A novel method for deinterleaving radar signals: First‐order difference curve based on sorted TOA difference sequence

IET Signal ProcessingVolume 17, Issue 1 e12162 ORIGINAL RESEARCHOpen Access A novel method for deinterleaving radar signals: First-order difference curve based on sorted TOA difference sequence Min Xie, Min Xie orcid.org/0000-0002-0990-1513 National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, ChinaSearch for more papers by this authorChuang Zhao, Corresponding Author Chuang Zhao [email protected] orcid.org/0000-0003-3932-8786 National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, China Correspondence Chuang Zhao, National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan 450002, China. Email: [email protected]Search for more papers by this authorYongjun Zhao, Yongjun Zhao National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, ChinaSearch for more papers by this authorDexiu Hu, Dexiu Hu orcid.org/0000-0002-6139-6489 National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, ChinaSearch for more papers by this authorZewen Wang, Zewen Wang orcid.org/0000-0001-9436-2456 National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, ChinaSearch for more papers by this author Min Xie, Min Xie orcid.org/0000-0002-0990-1513 National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, ChinaSearch for more papers by this authorChuang Zhao, Corresponding Author Chuang Zhao [email protected] orcid.org/0000-0003-3932-8786 National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, China Correspondence Chuang Zhao, National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan 450002, China. Email: [email protected]Search for more papers by this authorYongjun Zhao, Yongjun Zhao National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, ChinaSearch for more papers by this authorDexiu Hu, Dexiu Hu orcid.org/0000-0002-6139-6489 National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, ChinaSearch for more papers by this authorZewen Wang, Zewen Wang orcid.org/0000-0001-9436-2456 National Digital Switching System Engineering & Technological R & D Center, Zhengzhou, Henan, ChinaSearch for more papers by this author First published: 28 September 2022 https://doi.org/10.1049/sil2.12162AboutSectionsPDF 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 onFacebookTwitterLinkedInRedditWechat Abstract Radar signal deinterleaving is used to separate interleaved pulse streams in the electronic support measure (ESM) systems. The histogram methods based on the difference in time of arrival (TOA) are more mature in engineering applications than other methods because of their more effortless implementation. In the increasingly complex electronic battlefield, pulse repetition interval (PRI) jitter, and pulse missing are inevitable in the interleaved pulse stream. The TOA differences of pulse trains with PRI jitter distributes in multiple adjacent PRI bins, which results in the significant reduction of the corresponding histogram value. Furthermore, the histogram values of real PRIs vary with pulse missing, which indicates that the histogram lacks robustness to pulse missing. First-order difference curve based on sorted TOA difference sequence, denoted as FDC-DTOA, is proposed in this paper to overcome the shortcomings mentioned above. In this method, multi-order TOA differences are calculated and then sorted in ascending order. Then, FDC-DTOA is obtained after a first-order backward difference operation. Finally, an adaptive threshold is applied to FDC-DTOA to extract candidate PRIs. The simulation results show that the proposed method has remarkable robustness to pulse missing and excellent performance in deinterleaving the pulse stream with PRI jitter. 1 INTRODUCTION Since being used for military purposes, radar has played an essential role in modern warfare. Meanwhile, electronic reconnaissance, which takes radar radiation source signals as the main reconnaissance object, has become an important part of electronic warfare. In electronic warfare, ESM systems perform various reconnaissance tasks, such as searching, intercepting, analysing, and identifying enemy radar signals. With the increasingly complex battlefield electromagnetic environment, the radar pulse stream intercepted by the ESM receiver often consisted of multiple pulse sequences emitted by different radar emitters. To analyse enemy radar signals, the ESM system should first separate the interleaved radar pulse stream, and the technique to solve the problem is called deinterleaving. At present, deinterleaving methods can be broadly divided into two categories: 'classification' and 'identification'. The former is called source classification and be used to distinguish different types of radar emitter sources, while the latter is called specific emitter identification (SEI) and can be used to identify different individual emitter sources of the same radar type by extracting features from the corresponding signal sources. Graphical representation of the distribution of basic parameters has been used in the process of SEI [1]. However, the technique based on the statistical analysis of basic parameters is not sufficient for SEI problems, and J. Dudczyk [2] adopts the method of hierarchical clustering to the SEI process. Ł. Rybak and J. Dudczyk examine the efficiency of the Geometrical Divide method in the unbalanced datasets classification [3]. In addition, intra-pulse subtle features such as radio frequency (RF) fingerprints [4], spectral features [5], and pulse envelopes [6] etc. have also been successfully applied to the process of SEI and achieved promising effects. The methods for source classification can also be classified into two categories: feature clustering and PRI analysis. The main feature parameters currently used for source classification include direction of arrival (DOA), RF, pulse width (PW), pulse amplitude (PA), and TOA. The first four parameters are generally used for pre-deinterleaving in multiple parameters joint deinterleaving. The conventional method adopted in multiple parameters joint deinterleaving is feature clustering [7]. With density-based has emerged as a practical clustering method, the adaptive density-based clustering method has also been applied to radar signal deinterleaving [8]. Compared with the conventional clustering method, the fuzzy adaptive resonance theory (ART) has distinct advantages in intelligent processing [9]. The last parameter, TOA, is used for PRI analysis to deinterleave the interleaved pulse stream after pre-deinterleaving. As the final part of deinterleaving, PRI analysis is used to separate the interleaved TOA stream into multiple TOA sequences corresponding to different sources. This paper mainly studies the deinterleaving method based on PRI analysis, which is also called PRI deinterleaving. A large number of scholars have conducted relevant studies on PRI deinterleaving in recent years, such as statistic association pulse deinterleaving (SAPD) [10], square sine wave interpolating (SSWI) [11], isomorphic subsequence (IS) [12], geometrical approach (GA) [13], artificial intelligence methods [14-16], correlation matching method (CMM) [17], and histogram methods. SAPD uses Kalman filtering and multi-hypothesis tracking to deinterleave, but it is prone to model mismatch. SSWI, IS, and GA can only deinterleave the pulse train with slight PRI jitter. Z. Liu et al. [14] introduce recurrent neural networks (RNNs) to solve problems of sequential pattern classification, denoising, and deinterleaving of pulse streams. X. Li et al. [15] develop a novel approach based on denoising autoencoders for TOA deinterleaving. M. Zhu et al. [16] propose a novel method based on a supervised neural machine translation (NMT) network. The artificial intelligence methods have a remarkable deinterleaving effect, but they face tremendous challenges in training sample acquisition and real-time. CMM determines PRI values by comparing the projected energy of pulse sequences on different orthogonal period subspaces. However, CMM has some non-negligible problems in the threshold calculation method and period identification rules. Histogram methods divide the statistical range of PRI values into multiple PRI bins. The number of TOA differences that are distributed in each PRI bin is counted and then the histogram is built. The basic principle of histogram methods is to approximate the shape of the probability density distribution curve of TOA differences with the histogram and then set a detection threshold to extract the candidate PRIs. Cumulative difference (CDIF) histogram [18] accumulates the TOA differences of the current and all previous orders, sequence difference (SDIF) histogram [19] only accumulates the TOA differences of the current order, and PRI transform [20, 21] accumulates all the TOA differences within the preset reasonable range. While dealing with the pulse train with PRI jitter, CDIF, SDIF, and PRI transform are faced with the problem that TOA differences are distributed into multiple adjacent PRI bins. Once the quantity of TOA differences distributed in the PRI bin corresponding to real PRIs is reduced to below the threshold, there is a failure in extracting the candidate PRIs. To deal with this problem, the improved SDIF methods [22] and the improved PRI transform methods [23, 24] adopt overlapped PRI bins to histogram. The centers of overlapped PRI bins are the same as the original PRI bins, while the widths are required to be greater than the PRI jitter value in general. This strategy improves the peak level of jittered PRIs, but also widens the peak width of fixed PRIs and then increases the risk of overlapping the peak region corresponding to different real PRIs. M. Bagheri et al. improved PRI estimation accuracy by adopting novel overlapped PRI bins to histogram [25], but the negative impact of overlapping PRI bins is not fundamentally changed. Z. Ge et al. [26] introduce the mean filter to process the histogram and then adopt the interquartile range algorithm to determine fliers, which improves the processing ability for the pulse train with PRI jitter. However, the mean filter will also widen the peak width of fixed PRIs to a certain degree. Due to the histogram values of real PRIs vary with pulse missing, the histogram methods with a fixed threshold are not robust to pulse missing. Once the histogram values of real PRIs below the threshold, with similar histogram values and smaller thresholds, the sub-harmonics are easier to be extracted instead. In CDIF and SDIF, when there are a lot of missing pulses, the sub-harmonics of the real PRIs can cause severe interference. PRI transform methods can efficiently avoid the influence of sub-harmonics by adopting the complex-valued phase to histogram, but the problem that is the histogram values of the real PRIs below the threshold is not settled. To strengthen the robustness of histogram to pulse missing, Z. Ge et al. [26] design an adaptive adjustment threshold, which limits the proportion of the frequency of PRI bins whose frequencies are below the threshold in the total frequency. M. Bagheri et al. [27] design an adaptive threshold based on the constant false alarm rate (CFAR) theory, where the TOA difference histogram is modelled as received signal and noise. In addition, some scholars have also proposed some synthetic methods based on the histogram method to overcome the existence of PRI jitter and pulse missing. Y. Liu et al. [28] build histogram based on clustering in SDIF to avoid the reduction of the histogram peak caused by PRI jitter. In order to suppress the influence of clustering error, this method selects the interval around the potential PRI as the range of PRI value and then adopts PRI transform to extract the real PRI value. W. Cheng et al. [29] propose a novel method combined with the improved CMM and PRI transform, where the improved CMM is used to deinterleave fixed PRIs and staggered PRIs, and the improved PRI transform is used to deinterleave jitter PRIs. C. Zhang et al. [30] propose a method combined the improved SDIF with PRI transform, where a double-threshold is set in SDIF to enhance the detection ability in the case of pulse missing, and a new PRI bin structure is used for PRI transform to improve the PRI values estimation accuracy. A novel method based on FDC-DTOA is proposed to deinterleave the interleaved pulse stream in the presence of PRI jitter and pulse missing in this paper. Simulation results have proved that the proposed method has excellent deinterleaving performance and robustness to pulse missing. The rest of this paper is organized as follows. In Section 2, the TOA differences histogram is reviewed. Section 3 analyzes the disadvantages of the histogram. The details of the proposed method are introduced in Section 4. Experimental results and comparison with other deinterleaving methods are demonstrated in Section 5. Finally, the conclusion is drawn in Section 6. 2 HISTOGRAM METHOD PRI deinterleaving based on TOA difference histogram is a dynamic processing process where PRI extraction and sequence retrieval are operated alternately [19]. TOA difference histogram can be used to extract candidate PRI values, and the estimated PRI value is judged in sequence retrieval process. The candidate PRI value is true only if the corresponding pulse train is separated from the interleaved pulse stream. In this section, we will review TOA difference histogram and its threshold for extracting candidate PRI values. TOA difference histogram Multi-order TOA differences are calculated, and τ min , τ max $\left[{\tau }_{\mathrm{min}},{\tau }_{\mathrm{max}}\right]$ is the statistical range of PRI values. Then, divide the range uniformly into K $K$ different Sections, each of which is a PRI bin. The center of k t h $k\mathrm{t}\mathrm{h}$ PRI bin is τ k = k − 1 / 2 K τ max − τ min + τ min , k = 1 , . . . , K ${\tau }_{k}=\frac{k-1/2}{K}\left({\tau }_{\mathrm{max}}-{\tau }_{\mathrm{min}}\right)+{\tau }_{\mathrm{min}},k=1,...,K$ (1) The width of each PRI bin is b = τ max − τ min / K $b=\left({\tau }_{\mathrm{max}}-{\tau }_{\mathrm{min}}\right)/K$ . TOA sequence can be modelled as Ref. [23] g t = ∑ n = 1 L δ t − t n $g\left(t\right)=\sum\limits _{n=1}^{L}\delta \left(t-t{}_{n}\right)$ (2)where t n ${t}_{n}$ is the TOA of n t h $n\mathrm{t}\mathrm{h}$ pulse, L $L$ is the length of the pulse stream, and δ ( ⋅ ) $\delta (\cdot )$ is the unit impulse function. The number of TOA differences distributed in its corresponding PRI range is the histogram value of each PRI bin. The histogram value of k t h $k\mathrm{t}\mathrm{h}$ PRI bin is Ref. [23] C ( k ) = ∫ τ k − b / 2 τ k + b / 2 ∫ − ∞ ∞ g ( t ) g ( t + τ ) d t d τ = ∫ τ k − b / 2 τ k + b / 2 ∑ u = 2 N ∑ v = 1 u − 1 δ τ − t u + t v d τ $C(k)=\int \nolimits_{{\tau }_{k}-b/2}^{{\tau }_{k}+b/2}\int \nolimits_{-\infty }^{\infty }g(t)g(t+\tau )dtd\tau =\int \nolimits_{{\tau }_{k}-b/2}^{{\tau }_{k}+b/2}\sum\limits _{u=2}^{N}\sum\limits _{v=1}^{u-1}\delta \left(\tau -{t}_{u}+{t}_{v}\right)d\tau $ (3) Threshold for extracting the candidate PRIs After getting histogram values of different PRI bins, if C ( k ) $C(k)$ exceeds the corresponding threshold T h r e s h o l d τ k $Threshold\left({\tau }_{k}\right)$ , τ k ${\tau }_{k}$ is set as a candidate PRI value [18]. T h r e s h o l d τ k = α T τ k $Threshold\left({\tau }_{k}\right)=\alpha \frac{T}{{\tau }_{k}}$ (4)where T is the time length of the TOA sequence, α $\alpha $ is a tunable parameter from 0 to 1, and T / τ k $T/{\tau }_{k}$ is the length of the corresponding pulse train without pulse missing. 3 DISADVANTAGES OF THE HISTOGRAM In this section, we will introduce and analyze the disadvantages of TOA difference histogram when faced with PRI jitter and pulse missing. Influence of PRI jitter The TOA sequence of a pulse train can be represented as T O A n = T O A n − 1 + PR I n − 1 $TO{A}_{n}=TO{A}_{n-1}+\mathit{PR}{I}_{n-1}$ (5)Here, n $n$ is the sequential number of TOAs and PR I n − 1 $\mathit{PR}{I}_{n-1}$ represents the changing difference between adjacent TOAs. The instantaneous PRIs can be formulated as: PR I n = PR I 0 + j v , j v ∈ − PR I 0 ε , PR I 0 ε $\mathit{PR}{I}_{n}=\mathit{PR}{I}_{0}+jv,jv\in \left[-\mathit{PR}{I}_{0}\varepsilon ,\mathit{PR}{I}_{0}\varepsilon \right]$ (6) Here, PR I 0 $\mathit{PR}{I}_{0}$ is the central value of PRIs, j v $jv$ is the random jitter value of PRIs relative to PR I 0 $\mathit{PR}{I}_{0}$ , and ε $\varepsilon $ is the jitter bound of PRIs relative to PR I 0 $\mathit{PR}{I}_{0}$ . The PRI type of the pulse train can be regarded as fixed when ε ≤ 0.01 $\varepsilon \le 0.01$ , and jittered when 0.01 ≤ ε < 0.3 $0.01\le \varepsilon < 0.3$ [31]. While processing a pulse train with PRI jitter, its limit of PRI jitter value is far much than the width of PRI bins. As a result, TOA differences are distributed among multiple contiguous PRI bins, which significantly reduces the histogram values corresponding to real PRIs. Then, we analyse the effect of PRI jitter on histogram through a simple simulation experiment. The observation time is 5 × 104 us. There are two pulse trains in the interleaved pulse stream. Their missing rate of pulse is 0.05, central PRI values are 250 us and 370 us, and PRI jitter bounds are 0.005 and 0.1, respectively. As shown in Figure 1a, the histogram values of ε = 0.1 $\varepsilon =0.1$ is far less than ε = 0.005 $\varepsilon =0.005$ , which indicates that the performance of the histogram degrades greatly when faced with PRI jitter. FIGURE 1Open in figure viewerPowerPoint Influence of PRI jitter and pulse missing on histogram. (a) ε = 0.005 $\varepsilon =0.005$ and ε = 0.1 $\varepsilon =0.1$ , (b) σ = 0.05 $\sigma =0.05$ and σ = 0.3 $\sigma =0.3$ Influence of pulse missing For each real PRI, its histogram value C ( k ) $C(k)$ varies with the missing rate of pulse, while the threshold T h r e s h o l d τ k $Threshold\left({\tau }_{k}\right)$ is fixed. When plenty of pulse is missing, the histogram value of real PRIs less than the threshold. Because of similar histogram values and smaller thresholds, the sub-harmonics are easier to be detected instead. As a result, the sub-harmonics will mislead the sequence retrieval process. Then, we analyze the effect of pulse missing on histogram. The PRI jitter bound is 0.005, and the missing rates of pulse are 0.05 and 0.3 in the two pulse trains, respectively. Missing rate of pulse is denoted as σ $\sigma $ . As shown in Figure 1b, with the same threshold coefficient, the histogram value of σ = 0.05 $\sigma =0.05$ exceeds the threshold, while σ = 0.3 $\sigma =0.3$ below the threshold, which shows that the values of the histogram are not robust to pulse missing. Disadvantages of the improved histogram To avoid the influence of sub-harmonics, PRI transform adopts a complex-valued phase for calculating the histogram [20, 23]. Furthermore, to prevent TOA differences from being distributed into multiple adjacent PRI bins when faced with PRI jitter, overlapped PRI bins whose widths are greater than original PRI bins have been adopted in histogram [23]. The widths of the overlapped PRI bins are 2 × τ k × ε max $2\times {\tau }_{k}\times {\varepsilon }_{\mathrm{max}}$ , where ε max ${\varepsilon }_{\mathrm{max}}$ is equal to the maximum of PRI jitter bound in the interleaved pulse stream in general. When there are pulse trains with small PRI jitter or fixed PRI in the interleaved pulse stream, overlapped PRI bin will widen their peak width significantly. Then, we analyze the disadvantages of improved histogram through a simple simulation experiment. Let ε max = 0.1 ${\varepsilon }_{\mathrm{max}}=0.1$ , the improved histograms of ε = 0.1 $\varepsilon =0.1$ and ε = 0.005 $\varepsilon =0.005$ are shown in Figure 2a. Overlapped PRI bins improve the histogram value of ε = 0.1 $\varepsilon =0.1$ , but they also widen the peak width of ε = 0.005 $\varepsilon =0.005$ significantly. When there are pulse trains with small PRI jitter or fixed PRI in the interleaved pulse stream, overlapped PRI bins will widen their peak width significantly, which will increase the risk of overlapping the peak region corresponding to different real PRIs. Let ε max = 0.005 ${\varepsilon }_{\mathrm{max}}=0.005$ , the improved histograms of σ = 0.05 $\sigma =0.05$ and σ = 0.3 $\sigma =0.3$ are shown in Figure 2b. Weather σ = 0.05 $\sigma =0.05$ or σ = 0.3 $\sigma =0.3$ , the histogram values of sub-harmonics are suppressed significantly, which indicates that the influence of sub-harmonics is eliminated. When σ = 0.3 $\sigma =0.3$ , however, the histogram values of real PRIs are still less than the threshold, which suggests that the poor robustness to pulse missing is not changed. FIGURE 2Open in figure viewerPowerPoint Influence of PRI jitter and pulse missing on improved histogram. (a) ε = 0.005 $\varepsilon =0.005$ and ε = 0.1 $\varepsilon =0.1$ , (b) σ = 0.05 $\sigma =0.05$ and σ = 0.3 $\sigma =0.3$ 4 DEINTERLEAVING METHOD BASED ON FDC-DTOA Flow chart of the whole method To avoid the reduction of the histogram value caused by PRI jitter and improve the robustness to pulse missing, we substitute FDC-DTOA for histogram, and the deinterleaving method based on FDC-DTOA is proposed. Firstly, we calculate multi-order TOA differences of the interleaved pulse stream, and the results are sorted in ascending order. Then, the model of FDC-DTOA is built based on the sorted TOA differences. After that, the peak regions of FDC-DTOA that exceed the detection threshold are detected. Next, the candidate PRI corresponding to the peak region is estimated and optimized. Finally, the candidate PRIs are determined by the sequence retrieval. Figure 3 illustrates the flow chart for the whole method. The calculation of sorted TOA differences is explained in Algorithm 1, and the construction of FDC-DTOA is explained in Algorithm 2. Considering the adaptability to different PRI jitter bounds, the mean value and maximum value of FDC-DTOA are employed to design an adaptative threshold. After the comparison of FDC-DTOA with the threshold, the peak regions are obtained. To decrease false alarm and optimize the candidate PRIs, only the PRI value of the peak region whose width exceeds the preset value is estimated and then optimized with a weighted operator. FIGURE 3Open in figure viewerPowerPoint Flow chart of the whole method Construction and analysis of FDC-DTOA 4.2.1 Construction of FDC-DTOA Firstly, we calculate multi-order TOA differences of the interleaved pulse stream, and the calculations are sorted in ascending order. The details are elaborated as a pseudo-code in Algorithm 1, where T D $TD$ is the sorted TOA differences, and M $M$ is the length of T D $TD$ . Then, the model of FDC-DTOA is built based on T D $TD$ . The details are elaborated as a pseudo-code in Algorithm 2, where W 1 ${W}_{1}$ is the length of the filter, D T D $DTD$ is the first-order backward difference of T D $TD$ , and S D T D $SDTD$ is the smoothed result of D T D $DTD$ . With T D $TD$ as the abscissa and S D T D $SDTD$ as the ordinate, FDC-DTOA is modelled as: FDC − DTOA = { ( T D ( i ) , S D T D ( i ) ) | i = = 1 , . . . , M − 1 } $\mathrm{FDC}-\mathrm{DTOA}=\left\{(TD(i),SDTD(i))\vert i==1,...,M-1\right\}$ (7) Suppose there is a pulse train whose length is 2 N + 2 $2N+2$ in the interleaved pulse stream. This pulse train is denoted as P u l $Pul$ . Its central PRI value is p r i $pri$ and PRI jitter bound is ε P u l ${\varepsilon }_{Pul}$ . While getting T D $TD$ of the interleaved pulse stream, the first order TOA differences of P u l $Pul$ is also obtained and denoted as T D P u l $T{D}_{Pul}$ , where d i = T D P u l ( i + 1 ) − T D P u l ( i ) ${d}_{i}=T{D}_{Pul}(i+1)-T{D}_{Pul}(i)$ . Within p r i 1 − ε P u l , p r i 1 + ε P u l $\left[pri\left(1-{\varepsilon }_{Pul}\right),pri\left(1+{\varepsilon }_{Pul}\right)\right]$ , the corresponding probability density distribution curve and the FDC-DTOA before filtering are shown in Figure 4. Similar to the probability density distribution curve, the peak region of FDC-DTOA also has the function of representing p r i $pri$ . This similarity inspires an alternative idea to the histogram, that is, to extract the candidate PRIs based on FDC-DTOA. FIGURE 4Open in figure viewerPowerPoint Probability density distribution curve and the FDC-DTOA before filtering Apart from P u l $Pul$ , there are other pulse trains in the interleaved pulse stream. Due to the interaction between different pulse trains, a certain amount spurious TOA differences may be distributed into p r i 1 − ε P u l , p r i 1 + ε P u l $\left[pri\left(1-{\varepsilon }_{Pul}\right),pri\left(1+{\varepsilon }_{Pul}\right)\right]$ . Besides, spurious pulses can also create spurious TOA differences in this PRI range. Spurious TOA differences will make the upward burr exists on FDC-DTOA. Meanwhile, pulse missing will make the downward burr exists on FDC-DTOA. However, the impact of these burrs is too limited to destroy the characteristic of FDC-DTOA and can be eliminated by mean filtering. After mean filtering, each pulse train will create a smoothed peak region on FDC-DTOA. Algorithm 1. Calculation of sorted TOA differences Input: Arrival time of pulses T O A s $TOAs$ ; Range of PRI values τ min , τ max $\left[{\tau }_{\mathrm{min}},{\tau }_{\mathrm{max}}\right]$ ; Output:Sorted TOA differences T D $TD$ ; 1 Initialise the related parameter Length of L $L$ , T D = [ ] $TD=[]$ ; 2 Calculate TOA differences for i = L : 2 $i=L:2$ do for j = L − 1 : 1 $j=L-1:1$ do τ = T O A i − T O A j $\tau =TO{A}_{i}-TO{A}_{j}$ ; if τ > τ max $\tau > {\tau }_{\mathrm{max}}$ , break; elseif τ < τ min $\tau < {\tau }_{\mathrm{min}}$ , continue; else, T D = [ T D , τ ] $TD=[TD,\tau ]$ ; end end end 3 T D $TD$ is sorted in ascending order T D ( i ) ≤ T D ( i + 1 ) , i = 1,2 , . . . , M $TD(i)\le TD(i+1),i=1,2,...,M$ ; Algorithm 2. Construction of FDC-DTOA Input: Sorted TOA differences T D $TD$ ; Output: FDC-DTOA; 1 Initialise the related parameter Filter points W 1 ${W}_{1}$ , Length of T D $TD$ M $M$ ; 2 Calculate the first-order backward difference of T D $TD$ D T D = { T D ( i ) − T D ( i + 1 ) | i = 1 , . . . , M − 1 } $DTD=\left\{TD(i)-TD(i+1)\vert i=1,...,M-1\right\}$ ; 3 Smooth D T D $DTD$ with the mean filter M e a n ( D T D ) = 1 / ( M − 1 ) ⋅ ∑ j = 1 M − 1 D T D ( j ) $Mean(DTD)=1/(M-1)\cdot \sum \nolimits_{j=1}^{M-1}DTD(j)$ ; for i = 1 : M − 1 $i=1:M-1$ do if i < W 1 + 1 / 2 $i< \left({W}_{1}+1\right)/2$ or i > ( M − 1 ) − W 1 + 1 / 2 + 1 $i > (M-1)-\left({W}_{1}+1\right)/2+1$ , S D T D ( i ) = M e a n ( D T D ) $SDTD(i)=Mean(DTD)$ ; else, S D T D ( i ) = 1 / W 1 ⋅ ∑ j = i − W 1 − 1 / 2 j = i + W 1 − 1 / 2 D T D ( j ) $SDTD(i)=1/{W}_{1}\cdot \sum \nolimits_{j=i-\left({W}_{1}-1\right)/2}^{j=i+\left({W}_{1}-1\right)/2}DTD(j)\hspace*{.5em}$ ; end end 4 Construct FDC-DTOA F D C − T D O A = { ( T D ( i ) , S D T D ( i ) ) | i = 1 , . . . , M − 1 } $\mathrm{F}\mathrm{D}\mathrm{C}-\mathrm{T}\mathrm{D}\mathrm{O}\mathrm{A}=\left\{(TD(i),SDTD(i))\vert i=1,...,M-1\right\}$ ; 4.2.2 Analysis of FDC-DTOA As shown in Figure 5, probability density distribution curve is on the left, and C u r 1 $Cu{r}_{1}$ , modelled as { ( i , T D ( i ) ) } $\left\{(i,TD(i))\right\}$ , is on the right. Two real PRIs, denoted as pr i 1 $\mathit{pr}{i}_{1}$ and pr i 2 $\mathit{pr}{i}_{2}$ , are located at the extreme value of probability density distribution curve. FIGURE 5Open in figure viewerPowerPoint Probability density distribution curve and C u r 1 $Cu{r}_{1}$