Persymmetric Rao and Wald tests for adaptive detection of distributed targets in compound‐Gaussian noise
2016; Institution of Engineering and Technology; Volume: 11; Issue: 3 Linguagem: Inglês
10.1049/iet-rsn.2016.0251
ISSN1751-8792
AutoresXiaolu Guo, Haihong Tao, Hongyan Zhao, Jun Liu,
Tópico(s)Direction-of-Arrival Estimation Techniques
ResumoIET Radar, Sonar & NavigationVolume 11, Issue 3 p. 453-458 Research ArticleFree Access Persymmetric Rao and Wald tests for adaptive detection of distributed targets in compound-Gaussian noise Xiaolu Guo, Xiaolu Guo National Laboratory of Radar Signal Processing, Xidian University, Xi'an, People's Republic of China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this authorHaihong Tao, Haihong Tao National Laboratory of Radar Signal Processing, Xidian University, Xi'an, People's Republic of China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this authorHong-Yan Zhao, Hong-Yan Zhao National Laboratory of Radar Signal Processing, Xidian University, Xi'an, People's Republic of China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this authorJun Liu, Corresponding Author Jun Liu junliu@xidian.edu.cn National Laboratory of Radar Signal Processing, Xidian University, Xi'an, People's Republic of China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this author Xiaolu Guo, Xiaolu Guo National Laboratory of Radar Signal Processing, Xidian University, Xi'an, People's Republic of China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this authorHaihong Tao, Haihong Tao National Laboratory of Radar Signal Processing, Xidian University, Xi'an, People's Republic of China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this authorHong-Yan Zhao, Hong-Yan Zhao National Laboratory of Radar Signal Processing, Xidian University, Xi'an, People's Republic of China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this authorJun Liu, Corresponding Author Jun Liu junliu@xidian.edu.cn National Laboratory of Radar Signal Processing, Xidian University, Xi'an, People's Republic of China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this author First published: 01 March 2017 https://doi.org/10.1049/iet-rsn.2016.0251Citations: 8AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The problem of detecting a distributed target in the presence of compound-Gaussian noise with unknown covariance matrix is studied in this paper. Since no uniformly most powerful test exists for the problem at hand, two detectors based on the Rao and Wald tests are devised. Remarkably, the persymmetric structure of the covariance matrix is exploited in the design of the proposed detectors. Simulation results show that the proposed detectors outperform the traditional detectors, especially in training-limited scenarios. 1 Introduction The problem of adaptively detecting distributed targets has drawn considerable attention recently [1-4]. A high-resolution radar (HRR) can resolve a target into a group of scattering centres, depending on the range extent of the target and the range resolution capabilities of the radar. Moreover, it is well known that the point-like target model may fail in many practical scenarios wherein a low/medium resolution radar is employed [5]. When the target is separated into a number of range cells, it is referred to as a distributed or range-spread target. Much work has been directed toward the detection of distributed targets. For example, this detection problem has been addressed in white Gaussian noise of known spectral level [6] or known spectral structure [7]. In [8], Liu et al. examine the detection of distributed targets in the presence of subspace interference and Gaussian noise. Additionally, a model-based adaptive detection approach for the range-spread targets has been proposed in [9]. Most solutions to the distributed target detection assume a homogeneous environment, i.e. the training samples (secondary data) used for adaptation share the same covariance matrix as the noise in the test cell (primary data). Note that the homogeneity might not be met in realistic scenarios [3, 10]. Therefore, the target detection in a non-homogeneous background is actually a problem of primary concern. In fact, the Gaussian assumption can be no longer met in HRRs and/or at low grazing angles. More specifically, the noise is usually described as a compound-Gaussian process [11]. This process is mathematically expressed as the product of a texture and a speckle. The texture is a temporally and spatially 'slowly varying' component, accounting for the reflectivity of the illuminated patch. While speckle is a 'more rapidly' varying process [12, 13]. Note that the compound-Gaussian distribution includes as special cases many distributions such as Weibull, log-Weibull and K-distribution [14, 15]. Recently, many studies have been discussed on the distributed target detection in the compound-Gaussian noise. On the basis of the generalised likelihood ratio test (GLRT) principle, adaptive detection schemes of range-spread targets in the compound-Gaussian noise have been addressed partly in [16-18]. Furthermore, [19] provided the performance analysis of GLRT-based adaptive detector for distributed targets in the compound-Gaussian noise. The Rao and Wald tests for detecting distributed targets in the compound-Gaussian noise has been discussed in [5], where the texture is modelled as an unknown determinate parameter. Some other related works can be seen in [20-22] and the references therein. In these studies, sufficient training data are usually assumed to be available. On the other hand, there is a requirement of reducing the secondary data in adaptive detection. In [23], the Bayesian Rao and Wald tests are proposed when only a small number of training data is available. Both [24, 25] derived Bayesian tests in non-homogeneous environments or heterogeneous noise. In [26], a Bayesian space time adaptive programming (STAP) algorithm is developed with knowledge-aided in heterogeneous noise. It is worth noting that the demanding requirement on the number of secondary data maybe prohibitive in realistic scenarios. A possible way to reduce the sample requirement is to exploit the structural information of the noise covariance matrix. In radar system, the noise covariance matrix has a persymmetric property if a linear array is symmetrically spaced with respect to the phase centre or the pulse train is uniformly spaced [27]. Several detection schemes that take into account the persymmetric property are developed [28-33]. A GLRT adaptive detector with persymmetric property in homogeneous environments is proposed in [28]. In partially homogeneous environments, the GLRT, Rao and Wald tests have been proposed with exploiting persymmetry [29, 30]. For compound-Gaussian noise with unknown deterministic texture, a persymmetric normalised matched filter detector is given in [32, 33]. A persymmetric GLRT detector in compound-Gaussian noise with random texture has been proposed in [10]. Note that the persymmetric detectors in [10, 28-33] are designed for the detection of point-like targets. A persymmetric detector for distributed targets in a partially homogeneous environment is presented in [31]. An invariant approach which deals with detection in Gaussian noise with a persymmetric covariance is proposed in [34]. Persymmetric Rao and GLRT detectors are devised for point-like target in Gaussian noise [35]. Nevertheless, no persymmetric detector has been proposed for the distributed target detection problem in compound-Gaussian noise. In this paper, we propose two persymmetric detectors based on the Rao and Wald tests for the problem of detecting a distributed target in compound-Gaussian noise. The texture component in the compound-Gaussian noise is assumed to follow an inverse-Gamma distribution. This assumption has been verified in the real data [36]. Note that the persymmetric structure has been employed in our proposed detectors. Simulation results show that the proposed persymmetric detectors outperform the counterparts, especially in training-limited scenarios. Note that the Rao and Wald tests are widely used, which are asymptotically equivalent to the GLRT. Compared with the GLRT, the Rao and Wald tests have low computation load, since they need less unknown parameters to estimate. The rest of this paper is organised as follows. In Section 2, we state the problem to be addressed and introduce the distributed target and the noise models. In Section 3, we derive the adaptive detectors based on persymmetric Rao and Wald tests. Subsequently, the performance evaluations are given in Section 4. Finally, Section 5 is the discussion and conclusions. 2 Signal model We deal with the problem of detecting the presence of a target across H range cell. As customary, we define the data set as primary data, which include the entire target, while we define, as secondary data set, which is free of the target and share the same structure of the covariance matrix as the primary data. The detection of a distributed target in compound-Gaussian noise can be formulated in terms of the following binary hypotheses test: (1) where is an unknown target amplitude, is an N × 1-dimensional nominal steering vector with N denoting the dimension of the received data, represents a compound-Gaussian noise vector represented as the following product model: (2) where denotes the speckle component accounting for local backscattering and stands for the texture component, representing the variations of the local reflected power. In (2), is a zero-mean complex Gaussian vector with covariance matrix R and is a non-negative random variable that is assumed to follow an inverse-Gamma distribution [37]. The probability density function (PDF) of is given by (3) where and are shape and scaling parameters, respectively. 3 Detection scheme 3.1 Step 1: Exploiting the persymmetric property In this section, we investigate the detection problem of distributed targets in compound-Gaussian noise, by exploiting the persymmetric structure to improve the detection performance. Note that the persymmetry exists in the covariance matrix and the steering vector, when the system is equipped with a symmetrically spaced linear array or symmetrically spaced pulse trains [28]. The covariance matrix with the persymmetric property means (4) where denotes the complex conjugate operator and J is the permutation matrix whose cross-diagonal elements are ones and others are zeros. In addition, the persymmetry in the steering vector means that . On the basis of these persymmetric properties, we introduce a unitary matrix to transform the persymmetric structures to real ones. Proposition 1 [38].Let I be the identity matrix and T be the unitary matrix defined as (5) Persymmetric vectors and Hermitian matrix are characterised by the properties: (i) v is persymmetric vector if and only if Tv is a real vector and (ii) R is a persymmetric Hermitian matrix if and only if is a real symmetric matrix, where is the conjugate transpose operator. Define (6) According to Proposition 1, is a zero-mean complex Gaussian vector with a real covariance matrix . Therefore, the problem (1) is equivalent to (7) In the following, we propose two adaptive detectors according to the Rao and Wald tests by exploiting the persymmetry. 3.2 Step 2: The persymmtric Rao and persymmtric Wald tests For simplicity, we denote by is a 2H-dimensional vector, where and are the real and imaginary parts of , respectively; T denotes the transpose operator; is a H-dimensional vector, where are independent random variables; and is a 3H-dimensional vector. 3.2.1 Rao test The Rao test can be obtained by (8) where is the natural logarithm and is the partial derivative with respect to ; stands for the detection threshold to be set in order to ensure the desired probability of false alarm ; denotes the PDF of the data under H1 hypothesis; J(θ) = J(θr, θs) is the Fisher information matrix which can be partitioned as (9) ; and is maximum likelihood (ML) estimate of θ under H0 hypothesis. For obtaining the Rao test, we have to specify the PDF of and evaluate its gradient with respect to . Previous assumptions imply that the aforementioned PDF can be written as (10) where denotes the determinant. Moreover, it can be shown that (11) (12) where and denote the real and imaginary parts of the argument, respectively. Hence was can obtain (13) as shown at the bottom of this page. As to the blocks of the Fisher information matrix, we have where 0 denotes a matrix of zeros, stands for a square diagonal matrix with the elements of a given vector on the diagonal. It follows that: (14) Moreover (15) with is the value of which maximises the argument. Recall the definition of , we can obtain . Inserting (13), (14) and into (8), after some algebra, yields (16) where is the appropriate modification of the original threshold in (8). In practice, the covariance matrix is always unknown and should be estimated from the secondary data. Now, we utilise a persymmetric FP covariance estimator to finish the design of the Rao test. Specifically, the persymmetric FP covariance matrix can be obtained by [32] (17) Substituting into (16), we can write the Rao test as (18) For ease of reference, the proposed Rao test is called the Per-Rao test. 3.2.2 Wald test The Wald test can be obtained as (19) where stands for the detection threshold to be set in order to ensure the desired and is the ML estimate of θ under H1 hypothesis. For our problem, can be obtained as (20) shown at the bottom of this page, where (21) The estimate of under H1 hypothesis can be obtained as (22) shown at the bottom of the next page. Recall that . We can obtain by substituting (20) and (22) into the definition. Inserting and (14) into (19), we conclude that (23) where is the appropriate modification of the original threshold in (19). The persymmetric Wald test can be obtained by substituting into (23), i.e. (24) For ease of reference, the proposed Wald test is called the Per-Wald test. 4 Performance assessment In this section, we present numerical examples to show the performance of the proposed Per-Rao and Per-Wald tests. For comparison, we consider the traditional Rao test [5], traditional Wald test [5] and the generalised adaptive subspace detector (GASD) [1]. The analysis is carried out by resorting to the standard Monte Carlo counting techniques. As to the speckle component, it is modelled as an exponentially-correlated complex normal vector with one-lag correlation coefficient with . The signal-to-noise ratio (SNR) is defined as . We set , where fd = 0.1 and in the following simulations. In Fig. 1, the Pd's of the proposed persymmetric detectors versus SNR are plotted with N = 8 and H = 4 for different K. It is shown that: (i) the proposed detectors obviously outperform the traditional detectors especially in the case of limited training data; (ii) the detection performance of the Per-Rao test is slightly better than the performance of the Per-Wald test; and (iii) the performance of all detectors improves as K increases. When the training data is sufficient, all the considered detectors have close detection performance. This is expected, because the accuracy of the covariance matrix estimate can be achieved at a very high level with sufficient training data, even without exploiting the persymmetric structure. Fig. 1Open in figure viewerPowerPoint Pd versus SNR of the Per-Rao (18), Per-Wald (24), the GASD, the GLRT, the Rao and Wald tests for N = 8, H = 4 (a) K = 8, (b) K = 16, (c) K = 24 Furthermore, Fig. 2 presents the Pd against K for the Per-Rao, Per-Wald, Rao and Wald detectors for SNR = 8 dB, N = 8 and H = 4. The proposed Per-Rao and Per-Wald tests demonstrate good performance in low size of training data scenario. Nevertheless the traditional Rao and Wald tests would become invalid when K varies during the interval [N/2, N]. Fig. 2Open in figure viewerPowerPoint Pd versus K of the Per-Rao (18), Per-Wald (24), the Rao and Wald tests for SNR = 8 dB, N = 8 and H = 4 In Fig. 3, we plot contours of constant detection probability for the Per-Rao, Per-Wald and Per-GLRT algorithms. Such plots are referred to as mesa plots [30]. The curves in this figure show the probability of detecting a signal deteriorates as the degree of the mismatch increases, where the definition of the mismatch angle θ is (25) with θ denoting the angle in the whitened space between the nominal steering vector v used in the detectors and the actual steering vector vm. Fig. 3Open in figure viewerPowerPoint Contours of constant Pd for K = 8 (a) H = 1(point-like target), (b) H = 2 (distributed target) From Fig. 3a, we observe that, the curves of the Per-Rao and Per-Wald almost coincide. It means that, in point-like target situation, those two proposed detectors have the same selectivity. Secondly, the Per-Rao and Per-Wald tests are more robust than the Per-GLRT [10]. From Fig. 3b, it can be seen in the case of distributed target that the Per-Wald ensures the better performance than the Per-Rao test in terms of rejection of mismatched signals. Figs. 4a and b depict the detection performance of the Per-Rao and Per-Wald tests for different values of shape parameter, respectively. It is indicated that the two proposed detectors have better performance with higher shape values. Fig. 4Open in figure viewerPowerPoint Pd versus SNR of the Per-Rao (18), Per-Wald (24) for N = 8, H = 4 and K = 8 (a) Per-Rao, (b) Per-Wald 5 Conclusion In this paper, we have considered the problem of detecting a distributed target in the presence of compound-Gaussian noise with unknown covariance matrix, where the texture component is assumed to follow an inverse-Gamma distribution. We proposed two adaptive detectors (i.e. the Per-Rao and Per-Wald tests) by exploiting the persymmetric structure. Simulation results show that the Per-Rao and Per-Wald detectors outperform their counterparts, especially in training-limited scenarios. In this paper, we assume that the texture component follows an inverse-Gamma distribution. In the future, we will examine other situations where the texture component has different distributions. 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