A Doubly Parameterized Detector for Mismatched Signals

Chinese Journal of Electronics Vol.24, No.1, Jan. 2015 A Doubly Parameterized Detector for Mismatched Signals∗ LIU Weijian1,2 , XIE Wenchong2 , ZHANG...
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Chinese Journal of Electronics Vol.24, No.1, Jan. 2015

A Doubly Parameterized Detector for Mismatched Signals∗ LIU Weijian1,2 , XIE Wenchong2 , ZHANG Qianping3 , LI Rongfeng2 , DUAN Keqing2 and WANG Yongliang2 (1. College of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China) (2. Key Research Laboratory, Air Force Early Warning Academy, Wuhan 430019, China) (3. College of Computer Science and Technology, Wuhan University of Technology, Wuhan 430070, China) Abstract — In this paper, we consider the problem of adaptive multichannel signal detection in the presence of signal mismatch, and introduce a novel tunable detector, which is parameterized by two tunable parameters. It has the Constant false alarm rate (CFAR) property and covers Kelly’s generalized likelihood ratio test (KGLRT), Adaptive matched filter (AMF), and Adaptive coherence estimator (ACE) as its three special cases. The novel detector controls the mismatched signal by adjusting two tunable parameters. Remarkably, it can achieve improved detection performance for matched signals, enhanced rejection of seriously mismatched signals, and better robustness to slightly mismatched signals than its natural competitors. Key words — Constant false alarm rate (CFAR), Mismatched signal, Multichannel signal, Tunable detector.

I. Introduction Detecting a multichannel signal in Gaussian or nonGaussian noise with unknown covariance matrix is a fundamental task of any radar system, and a large number of approaches were proposed in the literature, such as Refs.[1–13]. Among these detectors the most pioneering and prominent ones are Kelly’s generalized likelihood ratio test (KGLRT)[1] , the Adaptive matched filter (AMF)[2] , and the Adaptive coherence estimator (ACE)[3] . However, theses detectors are designed without taking the possibility of the signal mismatch into account, and they behave quite differently in the situation. In practice, the actual signal backscattered from a target can be different from the nominal one. The mismatch may arise due to imperfect array calibration, spatial multipath, pointing errors, and so on[14,15] . In order to improve the mismatched signals rejection, the Adaptive beamformer orthogonal rejection test (ABORT)[16] , and Whitened ABORT (W-ABORT)[17] are proposed. The null hypothesis is modified by adding a fictitious signal which is orthogonal to the signal steering vector in the whitened or quasi-whitened space. If a target exists in

a direction different from the nominal one, the detector will incline to the null hypothesis. In Refs.[18,19], a different decision scheme is introduced. Precisely, at the design stage it is assumed that the Cell under test (CUT) contains a noise-like interferer, which is not accounted for in the secondary data. Consequently, the Double-normalized AMF (DN-AMF) is proposed therein. In Refs.[20,21], two-stage detectors, i.e., the Adaptive sidelobe blanker (ASB) and its improved versions, are proposed. These detectors usually contain two individual detectors with converse capabilities of mismatched signals rejection. The detectors declare the presence of a target in the CUT only when data survive both detection thresholds. Another scheme is modeling the actual signal as a vector belonging to a proper cone[22] , and then devising the detector by the theory of convex optimization. The other effective approach is resorting to tunable detectors[23−25] . By tuning a scalar parameter, they can control the level to which the mismatched signals are rejected. The ABORT, W-ABORT, and DN-AMF all have enhanced mismatched signals rejection, but lack of robustness to slightly mismatched signals. The capabilities of rejection and robustness to mismatched signals of the ASB-like detectors are confined by these capabilities of their individual detectors. As for the techniques of modeling the actual signal as a vector belonging to a proper cone, it only possesses the robustness to the mismatched signals, and it usually has no closed-form solution. Moreover, the tunable detector in Ref.[23] has limited capabilities of mismatched signals rejection, while the tunable detectors in Refs.[24,25] have limited robustness. In this paper, by comparing the similarity of the KGLRT, AMF, and ACE, we introduce a novel parametric detector, which encompasses the three aforementioned detectors as its special cases. For this reason, the novel detector is denoted as the KMACE. The first and second letter in the acronym above stand for the KGLRT and AMF, respectively, while the last

∗ Manuscript Received May 2013; Accepted May 2014. This work is supported by the National Natural Science Foundation of China (No.61102169).

A Doubly Parameterized Detector for Mismatched Signals three letters is the ACE. Unlike the single-parameter tunable detectors in Refs.[23–25], the KMACE is parameterized by two tunable parameters. Precisely, a parameter is referred to the additive parameter and the other is denoted as the exponential parameter. Remarkably, the KMACE has a superior flexibility of controlling the degree to which the mismatched signals are rejected. Increasing the additive parameter and/or decreasing the exponential parameter makes the KMACE much robust to the signal mismatch. In contrast, decreasing the additive parameter and/or increasing the exponential parameter makes the KMACE much selective (less tolerant to the signal mismatch). Moreover, the KMACE can also provide improved detection performance for matched signals than the KGLRT, AMF, and ACE. This paper is organized in the following fashion. Section II briefly describes the adaptive detection problem and presents the new parametric detector, while Section III shows the performance assessment. The numerical examples are given in Section IV. Finally, Section V concludes the paper.

II. Problem Formulation and Design Issues Denote the primary data by an N -dimensional column vector x. We want to discriminate between hypothesis H0 that x only contains the disturbance and hypothesis H1 that x contains the disturbance and a useful signal. Hence, the detection problem can be formulated as the following binary hypothesis test: ( H0 : x = n (1) H1 : x = as + n where a is the signal amplitude, s is the (spatial, temporal, or spatial-temporal) signal steering vector, and n is the disturbance, containing clutter and white noise, with a positive definite covariance matrix R. Note that the value of a and R are both unknown. As customary, we assume that a set of training data, yl , l = 1, 2, . . . , L, are available, which are statistically independent and share the same statistical property with the noise n in the primary data x. The AMF[2] , KGLRT[1] , and ACE[3] for the detection problem in Eq.(1) are ˛ H −1 ˛2 ˛s S x ˛ tAMF = (2) sH S −1 s tAMF (3) tKGLRT = 1 + xH S −1 x − tAMF and tAMF tACE = H −1 (4) x S x respectively, where S = Y Y H is L times the Sample covariance matrix (SCM) with Y = [y1 , y2 , . . . , yL ] being the secondary data matrix. By comparing Eqs.(2)–(4), we introduce the KMACE as tAMF tKMACE = (5) α + (xH S −1 x − tAMF )γ ∗ By

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where the additive parameter α and the exponential parameter γ are two positive scalars, referred to as the tunable parameters. Remark 1 When α = γ = 0, Eq.(5) reduces to the AMF. When α = γ = 1, Eq.(5) degenerates into the KGLRT. Furthermore, when α = 0 and γ = 1, Eq.(5) becomes tKMACE =

tAMF 1 = −1 xH S −1 x − tAMF tACE − 1

(6)

where tACE is given in Eq.(4). Note that Eq.(6) can be taken as a monotonically increasing function of tACE in the interval (0, 1). Moreover, tACE ∈ (0, 1). Hence, Eq.(6) is statistically equivalent∗ to tACE . In summary, when α = 0 and γ = 1, Eq.(5) turns into the ACE. eH Psex e, while the Remark 2 tAMF can be expressed as x H −1 eH Pse⊥ x e, where quantity x S x − tAMF can be recast as x e = S −1/2 x, se = S −1/2 s, S 1/2 is the square-root matrix∗∗ of x S, S −1/2 is the inversion of S 1/2 , Pse is the orthogonal projection matrix (projector) onto the column space of se, and Pse⊥ is the orthogonal complement of Pse, i.e., Pse⊥ = I −Pse. It follows that tAMF can be taken as the energy of the quasi-whitened e projected onto the quasi-whitened signal subprimary data x space spanned by se, denoted as < se >, while xH S −1 x − tAMF e projected onto the complementary subspace is the energy of x of < se >, i.e., < se >⊥ . Hence, we can reasonably conjecture that increasing the value of α and fixing the value of γ, will weaken the effect of the term xH S −1 x − tAMF . Consequently the detector turns more and more robust to mismatched signals. On the other hand, increasing the value of γ and fixing the value of α, the capabilities of the rejection of mismatched signals will increase. In fact, this is indeed the case; see Section IV.

III. Performance Evaluation We now proceed to analytically assess the performance of the KMACE in terms of Probability of false alarm (PFA) and Probability of detection (PD). The PD of the KMACE can be expressed as (7) Pr(tKMACE > η; H1 ) where η is the threshold to be assigned to ensure a presumed PFA. We consider the case of signal mismatch. When this phenomenon arises, the actual signal becomes asm , where sm is the actual signal steering vector, which is not necessarily aligned with the nominal array steering vector s. To quantify the signal mismatch, we introduce the quantity[16−18] cos2 φ =

|sH R−1 sm |2 −1 s · sH m mR

sH R−1 s

(8)

which is the cosine squared of the array steering s and the actual signal steering sm in the whitened space.

saying “the detector t1 is statistically equivalent to t2 ”, we means that the detectors t1 and t2 have the same detection performance[26] . Statistically equivalent detectors are usually related with monotonically increasing functions. ∗∗ A method to generate the square-root matrix is to resort to the Eigenvalue decomposition (EVD). Precisely, let the EVD of S be S = V ΛV H , where V is the unitary matrix whose columns are the eigenvectors of S, and Λ is the diagonal matrix with the diagonal components being the eigenvalues of S. Then the square-root matrix of S is S1/2 = V Λ1/2 V H , where Λ1/2 is a diagonal matrix whose diagonal components are the square-root of those of Λ.

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It is straightforward to verify that Eq.(5) can be recast as tKMACE = where

tKGLRT β γ−1 αβ γ + (1 − β)γ

β = (1 + xH S −1 x − tAMF )−1

(9)

(10)

and we have used the fact tKGLRT = tAMF β, with tKGLRT being the KGLRT given in Eq.(3). The conditional statistical distribution of tKGLRT with β fixed under H1 , is a complex noncentral F-distribution with 1 and L−N +1 Degrees of freedom (DOFs), and a noncentrality parameter ρφ β [27] , where ρφ = ρ cos2 φ and −1 ρ = |a|2 sH sm mR

(11)

is the output Signal-to-clutter-plus-noise ratio (SCNR). Hence, plugging Eq.(9) into Eq.(7) results in ˜ ˆ P D = P r tKGLRT > ηβ 1−γ [αβ γ + (1 − β)γ ]; H1 Z 1 ¯´ ` ˘ = 1 − P1|β ηβ 1−γ [αβ γ + (1 − β)γ ] f1 (β)dβ

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obtained by Monte Carlo simulations, for which 2 × 104 and 100/P F A independent data realizations are run to calculate the PD and the detection threshold necessary to ensure a preassigned PFA. The dimension of the primary data is set to be N = 12, the number of the secondary data is chosen as L = 2N , and the PFA is selected as P F A = 10−3 . Fig.1 depicts the PD of the KMACE with different tunable parameters for matched signals. The plane, drawn in Fig.1, is the PD of the KGLRT. For clear display, the PDs of the KMACE less than 0.9 are set to be 0.9. It is shown that it is about one-third of the number of the couples of γ and α, with which the PD of the KMACE is higher than that of the KGLRT. Furthermore, for fixed γ with small value, the value of α does not significantly affect the PD of the KMACE, while for fixed γ with large value, with the increase of the value of α, the PD turns higher and higher. On the other hand, when the value of α is fixed, the increase of the value of γ results in lower PD, but it will bring enhanced rejection of mismatched signals; see the following simulations figures.

0

(12) where P1|β (·) is the Cumulative distribution function (CDF) [27] of tKGLRT for given β under H1 , i.e., ! L−N X η L−N +1 ηk P1|β (η) = 1+k (1 + η)L−N+1 k=0 « „ ρφ β (13) · IGk+1 1+η where IGm+1 (a) is the incomplete Gamma function, given by P k IGm+1 (a) = e−a m k=0 a /k!. Moreover, β is ruled by a complex noncentral Beta distribution with L − N + 2 and N − 1 DOFs, and a noncentrality parameter δ 2 = ρ sin2 φ[27] . The quantity f1 (β) in Eq.(12) is the Probability density function (PDF) of β under H1 , which is found to be[27] ! L−N+2 X L−N +2 L!δ 2m −δ 2 β f1 (β) = e m (L + m)! m=0 ·fL−N+2,N+m−1 (β)

(14)

where fm,n (β) is the PDF of the complex central Beta distribution with m and n DOFs. Setting ρφ = 0 in Eq.(12), we have the PFA as Z 1 ˆ ˜−(L−N+1) PFA = f0 (β)dβ 1 + ηαβ + ηβ 1−γ (1 − β)γ 0

(15) where we have used the identity 1 − P0|β (x) = (1 + x)−(L−N+1)[24] , where in turn P0|β (·) is the CDF of tKGLRT for given β under H0 , and f0 (β) is the PDF of β under H0 , which can be obtained by setting δ 2 = 0 in Eq.(14). Remark 3 From Eq.(15), we know the PFA is not dependent on the noise covariance matrix R. Hence, the KMACE is Constant false alarm rate (CFAR).

IV. Numerical Examples In this section we evaluate the detection performance of the KMACE both for matched and mismatched signals. For independent confirmation, we also show the PD of the KMACE

Fig. 1. PDs of the KMACE with different tunable parameters matched signals (SCNR=16dB)

Fig.2 plots the PD of the KMACE for matched signals under different SCNRs, also in comparison with those of the KGLRT, AMF, and ACE. The values of α and γ are set to be 2.3 and 1, respectively. The subscript “MC” in the legend indicates that the corresponding PD is obtained by Monte Carlo simulations. Note that the theoretical results perfectly match the simulation results. It can be seen that for the specific setting, when the SCNR is high enough (say, SCNR>14dB), the KMACE has slightly higher PD than those of the KGLRT, AMF, and ACE. Moreover, the PD of the KMACE is higher than those of the AMF and ACE for nearly all SCNR in the interval from 7 to 17. The detection performance of the KMACE for mismatched signals is evaluated in Figs.3–5. Precisely, Fig.3 demonstrates the PD of the KMACE under different parameter settings. The results imply that by tuning the additive parameter α or the exponential parameter γ, the KMACE can flexibly control mismatched signals. Moreover, it is shown that the PD of the KMACE is significantly affected by the values of α, γ, and SCNR. Fig.4 plots contours of constant PDs, represented as functions of SCNR and cos2 φ. Such plots are referred to as the mesh-plot[16] . For the KMACE, α = 0 and γ = 1.5. It is seen that the KMACE has the best performance in term of mismatched signals rejection, followed in sequence by the ACE,

A Doubly Parameterized Detector for Mismatched Signals

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KGLRT, and AMF.

Fig. 2. PDs of the detectors for matched signals Fig. 5. PD versus cos2 φ for mismatched signals (SCNR = 20dB)

V. Conclusions

Fig. 3. PD of the KMACE for mismatched signals under different parameters (SCNR=20 dB). (a) Different cos2 φ − α; (b) Different cos2 φ − γ

In this paper, for the mismatched signal detection we have proposed the novel tunable detector KMACE, which is doubly parameterized by additive and exponential parameters. The KMACE is more flexible in controlling the degree to which the mismatched signals are rejected, compared to the existing detectors. Moreover, the KMACE, with appropriate tunable parameters, can also provide higher PD for matched signals than the existing detectors. References

Fig. 4. Contours of constant PDs for the AMF (a), KGLRT (b), ACE (c), and KMACE (d)

Fig.5 illustrates the PDs of the detectors versus cos2 φ. The results indicate that by tuning the additive parameter α or/and the exponential parameter γ, the KMACE can achieve flexible control of the degree to which the mismatched signal to be rejected. Precisely, fixing the value of γ to be 1.5, increasing the value of α form 2 to 6, the KMACE becomes more robust, while fixing the value of α to be 2, increasing the value of γ form 1.5 to 3, the KMACE turns more selective, and it is even more selective than the ACE. Besides, with α = γ = 0, the KMACE degenerates to the AMF, and it has the least mismatch discrimination capabilities, or, equivalently, it is most robust to mismatched signals.

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LIU Weijian was born in Shandong Province, China, in 1982. He received the B.S. and M.S. degrees from Air Force Radar Academy, Wuhan, China, in 2006 and 2009, respectively. He is currently working toward the Ph.D. degree at National University of Defense Technology (NUDT), Changsha, China. His current research interests include multichannel signal detection, statistical and array signal processing. (Email: [email protected]) XIE Wenchong was born in Shanxi Province, China, in 1978. He received the B.S. and M.S. degrees from Air Force Radar Academy, Wuhan, China, and the Ph.D. degree from NUDT in 2000, 2003, and 2006, respectively. He is an associate professor with the Key Research Laboratory, Air Force Early Warning Academy. His research interests include space time adaptive processing, radar signal processing, and adaptive signal processing. ZHANG Qianping was born in 1985. She received the B.S. degree from Air Force Engineering University, Xi’an, China, in 2007. She is currently working toward the M.S. degree at Wuhan University of Technology, Wuhan, China. Her current research interests include computer application and computer graphics. LI Rongfeng was born in 1971. He received the B.S. degree from NUDT in 1993, M.S. degree from Air Force Radar Academy, Wuhan, China in 1997, and Ph.D. degree from Air Force Engineering University, Xi’an, China in 2002. He is currently a professor in Air Force Early Warning Academy, Wuhan, China. His research interests include adaptive array signal processing and radar signal processing. DUAN Keqing was born in Hebei Province, China, in 1981. He received the B.S. and M.S. degrees from Air Force Radar Academy, Wuhan, China, in 2003 and 2006, and the Ph.D. degree from NUDT in 2010. Now he is a lecturer in the Key Research Laboratory, Air Force Early Warning Academy, China. His current research interests include array signal processing and space-time adaptive signal processing. He has authored or coauthored more than 20 papers. WANG Yongliang (corresponding author) was born in Zhejiang Province, China, in 1965. He received the B.S. degree in electrical engineering from Air Force Radar Academy, Wuhan, China, in 1987 and the M.S. and Ph.D. degrees in electrical engineering from Xidian University, Xi’an, China, in 1990 and 1994, respectively. From Jun. 1994 to Dec. 1996, he was a post-doctoral fellow with the Department of Electronic Engineering, Tsinghua University, Beijing, China. Since Jan. 1997, he has been a full professor and the director of the Key Research Laboratory, Air Force Early Warning Academy. His recent research interests include radar systems, space-time adaptive processing, and array signal processing. He has authored or coauthored three books and more than 100 papers. Dr. Wang was the recipient of the China Postdoctoral Award in 2001 and the Outstanding Young Teachers Award of the Ministry of Education, China, in 2001. (Email: [email protected])