Markov Chain of Active Tracking in a Radar System and Its Application to Quantitative Analysis on Track Formation Range

Abstract

Markov chains for active tracking which assigns additional track illuminations evenly between search illuminations for a radar system are presented in this article. And some quantitative analyses on track formation range are discussed by using them. Compared with track-while-search (TWS) tracking that uses scan-to-scan correlation at search illuminations for tracking of a target, active tracking has shown the maximum improvement in track formation range of about 27.6%. It is also shown that the number and detection probability of additional track beams have impact on the track formation range. For the consideration of radar resource management at the preliminary radar system design stage, the presented analysis method can be used easily without the need of Monte Carlo simulation.

1. Introduction

Compared with the mechanically scanned array (MSA) radar that uses hydraulically or electrically controlled gimbals to physically move the array for search, the electronically scanned array (ESA) radar implements rapid beam steering by independently setting the phase of phase shifters connected to each array element [1].

An ESA radar has improved search and track performance because of the beam agility [2]. However, consideration for resource management in an ESA radar is necessary to have the optimized system performance.

In a literature survey, there have been many researches to allocate radar resource optimally. Reference [3] investigated parameter optimization in the search function with the aim to minimize the required average power for a given search performance. Reference [4] examined the beam overlap impact on probability of detection during a single scan of a phased array radar. Reference [5] showed optimum parameters and performance in constraints applicable for air-to-air surveillance. Reference [6] investigated a particular problem of track maintenance under minimum radar energy conditions. Recently, the problem of target tracking with adaptive update rate has been addressed by many authors [7-9].

Compared with the track-while-search (TWS) tracking that uses scan-to-scan correlation at search illuminations for tracking targets, an ESA radar can use the active tracking which assigns additional track illuminations between search beams and the tracking performance can be improved as a result. Reference [10] showed the performance improvements in active tracking compared to TWS tracking using surveillance performance measure. Reference [11] presented that the active tracking of an ESA radar can provide significantly improved performance over TWS tracking of a MSA radar using Monte Carlo simulation.

In this article, improvement on track formation range in active tracking which assigns additional track illuminations evenly between search illuminations compared to TWS tracking is presented quantitatively. Analytical method using Markov chain with less computational effort is used rather than time-consuming Monte Carlo simulation to be required detailed radar definition. Firstly, Markov chain and track formation range of TWS tracking are shown with the brief explanation of basic theory. Secondly, Markov chain and track formation range of active tracking with one additional track beam are presented with them of the TWS tracking for comparison. It is also given the comparison with respect to the change on the detection probability of the track beam with the proper parameter selection. Moreover, the track formation ranges with various number of additional track beams are compared. In addition, impact on track formation range by the number and detection probability of additional track beams is analyzed simultaneously. Finally, a discussion is given of the proper parameter selection with taking into consideration the number and detection probability of additional track beams.

 

2. Analysis on Track Formation Range of TWS Tracking

The analysis on detection and tracking range of a radar system can be carried out conveniently using a Markov chain without using Monte Carlo method at the preliminary system design stage [12]. The Markov chain can be represented generally as follows :

where x(k) is a vector of probabilities (Pi) associated with being in various discrete states(Si) and F(k) is the transition matrix at time k, respectively. The transition matrix will be a known function of the probability of detection and this probability does not depend upon which states the chain was in before the current state. If the chain is currently in state Si, then it moves to state Sj at the next step with the corresponding probability in the transition matrix. Thus, given an initial condition vector x(0), succeeding values of x(k+1) can be calculated given x(k). In other words, whether a target is detected after k search scans depends only on detection status after the previous scan and the probability of detection. The Markov chain used for the computation of cumulative detection range in a radar system is shown in Fig. 1 for brief explanation of the basic theory.

Fig. 1.Markov chain used for the computation of cumulative detection range

S1 and S2 represent the state where no detection has taken place and at least one detection has been made, respectively. The single scan detection probability (p) and q=1-p are functions of time, and thus could be indexed by time instant k, but for notational convenience p and q will not be indexed. The Markov chain can be written as follows:

where x(k) is a state vector, containing all state probabilities at time k and F(k) is the transition matrix, describing the transition probabilities between all states at time k.

The cumulative detection range (Rd,acc,85), where the cumulative detection probability (Pd,acc) is 85% in the scenario with a decreasing target range from a long distance, can be calculated from (2) as follows:

• The single scan detection probability (p) at time instant k, corresponding to a Swerling I target at the range Rk, is obtained by (3).

where Pfa is the probability of false alarm, R50 is the range with the detection probability of 50%, and SNR50 is the signal to noise ratio at R50 [12].

• x(k) at time instant k is calculated by using (2) and (3) and its second element (x2(k)) is the cumulative detection probability. Then, Rd,acc,85 is obtained from the corresponding range and the time instant k when x2(k) is 85%.

Fig. 2 shows the single scan and cumulative detection probability versus range applying above explanation with radar system parameters given in Table 1. The values in Table 1 are used as typical examples for illustration. As shown in Fig. 2, Rd,acc,85 is about 135km. These single scan and cumulative detection probability are represented as the reference of performance in following comparison figures.

Fig. 2.Single scan and cumulative detection probability versus range

Table 1.Assumed parameters in a radar system

For computation of track formation range of TWS tracking with same parameters in Table 1, the track confirmation criterion is required. In this article, we require 2 detections out of 4 trials followed by 1 detection out of 4 trials. Markov chain used for the computation of track formation range on TWS tracking with this confirmation criterion is shown in Fig. 3 and every trial from S1 to S9 is performed by search illumination. p and q can be obtained from same method using (3) and the transition matrix and the initial condition vector of this Markov chain can be written as follows:

Fig. 3.Markov chain used for the computation of track formation range on TWS tracking with confirmation criterion of 2/4 and 1/4

Thus, x(k), the 9 by 1 state vector at time instant k, is calculated by using iteratively (3) and (4) for the corresponding range and k. The state variables in (4) are described in Table 2. Because the last element the last element (x9(k)) of the state vector is the track probability (Pt), the track probability of TWS tracking versus range can be depicted as shown in Fig. 4. The track formation range (Rt,acc,85) of TWS tracking, where the track probability is 85%, is about 105km from Fig. 4.

Table 2.Elements of x(k) and corresponding states in (4)

Fig. 4.Track probability of TWS tracking versus range

 

3. Analysis on Track Formation Range of Active Tracking

For quantitative comparison with track formation range of TWS tracking, Markov chain of active tracking with one track beam between search beams is proposed as shown in Fig. 5. In this article, active tracking that additional track illuminations are evenly distributed between search illuminations is considered. The same track confirmation criterion for TWS tracking (2 detections out of 4 trials followed by 1 detection out of 4 trials) is used and each trial is executed sequentially by search and track illumination, that is, Si states and Ai states. The W1 represents the wait state, an intermediate state in which no action occurs, but that serves to delay the next time an action does occur [12]. T0 is state of having achieved track confirmation. Ps is the detection probability by the search illumination and Qs is 1-Ps. The calculating method for them is identical to that for p and q. Pa is the detection probability by the track illumination and Qa is 1-Pa. The transition matrix and the initial condition vector of this Markov chain can be written as (5). The state variables in (5) are described in Table 3.

Fig. 5.Markov chain used for the computation of track formation range on active tracking with one track beam between search beams

Table 3.Elements of x(k) and corresponding states in (5)

Fig. 6 shows the comparison of track formation range between TWS and active tracking with one track beam between search beams. The last element (x18(k)) of x(k), the 18 by 1 state vector, is used for Fig. 6 because it represents the track probability in this Markov chain relationship. And Pa identical to Ps is also applied in consideration of using same waveform for search and track illumination. Compared with TWS tracking, the improvement in track formation range of about 3.8% is presented because Rt,acc,85 of this active tracking is about 109km.

Fig. 6.Comparison of track formation range between TWS and active tracking with one track beam between search beams and Pa identical to Ps

The higher Pa than the Ps at the same range can be expected by using different waveform for a candidate target [13]. Rt,acc,85 of active tracking with one track beam between search beams in cases with different Pa from Ps is easily analyzed by using the same Markov chain and the results are shown in Fig. 7. Compared with TWS tracking, the maximum improvement in track formation range of about 26.3% is achieved for Pa of 0.9 because Rt,acc,85 in this case is about 132.6km.

Fig. 7.Comparison of track formation range between TWS and active tracking with one track beam between search beams versus different Pa

The track formation ranges of active tracking versus the number of additional track beams between search beams are also analyzed. For analysis, the active tracking with two track beams and three track beams between search beams are considered. The corresponding Markov chains are shown in Figs. 8 and 9, respectively. The same track confirmation criterion for TWS tracking is used for comparison. Si, Ai, and Wi states for search, track, and wait illuminations are shown according to the beam assignment rule for each tracking. The transition matrices and the initial condition vectors of each Markov chain can be written as (6) and (7). The state variables in (6) and (7) are described in Table 4 and 5, respectively.

Fig. 8.Markov chain used for the computation of track formation range on active tracking with two track beams between search beams

Fig. 9.Markov chain used for the computation of track formation range on active tracking with three track beams between search beams

Table 4.Elements of x(k) and corresponding states in (6)

Table 5.Elements of x(k) and corresponding states in (7)

The last element (x19(k)) of x(k), the 19 by 1 state vector in (6), represents the track probability of active tracking with two track beams between search beams and the last

element (x20(k)) of x(k), the 20 by 1 state vector in (7), is that of active tracking with three track beams between search beams. Fig. 10 shows the comparison of track formation range of active tracking versus the number of track beams between search beams in consideration of using same waveform for search and track illumination. The track formation ranges of three active tracking with Pa of 0.6 are also presented in Fig. 10. As depicted in Fig. 10, the number of track beams between search beams has no great impact on the improvement of the track formation range when we use same waveform for search and track illumination. However, the track formation range of active tracking with Pa of 0.6 is improved gradually with respect to the number of track beams between search beams. Compared with TWS tracking, the improvement in track formation range of about 20.8%, 22.9%, and 25.2% is achieved for three active tracking with Pa of 0.6 as given in Table 6.

Fig. 10.Comparison of track formation range between TWS and active tracking versus the number of track beams between search beams and Pa

Table 6.Track formation range of TWS and active tracking with Pa of 0.6 versus the number of track beams between search beams

So far, impact on track formation range by the number and detection probability of additional track beams is analyzed separately. Finally, we take into consideration the number and detection probability of additional track beams simultaneously for discussion on optimized parameter selection. Fig. 11 shows the normalized track formation range of active tracking based on that of TWS tracking. All active tracking cases have the improved track formation range as against that of TWS tracking and maximum improvement of 27.6% is achieved for active tracking with three track beams and Pa of 0.9. It is shown that larger track formation range can be obtained with a given Pa by increasing the number of track beams between search beams. However, it is also shown that track formation range of active tracking with one track beams and Pa of 0.9 is similar to that of active tracking with three track beams and Pa of 0.6. It means the number of track beams becomes less influential as Pa increases.

Fig. 11.Normalized track formation range of active tracking based on that of TWS tracking

Generally, more resource is required for a higher detection probability. And the number of track beams is also one of parameters to be needed optimization in radar resource management. Thus, the proper waveform selection for active tracking with a given number of track beams must be carried out from efficient radar resource management perspective. Using the presented analysis method, the consideration of radar resource management can be performed roughly and the range of parameter selection for the optimization can be narrowed to some extent at the preliminary radar system design stage.

 

4. Conclusion

Compared with the TWS tracking that uses scan-to-scan correlation at search illuminations for tracking targets, a phased array radar can use active tracking which assigns additional track illuminations and the track formation range can be improved as a result. In this article, Markov chains for active tracking which assigns additional track illuminations evenly between search illuminations are presented to show the improved performance quantitatively. Three Markov chains with one, two, and three additional track beams are presented and the performances are obtained without using Monte Carlo simulation. The analysis is carried out with consideration of the detection probability of additional track beams and it is shown that all active tracking cases have the improved track formation range compared with TWS tracking. In addition, it is indicated that analytic selection of the number and detection probability of additional track beams must be carried out for meeting the required track formation range with limited radar resource. Thus, the presented analysis method can be used for reduction to range of parameter selection for the optimization at the preliminary radar system design stage.