The adaptive coherence estimator is the ... - Semantic Scholar
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
281
The Adaptive Coherence Estimator is the Generalized Likelihood Ratio Test for a Class of Heterogeneous Environments Stéphanie Bidon, Student Member, IEEE, Olivier Besson, Senior Member, IEEE, and Jean-Yves Tourneret, Member, IEEE
Abstract—The adaptive coherence estimator (ACE) is known to be the generalized likelihood ratio test (GLRT) in partially homoof the geneous environments, i.e., when the covariance matrix secondary data is proportional to the covariance matrix of the vector under test (or ). In this letter, we show that ACE is indeed the GLRT for a broader class of nonhomogeis a random matrix, neous environments, more precisely when with inverse complex Wishart prior distribution whose mean only is proportional to . Furthermore, we prove that, for this class of heterogeneous environments, the ACE detector satisfies the con. stant false alarm rate (CFAR) property with respect to and
=
Index Terms—Adaptive coherence estimator, adaptive detection, generalized likelihood ratio test, nonhomogeneous environments.
I. INTRODUCTION N many applications such as sonar and radar, it is desired to detect the presence of a signal of interest embedded in colored noise. The standard detection problem is formulated as a binary hypothesis test
I
(1a) (1b) where and stand for the amplitude and signature of the target in the cell under test, respectively. In (1), denotes the complex normal distribution of a -dimensional vector with mean and covariance matrix , and is the primary data vector with covariance matrix . The training samples are assumed to be independent with covariance matrix . The primary data vector is also independent from the training samples. When the signature and the structure of the covariance maare both known (i.e., is known up to a scaling trix factor), the generalized likelihood ratio test (GLRT) was shown to be the constant false alarm rate (CFAR) matched subspace
Manuscript received october 18, 2007; revised December 10, 2007. This work was supported in part by the Délégation Générale pour l’Armement (DGA) and in part by Thales Systèmes Aéroportés. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Xiaoli Ma. S. Bidon and O. Besson are with the University of Toulouse, ISAE, Department of Electronics, Optronics and Signal, 31055 Toulouse, France (e-mail: [email protected]; [email protected]). J.-Y. Tourneret is with IRIT/ENSEEIHT, 31071 Toulouse, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2007.916044
detector [1], [2]. The detector makes no use of the secondary data but requires prior knowledge on the noise statistics. To circumvent this drawback, an adaptive version of the detector was introduced in [3] and referred to as the adaptive coherence estimator. It consists in replacing the primary covariance matrix of the CFAR matched subspace detector by the estimate , which is the sample covariance matrix (SCM), and . The detection test then reduces to (2) The test (2) appeared to be very attractive for radar, sonar, and communication applications as it is invariant to data scaling. Later the adaptive coherence estimator (ACE) was shown to be the GLRT in a partially homogeneous environment [4], i.e., when the secondary covariance matrix is proportional to the primary covariance matrix (3) with and unknown. Additionally, in [5], the authors proved the ACE to be the uniformly most powerful invariant (UMPI) detection test. References [6]–[8] provide a performance analysis of the ACE under homogeneous and nonhomogeneous environment. The ACE test was also presented independently for detecting targets embedded in compound-Gaussian clutter [9]. It emerges naturally as the adaptive counterpart of an asymptotic approximation of the GLRT when the structure of the clutter covariance matrix is known. However, real-world environments lead to a large amount of heterogeneities which cannot be all embraced by the model (3). In the context of space time adaptive processing, Melvin proposed in [10] models of amplitude and spectral clutter heterogeneities (e.g., due to clutter edges or intrinsic clutter motion) that cause profound structural mismatches between primary and secondary covariance matrices. In a previous work [11], we proposed a new model of heterogeneous environments in a Bayesian framework. Our aim was to have a model that allows one to keep around while having mathematical tractability to derive new detectors. More precisely, the secondary covariance matrix was assumed to be a random matrix distributed according to a complex inverse Wishart distribution with mean , i.e., (4) This model ensures that with probability one. Furthermore, the parameter scales the distance between and : the larger , the more homogeneous the environment [11].
1070-9908/$25.00 © 2008 IEEE
282
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
We now extend this Bayesian model by introducing a power scaling factor
with (13)
(5) In this way, the heterogeneity level is increased compared to (4). However, on average, we recognize the partially homogenous deterministic environment
Using (11) and (12), the distribution of as (see [11] for similar derivations)
can thus be expressed
(6) In this letter, we show that the GLRT for the detection problem of (1) with the environment described by (5) is also the ACE. The test is shown to be independent of the primary covariance and the power ratio and hence possesses the CFAR matrix property.
(14) Finally, the joint distribution of is given by
conditioned on ,
(15)
II. GENERALIZED LIKELIHOOD RATIO TEST In this section, we show that the GLRT for the partially homogeneous Bayesian environment described by (5) is the ACE. The GLRT is classically defined as follows: (7)
where stands for the joint distribution of hypothesis , .
and
B. Maximum Likelihood Estimate (MLE) of Differentiating the logarithm of (15) and setting the derivative to zero implies that the MLE of verifies
(16)
under
A. Distributions This section derives the distributions required in (7). Let us denote the centered primary snapshot. Since and are independent, the joint distribution of is
on the left-hand By multiplying this equation by side and by on the right-hand side, one recognizes a quadratic matrix equation
(17) (8) with The density of conditional to and and thus can be written as
was set to
(18) (19) (9)
where and stand for the determinant and the exponential of the trace of a matrix, respectively. In order to derive the density of conditioned on and , note that
We proceed as in [11] to solve (17). Note that the matrix is Hermitian positive definite. Consider one of its eigenvectors associated with the eigenvalue . Multiplying (17) by , we obtain
(10)
(20)
Using the independence of the is given by
’s, the conditional density of
(11)
, and is an eigenvalue associated with . As is a rank one Hermitian of positive matrix, there exists a unitary matrix such that Hence,
is necessarily an eigenvector of
(21) The conditional distribution of given the heterogeneity model (5), i.e.,
is defined by Thus, has two different eigenvalues, and the following equations: (12)
verifies one of
(22)
BIDON et al.: ADAPTIVE COHERENCE ESTIMATOR IS THE GENERALIZED LIKELIHOOD RATIO TEST
The first equation yields a value of
proportional to :
283
Plugging (33) in (24), we obtain the following expression for the MLE of : (23) (34)
The second equation is a second-order polynomial equation (24) where and . This equation has a unique real positive solution whose explicit form is not required for our analysis. The remaining derivations will use (24) only. The solution of the quadratic matrix equation (17) is thus given by
D. MLE of Noticing that is proportional to , see (33), and that the is constant [cf (34)], we have product (35) under
The MLE of
amounts to minimizing the quantity
(25) with (26) Finally, using (19) and (25), the MLE of on only through the matrix
(36) The minimum is well known to be achieved for [12] (37)
is shown to depend and is equal to (27)
(38) C. MLE of Let us denote
E. GLRT Statistic (28)
The
th root of the GLR can be expressed as (39)
Using the previous expressions (27) and (15), one obtains and hence the GLRT is the ACE defined in (2). (29) Then noticing that does not depend on [use (23) and (26)], the above expression can be simplified to
(30) Differentiating the logarithm of and equating the derivative to zero implies that the MLE of verifies
III. DETECTOR PERFORMANCES A. CFAR Behavior , the distribution of the We show in this section that under and , and hence that test statistic (2) is independent of ACE has the enjoyable CFAR property. We proceed as in [12] and consider the unitary matrix such that (40) (41)
(31)
Let us define (42)
Then, gathering the terms which depend on plying by , one obtains
and multi-
(43) (44) Then the test statistic can be rewritten as
(32) Using (24), we observe that the coefficient of zero. So the MLEs of and are proportional
(45)
is equal to Under (33)
,
has a complex normal distribution (46)
284
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
. Fig. 1 displays the probability of deas SNR runs, versus SNR for different tection , obtained from values of . From inspection of this figure, we observe that, as expected, the more homogenous the environment (i.e., the larger ), the better the detector performances. There is almost 3 dB difference between the case and the case which corresponds to the partially homogeneous model (3). This shows that the Bayesian environment (5) models a larger degree of heterogeneity than the deterministic environment. Furthermore, the figure shows that the heterogeneity level does not depend linearly on . Indeed when is small, a slight variation in results in a large variation of the homogeneity level. The trend is inverted for larger values of . IV. CONCLUSIONS Fig. 1. Probability of detection versus SNR—Influence of .
Conditioned on , , and , the matrix has a complex Wishart distribution with degrees of freedom
(47) Therefore, the distribution of
given
is
The adaptive coherence estimator is a well-known detection scheme which has proved to be effective in a number of nonhomogeneous environments. It is the UMPI test in partially homogeneous environments and is known to perform well also in compound-Gaussian clutter with fully correlated texture. In this letter, we showed that it is also the GLRT in nonhomogeneous environments such that the conditional distribution of is a complex inverse Wishart distribution with . This seems to indicate that ACE is (close to) optimum for a large class of nonhomogeneous environments, at least it is rather robust to covariance mismatches characterized by scaling ambiguities. This is to be contrasted with the marked selectivity of ACE with respect to steering vector mismatches. Indeed, ACE is known to possess strong rejection capabilities for signals whose signatures differ from the presumed ones. REFERENCES
(48) which is recognized as a multivariate -distribution. We first observe that in (41) is a fixed vector. Moreover, and are independent —since and are independent— and the distributions of in (46) and in (48) are parameter free. Since the test statistic in (45) depends only on , , and , it follows that its distribution under does not depend on or , which proves that ACE is CFAR. B. Numerical Simulations In this section, we study the influence of on the detector performances. The dimension of the observation space is set to , and training samples are available. The . The detector probability of false alarm is set to threshold is computed from Monte Carlo runs, with a different value of at each run, drawn from the conditional distribution in (12). The signal-to-noise ratio (SNR) is defined
[1] L. L. Scharf, Statistical Signal Processing: Detection, Estimation and Time Series Analysis. Reading, MA: Addison Wesley, 1991. [2] L. L. Scharf and B. Friedlander, “Matched subspace detectors,” IEEE Trans. Signal Process., vol. 42, no. 8, pp. 2146–2157, Aug. 1994. [3] L. L. Scharf and T. McWhorter, “Adaptive matched subspace detectors and adaptive coherence estimators,” in Proc. 30th Asilomar Conf. Signals Systems Computers, Pacific Grove, CA, Nov. 3–6, 1996, pp. 1114–1117. [4] S. Kraut and L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Process., vol. 47, no. 9, pp. 2538–2541, Sep. 1999. [5] S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherence estimator: A uniformly most powerful invariant adaptive detection statistic,” IEEE Trans. Signal Process., vol. 53, no. 2, pp. 427–438, Feb. 2005. [6] S. Kraut, L. L. Scharf, and T. McWhorter, “Adaptive subspace detectors,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 1–16, Jan. 2001. [7] C. D. Richmond, “Performance of a class of adaptive detection algorithms in nonhomogeneous environments,” IEEE Trans. Signal Process., vol. 48, no. 5, pp. 1248–1262, May 2000. [8] K. F. McDonald and R. S. Blum, “Exact performance of STAP algorithms with mismatched steering and clutter statistics,” IEEE Trans. Signal Process., vol. 48, no. 10, pp. 2750–2763, Oct. 2000. [9] E. Conte, M. Lops, and G. Ricci, “Asymptotically optimum radar detection in compound-Gaussian clutter,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, no. 2, pp. 617–625, Apr. 1995. [10] W. L. Melvin, “Space-time adaptive radar performance in heterogeneous clutter,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, no. 2, pp. 621–633, Apr. 2000. [11] S. Bidon, O. Besson, and J.-Y. Tourneret, “A Bayesian approach to adaptive detection in non-homogeneous environments,” IEEE Trans. Signal Process., vol. 56, no. 1, pp. 205–217, Jan. 2008. [12] E. J. Kelly, “An adaptive detection algorithm,” IEEE Trans. Aerosp. Electron. Syst., vol. 22, no. 1, pp. 115–127, Mar. 1986.
281
The Adaptive Coherence Estimator is the Generalized Likelihood Ratio Test for a Class of Heterogeneous Environments Stéphanie Bidon, Student Member, IEEE, Olivier Besson, Senior Member, IEEE, and Jean-Yves Tourneret, Member, IEEE
Abstract—The adaptive coherence estimator (ACE) is known to be the generalized likelihood ratio test (GLRT) in partially homoof the geneous environments, i.e., when the covariance matrix secondary data is proportional to the covariance matrix of the vector under test (or ). In this letter, we show that ACE is indeed the GLRT for a broader class of nonhomogeis a random matrix, neous environments, more precisely when with inverse complex Wishart prior distribution whose mean only is proportional to . Furthermore, we prove that, for this class of heterogeneous environments, the ACE detector satisfies the con. stant false alarm rate (CFAR) property with respect to and
=
Index Terms—Adaptive coherence estimator, adaptive detection, generalized likelihood ratio test, nonhomogeneous environments.
I. INTRODUCTION N many applications such as sonar and radar, it is desired to detect the presence of a signal of interest embedded in colored noise. The standard detection problem is formulated as a binary hypothesis test
I
(1a) (1b) where and stand for the amplitude and signature of the target in the cell under test, respectively. In (1), denotes the complex normal distribution of a -dimensional vector with mean and covariance matrix , and is the primary data vector with covariance matrix . The training samples are assumed to be independent with covariance matrix . The primary data vector is also independent from the training samples. When the signature and the structure of the covariance maare both known (i.e., is known up to a scaling trix factor), the generalized likelihood ratio test (GLRT) was shown to be the constant false alarm rate (CFAR) matched subspace
Manuscript received october 18, 2007; revised December 10, 2007. This work was supported in part by the Délégation Générale pour l’Armement (DGA) and in part by Thales Systèmes Aéroportés. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Xiaoli Ma. S. Bidon and O. Besson are with the University of Toulouse, ISAE, Department of Electronics, Optronics and Signal, 31055 Toulouse, France (e-mail: [email protected]; [email protected]). J.-Y. Tourneret is with IRIT/ENSEEIHT, 31071 Toulouse, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2007.916044
detector [1], [2]. The detector makes no use of the secondary data but requires prior knowledge on the noise statistics. To circumvent this drawback, an adaptive version of the detector was introduced in [3] and referred to as the adaptive coherence estimator. It consists in replacing the primary covariance matrix of the CFAR matched subspace detector by the estimate , which is the sample covariance matrix (SCM), and . The detection test then reduces to (2) The test (2) appeared to be very attractive for radar, sonar, and communication applications as it is invariant to data scaling. Later the adaptive coherence estimator (ACE) was shown to be the GLRT in a partially homogeneous environment [4], i.e., when the secondary covariance matrix is proportional to the primary covariance matrix (3) with and unknown. Additionally, in [5], the authors proved the ACE to be the uniformly most powerful invariant (UMPI) detection test. References [6]–[8] provide a performance analysis of the ACE under homogeneous and nonhomogeneous environment. The ACE test was also presented independently for detecting targets embedded in compound-Gaussian clutter [9]. It emerges naturally as the adaptive counterpart of an asymptotic approximation of the GLRT when the structure of the clutter covariance matrix is known. However, real-world environments lead to a large amount of heterogeneities which cannot be all embraced by the model (3). In the context of space time adaptive processing, Melvin proposed in [10] models of amplitude and spectral clutter heterogeneities (e.g., due to clutter edges or intrinsic clutter motion) that cause profound structural mismatches between primary and secondary covariance matrices. In a previous work [11], we proposed a new model of heterogeneous environments in a Bayesian framework. Our aim was to have a model that allows one to keep around while having mathematical tractability to derive new detectors. More precisely, the secondary covariance matrix was assumed to be a random matrix distributed according to a complex inverse Wishart distribution with mean , i.e., (4) This model ensures that with probability one. Furthermore, the parameter scales the distance between and : the larger , the more homogeneous the environment [11].
1070-9908/$25.00 © 2008 IEEE
282
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
We now extend this Bayesian model by introducing a power scaling factor
with (13)
(5) In this way, the heterogeneity level is increased compared to (4). However, on average, we recognize the partially homogenous deterministic environment
Using (11) and (12), the distribution of as (see [11] for similar derivations)
can thus be expressed
(6) In this letter, we show that the GLRT for the detection problem of (1) with the environment described by (5) is also the ACE. The test is shown to be independent of the primary covariance and the power ratio and hence possesses the CFAR matrix property.
(14) Finally, the joint distribution of is given by
conditioned on ,
(15)
II. GENERALIZED LIKELIHOOD RATIO TEST In this section, we show that the GLRT for the partially homogeneous Bayesian environment described by (5) is the ACE. The GLRT is classically defined as follows: (7)
where stands for the joint distribution of hypothesis , .
and
B. Maximum Likelihood Estimate (MLE) of Differentiating the logarithm of (15) and setting the derivative to zero implies that the MLE of verifies
(16)
under
A. Distributions This section derives the distributions required in (7). Let us denote the centered primary snapshot. Since and are independent, the joint distribution of is
on the left-hand By multiplying this equation by side and by on the right-hand side, one recognizes a quadratic matrix equation
(17) (8) with The density of conditional to and and thus can be written as
was set to
(18) (19) (9)
where and stand for the determinant and the exponential of the trace of a matrix, respectively. In order to derive the density of conditioned on and , note that
We proceed as in [11] to solve (17). Note that the matrix is Hermitian positive definite. Consider one of its eigenvectors associated with the eigenvalue . Multiplying (17) by , we obtain
(10)
(20)
Using the independence of the is given by
’s, the conditional density of
(11)
, and is an eigenvalue associated with . As is a rank one Hermitian of positive matrix, there exists a unitary matrix such that Hence,
is necessarily an eigenvector of
(21) The conditional distribution of given the heterogeneity model (5), i.e.,
is defined by Thus, has two different eigenvalues, and the following equations: (12)
verifies one of
(22)
BIDON et al.: ADAPTIVE COHERENCE ESTIMATOR IS THE GENERALIZED LIKELIHOOD RATIO TEST
The first equation yields a value of
proportional to :
283
Plugging (33) in (24), we obtain the following expression for the MLE of : (23) (34)
The second equation is a second-order polynomial equation (24) where and . This equation has a unique real positive solution whose explicit form is not required for our analysis. The remaining derivations will use (24) only. The solution of the quadratic matrix equation (17) is thus given by
D. MLE of Noticing that is proportional to , see (33), and that the is constant [cf (34)], we have product (35) under
The MLE of
amounts to minimizing the quantity
(25) with (26) Finally, using (19) and (25), the MLE of on only through the matrix
(36) The minimum is well known to be achieved for [12] (37)
is shown to depend and is equal to (27)
(38) C. MLE of Let us denote
E. GLRT Statistic (28)
The
th root of the GLR can be expressed as (39)
Using the previous expressions (27) and (15), one obtains and hence the GLRT is the ACE defined in (2). (29) Then noticing that does not depend on [use (23) and (26)], the above expression can be simplified to
(30) Differentiating the logarithm of and equating the derivative to zero implies that the MLE of verifies
III. DETECTOR PERFORMANCES A. CFAR Behavior , the distribution of the We show in this section that under and , and hence that test statistic (2) is independent of ACE has the enjoyable CFAR property. We proceed as in [12] and consider the unitary matrix such that (40) (41)
(31)
Let us define (42)
Then, gathering the terms which depend on plying by , one obtains
and multi-
(43) (44) Then the test statistic can be rewritten as
(32) Using (24), we observe that the coefficient of zero. So the MLEs of and are proportional
(45)
is equal to Under (33)
,
has a complex normal distribution (46)
284
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
. Fig. 1 displays the probability of deas SNR runs, versus SNR for different tection , obtained from values of . From inspection of this figure, we observe that, as expected, the more homogenous the environment (i.e., the larger ), the better the detector performances. There is almost 3 dB difference between the case and the case which corresponds to the partially homogeneous model (3). This shows that the Bayesian environment (5) models a larger degree of heterogeneity than the deterministic environment. Furthermore, the figure shows that the heterogeneity level does not depend linearly on . Indeed when is small, a slight variation in results in a large variation of the homogeneity level. The trend is inverted for larger values of . IV. CONCLUSIONS Fig. 1. Probability of detection versus SNR—Influence of .
Conditioned on , , and , the matrix has a complex Wishart distribution with degrees of freedom
(47) Therefore, the distribution of
given
is
The adaptive coherence estimator is a well-known detection scheme which has proved to be effective in a number of nonhomogeneous environments. It is the UMPI test in partially homogeneous environments and is known to perform well also in compound-Gaussian clutter with fully correlated texture. In this letter, we showed that it is also the GLRT in nonhomogeneous environments such that the conditional distribution of is a complex inverse Wishart distribution with . This seems to indicate that ACE is (close to) optimum for a large class of nonhomogeneous environments, at least it is rather robust to covariance mismatches characterized by scaling ambiguities. This is to be contrasted with the marked selectivity of ACE with respect to steering vector mismatches. Indeed, ACE is known to possess strong rejection capabilities for signals whose signatures differ from the presumed ones. REFERENCES
(48) which is recognized as a multivariate -distribution. We first observe that in (41) is a fixed vector. Moreover, and are independent —since and are independent— and the distributions of in (46) and in (48) are parameter free. Since the test statistic in (45) depends only on , , and , it follows that its distribution under does not depend on or , which proves that ACE is CFAR. B. Numerical Simulations In this section, we study the influence of on the detector performances. The dimension of the observation space is set to , and training samples are available. The . The detector probability of false alarm is set to threshold is computed from Monte Carlo runs, with a different value of at each run, drawn from the conditional distribution in (12). The signal-to-noise ratio (SNR) is defined
[1] L. L. Scharf, Statistical Signal Processing: Detection, Estimation and Time Series Analysis. Reading, MA: Addison Wesley, 1991. [2] L. L. Scharf and B. Friedlander, “Matched subspace detectors,” IEEE Trans. Signal Process., vol. 42, no. 8, pp. 2146–2157, Aug. 1994. [3] L. L. Scharf and T. McWhorter, “Adaptive matched subspace detectors and adaptive coherence estimators,” in Proc. 30th Asilomar Conf. Signals Systems Computers, Pacific Grove, CA, Nov. 3–6, 1996, pp. 1114–1117. [4] S. Kraut and L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Process., vol. 47, no. 9, pp. 2538–2541, Sep. 1999. [5] S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherence estimator: A uniformly most powerful invariant adaptive detection statistic,” IEEE Trans. Signal Process., vol. 53, no. 2, pp. 427–438, Feb. 2005. [6] S. Kraut, L. L. Scharf, and T. McWhorter, “Adaptive subspace detectors,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 1–16, Jan. 2001. [7] C. D. Richmond, “Performance of a class of adaptive detection algorithms in nonhomogeneous environments,” IEEE Trans. Signal Process., vol. 48, no. 5, pp. 1248–1262, May 2000. [8] K. F. McDonald and R. S. Blum, “Exact performance of STAP algorithms with mismatched steering and clutter statistics,” IEEE Trans. Signal Process., vol. 48, no. 10, pp. 2750–2763, Oct. 2000. [9] E. Conte, M. Lops, and G. Ricci, “Asymptotically optimum radar detection in compound-Gaussian clutter,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, no. 2, pp. 617–625, Apr. 1995. [10] W. L. Melvin, “Space-time adaptive radar performance in heterogeneous clutter,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, no. 2, pp. 621–633, Apr. 2000. [11] S. Bidon, O. Besson, and J.-Y. Tourneret, “A Bayesian approach to adaptive detection in non-homogeneous environments,” IEEE Trans. Signal Process., vol. 56, no. 1, pp. 205–217, Jan. 2008. [12] E. J. Kelly, “An adaptive detection algorithm,” IEEE Trans. Aerosp. Electron. Syst., vol. 22, no. 1, pp. 115–127, Mar. 1986.
