Abstract
A nonlinear transformation is introduced, which can be used to compress a series of random variables. For a certain class of random variables, the compression results in the removal of unknown distributional parameters from the resultant series. Hence, the application of this transformation is investigated from a radar target detection perspective. It will be shown that it is possible to achieve the constant false alarm rate property through a simple manipulation of this transformation. Due to the effect the transformation has on the cell under test, it is necessary to couple the approach with binary integration to achieve reasonable results. This is demonstrated in an X-band maritime surveillance radar detection context.
Keywords
- transformations
- random variable properties
- radar detection
- mathematical statistics
- radar
1. Introduction
The fundamental problem to be examined in this chapter is the detection of targets embedded within the sea surface, from an airborne maritime surveillance radar. Artifacts of interest could be lifeboats or aircraft wreckage resulting from aviation or maritime disasters. From a military perspective, one may be interested in the detection and tracking of submarine periscopes. Another scenario may be the detection of illegal fishing vessels or small boats used for smuggling of people or contraband. An airborne maritime surveillance radar has a difficult task in the detection of such objects from high altitude, while surveying a very large surveillance volume.
Such radars operate at X-band and are high resolution, and as such are affected by backscattering from the sea surface, which is referred to as clutter. This backscattering tends to mask small targets and makes the surveillance task extremely difficult. One of the major issues with the design of radar detection schemes is the minimization of the detection of false targets, while maximizing the detection of real targets. As a statistical hypothesis test, one can apply the Neyman-Pearson Lemma to produce a decision rule that achieves these objectives. However, in many cases, such a decision rule requires clutter model parameter approximations as well as estimates of the target strength based upon sampled returns. An issue, well known within the radar community, is that small variations in the clutter power level can result in huge increases in the number of false alarms. Since clutter power is a function of the underlying clutter model’s parameters, approximations of the latter will have an inevitable effect on the former. Hence a large body of research has been devoted to designing radar detection strategies that maintain a fixed level of false alarms. A detector that achieves this objective is said to have the constant false alarm rate (CFAR) property [1].
In order to maintain a fixed rate of false alarms, sliding window decision rules were examined in early studies of radar detection strategies [2–6]. These investigations have been extended to account for different clutter models and to address issues with earlier detector design in a number of subsequent analyses [7–15]. Such decision rules can be formulated as follows. Suppose that the statistic
where
If
This chapter examines an alternative approach to achieve the CFAR property, based upon a nonlinear transformation that is used to compress the original clutter sequence. The consequence of this is that the resulting transformed series of random variables will have a fixed clutter power level and so permits a CFAR detector to be proposed. It is then shown how this transformation can be used to produce a practical radar detection scheme.
The chapter is organized as follows. Section 2 introduces the nonlinear mapping and formulates a decision rule. Section 3 specializes this to the case of Pareto distributed sequences, since the Pareto model is suitable for X-band maritime surveillance radar clutter returns. Section 4 demonstrates detector performance in homogeneous clutter, while Section 5 applies the decision rules directly to synthetic target detection in real X-band radar clutter.
2. Transformations and decision rule
2.1. Mapping
In X-band maritime surveillance radar, the Pareto distribution has become of much interest as a clutter intensity model due to its validation relative to real radar clutter returns [16–18]. This model arises as the intensity distribution of a compound Gaussian model with inverse Gamma texture. Consequently, the Pareto distribution fits into the currently accepted radar clutter model phenomenology [19]. Hence, there have been a number of recent advances in the design of CFAR processes under a Pareto clutter model assumption [20–25].
A random variable
for
for
Other random variables of interest in radar signal processing, such as the Weibull, can also be expressed in a form similar to Eq. (5). Hence, for the purposes of generality, suppose {
The sequence produced via Eq. (6) is a generalization of the Pareto model (3). Next define a nonlinear mapping
where each
The proof of Lemma 2.1 is now outlined. Supposing that
where properties of the logarithmic function have been utilized. Since Eq. (8) does not depend on
Lemma 2.1 suggests that if the original sequence of random variables is processed in 4-tuples, the compressed sequences’ statistical structure is only dependent on the random variables
2.2. Decision rule
In order to propose a decision rule exploiting the transformation introduced in Lemma 2.1, it is necessary to focus first on a series of four returns. Hence, suppose we have a CUT statistic
where
The test in Eq. (9) can also be re-expressed in terms of the preprocessed clutter statistics. In particular, it can be shown to be equivalent to rejecting
with the appropriate choice for
Observe that this test is not of the usual form found in the radar signal processing literature, since it compares a CUT with a measurement of clutter based upon three statistics, and not upon a sample of predetermined size. This will be discussed subsequently in terms of practical implementation of the test in Eq. (10). The next section discusses the application of Eq. (10) to the Pareto clutter case, enabling the determination of
3. Specialization to the Pareto Clutter model
3.1. Distributions under H 0
Since the motivation of the work developed here is the design of radar detection schemes for maritime surveillance radar, the results of the previous section are specialized to the Pareto case. In order to apply Lemma 2.1, it is necessary to determine the distribution of the resultant sequence produced by
for
This can be recognized as a Pareto distribution, with support the nonnegative real line and shape and scale parameter unity. More specifically,
To prove Corollary 3.1, suppose that
which is that of a Laplace distribution. Then it follows that
where Eq. (12) has been applied, and
Supposing that
and an application of Eqs. (4)–(14) shows that the ratio has cdf Eq. (11) with an evaluation of the integral. This establishes the result in Corollary 3.1, as required.
3.2. Thresholds and the CUT
Based upon Corollary 3.1 the univariate threshold for the Pareto case is given by
The threshold (Eq. (15)) illustrates the issues with the nonlinear mapping, as this threshold will be quite large for appropriate Pfa. Note that for a Pfa of 10− 6,
To explore this further, it is informative to examine the detection scheme in Eq. (10) when there is a target model present. Suppose
where each
where the change of variables
4. Performance in homogeneous clutter
4.1. Methodology and data
In order to examine the performance of the proposed detection scheme (9), clutter is simulated under the assumption of a Pareto clutter model, which has been found to fit Defence Science and Technology Group’s (DSTG’s) real X-band maritime surveillance radar data sets. Ingara is an experimental X-band imaging radar which has provided real clutter for the analysis of detector performance [28]. A trial in 2004 produced a series of clutter sets that have been analyzed from a statistical perspective in Ref. [29]. During the trial, the radar operated in a circular spotlight mode, surveying the same patch of the Southern Ocean at different azimuth and grazing angles. Additionally, the radar provided full polarimetric data. For the purposes of the numerical work to follow, focus is restricted to one particular data set. This is run 34683, at an azimuth angle of 225
For performance analysis in homogeneous clutter, the data is simulated with distributional parameters matched to those obtained from the Ingara data set. The data consists of 821 pulses with 1024 range compressed samples, from which maximum likelihood estimates of the distributional parameters can be obtained from the intensity measurements. Under the Pareto model assumption, the estimates are
As remarked previously, it is necessary to couple (10) with an integration scheme to enhance its performance. The integration scheme used for this purpose is binary integration, which is well-described in Ref. [30], and an application of it in a Pareto distributed clutter environment can be found in Ref. [31]. Such a process applies a series of
If PfaBI denotes the Pfa for binary integration, then it can be expressed in terms of the univariate detection processes Pfa through the equation
The threshold
To simulate detection performance, the probability of detection (Pd) is estimated, using 106 Monte Carlo runs based upon a Swerling 1 target model assumed for the CUT. For each SCR, the binary integration process is run using
4.2. Receiver operating characteristic curves
Receiver operating characteristic (ROC) curves are used to examine the performance, which plots the probability of detection as a function of the false alarm probability, when the target in the CUT is at a fixed SCR. Figures 2–4 provide examples of the performance of the new detector Eq. (10) with binary integration and compares it to the performance of some of the recently introduced detectors designed for operation in a Pareto clutter model environment. For a CUT
which is shown in Ref. [20] to have its threshold set via
which has its threshold multiplier
where the OS index 1 ≤
Figure 2 compares the performance of these decision rules, where the detection process (10) coupled with binary integration is denoted as the nonlinear mapping (NLM). In this case, the CUT SCR is 5 dB, representing a small target. As can be observed, the new decision rule has superior performance. The same experiment is repeated in Figure 3, where the CUT SCR is 15 dB, and then it is increased to 20 dB in Figure 4. These results show that the new detection process has superior performance, while not requiring
It is interesting to note that as
4.3. Effect of interference
Next the cost of interference on the new decision rule is examined, and for brevity, only this decision rule is considered. Figure 6 shows the case where the CUT has SCR of 5 dB, and the decision rule (10) coupled with binary integration is denoted BI, while the three interference cases are marked appropriately. Here we observe quite good performance that decreases with the interference. Figure 7 shows the result of increasing the SCR in the CUT to 20 dB. The result is an expected detection performance improvement as shown.
5. Performance in real data
As a final test of the proposed detection scheme, it was run directly on the Ingara data set under consideration, with the insertion of synthetic Swerling 1 target and interference as for the homogeneous case. A sliding window was run across the data sequentially, and detection performance was estimated by running the 3 out of 8 detection scheme, resulting in a run length of 840,672. The Ingara data is slightly correlated from cell to cell and so the detector Eq. (9), which has threshold set via an independence assumption, becomes a suboptimal decision rule. Detection performance under both clutter model assumptions is plotted on the same ROC curve to compare performance on the real data more easily. The same scenario is repeated as for the analysis under homogeneous independent clutter.
Figure 8 shows detection performance with the CUT SCR of 5 dB, while Figure 9 repeats the same numerical experiment as for Figure 8, except the CUT has SCR of 20 dB. Comparing Figure 8 with Figure 6 we observe that the effects of correlation are having an effect on the performance in real data. The new decision rule is designed to operate in independent homogeneous clutter returns, and so there is a serious variation in performance. The same situation is observed at a larger CUT SCR (comparing Figures 9 and 7).
6. Conclusions
A nonlinear transformation was introduced and shown to remove clutter parameter dependence for a class of statistical models. This was used to formulate a simple linear threshold detector in the transformed clutter domain. Due to issues with the magnitude of detection thresholds, it was necessary to couple the approach with binary integration.
Analysis of detection performance in simulated clutter showed good detection performance. Interference had a strong impact on performance as expected. When the detection process was applied directly to real data, similar results were observed. Nonetheless, the nonlinear transformation, coupled with binary integration, resulted in reasonable detection performance while guaranteeing the CFAR property is preserved.
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