This paper presents an improved algorithm for estimating pulse repetition intervals (PRIs) of an interleaved pulse train which consists of several independent ...
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, 2000. IEEE Log No. T-AES/36/2/05217. Refereeing of this contribution was handled by J. P. Y. Lee. Authors’ addresses: K. Nishiguchi, Advanced Technology R&D Center, Mitsubishi Electric Corporation, 8-1-1 Tsukaguchi-Honmachi, Amagasaki-shi, Hyogo, Japan; M. Kobayashi, Communication Systems Center, Mitsubishi Electric Corporation, 8-1-1 Tsukaguchi-Honmachi, Amagasaki-shi, Hyogo, Japan.
c 2000 IEEE 0018-9251/00/$10.00 °
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
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from the spectrum of the improved algorithm as well as the detection performance of the algorithm. Finally, Section VI draws some conclusions. II.
In this section we review the basic algorithm for subharmonic suppression, which we refer to as the PRI transform [12].
Let tn , 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) =
N¡1 X n=0
±(t ¡ tn )
(1)
where ±(¢) is the Dirac delta function. We consider the following integral transformation of g(t) [8, 11—13], Z 1 g(t)g(t + ¿ ) exp(2¼it=¿ ) dt (2) D(¿ ) = ¡1
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 R 1also similar to the Fourier transform F[g](¡1=¿ ) = 1 g(t) exp(2¼it=¿ ) dt. Substituting (1) into (2) and (3) yields
n=1 m=0
C(¿ ) =
n¡1 N¡1 XX
n=1 m=0
±(¿ ¡ tn + tm ) exp[2¼itn =(tn ¡ tm )] (4) ±(¿ ¡ tn + tm ):
(5)
The difference between the PRI transform and the autocorrelation function is that the former has the phase factor exp(2¼it=¿ ) or exp[2¼itn =(tn ¡ tm )], 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 408
n = 0, 1, 2, : : :
(6)
µ = 2¼´ mod 2¼:
A. Definition and Principle
n¡1 N¡1 XX
tn = (n + ´)p,
where p is the PRI and ´ is a constant. We define the phase of the pulse train by
PRI TRANSFORM
D(¿ ) =
PRI, which we refer to as a single pulse train, can be written as
(7)
Two phases, µ1 and µ2 , are equivalent if they satisfy µ1 = µ2 mod 2¼, or exp(iµ1 ) = exp(iµ2 ). In symbols we write µ1 ´ µ2 . The phase of a single pulse train with the PRI p can also be obtained by µ ´ 2¼tn =p = 2¼tn =(tn ¡ tn¡1 )
(8)
for all tn , 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¡1 X l=1
(N ¡ l)±(¿ ¡ lp):
(9)
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 µ1 = µ=l, µ2 = (µ + 2¼)=l, : : : , µl = (µ + 2¼(l ¡ 1))=l, where µ ´ 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 (tm , tn ) as adjacent pulses is given by 2¼tn =(tn ¡ tm ). This implies that if we multiply each term on the right-hand side (RHS) of (5) by the phase factor exp[2¼itn =(tn ¡ tm )] 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). The width of a PRI bin is b = (¿max ¡ ¿min )=K, and its center is ¿k = (k ¡ 1=2)b + ¿min ,
k = 1, 2, : : : , K:
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Obviously, the following inequality holds for every k jDk j · Ck :
(13)
The discrete PRI transform is easily calculated by the following procedure. 1) Initialization. Let Dk = 0 for 1 · k · K and let n = 1. 2) Let m = n ¡ 1. 3) Let ¿ = tn ¡ tm . 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. Dk = Dk + exp(2¼itn =¿ ). 5) Substitute m = m ¡ 1. If m < 0 go to 6. Else, go to 3. 6) Substitute n = n + 1. If n > N ¡ 1 stop. Else, go to 2.
III. PERFORMANCE OF ORIGINAL PRI TRANSFORM A. Application to Single Pulse Train with Constant PRI 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. µ1 = µ=l, µ2 = (µ + 2¼)=l, : : : , µl = (µ + (l ¡ 1)¼)=l.
Let us calculate the PRI transform of a single pulse train. Substituting (6) into (4) yields D(¿ ) =
n¡1 N¡1 XX
n=1 m=0
=
N¡1 X l=1
±(¿ ¡ (n ¡ m)p) exp
±(¿ ¡ lp) exp
2¼i(´ + n) (n ¡ m)
N¡l¡1 2¼i´ X 2¼in exp l l n=0
= (N ¡ 1)±(¿ ¡ p) exp(2¼i´) Fig. 2. PRI bins.
+ We define the discrete PRI transform as follows: Z ¿k +b=2 Dk = D(¿ ) d¿ ¿k ¡b=2
=
X
f(m,n);¿k ¡b=2 (1 + ²)¿max go to 6. 4) Calculate the range of PRI bins: ·³ ´Á ¸ ¿ k1 = ¢¿ + 1, ¡ ¿min 1+² ·µ ¶Á ¸ ¿ k2 = ¢¿ + 1 ¡ ¿min 1¡² Fig. 7. Shift of time origins. (a) When PRI bin includes PRI component. (b) When PRI bin includes subharmonic component of PRI.
where ¢¿ = (¿max ¡ ¿min )=K. 5) Repeat the next 5 steps (from 6 to 10) for k = k1 , : : : , k2 . 6) Initialization of the time origin. If the kth PRI bin is used for the first time, then let Ok = tn . 7) Calculate the preliminary phase and decompose it: ´0 = (tn ¡ Ok )=¿k , º = [´0 + 0:4999 : : :], ³ = ´0 =º ¡ 1:
Fig. 8. PRI bins for modified PRI transform.
B. Overlapped PRI Bins 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 (¿max ¡ ¿min ) + ¿min , K
k = 1, 2, : : : , K (36)
8) Shift of the time origin. If either of the following conditions are satisfied then let Ok = tn . a) º = 1 and tm = Ok . b) º ¸ 2 and j³j · ³0 . 9) Calculate the phase: ´ = (tn ¡ Ok )=¿k . 10) Update the PRI transform. Dk = Dk + exp(2¼i´). 11) Substitute m = m ¡ 1. If m < 1 go to 12. 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 (tm , tn )s is N(N ¡ 1), only those that satisfy ¿min · tn ¡ tm · ¿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
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TABLE II Parameters of Modified PRI Transform (Figs. 9—15) Parameter
number of pulses
8 1000, > > < 200,
N
Value
> 100, > :
Figs.9, 12 Fig.11 Figs.13, 15
50,
range of PRI number of PRI bins mobility of time origins
width parameter of PRI bins
[¿min , ¿max ] K ³0
²
8 0:001, > > > > 0:01, > > > > < 0:05, > > > 0:1, > > > > > : 0:15,
Fig.14 [0, 10] 201 0.03 Fig. 9a Fig. 9b Figs. 12a, 12c, 12e, 13a, 13c, 13e, 14a, 14c, 14e Figs. 9c, 11b, 15b 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 2.695 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. p Inputpdata is same as in Fig. 5. (Values of mean PRIs are 1, 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. 414
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 . On the other hand, jDk j denotes the number of pulses of the pulse train with a PRI ¿k , and thus ideally it becomes jDk j = T=¿k . In actual situations, each pulse train does
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not always exist throughout the entire observation time and there are some missing pulses; accordingly, we make the following criterion with a margin: jDk j ¸ ®
T ¿k
(38)
where ® is a tunable parameter. Criterion for Eliminating Subharmonics: If ¿k is the PRI of a pulse train, then ideally jDk j = Ck + number of pulses of the single pulse train. Otherwise, if ¿k is the subharmonics of the PRI of some single pulse train, then jDk j ¿ Ck . Therefore, we can judge whether ¿k is a PRI or its subharmonics by the criterion: jDk j ¸ ¯Ck (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 Dk is a noise component of the original PRI transform, then the variance of jDk j is less than T½2 bk , where ½ is the pulse density and bk 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 jDk j ¸ ° T½2 bk (40) where ° is a tunable parameter. If Dk 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: ¾ ½ q T Ak = max ® , ¯Ck , ° T½2 bk (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 p all orppartpof five p emitters with average PRIs of 1, 2, 3, 5, and 19. All emitters obey the uniform jitter with the same peak-to-peak jitter width of 10%, 20%, or 30%. Figs. 12—14 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. 12—14 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.
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Fig. 12. PRI spectra and PRI detection results when number of input pulses is 1000. (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. 416
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,
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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
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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. 418
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.
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TABLE III Detection Results Number of Detected Emitters N
Number of Emitters
10% Jitter (a = 0:05)
20% Jitter (a = 0:1)
30% Jitter (a = 0:15)
1000 1000 1000 1000 1000
1 2 3 4 5
1/1 2=2 + 2 False 3=3 + 1 False 4/4 5/5
1/1 2/2 3/3 3/4 3/5
1/1 2/2 3/3 3/4 4/5
100 100 100 100 100
1 2 3 4 5
1/1 2/2 3/3 4/4 5/5
1/1 2/2 3/3 4/4 4/5
1/1 2=2 + 1 False 2=3 + 1 False 4=4 + 2 False 5/5
50 50 50 50 50
1 2 3 4 5
1/1 2/2 2/3 2/4 2/5
1/1 2/2 3/3 2/4 4/5
1/1 2/2 2/3 2/4 4/5
30 30 30 30 30
1 2 3 4 5
1/1 2/2 2/3 2/4 2/5
1/1 2/2 2/3 1/4 1/5
1/1 1/2 2/3 2/4 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 threepemitters pthat have 20% jitter and whose average PRIs are 1, 2, and 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
APPENDIX.
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, tm and tn , arbitrarily from N pulses. The two pulses are mutually independent and
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have the same probability density function: ½ 1=T, 0·t·T Ã(t) = 0, otherwise: The probability that tn ¡ tm · ¿ becomes Z T Z T P(tn ¡ tm · ¿ ) = dtn ¢ Ã(tn )
maxf0,tn ¡¿ g
0
=
(42)
dtm ¢ Ã(tm )
T2 ¡ ¿ 2 + 2T¿ : 2T2
(43)
Consequently, the probability that tn ¡ tm is included in the kth PRI bin, i.e., the probability that ¿k ¡ bk =2 < tn ¡ tm · ¿k + bk =2 becomes P(¿k ¡ bk =2 < tn ¡ tm · ¿k + bk =2)
Therefore, taking the average of the PRI transform with respect to phases while Ck remains fixed yields
T2 ¡ (¿k + bk =2)2 + 2T(¿k + bk =2) = 2T2 T2 ¡ (¿k ¡ bk =2)2 + 2T(¿k ¡ bk =2) ¡ 2T2 (T ¡ ¿k )bk = : T2
Fig. 16. Noise levels of PRI transform. Number ofp input pulses if N = 1000. Solid horizontal lines indicate level of N½b pk (RHS of (53)) and dashed horizontal lines indicate level of 3 N½bk . (a) When input is Poisson arrival process. (b) When p input isp superimposition of 3 single pulse trains with PRIs 1, 2, and 5.
hDk i = (44)
N(N ¡ 1)(T ¡ ¿k )bk : T2
(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 (tm , tn ) is not N(N ¡ 1) but N(N ¡ 1) ¡ N1 (N1 ¡ 1) ¡ ¢ ¢ ¢ ¡ NM (NM ¡ 1) =N
2
¡ N12
¡ ¢ ¢ ¢ ¡ NM2
(46)
where Nl is the number of pulses of the lth single pulse train (N1 + ¢ ¢ ¢ + NM = N). Thus the average of Ck becomes hCk i =
(N 2 ¡ N12 ¡ ¢ ¢ ¢ ¡ NM2 )(T ¡ ¿k )bk T2
(47)
which can be evaluated by hCk i
