A NEW PULSE DEINTERLEAVING ALGORITHM BASED ON MULTIPLE HYPOTHESIS TRACKING Jingyao Liu Huadong Meng
Xiqin Wang
Department of Electronic Engineering, Tsinghua University Beijing 100084, P. R. China E-mail:
[email protected] Abstract—The main function of ESM is to receive, measure, deinterleave pulses and then identify alternative threaten emitters, among which pulse deinterleaving process is significantly difficult and important. Dense electromagnetic environments cause an ESM to receive a seemingly total random pulse stream with large noise. The traditional algorithms have behaved weak and unstable under the rather severe situation, so we propose a new pulse deinterleaving approach based on Multiple Hypothesis Tracking method in target tracking system. Simulation results have shown that the algorithm can perform PRI estimation and pulse recognition with high precision. Moreover, the algorithm is proved to be considerably robust and reliable even with large observation noise and PRI jitter at a missing rate up to 25%. Keywords- pulse deinterleave; tracking system; Kalman filter; MHT method
I.
INTRODUCTION
From 1990s last century, Radar Electronic War has played a more and more important role in the local wars. The radar reconnaissance equipment, named Electronic Support Measure (ESM) [1][2] is one of the most important parts in the Electronic War, performing threat detection and area surveillance to determine the bearing and identity of the surrounding radar emitters. In the dense electromagnetic environments encountered in warfare, the large number of surrounding emitters will cause an ESM to receive a seemingly total random pulse stream consisting of interleaved pulse chains with large noise. Only when we segregate the different radar pulse chains from the pulse stream, can we proceed with the measurement, analysis, identification of the signal parameters and types, and then impose all kinds of interference on the radar threatening emitters, including blanket jamming and deceptive jamming, etc.
In time sequence, people have already proposed time of arrival (TOA) difference histogram [3], sequence search algorithm [4], cumulative difference histogram (CDIF) [3], sequential difference histogram (SDIF) [4], plane transformation algorithm [5] and joint recognition algorithm [6] to resolve the issue of pulse recognition and PRI estimation. Most of them are based on TOA difference histogram, the core element of which is to calculate the interval between every two pulses received and plot them in a histogram, and then set a corresponding threshold to determine the potential PRI. So one of the vital defects is that it is so sensitive to the deciding threshold that we can hardly find out a threshold applicable to all kinds of complicated circumstances. At the same time, it is not robust to parameter agility and pulse missing. All in all, the traditional radar pulse deinterleaving algorithm behaves weak and unstable when confronts the dense pulse stream with large variation and huge amounts of missing pulses. Just for these reasons, we need to make a great improvement on the algorithm performance. In this paper we propose a new approach directing at the rather severe environment. It still merely makes use of the information of pulse TOA but the central idea of this method is to calculate the likelihood rate of a series of pulses belonging to a cognate pulse chain, which means that they are emitted from the same radar, and then compare the likelihood rate to make the decision. We have compared the issue of pulse deinterleaving to the traditional target tracking system and made use of the techniques of the latter to resolve it. II.
PULSE DEINTERLEAVING BASED ON MHT
In our problem, comparing the deinterleaving process to the tracking system, it is obviously that every TOA can be regarded as an observation and every emitter as a target. Then we can roughly model the target dynamic process as a uniform rectilinear motion with some random noise in position (TOA) and velocity (PRI). Our aim is to find out the actual motion path of different target, which in other words, equals to the most probable hypothesis that associates a series of pulses to an individual emitter. And now we will introduce the detail procedure to perform pulse interleaving making use of the techniques in target tracking.
A.
the Main Procedure The basic procedure of pulse interleaving is as follows: First, among the whole received pulse stream arrayed in the order of TOA, begin a search from the first one and initial a number of alternative hypotheses, pulses of which may be cognate. Second, continue the hypotheses with the picked pulses till to the end of the pulse stream and choose the hypothesis with the maximum probability, which indicates the real pulse chain including the first pulse. Third, make a marker on each one of this chain. Fourth, begin another search from the first pulse unmarked, and repeat the three steps above again and again until all the pulses received are marked. Until now, we have segregated the pulse stream into several individual pulse chains with estimated PRI. B. the Tracking Algorithm Next, we have some details to discuss, first of which is how to pick the alternative pulse to continue the hypotheses existed. Just like the tracking system, we first predict next pulse from the previous ones and compare it with the observed TOA falling in the gate to get the estimated value to go on the hypothesis. The score representing the probability of the estimated one belonging to the pulse chain is calculated at the same time. The score calculation and gate determination as well as filtering and predicting method are elaborated below. 1) Filtering and predicting: As we all know, the Kalman filter provides an optimal solution to the recursive minimized mean square estimation problem within the class of linear estimators as long as the target dynamics and the measurement noise are accurately modeled. And for the case where the target dynamics and measurement process are stationary, the Kalman error covariance matrix and the resulting gain matrix will reach constant values. So, our model can be considered as a case in which TOA and PRI are the state vector to be estimated but only TOA is the observation, then the Kalman filter equations become: ˆ ( k | k ) ⎤ ⎡TOA ˆ ( k | k − 1) ⎤ ⎡α ⎤ ⎡ ˆ ( k | k − 1) ⎤ ⎤ ⎡TOA ⎡TOA ⎢ ⎥=⎢ ⎥ + ⎢ ⎥ ⎢TOA0 ( k ) − [1 0] ⎢ ⎥⎥ ˆ ( k | k ) ⎥ ⎢ PRI ˆ ( k | k − 1) ⎥ ⎣ β ⎦ ⎢ ˆ ( k | k − 1) ⎥ ⎥ ⎢⎣ PRI ⎢⎣ PRI ⎦ ⎣ ⎦ ⎦⎦ ⎣
ˆ (k + 1| k ) ⎤ ⎡1 1⎤ ⎡TOA ˆ (k | k ) ⎤ ⎡TOA ⎢ ⎥=⎢ ⎥ ⎥⎢ ˆ ˆ 0 1 ⎦ ⎣⎢ PRI (k | k ) ⎦⎥ ⎣⎢ PRI (k + 1| k ) ⎦⎥ ⎣
(1) (2)
ˆ (k | k − 1) and where, TOA0 represents observation, TOA ˆ (k | k ) ˆ ( k | k − 1) represent the predicted value, while TOA PRI ˆ and PRI (k | k ) represent the value after filtering. The choice
of the fixed gain α , β can be made from experiments. For initialization, we replace the first two estimations by the observations. 2) Calculating the gate: Gating is a technique for eliminating unlikely observation-to-track pairings. A gate is formed about the predicted value( yˆ ) and all observations( y ) that satisfy the gating relationship (fall within the gate) are considered for track update. Here, we adopt the simplest gating technique, which defines an interval such that an
y is said to satisfy the gate of a given track if the residual vector y satisfies the relationship y − yˆ = y ≤ KGlσr ,
observation
where σ r is the residual standard deviation as defined in terms of the measurement ( σ o2 ) and the prediction ( σ p2 ) variances σr = σp2 +σo2 . In practice, a typical choice of the gating coefficients K Gl will be K Gl ≥ 3.0 . This large choice of gating coefficient is typically made in order to compensate for the agility of PRI as well as the noise at the Receiver. 3) Calculating the scores: The evaluation of alternative pulses formation hypothesis requires a probabilistic expression. We define a likelihood ratio (LR) for a pulse into a chain to be
LR =
p( D H1 ) P0 ( H1 ) PT = p( D H 0 ) P0 ( H 0 ) PF
(3)
Hypotheses H1 and H 0 are the true target (i.e., emitter) and false alarm hypotheses with probabilities PT and PF , respectively and D is pulse TOA, so that p( D H i ) represents probability density function evaluated with the received data under the assumption that H i is correct. P0 ( H i ) represents a priori probability of H i . It is convenient to use the log likelihood ratio ( LLR ) such that
LLR = ln [ PT / PF ]
(4)
Then, LLR can be directly converted to the probability of a true target through
PT = e LLR 1 − PT
(5)
PT = e LLR / ⎡⎣1 + e LLR ⎤⎦
(6)
PT / PF =
Given K scans of pulse, and assuming scan-to-scan independence of the measurement error, LR can be partitioned into a product of terms LR (k ) , for each of the K scans. Thus, the log likelihood ratio, or the score function, for a given track is the sum of K terms. The track score L( K ) is thus defined to be K
L( K ) = ln [ LR( K )] = ∑ [ LLR(k ) ]
(7)
k =1
Assume a Gaussian distribution for true target returns and a uniform distribution over the measurement volume, VC, for false returns. Then, LR =
e( − d
2
/ 2)
/[(2π ) M / 2 S ] 1/ VC
=
VC e( − d (2π )
2
M /2
/ 2)
(8)
S
where, M represents measurement dimension; VC represents measurement volume element such that independent true target detection and false alarm events occur within each element; S represents measurement residual covariance matrix;
d 2 represents normalized statistical distance for the measurement defined in terms of the measurement residual vector y and covariance matrix S ,
−1 y d 2 = yS
(9)
Using formulas above, we can give a recursive form for the computation of the track score
L(k ) = L(k − 1) + ΔL(k )
(10)
⎡ V ⎤ M ln [ 2π ] d 2 ΔL(k ) = ln ⎢ C ⎥ − − 2 2 ⎢⎣ S ⎥⎦
(11)
Assume S is a constant and thus d 2 can just be simplified ˆ (k | k − 1) as proportional to the square difference between TOA and TOA0 (k ) . So, we can calculate the increment score of every alternative pulse as the form a − b ⋅ d (a, b is constant). 2
C. the Data Association Algorithm With the score of every alternative pulse, representing the probability of belonging to a single emitter, how to associate the alternative pulses to form a cognate pulse chain lies ahead. There are many data association methods to evaluate alternative hypotheses, the most common of which is Global Nearest Neighbor (GNN) method [7]. In this method, we just choose the pulse with the highest score in the gate to update the track every scan, thus it is so sensitive to the jitter of PRI and noise because there are so many pulses crowding in the gate and their scores differ a little. Such tiny deviation may lead a wrong pulse to be associated. Then multiple hypotheses tracking (MHT) [7] method is proposed, as the most exact and comprehensive, especially applicable to complicated situation. It is a deferred decision logic in which alternative data association hypotheses are formed whenever there are observation-to-track conflict situations. Rather than combining these hypotheses, the hypotheses are propagated in anticipation that subsequent data will resolve the uncertainty at the current time. So there is clearly a potential explosion in the number of hypotheses that an MHT system can generate. Our issue just accords with this method for the high density of pulses with large deviation and missing. We just would not like to give a decision every scan because it’s hard and cursory to do this, but would like to make use of more scans to compensate for the error and pick out the right pulse chain. . D. Another Key Improvement Another important advantage of our algorithm than the traditional is a trick to deal with the missing pulses. Tranditional algorithms such like CDIF [3] and SDIF [4] just utililize pulse interval to determine PRI and behave weak and wrong when lots of missing pulses exist. They could not know whether there are missing pulses. In our algorithm stated above, during the tracking process based on the Kalman filter, if no pulse falls in the gate, we usually end the hypothesis and a final score of it is given. But considering that the
corresponding pulses belonging to the pulse chain may be lost at the Receiver, we allow the predicted value to be regarded as another observation to update the track, no matter whether there are other observations in the gate. But the score of this pulse must be set much lower, for they are just used to continue the hypothesis temporarily and the hypothesis with too many virtual pulses will still be abondoned for its low score. This method aiming at the missing pulses is proved to be effective at section 3 and make a great improvement on the bottleneck of pulse deinterleaving brought by missing pulses. Until now, we have introduced the central idea of the new pulse deinterleaving algorithm with some important parts elaborated in detail. It is put forward directed against the rather severe circumstances and then we will examine its feasibility and performance with the simulated pulse stream. III.
SIMULATION AND ANALYSIS
Usually, radar pulse signal is classified into three types according to its PRI mode: constant PRI, staggered PRI and jittered PRI. Constant PRI means the interval between every two adjacent pulses is a constant, while staggered PRI means the interval is a repetition of several fixed value, and jittered PRI means the interval is about the mean value but with a random jitter satisfying homogeneous distribution, as shown in Fig. 4. The latter two types are both designed to reduce the intercepted probability since their intervals are not stable, compared to the conventional pulse radar signals with constant PRI.
Figure 4. Three types of PRI
Actually, the pulses intercepted will never belong to only one type, so it’s more reasonable for us to simulate a pulse stream consisted of three kinds of chains, along with large noise and a lot of missing. Here, the first type is constant PRI (PRI1=7.8); the second one is jittered PRI (PRI2=10), the jittered bound is 10% of PRI; the third is staggered PRI (PRI1=3, PRI2=10); the standard deviation of zero-mean white Gaussian noise is 10% of the minimum PRI and the missing rate is 25%, referring to Fig. 5. Abscissa just represents the TOA of pulses received at Receiver in time sequence while longitudinal coordinates represents its amplitude which is just a tag to distinguish its emitter and calculate the deinterleaving accuracy for convenience, with no meaning in itself. We can find the missing pulses obviously in this figure.
PRI are estimated with quite small deviation. Also, we can associate every received pulse with its corresponding emitter with an accuracy of more than 87% even under 10% noise deviation, 10% jittered bound and 25% missing rate, which is so inconceivable to the traditional TOA difference algorithm. At the place where pulse is missing, it either associates some wrong pulse belonging to other emitter, or behaves vacant just as it should be, both of which just help to continue the due pulse chain and have no influence to pulse deinterleaving and PRI estimation. IV. Figure 5. TOA with its source tag
However, part of the output of our algorithm is shown as Fig. 7. It can be seen that we have interleaved three types of pulse chains into four. That is because the staggered pulse chain is deinterleaved into two individual pulse chains with similar PRI equaling to the sum of each staggered PRI but different initial phases. It attributes to the basic principle that constant PRI and staggered PRI are uniform essentially. So this reminds us that when we get several pulse chains with similar PRI but different initial phases, we should compare the other pulse parameters simultaneously to make sure whether they belong to the same emitter or not, and then calculate each staggered PRI according to the relative position.
CONCLUSIONS
Aiming at the traditional pulse interleaving algorithm based on TOA difference histogram [3][4], which holds low robustness to parameter agility, lots of missing and large noise, a new pulse deinterleaving method is proposed and analyzed in this paper. The new approach finds out the similarity between pulse interleaving and target tracking and then modifies the filtering, predicting and data association method used in tracking system to resolve our issue. It replaces the traditional batch process by the sequential processing method and uses probability to scale the reliability of alternative pulse chain, which is more applicable to the complicated electronic environment and various signal modes, as well as provides an important reference for the judgment of radar type and estimation of threat. Simulation results have shown that the proposed approach can successfully identify constant, jittered, staggered PRI radar with high precision of PRI estimation and the pulse identification accuracy is high up to 87% even under the rather severe environment with large noise, jitter, missing. The approach in itself is steady and adaptable to every condition due to the optimality of Kalman filter and MHT method. Thus, it can provide excellent deinterleaving results to the main processor of ESM system, which can take appropriate actions against the identified radars and avoid wasting available resources against false one. REFERENCES [1]
Figure 7. The associated emitter of every pulse
[2]
Table I summarizes the pulse deinterleaving result above. The first column is the emittered type while the first low represents the deinterleaved type. TABLE I.
DEINTERLEAVING RESULT OF PULSE STREAM
PRI type
constant
staggered
jittered
constant
95.5%
4.5%
0%
staggered
7.8%
87%
5.2%
jittered
1.83%
9.17%
11%
[3] [4]
From the figures and table above, we can come to the conclusion that all three emitters are correctly identified and
[5] [6] [7]
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