1086
IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 11, NOVEMBER 2013
Rao and Wald Tests for Distributed Targets Detection With Unknown Signal Steering Weijian Liu, Wenchong Xie, and Yongliang Wang, Member, IEEE
Abstract—In this letter, we consider the problem of detecting a distributed target with unknown signal steering in the Gaussian noise. We derive the Rao and Wald tests. It is found that the Rao test coincides with the so-called modified two-step generalized likelihood ratio test (M2S-GLRT), while the Wald test is equivalent to the plain two-step GLRT (2S-GLRT). We also give some intuitive interpretations about the Rao and Wald tests, as well as other existing detectors. Index Terms—Distributed targets, Rao test, unknown signal steering, Wald test.
I. INTRODUCTION
A
COMMON problem in signal processing is the detection of a signal in the background of colored noise with unknown statistical properties. This is an active field of research ever since Kelly introduced his famous generalized likelihood ratio test (GLRT) [1]. Other well-known detectors include the adaptive matched filter (AMF) [2], [3], adaptive coherence estimator (ACE) [4], adaptive beamformer orthogonal rejection test (ABORT) [5], De Maio’s Rao test [6], double-normalized AMF (DN-AMF) [7], [8], adaptive energy detector (AED) [9], and so on. All these detectors are devised for the detection of point targets. In some situation of practical interest, the targets are naturally distributed, or the signals are multiband; refer to [10] for more details. Otherwise stated, the signals are multidimensional. The receivers for distributed targets (or multidimensional signals) include the multiband GLRT [11] (MBGLRT), generalized adaptive subspace detector (GASD) [12], generalized AMF (GAMF) [12], generalized adaptive direction detector (GADD) [13], and others [14], [15]. Among the problems of distributed target detection, one class is that the signature of the signal is completely unknown. In this situation, several detectors are derived in [10]; extensive simulation analysis is given for the Gaussian random signals therein. Another significant contribution of [10] is that it is shown there is no uniformly most powerful invariant (UMPI) detector. Hence, it is reasonable to design different detectors Manuscript received April 09, 2013; revised June 09, 2013; accepted July 30, 2013. Date of publication August 07, 2013; date of current version September 16, 2013. This work was supported in part by the National Natural Science Foundation of China under Grants 61102169 and 60925005. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Keith Davidson. W. Liu is with the College of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China (e-mail:
[email protected]). W. Xie and Y. Wang are with the Wuhan Radar Academy, Wuhan 430019, China (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LSP.2013.2277371
with improved detection performance, lower computational complexity, or with other valuable features. This is the main motivation of this letter. Precisely, we resort to the Rao and Wald tests to design constant false alarm rate (CFAR) adaptive detectors for the same problem in [10]. Surprisingly, it is found that the Rao test is identical to the modified two-step GLRT1 (M2S-GLRT) in [10], while the Wald test coincides with the plain two-step GLRT (2S-GLRT) in [10]. Moreover, we provide some further insights into the Rao and Wald tests and other existing detectors. This letter is organized as follows. Section II shows the problem formulation. Section III presents the Rao test, while Section IV exploits the Wald test. Further investigations about the Rao and Wald tests, as well as other existing detectors, are given in Section V. Finally, Section VI summarizes the letter. II. PROBLEM FORMULATION Given an -dimensional data matrix , we want to discriminate between hypothesis that contains only colored noise and hypothesis that contains colored noise and a useful signal . The colored noise (including clutter and white noise) in each column of is independent and identically distributed (IID), zero-mean, complex circular Gaussian random vector, with a positive definite covariance matrix . As customary, a secondary data matrix , of dimension , is available, which only contains noise and shares the same statistical property with the primary data. Summarizing, the problem to be solved can be formulated in terms of the following binary hypothesis test: (1) As in [10], we assume that both the signal matrix covariance matrix are unknown.
and the
III. THE RAO TEST Suppose
is the parameter vector, namely, (2)
, and , the notation where is the stands for vectorization operator [16]. Note that so-called nuisance parameter. 1The 2S- or M2S- GLRT has the following two steps. First, it is assumed that the covariance matrix is known and the corresponding GLRT is derived based on the primary data. Then, the unknown covariance matrix is replaced with a proper estimate based on the secondary (or both the secondary and primary) data [2], [3], [12]. The 2S- or M2S- Rao and Wald tests have the similar manners.
1070-9908 © 2013 IEEE
LIU et al.: RAO AND WALD TESTS FOR DISTRIBUTED TARGETS DETECTION
The Fisher information matrix (FIM) for real-valued signal is well-known; see, for example, [17]. For complex-valued signal, the FIM is given in [18], described as
1087
under hypothesis ; substituting it into (10) ignoring the constant scalar, yields the final Rao test The
MLE
of
is and (11)
(3) which is called the M2S-GLRT in [10]. Then we partition the FIM
as follows:
IV. THE WALD TEST (4)
The Rao test for complex-valued signal is given by (5) [19], is the maximum shown at the bottom of the page, where likelihood estimate (MLE) of under hypothesis . The joint probability density function (PDF) of and , under hypothesis , is found to be
For complex-valued signal the Wald test is found to be [19] (12) is the MLE of under hypothesis where of under hypothesis , and complement of , evaluated at , namely,
is the value is the Schur
(13) is found to be a null matrix, it follows that . Taking the derivative of (6) w.r.t. and equating the result to zero, produces the MLE of Since
(6) where is times the sample covariance matrix (SCM) based on the secondary data. According to the knowledge of the complex-valued matrix derivatives [16], we have the following two results (7)
(14) Plugging (14) and (9) into (12), yields the intermediate Wald test, for fixed
(8) (15) A substitution of (7) and (8) into (3), yields Inserting (14) into (6), then taking the derivative w.r.t. , and equating the result to zero, yields the MLE of under hypothesis (16) (9) where , the notation stands for the Kronecker product [16]. Taking the derivative of (7) with respect to (w.r.t.) , then performing the expectation over under hypothesis and setting , yields the fact that is a null matrix. As a consequence, we have . Setting in (7) and (8), then inserting them and (9) into (5), yields the Rao test for given (10)
Plugging (16) into (15), and dropping the constant scalar, yields the final Wald test (17) which is derived in [10] according to the 2S-GLRT design criterion. V. FURTHER INVESTIGATIONS OF THE DETECTORS This section is devoted to further investigation about the Rao and Wald tests, as well as the GLRT and the so-called spectral
(5)
1088
IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 11, NOVEMBER 2013
norm test (SNT) introduced in [10]. For clarity, the statistics of the GLRT and the SNT are given below (18) (19) , and denotes the -th nonzero where eigenvalue of the corresponding matrix argument. It is shown in [10], for random signal array , when the signal energy is uniformly distributed among the columns of (referred to as the uniform signal model for convenience), the GLRT, Rao and Wald tests roughly exhibit the same probabilities of detection (PD’s), provided the number of the secondary data is large. On the other hand, when the number of the secondary data is small, the Rao test exhibits the best detection performance. Furthermore, for uniform signal model, the SNT has the least PD, irrespective of the number of the secondary data. In contrast, when most of the signal energy of signal array is occupied by one component (denoted as dominant signal model for simplicity), the SNT provides the highest PD, both in the cases of low and high sample support. Besides, for dominant signal model, the Rao test has the least detection performance. It is worth pointing out that for nonrandom signal array , the above remarks are still valid. Due to space limitations, it is not shown here. Now we proceed to do some theoretical analysis about the detectors, which can give some interpretation about the detectors’ behavior mentioned above. Defining , and arranging the eigenvalues of in decreasing order . Then we can rewrite the Rao test as (20) where we have used the matrix inversion lemma and the spectral mapping theorem [20]. In a similar fashion, we can cast the Wald test and SNT in the following equivalent forms (21) (22) In [10], it is shown that the GLRT can be expressed as (23) It is helpful to introduce the optimum matched filter (OMF) for nonrandom signal array and known covariance matrix (24) which is referred to as the asymptotic GLRT (AS-GLRT) in [10]. In (24), is the th eigenvalue of the matrix .
These equivalent forms above are very powerful. According to them, we can gain some useful insights into the detectors. • From (20), (21), and (23), we see that the Rao test, Wald test, and the GLRT all approach the OMF, as the number of the secondary data is large enough, written symbolically as, , respectively. is also shown in [10]. This explains why as the number of the secondary data becomes large, the GLRT, Rao and Wald tests nearly have the same PD’s as the OMF. • Using the SNT’s equivalent form (22), we can give the reason why the SNT has the least PD for the uniform signal model. In fact, for uniform signal model, all eigenvalues ’s have roughly the same values, and they provide nearly the same information. Thus, if one only chooses the biggest eigenvalue as a detector, it inevitably brings certain performance loss. • According to the Rao test’s equivalent form (20), we can also explain why the Rao test exhibits the highest PD for uniform signal model when the number of the secondary data is small. In this case, there will be some relatively large perturbation about the eigenvalues w.r.t. . Only the Rao test weakens the perturbation effect through the normalization factor . • In light of (22), we can show why the SNT provides the highest PD for dominant signal model, especially when the number of the secondary data is large. In this situation, the value of is very larger than others. Besides, since the covariance matrix is unknown, it needs to be estimated. There unavoidably introduces perturbation about the eigenvalues. The smaller the eigenvalue is, the larger the perturbation it has. Therefore, it is reasonable to choose the biggest eigenvalue as a detector. In contrast, the Rao test reduces the effect of the biggest eigenvalue , thus this results in the least detection performance. Before closing this section, we would like to give the following four remarks. • When , all the detectors coincide with the detector , which is the GLRT for point target [9]. • A notable feature about the Rao test is that it can work even the SCM is singular, provided the following inequality holds . This is due to the fact that the matrix in (11) is invertible with probability 1, if the inequality holds. • Both the two-step Rao test and the two-step Wald test coincide with the Wald test. One can verify this result by substituting the SCM into (10) and (15), respectively. • All the detectors possess the CFAR property, which is proved by the principle of invariance in [10]. In fact, if one proves that, under hypothesis , the matrix is not dependent on the covariance matrix , then the CFARness follows. We do this by rewrite as , where , and is the square-root matrix of . Under hypothesis , the columns of are distributed as zero-mean complex circular Gaussian random vectors with covariance
LIU et al.: RAO AND WALD TESTS FOR DISTRIBUTED TARGETS DETECTION
matrix ; and is ruled by a complex Wishart matrix with degrees of freedom (DOF’s), and a covariance matrix . This completes the proof. VI. CONCLUSION In this letter, we have derived the Rao and Wald tests for the problem of detection of a distributed target with unknown signature. Some useful interpretations about the Rao and Wald tests, as well as the GLRT and SNT are given. Since no UMPI detector exists, it seems that if the characteristic of the signal energy distribution is known in advance, one can reasonably chooses the Rao test (for uniform signal model) or the SNT (for dominant signal model). REFERENCES [1] E. J. Kelly, “An adaptive detection algorithm,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-22, no. 1, pp. 115–127, 1986. [2] F. C. Robey et al., “A CFAR adaptive matched filter detector,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 1, pp. 208–216, 1992. [3] A. De Maio, “A new derivation of the adaptive matched filter,” IEEE Signal Process. Lett., vol. 11, no. 10, pp. 792–793, 2004. [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, 1999. [5] N. B. Pulsone and C. M. Rader, “Adaptive beamformer orthogonal rejection test,” IEEE Trans. Signal Process., vol. 49, no. 3, pp. 521–529, 2001. [6] A. De Maio, “Rao test for adaptive detection in Gaussian interference with unknown covariance matrix,” IEEE Trans. Signal Process., vol. 55, no. 7, pp. 3577–3584, 2007. [7] D. Orlando and G. Ricci, “A Rao test with enhanced selectivity properties in homogeneous scenarios,” IEEE Trans. Signal Process., vol. 58, no. 10, pp. 5385–5390, 2010.
1089
[8] W. Liu, W. Xie, and Y. Wang, “A Wald test with enhanced selectivity properties in homogeneous environments,” EURASIP J. Adv. Signal Process., vol. 2013, no. 14, 2013. [9] R. S. Raghavan, H. F. Qiu, and D. J. Mclaughlin, “CFAR detection in clutter with unknown correlation properties,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, no. 2, pp. 647–657, 1995. [10] E. Conte, A. De Maio, and C. Galdi, “CFAR detection of multidimensional signals: An invariant approach,” IEEE Trans. Signal Process., vol. 51, no. 1, pp. 142–151, 2003. [11] H. Wang and L. Cai, “On adaptive multiband signal detection with GLR algorithm,” IEEE Trans. Aerosp. Electron. Syst., vol. 27, no. 2, pp. 225–233, 1991. [12] E. Conte, A. D. Maio, and G. Ricci, “GLRT-based adaptive detection algorithms for range-spread targets,” IEEE Trans. Signal Process., vol. 49, no. 7, pp. 1336–1348, 2001. [13] O. Besson, L. L. Scharf, and S. Kraut, “Adaptive detection of a signal known only to lie on a line in a known subspace, when primary and secondary data are partially homogeneous,” IEEE Trans. Signal Process., vol. 54, no. 12, pp. 4698–4705, 2006. [14] C. Hao et al., “Adaptive detection of distributed targets in partially homogeneous environment with Rao and Wald tests,” Signal Process., vol. 92, pp. 926–930, 2012. [15] C. Hao, J. Yang, and X. Ma et al., “Adaptive detection of distributed targets with orthogonal rejection,” IET Radar, Sonar Navig., vol. 6, no. 6, pp. 483–493, 2012. [16] A. Hjørungnes, Complex-Valued Matrix Derivatives: With Applications in Signal Processing and Communications. New York, NY, USA: Cambridge Univ. Press, 2011. [17] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory. Englewood Cliffs, NJ, USA: Prentice-Hall, 1998. [18] S. Pagadarai, A. M. Wyglinski, and C. R. Anderson, “An evaluation of the Bayesian CRLB for time-varying MIMO channel estimation using complex-valued differentials,” in Proc. IEEE Pacific Rim Conf. Communications, Computers and Signal Processing, Victoria, BC, Canada, 2011, pp. 818–823. [19] W. Liu, W. Xie, and Y. Wang, “Fisher information matrix, Rao test, and Wald test for complex-valued signals and their applications,” Signal Process., vol. 94, pp. 1–5, 2013. [20] S. Treil, Linear Algebra Done Wrong, 2009 [Online]. Available: http:// www.math.brown.edu/~treil/papers/LADW/LADW.pdf
