Weijian Liu,Wenchong Xie,Haibo Tong,Honglin Wang,

CuiZhou2,and Yongliang Wang2,*

1.College of Electronic Science and Engineering,National University of Defense Technology,Changsha 410073,China; 2.Air Force Early Warning Academy,Wuhan 430019,China

Adaptive coherence estimatorbased on the Krylov subspace technique for airborne radar

Weijian Liu1,2,Wenchong Xie2,Haibo Tong1,Honglin Wang2,

CuiZhou2,and Yongliang Wang2,*

1.College of Electronic Science and Engineering,National University of Defense Technology,Changsha 410073,China; 2.Air Force Early Warning Academy,Wuhan 430019,China

A noveladaptive detector for airborne radar space-time adaptive detection(STAD)in partially homogeneous environments is proposed.The noveldetector combines the numerically stable Krylov subspace technique and diagonalloading technique,and it uses the framework of the adaptive coherence estimator(ACE).It can effectively detecta targetwith low sample support.Compared with its natural competitors,the noveldetector has higher probability ofdetection(PD),especially when the number ofthe training data is low.Moreover,it is shown to be practically constant false alarm rate(CFAR).

airborne radar,space-time adaptive detection(STAD), constant false alarm rate(CFAR),adaptive coherence estimator (ACE),Krylov subspace,numerical stability,partially homogeneous environments.

1.Introduction

The problem of detecting a multichannel signal in Gaussian or non-Gaussian noise has received considerable attention more than three decades[1–11].Three most prominent and pioneering adaptive detectors are Kelly’s generalized likelihood ratio test(KGLRT)[1],adaptive matched filter(AMF)[2]and adaptive coherence estimator(ACE)[3].These detectors behave quite well,provided thatsufficientindependentand identically distributed(IID) training data are available.However,in some situations of practical interest,there are insufficient IID training data. Forexample,forthe airborne radarspace-time adaptive detection(STAD),there are usually limited IID training data due to the environmentaland instrumentalfactors[12,13]. In the low sample support environments,all the detectors cited above suffera significantperformance loss.

The reduced-rank technique is a simple buteffective approach to deal with the problem mentioned above[13]. Other effective techniques include the diagonal loading [14,15],structured covariance matrix based Doppler distributed clutter(DDC)model[16],indirectdominantmode rejection(IDMR)[17],joint iterative optimization(JIO) [18],and so on.Recently,the Krylov subspace technique, originally as a tool in the field of numerical analysis,is successfully adopted in the area of signalprocessing[19–23],which can serve as an effective approach to solve the low sample support problem for the airborne radar. The Krylov subspace technique does not need the eigenvalue decomposition(EVD)or the inversion of the covariance matrix.Moreover,it can reduce the negative effect of the perturbation of small eigenvalues of the sample covariance matrix(SCM).This is very important,especially for the case of the low sample support.The most famous Krylov subspace techniques adopted in signal processing, are the multistage Wienerfilter(MWF)[19],the auxiliaryvector filter(AVF)[21],and the conjugate gradient filter (CGF)[22].They allbelong to the field of adaptive filtering.Recently,the CGF is adopted in the area ofmultichannel detection under the assumption of the known covariance matrix[20],while MWF and AVF are employed in adaptive detection underthe assumption ofno exactknowledge of the covariance matrix[23].In[23],the equivalence of MWF and AVF in a directmannerare proved,and MWF and AVF with AMF and KGLRT are combined to constructreduced-rank detectors in homogeneous environments.Furthermore,itis shown thatthe Krylov subspacebased methods provide improved detection performance than the traditionalreduced-rank approaches in[13].

In this paper,we consider the STAD for airborne radar with limited sample supportin partially homogeneous environments,where the covariance matrix of the cell under test(CUT)and thatin the training data have the same structure butwith unknown powermismatch[3].Itis found thatthe partially homogeneous model well describes the environmentsforthe airborne radar,especially with low sample support[24].This paperis an outgrowth of[23].The main contribution of the presentpaper is threefold.

(i)We employ a numericalstable Krylov subspace technique to the ACE for the STAD in the case of low sample support.Note that only the ACE,among the detectors cited above,possesses the constant false alarm rate (CFAR)property in partially homogeneous environments. Moreover,it is worth pointing out that a serious problem usually encountered but often neglected in practice is the rounding error.None of the aforementioned methods,both for the filtering and detection,has considered the effectof the rounding error.It is noteworthy that the essential step of all kinds of the Krylov subspace is the calculation of the orthonormalbasis.The CGF,MWF,and AVF allcompute the orthonormalbasis by the classical Gram-Schmidt (CGS),which has a poor numerical stability[25,26].In this paper we introduce a very stable method to calculate the orthonormalbasis of the Krylov subspace,namely,the CGS of reorthogonalization done by iterative refinement (CGSI)[25].

(ii)We also adoptthe diagonalloading technique to design the noveladaptive detector,since the diagonalloading is a simple but effective method when no sufficient training data exists.Thus,the noveldetectoris referred to as the KryCGSI-ACE-DL.

(iii)We derive the asymptotic statisticaldistribution and probability of detection(PD)of the noveldetector,which is verified by Monte Carlo simulations.

Remarkably,itis found thatthe KryCGSI-ACE-DL significantly improves the detection performance,especially when the number ofthe training data is small.Moreover,it guarantees the practical CFAR property.

The rest of the paper is organized as follows.Section 2 presents the problem formulation.The design of the adaptive detector is given in Sections 3.Section 4 analyzes the detection performance.Numericalexamples are shown in Section 5.Finally,Section 6 concludes the paper.

2.Problem formulation

For simplicity,we assume the data is received by a linear array with Naantennas,and each antenna collects Npsamples.Denote the received data in the CUT by an N×1 dimensionalcolumn vector x,with N=NaNp.We want to decide whether there exists a useful signal in x or not, exceptfor the disturbance n,including clutterand thermal noise.As customary,we also assume that a set of training data xl(l=1,2,...,L),is available.xlonly contains the disturbance nl,and shares the same structure of the covariance matrix as the disturbance d in the CUT.n and nl’s are modeled as independent,zero-mean,complex circular Gaussian vectors,with covariance matrices=R,respectively,where E[·]denotes the statisticalexpectation and(·)Hrepresents the conjugate transpose.The quantityσ2stands forthe unknown power mismatch between the covariance matrix of the CUT and thatof the training data[3].

To sum up,the detection problem to be solved can be formulated in the following binary hypothesis test:

where s is the signal steering vector and a is the signal amplitude,which accounts for the target reflectivity and channelpropagation effects.Note thatthe quantities a,σ2and R are allunknown.

3.Detector design

The ACE is the uniformly most-powerful-invariant (UMPI)detector[27]forthe problem in(1),with the statistic

as the weightvector of the ACE,then(2)can be recastas

When the number of the training data approaches infi-is equalto R.Therefore,(3)turns into

As a consequence,(4)becomes

which is the optimum detector for the problem in(1)for the partially homogeneous environments,and it is known as the normalized matched filter(NMF)[28].However,for the sample-starved scenario,(3)suffers a significant performance loss with respect to(5).The Krylov subspace technique is an effective approach to deal with this problem.Its weightvector for the problem at hand is given by [23]where Tcgsis the basis of the Krylov subspaceis defined as

where Sp{·}denotes the subspace spanned by the corresponding arguments,D=r+1 is defined as the dimension of the Krylov subspace,and r is the clutter rank,which is the number of the dominant eigenvalue of the covariance matrix[13].

Itis worth pointing outthatalthough the weightvectors forthe MWF,AVF,and CGF have forms differentfrom(7), they are equivalentto(7)indeed[23].Otherwise stated,(7) can be taken as the common weight vector for the MWF, AVF,and CGF.Moreover,all the aforementioned weight vectors are usually computed by the CGS orthogonalization.

Compared with(3),(7)can significantly alleviate the performance loss in the case oflow sample support.Therefore,in a similar manner of(4),we can reasonably introduce the following detector

which,for convenience,is denoted as the KryCGS-ACE. The diagonalloading is an effective technique for the case of low sample support[14].When the diagonal loading is adopted,(9)becomes

is the diagonally loaded SCM,and the positive scalathe loading factor[14].

Note thatthe weightvector wcgsin(10)is differentfrom that in(9).The former is calculated according to(7)with ?R is replaced by?Rd.For simplicity,we use the same notation.Their differences can be identified from the context. Precisely,the weightvector wcgsin(10)is the orthonormal basis of the following Krylov subspace

For the comparison purpose,we name the detector in(10) as the KryCGS-ACE-DL.

Due to the simple substitution of waceby wcgsin(9) or(10),substantial improvement on the detection performance is achieved.This is due to the fact that the negative effect of the small eigenvalue of the SCMmatically reduced.However,a practical effect,which degrades the performance of the detectors in(9)and(10)is the rounding error.An effective approach to dealwith this problem is the numerically stable orthonormalbasis calculation method of the CGSI[25].Moreover,since the diagonalloading is also a robusttechnique for the case of the low sample support,we combine the CGSI and the diagonal loading to calculate the orthonormalbasis.The procedure of the CGSI for the calculation of the orthonormal basis of the Krylov subspace in(12)is described as follows,with some modification with respect to the original version in[25].

Initialization:

Notice thatthe value ofα0is setto be 2 in[26].However,as pointed outin[25],itbringslittle effectforthe final result.In orderto assess the orthogonality of the computed orthonormal basis,it is necessary to calculate the loss of orthogonality[30],defined as

which is called the orthogonality of Tr+1in[25].

In a manner analogous to(9),we can obtain the novel detector with improved numericalstability,given by

which is referred to as the KryCGSI-ACE-DL.The quantity wcgsiin(14)has the same form as those in(9)and (10),but the orthonormalbasis is computed by the CGSI approach described as mentioned above.

In summary,the KryCGSI-ACE-DL has the following three main steps:(i)form the Krylov subspace KD(?Rd,s) in(12)with a proper diagonal loading factor,(ii)calculae the orthonormalbasis wcgsiof KD(?Rd,s)by the numericalstable CGSI,(iii)form the finaldetector shown in(14).

4.Performance evaluation

4.1 Asymptotically statisticaldistribution

The statistical characteristics of the new detector KryCGSI-ACE-DL is rather involved,and the explicitderivation of the statistical distribution is intractable. Therefore,we analyze its asymptotic distribution.Consequently,we derive the asymptotic PD,which is validated by Monte Carlo trials in Section 5.

Itis worth noting that when the covariance matrix R is known as a priori,the three detectors in(9),(10),and(14) have almostthe same detection performance,and they all approach the detection performance of the NMF in(6).Because of their common forms and asymptotic versions,the asymptotic detectors and the weights are simply denoted as taand wa,respectively.By asymptotic,we mean thatis approximately equal to R.As a consequence,tacan be represented by

where wais equal to woptin(5),i.e.,wa=R?1s.Let ,then we have

where E[·]denotes the statisticalexpectation,and Tais the asymptotic version of Tcgsand Tcgsi.Therefore,under hypothesis H1,itfollows that

One can verify thatthe distribution of(18)is a complex doubly noncentral F-distribution with 1 and N DOFs and noncentrality parametersρa(bǔ)andρ[31],denoted as

Equation(18)can be furtherexpressed as

It is shown in[23]thatρa(bǔ)is smaller thanρ,however the differenceΔρ=ρ?ρa(bǔ),which is the output SCNR loss for the Krylov subspace technique compared with the optimum processor,is smallenough and can be neglected. Thereby,underthe hypothesis H1,t3is asymptotically dis-.Therefore,(18)can be approximated asIt can be readily shown that the distribution of(23)is a complex noncentral Beta distribution with 1 and N?1 DOFs and a noncentrality parameterρa(bǔ), which is written symbolically as

4.2 Asymptotic PD

In orderto assess the detection performance ofthe new detectors,itis necessary to analyze the upperbound on detection performance ofthatof the ACE and the new detectors, which is the NMF in(6).Under the hypothesis H1,the NMF is ruled by a complex noncentral Beta distribution with 1 and N DOFs and a noncentrality parameterρ[28], as denoted as tNMF～Cβ1,N?1(ρ),whereρis given in (19).

It should be emphasized that a common feature of the detectors in(9),(10),and(14)is that they suffer certain SCNR loss,due to the rank reduction process,which,in turn,alleviates the perturbation caused by the SCM.Fortunately,the latter effectdominates the former one,and it brings significantly performance improvement.

Under the hypothesis H0,the probability density function(PDF)of tain(24)is f(t;H0)=N(1?t)N?1[32]. Itfollows thatthe probability of false alarm(PFA)is

Therefore,the threshold is found to beAccording to the cumulative distribution functions(CDF)ofthe complex noncentralBeta distribution[33],the PDof tais calculated as

Before closing this section,we would like to give some discussions on the computationalcomplexity forthe detectors.The mosttime consuming terms are the matrix inversion operation and the computation of the orthonormalbasis of Krylov subspace.For the conventional ACE,it only needs the matrix inversion,whereas forthe KryCGS-ACEDL and KryCGSI-ACE-DL they need both the matrix inversion and the orthonormalbasis of the Krylov subspace. Note that the computational complexity of the matrix inversion operation is of order of O(N3).Moreover,the computationalcomplexity of the orthonormalbasis of the Krylov subspaceeither by the CGS or by the CGSI,is of order of O(Nr2)[26].Hence, when r is much smallerthan N,the complexity ofthe computation of the orthonormalbasis can be ignored,as compared with the computation of the matrix inversion.Therefore,the ACE,KryCGS-ACE-DL and KryCGSI-ACE-DL almosthave the same computationalcomplexity.

5.Numericalexperiments

It has been shown in[23]that the Krylov subspace-based approaches with the diagonalloading have superiorperformance to the methods employed Krylov subspace-based, diagonal loading technique,or principal component[34]. In otherwords,the KryCGS-ACE-DLand KryCGSI-ACEDL have improved detection performance compared with the KryCGS-ACE,KryCGSI-ACE,diagonal loaded ACE (DL-ACE)[28],reduced-rank ACE(RR-ACE)[13],and the ACE.Therefore,for clarification and simplicity,we mainly focus on the comparison of the KryCGS-ACE-DL and KryCGSI-ACE-DL in this section.

The parameters are given below.The number of antennas and pulses are Na=4 and Np=6,respectively.Thus, we have N=NaNp=24.The array is assumed to work in the side-looking mode.For simplification,the slope of the clutter line is set to be unity.Thus the clutter rank is r=Na+Np?1=9.To reduce the computationalcomplexity,the PFA is set to be Pfa=10?3and the clutterto-noise ratio(CNR)is chosen as 60 dB.The normalized Dopplerfrequency and normalized spatialfrequency ofthe signalare setto berespectively.

Fig.1 shows the loss of the orthogonality of the CGS and CGSI for the orthonormal basis computation of the Krylov subspaceThe working precision is 2.220 4×10?16(i.e.,2?52).As shown in Fig.1,the loss of orthogonality ofthe CGSIis of order O(10?16).However, the loss of the orthogonality of the CGS is much larger, i.e.,of the order of O(100).In other words,the CGSI has less loss than the CGS.Therefore,we can reasonably expect that the KryCGSI-ACE-DL has improved detection performance than thatofKryCGS-ACE-DL.This is indeed the case,as shown in the following simulations.

Fig.1 Loss of orthogonality of the CGS and CGSI

Fig.2 displays the PD of the KryCGSI-ACE-DL,also in comparison with those of the KryCGS-ACE-DL and ACE when the number of the sample is L=2N.The label Asympt denotes“the asymptotical performance of the novel detector”.The superscript an stands for“analytical”,while mc indicates“the PDs are obtained by Monte Carlo simulations”.The thresholds for the detectors are determined through NPfa=100/PfaMonte Carlo trials.The PDs are obtained using NPd=10 000 independent data realizations.The results in Fig.2 highlight that the asymptotically analyticalPDs ofthe KryCGS-ACE-DL and KryCGSI-ACE-DL approximate that of the NMF.Essentially,itis due to the factthatthey share the same type of statistical distributions and nearly have the same noncentrality parameters.Moreover,the PD of the KryCGSIACE-DL is higher than thatof the KryCGS-ACE-DL,and they are both higherthan thatofthe conventionalACE.For example,the performance improvementof the KryCGSIACE-DL with respect to the KryCGS-ACE-DL is about 0.5 dB in terms of the SCNR when PD=0.9.Compared with the optimum detector NMF,the aforementioned loss of the KryCGSI-ACE-DL is only about 1 dB when PD= 0.9.

Fig.2 PDs of the detectors when L=2N

Fig.3 illustrates the PDs of the detectors with L= [1.2N],where the notation[·]represents the rounding operation.It is shown that the PD of the KryCGSI-ACEDL is higher than that of the KryCGS-ACE-DL,which in turn is significantly higher than that of the ACE.Particularly,when PD=0.8,the performance improvements of the KryCGSI-ACE-DL with respectto the KryCGS-ACEDL and ACE in terms of the SCNR are 1 dB and 8 dB, respectively.

Fig.3 PDs of the detectors when L=[1.2N]

Fig.4 displays the PDs of the KryCGS-ACE-DL and KryCGSI-ACE-DL when L＜N.When the numberofthe training data is lower than the dimension of the data,i.e., L＜N,the ACE is invalid due to the singularity of the SCM.In contrast,the KryCGS-ACE-DL and KryCGSIACE-DL are still effective.The results highlight that the KryCGSI-ACE-DLexhibits a higherPDthan the KryCGSACE-DL.This is due to the numerically stable calculation of the orthonormalbasis of the Krylov subspace.

Fig.4 PDs of the detectors when L=2r

Fig.5 depicts the effect of the number of the training data on the detection performance of the detectors.It is shown that when the number of the training data is low (e.g.,less than 35 for the specific parameter setting),the traditionalACE cannotdetecta targetwith PD higherthan 0.5.Instead,the Krylov subspace-based methods are still valid.In addition,when the number of the training data is large enough(e.g.,L=70),the PDs of the detectors are almostthe same.

Fig.5 PDs of the detectors versus the number of the training data

Fig.6 plots the PDs of the KryCGS-ACE-DL and KryCGSI-ACE-DL with different loading factors when L=2r.The results indicate that for a wide range of the loading factor,i.e.,[–2.8 dB,22 dB](orequivelently[0.52,170]),the detection performance ofthe detectorsis notdramatically affected by the loading factors.This resultcoincides with the results in[28].

Fig.6 PDs of the detectors versus the loading factor

The detection performance of the detectors is characterized in a different way in Fig.7.Precisely,it shows the CFAR property of these new detectors.More precisely, it plots the thresholds of these detectors under different CNRs when the numberofthe sample is setto be L=2N. The thresholds of the detectors vary little for different CNRs.This reveals the factthatthey are practically CFAR. Moreover,the KryCGSI-ACE-DL is mostrobust.

Fig.7 Thresholds versus CNR

6.Conclusions

In this paper,we adopt the Krylov subspace technique for the airborne radar STAD,and propose a noveldetector in partially homogeneous environments,i.e.,the KryCGSIACE-DL.The new detector is numerically stable and improves detection performance,especially when the number of the training data is small.Additionally,the KryCGSIACE-DL has the negligible costin the computationalcomplexity,compared with the traditionalACE.

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Biographies

Weijian Liuwas born in 1982.He is currently working toward the Ph.D.degree at National University of Defense Technology,Changsha,China. His currentresearch interests include multichannel signaldetection,statisticaland array signalprocessing.

E-mail:liuvjian@163.com

Wenchong Xiewas born in 1978.He received his Ph.D.degree in 2006 from National University of Defense Technology,Changsha,China.Now he is an associate professor in Air Force Early Warning Acadamy.His current research interests indude space-time adaptive processing and ariborne radar signalprocessing.

E-mail:xwch1978@aliyun.com

Haibo Tongwas born in 1984.He is currently working toward his Ph.D.degree at National University of Defense Technology,Changsha,China.His currentresearch interests include the weak signaldetection and tracking for the globalsatellite system.

E-mail:hbo.tong@gmail.com

Honglin Wangwas born in 1982.He received his M.S.degree on military equipmentin 2006.Now he is a lecture of Scientific Research Departmentin Air Force Earlying Warning Academy,China.His current research interests indude radar technology and its applications in military affairs.

E-mail:ly whl@163.com

Cui Zhouwas born in 1983.She received her M.S. degree in communication and information system from China University of Geosciences(Wuhan)in 2008.Her currentresearch interest is FPGA.

E-mail:cuizhou2014@163.com

Yongliang Wangwas born in 1965.He received his Ph.D.degree in 1994 from Xidian University. Now he is a professor in Air Force Early Warning Acadamy.His current research interests indude space-time adaptive processing and ariborne radar signalprocessing.

E-mail:ylwangkjld@163.com

10.1109/JSEE.2015.00078

Manuscript received November 06,2013.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(60925005;61102169;61501505).