WAN Tao，JIANG Kaili，LIAO Jingyi，TANG Yanli，and TANG Bin

School of Information and Communication Engineering，University of Electronic Science and Technology of China，Chengdu 611731，China

Abstract: The detection and recognition of radar signals play a critical role in the maintenance of future electronic warfare (EW).So far，however，there are still problems with signal detection and recognition，especially in the low probability of intercept(LPI) radar. This paper explores the usefulness of such an algorithm in the scenario of LPI radar signal detection and recognition based on visibility graphs (VG). More network and feature information can be extracted in the VG two-dimensional space，this algorithm can solve the problem of signal recognition using the autocorrelation function. Wavelet denoising processing is introduced into the signal to be tested，and the denoised signal is converted to the VG domain. Then，the signal detection is performed by using the constant false alarm of the VG average degree. Next，weight the converted graph. Finally，perform feature extraction on the weighted image，and use the feature to complete the recognition. It is testified that the proposed algorithm offers significant improvements，such as robustness to noise，and the detection and recognition accuracy，over the recent researches.

Keywords: detection，recognition，visibility graph (VG)，support vector machine (SVM)，k-nearest neighbor (KNN).

1. Introduction

Low probability of intercept (LPI) [1]radar signal detection and recognition technology is a research hotspot in recent years. Signal detection is the premise of electronic intelligent (ELINT)，therefore，it is the basic content of radar signal reconnaissance theory research. And the signal modulation type has become an important direction of the electronic warfare (EW) research since 1980 [2]，it is an important parameter for recognizing the working state of the radar. LPI radars have very low peak power，wide spectrum，high duty cycle，and low signal-to-noise ratio(SNR)，making it difficult for EW receivers to detect and recognize them [3]. Therefore，the research on LPI radar signal detection and recognition is very important for future EW.

In recent years，many methods for recognizing LPI radar signals have been preprocessed by using time-frequency analysis. The purpose is to use new machine learning methods [4]in the time-frequency domain and recognize the generated signals [5-8]，the recognition probability and the robustness to noise are getting good performance. However，the detection of LPI radar signals has rarely been proposed，and the basic idea is to recognize the detected signals. Moreover，the visibility graph (VG)research is also one of the hotspots in recent years. References [9，10]proposed the VG and horizontal VG (HVG)method to convert time signals into complex networks.In [11]，a method for rapidly converting time series into VG was proposed. In [12]，the VG theory was introduced into the analysis of wall turbulence time series，which provides the strong support for accurate time series analysis of uneven turbulence. References [13，14]introduced a weighted complex network method based on VG，assigning a weight to each adjacent edge，and signal detection and recognition processing were performed on the obtained weighted complex network.

VG is periodic and it has a linear relationship with fractional Brownian motion and fractional Gaussian distribution [15，16]. For simple random sequences，HVG presents analytical solutions [10]. The reliability of converting white noise random sequences into VG was proved in [17]. The VG converted into the band-limited signal detection was introduced in [18]，and the authors extracted the eigenvalues of the VG adjacency matrix for signal detection. In [19-22]，VG theory was introduced into various types of signal recognition，the main idea is to convert the original random sequence into another domain that is easy to be processed by the matrix. The result is to promote the analysis and processing accuracy of the original sequence [22].

This paper discusses the case of the LPI radar signals processing method based on the VG theory. Firstly，the signal received by the EW receiver is preprocessed. Preprocessing is a filtering algorithm based on the wavelet denoising principle (WDP). Then we obtain the power spectrum of the signal to be tested and compare it with the power spectrum of the additive white Gaussian noise(AWGN) signal. The real signal is detected. Secondly，the autocorrelation function is obtained for the detected signal，and the autocorrelation signal is converted into a VG network. Thirdly，we generate the weighted VG network and perform feature extraction. Finally，the support vector machine (SVM) and thek-nearest neighbor(KNN) [4]are used for signal modulation type recognition. The machine learning algorithm classifies the extracted features，including the modulation types of 14 LPI radar signals，which are monopulse (MP)，linear frequency modulation (LFM)，binary phase shift keying(BPSK)，quadrature phase shift keying (QPSK)，frequency shift keying (FSK)，V-Shaped，LFM-BPSK，LFMQPSK，Barker，Frank，and polyphase code (P1，P2，P3，P4).

The rest of this paper is organized as follows. Section 2 gives the expressions of the modulation types of each LPI radar signal and introduces the viewable method. The signal preprocessing algorithm and the VG weighting algorithm are overviewed in Section 3. Section 4 briefly describes the feature extraction. Section 5 is the validation of this paper including simulation experiment setup，results，and discussion.

2. Signal model

2.1 LPI radar signals

Assume that the radar pulse signal observation equation can be expressed as

wherey[k]represents the discrete-time signal intercepted by the EW receiver，x[k]is the discrete-time complex radar signal，n[k]is the complex AWGN，Vis the signal amplitude which is constant within a pulse width，fc[k]is the signal carrier frequency，kis the sample index for eachTsincrease for the sampling frequencyfs，for a given pulse time interval τpw，0 ≤kTs≤ τpw，and φ [k]is the phase modulation function.

In practice，LPI radar signals are generally classified into three types: frequency modulation，phase modulation，and combined modulation. When the radar signal type is frequency modulationfc[k]，the φ [k]phase portion is constant; when the radar signal type is phase modulation φ [k]，thefc[k]frequency portion is constant [6]，and the third one is frequency modulation and phase modulation combination. The 14 modulation types are shown in Table 1，including simple pulses，frequency modulation，phase modulation，and combined modulation.

Table 1 Radar signal modulation type

2.2 VG and HVG

For VG，considering that the time seriesS={t1，t2，···，tn}is the sampling time ofnradar signals，without loss of generality，there are three arbitrary momentsa，b，c，anda＜c＜bis satisfied between them. If and only if one node can draw a straight line connectingyaandyb，and a line does not intersect any intermediate reference “height”yc，the two nodesaandb(assuminga＜band without loss of generality) can be connected by a (non-directional) link line. According to the three criteria stated in [8]，the following inequation is satisfied:

For HVG，similarly，if and only if a node can draw a horizontal line connectingyaandyb，not intersecting any intermediate referenceyc，the two nodesaandb(assuminga＜band without loss of generality) can pass one(non-directional) link line connection. According to the three criteria set out in [9]，the following inequation is satisfied:

Table 2 Sequential VG/HVG motifs sequence of three points

We can know that VG has periodicity from the definitions of VG and HVG，and at least one adjacent edge of the VG can be connected to the vertex. There are no isolated points，and boundary points will appear at the beginning and the end of the sequence [23-25]. The 14 kinds of LPI radar wave length determine the size，and VG and HVG adjacency matrix. If the time series hasnnodes and the adjacency matrix isn×n，the adjacent matrix form is naturally different. For simple time series，HVG can obtain an analytical solution，assuming that the probability distribution of a random noise sequence isp(n)(n∈ [0，1])，thenFor more general sequences，such as Markov sequences，p(n) can also obtain analytical solutions [24]. Therefore，for LPI radar signals，under different sequence lengths，the VG adjacency matrix behaves differently，but they all have their own distribution methods. Within a certain range，the length of the size of VG or HVG matrix will make a little difference on recognition.

3. Preprocessing techniques

The algorithm illustrates the overall LPI radar signal detection and recognition technique proposed in this paper.The output is successful detection probability (SDP) and successful recognition probability (SRP).

AlgorithmLPI radar signal detection and recognition algorithm

InputSignal to be tested through the AWGN channel

OutputSDP，SRP

Step 1Give LPI radar parameters including sampling frequencyfs，code ratefb，the number of symbolsN，and {f1，f2，···，fNF} of FSK.

Step 2Wavelet denoising for the signal to be tested through the AWGN channel.

Step 3Convert the signal to be tested into VG or HVG.

Step 4Set the dynamic threshold for signal detection.

Step 5Perform autocorrelation for the detected real signal.

Step 6Weight the VG or HVG network.

Step 7Perform feature extraction on weighted complex networks.

Step 8Classify features by using SVM and KNN algorithms.

3.1 Signal preprocessing

LPI radar signals generally have low SNR characteristics，so reasonable noise reduction is one of the key technologies for signal detection and recognition. The essence of wavelet denoising is equivalent to signal filtering，and the interference noise is removed as much as possible under the premise of retaining LPI radar signals，which combines feature extraction and low-pass filtering. Set an appropriate threshold. When the high-frequency wavelet

When the adjacent edges are connected，there are only two connections between M0 and M1 between any three nodes，which can be summarized as shown in Table 2 [15].coefficients extracted after the time-domain signal processing are fewer than the threshold，the noise signal is considered to be removed; otherwise，the signal data greater than the threshold is retained as an effective LPI radar signal. Therefore，the suppression of LPI radar noise by wavelet transform is reasonable. In this paper，a signal detection method based on wavelet denoising is designed. Assume that the signal after through the AWGN channel can be expressed as

whereYiis the signal obtained after through the AWGN channel，Siis the original signal to be tested，σ is the noise level，andZiis the noise signal. The contaminated signalYiis mostly the high-frequency signal. Using the principle that signals and noise have different characteristics under wavelet transform，a series of coefficients obtained after wavelet decomposition are processed，and the noise and signals are separated.

whereDiis the high-frequency component of the decomposed signal，andAiis the low-frequency component of the decomposed signal. Usually，the low-frequency component is the most important，which can roughly reflect the characteristics of the signal，and the noise is generally distributed in the high-frequency component so that the signal and noise are separated. With the increase of the number of layers，the better noise suppression effect of theAiafter wavelet decomposition is，the better the signal is obtained after wavelet inverse transformation.However，considering the amount of calculation and time cost，the number of layers is generally not too much. The essence of the algorithm is to reduce the wavelet coefficients generated by noise and retain the information of the original signal as much as possible.

3.2 Weighted complex network

The weighting algorithm [26]is to use the following equation for all adjacent edges of a vertex in a complex network，it can be expressed as

4. Feature extraction

4.1 Average weighted degree

Consider the average weighting of a complex network as one of the best statistical attributes，because it is the most direct and most reflective factor of network complexity. It can be expressed as

whereB(i) is the field of the nodei. The average weighting of the network is the average of the total weight of all vertex connections in the network.

4.2 Average clustering coefficient

The average clustering coefficient reflects the clustering of the connection nodes，and its definition [26，27]can be written as

whereCiis the clustering coefficient of the nodei，Mis the VG or HVG matrix，andis the average clustering coefficient. This coefficient is directly processed by the network as a whole，therefore it is universal.

In addition，there are other features that can be used for LPI radar modulation type recognition，such as Newman coordination coefficient [28，29]，normalized network structure entropy [30-32]and other features. However，in the application of this paper，adding more features has little effect on the improvement of the recognition effect，so we comprehensively consider the selection of the above two features.

5. Experimental results and discussion

5.1 Experimental setup

5.2 Detection

Convert the measured signal after wavelet denoising [33]into a VG adjacency matrix，and find its average degree.Similarly，we obtain the average degree of the noise signal. The false alarm probabilitypfais set to 0.005，500 Monte Carlo independent experiments are carried out，and the results are shown in Fig. 1.

Fig. 1 Signal detection simulation analysis

The figure shows the average VG degree of signal passing through AWGN channel and after wavelet denoising processing. After setting the false alarm probabi lity，each threshold value in the SNR range of (-20，0) is obtained as shown in Fig. 1(a)，and the detection probability of the signal is expected to be high. To solve the SDP，500 times Monte Carlo experiments verify the detection probability as shown in Fig. 1(b).

In Fig. 1，when the SNR is -10 dB，the SDP is 90.9%，and as the SNR increases，the SDP also increases，especially when SNR＞-8 dB，the SDP is basically 100%.Compared with [34]，when the false alarm probabilitypfais lower，the detection performance is still significantly improved. Therefore，we have enough reasons to believe that the detection algorithm is robust to noise. Also，the algorithm is currently only for LPI radar signals，and it is still applicable to the scenarios of acoustic signal processing and other signal processing.

5.3 Recognition

Modulation type recognition is performed on the detected signal. First，find the autocorrelation function of the signal and convert it to VG or HVG. The inverse Fourier transform of the signal is used to obtain the autocorrelation function，which is converted into VG or HVG. Then the VG complex network is weighted，and then features of the weighted network are extracted. Finally，the machine learning algorithm is used to recognize the features，including SVM and KNN [18]. The indicator for judging the classification performance mainly is the SRP. The performance of the different kernel functions of the SVM classifier is also different in the classification. Therefore，to evaluate the classification performance of these feature sets，we apply an additional SVM with radial basis function (RBF) kernel functions and an SVM with polynomial kernel functions. Similarly，consider KNN to experiment with different values ofK(K=3 andK=10) on the combined feature vector set. Their classification performance for 14 LPI radar signals is shown in Fig. 2.

Fig. 2 Classification performance of different classifiers

Table 3 shows that the proposed algorithm is more accurate than the previous research. Among them，we only compare the overall experimental results with those using other existing methods and properly evaluate the SRP function.

The overall SRP of the algorithm proposed in this paper is 0.974，which is greatly improved compared with the previously proposed LPI radar waveform recognition technology. It is interesting to compare the following two situations: one is to improve the accuracy obtained by analyzing the time-frequency data processed manually，and the other is to improve the accuracy obtained by using the VG domain with complex network properties.Note that time-frequency analysis and deep learning re-quire a lot of work and time，while the latter method requires very little engineering work and adds minimal computational cost. Introducing the VG idea into radar signal recognition is not only an attempt but also a new idea for the signal processing field. However，VG still has certain limitations. For example，because it is an undirected graph，if the signal amplitude and phase are the same，the amplitude linearly increases or decreases，and the converted VG is the same，such signals cannot be identified.

Table 3 Comparison between the proposed algorithm and related studies

6. Conclusions

This paper proposes an LPI radar detection and recognition algorithm based on the VG theory. The algorithm first performs signal noise reduction processing，and converts the noise-reduced signal into a VG complex network. Secondly it obtains the dynamic threshold by setting the false alarm probability of the VG average degree to complete signal detection. Then，the network extracts the features and uses machine learning algorithms to complete the LPI radar signal recognition. When the SNR is greater than -8 dB，the SDP is approximately 100%;and the SRP is nearly 100% when the SNR is greater than-2 dB. The experimental results also show that the average weighting and the average clustering coefficient are the most promising features for revealing LPI radar waveform hidden information when considering the weighted complex network theory. In future research，we will extend the VG theory to radar signal sorting and radar working pattern recognition applications.