On Persymmetric covariance matrices in adaptive detection

ON PERSYMMETRIC COVARIANCE MATRICES IN ADAPTIVE DETECTION G. Pailloux12 , P. Forster2, J.P. Ovarlez1 and F. Pascal3, 1 ONERA - DEMR/TSI, Chemin de la Hunière, F-91120 Palaiseau, France 2 GEA, 1 Chemin Desvallières, F-92410 Ville d’Avray, France 3 SATIE, ENS Cachan, CNRS, UniverSud, 61, av President Wilson, F-94230 Cachan, France ABSTRACT In the general area of radar detection, estimation of the clutter covariance matrix is an important point. This matrix commonly exhibits a persymmetric structure: this is the case for instance for active systems using a symmetrically spaced linear array or pulse train. In this context, this paper provides a new Gaussian adaptive detector called the Persymmetric Adaptive Matched Filter (P-AMF). Its theoretical distribution is derived allowing adjustment of the detection threshold for a given Probability of False Alarm (PFA). Simulations results highlight the improvement in term of probability of detection (PD) of the P-AMF in comparison with the classical Adaptive Matched Filter (AMF). Index Terms— Adaptive signal detection, Parameter estimation, Maximum Likelihood Estimation, Covariance matrices, Radar detection. 1. INTRODUCTION (GLRT) is referred to as the Optimum Gaussian Detector (OGD: |pH M−1 y|2 H1 ≷ λ, (2) ΛOGD = H −1 p M p H0 where the detection threshold λ is related to the PFA by: λ = p − ln(Pf a ). However, the clutter covariance matrix M is generally unknown and has to be estimated. For that purpose, the Maximum Likelihood theory provides the well-known Sample Covariance Matrix (SCM) built from the secondary data and defined by: K X b SCM = 1 yk yH M k . K (3) k=1 b SCM for M in (2) leads to the so-called Then, substituting M Adaptive Matched Filter (AMF) test [1]: ΛAMF = 2 H1 b −1 |pH M SCM y| ≷ λ. −1 b SCM p H0 pH M (4) One of the major problems in radar detection consists in detecting a known signal p ∈ Cm corrupted by an additive clutter c. Classically, this problem can be stated as the following binary hypothesis test:  H0 : y = c, yk = ck , for 1 ≤ k ≤ K , (1) H1 : y = A p + c, yk = ck , for 1 ≤ k ≤ K , The AMF has the Constant False Alarm Rate (CFAR) property and its distribution is known (see e.g. [1, 2, 3]). However, the AMF exhibits a detection loss in comparison with the b SCM defined by (3) does not take into acOGD. Moreover, M count any prior knowledge on the clutter covariance structure. where y is the complex m-vector of the received signal, A is an unknown complex target amplitude and c is a complex zero-mean Gaussian m-vector with covariance matrix M = E[c cH ]. Under both hypotheses, it is assumed that K signalfree data yk are available for clutter parameters estimation. The yk ’s are the so-called secondary data. They are independent and identically distributed (i.i.d) with the same distribution as c. In the sequel, the real (resp. complex) Gaussian distribution with zero-mean and covariance matrix M is denoted by N (0, M) (resp. CN (0, M)), E[.] stands for the expectation operator, H denotes the conjugate transpose, ∗ the conjugate and ⊤ the transpose operator, k . k is the usual L2 -norm, Im is the m-th order identity matrix and ∼ means ”distributed as”. When M is known, the Generalized Likelihood Ratio Test M = Jm M∗ Jm , Many applications can result in a clutter covariance matrix that exhibits some particular structure. Such a situation is frequently met in radar systems using a symmetrically spaced linear array for spatial domain processing, or symmetrically spaced pulse train for temporal domain processing [4, 5]. In these systems, the clutter covariance matrix M has the persymmetric property: (5) where Jm is the m-dimensional antidiagonal matrix having 1 as non-zero elements. The signal vector is also persymmetric, i.e. it satisfies: p = Jm p∗ . (6) The persymmetric structure of M can be exploited to improve its estimation accuracy compared to the SCM. This approach has been followed by [4] in the framework of Kelly’s GLRT [6]. Our paper is based on another widespread detection test, the AMF. It studies its statistical performance when using in (4) the persymmetric Maximum Likelihood (ML) estimate of the clutter covariance matrix instead of the SCM. The resulting detection test is called P-AMF for Persymmetric Adaptive Matched Filter. One of the main contribution is the derivation of the P-AMF distribution under hypothesis H0 . This allows the theoretical setting of the detection threshold for a given PFA. This paper is organized as follows. Section 2 shows how the persymmetric structure of the covariance matrix can be exploited to provide the new P-AMF. Section 3 derives the statistical distribution of the P-AMF under hypothesis H0 . Section 4 presents simulation results which illustrate the improvement in terms of detection performance of the P-AMF on the conventional AMF. 2. PROBLEM STATEMENT AND PRELIMINARIES In the context of persymmetric M and p, the problem defined by (1) will be first reformulated thanks to the following proposition. Proposition 2.1 Let T be the unitary matrix defined as: T=                √1 2   Im/2 iIm/2 Jm/2 −iJm/2 I(m−1)/2 √1  0 2 iI(m−1)/2   J(m−1)/2 √0  2 0 0 −iJ(m−1)/2 From proposition 2.1, the transformed signal vector s and the transformed clutter covariance matrix are both real. Then, the original problem (1) is equivalent to:  H0 : x = n xk = nk , for 1 ≤ k ≤ K , (8) H1 : x = A s + n xk = nk , for 1 ≤ k ≤ K , where x ∈ Cm , n ∼ CN (0, R), s is a known real vector, R is an unknown real symmetric matrix. The K transformed secondary data xk are i.i.d and share the same CN (0, R) distribution as n. From now on, the problem under study is the problem defined by (8). Let us now investigate the ML estimate of the real covariance matrix R from the K secondary data xk . The main motivation for introducing the transformed data is that the resulting distribution of the ML estimate of R is very simple. This was not the case in [7] when dealing with the original secondary data yk with persymmetric covariance matrix. Proposition 2.2 b of real matrix R is unbiased and is given The ML estimate R by: b = Re(R b SCM ) , R (9) where Re(.) stands for the real part and where: K X H b SCM = 1 b R xk xH k = T MSCM T . K (10) k=1 for m even for m odd. (7) Persymmetric vectors and Hermitian matrices are characterized by the following properties: • p ∈ Cm is a persymmetric vector if and only if T p is a real vector. • M is a persymmetric Hermitian matrix if and only if T M TH is a real symmetric matrix. Proof 2.1 The proof is straightforward and involves elementary algebraic manipulations. Using previous proposition, the original problem (1) can be equivalently reformulated as follows. Let us introduce the transformed primary data x, the transformed secondary data xk , the transformed clutter vector n, the transformed signal vector s and the transformed clutter covariance matrix R defined as: x = T y, xk = T yk , s = T p, n = T c, nk = T ck H and R = E(n nH ) = E(nk nH k ) = TMT . b is real Wishart distributed with 2K degrees of freedom. 2KR Proof 2.2 The proof is straightforward and is omitted. Actually, taking into account the real structure of R (or equivalently the persymmetric structure of M) in the ML estimation procedure allows to virtually double the number of secondary data. Let us now consider the AMF for the detection problem b defined by (9). This leads to the (8) based on the estimate R following detection test, called the P-AMF, ΛP AMF = −1 b x|2 H1 |s⊤ R ≷ λ, b −1 s H0 s⊤ R (11) or equivalently, in terms of the original data, ΛP AMF = b SCM TH )]−1 Ty|2 H1 |pH TH [Re(TM ≷ λ . (12) b SCM TH )]−1 Tp H0 pH TH [Re(TM b are real in (11), while x is complex. The Note that s and R aim of this paper is twofold: • to derive the statistical distribution of ΛP AMF under hypothesis H0 in order to set the detection threshold λ for a given PFA; • to emphasize the improvement of the P-AMF on the AMF in terms of probability of detection. 3. STATISTICAL ANALYSIS OF THE P-AMF The purpose of this section is to derive the statistical distribution of the P-AMF and, as a consequence, to establish the relationship between the PFA and the detection threshold λ. Proposition 3.1 • Under H0 , the Probability Density Function (PDF) of ΛP AMF , defined by (11), is: (2K − m + 1) (2K − m + 2) 2 K (2K + 1)   2K − m + 3 2K − m + 4 2K + 3 z , , ; ;− × 2 F1 2 2 2 K (13) where 2 F1 is the hypergeometric function [8] defined by Z 1 b−1 Γ(c) t (1 − t)c−b−1 dt . 2 F1 (a, b; c; λ) = Γ(b)Γ(c − b) 0 (1 − tλ)a p(z) = • The relationship between the PFA and the detection threshold λ is thus:   2K − m + 1 2K − m + 2 2K + 1 λ P F A = 2 F1 . , ; ;− 2 2 2 K Proof 3.1 Due to limited space, the proof is just outlined. Let e1 = (1, 0, ..., 0)⊤ , let R = R1/2 R⊤/2 be a standard factorization of R and let Q be a real unitary matrix such that e1 = Q R−1/2 s. Let us set ′ vector u′1,k must satisfy u′⊤ 1,k uk,k + u1,k+1 uk+1,k+1 = 0 , for 1 < k < m. This leads to: u′2 1,k+1 = ′ 2 2 |u′⊤ 1,k uk,k | k u1,k k k u′1,k k2 u2k+1,k+1 = αk k u′1,k k2 u2k+1,k+1 . (15) Conditioned to u′1,k , αk is χ21 -distributed and is independent of u′1,k . By using the relation U−1 e1 = u′1,1 e1 , the random variable b can be written as: −1 −⊤ −1 2 k =k u′1,m k2 . b = e⊤ U e1 =k e⊤ 1 U 1 U (16) Now, notice that k u′1,m k2 =k u′1,m−1 k2 +u′2 1,m . From (15), one has: !   m Y α 1 α m−1 k−1 b =k u′m−1 k2 1 + 2 = 2 , 1+ 2 um,m u1,1 uk,k k=2 where αk are all independent, are independent of u2k,k and −1 
∼ β1 2K−m+k , 12 are χ21 -distributed. Since 1 + αuk−1 2 2 k,k and m Y k=2 1 β1 2K−m+k 1 ,2 2
∼ 1 β1 2K−m+2 m−1 , 2 2
, where the β1 ’s are (first kind) Beta distributed random variables. This leads to the distribution of b and consequently to the PDF (13). Z +∞ Moreover, P f a = p(z) dz which concludes the proof. λ b = 2K Q R−1/2 R b R−⊤/2 Q⊤ and z = QR−1/2 x . W b is real Wishart distributed with 2K degrees of freedom and W parameter matrix Im , z ∼ CN (0, Im ). Then, the test statistic ΛP AMF (11) may be rewritten as: ! −2 −1 2 e⊤ e1 |e⊤ x| 1 Ŵ 1 Ŵ ΛP AMF = 2K = 2K ba. −1 −2 ⊤ ⊤ e1 Ŵ e1 e1 Ŵ e1 (14) b is a Chi square The conditional distribution of 2a given W distribution with 2 degrees of freedom denoted by χ22 . This b and a ∼ 1 χ22 is therefore distribution does not involve W, 2 independent of b. Now, to derive the PDF of b, we use the Bartlett matrix decomposition [9] Ŵ = U.U⊤ where U = (ui,j )1≤i≤j≤m is an upper triangular matrix whose random elements are independent and distributed as: u2i,i ∼ χ22K+i−m and ui,j ∼ N (0, 1) for i < j. Let u′i,j be the elements of the matrix U−1 , u′1,k the vector built with the k ≤ m first non-zero elements of the first row of U−1 and uk,k the vector built with the k ≤ m first non-zero elements of the k -th column of U. Given that U−1 U = Im , the upper-left corner element of U−1 is u′1,1 = u−1 1,1 and the 4. SIMULATIONS This section presents some simulations with vector data of order m = 20. To illustrate previous results, we first plot on figure 1 the PFA versus the P-AMF detection threshold. These plots show the perfect agreement between the theory (circles) and the Monte-Carlo trials (solid lines) for different values of K. Figure 2 shows the threshold decrease brought by the P-AMF compared to the AMF whose theoretical distribution can be found in [1]. The threshold is closer to the optimal OGD’s one for a given PFA. This explains the result obtained in figure 3 where it can be observed an improvement of about 6dB in the detection performance between AMF and P-AMF. 5. CONCLUSION In this paper, we introduce a new adaptive detection test which takes into account the persymmetric structure of the clutter covariance matrix. This one is estimated by a Maximum Likelihood procedure. The corresponding modified Adaptive Matched Filter (AMF) is called the Persymmetric AMF. We derived the analytical distribution of the P-AMF 0 1 10 0.9 AMF OGD PAMF 0.8 −1 0.7 Detection probability PFA 10 −2 10 0.6 0.5 0.4 0.3 PAMF simulated K=25 PAMF theoretical K=25 PAMF simulated K=50 PAMF theoretical K=50 PAMF simulated K=100 PAMF theoretical K=100 −3 10 −4 10 0 2 4 6 Detection Threshold λ (linear) 0.2 0.1 0 −30 8 10 Fig. 1. PFA versus P-AMF detection threshold for different values of K: theory (circle) and Monte-Carlo (solid line). test statistic. This result allows to set the detection threshold for a given PFA. Simulations validate theoretical results and show significant improvement of the P-AMF detection performance on the conventional AMF. −20 −10 0 10 Signal to Noise Ratio (dB) 20 30 Fig. 3. Probability of detection versus Signal to Noise Ratio for P f a = 10−3 , K = 25 and m = 20. Trans. Aerosp. Electron. System, vol. 23, pp. 208–216, January 1992. [2] I. Reed, Y.S.Gau, and T.K. Truong, “CFAR Detection and Estimation for STAP Radar,” IEEE Trans. Aerosp. Electron. System, vol. 34, pp. 722–735, July 1998. [3] S. Kraut, L.L. Scharf, and L.T. Mc Whorter, “Adaptive Subspace Detector,” IEEE Transactions on signal processing, vol. 49, January 2001. 6. REFERENCES [1] F.C. Robey, D.R. Fuhrmann, E.J. Kelly, and R. Nitzberg, “A CFAR Adaptive Matched Filter Detector,” IEEE [4] L. Cai and H. Wang, “A Persymmetric Multiband GLR Algorithm,” IEEE trans. on Aerosp. Electron. System, pp. 806–816, July 1992. −1 [5] E. Conte and A. De Maio, “Exploiting Persymmetry for CFAR Detection in Compound-Gaussian Clutter,” IEEE trans. on Aerosp. Electron. System, vol. 39, pp. 719–724, April 2003. −2 [6] E.J. Kelly, “An Adaptive Detection Algorithm,” IEEE Trans. on Aerosp. Electron. System, vol. 22, pp. 115–127, March 1986. 0 10 PFA 10 10 [7] R. Nitzberg and J.R. Burke, “Application of Maximum Likelihood Estimation of Persymmetric Covariance Matrices to Adaptive Detection,” IEEE Trans. Aerosp. Electron. System, vol. 25, pp. 124–127, January 1980. −3 10 OGD AMF PAMF −4 10 0 2 4 6 Detection Threshold λ (linear) 8 10 Fig. 2. PFA versus threshold for the OGD, AMF and P-AMF, for K = 25 and m = 20. [8] M. Abramovitch and I.A. Stegun, “Handbook of Mathematical Functions,” National Bureau of standard, vol. AMS 55, June 1964. [9] R.J. Muirhead, “Aspects of Multivariate Statistical Theory,” Wiley, New-York, 1982.