Improved Algorithm for Estimating Pulse Repetition Intervals

Transcription

1 I. INTRODUCTION Improved Algorithm for Estimating Pulse Repetition Intervals KEN ICHI NISHIGUCHI, Member, IEEE MASAAKI KOBAYASHI, Member, IEEE Mitsubishi Electric Corporation This paper presents an improved algorithm for estimating pulse repetition intervals (PRIs) of an interleaved pulse train which consists of several independent radar signals with different PRIs. The original version of this algorithm is a complex-valued autocorrelation-like integral, which leads to a kind of PRI spectrum wherein the locations of the spectral peaks indicate the PRI values. The original algorithm, however, has a serious drawback in that it is vulnerable to timing jitter (PRI jitter). We analyze the cause of this vulnerability and propose an improved algorithm using overlapped PRI bins which have shifting time origins. The improved algorithm has proven to be quite effective in obtaining the PRI spectrum for jittered pulse trains, which enables detection of mean PRIs by thresholding. Manuscript received March 4, 1998; revised May 28 and November 29, 1999; released for publication January 3, IEEE Log No. T-AES/36/2/ Refereeing of this contribution was handled by J. P. Y. Lee. Authors addresses: K. Nishiguchi, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Tsukaguchi-Honmachi, Amagasaki-shi, Hyogo, Japan; M. Kobayashi, Communication Systems Center, Mitsubishi Electric Corporation, Tsukaguchi-Honmachi, Amagasaki-shi, Hyogo, Japan /00/$10.00 c 2000 IEEE We deal here with the problem of estimating pulse repetition intervals (PRIs) of an interleaved pulse train, which is a superimposition of several independent radar signals with different PRIs. This problem arises in such areas as radar and electronic support measures (ESM) signal processing, where the estimated PRIs as well as the instantaneous pulse parameters, e.g. RF and direction of arrival (DOA), constitute important parameters for deinterleaving pulse trains that are interleaved [1 10]. Various algorithms have been developed to estimate the PRIs of an interleaved pulse train. A comprehensive review of these algorithms is given in [8]. The common base of these algorithms is the autocorrelation function of the pulse train, which is called the delta- histogram or time of arrival (TOA) difference histogram. In the autocorrelation function peaks are yielded at the locations of the PRIs contained in the original pulse train; however, many peaks are also yielded at the locations of integer multiples of the fundamental PRIs, i.e., subharmonics. To detect fundamental PRIs, such algorithms as the cumulative difference (CDIF) histogram [9] and the sequential difference (SDIF) histogram [10] have been proposed. These algorithms intend to avoid the subharmonics by calculating the autocorrelation function partially and sequentially. On the other hand, there is an algorithm that can suppress the subharmonics in the autocorrelation function almost completely. One of the present authors [11, 12] proposed a complex-valued autocorrelation-like integral, which yields a kind of spectrum whose peak locations indicate the fundamental PRIs. Nelson [13] also proposed the same integral formula independently. Our original algorithm for subharmonic suppression works well for detecting PRIs from an interleaved pulse train with constant PRIs [12]. In a practical situation, however, the original algorithm has a serious drawback in that it is vulnerable to PRI jitter due to measurement noise, quantization error or intentional variation [8], even when it is not very large. We analyze the cause of the vulnerability of the original algorithm and propose an improved algorithm using overlapped PRI bins which have shifting time origins. The improved algorithm has proven to be quite effective in obtaining the PRI spectrum for jittered pulse trains, which enables detection of mean PRIs by thresholding. The organization of this paper is as follows. In Section II, the basic algorithm for subharmonic suppression is reviewed. In Section III, the performance of the basic algorithm and the degradation due to PRI jitter are analyzed. In Section IV, an improved algorithm is proposed. Section V discusses methods of automatically detecting PRIs IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL

2 from the spectrum of the improved algorithm as well as the detection performance of the algorithm. Finally, Section VI draws some conclusions. II. PRI TRANSFORM In this section we review the basic algorithm for subharmonic suppression, which we refer to as the PRI transform [12]. A. Definition and Principle Let t n, n =0,:::,N 1 be pulse arrival times, where N is the number of pulses. If we consider the TOA as the only parameter of each pulse, the pulse train can be modeled as a sum of unit impulses, g(t)= ±(t t n ) (1) n=0 where ±( ) is the Dirac delta function. We consider the following integral transformation of g(t) [8, 11 13], D( )= Z 1 1 g(t)g(t + )exp(2¼it= )dt (2) where the domain of is >0. This integral is referred to as the harmonics rejecting correlation function in [8] or as the Nelson TDOA histogram in [14]. However we give it the brief name of the PRI transform since its absolute value gives a kind of PRI spectrum wherein the locations of the spectral peaks indicate the PRI values [11, 12]. The PRI transform is similar in its form to the autocorrelation function defined by Z 1 C( )= g(t)g(t + )dt (3) 1 and also similar to the Fourier transform F[g]( 1= ) = R 1 1 g(t)exp(2¼it= )dt. Substituting (1) into (2) and (3) yields n 1 D( )= ±( t n + t m )exp[2¼it n =(t n t m )] n=1 m=0 (4) n 1 C( )= ±( t n + t m ): (5) n=1 m=0 The difference between the PRI transform and the autocorrelation function is that the former has the phase factor exp(2¼it= ) orexp[2¼it n =(t n t m )], and this factor plays an important role in suppressing the subharmonics which appear in the autocorrelation function. To explain the effect of the phase factor of the PRI transform, let us define the phase of a pulse train. The pulse arrival times of a pulse train with a single PRI, which we refer to as a single pulse train, can be written as t n =(n + )p, n =0,1,2,::: (6) where p is the PRI and is a constant. We define the phase of the pulse train by µ =2¼ mod2¼: (7) Two phases, µ 1 and µ 2, are equivalent if they satisfy µ 1 = µ 2 mod2¼, or exp(iµ 1 ) = exp(iµ 2 ). In symbols we write µ 1 µ 2. The phase of a single pulse train with the PRI p can also be obtained by µ 2¼t n =p =2¼t n =(t n t n 1 ) (8) for all t n, n =1,2,:::. Therefore, the phase is calculated in terms of every two adjacent pulses. Next, we consider the autocorrelation function of a single pulse train. Substituting (6) into (5), we obtain C( )= (N l)±( lp): (9) l=1 Although the impulses located at = lp, l =2,3,::: are the subharmonics of the PRI p, from another viewpoint these impulses can be considered indicators of the pulse trains with PRI lp. Actually, the single pulse train with pulse TOAs given by (6) can be decomposed to l single pulse trains with the same PRI lp as shown in Fig. 1(a). By definition, the phases of these l pulse trains become µ 1 = µ=l, µ 2 = (µ +2¼)=l,:::,µ l =(µ +2¼(l 1))=l, whereµ 2¼, 0 µ<2¼. If we represent these phases by points on the unit circle as in Fig. 1(b), the vector sum of these points become zero except when l = 1. The phase of the pulse train that includes the pulse pair (t m,t n )as adjacent pulses is given by 2¼t n =(t n t m ). This implies that if we multiply each term on the right-hand side (RHS) of (5) by the phase factor exp[2¼it n =(t n t m )] and take the summation as in (4), the subharmonics appearing in the autocorrelation function would be suppressed. B. Discrete PRI Transform The PRI transform defined by (2) or (4) has the form of the sum of the impulses, and hence it is inappropriate to calculate it numerically. We must obtain a discrete version of the PRI transform, which takes some finite values at discrete points on the -axis. Let [ min, max ] be the range of the PRI to be investigated. We separate this range into K small intervals, which we refer to as PRI bins (see Fig. 2). ThewidthofaPRIbinisb =( max min )=K, andits center is k =(k 1=2)b + min, k =1,2,:::,K: (10) 408 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000

3 Obviously, the following inequality holds for every k jd k j C k : (13) The discrete PRI transform is easily calculated by the following procedure. 1) Initialization. Let D k =0for1 k K and let n =1. 2) Let m = n 1. 3) Let = t n t m.if min go to 5. Else, if > max go to 6. 4) Processing for each pair (m,n). a) Choose k such that k b=2 < k + b=2. b) Update the PRI transform. D k = D k + exp(2¼it n = ). 5) Substitute m = m 1. If m<0goto6.else,go to 3. 6) Substitute n = n +1. If n>n 1stop.Else,go to 2. III. PERFORMANCE OF ORIGINAL PRI TRANSFORM Fig. 1. Subharmonic components of pulse train. (a) Decomposition. Single pulse train with PRI = p can be decomposed into l subharmonic components with PRI = lp. (b) Phases of l subharmonic components. µ 1 = µ=l, µ 2 =(µ +2¼)=l,:::,µ l =(µ +(l 1)¼)=l. Fig. 2. PRI bins. We define the discrete PRI transform as follows: D k = = Z k +b=2 k b=2 D( )d f(m,n); k b=2<t n t m k +b=2g 2¼itn exp : (11) t n t m Further we define a PRI spectrum by jd k j. We note that if b! 0, then D k =b! D( ) in the sense of distribution. For the sake of comparison, we calculate a discrete version of the autocorrelation function (4) as follows: C k = Z ¼k +b=2 k b=2 C( )d = number of pairs (t m,t n )thatsatisfy k b=2 <t n t m k + b=2, k =1,2,:::,K: (12) A. Application to Single Pulse Train with Constant PRI Let us calculate the PRI transform of a single pulse train. Substituting (6) into (4) yields n 1 D( )= n=1 m=0 = l=1 2¼i( + n) ±( (n m)p)exp (n m) ±( lp)exp 2¼i l N l 1 n=0 =(N 1)±( p)exp(2¼i ) + l=2 ±( lp) sin(n¼=l) sin(¼=l) exp 2¼in l ¼i(N 1+2 ) exp : l (14) Similarly, the autocorrelation function of g(t) is obtained by substituting (6) into (5) as follows: C( )=(N 1)±( p)+ (N l)±( lp): (15) The first term on the RHS of (14) represents an impulse located at = p, and the absolute value of its coefficient is N 1, which is the same as that of the autocorrelation function. The second term on the RHS of (14) is the sum of impulses located at = lp, l =2,:::,N 1, and the absolute value of each coefficient is evaluated by sin(n¼=l) sin(¼=l) 1 sin(¼=l) l 2 = 2p : (16) We note that the RHS does not depend on N, so that the ratio between the peak level at the locations l=2 NISHIGUCHI & KOBAYASHI: IMPROVED ALGORITHM FOR ESTIMATING PULSE REPETITION INTERVALS 409

4 Fig. 3. Comparison between PRI spectrum and autocorrelation function for pulse train with constant PRIs. (a) Input pulse train, which is superimposition of 3 single pulse trains with PRIs 1, p 2, and p 5, (b) PRI spectrum. (c) Autocorrelation function. corresponding to the PRI and the noise level decreases as N becomes larger. B. Application to Interleaved Pulse Train with Constant PRIs In order to analyze the PRI transform of an interleaved pulse train, we represent the interleaved pulse train as follows: M g(t)= g ¹ (t) (17) where ¹=1 N ¹ 1 g ¹ (t)= ±(t t n¹ ), ¹ =1,:::,M (18) n ¹ =1 are single pulse trains with PRI p ¹, ¹ =1,:::,M. The PRI transform of (17) becomes M M D( )= D ¹¹ ( )+ ¹=1 M ¹=1 º=1(º6=¹) D ¹º ( ) (19) where Z 1 D ¹º ( )= g ¹ (t)g º (t + )exp(2¼it= )dt: (20) 1 The first term of the RHS of (19) is the sum of the PRI transforms of single pulse trains with constant PRIs. Hence, if the PRIs are different, this term has notable peaks only at the locations corresponding to the PRIs of the single pulse trains as in the case of a single pulse train described in the previous section. The second term on the RHS of (19) is caused by the mutual interference between different single pulse trains. This interference gives rise to a noise-like spectral shape in the PRI transform and its level can be evaluated probabilistically using the Poisson arrival TABLE I Parameters of PRI Transform (Figs. 3 and 5) Parameter Value number of pulses N 1000 range of PRI [ min, max ] [0,10] number of PRI bins K 201 model, shown in the Appendix. As a result, the value of the discrete PRI transform at the PRI bins that do not correspond to any true PRIs or their integer multiples are evaluated by q hjd k j 2 i < p N½b (21) where h i means the sample average, and ½ denotes the pulse density. Fig. 3 shows an example of the PRI spectrum of an interleaved pulse train, where the PRIs of the input pulse train are 1, p 2, and p 5. The parameters used are shown in Table I. As is apparent from the figure, the subharmonics that appeared in the autocorrelation function are suppressed almost completely by the PRI transform. C. Application to Jittered Pulse Trains We assume that the input pulse train is a single pulse train and the TOAs are represented as t 0 =0 (22) t n = t n 1 + p(1 + ² n ), n =1,2,:::,N 1 (23) where p is the average PRI and ² n is the relative deviation of the adjacent pulse interval from the average PRI. We further assume that ² n sare independently and identically distributed random variables with mean 0 and the standard deviation 410 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000

5 ¾ = p h² 2 ni. Under these assumptions, the phase of the pairs of adjacent pulses is given by µ n =2¼t n =(t n t n 1 ) =2¼ n + ² ² n 1+² n 2¼ ² ² n n² n 1+² n +2¼(² ² n n² n ): (24) When n is large ² ² n is the order of p n while n² n is the order of n, so that the latter is significant and the phase is approximated by µ n + 2¼n² n (25) which means that the phase error increases in proportion to n. We suppose that all the pulse pairs (t n 1,t n ), n =1,2,:::,N 1 are gathered into the kth PRI bin. Then the PRI transform at the bin is given by D k = e iµ n + e 2¼in² n : (26) n=1 n=1 Let q(²) be the probability density function of ² n,so the expectation of D k can be written as hd k i+ N n=2 Z 1 1 e 2¼in² q(²)d²: (27) We calculate this expectation for the following two cases. 1) Uniform distribution. If q(²) isgivenby ½ 1=2a, a ² a, q(²)= (28) 0, otherwise, where 0 <a<1=2, then Z a hd k i = cos2¼n² 1 2a d² n=1 a 1 = 2¼na sin2¼na n=1! 1 4a 1, as N!1 (29) 2 (see Fig. 4(a)). When a is sufficiently small hd k i can be approximated by hd k i+ 1 2¼a Z 2¼aN 0 sinµ dµ (30) µ which takes the maximum value (1=2¼a) R ¼ 0 (1=µ) sinµdµ=0:295=a at N =1=2a. 2) Gaussian distribution. If q(²) isgivenby q(²)= 1 p 2¼¾ 2 e ²2 =2¾ 2 (31) Fig. 4. then hd k i versus N. (a) In the case of uniform distribution. (b) In the case of Gaussian distribution. hd k i = n=1 Z a a = e (2¼n¾)2 =2 n=1 1 =2¾ cos2¼n² p 2 d² 2¼¾ 2 e ²2 (32) which is a monotonically increasing function of N (see Fig. 4(b)). When ¾ is sufficiently small hd k i can be approximated by hd k i+ 1 Z 2¼¾N e µ2 =2 dµ (33) 2¼¾ 0 which approaches 1=2 p 2¼¾ =0:1995=¾ as N!1. As is seen by the above examples, though hd k i increases in proportion to N when N is small, it does not exceed an upper bound. The upper bound decreases as the PRI jitter becomes larger. On the other hand the noise level of the PRI transform increases in proportion to p N,sothat the ratio between the peak level at the locations corresponding to the PRI and the noise level decreases as N becomes larger. In Fig. 5 the results of the PRI transform applied to an interleaved pulse train which includes 3 single pulse trains with PRI jitter are shown. The parameters used are shown in Table I. As is shown in the figure, the spectral peaks corresponding to true PRIs were submerged in the noise even in the case of a =0:01 (Fig. 5(b)). NISHIGUCHI & KOBAYASHI: IMPROVED ALGORITHM FOR ESTIMATING PULSE REPETITION INTERVALS 411

6 Fig. 6. Subharmonic components of PRI. origin. The role of the phase factor exp[2¼it n =(t n t m )] in the PRI transform (4) is to suppress the subharmonics which appeared in the autocorrelation function, while keeping the peak levels at the PRIs. To do so, it is not necessary that the phases of all pulse pairs are determined using a common time origin. For example, in the case of Fig. 6, it is sufficient that the phases of 3 pairs (t 1,t 4 ), (t 2,t 5 ), (t 3,t 6 ) differ from each other by 2¼=3. Consequently, we may update the time origin in the period 3 PRI. In general to eliminate the k PRI components, we may update the time origin in the period k PRI. On the other hand to keep the peak levels at the PRIs we may update the time origin in the period of a PRI. Hence, in all cases we may update the time origin in the period k in the kth PRI bin; therefore, a different time origin is necessary for each PRI bin. If there are no missing pulses, the condition of the time shift is that t m of the pair (t m,t n ) agrees with the previous time origin. However, we have to consider the missing pulses which occur in practical situations. It is also necessary to shift the time origin when t m O k takes a value near some integer multiple of k. Considering the above, we shift the time origins as follows. First we calculate a preliminary phase by Fig. 5. PRI spectrum by PRI transform. Input data is an interleaved pulse train with PRI jitter. Mean PRIs are 1, p p 2, and 5, and jitter follows uniform distribution with width 2a as in (28). (a) a =0:001. (b) a =0:01. (c) a =0:1. IV. MODIFICATION OF PRI TRANSFORM In the previous section we saw that the peaks of the PRI spectrum derived from the original PRI transform are reduced in the case of jittered pulse trains. There are two factors that cause this reduction. One is that the phase error of the phase factor of the PRI transform is enlarged as the TOAs grow apart from the time origin. The other is that the pulse pairs, concentrated in a PRI bin when the PRI is constant, are distributed in several bins around the average PRI. In this section we describe a modified algorithm of the PRI transform to overcome these drawbacks. A. Shifting Time Origins To avoid the enlargement of the phase error caused by the TOAs becoming large, we may change the time 0 = t n O k (34) k where O k denotes the previous time origin of the kth PRI bin. Here we use k instead of t n t m to accommodate the influence of the PRI jitter. Then we decompose the phase as 0 = º(1 + ³) (35) where º is an integer and ³ is a real number such that 1=2 <³<1=2. Finally we decide whether to shift the time origin or not according to the following conditions: 1) when º = 0, do not shift the time origin, 2) when º =1,ift m = O k,thenlett n be the new time origin, 3) when º 2, if j³j ³ 0,thenlett n be the new time origin, where ³ 0 is a positive parameter that determines the mobility of the time origins. In Fig. 7 examples of the shift of time origins are shown. 412 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000

7 where [ min, max ] is the range of PRI to be investigated and K is the number of PRI bins. Then the width of the PRI bin may be set as b k =2² k : (37) C. Modified PRI Transform Fig. 7. Shift of time origins. (a) When PRI bin includes PRI component. (b) When PRI bin includes subharmonic component of PRI. Fig. 8. B. Overlapped PRI Bins PRI bins for modified PRI transform. To avoid the reduction of the peaks by the distribution of the pulse pairs, the width of the PRI bins must be greater than the width of the PRI jitter. However this causes the degradation of the resolution of the estimated PRIs and makes it difficult to deinterleave an interleaved pulse train. To resolve this dilemma we may use overlapped bins (see Fig. 8). Let ² be the upper limit of the PRI jitter. Let K be the number of PRI bins. We determine the center of each PRI bin in the same way as before, i.e., = k 1=2 K ( max min )+ min, k =1,2,:::,K (36) By combining shifting time origins and overlapped PRI bins we obtain the following modified PRI transform algorithm. 1) Initialization. Let D k =0for1 k K and let n =2. 2) Let m = n 1. 3) Let = t n t m.if (1 ²) min go to 5. Else, if >(1 + ²) max go to 6. 4) Calculate the range of PRI bins: ³ Á k 1 = 1+² min µ k 2 = 1 ² min +1, Á +1 where =( max min )=K. 5) Repeat the next 5 steps (from 6 to 10) for k = k 1,:::,k 2. 6) Initialization of the time origin. If the kth PRI bin is used for the first time, then let O k = t n. 7) Calculate the preliminary phase and decompose it: 0 =(t n O k )= k, º =[ 0 +0:4999:::], ³ = 0=º 1: 8) Shift of the time origin. If either of the following conditions are satisfied then let O k = t n. a) º =1andt m = O k. b) º 2andj³j ³ 0. 9) Calculate the phase: =(t n O k )= k. 10) Update the PRI transform. D k = D k + exp(2¼i ). 11) Substitute m = m 1. If m<1goto12.else, go to 3. 12) Substitute n = n +1. If n>n stop. Else, go to 2. In Fig. 9 the results of the modified PRI transform are applied to the same pulse train as in Fig. 5. The parameters used are shown in Table II. As is apparent from the figure, the spectral peaks corresponding to the true PRIs are recovered. Although the number of all pulse pairs (t m,t n )s is N(N 1), only those that satisfy min t n t m max are processed by the modified PRI transform, so that the processing time of the modified PRI transform is proportional to N½( max min ), where ½ is the pulse density. Fig. 10 shows the CPU time of the modified PRI transform on an Intel Pentium III NISHIGUCHI & KOBAYASHI: IMPROVED ALGORITHM FOR ESTIMATING PULSE REPETITION INTERVALS 413

8 number of pulses Parameter TABLE II Parameters of Modified PRI Transform (Figs. 9 15) N 8 >< >: Value 1000, Figs.9,12 200, Fig , Figs.13,15 50, Fig.14 range of PRI [ min, max ] [0,10] number of PRI bins K 201 mobility of time origins ³ :001, Fig. 9a 0:01, Fig. 9b >< 0:05, Figs. 12a, 12c, 12e, 13a, width parameter of PRI bins ² 13c, 13e, 14a, 14c, 14e 0:1, Figs. 9c, 11b, 15b >: 0:15, Figs. 10, 12b, 12d, 12f, 13b, 13d, 13f, 14b, 14d, 14f Fig. 10. Processing time of modified PRI transform as measured on Intel Pentium III 550 MHz processor (densities ½ =1:0, 2.150, and correspond to 1, 3, and 5 emitters, respectively). 550 MHz processor. The parameters are shown in Table II. Fig. 10 exhibits the linear dependence of the computational load on N, which is common to a broad class of pulse deinterleaving algorithms [15]. V. DETECTION OF PRIS USING MODIFIED PRI TRANSFORM A. Threshold for Detection of PRIs Fig. 9. PRI spectrum by modified PRI transform. Input data is same as in Fig. 5. (Values of mean PRIs are 1, p 2, and p 5, and jitter follows uniform distribution with width 2a as in (28).) (a) a =0:001. (b) a =0:01. (c) a =0:1. To detect PRIs from the result of the modified PRI transform, the PRI bins that correspond to the correct PRIs must be distinguished from the other PRI bins. This discrimination can be achieved by using three criteria: a criterion by observation time, a criterion for eliminating subharmonics, and a criterion for eliminating noise. The threshold used to detect PRIs can be established by these criteria. Criterion by Observation Time: If a single pulse train with a PRI k exists in the entire observation time T, then the number of pulses is T= k.onthe other hand, jd k j denotes the number of pulses of the pulse train with a PRI k, and thus ideally it becomes jd k j = T= k. In actual situations, each pulse train does 414 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000

9 not always exist throughout the entire observation time and there are some missing pulses; accordingly, we make the following criterion with a margin: jd k j T k (38) where is a tunable parameter. Criterion for Eliminating Subharmonics: If k is the PRI of a pulse train, then ideally jd k j = C k +number of pulses of the single pulse train. Otherwise, if k is the subharmonics of the PRI of some single pulse train, then jd k j C k. Therefore, we can judge whether k is a PRI or its subharmonics by the criterion: jd k j C k (39) where is a tunable parameter. This criterion is effective for jittered pulse trains, which has incomplete suppression of the subharmonics by the modified PRI transform. Criterion for Eliminating Noise: To detect PRIs from the result of the PRI transform, it is necessary that the levels of the PRI bins that correspond to the correct PRIs are much larger than the noise level, i.e., the level of the PRI bins other than those including the PRIs or their integer multiples. In the case of the modified PRI transform, however, it is not easy to estimate the noise level because of the shifting time origins. Therefore, we have devised a criterion that uses the estimate of the noise level of the original PRI transform. As is shown in the Appendix, if D k is a noise component of the original PRI transform, then the variance of jd k j is less than T½ 2 b k,where½ is the pulse density and b k is the width of the kth PRI bin. Using this variance, we can judge that the kth PRI bin includes some component other than noise by the following criterion: q jd k j T½ 2 b k (40) where is a tunable parameter. If D k is the value of the original PRI transform, then = 3 is adequate by the three-¾ criterion. Since the noise level of the modified PRI transform is greater than that of the original PRI transform, it is necessary to choose a value of not less than 3. Combining the above three criteria we can establish the threshold as follows: ½ A k =max T ¾, C k, qt½ 2 b k (41) k where three tunable parameters are,, and. We tuned the values of these parameters through simulations under various conditions to increase the detection probabilities and to reduce the false alarm probability. All numerical examples in this paper were calculated by the following common values: =0:3, =0:15, =3: Fig. 11. Determination of threshold and detection of PRIs. (a) Components of threshold (RHS of (41) with =0:3, =0:15, = 3). (b) Detection of PRIs based on threshold. In Fig. 11, an example of the components of the above threshold and the detection by the threshold is shown. As the figure clearly shows, we can easily detect correct PRIs by finding the peaks that exceed the threshold. B. Detection Performance There are mainly three factors that affect the detection performance: number of input pulses, number of emitters (single pulse trains), and jitter width. To investigate the influence of these factors, computer simulation was performed under various conditions. The input data was generated by the superimposition of all or part of five emitters with average PRIs of 1, p 2, p 3, p 5, and p 19. All emitters obey the uniform jitter with the same peak-to-peak jitter width of 10%, 20%, or 30%. Figs show the PRI spectrum and the detection results using the threshold described in the preceding section. The parameters of the modified PRI transform are shown in Table II. The detection results shown in Figs as well as others are summarized in Table III. When the number of input pulses is 1000 (Fig. 12), up to 5 emitters with a 10% PRI jitter can be detected. If the jitter width is expanded to 30%, the number of detected emitters is reduced to three or four. In some cases, there are false detections. It seems, however, that these false detections are caused by the nonoptimality of the current threshold. NISHIGUCHI & KOBAYASHI: IMPROVED ALGORITHM FOR ESTIMATING PULSE REPETITION INTERVALS 415

10 Fig. 12. PRI spectra and PRI detection results when number of input pulses is (a) 1 emitter with 10% PRI jitter. (b) 1 emitter with 30% PRI jitter. (c) 3 emitters with 10% PRI jitter. (d) 3 emitters with 30% PRI jitter. (e) 5 emitters with 10% PRI jitter. (f) 5 emitters with 30% PRI jitter. When the number of input pulses is reduced to 100 (Fig. 13), the spectral shape become less clear; however, the detection results are not so different from the case of 1000 pulses, i.e., up to 5 emitters can be detected when the jitter width is 10% and up to 2 or 3 emitters can be detected when the jitter width is 30%. Even if the number of input pulses is reduced to 50 (Fig. 14), detection of PRIs is possible to some extent, but uncertainty is increased. If the number of input pulses is reduced to 30, the detection results become very poor. The number of detected emitters are restricted to one or two in any case. Some of the findings from the above results are as follows. 1) To detect PRIs of multiple emitters at the same time, at least 100 pulses are needed. In that case, up to 2 or 3 emitters with a 30% PRI jitter can be detected. 2) The current method to determine the detection threshold leaves some room for improvement to make the detection of PRIs more certain. C. Robustness Against Missing Pulses In an actual situation, each emitter might not exist throughout the entire observation time. Moreover, 416 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000

11 Fig. 13. PRI spectra and PRI detection results when number of input pulses is 100. (a) 1 emitter with 10% PRI jitter. (b) 1 emitter with 30% PRI jitter. (c) 3 emitters with 10% PRI jitter. (d) 3 emitters with 30% PRI jitter. (e) 5 emitters with 10% PRI jitter. (f) 5 emitters with 30% PRI jitter. there might be missing pulses. For these incomplete input data, the modified PRI transform is very robust due to its statistical nature. Roughly speaking, if an emitter exists during a time interval whose duration is x percent of total observation time, then the peak level would be reduced to x percent of that of complete data. Also, if y percent of the input pulses are missing, which means a loss of 2y percent of pulse pairs, then the peak level would be reduced by 2y percent. Therefore, if the peak level for the complete data is sufficiently high, PRIs are detectable from such incomplete data. Fig. 15 shows an example of the detection of PRIs from incomplete data. In this example, the input data are the superimposition of three emitters with a 20% PRI jitter, each of which only exists during a part of the entire observation time and 10 percent of its pulses are missing. The numbers of pulses of the three emitters are 44, 25, and 31 for a total number of 100. It is apparent from the figure that PRI detection from such incomplete data is possible by using the modified PRI transform. D. Remarks on Application of Modified PRI Transform to Deinterleaving Problem Like any algorithm, the PRI transform has both merits and demerits. Its major merit is the ability to NISHIGUCHI & KOBAYASHI: IMPROVED ALGORITHM FOR ESTIMATING PULSE REPETITION INTERVALS 417

12 Fig. 14. PRI spectra and PRI detection results when number of input pulses is 50. (a) 1 emitter with 10% PRI jitter. (b) 1 emitter with 30% PRI jitter. (c) 3 emitters with 10% PRI jitter. (d) 3 emitters with 30% PRI jitter. (e) 5 emitters with 10% PRI jitter. (f) 5 emitters with 30% PRI jitter. detect multiple PRIs at the same time. This ability is extended to jittered PRIs by the modified PRI transform. On the other hand, the demerit of the PRI transform applied to deinterleaving problems is its inability to detect staggered pulse trains. Because a staggered pulse train is regarded as a superimposition of multiple pulse trains with the same PRI but different phases, the PRI peak of the PRI spectrum is suppressed by the same principle that suppress subharmonics. Considering this limitation, it is necessary to prepare other methods to detect and separate staggered pulse trains before the detection of the jittered pulse trains. In addition to staggered pulse trains, there arise multiple pulse trains with the same PRI but with different phases, such as interference due to a multipath. It is, however, difficult to determine how many pulse trains with the same PRI are included in the input data only from the result of the PRI transform. To do so, it is necessary to analyze the pulse train separately from the input data based on the detected PRI. Although the detection of staggered pulse trains and multiple pulse trains with the same PRI but with different phases is, in itself, an interesting problem, it is beyond the scope of this paper and will be presented elsewhere. 418 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000

13 TABLE III Detection Results Number of Detected Emitters Number of 10% Jitter 20% Jitter 30% Jitter N Emitters (a =0:05) (a =0:1) (a =0:15) /1 1/1 1/ =2+2 False 2/2 2/ =3+1 False 3/3 3/ /4 3/4 3/ /5 3/5 4/ /1 1/1 1/ /2 2/2 2=2+1 False /3 3/3 2=3+1 False /4 4/4 4=4+2 False /5 4/5 5/ /1 1/1 1/ /2 2/2 2/ /3 3/3 2/ /4 2/4 2/ /5 4/5 4/ /1 1/1 1/ /2 2/2 1/ /3 2/3 2/ /4 1/4 2/ /5 1/5 2/5 main feature of this algorithm is that it gives a kind of spectrum in a PRI domain and facilitates the detection of PRIs included in the input pulse trains even if there are timing jitters. This algorithm is a modification of the algorithm based on a complex-valued autocorrelation-like integral with a phase term, which is called in this paper, the PRI transform. Though the original PRI transform has the effect of suppressing the subharmonics of the constant PRIs, it suffers serious degradation when timing jitters are included. The modified PRI transform, the improved algorithm proposed in this paper, resolves this difficulty by introducing the notion of shifting time origins and overlapped PRI bins. It has been shown that it is possible to detect the average PRIs of a jittered pulse train using the modified PRI transform. For example, the simulation results shows that up to 2 or 3 single pulse trains with a 30% PRI jitter can be detected from 100 pulses. Fig. 15. PRI spectrum applied to sporadic emitters with jittered PRI. Input data is interleaved pulse train of three emitters that have 20% jitter and whose average PRIs are 1, p 2, and p 5. Number of input pulses is 100. Each emitter is sporadic, and 10% of pulses are missing. (a) Input data. (b) PRI spectrum and detection result. VI. CONCLUSIONS An improved algorithm for estimating PRIs from an interleaved pulse train has been presented. The APPENDI. NOISE LEVEL OF PRI TRANSFORM We analyze the noise level of the PRI transform using the Poisson arrival model. We first investigate the autocorrelation function of a pulse train, the TOAs of which are randomly distributed in time. Let T be the time length and N the number of pulses. We choose two pulses, t m and t n, arbitrarily from N pulses. The two pulses are mutually independent and NISHIGUCHI & KOBAYASHI: IMPROVED ALGORITHM FOR ESTIMATING PULSE REPETITION INTERVALS 419

14 have the same probability density function: ½ 1=T, 0 t T Ã(t)= 0, otherwise: The probability that t n t m becomes (42) P(t n t m )= Z T 0 dt n Ã(t n ) Z T maxf0,t n g dt m Ã(t m ) = T2 2 +2T 2T 2 : (43) Consequently, the probability that t n t m is included in the kth PRI bin, i.e., the probability that k b k =2 < t n t m k + b k =2 becomes P( k b k =2 <t n t m k + b k =2) = T2 ( k + b k =2) 2 +2T( k + b k =2) 2T 2 T2 ( k b k =2) 2 +2T( k b k =2) 2T 2 = (T k )b k T 2 : (44) Since the number of pairs (t m,t n )isn(n 1), the mean of the autocorrelation function C k is given by hc k i = N(N 1)(T k )b k T 2 : (45) In the case of an interleaved pulse train of M single pulse trains, the level of the autocorrelation function at the PRI bins, not including PRIs or their integer multiples, is the same as that of the Poisson arrival, providing that the number of pulse pairs (t m,t n ) is not N(N 1) but N(N 1) N 1 (N 1 1) N M (N M 1) = N 2 N1 2 N2 M (46) where N l is the number of pulses of the lth single pulse train (N N M = N). Thus the average of C k becomes hc k i = (N2 N1 2 N2 M )(T k )b k T 2 (47) which can be evaluated by hc k i < N2 b k T = N½b k (48) where ½ = N=T denotes the pulse density. Next we evaluate the noise level of the PRI transform. We rewrite the discrete PRI transform as C k D k = e 2¼i j : (49) j=1 If the kth PRI bin does not include any PRIs or their subharmonics, we can assume that the phases of the pulse pairs coming into the PRI bin are random. Fig. 16. Noise levels of PRI transform. Number of input pulses if N = Solid horizontal lines indicate level of p N½b k (RHS of (53)) and dashed horizontal lines indicate level of 3 p N½b k. (a) When input is Poisson arrival process. (b) When input is superimposition of 3 single pulse trains with PRIs 1, p 2, and p 5. Therefore, taking the average of the PRI transform with respect to phases while C k remains fixed yields C k hd k i = he 2¼i j i =0: (50) j=1 Similarly the average of the square of the absolute value becomes C k hjd k j 2 i = 1+ he 2¼i( j 1 j2 ) i = C k : (51) j 1 6=j 2 j=1 Further taking its average with respect to C k yields hjd k j 2 i = hc k i (52) and by using (48), we can obtain the evaluation of the noise level of the PRI transform as q hjd k j 2 i < p N½b k : (53) Fig. 16(a) shows the result of the PRI transform applied to a Poisson arrival process. Fig. 16(b) shows the result of the PRI transform applied to an interleaved pulse train of three emitters. The numbers of input pulses and pulse densities are the same in both figures. As the figures clearly show, the shapes of the noise spectrum are very similar and the noise level for the interleaved pulse train is only slightly less than that for the Poisson arrival process. Therefore, we can use the Poisson arrival model to evaluate the noise level of the PRI transform in the form of (53). REFERENCES [1] Schmidt, R. O. (1974) On separating interleaved pulse trains. IEEE Transactions on Aerospace and Electronic Systems, AES-10 (Jan. 1974), [2] Hawkes, R. M. (1979) Radar emitter recognition using pulse repetition interval. In Proceedings of the International Conference on Information Systems and Science, Patras, Greece, [3] Davies, C. L., and Hollands, P. (1982) Automatic processing for ESM. IEE Proceedings, Pt.F,129, 3 (June 1982), IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 2 APRIL 2000

15 [4] Whittall, N. J. (1985) Signal sorting in ESM systems. IEE Proceedings, Pt.F,132, 4 (July 1985), [5] Wilkinson, D. R., and Watson, A. W. (1985) Use of metric techniques in ESM data processing. IEE Proceedings, Pt.F,132, 4 (July 1985), [6] Rogers, J. A. V. (1985) ESM processor system for high pulse density radar environment. IEE Proceedings, Pt.F,132, 7 (Dec. 1985), [7] Kofler, E. T., and Leondes, C. T. (1989) New approach to the pulse train deinterleaving problem. International Journal of Systems Science, 20, 12 (1989), [8] Wiley, R. G. (1993) Electronic Intelligence: The Analysis of Radar Signals (2nd ed.). Boston: Artech House, [9] Mardia, H. K. (1989) New techniques for the deinterleaving of repetitive sequences. IEE Proceedings, Pt.F,136, 4 (Aug. 1989), [10] Milojevic, D. J., and Popovic, B. M. (1992) Improved algorithm for the deinterleaving of radar pulses. IEE Proceedings, Pt.F,139, 1 (Feb. 1992), [11] Nishiguchi, K. (1979) Detection method of pulse trains. Japanese Patent 1,419,001; application data in Japan: Dec. 4, [12] Nishiguchi, K. (1983) A new method for estimation of pulse repetition intervals. National Convention Record of IECE of Japan, Information and Systems Section (Sept. 1983), 1-1 (in Japanese). [13] Nelson, D. J. (1993) Special purpose correlation functions for improved signal detection and parameter estimation. In Proceedings of International Conference on Acoustics, Speech, and Signal Processing (ICASSP 93), 4 (1993), [14] Balin, M. (1996) IDEA Interactive deinterleaver for ELINT analysis. In Proceedings of ICSPAT, Oct. 1996, [15] Maier, M. W. (1998) Processing throughout estimation for radar intercept receivers. IEEE Transactions on Aerospace and Electronic Systems, 34 (Jan. 1998), Ken ichi Nishiguchi (M 91) was born in Nagasaki, Japan, in He received the B.S. degree in mathematics from Kyoto University, Kyoto, Japan, in 1974, and the Ph.D. degree in engineering from Osaka University, Osaka, Japan, in In 1974 he joined Mitsubishi Electric Corporation, Central Research Laboratory (reorganized Advanced Technology R&D Center), where he has been studying modeling, estimation, and optimization in such areas as radar and electronic warfare signal processing, IR image processing, reflective optical systems design, and optical navigation of spacecraft. His current research interests are statistical signal and image processing, and navigation using optical images. Dr. Nishiguchi is a member of the Institute of Electronics, Information and Communication Engineers of Japan, and the Society of Instrument and Control Engineers of Japan. Masaaki Kobayashi (S 69 M 75) was born in Osaka, Japan, in He received the B.E., M.E., and Ph.D. degrees in electrical communication engineering from Osaka University, Osaka, Japan, in 1969, 1971, and 1974, respectively. Since 1974, he has been with the Communication Systems Center of Mitsubishi Electric Corporation, where he is now a Chief Engineer. Since 1993, he also serves as a part-time lecturer at Kobe University, Kobe, Japan. His main research interests are in the fields of signal detection, direction finding, and identification for radar and communications electronic warfare applications. Dr. Kobayashi is a member of the Institute of Electronics, Information and Communication Engineers of Japan, and a member of the Information Processing Society of Japan. NISHIGUCHI & KOBAYASHI: IMPROVED ALGORITHM FOR ESTIMATING PULSE REPETITION INTERVALS 421