UNCLASSIFIED A Bayesian-Based CFAR Detector for Pareto Type II Clutter Graham V. Weinberg, Stephen D. Howard and Charlie Tran
UNCLASSIFIED Overview The problem space is determination of non-coherent sliding window detectors, for operation in an X-band high 1 resolution maritime surveillance radar environment; We want these detectors to have the constant false alarm rate (CFAR) property; 2 It is assumed that the clutter is modelled by a Pareto Type II model (with an effective shape parameter if desired); 3 A novel Bayesian approach is used to produce the decision rule; 4 Prior distributions are selected based upon the Fisher Information matrix; 5 A Bayes predictive distribution, for the cell under test conditioned on the clutter statistics, is produced; 6 This permits the determination of a Bayes decision rule; 7 The Bayesian detector does not require a priori knowledge of the Pareto clutter parameters and so is a legitimate 8 CFAR detector. 2
UNCLASSIFIED Overview (Cont.) This Bayesian approach has also been applied in the context of clutter modelled by scale and power invariant 1 distributions; In the latter case, it is possible to define the concept of optimality of the detector; 2 Optimality is related to the concept of strong consistency; 3 As a result of this, it has been possible to generalise the results of Gandhi and Kassam (1994) so that the cell 4 under test target model is not Swerling I; In addition to this, Howard and Weinberg (2018) demonstrated optimality of the geometric mean CFAR introduced 5 in Weinberg (2017) for target detection in a Pareto Type I environment. Gandhi, P . P . and Kassam, S. A. (1994). Optimality of the cell averaging CFAR detector IEEE Transactions on Information Theory, 40, 1222-1228. Weinberg, G. V. (2017). On the Construction of CFAR Decision Rules via Transformations. IEEE Transactions on Geoscience and Remote Sensing, 55, 1140-1146. Howard, S. D. and Weinberg, G. V. (2018). Optimal Predictive Inference and Non-Coherent CFAR Detectors. IEEE Transactions on Signal Processing (submitted). 3
UNCLASSIFIED Sliding Window Detection Process · · · · · · Z 1 Z m G 1 G 2 Z 0 G 3 G 4 Z m + 1 Z N cell under test (CUT) clutter range profile (CRP) g ( h 1 , h 2 ) scale-invariant function g normalisation by multiplicative factor τ τ g ( h 1 , h 2 ) Detector Output detection decision 4
UNCLASSIFIED Pareto Distribution for X-Band Maritime Surveillance Radar Clutter Validation of the Pareto model for X-band maritime surveillance radar clutter has been documented in three independent studies: Balleri, A. Nehorai, A. and Wang, J. (2007). Maximum Likelihood Estimation for Compound-Gaussian Clutter with 1 Inverse-Gamma Texture. IEEE Transactions on Aerospace and Electronic Systems, 43, 775-779. Farshchian, M. and Posner, F. L. (2010). The Pareto Distribution for Low Grazing Angle and High Resolution 2 X-Band Sea Clutter. IEEE Radar Conference Proceedings, 789-793. Weinberg, G. V. (2011). Assessing Pareto Fit to High Resolution High Grazing Angle Sea Clutter. IET Electronics 3 Letters, 47, 516-517. Accounting for receiver thermal noise with a Pareto effective shape parameter has also been examined: Rosenberg, L. and Bocquet, S. (2013). The Pareto distribution for high grazing angle sea-clutter. IEEE 1 International Geoscience and Remote Sensing Symposium - IGARSS, 4209-4212. Alexopoulos, A. and Weinberg, G. V. (2015). Fractional order Pareto distributions with application to X-band 2 maritime radar clutter. IET Radar, Sonar and Navigation, 9, 817-826. 5
UNCLASSIFIED Pareto Distributions The Pareto Type II model has distribution function: � α � β P ( Z ≤ t ) = 1 − , for t ≥ 0 , I t + β where α > 0 is the shape parameter and β > 0 is the distribution’s scale parameter. The latter has been validated as a model for X-band maritime surveillance radar clutter returns. It is also the intensity model of a compound Gaussian process with inverse Gamma texture (K-distribution has root Gamma as texture). This distribution is closely related to the Pareto Type I model, which has distribution function: � α � β I P ( Z ≤ t ) = F Z ( t ) = 1 − , for for t ≥ β . t Fits of the Pareto Type II model to real data show that β << 1, which implies a Pareto Type II distribution can be approximated by a Pareto Type I, which facilitates the construction of sliding window detection processes with the CFAR property. Weinberg, G. V. (2017). Radar Detection Theory of Sliding Window Processes. CRC Press, Boca Raton, Florida. 6
UNCLASSIFIED Problem specification in statistical terms Assume a series of nonnegative clutter measurements are available Z 1 , Z 2 , . . . , Z N ; 1 These are referred to as the clutter range profile (CRP) or clutter training cells; 2 Assumed independent and identically distributed with a common distribution function F Z j ( t ) = I P ( Z j ≤ t ) ; 3 A scale-invariant function g is used to produce an estimate of the clutter level: g ( Z 1 , Z 2 , . . . , Z N ) ; 4 g ( λ ( Z 1 , Z 2 , . . . , Z N )) = λ g ( Z 1 , Z 2 , . . . , Z N ) ∀ λ > 0; 5 A cell under test (CUT) is assumed to exist and based upon the measurement of clutter level one wants to decide 6 whether the CUT is also clutter or whether it contains a target embedded within clutter; The measurement of clutter is normalised by a constant τ > 0 which will allow adaptive control of the false alarm 7 probability in ideal situations; Let H 0 be the hypothesis that the CUT does not contain a target, and H 1 the alternative that it does contain a 8 target in clutter; The test is to reject H 0 if Z 0 > τ g ( Z 1 , Z 2 , . . . , Z N ) . 9 7
UNCLASSIFIED Problem specification (continued) Such a test can be written in the compact form H 1 > H 0 τ g ( Z 1 , Z 2 , . . . , Z N ) . Z 0 < The probability of false alarm is Pfa = I P ( Z 0 > τ g ( Z 1 , Z 2 , . . . , Z N ) | H 0 ) ; 1 If the threshold multiplier τ can be determined so that it does not vary with the clutter power (or equivalently 2 clutter parameters) then the test is said to achieve the constant false alarm rate (CFAR) property; This means the detector can be used so that it maintains a steady number of false alarms in practice; 3 Severe variation in the Pfa can be a disaster for a (maritime surveillance) radar; 4 The probability of detection is Pd = I P ( Z 0 > τ g ( Z 1 , Z 2 , . . . , Z N ) | H 1 ) ; 5 To evaluate this one must combine a target model in the complex domain with a compound Gaussian model with 6 appropriate texture which generates the desired intensity clutter model. 8
UNCLASSIFIED Construction of the Bayesian Detector The idea of the Bayesian approach is to construct the Bayes predictive distribution of Z 0 | Z 1 , Z 2 , . . . , Z N . Then the Pfa = I P ( Z 0 > τ | Z 1 = z 1 , . . . , Z N = z N ) � ∞ = f Z 0 | Z 1 ,..., Z N ( z 0 | z 1 , . . . , z N ) dz 0 τ where f Z 0 | Z 1 ,..., Z N ( z 0 | z 1 , . . . , z N ) is the density of the predictive distribution. This expression can then be inverted for τ to obtain the corresponding detector with the desired Pfa. It is necessary to specify a prior distribution for the unknown clutter parameters so that the predictive distribution can be constructed. 9
UNCLASSIFIED Construction of the Bayesian Detector (Cont.) Based upon Bayes’ Theorem, the Bayesian predictive density is given by � ∞ � ∞ f Z 0 | Z 1 , Z 2 ,..., Z N ( z 0 | z 1 , z 2 , . . . , z N ) = f Z 0 |A , B ( z 0 | α, β ) × 0 0 f A , B| Z 1 , Z 2 ,..., Z N ( α, β | z 1 , z 2 , . . . , z N ) f A ( α ) f B ( β ) d α d β. A and B are the distributions of the clutter parameters; 1 f Z 0 |A , B ( z 0 | α, β ) is the density of the CUT conditioned on the clutter parameters; 2 f A , B| Z 1 , Z 2 ,..., Z N ( α, β | z 1 , z 2 , . . . , z N ) is the posterior density of the parameters A and B conditioned on the CRP; 3 f A ( α ) and f B ( β ) are the prior distributions of the clutter parameters. 4 10
UNCLASSIFIED Construction of the Bayesian Detector (Cont.) To select suitable prior distributions for the Pareto clutter parameters, one can resort to the Jeffreys prior, which is proportional to the square root of the determinant of the Fisher information matrix. The Jeffreys prior can be shown to be proportional to 1 1 f ( α, β ) = β . � α ( α + 1 ) 2 ( α + 2 ) The interesting thing about the form of the Jeffreys prior is that d β β is the right Haar measure; the Pareto Type II distribution is scale-invariant and so Howard and Weinberg (2018) argue that in the Pareto Type I case this is the most appropriate improper prior for β . This can be repeated for the Pareto Type II case. The problem with the Jeffreys prior above is that the resultant detector has computational complexity. Hence an approximate prior is taken to be 1 1 f ( α, β ) = β . α This is the prior used for the Pareto Type I case; the right Haar measure can be used as a non-informative prior for α due to the invariance of the Pareto Type I model. 11
UNCLASSIFIED Construction of the Bayesian Detector (Cont.) By working through the construction of the posterior density, and doing some evaluations, it can be shown that the Bayesian test is to reject H 0 if the test statistic T ( Z 0 , Z 1 , . . . , Z N ) is negative, where T ( Z 0 , Z 1 , . . . , Z N ) = � − N � − N � ∞ � N � N φ N − 1 � � d φ. log( Z 0 φ + 1 ) + log( φ Z j + 1 ) − Pfa log( φ Z j + 1 ) � N j = 1 ( φ Z j + 1 ) 0 j = 1 j = 1 This test is CFAR with respect to both Pareto clutter parameters. 12
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