Automatic pulse repetition pattern reconstruction of conventional radars

IET Radar, Sonar & NavigationVolume 15, Issue 5 p. 500-509 ORIGINAL RESEARCH PAPEROpen Access Automatic pulse repetition pattern reconstruction of conventional radars Zhang-Meng Liu, Corresponding Author liuzhangmeng@nudt.edu.cn orcid.org/0000-0001-9472-279X School of Electronic Science and Technology, National University of Defense Technology, Changsha, China Correspondence Zhang-Meng Liu, School of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China. Email: liuzhangmeng@nudt.edu.cnSearch for more papers by this authorShi-Qian Kang, School of Electronic Science and Technology, National University of Defense Technology, Changsha, ChinaSearch for more papers by this authorXian-Ming Chai, DFH Satellite Co. Ltd, Beijing, ChinaSearch for more papers by this author Zhang-Meng Liu, Corresponding Author liuzhangmeng@nudt.edu.cn orcid.org/0000-0001-9472-279X School of Electronic Science and Technology, National University of Defense Technology, Changsha, China Correspondence Zhang-Meng Liu, School of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China. Email: liuzhangmeng@nudt.edu.cnSearch for more papers by this authorShi-Qian Kang, School of Electronic Science and Technology, National University of Defense Technology, Changsha, ChinaSearch for more papers by this authorXian-Ming Chai, DFH Satellite Co. Ltd, Beijing, ChinaSearch for more papers by this author First published: 26 March 2021 https://doi.org/10.1049/rsn2.12053AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract Signals of radars with fixed, stagger and other conventional pulse repetition intervals (PRI) occupy a large proportion in most radar reconnaissance datasets. Automatic and accurate reconstruction of the PRI patterns of these radars is an important task in realizing intelligent processing of radar reconnaissance data. In order to address this problem, this article proposes an automatic method based on the extraction and extension of frequent differential time-of-arrival (DTOA) items, which can be combined with other pulse parameters such as pulse width. The values and ordering of the frequent items are determined to reconstruct a complete PRI cycle, and the radar pulses are deinterleaved at the same time as a by-product. Compared with the commonly used one-dimensional radar reconnaissance parameters such as framework PRI or typical PRI set, the reconstructed PRI pattern contains more information about radar operational modes and can be applied to realize online and parallel pulse deinterleaving and more accurate radar type classification. 1 INTRODUCTION Pulse repetition interval (PRI) is an important parameter of pulsing radars, and it is closely related to radar coverage area and operating function. According to different requirements of various radar functions, PRI patterns are designed to be fixed, stagger, jitter and so on. [1]. Generally speaking, advanced multi-function radars (MFR) developed in recent years, such as electronic phase-shifting array radars, usually adopt jitter and agile PRIs [2-4] to realize multiple functions and enhance anti-reconnaissance and anti-jamming capabilities, whereas traditional mechanically scanning radars widely adopt simpler PRI patterns, for example, fixed and stagger ones. A variety of civilian radar systems also use such simple PRI patterns to reduce the cost. Conventional radars are referred as radars that transmit signals with regular PRI patterns such as fixed and stagger ones, and the other signal parameters like pulse width (PW) have similar agility. Signals of these radars occupy a large proportion of reconnaissance datasets. It is of great significance to analyse these signals automatically to improve the utility of the big reconnaissance data. The PRI pattern of conventional radar signals refers to the value and ordering of multiple adjacent inter-pulse intervals within a cycle, and the other pulse parameters like PW and frequency may also be included. Precise analysis of radar PRI pattern plays an important role in many radar reconnaissance problems such as PRI modulation recognition [5-7] and pulse deinterleaving [8-12]. The PRI modulation recognition methods extract features from sequential differential time-of-arrival (DTOA) to distinguish between different types of PRIs, for example, constant, stagger and jitter PRI. Radar pulse deinterleaving methods make use of DTOA-based state transition rules, which describe the relationship between adjacent pulses of a radar to distinguish radar signals from noises, and cluster pulses that conform to the same rule to separate radar signals from mixed pulse train. The PRI pattern of radar signals contains complete information about the state transition rules used in deinterleaving [8], and therefore PRI pattern analysis is helpful in addressing the problem of radar pulse deinterleaving. Moreover, in the field of radar type recognition, a common method is pulse description word matching. This method matches statistical radar signal parameters, including carrier frequency, PW, PRI, and so forth that are obtained from reconnaissance data with the emitter parameter library, and determines the radar type according to the highest matching degree [1]. If data noises such as missing pulses and interferential pulses exist, the matching method is difficult to obtain a satisfying result. That is because the statistical parameter estimates may deviate from true radar parameters, and the parameter library used for emitter recognition may also contain many incorrect PRI values due to missing and interferential pulses. In the field of radar intelligence, most of the existing methods artificially reduce the dimension of radar PRI pattern to facilitate signal analysis. The pulse deinterleaving problem is taken as an example, it is usually addressed based on the one-dimensional framework PRI. The most widely used radar signal deinterleaving methods include cumulative difference (CDIF) histogram method [9], sequential difference (SDIF) histogram method [10] and PRI transform method [11]. First, these methods estimate the framework PRI of radar pulses, and then exploit the PRI estimate to separate interleaved pulses of multiple radars. In these methods, pulses of radars with stagger or group-stagger PRIs will be split into multiple pulse sequences, and a splicing step is required to join the subsequences into a complete pulse sequence. Reference [8] models the high-dimensional PRI pattern of radar pulse train concretely, and then exploits the PRI pattern to realize online and parallel deinterleaving of staggered pulse train directly. When compared with its counterparts based on low-dimensional (mostly one-dimensional) PRI parameters, this method gains large performance improvement in deinterleaving accuracy. However, this method requires prior information about the high-dimensional PRI pattern of the radar pulse train, which is hardly available based on existing radar signal analysis methods. Deep learning-based radar signal deinterleaving [13] and recognition [14] methods also extract and utilize the pulse repetition pattern implicitly. However, the pattern is hidden in the parameters of deep neural networks, which is very unintuitive for human beings. With the continuous increase of radar reconnaissance data, it has become a difficult task to analyse all radar signals manually. Moreover, only if coarse-grained knowledge is extracted, for example, framework PRIs, from reconnaissance data, the efficiency and accuracy of radar intelligence will be greatly reduced. Automatic PRI pattern analysis of conventional radars is a key technology to explore the potential of big radar reconnaissance data and promote the development of automation of radar intelligence analysis. The machine learning technique of frequent item mining [15] to the analysis of radar reconnaissance data is introduced and an automatic PRI pattern reconstruction method is proposed that is suitable for radars with conventional PRI types, such as constant, stagger and group stagger. The rest of this article mainly consists of four parts. Section 2 illustrates the purpose of PRI pattern analysis and highlights its role in boosting radar intelligence. Section 3 presents the procedures of PRI pattern reconstruction in detail. Section 4 carries out simulations to verify the performance of the proposed method. Section 5 concludes the whole article. 2 ILLUSTRATION OF RADAR PRI PATTERN RECONSTRUCTION Radar signals are usually arranged temporally according to particular PRIs. The PRI parameters are closely related to radar performances such as operating range and ambiguity resolution ability. The ordering of sequential signals constitutes the PRI pattern of radar. Radar PRI pattern is preserved in the DTOA of reconnaissance signals, except that it is contaminated with data noises such as missing pulses and interferential pulses. If emitter location information is available to separate signals from different radars, the influence of missing pulses on the observed pulse train is more prominent than that of interferential ones, and the DTOAs calculated from reconnaissance pulses may deviate from real PRI values. Figure 1 shows a schematic diagram of the relationship between DTOA of reconnaissance pulses and PRI of transmitted pulses. The parameter of PW is also included to represent the influence of other parameters. In the figure, solid rectangles represent observed radar pulses with a third-order stagger PRI pattern, and dotted rectangles represent missing pulses caused by various factors such as antenna scanning, signal propagation and so forth. FIGURE 1Open in figure viewerPowerPoint Relationship between differential time-of-arrival (DTOA) in radar reconnaissance pulses and pulse repetition interval (PRI) of radar transmitted pulses The PRI pattern of the pulse train in Figure 1 is s 1 → PRI 1 s 2 → PRI 2 s 3 → PRI 3 s 1 , where s1, s2 and s3 indicate cyclically changing pulse states in the pulse train with a third-order stagger PRI pattern. Each state is determined by the context of PRIs and the width of the pulse. Some other signal parameters, such as carrier frequency and pulse amplitude, can also be included as part of radar pulse state. In order to facilitate presentation, it is assumed in this article that the other pulse parameters except the time of arrival (TOA) and PW are fixed, and thus the periodically repeating pulse states depend on the value of the PRIs surrounding the pulse and the PW. Accordingly, the PRI pattern of the pulse train in Figure 1 can be abbreviated as (PRI1〈PW2〉, PRI2〈PW3〉, PRI3〈PW1〉). In order to facilitate representation, the PW is left out from the pattern when no confusion is introduced, which simplifies the PRI pattern to (PRI1, PRI2, PRI3). Compared with the discrete PRI set of {PRI1, PRI2, PRI3}, the PRI pattern retains the ordering information of different PRIs, which makes the description of radar pulse repetition law more accurate. The PRI pattern can also promote an online and parallel solution of the radar signal deinterleaving problem [8], which is very difficult or even impossible based on PRI set. The PRI pattern analysis process is a process of automatic PRI pattern reconstruction from radar reconnaissance signals, as shown in Figure 2. FIGURE 2Open in figure viewerPowerPoint Illustration of PRI pattern analysis based on radar reconnaissance signals After receiving radar signals containing TOA information and directly calculating DTOA between adjacent pulses, the resulting DTOA set contains not only the real PRI of radar, such as PRI1 and PRI3 in Figure 1, but also false DTOA items caused by missing pulses, such as PRI2 + PRI3 and PRI1 + PRI2 in Figure 1. At the same time, PWs of all pulses are recorded, which are denoted by PW1, PW2 and PW3. These directly calculated DTOAs, which contain both true and false PRI values, may deliver spurious information to subsequent pulse analysis processes, for example, radar pulse deinterleaving and radar type identification. Among the most widely known contributions in the area of radar reconnaissance signal analysis, people may always mention some pulse deinterleaving methods, including those based on the DTOA histogram [9, 10] and the PRI transform [11]. The methods weaken the impact of data noises in pulse train by exploiting parameter statistics. For the pulse train in Figure 1, these methods first estimate the framework PRI, that is, PRI = PRI1 + PRI2 + PRI3, and then use this parameter to deinterleave radar pulses. From the statistical point of view, the framework PRI has stronger noise adaptability than that of PRIs between adjacent pulses. However, the statistical parameter estimation process artificially reduces the dimension of high-order PRI pattern to one which introduces a large amount of information loss to the temporal ordering of radar signals, and also increases the difficulties of radar signal deinterleaving, identification and other applications. For example, in the application of radar signal deinterleaving, the methods based on framework PRI can only be realized by offline processing after accumulating a certain amount of data, and they are also difficult to separate interleaved pulse trains with similar PRIs. On the contrary, the method based on complete PRI pattern can realize parallel deinterleaving of multiple pulse trains with the same framework PRI, and it is more robust to signal measurement and parameter estimation errors [8]. With the participation of experts, the analysis of PRI pattern based on reconnaissance signals of conventional radar is not very difficult, but the manual analysis process has significant limitations when compared with automatic process in terms of calculation efficiency and analysis ability of big data. Automatic PRI pattern reconstruction is a very important problem in the intelligent processing of radar reconnaissance big data, especially in the existence of various data noises such as missing pulses and interferential pulses. 3 AUTOMATIC PRI PATTERN RECONSTRUCTION In view of the significant advantages of the PRI pattern in radar signal feature representation and reconnaissance signal processing applications, an automatic PRI pattern reconstruction method is proposed in this section. 3.1 Principle of PRI pattern reconstruction: extraction and extension of frequent DTOA items The process of PRI pattern reconstruction is to extract the stable temporal mode of radar signals, for example, (PRI1, PRI2, PRI3) in Figure 2, from reconnaissance data with various noises. In the absence of data noises, it becomes a formal representation problem of the periodic pulse repetition pattern, which can be solved by data compression [16]. The existence of data noises introduces outlier DTOAs to the pulse train, such as PRI2 + PRI3 and PRI1 + PRI2 in Figure 1. These outliers and their influence on context cause great diversity to the PRI sequence of pulse train. It is a key problem to eliminate the influence of data noises in stable PRI pattern reconstruction. The measurement errors of pulse TOAs and PWs are not considered in this article except in the simulations, because they can be well smoothed by introducing error tolerance during parameter clustering. Although observation noises such as missing pulses and interferential pulses temporally destroy the stable PRI mode of the pulse train, the disturbed PRI mode will quickly recover to normality after instantaneous disturbances, and the overall PRI law will be retained in the whole pulse train. From the perspective of frequency of emergence, DTOAs (together with widths of their head and tail pulses) between adjacent radar pulses constitute a frequent DTOA set, while spurious DTOAs caused by interferential pulses have random values and emerge infrequently. A frequent item set is defined as DTOAs folded by a particular pair of PW, and the frequency of item emergence exceeds a preset threshold [15], for example, 〈PW1〉PRI1〈PW2〉, 〈PW2〉PRI2〈PW3〉, 〈PW3〉PRI3〈PW1〉, 〈PW2〉[PRI2 + PRI3]〈PW1〉 and 〈PW1〉[PRI1 + PRI2 + PRI3]〈PW1〉 in the pulse train in Figure 1 when the reconnaissance time is appropriately long. In the frequent DTOA set, large DTOAs arise when the pulse train switches between non-adjacent states, and they have values equal to the sum of multiple adjacent unit PRIs. The key to reconstruct radar PRI pattern from noise-contaminated reconnaissance signals is to eliminate the influence of data noises such as missing and interferential pulses on pulse repetition rule and use a unified temporal pattern to describe diversified reconnaissance data, as shown in Figure 3. In noise-contaminated reconnaissance pulse trains, split PRIs caused by interferential pulses have strong randomness, as illustrated in the value of PR'2 and PR'3 in the second pulse segment on the left of Figure 3, in which the filled rectangle represents interferential pulse, while large DTOAs caused by missing pulses match the sum of several adjacent PRIs in the accurately reconstructed PRI pattern, for example, the pulse interval PRI2 + PRI3 in the last pulse train on the left of Figure 3 is the sum of two adjacent unit PRI. Based on these evidence, reconstruction of the radar PRI pattern based on the extraction and extension of frequent 〈PW〉PRI〈PW〉 items was proposed. The influence of interferential pulses will be eliminated by the statistical significance of true DTOA values, and the influence of missing pulses will be eliminated by exploiting the matching between large DTOAs and the sum of the successive unit PRIs. FIGURE 3Open in figure viewerPowerPoint Schematic diagram of a single iteration during PRI pattern reconstruction via extension of frequent DTOAs Figure 3 gives a brief illustration of the PRI pattern reconstruction process in the presence of various data noises. It is assumed that the first PRI element of the PRI pattern, that is, PRI1, has been obtained, and the following PRI element PRI2 is observed in some pulse segments but destroyed by interferential pulses and missing ones in other pulse segments. The principle of extending the radar PRI pattern based on PRI1 is as follows, P R I 2 = max D T O A f ( { P R I 1 } , Δ ) (1)where {PRI1} represents the index sets of the head and tail pulses corresponding to DTOAs equalling PRI1 in the observed signals, and Δ represents forward DTOA extension beginning with the tail pulses of the previous DTOA equalling PRI1. The PW of the head pulse of Δ is required to be consistent with the PW of the tail pulse of PRI1. The function f(·,·) is a count-dependent function that depends on the number of pulses shared by the tail pulses of {PRI1} and head pulses of Δ. In the four pulse segments shown on the left side of Figure 3, the interval PRI1 is extended forward. When the value of DTOA is taken in the set {PRI2, PR'2, PRI2 + PRI3}, the counter f(gg) in Equation (1) takes non-zero values. With the increase in the number of pulse segments, the frequencies of true pulse intervals such as PRI2 and PRI2 + PRI3 increase continuously, while the split PRI value of PR'2 caused by interferential pulse does not form a frequent item due to its randomness. During automatic PRI pattern reconstruction, one needs to design a reasonable counting function f(gg), so that the correct unit PRI of PRI2 will be highlighted and finally be selected as the extension PRI in this iteration. As the PRI pattern is extended from PRI1 to (PRI1, PRI2, PRI3) in two iterations, an effective criterion is needed to terminate the iterative process, and the PRI sequence is output as the reconstructed PRI pattern. Based on the abovementioned major idea of PRI pattern reconstruction, the process mainly consists of two steps, that is, frequent DTOA extraction and correlation-based pattern extension. The process flow is shown in Figure 4. FIGURE 4Open in figure viewerPowerPoint Schematic flow of PRI pattern reconstruction 3.2 Extraction of frequent DTOAs In order to eliminate the influence of interferential pulses and avoid repeated scanning of the original pulse train in the process of PRI pattern extension, first, the DTOAs (adjacent or non-adjacent) that emerge frequently in the pulse train are extracted and their head and tail pulse indexes are recorded in the original pulse train, together with their PWs. The following frequent 〈PW〉DTOA〈PW〉 set is obtained in this step, { ( P W i ( h ) , D T O A i , P W i ( t ) ) , M i , { h m ( i ) } m = 1 , ⋯ , M i , { t m ( i ) } m = 1 , ⋯ , M i } i = 1 , ⋯ , I (2)where I represents the number of frequent items in the reconnaissance pulse train whose occurrence time exceeds a preset threshold M0. The pattern of each item is represented by a DTOA and two PWs folding it, where DTOAi represents the DTOA value corresponding to the ith frequent item that is obtained by numerical clustering; PWi(h) and PWi(t) represent the widths of the head and tail pulses, respectively. The ith frequent item appears Mi (Mi ≥ M0) times in the pulse train, the head pulse indexes corresponding to the frequent item DTOAi constitute a set {hm(i)}m=1,⋯,Mi, and the tail indexes constitute a set {tm(i)}m=1,⋯,Mi. As each unit PRI will appear repeatedly in a reconnaissance pulse train that is appropriately long, they will always be included in the frequent item set in Equation (2). The spurious PRIs caused by interferential pulses are excluded from the frequent item set because of their random values, and the influence of interferential pulses is thus smoothed during frequent extraction. However, the DTOAs caused by non-adjacent pulses, such as PRI1 + PRI2, are still retained in the frequent item set. Therefore, the difficulty of PRI pattern reconstruction based on the frequent set is mainly two-fold. One is eliminating the interference of DTOAs between non-adjacent pulses, and the other is determining the ordering and dimension of the unit PRIs. 3.3 Pattern extension based on frequent DTOAs In the frequent item set given by Equation (2), it is incorrect to judge whether an interval DTOAi constitutes the PRI pattern according to its occurrence frequency Mi. This is because the occurrence frequency of non-adjacent pulse intervals, for example, PRI1 + PRI2, is not necessarily lower than that of its PRI components of PRI1 and PRI2. In particular, the occurrence frequency of the framework PRI, that is, PRI1 + PRI2 + PRI3 in Figure 1, and its harmonics is usually higher than that of other DTOAs. Based on this consideration, a DTOA selection criterion is set during PRI pattern extension according to the ratio of its occurrence time to its value. Denote the already reconstructed PRI pattern by the current time instant as Ξ, it contains 0 or several ordered PRI units. The indexes of pulse segments in the observed pulse train that match the reconstructed PRI pattern constitute matrix C, with each row in C corresponding to the index sequence of each pulse segment. The indexes of tail pulses in the pulse segments constitute set T. Define the DTOA selection parameter as α i = M ′ i / D T O A i (3)where M'i represents the number of pulse pairs separated by an interval of DTOAi that can be used to extend any pulse segment matching PRI pattern Ξ, and M'i = Mi if Ξ is empty. If Ξ is not empty, is defined as the number of head pulses of frequent item DTOAi having the same pulse indexes and widths with the tal pulses contained in the index set T, that is, M ′ i = ∑ m = 1 M i δ ( h m ( i ) , T ) (4)where δ(hm(i), T) is an indicative function, δ(hm(i),T) = 1 when hm(i) ∈ T, and δ(hm(i),T) = 0 otherwise. The criterion of PW consistence is not included in Equation (4) to facilitate presentation. By checking each frequent DTOA in Equation (2) and calculating their selection parameters defined in Equation (3), the index of frequent DTOA selected for the PRI pattern extension is determined as follows, j = arg max i { α i } i = 1 , ⋯ , I (5) This frequent item is used to extend the already reconstructed pattern Ξ, and an extended variant of (Ξ, DTOAj) is obtained. The indexes of the head pulses of the selected frequent item DTOAj, denoted by {hm(j)}m=1,⋯,Mj, overlap largely with the tail pulse indexes T of the already reconstructed pattern. During the pattern extension procedure, the tail pulse index tm(j) corresponding to each overlapped head pulse hm(j) is added to the end of the corresponding row in C, and null indexes are added to the end of the other rows whose tail pulse index does not overlap with {hm(j)}m=1,⋯,Mj. An extended index matrix C˜ is obtained at the end of this procedure, as is shown in Figure 5. The PWs of the pulses in each column of C˜ are consistent. FIGURE 5Open in figure viewerPowerPoint Extension of the pulse segments in the existence of missing pulses FIGURE 6Open in figure viewerPowerPoint Influence of observed pulse number and pulse missing rate on PRI pattern reconstruction performance (a) PRI pattern: [300, 330, 370] µs, (b) PRI pattern: [300, 330, 300, 330, 370] µs The DTOA selection parameter defined in Equation (3) also helps to solve the problem of ambiguity in the PRI pattern caused by its rotation symmetry. Taking the third-order stagger PRI pulse train in Figure 1 as an example (it is assumed that the three unit PRIs are not equal), the PRI pattern of the pulse train can be reasonably described as {PRI1, PRI2, PRI3}, {PRI2, PRI3, PRI1} or {PRI3, PRI1, PRI2}. After defining the DTOA selection parameter in Equation (3), the minimal PRI in {PRI1, PRI2, PRI3} will probably be automatically selected as the starting element of the reconstructed PRI pattern, while the other possibilities are eliminated and the uniqueness of the constructed pattern is guaranteed. 3.4 Adaptation to missing pulses The negative influence of interferential pulses on PRI pattern reconstruction can be well smoothed in the process of frequent item extraction, but the influence of missing pulses still exists. The specially designed DTOA selection parameter in Equation (3) helps true PRIs to outperform non-adjacent DTOAs during PRI pattern extension, which largely reduces the effect of missing pulses. However, complete exclusion of pulse segments containing missing pulses from C will result in a large reduction of the available data, and the performance of pattern extension will probably be heavily deteriorated. Taking the last pulse segment on the left side of Figure 3 as an example, although the third pulse in the PRI cycle is missing, the pulse TOAs before and after this pulse still match the PRI pattern of PRI1, PRI2, PRI3 well. If these incomplete pulse segments are excluded arbitrarily, no complete PRI pattern may be found in pulse trains with large missing rate and high-dimensional PRI pattern, leading to unsuccessful PRI pattern reconstruction processes. An effective way to deal with the pulse segments with missing pulses is to store the partially matched segments in C and perform normal extension steps on them. Figure 5 illustrates the PRI pattern matching procedure in the existence of missing pulses. The first pulse segment represents the reconstructed PRI pattern obtained by the proposed method, and in the latter, several segments represent the observed pulse segments stored in the sequence matrix C. In the latter segments, the unfilled rectangles correspond to the pulses stored in C before this iteration, and the filled rectangles represent the newly added pulses in this iteration. In the second and third pulse segments, a tail pulse is added according to the newly extended pattern of PRI3. The fourth pulse segment only matches partially with the PRI pattern before the current extension process, and a tail pulse is added according to the frequent DTOA PRI2 + PRI3 in this iteration to ensure that the segment still matches the reconstructed PRI pattern in case of missing pulses. In the fifth segment, the head and tail pulses corresponding to the frequent DTOA of PRI1 + PRI2 + PRI3 are added. The pulse segment extension idea illustrated in Figure 5 well adapts to inter-pulse DTOA variants caused by missing pulses, and fully covers pulse segments with completely and partially matching PRI patterns. Following similar processing procedures, the pulse segments can be fully extracted from the observed pulse train when the order of the reconstructed PRI pattern increases gradually. 3.5 Termination criterion for PRI pattern extension The dimension of the reconstructed PRI pattern increases gradually as the pattern extension process is carried out continuously. If no termination criterion is designed for this process, a repeated PRI pattern will be obtained, for example, (PRI1, PRI2, PRI3, PRI1,⋯) for the pulse sequence in Figure 1. Only by terminating the iterative process appropriately, the accurate PRI pattern will be obtained, that is, (PRI1, PRI2, PRI3). By observing the radar pulse train with conventional PRI patterns, such as constant, stagger and group-stagger, it is easy to find that when the reconstructed PRI pattern forms a complete period, the tail pulses of each row in the pulse index matrix C and the first pulses of other rows overlap with each other or are separated from each other by an integer multiple of the framework period. But a precondition of this property is that the pulse states of each row in C are consistent. Take the pulse segments in Figure 5 for example, the pulse states of each row should correspond to the PRI pattern (PRI1, PRI2, PRI3) or {PRI2, PRI3, PRI1} or {PRI3, PRI1, PRI2} simultaneously, and there is no state dislocation between them. The state dislocation of the pulse segments is partly exemplified in Figure 5, in which the pulse segments that match the reconstructed PRI pattern are labelled with pulse indexes. The PRI pattern of the pulse train is a three-dimensional stagger, the pulse states of the other four segments in the middle correctly match the reconstructed pattern. The pulse segments having mismatched states should be removed from C, so that the PRI pattern extension process can be terminated more accurately. The method of eliminating pulse segments with mismatched states is to define a confidence degree according to the number of matched pulses between the pulse segment and the reconstructed PRI pattern (the more the number of matched pulses, the higher the confidence degree), and select pulse segments in C with higher confidence degrees as references to verify the validity of segments with lower degrees sequentially. If the latter pulse segments overlap any reference segment (an exclusion is that the head pulse of one segment overlaps the tail pulse of another), the latter is considered to be mismatching with the reconstructed PRI pattern and thus is eliminated from C. After aligning the index matrix C according to pulse states, a termination criterion for the iteration process can be set according to the temporal relationships between the pulse segments in C as follows, max k { r e m ( T O A c 1 ( k + 1 ) − T O A c e n d ( k ) , s u m ( Ξ ) ) } k = 1 , ⋯ , K − 1 < ε (6)where K represents the number of rows of C after eliminating overlapped pulse segments, c1(k+1) and cend(k) represent the indexes of the first pulse in the (k+1)th row of C and the last pulse in the kth row, TOAc1(k+1) and T O A c e n d ( k ) represent the TOA of the two pulses, sum (Ξ) stands for the sum of the PRIs in the reconstructed pattern Ξ, and rem(a,b) represents the minimum deviation between a and the integer times of b, that is, rem(a,b) = |a−b × round(a/b)|, ε is a pre-defined threshold that depends on the measurement error of pulse TOAs. Once the iteration terminates, the pulse segments retained in C can be considered to be consisting of pulses of the same radar, and they can be gathered together to realize coarse pulse deinterleaving. The deinterleaving result can be refined using the FSA-based method [8] based on the reconstructed PRI pattern. 4 SIMULATIONS The difficulty of radar PRI pattern reconstruction mainly depends on the number of observed radar pulses, the complexity of PRI pattern, pulse missing rate and ratio of interferential pulses. In this section, simulations are carried out to verify the performance of the proposed method and investigate the influence of different factors on its performance. In the simulations, it is assumed that the standard deviation of pulse TOA measurement error is 0.2 μs, the range of frequent DTOAs is 200 to 2000 μs, and their frequency threshold is taken as M0 = 10. The widths of all pulses are fixed at 2 μs to simplify simulations. When the PRI pattern reconstruction result is consistent with a particular rotation symmetric representation of the true PRI pattern, for example, (PRI1, PRI2, PRI3), (PRI2, PRI3, PRI1) or (PRI3, PRI1, PRI2) for the pulse train in Figure 2, the reconstructed pattern is considered to be correct. The error threshold of the iteration termination criterion in Equation (6) is set to ε = 1 μs. 4.1 Influence of observed pulse number and missing pulse rate In this group of simulations, it is assumed that no interferential pulse is contained in the observed pulse train, and different pulses are missed independently and randomly with a particular probability. The pulse missing rate is set to 0%, 20%, 40% and 60%, and the number of pulses N in the observed pulse train is set to 100, 200 and 300. When the PRI pattern is set to [300, 330, 370] μs and [300, 330, 300, 330, 370] μs, the pattern reconstruction accuracies obtained in 300 simulations in cases of different parameters are shown in Figure 6a,b It can be seen from Figure 6 that the proposed PRI pattern reconstruction method can accurately reconstruct the radar PRI pattern in cases of appropriate radar pulse number and pulse missing rate. When the number of radar pulses exceeds 100 and pulse missing rate does not exceed 40%, the PRI pattern reconstruction accuracy is very close to 100%. The pattern reconstruction performance deteriorates with the decrease of radar pulse number and the increase of pulse missing rate. Moreover, the increase of PRI pattern complexity also leads to the decline of the reconstruction performance. 4.2 Influence of interferential pulses Based on the previous group of simulations, the pulse missing rate is fixed at 30%, set the number of observed pulses to 100 and 200, and increase the ratio of interferential pulse number with respect to the radar pulse number gradually from 0 to 1. The TOAs of interferential pulses are set randomly within the observation time, and the number of interferential pulses between each pair of adjacent radar pulses is set according to Poisson distribution, whose mean equals the preset interferential pulse ratio. When the radar PRI pattern are set to [300, 330, 370] μs and [300, 330, 300, 330, 370] μs, the PRI pattern reconstruction accuracies of the proposed method obtained in 300 simulations in different scenarios are shown in Figure 7a,b. FIGURE 7Open in figure viewerPowerPoint Influence of observed pulse number and interferential pulse rate on PRI pattern reconstruction performance (a) PRI pattern: [300, 330, 370] µs, (b) PRI pattern: [300, 330, 300, 330, 370] µs On the whole, the PRI pattern reconstruction performance deteriorates with the decrease of observed pulse number, the increase of interferential pulse number and the increase of PRI pattern complexity in Figure 7. The influence of these factors is particularly significant when the number of radar pulses is small, and the PRI pattern has a high dimension. That is because when the number of obtained pulses is adequately large, the statistical process of frequent DTOA extraction helps to eliminate the influence of interferential pulses. 5 CONCLUSIONS An automatic PRI pattern reconstruction method is proposed, and it is based on the extraction and extension of frequent DTOA items. The basic PRI pattern of radar pulses is reconstructed automatically, and the radar pulses are deinterleaved as a by-product. Simulation results show that the proposed method can stably reconstruct the PRI pattern from noise-contaminated radar pulse train under conditions of appropriate pulse number, missing the pulse rate and the interferential pulse ratio, and a better PRI pattern reconstruction performance can be obtained in cases of more observed radar pulses, lower missing and interferential pulse rates. ACKNOWLEDGEMENTS The work is partially supported by the Hunan Provincial Natural Science Foundation of China (2020JJ2037), Hunan Provincial Support Program for Outstanding Youth (2019RS2026), and Hunan Provincial Innovation Research Group Program (2019JJ10004). CONFLICT OF INTEREST None. REFERENCES 1Wiley, R.G.: ELINT: The Interception and Analysis of Radar Signals. 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