Sequential Detection of RGPO in Target Tracking ... - Semantic Scholar
Sequential Detection of RGPO in Target Tracking by Decomposition and Fusion Approach Jing Hou1 , X. Rong Li2 , Vesselin P. Jilkov2, Zhanrong Jing1 1 Department of Electronic and Information Northwestern Polytechnical University, Xi’an, China Email: [email protected], [email protected] 2 Department of Electrical Engineering University of New Orleans, New Orleans, LA, U.S.A Email: [email protected], [email protected]
Abstract—A modified decomposition-and-fusion approach for target tracking in the presence of range-gate-pull-off (RGPO) is proposed. The RGPO detection problem consists of two parts: onset detection and termination detection. The likelihood ratio test used in the decomposition-and-fusion approach is replaced with sequential change detection, such as the cumulative sum test and Shiryayev’s sequential probability ratio test. The proposed approach overcomes the deficiencies of the likelihood ratio test, such as uncontrollable detection probability and neglect of old information, and fits well with the RGPO detection problem. These detectors are evaluated. Simulation results show that the proposed solution substantially outperforms the original solution since miss detection rate is greatly reduced by sequential detection.
Keywords: target tracking, RGPO detection, sequential detection, ECM. I. I NTRODUCTION Many modern military aircraft are equipped with Electronic Countermeasure (ECM) capabilities aiming to interfere with the tracking radar from obtaining true information. An effective category of range deception ECM techniques is the so-called range gate pull off (RGPO). It mimics the realistic target and retransmits a deception signal with a progressive time delay, thereby trying to pull the tracker’s range gate off of the true target return as time progresses. When the range gate is sufficiently removed from the actual target, the RGPO is turned off, thus forcing the radar to reacquire the target. By repeated reacquisition of the target, the track quality is degraded, causing the launch of missiles to be prevented or delayed, or target loss for a launched missile. The problem of tracking in the presence of RGPO draws increasing attention since the second benchmark problem for radar target tracking [1], [2] was proposed, which provides a standardized radar system model for tracking maneuvering targets in the presence of ECM and false alarms in order to compare and evaluate different algorithms. Two types of ECMs were included: standoff noise jamming and RGPO. A method for neutralizing standoff noise jamming was proposed in [3]. In this paper, we focus on defeating RGPO countermeasures Research supported in part by ARO through grant W911NF-08-1-0409, ONR-DEPSCoR through grant N00014-09-1-1169, and Louisiana BoR through grant LEQSF(2009-12)-RD-A-25. This work was completed during the first author’s visit to the University of New Orleans.
at the data processing (tracking) level. Several algorithms on this topic have been developed. The algorithm of [3] declares a RGPO if there exists two measurements with signal-tonoise-ratio (SNR) above a threshold, and then an “anti-RGPO discriminating function” is used to reduce the association probabilities of the more distant measurement to discount the RGPO measurements. In [4], a chi-square test was suggested. All measurements were compared pairwise, and then a RGPO was declared if the normalized angle difference squared of the measurement pair was smaller than a threshold. Reference [5] identified a RGPO from the occurrence of two or more closely spaced (in angle), high SNR observations that both fell within track gates for a given target family. Thereafter, the tracks on the true target and the RGPO were both maintained until there were three consecutive looks on which the RGPO criteria were not satisfied. Other methods are also available in [6], [7]. However, most of them are more or less heuristic, and lack a solid theoretical support. Li et al. [8] proposed a general and systematic approach, called decomposition-and-fusion (DF) approach, which is effective against RGPO. It takes full advantage of the fact that the deception measurements for many radar systems (e.g., monopulse radars) have virtually the same angles as the target measurement. This approach has four fundamental components: (a) detection/determination of range deception measurements by hypothesis testing; (b) running one or more tracking filters using the detected range deception measurements only; (c) running a conventional tracking-in-clutter filter for the remaining measurements in the track gate; (d) fusing the tracking filters by a probabilistically weighted sum of their estimates. The implementation and analysis using the second benchmark problem were given in [9]. Simulation results showed that the tracking performance was indeed improved compared with the standard tracking algorithms. The detection of RGPO was formulated as a binary simple hypothesis test and the likelihood ratio test was adopted in [8], [9]. However, three major problems have yet to be addressed. First, only current information is used in the likelihood ratio test. A small data size for hypothesis testing may cause performance degradation. Second, the detection probability cannot be improved given the type I error rate, which may cause severe miss detection of RGPO measurements. Third, the subsequent test after detection of RGPO is unclear. A
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straightforward way is to repeat the test as before. Obviously, on account of the generation pattern of RGPO, it is unadvisable to ignore the previous decision and just continue the test independently at the next circle, especially when RGPO was detected. Actually, the RGPO measurement and non-RGPO measurement follow different statistical distributions, which implies a change occurring in the measurement distribution after the RGPO onset or termination. Moreover, the status will continue over a period of time after the change occurs. In other words, it is not the case that the change occurs frequently with alternate RGPO onset and termination. Considering these features, change detection should be well suited to this problem and should overcome the deficiencies mentioned above. Based on this idea, the problem in this paper is reformulated into two parts: RGPO onset detection and RGPO termination detection. They are both simple hypothesis testing problems since all parameters are attainable. So, abundant detection methods can be used here [10]. We choose sequential hypothesis testing methods for the following reasons [11]: (a) the measurements in the tracking systems are usually available sequentially; (b) a sequential test usually makes a decision faster than a nonsequential test on average under the same decision error rates; (c) a sequential test does not need to determine the sample size in advance; (d) the thresholds of SPRT-based sequential tests can be approximately determined without knowing the distribution of the test statistics. Cumulative sum (CUSUM) [12] test and Shiryayev’s sequential probability ratio test (SSPRT) [13] are two of the most popular approaches for sequential change detection. CUSUM is non-Bayesian while SSPRT is Bayesian. CUSUM minimizes the average detection delay given the decision error probabilities, and SSPRT minimizes a risk function at each time assuming that the change point has a geometric distribution a priori. The paper is organized as follows. Section II gives statistical models of the measurement in order to utilize the DF approach. Hypotheses and distribution of the measurements are also derived. In section III, the problem is reformulated as a change detection problem. Sequential detection methods for RGPO onset and termination detection are presented in section IV. Section V provides simulation results and analysis of the detection and tracking performance, and a summary is given in the last section. II. S TATISTICAL MODELING OF MEASUREMENTS A. Measurement model We consider a phased array monopulse radar for single target tracking in this problem. Multiple measurements can be obtained at a given time point. Let the measurements at time k be defined as Zk = [zk1 , zk2 , ..., zkMk ]
(1)
where Mk is the number of measurements, not larger than the total number of range cells. Each measurement is zki = [rki , θki , ψki ]
(2)
where rki , θki , ψki denote range, bearing and elevation, respectively. The radar also provides the SNR for each measurement. Let zkc , zkt , zkf denote clutter, true target measurement and false target (RGPO) at time k, respectively.
Each individual measurement must be a return from a true target, clutter or false target. Here, the term “clutter” includes “clutter returns” (scattering off of objects in the environment such as land, water and buildings) and “false alarms” resulting from thermal noise in the radar receiver [8]. In order to perform a statistical hypothesis test in the DF approach, the following assumptions are made for the angle measurement of each type. It is reasonable to assume that bearing and elevation are independent for a given type of measurement. Assumption 1. The bearing and elevation of a clutter measurement are independent of all other measurements, clutter or not. For a clutter return, bearing θkc and elevation ψkc are uniformly distributed over (−Bb /2 + θˆk|k−1 , Bb /2 + θˆk|k−1 ) and (−Be /2 + ψˆk|k−1 , Be /2 + ψˆk|k−1 ), respectively, where Bb and Be are the beamwidth in bearing and elevation, θˆk|k−1 and ψˆk|k−1 are the predicted bearing and elevation. For a monopulse radar, the bearing and elevation of a false alarm have Gaussian distributions: θkf a ∼ N (θˆk|k−1 , σ ˆθ2f a ), k ψ f a ∼ N (ψˆk|k−1 , σ ˆ 2 f a ), where σ ˆ 2f a and σ ˆ 2 f a are obtainable k
ψk
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from the measured SNR, monopulse slope and beamwidth [14]. Assumption 2. The bearing θkt and elevation ψkt of the true target measurement have Gaussian distributions: θkt ∼ ˆθ2t ), ψkt ∼ N (ψˆk|k−1 , σ ˆψ2 t ), where σ ˆθ2t and σ ˆψ2 t N (θˆk|k−1 , σ k k k k are the variance of the predicted target bearing and elevation, often determined by radar signal processing, which depend on the SNR of the particular waveform used. Assumption 3. The bearing θkf and elevation ψkf of a false-target (RGPO) measurement have Gaussian distributions: θkf ∼ N (θkt , σ ˆθ2f ), ψkf ∼ N (ψkt , σ ˆψ2 f ), where σ ˆθ2f and σ ˆψ2 f are k k k k obtained from the signal processor. For a RGPO measurement, the bearing and elevation are nearly the same as the true target, but the range is progressively walking away from the true target return, denoted as rkf = rkt + Δr. The range walkoff Δr is determined by the time delay τ of the RGPO program, that is, Δr = cτ /2. In general, there are two basic models to generate the time delay [9] v0 (tk −t0 ) , Linear c τ= (3) a0 (tk −t0 )2 , Parabolic 2c where t0 is the start time of the walk-off program, c is the speed of light, v0 is the velocity, and a0 is the acceleration. B. Hypotheses In [8], it was assumed that the measurements always come in pairs when interfered with RGPO, one as a false target and the other as the true target or cover pulse (a RGPO technique with zero time delay, attempting to cover the actual target return), which is referred to as target and false-target (TFT) measurement pair. In a single-target environment, there are five possibilities, and thus five hypotheses, for any given pair of measurements with very close angles, referred to as the “common-angle measurements”: (a) a pure clutter pair (hypothesis Hcc ); (b) a clutter and target measurement pair (Hct ); (c) a false-target measurement pair (including the cover pulse) or a target and false-target measurement pair (Htf ), both referred as T-FT
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measurement pair; (d) a multiple false-target measurement pair, excluding the cover pulse (Hf f ); and (e) a clutter and false-target measurement pair (Hcf ). Hypothesis Hcc actually consists of three sub-hypotheses Hcc (a clutter return pair), Hcc (a clutter return and a false alarm), and Hcc (a false alarm pair). Hypothesis Hct actually consists of two sub hypotheses Hct (a clutter return and a target), Hct (a false alarm and a target). Hypothesis Htf actually consists of two sub-hypotheses Htf (a target and false-target measurement pair) and Htf (a false-target pair, one being the cover pulse). We neglect Hcf due to its extremely small probability, which occurs when neither the target measurement nor the cover pulse is detected and a clutter measurement happens to have a “common” angle with the other false-target measurement.
C. Probability distributions The RGPO hypothesis test in the DF approach is performed based on the bearing and elevation difference for a given common-angle measurement pair, Δθ = θ1 − θ2 ,
Δψ = ψ1 − ψ2
(4)
Therefore, with the above assumptions of angle measurements, the hypotheses become Hcc Hct Htf Hf f
: f (Δθ, Δψ|Hcc ) = fcc (Δθ)fcc (Δψ) : f (Δθ, Δψ|Hct ) = fct (Δθ)fct (Δψ) : f (Δθ, Δψ|Htf ) = ftf (Δθ)ftf (Δψ) : f (Δθ, Δψ|Hf f ) = ff f (Δθ)ff f (Δψ)
(5) (6) (7) (8)
By using the total probability theorem and the independence of θ1 and θ2 in Eq. (4), we can derive the probability density function (pdf) of bearing difference under each hypothesis as:
fcc (Δθ) = T ri(Δθ; −Bb , Bb )P {Hcc |Hcc } + U (Δθ; −Bb + θˆk|k−1 , Bb + θˆk|k−1 ) ∗ N (Δθ; θˆk|k−1 , σ ˆθ2f a )P {Hcc |Hcc } + N (Δθ; 0, 2ˆ σθ2f a )P {Hcc |Hcc }
⎡ ⎛ ⎞ ⎛ ⎞⎤ Δθ + B Δθ − B /2 /2 1 b b ⎣Φ ⎝ ⎠ − Φ⎝ ⎠⎦ P {Hcc = T ri(Δθ; −Bb , Bb )P {Hcc |Hcc } + |Hcc } Bb σ ˆ 2f a σ ˆ 2f a k
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N (Δθ; 0, 2ˆ σθ2f a )P {Hcc |Hcc } k
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fct (Δθ) = U (Δθ; −Bb + θˆk|k−1 , Bb + θˆk|k−1 ) ∗ N (Δθ; θˆk|k−1 , σ ˆθ2t )P {Hct |Hct } + N (Δθ; 0, σ ˆθ2t + σ ˆθ2f a )P {Hct |Hct } k k k
Δθ + Bb /2 Δθ − Bb /2 1 Φ −Φ P {Hct = |Hct } + N (Δθ; 0, σ ˆθ2t + σ ˆθ2f a )P {Hct |Hct } (10) k k Bb σ ˆθ2t σ ˆθ2t k
k
ftf (Δθ) = N (Δθ; 0, σ ˆθ2f )P {Htf |Htf } + N (Δθ; 0, 2ˆ σθ2f )P {Htf |Htf } ≈ N (Δθ; 0, 2ˆ σθ2f ) k
ff f (Δθ) =
N (Δθ; 0, 2ˆ σθ2f ) k
k
k
(11) (12)
where the four pdfs are all mixtures, T ri(Δθ; −Bb , Bb ) stands for a symmetric triangular pdf of a width 2Bb and height 1/Bb , U ∗ N is the convolution of an uniform pdf and a Gaussian pdf, and Φ is the standard Gaussian cumulative distribution function. P {H• |H• }, P {H• |H• }, P {H• |H• } in Eqs. (9) and (10) are obtainable depending on the false alarm rate and clutter return density [8]. In Eq. (11), the approximation is based on P {Htf |Htf } ≈ 1 since the cover pulse rather than the skin return is usually detected. Similar results hold for the elevation difference Δψ of (4). We merge Hf f into Htf , since ff f (Δθ) ≈ ftf (Δθ). Finally, only the hypotheses Htf , Hcc , and Hct remain. III. P ROBLEM FORMULATION A. Neyman-Pearson binary hypothesis test In the original DF approach, the RGPO detection problem is formulated as a binary hypothesis test—the null hypothesis H0 (the pair is not a T-FT measurement pair) against the alternative hypothesis H1 (the pair is a T-FT measurement
pair): H0 : Δγ ∼ fc (Δγ) H1 : Δγ ∼ ftf (Δγ)
(13) (14)
where Δγ = [Δθ, Δψ] , and fc (Δγ) = f (Δγ|Hcc )P {Hcc |H0 } + f (Δγ|Hct )P {Hct |H0 } = fcc (Δθ)fcc (Δψ)P {Hcc |H0 } + fct (Δθ)fct (Δψ)P {Hct |H0 } (15) ftf (Δγ) = ftf (Δθ)ftf (Δψ) (16) The prior probability P {Hcc |H0 } and P {Hct |H0 } can be obtained (see [8] for details). Therefore, both hypotheses are actually simple hypotheses since all parameters of the distribution are specified. With the above formulation, the following optimum likelihood ratio test is performed in the original DF approach. • Decide on H1 if ftf (Δγk ) f (Δγk |H1 ) = ≥λ (17) Lk = f (Δγk |H0 ) fc (Δγk )
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• Decide on H0 if Lk < λ. In the above test, λ is an appropriate threshold, determined by the maximum allowable type I error probability α. The likelihood ratio test can also be extended to the case with multiple false-target measurements, and the key idea is to replace the angle difference Δγ with the maximum absolute value |Δγ|max of all the common-angle measurement pairs. See [8] for details.
once RGPO is detected. Similarly, we will restart the onset procedure right after RGPO is declared terminated, and then repeat this process. Note that Δγ in this formulation can also be replaced with the maximum absolute value |Δγ|max for multiple false target measurements, and the pdfs of |Δγ|max under two hypotheses can be derived in the same way as the LR test [8]. IV. S EQUENTIAL DETECTION FOR RGPO ONSET AND
B. Problem reformulation
TERMINATION
In the above likelihood ratio test, only the current measurement is used. It ignores old information. Moreover, the detection probability cannot be improved if the type I error rate is specified. This may cause serious miss detections, even track loss. In addition, this single-scan approach does not specify explicitly what should be done after the RGPO is detected. A simple way is to perform the test independently at every time point. However, it is not good to ignore the previous decision and just continue the test independently, especially when RGPO was detected, since the RGPO program tends to last for a period of time once it starts. Actually, both RGPO onset and termination represent a change in the measurement distribution. We can resort to change detection. Thus, this problem can be reformulated into two parts: RGPO onset detection and RGPO termination detection. It has several merits over the previous formulation: a) more measurements are used to detect the change; b) the detection probability can be improved at the expense of time delay; c) the previous decision is accounted for by conducting alternative RGPO onset detection and termination detection. 1) RGPO onset detection: Here, the null hypothesis is that there are no RGPO measurements up to the current time k, and the alternative hypothesis is that the RGPO has started at time no (no ≤ k). Mathematically, it is H0 : Δγi ∼ fc (Δγ) for i = 1, 2, . . . , k Δγi ∼ fc (Δγ) , for i = 1, 2, . . . , no − 1 H1 : Δγi ∼ ftf (Δγ), for i = no , . . . , k
(18)
A. CUSUM-based detector The typical behavior of the cumulative sum of the loglikelihood ratio shows a negative drift before change, and a positive drift after change, and thus the change time corresponds roughly to the time when cumulative sum reaches its minimum. The CUSUM test is in a non-Bayesian framework, which assumes the unknown change time n is deterministic. One of its derivations is based upon a repeated use of SPRT with the lower threshold log A equal to 0 and the upper threshold equal to B, depending on decision error rates. The key difference is to restart the SPRT whenever H0 is declared, which makes it fit to change detection problems. The cumulative sum of the log-likelihood ratio is
(19)
where no is the time at which RGPO starts, and fc (Δγ) and ftf (Δγ) were given by (15)-(16). 2) RGPO termination detection: Here, the null hypothesis is that the RGPO measurements continue to exist, and the alternative hypothesis is that they have stopped at time nt (nt ≤ k): H0 : Δγi ∼ ftf (Δγ) for i = 1, 2, . . . , k Δγi ∼ ftf (Δγ), for i = 1, 2, . . . , nt − 1 H1 : Δγi ∼ fc (Δγ) , for i = nt , . . . , k
Sequential change detection methods for binary hypothesis testing problems have been successfully applied to maneuver detection in target tracking [15]–[18], fault detection in control systems [13], [19] and many other practical problems. Wald’s sequential probability ratio test (SPRT) [20] is the basis of such methods. But the problem formulation of SPRT does not fit change detection. Therefore, extended versions of SPRT for change detection have been proposed, two most well known of which are the cumulative sum (CUSUM) test and the Shiryayev’s sequential probability ratio test (SSPRT). They are both optimal for simple binary hypothesis testing under different criteria.
(20) (21)
where fc (Δγ) and ftf (Δγ) were given in Eqs. (15) and (16). An important feature of sequential detection is that nothing needs to be done when there is no decision. But we need a decision to take action in target tracking. To handle this contradiction, we choose H0 as the default decision when there is no decision from hypothesis testing. It is rational because H0 usually has a much larger probability than H1 when there is not enough data to make a decision. In general, there is no RGPO at the beginning, so the RGPO onset detection is performed first, and then the termination test is activated
Lk = max{Lk−1 + log
f (Δγk |H1 , Δγ k−1 ) , 0}, f (Δγk |H0 , Δγ k−1
L0 = 0
(22) • If Lk ≥ λ, declare H1 and then the stopping time n ˆ = min{k : Lk ≥ λ} is taken as the RGPO onset (or termination) time. • Else, continue the test (k → k + 1) if Lk < λ. It has been proved that CUSUM asymptotically minimizes the worst-case expected detection delay for simple hypotheses subject to a lower bound on the mean time between type I errors. B. SSPRT-based detector The SSPRT minimizes an expected cost at each time. It is the quickest detection of a change in a sequence of conditionally independent measurements under the given decision cost. SSPRT is a Bayesian approach, which assumes the change time is random. It needs to know the prior probability p10 of the change at n = 0 and the transition probability π from H0 to H1 . The posterior probability, defined as p1k = P {n ≤ k|Δγ k }, stands for RGPO onset (termination) at an unknown time n no later than time k given the available measurements, Δγ k = {Δγ1 , · · · , Δγk }. Then p0k = 1 − p1k
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is the posterior probability that no change occurred up to time k. The test statistic of the SSPRT is obtained recursively by Pk =
p1k f (Δγk |H1 , Δγ k−1 ) Pk−1 + π p1 , P0 = 00 (23) = 0 k−1 pk f (Δγk |H0 , Δγ ) 1−π p0
and the decision rule is • Declare H1 , if Pk ≥ PT • Else, continue the test k → k + 1 where PT is an appropriate threshold. Note that it is difficult to obtain the prior information for the RGPO detection problems, but fortunately as shown by our simulation results ( Fig. 1 and Fig. 2), the performance of the tracking filter using the SSPRT is not sensitive to the prior information. See Sec. VB(3) for details of the tracking filter. Position RMSE 60
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Figure 1: Position RMSE of the tracking filter using the SSPRT detector with different prior probabilities of no RGPO at n = 0. The transition probability π from H0 to H1 for RGPO onset detection is set at 0.02. Position RMSE 80 π=0.02 π=0.1 π=0.2 π=0.3 π=0.4 π=0.5
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V. SIMULATION A. Simulation parameters To evaluate the performance of sequential detection of RGPO, we design the following simulation scenario. There is only one T-FT pair, but multiple clutter returns and false alarms during target tracking. We set the radar revisit time to constant at 0.1s. The entire track lasts 100s. The RGPO is active from time k = 201 to 400 and from k = 501 to 700. The measurement pair used in the test is recognized as the one which has the minimum angle difference (maximum Lik ) by comparing all the measurement pairs. Since the single-scan approach does not specify the subsequent test after detection of RGPO, we interpret it as reversing the null and alternative hypotheses after RGPO is declared and then performing the likelihood ratio test. A similar procedure is conducted if RGPO is declared terminated. The target state is X = [x, y, z, x, ˙ y, ˙ z] ˙ , representing the position and velocity in the Cartesian coordinate system, with the initial state X 0 = [10000, 20000, 10000, 300, 300, 200]. For the observation, beamwidth in bearing and elevation are set at 1.5◦ and 2◦ , respectively. The standard deviations of bearing, elevation and range measurements of the true target are σθkt = 1mrad, σψkt = 1mrad and σrkt = 25m, respectively. The false target measurement has the same standard deviations as the true target in bearing and elevation, but σrf = 30m in k range. The standard deviations of false alarm measurements are σθf a = 2mrad, σψf a = 2mrad and σrf a = 50m. For k k k the SSPRT detector for RGPO onset, the prior probability of H0 is 0.9 and transition probability from H0 to H1 is 0.02. The transition probability from H0 to H1 is set to 0.1 for RGPO termination detection. The thresholds for the detectors and the performance evaluation are obtained by a Monte Carlo simulation of 100 runs unless other specified. The thresholds decrease with the increase of Type I error rate Pf a , but the quantitative relationships are quite different for the three detectors. B. Simulation results Detection and tracking performance of the likelihood ratio (LR) test and the sequential tests are compared in terms of the average detection delay, receiver operating characteristics (ROC) curves and tracking accuracy. 1) Average detection delay: The average RGPO onset detection delays (ˆ n − n) of CUSUM, SSPRT, and the LR test are shown in Table I under different type I error rates. It is clear that the average detection delay decreases as the type I error rate increases for the three detectors. The detection delays of CUSUM and SSPRT are basically the same, but SSPRT is slightly faster than CUSUM. The LR test has “better” performance in detection delay, but this is at the price of a much worse miss detection rate than the other two detectors at the times they have decisions. Indeed, it is quick to use only one measurement in the LR test to make a decision, but it may miss detect the RGPO at the next time although the RGPO is declared before since the LR test is performed independently at each time. This causes serious miss detections during the entire tracking. While the sequential tests need to wait until enough measurements are collected to make a decision, resulting in a longer detection delay to ensure a
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Table II: Miss detections for RGPO onset at k = 202, 203, 204 with 1000 runs Detector LR test CUSUM test SSPRT test
LR test CUSUM test SSPRT test
1% 2.99 2.84 2.82
Type I error rate Pf a 2% 3% 4% 1.16 0.58 0.24 2.47 2.32 2.14 2.43 2.31 2.11
5% 0.11 2.06 2.06
2) ROC curves: The RGPO was turned on at time k = 201. The ROC curves of RGPO onset detection at k = 202, 203, 204 are shown in Fig. 3. Pd is computed as the percentage of the number of correct decision over the number of total trials. For the sequential tests, if no decision is made it is counted as H0 . Obviously, the LR detector outperforms the sequential tests at k = 202 since the onset detection delays
I error rate Pf a 5% 10% (73, 74, 59) (2, 10, 8) (909, 173, 2) (873, 28, 0) (907, 161, 1) (869, 27, 0)
Table III: Average delay for RGPO termination detection Detector LR test CUSUM test SSPRT test
Table I: Average delay for RGPO onset detection Detector
Type 1% (606, 469, 403) (976, 884, 48) (976, 874, 51)
1% 0.12 0.18 0.01
Type I error rate Pf a 2% 3% 4% 0.10 0.09 0.08 0.17 0.16 0.14 0 0 0
5% 0.08 0.13 0
of the sequential tests are greater than 2 time steps. At this time, most runs of the sequential tests can not make a decision. The plot only accounts for the correct decision part. But the weakness here is not serious because the track degradation is usually small at the beginning of RGPO. At k = 203, the sequential tests have comparable performance with the LR test. Later (e.g., at k = 204), the sequential detectors have greater detection probabilities than the LR test. It also verifies that SSPRT outperforms the CUSUM but the difference is insignificant, which can be observed from the enlarged ROC curve from Pf a = 0.01 to Pf a = 0.1 at k = 204 in Fig. 3.
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Figure 4: ROC curves of RGPO termination detectors at k = 401, 402, 403, 404
The ROC curves of RGPO termination detection at k = 401, 402, 403 and 404 are shown in Fig. 4. The RGPO was off at k = 401. It is clear that the SSPRT detector has the best performance for RGPO termination detection. It detects the change immediately after it happens. As time goes, the CUSUM detector has better and better performance until it surpasses the LR detector. Position RMSE
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Figure 5: Position and velocity RMSE with no detectors
3) Tracking performance: We utilized two scenarios to compare the tracking accuracy of the three detectors: Scenario 1 with a linear RGPO walk-off program with v0 = 200m/s and Scenario 2 with a parabolic walk-off RGPO with a constant acceleration a0 = 10m/s. We used the extended Kalman filter (EKF) to track the target using the commonangle measurements. The Strongest Neighbor (SN) filter was used before RGPO was detected and the Nearest Neighbor (NN) filter was adopted to deal with the remaining measurements after RGPO onset detection. The thresholds for RGPO onset detection and termination detection were determined by type I error Pf a = 0.1 and Pf a = 0.01, respectively. Fig. 5 shows position and velocity RMSE of tracking in the presence of RGPO but without any RGPO detector. The radar would lose the target without a RGPO detector. Hence, it is necessary to utilize a RGPO detector to track the target. For Scenario 1, the position and velocity RMSE of the tracks with three detectors are given in Fig. 6. The RMSE goes up gradually when RGPO starts and goes down quickly after RGPO ends for the LR detector. It is clear that the tracking accuracy of the LR detector is worse than the sequential tests because it has much more miss detections. This can be seen from Table II. Considering the linearly increasing nature of the range measurements of RGPO, the later a miss detection happens, the more accuracy loss will be. It can be made up partially by correct detection. That is why the LR detector
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as RGPO onset detection and RGPO termination detection. For change detection CUSUM and SSPRT have been applied. Simulation results show that CUSUM and SSPRT have comparable performance, but SSPRT is slightly better than CUSUM in terms of average detection delay and ROC curve, and they both substantially outperform the likelihood ratio test.
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Figure 7: Position and velocity RMSE for Scenario 2
is more accurate than without any detector but less than the sequential tests. For CUSUM and SSPRT, their tracking performance is close to the case without any RGPO. For Scenario 2, Fig. 7 shows the position and velocity RMSE of the detectors. Results and analysis similar to Scenario 1 hold for this case. The small fluctuations are due to detection delay and false alarms. In general, the tracking performance with CUSUM and SSPRT detectors is significantly better than that of the LR detector. VI. SUMMARY The decomposition-and-fusion approach for tracking in the presence of range deception ECM [8] is general and systematic, but its RGPO detection part as in [8] has drawbacks such as small data size, ignoring old decision and uncontrollable detection probability. In essence, the measurement distributions after RGPO onset and termination are greatly changed, and thus it is better to be formulated as a change detection problem. Based on this, in this paper the problem has been reformulated
[1] W. D. Blair, G. A. Watson, G. L. Gentry and S. A. Hoffman, “Benchmark Problem for Beam Pointing Control of Phased Array Radar against Maneuvering Targets in the Presence of ECM and False Alarms,” in Proceedings of 1995 American Control Conference, vol. 4, Seattle, WA and June, 1995, pp. 2601–2605. [2] W. D. Blair, G. A. Watson, T. Kirubarajan and Y. Bar-Shalom, “Benchmark for Radar Allocation and Tracking in ECM,” IEEE Trans. Aerospace and Electronic Systems, vol. 34, no. 4, pp. 1097–1114, 1998. [3] T. Kirubarajan, Y. Bar-Shalom, W. D. Blair and G. A. Watson, “IMMPDAF for Radar Management and Tracking Benchmark with ECM,” IEEE Trans. Aerospace and Electronic Systems, vol. 34, no. 4, pp. 1115– 1132, 1998. [4] B. J. Slocumb, P. D. West, T. N. Shirey and E. W. Kamen, “Tracking a Maneuvering Target in the Presence of False Returns and ECM Using a Variable State Dimension Kalman Filter,” in Proceedings of 1995 American Control Conference, vol. 4, Seattle, WA and June, 1995, pp. 2611–2615. [5] S. S. Blackman, R. J. Dempster, M. T. Busch and R. F. Popli, “IMM/MHT Solution to Radar Benchmark Tracking Problem,” IEEE Trans. Aerospace and Electronic Systems, vol. 35, no. 2, pp. 730–738, 1998. [6] H. Xu, G. Wang and J. Bai, “A Modified Anti-RGPO Jamming Algorithm,” in 2010 IEEE 10th International Conference on Signal Processing (ICSP), Beijing and Oct, 2010, pp. 2555–2559. [7] V. Behar, C. Kabakchiev and X. R. Li, “Excision Techniques for Reduction of the Flow of the RGPO When Solving the Benchmark Problem,” in Radar Conference, 2000, Alexandria, VA, and May, 2000, pp. 862–866. [8] X. R. Li, B. J. Slocumb and D. West, “Tracking in the Presence of Range Deception ECM and Clutter by Decomposition and Fusion,” in Proc, of SPIE, Signal and Data Processing of Small Targets 1999, vol. 3808, Denver, CO and July 1999, pp. 198–210. [9] B. J. Slocumb, P. D. West, and X. R. Li, “Implementation and Analysis of the Decomposition-Fusion ECCM Technique,” in Proc. 2000 SPIE Conf. on Signal and Data Processing of Small Targets, vol. 4048, Orlando, FL and April, 2000, pp. 486–497. [10] M. Basseville and I. Nikiforov, Detection of Abrupt Changes: Theory and Application. NJ: Prentice Hall, Englewood Cliffs, 1993. [11] X. R. Li, “Multiple-Model Estimation with Variable Structure - Part II: Model-set Adaptation,” IEEE Transaction on Automatic Control, vol. 45, no. 11, pp. 2047–2060, November 2000. [12] E. S. Page, “Continuous Inspection Schemes,” Biometrika, vol. 41, no. 2, pp. 100-115, 1954. [13] D. P. Malladi and J. L. Speyer,“A Generalized Shiryaev Sequential Probability Ratio Test for Change Detection and Isolation,” IEEE Transactions on Automatic Control, vol. 44, no. 8, pp. 1522–1534, April 1999. [14] W. D. Blair and M. Brandt-Pearce, “Statistical Description of Monopulse Parameters for Tracking Rayleigh Targets,” IEEE Transactions on Aerospace and Electronic Systems, vol. 34, no. 2, pp. 597–611, Apr 1998. [15] J. Ru, V. P. Jilkov, X. R. Li and A. Bashi, “Detection of Target Maneuver Onset,” IEEE Transactions on Aerospace and Electronic Systems, vol. 45, no. 2, pp. 536–554, April 2009. [16] J. Ru, X. R. Li and V. P. Jilkov, “Multiple Model Detection of Target Maneuvers,” in Proceedings of Signal and Data Processing of Small Target, vol. 5913, San Diego, CA and Oct, 2005, pp. 100–108. [17] Y. Liu and X. R. Li, “Sequential Multiple-Model Detection of Target Maneuver Termination,” in Proceedings of Conference on Information Fusion 2011, Chicago, IL and July, 2011, pp. 1–8. [18] X. R. Li and V. P. Jilkov, “A Survey of Maneuvering Target Tracking¡a Part IV: Decision-Based Methods,” in Proceedings of SPIE Conference on Signal and Data Processing of Small Targets, vol. 4728, Orlando, FL and April, 2002, pp. 511–534. [19] T. L. Lai, “Sequential Multiple Hypothesis Testing and Efficient Fault Detection-Isolation in Stochastic Systems,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 595–608, Mar 2000. [20] A. Wald, Sequential Analysis. New York: Wiley, New York, 1947.
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Abstract—A modified decomposition-and-fusion approach for target tracking in the presence of range-gate-pull-off (RGPO) is proposed. The RGPO detection problem consists of two parts: onset detection and termination detection. The likelihood ratio test used in the decomposition-and-fusion approach is replaced with sequential change detection, such as the cumulative sum test and Shiryayev’s sequential probability ratio test. The proposed approach overcomes the deficiencies of the likelihood ratio test, such as uncontrollable detection probability and neglect of old information, and fits well with the RGPO detection problem. These detectors are evaluated. Simulation results show that the proposed solution substantially outperforms the original solution since miss detection rate is greatly reduced by sequential detection.
Keywords: target tracking, RGPO detection, sequential detection, ECM. I. I NTRODUCTION Many modern military aircraft are equipped with Electronic Countermeasure (ECM) capabilities aiming to interfere with the tracking radar from obtaining true information. An effective category of range deception ECM techniques is the so-called range gate pull off (RGPO). It mimics the realistic target and retransmits a deception signal with a progressive time delay, thereby trying to pull the tracker’s range gate off of the true target return as time progresses. When the range gate is sufficiently removed from the actual target, the RGPO is turned off, thus forcing the radar to reacquire the target. By repeated reacquisition of the target, the track quality is degraded, causing the launch of missiles to be prevented or delayed, or target loss for a launched missile. The problem of tracking in the presence of RGPO draws increasing attention since the second benchmark problem for radar target tracking [1], [2] was proposed, which provides a standardized radar system model for tracking maneuvering targets in the presence of ECM and false alarms in order to compare and evaluate different algorithms. Two types of ECMs were included: standoff noise jamming and RGPO. A method for neutralizing standoff noise jamming was proposed in [3]. In this paper, we focus on defeating RGPO countermeasures Research supported in part by ARO through grant W911NF-08-1-0409, ONR-DEPSCoR through grant N00014-09-1-1169, and Louisiana BoR through grant LEQSF(2009-12)-RD-A-25. This work was completed during the first author’s visit to the University of New Orleans.
at the data processing (tracking) level. Several algorithms on this topic have been developed. The algorithm of [3] declares a RGPO if there exists two measurements with signal-tonoise-ratio (SNR) above a threshold, and then an “anti-RGPO discriminating function” is used to reduce the association probabilities of the more distant measurement to discount the RGPO measurements. In [4], a chi-square test was suggested. All measurements were compared pairwise, and then a RGPO was declared if the normalized angle difference squared of the measurement pair was smaller than a threshold. Reference [5] identified a RGPO from the occurrence of two or more closely spaced (in angle), high SNR observations that both fell within track gates for a given target family. Thereafter, the tracks on the true target and the RGPO were both maintained until there were three consecutive looks on which the RGPO criteria were not satisfied. Other methods are also available in [6], [7]. However, most of them are more or less heuristic, and lack a solid theoretical support. Li et al. [8] proposed a general and systematic approach, called decomposition-and-fusion (DF) approach, which is effective against RGPO. It takes full advantage of the fact that the deception measurements for many radar systems (e.g., monopulse radars) have virtually the same angles as the target measurement. This approach has four fundamental components: (a) detection/determination of range deception measurements by hypothesis testing; (b) running one or more tracking filters using the detected range deception measurements only; (c) running a conventional tracking-in-clutter filter for the remaining measurements in the track gate; (d) fusing the tracking filters by a probabilistically weighted sum of their estimates. The implementation and analysis using the second benchmark problem were given in [9]. Simulation results showed that the tracking performance was indeed improved compared with the standard tracking algorithms. The detection of RGPO was formulated as a binary simple hypothesis test and the likelihood ratio test was adopted in [8], [9]. However, three major problems have yet to be addressed. First, only current information is used in the likelihood ratio test. A small data size for hypothesis testing may cause performance degradation. Second, the detection probability cannot be improved given the type I error rate, which may cause severe miss detection of RGPO measurements. Third, the subsequent test after detection of RGPO is unclear. A
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straightforward way is to repeat the test as before. Obviously, on account of the generation pattern of RGPO, it is unadvisable to ignore the previous decision and just continue the test independently at the next circle, especially when RGPO was detected. Actually, the RGPO measurement and non-RGPO measurement follow different statistical distributions, which implies a change occurring in the measurement distribution after the RGPO onset or termination. Moreover, the status will continue over a period of time after the change occurs. In other words, it is not the case that the change occurs frequently with alternate RGPO onset and termination. Considering these features, change detection should be well suited to this problem and should overcome the deficiencies mentioned above. Based on this idea, the problem in this paper is reformulated into two parts: RGPO onset detection and RGPO termination detection. They are both simple hypothesis testing problems since all parameters are attainable. So, abundant detection methods can be used here [10]. We choose sequential hypothesis testing methods for the following reasons [11]: (a) the measurements in the tracking systems are usually available sequentially; (b) a sequential test usually makes a decision faster than a nonsequential test on average under the same decision error rates; (c) a sequential test does not need to determine the sample size in advance; (d) the thresholds of SPRT-based sequential tests can be approximately determined without knowing the distribution of the test statistics. Cumulative sum (CUSUM) [12] test and Shiryayev’s sequential probability ratio test (SSPRT) [13] are two of the most popular approaches for sequential change detection. CUSUM is non-Bayesian while SSPRT is Bayesian. CUSUM minimizes the average detection delay given the decision error probabilities, and SSPRT minimizes a risk function at each time assuming that the change point has a geometric distribution a priori. The paper is organized as follows. Section II gives statistical models of the measurement in order to utilize the DF approach. Hypotheses and distribution of the measurements are also derived. In section III, the problem is reformulated as a change detection problem. Sequential detection methods for RGPO onset and termination detection are presented in section IV. Section V provides simulation results and analysis of the detection and tracking performance, and a summary is given in the last section. II. S TATISTICAL MODELING OF MEASUREMENTS A. Measurement model We consider a phased array monopulse radar for single target tracking in this problem. Multiple measurements can be obtained at a given time point. Let the measurements at time k be defined as Zk = [zk1 , zk2 , ..., zkMk ]
(1)
where Mk is the number of measurements, not larger than the total number of range cells. Each measurement is zki = [rki , θki , ψki ]
(2)
where rki , θki , ψki denote range, bearing and elevation, respectively. The radar also provides the SNR for each measurement. Let zkc , zkt , zkf denote clutter, true target measurement and false target (RGPO) at time k, respectively.
Each individual measurement must be a return from a true target, clutter or false target. Here, the term “clutter” includes “clutter returns” (scattering off of objects in the environment such as land, water and buildings) and “false alarms” resulting from thermal noise in the radar receiver [8]. In order to perform a statistical hypothesis test in the DF approach, the following assumptions are made for the angle measurement of each type. It is reasonable to assume that bearing and elevation are independent for a given type of measurement. Assumption 1. The bearing and elevation of a clutter measurement are independent of all other measurements, clutter or not. For a clutter return, bearing θkc and elevation ψkc are uniformly distributed over (−Bb /2 + θˆk|k−1 , Bb /2 + θˆk|k−1 ) and (−Be /2 + ψˆk|k−1 , Be /2 + ψˆk|k−1 ), respectively, where Bb and Be are the beamwidth in bearing and elevation, θˆk|k−1 and ψˆk|k−1 are the predicted bearing and elevation. For a monopulse radar, the bearing and elevation of a false alarm have Gaussian distributions: θkf a ∼ N (θˆk|k−1 , σ ˆθ2f a ), k ψ f a ∼ N (ψˆk|k−1 , σ ˆ 2 f a ), where σ ˆ 2f a and σ ˆ 2 f a are obtainable k
ψk
θk
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from the measured SNR, monopulse slope and beamwidth [14]. Assumption 2. The bearing θkt and elevation ψkt of the true target measurement have Gaussian distributions: θkt ∼ ˆθ2t ), ψkt ∼ N (ψˆk|k−1 , σ ˆψ2 t ), where σ ˆθ2t and σ ˆψ2 t N (θˆk|k−1 , σ k k k k are the variance of the predicted target bearing and elevation, often determined by radar signal processing, which depend on the SNR of the particular waveform used. Assumption 3. The bearing θkf and elevation ψkf of a false-target (RGPO) measurement have Gaussian distributions: θkf ∼ N (θkt , σ ˆθ2f ), ψkf ∼ N (ψkt , σ ˆψ2 f ), where σ ˆθ2f and σ ˆψ2 f are k k k k obtained from the signal processor. For a RGPO measurement, the bearing and elevation are nearly the same as the true target, but the range is progressively walking away from the true target return, denoted as rkf = rkt + Δr. The range walkoff Δr is determined by the time delay τ of the RGPO program, that is, Δr = cτ /2. In general, there are two basic models to generate the time delay [9] v0 (tk −t0 ) , Linear c τ= (3) a0 (tk −t0 )2 , Parabolic 2c where t0 is the start time of the walk-off program, c is the speed of light, v0 is the velocity, and a0 is the acceleration. B. Hypotheses In [8], it was assumed that the measurements always come in pairs when interfered with RGPO, one as a false target and the other as the true target or cover pulse (a RGPO technique with zero time delay, attempting to cover the actual target return), which is referred to as target and false-target (TFT) measurement pair. In a single-target environment, there are five possibilities, and thus five hypotheses, for any given pair of measurements with very close angles, referred to as the “common-angle measurements”: (a) a pure clutter pair (hypothesis Hcc ); (b) a clutter and target measurement pair (Hct ); (c) a false-target measurement pair (including the cover pulse) or a target and false-target measurement pair (Htf ), both referred as T-FT
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measurement pair; (d) a multiple false-target measurement pair, excluding the cover pulse (Hf f ); and (e) a clutter and false-target measurement pair (Hcf ). Hypothesis Hcc actually consists of three sub-hypotheses Hcc (a clutter return pair), Hcc (a clutter return and a false alarm), and Hcc (a false alarm pair). Hypothesis Hct actually consists of two sub hypotheses Hct (a clutter return and a target), Hct (a false alarm and a target). Hypothesis Htf actually consists of two sub-hypotheses Htf (a target and false-target measurement pair) and Htf (a false-target pair, one being the cover pulse). We neglect Hcf due to its extremely small probability, which occurs when neither the target measurement nor the cover pulse is detected and a clutter measurement happens to have a “common” angle with the other false-target measurement.
C. Probability distributions The RGPO hypothesis test in the DF approach is performed based on the bearing and elevation difference for a given common-angle measurement pair, Δθ = θ1 − θ2 ,
Δψ = ψ1 − ψ2
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Therefore, with the above assumptions of angle measurements, the hypotheses become Hcc Hct Htf Hf f
: f (Δθ, Δψ|Hcc ) = fcc (Δθ)fcc (Δψ) : f (Δθ, Δψ|Hct ) = fct (Δθ)fct (Δψ) : f (Δθ, Δψ|Htf ) = ftf (Δθ)ftf (Δψ) : f (Δθ, Δψ|Hf f ) = ff f (Δθ)ff f (Δψ)
(5) (6) (7) (8)
By using the total probability theorem and the independence of θ1 and θ2 in Eq. (4), we can derive the probability density function (pdf) of bearing difference under each hypothesis as:
fcc (Δθ) = T ri(Δθ; −Bb , Bb )P {Hcc |Hcc } + U (Δθ; −Bb + θˆk|k−1 , Bb + θˆk|k−1 ) ∗ N (Δθ; θˆk|k−1 , σ ˆθ2f a )P {Hcc |Hcc } + N (Δθ; 0, 2ˆ σθ2f a )P {Hcc |Hcc }
⎡ ⎛ ⎞ ⎛ ⎞⎤ Δθ + B Δθ − B /2 /2 1 b b ⎣Φ ⎝ ⎠ − Φ⎝ ⎠⎦ P {Hcc = T ri(Δθ; −Bb , Bb )P {Hcc |Hcc } + |Hcc } Bb σ ˆ 2f a σ ˆ 2f a k
θk
+
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fct (Δθ) = U (Δθ; −Bb + θˆk|k−1 , Bb + θˆk|k−1 ) ∗ N (Δθ; θˆk|k−1 , σ ˆθ2t )P {Hct |Hct } + N (Δθ; 0, σ ˆθ2t + σ ˆθ2f a )P {Hct |Hct } k k k
Δθ + Bb /2 Δθ − Bb /2 1 Φ −Φ P {Hct = |Hct } + N (Δθ; 0, σ ˆθ2t + σ ˆθ2f a )P {Hct |Hct } (10) k k Bb σ ˆθ2t σ ˆθ2t k
k
ftf (Δθ) = N (Δθ; 0, σ ˆθ2f )P {Htf |Htf } + N (Δθ; 0, 2ˆ σθ2f )P {Htf |Htf } ≈ N (Δθ; 0, 2ˆ σθ2f ) k
ff f (Δθ) =
N (Δθ; 0, 2ˆ σθ2f ) k
k
k
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where the four pdfs are all mixtures, T ri(Δθ; −Bb , Bb ) stands for a symmetric triangular pdf of a width 2Bb and height 1/Bb , U ∗ N is the convolution of an uniform pdf and a Gaussian pdf, and Φ is the standard Gaussian cumulative distribution function. P {H• |H• }, P {H• |H• }, P {H• |H• } in Eqs. (9) and (10) are obtainable depending on the false alarm rate and clutter return density [8]. In Eq. (11), the approximation is based on P {Htf |Htf } ≈ 1 since the cover pulse rather than the skin return is usually detected. Similar results hold for the elevation difference Δψ of (4). We merge Hf f into Htf , since ff f (Δθ) ≈ ftf (Δθ). Finally, only the hypotheses Htf , Hcc , and Hct remain. III. P ROBLEM FORMULATION A. Neyman-Pearson binary hypothesis test In the original DF approach, the RGPO detection problem is formulated as a binary hypothesis test—the null hypothesis H0 (the pair is not a T-FT measurement pair) against the alternative hypothesis H1 (the pair is a T-FT measurement
pair): H0 : Δγ ∼ fc (Δγ) H1 : Δγ ∼ ftf (Δγ)
(13) (14)
where Δγ = [Δθ, Δψ] , and fc (Δγ) = f (Δγ|Hcc )P {Hcc |H0 } + f (Δγ|Hct )P {Hct |H0 } = fcc (Δθ)fcc (Δψ)P {Hcc |H0 } + fct (Δθ)fct (Δψ)P {Hct |H0 } (15) ftf (Δγ) = ftf (Δθ)ftf (Δψ) (16) The prior probability P {Hcc |H0 } and P {Hct |H0 } can be obtained (see [8] for details). Therefore, both hypotheses are actually simple hypotheses since all parameters of the distribution are specified. With the above formulation, the following optimum likelihood ratio test is performed in the original DF approach. • Decide on H1 if ftf (Δγk ) f (Δγk |H1 ) = ≥λ (17) Lk = f (Δγk |H0 ) fc (Δγk )
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• Decide on H0 if Lk < λ. In the above test, λ is an appropriate threshold, determined by the maximum allowable type I error probability α. The likelihood ratio test can also be extended to the case with multiple false-target measurements, and the key idea is to replace the angle difference Δγ with the maximum absolute value |Δγ|max of all the common-angle measurement pairs. See [8] for details.
once RGPO is detected. Similarly, we will restart the onset procedure right after RGPO is declared terminated, and then repeat this process. Note that Δγ in this formulation can also be replaced with the maximum absolute value |Δγ|max for multiple false target measurements, and the pdfs of |Δγ|max under two hypotheses can be derived in the same way as the LR test [8]. IV. S EQUENTIAL DETECTION FOR RGPO ONSET AND
B. Problem reformulation
TERMINATION
In the above likelihood ratio test, only the current measurement is used. It ignores old information. Moreover, the detection probability cannot be improved if the type I error rate is specified. This may cause serious miss detections, even track loss. In addition, this single-scan approach does not specify explicitly what should be done after the RGPO is detected. A simple way is to perform the test independently at every time point. However, it is not good to ignore the previous decision and just continue the test independently, especially when RGPO was detected, since the RGPO program tends to last for a period of time once it starts. Actually, both RGPO onset and termination represent a change in the measurement distribution. We can resort to change detection. Thus, this problem can be reformulated into two parts: RGPO onset detection and RGPO termination detection. It has several merits over the previous formulation: a) more measurements are used to detect the change; b) the detection probability can be improved at the expense of time delay; c) the previous decision is accounted for by conducting alternative RGPO onset detection and termination detection. 1) RGPO onset detection: Here, the null hypothesis is that there are no RGPO measurements up to the current time k, and the alternative hypothesis is that the RGPO has started at time no (no ≤ k). Mathematically, it is H0 : Δγi ∼ fc (Δγ) for i = 1, 2, . . . , k Δγi ∼ fc (Δγ) , for i = 1, 2, . . . , no − 1 H1 : Δγi ∼ ftf (Δγ), for i = no , . . . , k
(18)
A. CUSUM-based detector The typical behavior of the cumulative sum of the loglikelihood ratio shows a negative drift before change, and a positive drift after change, and thus the change time corresponds roughly to the time when cumulative sum reaches its minimum. The CUSUM test is in a non-Bayesian framework, which assumes the unknown change time n is deterministic. One of its derivations is based upon a repeated use of SPRT with the lower threshold log A equal to 0 and the upper threshold equal to B, depending on decision error rates. The key difference is to restart the SPRT whenever H0 is declared, which makes it fit to change detection problems. The cumulative sum of the log-likelihood ratio is
(19)
where no is the time at which RGPO starts, and fc (Δγ) and ftf (Δγ) were given by (15)-(16). 2) RGPO termination detection: Here, the null hypothesis is that the RGPO measurements continue to exist, and the alternative hypothesis is that they have stopped at time nt (nt ≤ k): H0 : Δγi ∼ ftf (Δγ) for i = 1, 2, . . . , k Δγi ∼ ftf (Δγ), for i = 1, 2, . . . , nt − 1 H1 : Δγi ∼ fc (Δγ) , for i = nt , . . . , k
Sequential change detection methods for binary hypothesis testing problems have been successfully applied to maneuver detection in target tracking [15]–[18], fault detection in control systems [13], [19] and many other practical problems. Wald’s sequential probability ratio test (SPRT) [20] is the basis of such methods. But the problem formulation of SPRT does not fit change detection. Therefore, extended versions of SPRT for change detection have been proposed, two most well known of which are the cumulative sum (CUSUM) test and the Shiryayev’s sequential probability ratio test (SSPRT). They are both optimal for simple binary hypothesis testing under different criteria.
(20) (21)
where fc (Δγ) and ftf (Δγ) were given in Eqs. (15) and (16). An important feature of sequential detection is that nothing needs to be done when there is no decision. But we need a decision to take action in target tracking. To handle this contradiction, we choose H0 as the default decision when there is no decision from hypothesis testing. It is rational because H0 usually has a much larger probability than H1 when there is not enough data to make a decision. In general, there is no RGPO at the beginning, so the RGPO onset detection is performed first, and then the termination test is activated
Lk = max{Lk−1 + log
f (Δγk |H1 , Δγ k−1 ) , 0}, f (Δγk |H0 , Δγ k−1
L0 = 0
(22) • If Lk ≥ λ, declare H1 and then the stopping time n ˆ = min{k : Lk ≥ λ} is taken as the RGPO onset (or termination) time. • Else, continue the test (k → k + 1) if Lk < λ. It has been proved that CUSUM asymptotically minimizes the worst-case expected detection delay for simple hypotheses subject to a lower bound on the mean time between type I errors. B. SSPRT-based detector The SSPRT minimizes an expected cost at each time. It is the quickest detection of a change in a sequence of conditionally independent measurements under the given decision cost. SSPRT is a Bayesian approach, which assumes the change time is random. It needs to know the prior probability p10 of the change at n = 0 and the transition probability π from H0 to H1 . The posterior probability, defined as p1k = P {n ≤ k|Δγ k }, stands for RGPO onset (termination) at an unknown time n no later than time k given the available measurements, Δγ k = {Δγ1 , · · · , Δγk }. Then p0k = 1 − p1k
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is the posterior probability that no change occurred up to time k. The test statistic of the SSPRT is obtained recursively by Pk =
p1k f (Δγk |H1 , Δγ k−1 ) Pk−1 + π p1 , P0 = 00 (23) = 0 k−1 pk f (Δγk |H0 , Δγ ) 1−π p0
and the decision rule is • Declare H1 , if Pk ≥ PT • Else, continue the test k → k + 1 where PT is an appropriate threshold. Note that it is difficult to obtain the prior information for the RGPO detection problems, but fortunately as shown by our simulation results ( Fig. 1 and Fig. 2), the performance of the tracking filter using the SSPRT is not sensitive to the prior information. See Sec. VB(3) for details of the tracking filter. Position RMSE 60
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Figure 1: Position RMSE of the tracking filter using the SSPRT detector with different prior probabilities of no RGPO at n = 0. The transition probability π from H0 to H1 for RGPO onset detection is set at 0.02. Position RMSE 80 π=0.02 π=0.1 π=0.2 π=0.3 π=0.4 π=0.5
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Figure 2: Position RMSE of the tracking filter using the SSPRT detector with different transition probabilities for RGPO onset detection. The prior probability of no RGPO at n = 0 is set to 0.9.
V. SIMULATION A. Simulation parameters To evaluate the performance of sequential detection of RGPO, we design the following simulation scenario. There is only one T-FT pair, but multiple clutter returns and false alarms during target tracking. We set the radar revisit time to constant at 0.1s. The entire track lasts 100s. The RGPO is active from time k = 201 to 400 and from k = 501 to 700. The measurement pair used in the test is recognized as the one which has the minimum angle difference (maximum Lik ) by comparing all the measurement pairs. Since the single-scan approach does not specify the subsequent test after detection of RGPO, we interpret it as reversing the null and alternative hypotheses after RGPO is declared and then performing the likelihood ratio test. A similar procedure is conducted if RGPO is declared terminated. The target state is X = [x, y, z, x, ˙ y, ˙ z] ˙ , representing the position and velocity in the Cartesian coordinate system, with the initial state X 0 = [10000, 20000, 10000, 300, 300, 200]. For the observation, beamwidth in bearing and elevation are set at 1.5◦ and 2◦ , respectively. The standard deviations of bearing, elevation and range measurements of the true target are σθkt = 1mrad, σψkt = 1mrad and σrkt = 25m, respectively. The false target measurement has the same standard deviations as the true target in bearing and elevation, but σrf = 30m in k range. The standard deviations of false alarm measurements are σθf a = 2mrad, σψf a = 2mrad and σrf a = 50m. For k k k the SSPRT detector for RGPO onset, the prior probability of H0 is 0.9 and transition probability from H0 to H1 is 0.02. The transition probability from H0 to H1 is set to 0.1 for RGPO termination detection. The thresholds for the detectors and the performance evaluation are obtained by a Monte Carlo simulation of 100 runs unless other specified. The thresholds decrease with the increase of Type I error rate Pf a , but the quantitative relationships are quite different for the three detectors. B. Simulation results Detection and tracking performance of the likelihood ratio (LR) test and the sequential tests are compared in terms of the average detection delay, receiver operating characteristics (ROC) curves and tracking accuracy. 1) Average detection delay: The average RGPO onset detection delays (ˆ n − n) of CUSUM, SSPRT, and the LR test are shown in Table I under different type I error rates. It is clear that the average detection delay decreases as the type I error rate increases for the three detectors. The detection delays of CUSUM and SSPRT are basically the same, but SSPRT is slightly faster than CUSUM. The LR test has “better” performance in detection delay, but this is at the price of a much worse miss detection rate than the other two detectors at the times they have decisions. Indeed, it is quick to use only one measurement in the LR test to make a decision, but it may miss detect the RGPO at the next time although the RGPO is declared before since the LR test is performed independently at each time. This causes serious miss detections during the entire tracking. While the sequential tests need to wait until enough measurements are collected to make a decision, resulting in a longer detection delay to ensure a
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Figure 3: ROC curves of RGPO onset detectors at time k = 202, 203, 204. low miss detection probability. This is verified by Table II, which reveals that the number of miss detections decreases dramatically with the time for sequential tests but the decrease is not significant for the LR test. Moreover, the LR test has much more miss detections than the sequential tests at k = 204, when the sequential tests have decisions. Table III shows the average RGPO termination detection delays. It is amazingly small compared with the onset detection delay. This is because the detection probability of RGPO termination is much higher than that of RGPO onset (see the ROC curves in Fig. 3 and Fig. 4 for details), thus it is easy to detect the termination.
Table II: Miss detections for RGPO onset at k = 202, 203, 204 with 1000 runs Detector LR test CUSUM test SSPRT test
LR test CUSUM test SSPRT test
1% 2.99 2.84 2.82
Type I error rate Pf a 2% 3% 4% 1.16 0.58 0.24 2.47 2.32 2.14 2.43 2.31 2.11
5% 0.11 2.06 2.06
2) ROC curves: The RGPO was turned on at time k = 201. The ROC curves of RGPO onset detection at k = 202, 203, 204 are shown in Fig. 3. Pd is computed as the percentage of the number of correct decision over the number of total trials. For the sequential tests, if no decision is made it is counted as H0 . Obviously, the LR detector outperforms the sequential tests at k = 202 since the onset detection delays
I error rate Pf a 5% 10% (73, 74, 59) (2, 10, 8) (909, 173, 2) (873, 28, 0) (907, 161, 1) (869, 27, 0)
Table III: Average delay for RGPO termination detection Detector LR test CUSUM test SSPRT test
Table I: Average delay for RGPO onset detection Detector
Type 1% (606, 469, 403) (976, 884, 48) (976, 874, 51)
1% 0.12 0.18 0.01
Type I error rate Pf a 2% 3% 4% 0.10 0.09 0.08 0.17 0.16 0.14 0 0 0
5% 0.08 0.13 0
of the sequential tests are greater than 2 time steps. At this time, most runs of the sequential tests can not make a decision. The plot only accounts for the correct decision part. But the weakness here is not serious because the track degradation is usually small at the beginning of RGPO. At k = 203, the sequential tests have comparable performance with the LR test. Later (e.g., at k = 204), the sequential detectors have greater detection probabilities than the LR test. It also verifies that SSPRT outperforms the CUSUM but the difference is insignificant, which can be observed from the enlarged ROC curve from Pf a = 0.01 to Pf a = 0.1 at k = 204 in Fig. 3.
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Figure 4: ROC curves of RGPO termination detectors at k = 401, 402, 403, 404
The ROC curves of RGPO termination detection at k = 401, 402, 403 and 404 are shown in Fig. 4. The RGPO was off at k = 401. It is clear that the SSPRT detector has the best performance for RGPO termination detection. It detects the change immediately after it happens. As time goes, the CUSUM detector has better and better performance until it surpasses the LR detector. Position RMSE
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Figure 5: Position and velocity RMSE with no detectors
3) Tracking performance: We utilized two scenarios to compare the tracking accuracy of the three detectors: Scenario 1 with a linear RGPO walk-off program with v0 = 200m/s and Scenario 2 with a parabolic walk-off RGPO with a constant acceleration a0 = 10m/s. We used the extended Kalman filter (EKF) to track the target using the commonangle measurements. The Strongest Neighbor (SN) filter was used before RGPO was detected and the Nearest Neighbor (NN) filter was adopted to deal with the remaining measurements after RGPO onset detection. The thresholds for RGPO onset detection and termination detection were determined by type I error Pf a = 0.1 and Pf a = 0.01, respectively. Fig. 5 shows position and velocity RMSE of tracking in the presence of RGPO but without any RGPO detector. The radar would lose the target without a RGPO detector. Hence, it is necessary to utilize a RGPO detector to track the target. For Scenario 1, the position and velocity RMSE of the tracks with three detectors are given in Fig. 6. The RMSE goes up gradually when RGPO starts and goes down quickly after RGPO ends for the LR detector. It is clear that the tracking accuracy of the LR detector is worse than the sequential tests because it has much more miss detections. This can be seen from Table II. Considering the linearly increasing nature of the range measurements of RGPO, the later a miss detection happens, the more accuracy loss will be. It can be made up partially by correct detection. That is why the LR detector
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as RGPO onset detection and RGPO termination detection. For change detection CUSUM and SSPRT have been applied. Simulation results show that CUSUM and SSPRT have comparable performance, but SSPRT is slightly better than CUSUM in terms of average detection delay and ROC curve, and they both substantially outperform the likelihood ratio test.
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Figure 7: Position and velocity RMSE for Scenario 2
is more accurate than without any detector but less than the sequential tests. For CUSUM and SSPRT, their tracking performance is close to the case without any RGPO. For Scenario 2, Fig. 7 shows the position and velocity RMSE of the detectors. Results and analysis similar to Scenario 1 hold for this case. The small fluctuations are due to detection delay and false alarms. In general, the tracking performance with CUSUM and SSPRT detectors is significantly better than that of the LR detector. VI. SUMMARY The decomposition-and-fusion approach for tracking in the presence of range deception ECM [8] is general and systematic, but its RGPO detection part as in [8] has drawbacks such as small data size, ignoring old decision and uncontrollable detection probability. In essence, the measurement distributions after RGPO onset and termination are greatly changed, and thus it is better to be formulated as a change detection problem. Based on this, in this paper the problem has been reformulated
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