Estimation of Radar Cross Section of a Target under Track
© Young-Hun Jung et al. 2010
Received: 19 April 2010
Accepted: 6 October 2010
Published: 7 October 2010
In allocating radar beam for tracking a target, it is attempted to maintain the signal-to-noise ratio (SNR) of signal returning from the illuminated target close to an optimum value for efficient track updates. An estimate of the average radar cross section (RCS) of the target is required in order to adjust transmitted power based on the estimate such that a desired SNR can be realized. In this paper, a maximum-likelihood (ML) approach is presented for estimating the average RCS, and a numerical solution to the approach is proposed based on a generalized expectation maximization (GEM) algorithm. Estimation accuracy of the approach is compared to that of a previously reported procedure.
Beam allocation in phased array radar tracking is a well-known problem, and it has been addressed in a large body of literature [1–6] and the references therein. A goal of the allocation is to minimize the use of radar resources while maintaining a target under track. In the allocation for a track update, one of the principal parameters to be adjusted is transmitted power. The transmitted power has an impact on the signal-to-noise ratio (SNR) of return signal from an illuminated target. The SNR is directly proportional to the transmitted power and the radar cross section (RCS) of the target [7, 8].
It is often attempted to maintain SNR close to an optimum value for efficient track updates. The transmitted power is adjusted in the attempt such that a desired SNR can be realized. Since the target RCS is unknown, it is required to adjust the power based on an estimate of the average RCS . An algorithmic procedure for estimating the average RCS was proposed in [1, 2]. The procedure, denoted by MED, adjusts the RCS estimate, depending on the median difference between the SNR measurements of return signals in a sliding window and their corresponding expected values. In this paper, a maximum-likelihood (ML) approach is presented for estimating the average RCS, and a numerical solution to the approach is proposed based on a generalized expectation maximization (GEM) algorithm. Numerical experiments were performed to compare estimation performance of the ML approach with that of MED. The experimental results show that ML estimation can perform successfully even for a low SNR target that MED fails to estimate.
2. RCS Model
where denotes the average SNR at scan . Note that is proportional to the average RCS, denoted by . That is, , where is a known constant that depends on target range and transmitted power at .
3. ML Estimation of RCS
An algorithmic procedure for estimating the average RCS was proposed in [1, 2]. The procedure (denoted by MED) adjusts the average RCS estimate by 0.5 dB whenever the median difference between SNR measurements of return signals in a sliding window and their corresponding expected values is 1 dB or greater. In the case of a missed detection, the estimate is decreased by 0.5 dB. We adopt the acronym MED to represent the procedure, since it uses the median as its statistic. In this section, a maximum-likelihood approach for estimating the average RCS and its numerical solution is presented. Along the context of the problem posed in [1, 2], we assume that the false alarm probability is small enough so that, if a detection occurs, it is from the target under track. Under this assumption, we obtain an estimate of from a sequence of detections and misses over a sliding window.
Note that each iteration is guaranteed to increase the log-likelihood function (4) and the EM algorithm leads ultimately to the value that maximizes the likelihood function .
where denotes the length of the sliding window and it is the sum of the number of detections ( ), and the number of misses ( ) in the window. Note that the function (14) corresponds to an average of the estimates of that are obtained based on the observations of the detections and misses in the window. To be more specific, in the first term of (14) is an (one-sample) unbiased estimate of with the observed signal strength of the detection at scan , and in the second term is also an one-sample estimate of with the unobserved but estimated signal strength of the missed detection at scan . Recall that the unobserved signal strength is estimated by (13) which evaluates the expected value of the undetected signal strength distributed with the pdf of (7) with .
It can be shown that the iteration with the mapping (14) forms an instance of a GEM algorithm  in the following cases: where 's are a constant over a sliding window with scans appearing in (14), and where 's and 's are sufficiently large. In these cases, the sequence of (14) satisfies the sufficient conditions (in Theorem of ) for convergence to a ML estimate. In the event that the mapping (14) fails to satisfy the inequality , a possible simplest step is to stop the iteration and return as an estimate. The GEM algorithm was implemented to compute a ML estimate, and its estimation performance is discussed in the following section.
4. Numerical Experiments
We performed numerical experiments to investigate RCS estimation performance of the ML approach. The ML estimates were evaluated with iterations of (14) that form an implementation of the GEM algorithm. The algorithmic procedure MED was also implemented and its performance was compared with that of the ML estimation. The procedure estimates the average RCS using the median difference between SNR measurements of return signals in a sliding window and their corresponding expected values. The procedure uses as the measurement of SNR at scan  and a sliding window with 5 detections. The experiments were performed for .
The table shows that the RMS errors of ML( ) and ML( ) decrease with . It appears counterintuitive, but the decrease is due to more information on gathered over a sliding window whose length increases in average as decreases. Note that as decreases, the probability of detection decreases, and more scans are required in average to retain a specified number of detections. In contrast, the RMS error of ML( ) decreases as increases. This implies that detections are more informative than misses in estimating , since they provide additional information on via signal strength observations. In all cases, the error decreases as the window size increases through or . The window size of ML( ) corresponds to the size of ML( ) in effect with . For instance, corresponds approximately to for and for . Table 1 shows that the errors of ML( ) and ML( ) are 0.278 and 0.273 for , respectively, and those of ML( ) and ML( ) are 0.290 and 0.297 for , respectively. The numbers are comparable to each other, and it implies that ML( ) does not have any particular advantages over ML( ) in practice.
The table also shows that MED outperforms ML( ) and ML( ) for higher signal strength, but it is less effective than ML( ) and ML( ). It should be remarked that MED failed to estimate for (the estimates of MED were close to zero). Note that MED uses a sliding window with 5 detections. We also performed numerical experiments to obtain the RMS errors of MED for a sliding window with 10 detections. It failed again to estimate for , and all the errors for higher 's were larger than those of MED with 5 detections. It appears that MED was designed based on 5 detections and it needs a modification for a different number of detections to assure its best performance. The experimental results indicate that ML estimation can perform successfully for a low SNR target that MED fails to estimate. The computational cost of ML estimation was not significant. The GEM terminated in 3.59 iterations of (14) in average and in maximum 5 iterations to yield a ML estimate.
Figures 1 to 3 show that the errors for each decrease eventually to the values slightly lower than those for . This observation is consistent with the results presented in Table 1. Note, furthermore, that ML ( ) maintains the error for to be smaller than that for at all scans. This confirms that the argument on the results of the table holds for ML ( ) over all scans, including the early stage of tracking. In contrast, MED decreases its error faster at the early stage for than for ; see Figures 1 and 3. This is the case where the initial estimate error is larger than the "steady-state" error. In this case, MED is activated and starts to correct the estimation error at an earlier scan for since the window can retain 5 detections faster when the probability of detection is higher. This correction allows to pass a more accurate initial estimate to the MED estimation of the following scan and causes the estimation error to decrease faster. Conversely, MED starts to miscorrect the error at an earlier scan for in Figure 2, where the initial estimate error is smaller than the "steady-state" error. This miscorrection passes a worse initial estimate to the MED estimation of the next scan. This affects adversely the estimation and causes the error to increase faster.
Figure 4 presents the RMS errors of ML( ) and ML( ), , 12, and 15, for . At the early scans, ML( ) and ML( ) use the same windows and the RMS errors coincide as a consequence. The "steady-state" errors are also consistent with the results presented in Table 1. The transient "dynamics" of ML( ) can be described as follows. ML( ) starts at scan 1 with three detections as it is activated according to the "3 out of 5" track initiation logic. The three detections according to the logic requires 3.9 and 3.6 scans in average for and 32, respectively. This implies that ML( ) requires and more scans in average for and 32, respectively, to stop expanding its window. Suppose that, for instance, . In this case, the stopping starts to occur at scan 8 and occurs in average between scans 9 and 10. Note that the statistical characteristics of the observations before and after are different due to the track initiation logic, since the logic intervenes in effect to select observations with a higher probability to retain more detections in the window at scans earlier than and at scan 1. The detections are more informative than misses in the RCS estimation. It is shown in Figures 4(a) and 4(b), respectively, that ML( ) begins to lose the better quality information by releasing the detections from the window and its errors start to increase at scan 8 and at scan 9. This explains the reason that the "undershoot" occurs at scan 8 and scan 9, respectively, for and 32.
The transient "dynamics" of ML( ) in Figures 1 to 4 can be explained based on the arguments similar to the case of ML( ). ML( ) starts at scan 1 with three detections according to the "3 out of 5" logic. It increases its window size until the window retains 7 more detections, which requires 10.5 and 8.6 additional scans in average for and 32, respectively. This implies that ML( ) begins to stop expanding its window at scan 8 and stops in average around at scan 11 for and at scan 9 for . As discussed earlier, the stopping causes to lose the quality information and increase the estimation errors. This explains why the error starts to increase at scan 8 and scan 9, and it increases until scan 11 and scan 10 to generate the "overshoot" for and 32, respectively. The transient "dynamics" of MED seems more complicated than in the case of ML( ). Its analysis is not pursued in this paper.
An ML approach has been presented for estimating the average RCS, and a numerical solution to the approach has been proposed based on a generalized expectation maximization algorithm. Numerical experiments were performed to compare the RCS estimation performance of the ML approach with that of a previously reported procedure MED. The experimental results show that the ML approach can perform successfully even for a low-SNR target that MED fails to estimate. The results also show that, in contrast to MED, the ML approach is not susceptible to the error of a preset initial value of the RCS estimate at the early stage of tracking. Extension to the case in the presence of false alarms is currently under investigation.
This work was supported by the BK-21 Program.
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