang Liu1*and Denghua Hu1
1
Air and Missile Defense College, Air Force Engineering University, Shanxi Xi’an
710051, China.
*
Corresponding author’s e-mail: dreamlq@15213060@bjtu.edu.cn
Abstract. Modulation recognition of radar signals is an important part of modern electronic
intelligence reconnaissance and electronic support systems. In this paper, to solve the problem
of low recognition accuracy and low noise resistance of radar signals under low signal-to-noise
ratio(SNR), a recognition method based on variational mode decomposition(VMD) and
bispectrum feature extraction is proposed. Based on the feature that bispectrum can suppress
Gaussian noise, the feasibility of signals modulation recognition under low SNR is analyzed and
the noise item is introduced. Due to the interference of noise item, the noise suppression effect
of bispectrum is worse under 0dB. An improved VMD algorithm based on artificial bee
colony(ABC) algorithm optimization and envelope entropy evaluation is proposed to preprocess
the signal to improve the SNR. Finally, we designed a convolution neural network(CNN)
classifier to recognize signals of different modulation types. The simulation results show that
this method has better noise resistance than traditional methods, and can effectively identify
different types of signals under low SNR.
1. Introduction
Modulation recognition of radar signals is an important part of electronic information technology. In the
complex electromagnetic environment, how to accurately identify radiation source signals determines
the performance of electronic information analysis[1]. Due to the complex and changeable
electromagnetic environment in the battlefield, the signals are subject to a lot of interference in the actual
transmission process, leading to a large distortion of the received signals and making it difficult to
distinguish the original signals. Therefore, it is of great significance to study radar signal recognition
under low SNR to improve the accuracy of electronic intelligence analysis under complex environment.
In the past decades, great progress have been made on modulation recognition of radar signals. In
the literature, radar signal recognition is divided into two categories, namely, likelihood-based (LB) and
signal statistical feature-based (FB) approaches[2]. The likelihood-based approach is a class of
hypothesis testing problems. The probability density distribution of the received signal is tested for
consistency with the hypothesis by assuming that the probability density function of the target signal is
known [3]. Finally, the optimal recognition result is given under the bayesian theory[4]. From theory,
the likelihood method shows optimal efficiency due to maximizing the average probability of correct
identification. But for complex modulated signals with low SNR, the calculation of prior probability
becomes extremely difficult due to the mismatching of the model, which degrades the classification
accuracy.
The alternative approach is signal statistical FB approaches. This method can be regarded as a
mapping relationship that maps time series signals to feature domains, and uses these feature parameters
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1
EEI 2021
Journal of Physics: Conference Series
1971 (2021) 012099
IOP Publishing
doi:10.1088/1742-6596/1971/1/012099
for classifier design[5]. Typical feature extraction methods include instantaneous amplitude, phase,
frequency ,wavelet transform (WT), higher-order moments and cumulants, cyclic cumulants , and signal
spectra[6]. The research of these feature extraction methods focuses on how to obtain the feature with
robustness, but it is always difficult to extract the feature with good resolution capability under low SNR.
For example, in reference[7], instantaneous amplitude, phase, and frequency are extracted from the
obtained signal. Although these features are simple to extract, they are very sensitive to noise and are
prone to estimation errors. Moreover, the extraction of instantaneous information is completely
dependent on the threshold value, and it is usually necessary to set the threshold in advance. In
reference[8], time-frequency image of signal is extracted by using smooth pseudo-Wigner timefrequency analysis, but the time-frequency image is easily affected by noise. In literature[9], wavelet
transform is used to extract signal features, but how to select the appropriate wavelet function is a
complicated problem. In literature[10], the signals bispectrum that is less affected by noise is taken as
the feature of signal recognition to improve the anti-noise performance. In summary, the bispectral
feature has the excellent performance in noise suppression. Therefore, this paper will continue studying
the anti-noise performance of bispectral features under low SNR to achieve superior resolution
capability.
In this paper, we present an enhanced recognition of radar signals modulation technique based on
bispectral features. First, we analyze the effectiveness of the bispectral features for suppression of
gaussian noise according to higher order spectral analysis. We reveal that the noise suppression effect
of bispectrum becomes worse with the decrease of SNR. Based on the bispectral expression of the signal,
we find and define the interference term that affects the noise suppression performance. Second, for the
proposed interference term, we use data-driven signasl decomposition technology to eliminate noise
interference. Recently, a variational method has been proposed for the decomposition of data-driven
signals, called variational mode decomposition (VMD), which adaptively obtains the principal mode of
the signal by solving a convex optimization problem[11]. In our work, an improved VMD algorithm
based on artificial bee colony(ABC) algorithm optimization and envelope entropy evaluation is
proposed. Finally, we designed a convolution neural network(CNN) classifier, which can learn twodimensional images features and realize automatic modulation classification. The simulation results
show that the proposed system has high performance. When the SNR is less than 0dB, the average
classification accuracy of this method is close to 90%.
The remainder of this paper is organized as follows. Section II introduces the model of received
signals and bispectral analysis. Section III describes the proposed scheme in detail, including system
structure, preprocess method, feature extraction and the classifier. Section IV shows simulation results
and Section V provides this paper.
2. SIGNAL MODEL AND BISPECTRUM ANALYSIS
2.1. Signal Model
Along with the development of radar technology, the modulation of radar signal becomes more and
more complicated, such as continuous wagve signals (CW), linear frequency modulation signals (LFM),
non-linear frequency modulation signals (NLFM), the binary phase shift keying (BPSK) signal, and so
on. This paper mainly studies the modulated signal affected by Additive White Gaussian Noise (AWGN).
Assuming that the discrete signal received by the radar reconnaissance receiver can be expressed as
y  n  s  n  g  n , 1 n  L
(1)
where 𝑦 𝑛 and 𝑠 𝑛 represent the discrete received signal and transmitted signal, respectively. The
𝑔 𝑛 is the noise,𝐿 is the length of the signal. The radar transmitted signal is as follows


s  n    Am  an g  t  nTs   cos  2  f c  f m  t  0  m 
n


(2)
where 𝐴 is the modulation amplitude ， 𝑓 and 𝑓 represent carrier frequency and modulation
frequency.𝜙 and 𝜙 denote the initial phase and modulation phase.
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EEI 2021
Journal of Physics: Conference Series
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doi:10.1088/1742-6596/1971/1/012099
1971 (2021) 012099
2.2. Bispectrum Analysis
Bispectrum is the Fourier transform of the third-order cumulant, which can completely retain the
amplitude, frequency and phase information of the signal, and has the characteristics of time-shift
invariability, scale variation and phase retention, etc., and can effectively reduce the Gaussian noise[10].
Suppose the random sequence of the signal is 𝑥 𝑛 , 𝑥 𝑛 𝑡 , ⋯ , 𝑥 𝑛 𝑡
, the higher-order
. Then the k-order spectrum of the signal can be expressed as
cumulant is expressed as 𝑐 𝜏 , ⋯ , 𝜏
S kx 1 ,  ,  k 1  
+
+
  

1 =-
k 1 =-
𝜏 ,⋯,𝜏
where the higher-order cumulant 𝑐
+
+
  

1 =-
k 1 =-
ckx  1 ,  , k 1  e
 j 1 1    k-1 k 1 
(3)
satisfy as
ckx  1 ,  , k 1   
(4)
then bispectrum is defined as:
Bx 1 , 2  
+
+
 

c3 x  1 , 2  e  j 11 2 2 
(5)
The third-order cumulant, can be computed throughcan be computed through
1 =-
2 =-
c3 x  1 ,  2   E  x  m  x  m   1  x  m   2 
(6)
Here, 𝐸 ∙ is the expectation operator. For a definite signal with finite energy in discrete time,
bispectrum is defined as
Bx 1 , 2   X 1  X 2  X  1  2 
(7)
where 𝑋 is the fourier transform of the signal 𝑥 𝑚 , 𝑋 ∗ denotes the conjugate of 𝑋. The 𝑋 𝜔 can be
derived as
X     x  me jm
m
(8)
There are several ways to estimate bispectrum. In this paper, we accept the non-parametric direct
estimation method for bispectrum calculation. In particular, we describe the bispectrum estimation
algorithm in the following[24].
1.
The radar signal,𝑥 0 , 𝑥 1 , ⋯ , 𝑥 𝑁 1 , begins to decompose into K segments 𝑥 𝑚 ,
when 𝑘 1,2, ⋯ , 𝐾 and 𝑚 0,1, ⋯ , 𝑀, 𝑀 is the length of each segment.
2.
Calculate the discrete Fourier Transform (DFT) coefficients
X k   
1
M
M 1
 x  me
k
 j 2 m
M
m0
(9)
where 𝑘 1,2, ⋯ , 𝐾 and 𝜆 0,1, ⋯ , 𝑀⁄2.
3.
Calculate DFT of three-order correlation coefficients,

L1
L1
k  ,   1   X k  i  X k  i 
1 2
1
1
2
2
2
b

0
i1 L1 i2 L1

 X k  1  2  i1  i2 
(10)
𝜆 ,𝜆
𝜆
𝑓 ⁄2, where 𝑓 is the
where 𝑘 1,2, ⋯ , 𝐾. 𝜆 and 𝜆 respectively satisfy 0 𝜆
⁄
sampling frequency, ∆ 𝑓 𝑁 . Here, 𝑁 and 𝐿 should be selected to satisfy the equation 𝑀
1 𝑁.
2𝐿
4.
The bispectrum can be estimated using all the K segments of radar signal as

B 1 , 2  
1
K
K

 b  ,  
k 1
2 f s
2 f s
1 and 2 =
2 .
where 1 =
N0
N0
3
k
1
2
(11)
EEI 2021
Journal of Physics: Conference Series
1971 (2021) 012099
IOP Publishing
doi:10.1088/1742-6596/1971/1/012099
The estimated signal bispectrum can well reflect the non-Gaussian distribution information of each
radiation source and obtain the individual characteristics of each radar radiation source.
2.3. Disadvantages Of Bispectrum Analysis
According to the theory in the first two sections, the third-order cumulant of the received signal is
calculated as
C3y 1,2 


 E s n  g  n 
s n 1   g  n 1 
s n 2   g  n 2 
 C3s 1,2  C3g 1,2   Es n  C2g 1  C2g 2  C2g 2 1 
E g  n  C2s 1 C2s 2  C2s 2 1 
(12)
Suppose the mean of signal 𝑠 𝑛 and gaussian noise 𝑔 𝑛 is zero, the third-order cumulant of the
received signal change as
C3 y 1 , 2   C3s 1 , 2   C3g 1 , 2 
(13)
where 𝑔 𝑛 is the Gaussian noise, so 𝐶 𝜏 , 𝜏
0. From formula (5), we have the bispectrum of the
received signal
By 1 , 2   Bs 1 , 2 
(14)
However, in general,because the transmitter output signal is modulated signal, the mean of 𝑠 𝑛
usually is not zero. Therefore, the third-order cumulant of the received signal 𝐶 𝜏 , 𝜏 is affected by
𝜏 . In this paper, we define the term 𝐶 𝜏
the term 𝐶 𝜏
𝐶 𝜏
𝐶 𝜏
𝐶 𝜏
𝜏 as noise interference term. When the SNR is relatively large, the interference of the noise
𝐶 𝜏
term is reduced and can be ignored. When the SNR is smaller, the interference of the noise term is more
obvious.
At low SNR, the bispectrum has obvious fluctuation fluctuations, which is caused by the interference
𝜏 in formula (12).The lower the SNR of Gaussian noise
𝑐 𝜏
𝑐 𝜏
of noise term 𝑐 𝜏
𝑔 𝑛 is, the greater the interference of the noise item is, so the bispectral fluctuation is more obvious.
This indicates that bispectrum does not suppress Gaussian noise infinitely, and the suppression effect is
𝜏 . Therefore, in
𝑐 𝜏
𝑐 𝜏
affected by the signal mean 𝐸 𝑠 𝑛 and noise terms 𝑐 𝜏
order to reduce the interference of bispectrum, only the original signal needs to be denoised to improve
the SNR.
3. MATH
3.1. System Structure
As analyzed in the previous section, the ability of bispectral features to suppress noise becomes poor
when the SNR is reduced. Therefore, it is necessary to preprocess the obtained signal to suppress the
influence of noise on bispectrum. A block diagram of the proposed system showing signal processing,
feature extraction, and CNN classifier of the proposed system is given in Figure 1. The main purpose of
this paper is to recognize accurately the signal in lower SNR. First, signal preprocessing is carried out,
and the parameters of VMD are optimized by artificial bee swarm algorithm. The optimized VMD is
used for signal decomposition to separate the signal from the noise. Next, the bispectral feature image
is extracted, and the 3D image is converted into grayscale image. Finally, The original signals
classification is realized by designing a classifier of convolutional neural network.
4
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Journal of Physics: Conference Series
1971 (2021) 012099
IOP Publishing
doi:10.1088/1742-6596/1971/1/012099
The Preprocessing
Variational Mode
Decomposition
Artificial Bee
Colony
Algorithm
Best 
Input signals
Feature Extraction
CNN
Classifier
Output
Fig.1 The proposed system Structure
3.2. Preprocess Method
As mentioned in the second section, the level of SNR affects the accuracy of bispectral features.
Therefore, it is necessary to preprocess the obtained signal to improve the SNR, so that the bispectral
feature can more accurately reflect the actual signal details and improve the accuracy of recognition.
This section proposed an advanced method based on variational mode decomposition (VMD) and
artificial bee colony (ABC) algorithm.
VMD is a method of variable-scale signal processing with the support of variational theory into a set
of intrinsic mode functions determined by the band. The decomposition is given by


min 
uk ,k  k



j 
  j t
t    t   uk  t   e k
t 


s.t. uk  f




2
2
（15）
k
Where 𝑢 𝑘 1,2, ⋯ , 𝐾 is the set of all modes and 𝜔 𝑘 1,2, ⋯ , 𝐾 represent their center
frequency, and 𝑓 is the input signal. We use both quadratic α penalty terms and Lagrangian multipliers
λ to turn the problem into an unconstrained optimization problem as follows [12].

j 
  jk t
L u  ,   ,      
u k  t   e
t    t  
t 


k
2
k
k

f  t   uk  t 
k
（16）
2
2
   t  , f  t    uk  t 
2
k
The solution of variational problems generally uses the method of the alternate direction method of
multipliers (ADMM). Detailed steps are given by Algorithm 1.
Algorithm 1: The solution of VMD problem.
Input: s  k 
Initialize u1k  ,  k1  ,  1 , n  0
Repeat
n  n 1
for k  1 : K do
Update uk :
5
EEI 2021
Journal of Physics: Conference Series
1971 (2021) 012099
IOP Publishing
doi:10.1088/1742-6596/1971/1/012099

u kn 1  arg min L uink1  , uin k  , in  ,  n
uk
end for
for k  1 : K do
Update k :

 kn 1  arg min L uin 1  , ink1  , in k  ,  n
k


end for
Dual ascent:


 n 1   n    f   ukn 1 


k
Until convergence:
n 1
n
 u k  uk
k
Output: ui  k 
2
ukn
2
2

2
Considering that the VMD is suitable for blind processing of received signal, the penalty factor 𝛼
need to be optimized and the number of components 𝐾 can be set to 2 by default. We adopt artificial
bee colony (ABC) algorithm to select the best the penalty factor 𝛼. Due to only the penalty to be
optimized, the solution space of the problem is one-dimension, in which the search time will be greatly
reduced.
In addition, it is necessary for the ABC algorithm to determine a fitness function. We select mean
envelope entropy that each component in VMD of food source as the fitness value. The envelope entropy
can reflect the different characteristics of the signal and noise according to quantify the fluctuation and
flatness of the received signal envelope. In IMF components, the signal component has smaller envelope
entropy and the noise component has bigger envelope entropy. Assuming the IMF component is 𝑢 𝑘 ,
envelope entropy of 𝑢 𝑘 can be given by
L
Es   u  k  lg u  k 
（17）
k 1
where 𝐿 is the length of 𝑢 𝑘 . Algorithm 2 shows the steps of VMD parameter optimization[13].
Algorithm 2: VMD parameter optimization algorithm
Input: s  k 
1:Initialize: xij :the initial solutions of population
2: fiti : the initial fitness value
3: c y c le  1
4:repeat
5:for each employer bee
vij  xij   ij  xij  x kj 
fiti 1 Es
Apply greedy selection process
Pi 

fiti
SN
i 1
6:for each onlooker bee
fiti
vij  xij   ij  xij  x kj 
6
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Journal of Physics: Conference Series
1971 (2021) 012099
IOP Publishing
doi:10.1088/1742-6596/1971/1/012099
fiti 1 Es
Apply greedy selection process
Pi 

fiti
SN
i 1
fiti
7:If exist an abandoned solution, replace it with a new solution
j
j
j
xij  xmin
 R   xmax
 xmin

8:Memorize the best solution so far
9: c y c le  c y c le  1
10:until c y c le  N
Output: best 
where SN is the number of employer bees and onlooker bees, 𝑖
1,2, ⋯ , 𝑆𝑁,𝑗
1 and 𝜙 and 𝑅 is a
and 𝑥
random number in 0，1 , 𝑘 is random generation number and 𝑘 ∈ 1,2, ⋯ 𝑆𝑁 ,𝑘 𝑗, 𝑥
are upper and lower bounds of solution space, respectively.
The parameters optimized by ABC algorithm are used into VMD. After decomposition of the original
noise-containing signal, signal component and noise component can be obtained. Therefore, the process
of removing the second component after VMD of the signal can be considered as filtering and noise
reduction. To sum up, the noise reduction process of the received signal is as follows:
Step1: The parameters of VMD were optimized by using envelope entropy as fitness function
Step2: The parameters optimized in step1 were used to decompose the received signal in VMD
Step3: Get rid of the second component signal
3.3. Feature Extration
As described in Section 2, the bispectral signature can be obtained by the Fourier transform of the thirdorder cumulant of the signal. Since the 3D feature map is directly obtained , in order to reduce the
redundancy of information and the computation of classifier, we transform the 3D feature map into
grayscale image. The depth of the gray level corresponds to the amplitude of the three-dimensional
image.
3.4. CNN classifier
In this section, a classifier based on convolutional neural network is designed. The processing process
of the convolutional neural network mainly includes initialization, convolution and pooling, flat, full
connection layer and reverse error adjustment. This classifier first extracts the gray image of bispectral
features as the input, and constructs the output layer of three convolutional layers, two pooling layers,
two full connection layers and N signal categories. The network structure of classifier based on
convolutional neural network is given by Figure 2.
7
EEI 2021
Journal of Physics: Conference Series
1971 (2021) 012099
IOP Publishing
doi:10.1088/1742-6596/1971/1/012099
Fig. 2 The network structure of classifier based on convolutional neural network
4. Simulation Results and Discussion
In this section, various simulations were performed to evaluate the performance of the proposed
algorithm. We selected six radar waveforms of conventional signal, include CW, LFM, BPSK, QPSK,
FSK NLFM, for testing and analysis. Each signal sampling frequency for 520 𝑀𝐻𝑧, sampling points are
256, 512, 1024 and 2048 respectively.
4.1. Performance Analysis of Preprocessing
In order to reflect the suppression effect of noise interference term in bispectrum estimation after
pretreatment, we extract the bispectral gray image before and after pretreatment. Six different signals
were preprocessed and bispectral extracted under -10dB. In FIG 3, the left side is the unpreprocessed
bispectral grayscale, and the right side is the preprocessed bispectral grayscale.
(a) CW signal
(b) LFM signal
(c) BPSK signal
(d) QPSK signal
(e) FSK signal
(f) NLFM signal
Fig. 3 Bispectrum after signal preprocessing under noise of - 5dB
As shown in FIG.6, in the absence of preprocessing, the bispectral features of the signal are
submerged in the surrounding interference, making it difficult to distinguish the bispectral features of
the signal. After preprocessing, the interference in the bispectral feature image is obviously reduced ,
8
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Journal of Physics: Conference Series
1971 (2021) 012099
IOP Publishing
doi:10.1088/1742-6596/1971/1/012099
and it is easy to distinguish the bispectral feature of the signal. For different kinds of signals, the
preprocessed bispectrum can completely extract the obvious features from the noisy interference, and
realize the suppression of the interference term in the bispectrum feature. This is because the SNR of
the original signal is improved and the noise is weakened, so the interference term 𝑐 𝜏
𝑐 𝜏
𝑐 𝜏
𝜏 in the bispectral feature is suppressed.
Compared with the feature image of the original signal, the preprocessed feature image is closer to
the original one, and the similarity between BPSK and FSK signals is higher. This is determined by the
complex mode of modulation, the more complex the modulation mode, the more obvious the bispectral
characteristics. In general, the preprocessed bispectrum can effectively reflect the characteristics of the
original signal, so the preprocessing method in this paper has a good inhibitory effect on the interference
term in the bispectrum feature.
Recognition accuracy/%
4.2. Performance Analysis of Signal Recognition
In this section, the recognition performance is evaluated by using the recognition rate as the evaluation
index. Six signals are identified under different SNR, and the methods based on the bispectrum feature
and convolutional neural network(BF-CNN) and the cyclic spectrum and convolutional neural
network(CS-CNN) are compared with the method in this paper. The simulation results are as follows:
Fig. 4 Recognition rate of various signals under different SNR
As shown in FIG 4, the recognition rate of the algorithm in this paper is different for different signal
types. In general, with the improvement of SNR, the recognition rate keeps increasing. The recognition
rate of various modulation signals is also different, in which the recognition rate of BPSK signal is the
largest and that of CW signal is the lowest. This is due to the different complexity of signals, resulting
in different performance of preprocessing and bispectral features, resulting in different final recognition
rate.
As shown in FIG. 5, when the SNR is greater than 0dB, the recognition rate of the method in this
paper is the same as the BF-CNN method, while the CS-CNN method is poor. This indicates that the
bispectral feature has a good performance in signal modulation recognition. When the SNR is less than
0dB, the average recognition rate of our method is close to 90%, which is much higher than the method
of BF-CNN and CS-CNN. Therefore, the algorithm proposed in this paper is best suited for the detection
of complex signals with low SNR.
9
1971 (2021) 012099
IOP Publishing
doi:10.1088/1742-6596/1971/1/012099
Recognition accuracy/%
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Journal of Physics: Conference Series
Fig. 5 Comparison of recognition rates of different methods
5. conclusion
This paper mainly attempts to enhance the recognition performance of bispectral features at low SNR
and proposes a recognition of radar signals modulation technique based on bispectral features. We
analyze the characteristics of bispectral features and propose the interference terms that affect the
performance of noise reduction. The focus of the follow-up research is to reduce the interference items.
We adopted the data-driven signal decomposition technology optimized by ABC algorithm to eliminate
the interference items. After the received signal is pretreated by VMD, the bispectrum feature is
extracted, which reduces the influence of interference terms on the bispectrum feature.Finally, using the
excellent learning ability of CNN, a classifier is designed to recognize six common radar signals. The
simulation results have revealed that a recognition of radar signals modulation technique based on
bispectral features achieves superior performance especially under a low SNR
In further work, the study can be continued in several directions. On the one hand, we study more
kinds of signal recognition and enhance the generalization ability of signal recognition methods. On the
other hand, signal recognition under multipath and interference conditions is studied to enhance
robustness.
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