Theoretical performance analysis of the W-ABORT detector

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(1)IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 5, MAY 2008. 2117. Correspondence Theoretical Performance Analysis of the W-ABORT Detector Francesco Bandiera, Member, IEEE, Olivier Besson, Senior Member, IEEE, Danilo Orlando, and Giuseppe Ricci, Member, IEEE. Abstract—In a recent paper we introduced a modification of the adaptive beamformer orthogonal rejection test (ABORT) for adaptive detection of signals in unknown noise, by supposing under the null hypothesis the presence of signals orthogonal to the nominal steering vector in the whitened observation space. We will refer to this new receiver as the whitened adaptive beamformer orthogonal rejection test (W-ABORT). Through Monte Carlo simulations this new detector was shown to provide better rejection capabilities of mismatched (e.g., sidelobe) signals than existing ones, like ABORT or the adaptive coherence estimator (ACE), but at the price of a certain loss in terms of detection of matched (i.e., mainlobe) signals. The aim of this paper is to provide a theoretical validation of this fact. We consider both the case of distributed targets and point-like targets. We provide a statistical characterization of the W-ABORT test statistic, under the null hypothesis, and for matched and mismatched signals under the alternative hypothesis. For distributed targets, the probability of false alarm and the probability of detection can only be expressed in terms of multi-dimensional integrals, and are thus very complicated to obtain; in contrast, for point-like targets, such probabilities can be easily calculated by numerical integration techniques. The theoretical expressions derived herein corroborate the simulation results obtained previously.. (AMF) [4]. In the original ABORT formulation the signal under H0 was assumed to be orthogonal to v , in the quasi-whitened space, i.e., after whitening by the sample covariance matrix of the training samples. In [5] we modified this assumption and addressed the adaptive detection of distributed targets embedded in homogeneous disturbance, by resorting to the generalized likelihood ratio test (GLRT) assuming that the signal under H0 is orthogonal to v in the whitened space, i.e., after whitening with the true covariance matrix. This seemingly minor modification led to significant differences compared to ABORT and, more particularly, to an enhanced rejection of sidelobe signals. In this paper we provide a statistical analysis of W-ABORT, with a view to provide a theoretical justification for the behavior. In order to keep the analysis as general as possible, we consider the case of distributed targets and view the case of point-like targets as a special case. In either cases, we provide a characterization of the W-ABORT test statistic in terms of random variables with known distribution. However, as it will become clear in the sequel, evaluating the probability of false alarm (Pfa ) and the probability of detection (Pd ) of the W-ABORT detector for distributed targets is a nearly intractable task. We will thus focus on point-like targets and derive theoretical expressions for Pfa and Pd , for both matched, i.e., p = v , and mismatched signals, i.e., p 6= v . This analysis enables one to obtain a theoretical validation of the results observed in [5].. II. PERFORMANCE ANALYSIS. Index Terms—Adaptive radar detection, constant false alarm rate (CFAR), sidelobe targets rejection, statistical analysis.. A. Background The W-ABORT detector originates from the following binary hypothesis test. I. PROBLEM STATEMENT In an adaptive detection problem one is usually confronted with a tradeoff between detection of matched signals and rejection of unwanted, mismatched signals [1], [2]. Ideally, one would indeed desire a high sensitivity to mainlobe targets and good rejection of sidelobe targets. In order to solve this dilemma, the ABORT detector was introduced in [3]. The idea of ABORT is to modify the null hypothesis H0 , which usually states that the vector under test contains noise only, so that it possibly contains a vector which, in some way, is orthogonal to the assumed target’s signature v . Doing so, if a signal with steering vector p 6= v is present, the detector will be less inclined to declare a detection, as the null hypothesis will be more plausible than in the case where, under H0 , the test vector contains noise only. For instance, if a sidelobe target is present, and hence p is not close to v , the ABORT detector will exhibit less false alarms than detectors which are rather sensitive to mismatched signals, such that the adaptive matched filter Manuscript received May 2, 2007; revised October 15, 2007. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Chong-Meng Samson See. The work of O. Besson was supported by the Mission pour la Recherche et l’Innovation Scientifique (MRIS) of the Délégation Générale pour l’Armement (DGA) under Grant no. 06-60-034. F. Bandiera, D. Orlando, and G. Ricci are with the Dipartimento di Ingegneria dell’Innovazione, Università del Salento, 73100 Lecce, Italy (e-mail: francesco. [email protected]; [email protected]; [email protected]). O. Besson is with the Department of Electronics, Optronics and Signal, ISAE, 31055 Toulouse, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2007.912269. ? H0 : z k = v k + n k ; z k = nk ; z = k v + n k ; H1 : k z k = nk ;. k 2 P k 2 S k 2 P k 2 S. where • the z k ’s, k 2 P = f1; . . . ; KP g, are the vectors under test, while the z k ’s, k 2 S = fKP + 1; . . . ; KP + KS g are the training samples—or secondary data—used to infer the noise statistics, with KS N ; • the noise vectors nk ’s, k 2 = P [ S , are zero mean, independent, complex normal, with unknown covariance matrix R [6]; • v is the presumed target’s steering vector and the v ? k ’s, k 2 P , are orthogonal to v in the whitened space, i.e., v y R 01v ? k = 0, k 2 P . The k ’s stand for the unknown deterministic signal’s amplitudes. The generalized likelihood ratio for the above problem was derived in [5] and is given by. tWA =. 1. (tK 0 1) det I K + Z yP S 01Z P 2. where det(1) is the determinant of a square matrix, I K is the KP -dimensional identity matrix, Z P = [z 1 1 1 1 z K ] 2 CN 2K is the pri-. 1053-587X/$25.00 © 2008 IEEE.

(2) 2118. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 5, MAY 2008. mary data matrix, S is KS times the sample covariance matrix of the y secondary data, i.e., S = k2 z k z k , and. 01 y 01 v yS 01Z P I K + Z yP S 01Z P ZP S v tK = 0 1 y v S v. Now, let us proceed by applying the whitening transformation R 01=2 to the vectors z k ’s, k 2 , in order to obtain (1) shown at the bottom of the page, where Z P w = R 01=2Z P , v w = R 01=2v , and S w = R01=2S R01=2 . Next, let U 2 CN 2N be a unitary matrix that rotates v w into the first elementary vector e1 , i.e.,. U R01=2v = v yR 01v e1 :. that. X yS 101X = y y CAAy + X By S 101 X B e1yS 101e1 = CAA X yS 101e1 = y y CAA with. y = xA 0 S 1 S 101 X B 2 C12K CAA = S1 0 S 1 S 101 S 1 01 :. y 1 0 (I +Z S Z ) v yS v. 01 y I K + X By S 101 X B y y t= 01 CAA 01. y. Zy S v. y. (I +Z y S Z ) v yS v. det2 I K + Z yP S 01Z P 0 Z yP S 01v v yS 01v. 01. Zy S v. det I K + Z yP w S w01Z P w. det(I K + X yS 101X ). 2. v yS 01Z P. 01 det2 I K + Z yP w S w01Z P w 0 Z yP w S w01v w v yw S w01v w v yw S w01Z P w. tWA =. 2C. and B = I K + X B S 101 X B 2 CK 2K . The representation of tWA in (3) is the basis for the analysis to be presented. Moreover, we will rely heavily on the results given by Kelly in [7] and [9] to obtain the distribution of tWA .. det I K + Z yP S 01Z P. tWA =. and. Substituting the above equations into (2) yields (3), shown at the bottom of the next page, where. det01 I K + Z yP S 01Z P. tWA =. =. xA ; where xA 2 C12K and X B 2 C(N 01)2K ; XB S S1 C C S1 = 1 ; and S 101 = AA AB S1 S1 C BA C BB where, denoted by K a generic L 2 L matrix, KAA 2 C, K AB 2 C12(L01) , K BA 2 C(L01)21 , and K BB 2 C(L01)2(L01) . Observe X=. is the well-known decision statistic of Kelly’s detector (GLRT) [7], [8]. In the sequel, we derive the distribution of the test statistic tWA under the noise-only hypothesis (referred to as H00 )—i.e., z k = nk , k 2 P —and under the hypothesis H10 that a signal is present with signature p , viz. z k = k p + nk , k 2 P . First, we will examine the case of a mismatched signal p 6= v , then we will derive the formulas related to the case of perfectly matched signal—p = v —as a straightforward consequence of the previous one. For a range-spread target, i.e., KP > 1, obtaining Pfa and Pd requires to evaluate probability integrals involving matrix differentials, which is a formidable task. Therefore, we assess the performance of the above receiver for the simpler case of a point-like target, i.e., KP = 1. The analysis to be presented proceeds along the same lines as in [7] and [9] and we refer the reader to these references for some details that may be omitted. As a first step, we apply linear transformations in order to rewrite tWA in a canonical and suitable form. First, it is straightforward to show that tWA can be recast as the first equation shown at the bottom of the page, where the last equality follows from identity [7]. det(I m + A1A2 ) = det(I n + A2A1 ) m where A1 2 C 2n and A2 2 Cn2m .. It follows that test statistic (1) can be rewritten as (2) shown at the bottom of the page, where X = U Z P w and S 1 = U S w U y . Following the lead of [7], we now decompose all vectors into two components. 01 det2 I K + X yS 101X 0 X yS 101e1 e1yS 101e1 e1yS 101X. (1). (2).

(3) IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 5, MAY 2008. 2119. with, in turn, 0(1) being the Gamma function [12]. As to B , it is the set of the positive semidefinite matrices with eigenvalues varying in the interval [0; 1]. Therefore, Pf a ( ), namely the probability to declare the presence of a target under H00 , can be written as. B. Statistical Characterization Under the Noise-Only Hypothesis We first assume that. H00 : Z P = N P where N P = [n1 1 1 1 n K ] 2 CN 2K is a matrix whose columns are independent and identically distributed complex normal random vectors with zero mean and covariance matrix R , i.e., N P CNN;K (0; R; I K ). Under this assumption, the vectors involved in (2) are characterized as follows:. 0 1; H00 Pf a ( ) = P tWA > ; H00 = 1 0 P t det(B ) 0 8 =1 0 P t 8) 0 1jB = 8; H0 p(8)d8 det(8. 8 P0 det(8 (5) =1 0 8) 0 1 p(8)d8. X = U R01=2Z P CNN;K (0; I N ; I K ) S 1 = U R01=2S R01=2U y CWN (KS ; I N ). where P0 (1) is the cumulative distribution function (CDF) of the random variable (rv) t, given the B components and under H00 , p(1) is given by (4), and is the threshold set in order to achieve the desired Pf a . However, the above expression for Pf a is difficult to evaluate as it stands since it involves matrix differentials. For this reason we proceed by giving a more suitable characterization of the random matrix B . More precisely, in [7, pp. 51–54], see also [11, pp. 107–108], it is proved that det(B ) is statistically equivalent to the product of N 0 1 independent random variables ruled by the complex central univariate beta distribution, i.e.,. where CWN (KS ; I N ) denotes the complex Wishart distribution with parameters N , KS , and I N . Observe that since X and S 1 do not depend on R , tWA ensures the constant false alarm rate (CFAR) property with respect to the unknown covariance matrix of the noise. It is possible to show that t, given the B components, is ruled by the complex central univariate F-distribution with KP ; KS 0 N +1 degrees of freedom, 1 while, for KP N 0 1, B is ruled by the complex central multivariate beta distribution with parameters KP , KS 0 N +1+ KP , and N 0 1, namely probability density function (pdf) given by2. 1 B (K 0 N + 1 + K ; N 0 1) 2 [det(B )] 0 +1 [det(I 0 B )] 010 B 2. p(B ) =. K. S. j. P. K. N. N. K. K. ;. 0 (n)0 (m) and 0 (n + m ) 01 0 (b) = ( 01) 2 0(b 0 i). Bp (n; m) =. p. bj. p. P f a ( ). p. a. a a. a. =2. where bj is a random variable subject to the complex central univariate beta distribution with KS 0 N + j; KP degrees of freedom. It follows that the Pf a can be recast as. (4). B. where. N. det (B ) . = P tWA > ; H00 = 1 0 P t . =. =0. i. 1Observe that the random vector y , given the B components (and consequently B ), is ruled by the complex multivariate normal distribution with zero mean and covariance matrix B . It follows that y B y is ruled by the complex central univariate -distribution with K degrees of freedom. Moreover, is subject to the complex central univariate -distribution with since C N degrees of freedom [7, pp. 149–152], we can conclude that the K ratio between the above random variables obeys to the complex central uniN degrees of freedom [10, p. 1244]. variate F-distribution with K ; K 2As a matter of fact the pdf of B comes from that of V I B which, in. = 10. 0 +1. =. y. 0. det. 0. .... 0. P t. =2 bj. . 0. N. 0. + X y S 101 X + y y C y tWA = det2 I + X y S 101 X 01 2 y 01 2 det I + I + X y S 101 X y C y I + X y S 101 X = det I + X y S 101 X 01 y y I + X y S 101 X y 1 = 1+ = (1 + t) det(B ) 0 1 C det I + X y S 101 X det I. K. B. B. K. AA. B. B. =. K. K. =. B. B. K. K. 0 1; H00. (6) where pb (1) is the pdf of the rv bj . Note that (5) is a multidimensional integral that, despite the fact that it can be evaluated with modern integration software, is still difficult to calculate. Therefore, we now turn. X , which is given by equations turn, can be calculated from that of X S (7.1.1) and (7.1.2) of [11], exploiting the fact that the Jacobian of the transforX in terms of V is I V mation which expresses X S [11, p. 108].. y. 1. j. 0 1jb2 =

(4) 2 ; . . . ; bN =

(5) N ; H0

(6) j j =2 2 pb (

(7) 2 ) . . . pb (

(8) N )d

(9) 2 . . . d

(10) N 1 1 = 1 0 . . . P0 0 1 p b (

(11) 2 ) N

(12) j 0 0 j =2 . . . pb (

(13) N )d

(14) 2 . . . d

(15) N. y. 0 +1. 1. . N. AA. K. B. B. B. B. B. B. AA. K. B. B. (3).

(16) 2120. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 5, MAY 2008. to the case of point-like targets, i.e., KP to a 1-D integral, namely. Pfa ( ) = 1 0. (). 1 0. = 1. Doing so, (5) boils down. this case, the random variable t, given the B components, is ruled by the complex noncentral univariate F-distribution with ; KS 0 N degrees of freedom and non-centrality parameter given by. 1. +1. 2 (SNR; ) = SNRcos2 . P0 0 0 1 p (0 )d0. where now P0 1 is the CDF of a rv ruled by the complex central degrees of freedom, univariate F-distribution with ; KS 0 N X yB S 01 1 X B , and p 1 is the pdf of the rv , which = is ruled by the complex central univariate beta distribution with KS 0 N ; N 0 degrees of freedom.. (for future reference we denote the quoted distribution by CF1;K 0N +1 ) where. C. Statistical Characterization Under the Signal-Plus-Noise Hypothesis. is a random variable ruled by the complex noncentral univariate beta distribution with KS 0 N ; N 0 degrees of freedom, noncentrality parameter. = 1 1+ 1 +2. 1. (). +1. In this section we derive a closed-form expression for the Pd of the W-ABORT, assuming that. ( ). =. 1 + X By S 101. XB. 01. +2. . det. IK. 1. + X By S 101. XB. 1. = SNRsin2 H10 : Z P = p T + N P and pdf [9, (2-9) and (5-28) with L = KS +1 0 (N 0 1), N = N 0 1, and c = 2 ] N 2 1 where p 2 C is the actual steering vector which is assumed to K 0N +2 have unitary norm, i.e., kpk = 1, T denotes transpose, and = KS 0 N + 2 [ 1 1 1 1 K ]T 2 CK 21 . In this case, the random vector X is dis- p (x) = p

(17) (x)e0 x ` `=0 tributed as [9, pp. 17–19] j e cos T (N 0 2)! 2 ` X CNN;K Ap ; IN; IK h sin (N 0 2 + `)! (1 0 x) where where p

(18) (x) is the pdf of a complex central univariate beta rv with KS 0 N +2; N 0 1 degrees of freedom. Now, it is possible to evaluate y 0 1 v R p j the Pd of the W-ABORT for a point-like target, i.e., e cos = 0 1 0 1 y y pR p v R v Pd (; ; SNR) = P tWA > ; H10 ( N 0 1) 2 1 0 1 y is a unit-norm vector, and Ap = p R p. h2C = 1 0 P t 0 1; H10 Notice that, due to the useful signal components, the random variable t, given the B. components, is now ruled by the complex noncentral univariate F-distribution with KP ; KS 0 N degrees of freedom and non-centrality parameter given by [10, p. 1244]. +1. 2 (SNR; B ) = SNRcos2 T1 B 31 where the quantity. SNR = k k2pyR01p is the total available SNR while 1 2 CK 21 is such that = k k 1 . Since the last (N 0 1) 2 KP components of the mean of X are not equal to zero, the random matrix B obeys to the complex noncentral multivariate beta distribution with parameters KP , KS 0 N +1+ KP , and N 0 1, whose pdf is given by [13, p. 523] p(B ) = BK (KS 0 N +11 + KP ; N 0 1) [det(I K 0 B )]N 010K 2 [det(B )]K 0N +1 e0tr[1 ] 1 F1 (KP + KS ; N 0 1; 12 (I K 0 B )); B 2 B where 1 F1 (1; 1; 1) is the generalized hypergeometric function with matrix argument and 1 2 CK 2K is the non-centrality parameter, defined as. 12 = SNR sin2 ( 31 T1 ): 1. Again, for KP > , evaluating the Pd is very difficult, since this calculation requires to solve integrals involving matrix differentials. When KP , the expression of the Pd becomes simpler and it can be evaluated by means of numerical integration techniques. More precisely, in. =1. 2. 1. P t 0 1j = ; H10 p ( )d 0 1 = 1 0 P (SNR; ) 0 1 p ( )d (7) 0 where P (1) is the CDF of a CF1;K 0N +1 ( ). In the case of a perfect match between v and p , it follows that = 0 and = 0 is ruled by the complex central univariate beta distribution with KS 0 N +2; N 0 1. =1 0. degrees of freedom.. III. NUMERICAL ILLUSTRATIONS In this section we assess the analytical performance of the W-ABORT for the case of point-like target KP . We compare the selectivity of the W-ABORT with that of the GLRT, the AMF, the ABORT, and the adaptive coherence estimator (ACE), proposed in [14] and [15]. The probability of false alarm is set to 06 ; the theoretical Pd ’s and the thresholds to obtain the preassigned Pfa have been evaluated resorting to standard Wolfram Mathematica routines. In Fig. 1 we plot the theoretical Pd , as given by (7), considering both the case of a perfect match between the actual steering vector and the 2 , and the case where there is a slight nominal one, namely misalignment between the two aforementioned vectors, more precisely 2 : . The figure confirms the marked selective behavior of the W-ABORT; indeed, note that, when the mismatch angle is not zero, Pd drops from 0.9 to 0.24 at SNR = 23.4 dB and decreases for higher values of SNR. 2 for the In Figs. 2 and 3, we plot theoretical Pd versus W-ABORT, the ABORT, the GLRT, the AMF, and the ACE. Fig. 2 and KS , while Fig. 3 refers to N and assumes N. (. = 1). 10. cos = 1. cos = 0 95. cos. =5. = 10. = 20.

(19) IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 5, MAY 2008. 2121. K. S = 40. In particular, we use the expressions derived in [3], [4], and [10] for the ABORT, the AMF, and the ACE, respectively. The curves reported in Fig. 2 show that when the number of sensors is small the ACE and the W-ABORT are the most selective detectors. On the other is sufficiently high the above hierarchy changes as shown hand, if in Fig. 3; in this case, the W-ABORT is definitely superior to the other detectors in terms of rejection capabilities of mismatched signals.. N. N. IV. CONCLUSION. Fig. 1. P versus SNR for the W-ABORT with P 1, and K = 10.. = 10. , N = 5, K. =. In this paper we provided a theoretical analysis of the recently introduced W-ABORT detector, for both matched and mismatched signals, at least in the case of point-like targets. For distributed targets, we provided a statistical characterization of the W-ABORT test statistic. The analysis corroborates the previously obtained empirical results and confirms that the W-ABORT has excellent mismatched signals rejection capabilities. This suggests the use of W-ABORT as the second stage of a two-stage detector [10], [16], where the first stage detector has good mainlobe sensitivity but poor sidelobe rejection capabilities.. REFERENCES. Fig. 2. P versus cos for GLRT (dotted line), AMF (dashed line), ABORT (cross marker), ACE (point marker), W-ABORT (solid line) with P = 10 , SNR = 35 dB, N = 5, K = 1, and K = 10.. Fig. 3. P versus cos for GLRT (dotted line), AMF (dashed line), ABORT (cross marker), ACE (point marker), W-ABORT (solid line) with P = 10 , SNR = 35 dB, N = 20, K = 1, and K = 40.. [1] E. J. Kelly, “Performance of an adaptive detection algorithm; rejection of unwanted signals,” IEEE Trans. Aerosp. Electron. Syst., vol. 25, no. 2, pp. 122–133, Apr. 1989. [2] S. Z. Kalson, “An adaptive array detector with mismatched signal rejection,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 1, pp. 195–207, Jan. 1992. [3] N. B. Pulsone and C. M. Rader, “Adaptive beamformer orthogonal rejection test,” IEEE Trans. Signal Process., vol. 49, no. 3, pp. 521–529, Mar. 2001. [4] F. C. Robey, D. R. Fuhrmann, E. J. Kelly, and R. Nitzberg, “A CFAR adaptive matched filter detector,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 1, pp. 208–216, Jan. 1992. [5] F. Bandiera, O. Besson, and G. Ricci, “An ABORT-like detector with improved mismatched signals rejection capabilities,” IEEE Trans. Signal Process., vol. 56, no. 1, pp. 14–25, Jan. 2008. [6] N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (An Introduction),” Ann. Math. Stat., vol. 34, no. 1, pp. 152–177, Mar. 1963. [7] E. J. Kelly and K. M. Forsythe, “Adaptive detection and parameter estimation for multidimensional signal models,” Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, MA, Tech. Rep. 848, 1989. [8] 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, Jul. 2001. [9] E. J. Kelly, “Adaptive detection in non-stationary interference—Part III,” Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, MA, Tech. Rep. 761, 1987. [10] C. D. Richmond, “Performance of the adaptive sidelobe blanker detection algorithm in homogeneous environments,” IEEE Trans. Signal Process., vol. 48, no. 5, pp. 1235–1247, May 2000. [11] C. G. Khatri, “Classical statistical analysis based on a certain multivariate complex Gaussian distribution,” Ann. Math. Stat., vol. 36, no. 1, pp. 98–114, Feb. 1965. [12] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graph, and Mathematical Tables. New York: Dover, 1972. [13] K. C. S. Pillai and G. M. Jouris, “Some distribution problems in the multivariate complex Gaussian case,” Ann. Math. Stat., vol. 42, no. 2, pp. 517–525, Apr. 1971. [14] E. Conte, M. Lops, and G. Ricci, “Asymptotically optimum radar detection in compound Gaussian noise,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, no. 2, pp. 617–625, Apr. 1995. [15] L. L. Scharf and T. M. Whorter, “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. [16] N. B. Pulsone and M. A. Zatman, “A computationally efficient two-step implementation of the GLRT,” IEEE Trans. Signal Process., vol. 48, no. 3, pp. 609–616, Mar. 2000..

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