Knowledge-aided covariance estimate via geometric mean for adaptive detection

Digital Signal Processing 97 (2020) 102616

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Digital Signal Processing www.elsevier.com/locate/dsp

Knowledge-aided covariance estimate via geometric mean for adaptive detection Zheran Shang a , Kai Huo a , Weijian Liu b,∗ , Yang Sun a , Yongliang Wang b a b

College of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China People’s Liberation Army Air Force Early Warning Academy, Wuhan 430019, China

a r t i c l e

i n f o

Article history: Available online 19 November 2019 Keywords: Knowledge-aided Adaptive detection Geometric mean Compound Gaussian clutter

a b s t r a c t This paper deals with the covariance estimation and its application in radar signal processing when the number of the secondary data is limited. We model the covariance estimation as a color loading version and use three different geometric means to derive three kinds of knowledge-aided (KA) covariance estimators, namely, KA Euclid (KA-E) metric estimator, KA Power-Euclid (KA-PE) metric estimator, and KA Log-Euclid (KA-LogE) metric estimator. Experimental results on simulation and measured data demonstrate that the proposed estimators achieve better performance than their natural competitors and keep good constant false alarm ratio (CFAR) property. Among the three proposed estimators, the KA-E estimator has the best performance. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Covariance matrix estimation is the key to advanced radar signal processing algorithms, such as space-time adaptive processing (STAP) [1] and adaptive signal detection [2–5]. In Gaussian clutter, the sample covariance matrix (SCM) estimator is the maximum likelihood estimation (MLE) [6]. In non-Gaussian clutter, the SCM estimator is sensitive to the texture component, so it can not estimate the covariance efficiently and, as a result, affects the performance of adaptive detectors. The normalized SCM (NSCM) estimator [7] and fixed point (FP) estimator [8] can solve this issue via normalizing the data vector. However, the covariance matrix will often be ill-conditioned when the number of secondary data is only slightly larger than the system degrees of freedom (DOF). In this paper, the DOF is the number of the coherent pulse train in one coherent processing interval (CPI). The DOF is also equal to the data dimension. To ensure an acceptable performance, the number of samples should be at least roughly twice the DOF [9]. In order to reduce the requirement of samples, several regularized covariance matrix estimators have been proposed. Diagonal loading, a class of regularized estimation methods, is first applied in [10,11]. The modified generalized likelihood ratio test (GLRT) and modified adaptive matched filter (AMF) detector given in [12]

*

Corresponding author. E-mail addresses: [email protected] (Z. Shang), [email protected] (K. Huo), [email protected] (W. Liu), [email protected] (Y. Sun), [email protected] (Y. Wang). https://doi.org/10.1016/j.dsp.2019.102616 1051-2004/© 2019 Elsevier Inc. All rights reserved.

and [13] apply the regularized SCM (RSCM) estimator to the Kelly’s GLRT (KGLRT) [2] and AMF detector [14,15] for dealing with the sample starvation effect. In [16], the authors derive the closedform of diagonal loading factor via minimizing the mean-square error (MSE) between the true covariance and the estimated covariance. The RSCM has robust estimation performance in Gaussian clutter but degrades in non-Gaussian clutter. To deal with regularized covariance estimation problem in non-Gaussian clutter, the shrinkage FP (SFP) estimators is derived in [17,18], which is a distribution-free method. Another class of regularized estimation methods to deal with the insufficient sample issue is the knowledge-aided (KA) covariance estimation. The KA covariance estimation methods can fall loosely into three categories based on the different kinds of knowledge. The first category of knowledge is the structure information of covariance matrix, such as symmetric power spectrum density (PSD) [19, 20] and persymmetric covariance [21–24]. The second category of knowledge is the prior statistical distribution of covariance matrix, such as the Wishart or inverse Wishart distribution [25–27]. The third category of knowledge is the data of environment, such as synthetic aperture radar (SAR) imagery [28], physics-based models [29], digital elevation model (DEM) [30] and history or simulation data of clutter [31,32]. The data from environment can be used to obtain a priori clutter covariance matrix and utilize this covariance matrix combining with the estimated covariance to obtain a more accurate one, which is usually in the form of color loading version.

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Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

The color loading methods model the estimation issue as the linear or nonlinear combination of the prior covariance matrix and the estimated covariance. The key of these methods are to determine the color loading factor. In [33], the authors minimize the MSE between estimation covariance and actual covariance to obtain the weighting factor via using convex optimization method, which is called the convex combination (CC) estimator. In [34], the authors leveraged a maximum likelihood (ML) approach to obtain the weighting factor which is an iterative method. However, these methods only use the SCM as the estimated covariance to obtain the color loading factor and assume the clutter follows the Gaussian distribution. Therefore, they can not exactly estimate the statistical property of the primary data in the non-Gaussian clutter and suffer from performance loss in the estimation and detection because of the model mismatch. Hence, the KA methods in nonGaussian clutter need to be further studied. Recently, covariance estimation methods based on the geometric mean and median are proposed in the open literature. In [35, 36], five geometric means (Euclid mean, Log-Euclid mean, RootEuclid mean, Power-Euclid mean, Cholesky mean) are proposed to obtain the corresponding covariance estimators without considering the clutter distribution. Two matrix CFAR detectors are derived in [37], which detect targets by measuring the symmetrized Kullback Leibler (KL) and total KL divergences distance of covariance matrix between the primary data and the secondary data. In [38], the authors use five geometric means and medians (KL divergence, Bhattacharyya distance, Log-Euclid distance, Log-Determinant divergence and Hellinger distance) to derive the covariance estimators and obtain the corresponding detectors. These studies show that geometric methods can obtain the non-linear model of covariance estimation that can improve the covariance estimation performance in non-Gaussian clutter. In this paper, our contribution is to combine the prior covariance matrix with different geometric mean estimators to obtain three new KA estimators, namely, KA Euclid metric (KA-E) estimator, KA Power-Euclid metric (KA-PE) estimator and KA Log-Euclid metric (KA-LogE) estimator, to further improve the performance of adaptive detectors in non-Gaussian clutter. We show that, compared with their natural competitors, the adaptive detectors with proposed estimators have better detection performance in nonGaussian clutter when secondary data are insufficient. Moreover, among three proposed KA estimators, the KA-E estimator is better than the KA-PE and KA-LogE estimators. Numerical experiments verify these results. The rest of this paper is organized as follows. Section 2 gives a brief description of the geometric means. The estimators proposed in this paper is detailed in Section 3. Results obtained from experiments are presented in Section 4. Section 5 concludes the paper.

In this section, we introduce three geometric means and the corresponding covariance matrix estimators. First, we formulate the data model. Then, we obtain the covariance matrix estimators via geometric means. 2.1. Data model Consider a set of received data where the primary (cell under test, CUT) and secondary data share the same covariance structure with different local power. Specifically, let c 0 and c k being N-dimensional random vectors, which follow a complex circular multivariate compound-Gaussian distribution, expressed as

c0 = ck =

√ √

τ0 g ,

τk g , (k = 1, 2, . . . , K ) ,

√





E g g H = ,

(2)

where (·)H denotes the conjugate transpose operator and E [·] denotes the statistical expectation. The covariance matrixes are R =    E c 0 c 0 H |τ 0 = τ 0  and R k = E c k c k H |τ k = τk , for given τ 0 and τ k. 2.2. Geometric mean In number field, for a set of K positive numbers {x1 , x2 , . . . , x K }, the mean x¯ is the minimum value of the sum of the squared distance to the given points

x¯ = arg min x>0

K 1 

K

| x − x i |2 .

(1)

(3)

i =1

In geometry field, the mean for a finite set of Hermitian positive definite (HPD) matrices origins from generalized mean for positive numbers. It is proved in [39] that the geometric mean for a set of HPD matrices exists and is unique. A simple and fast convergence algorithm to compute the geometric mean is using the FP algorithm [40]. The geometric mean associated with geometric distance, of a set of K HPD matrices, is defined by K   ¯ = arg min 1 ˆk  d2 , 

0

K

(4)

k =1

where, d (·, ·) denotes the geometric distance. As to ˆk covariance matrix estimates, we assume that each  tion of the single secondary datum k . In particular, choice could be the rank-one sample matrix related to

ˆ k = zk z H , where zk = c k / k , namely  k

1 c k 2 N

the set of is a funca possible the vector

is the normalized

sample [41] which can eliminate the effect of the texture component. However, it is not acceptable since the aforementioned definition of distance requires that the considered matrices are HPD. Hence, according to [38,42] we transform the normalized sample data into Toeplitz covariance matrices, namely

⎡

r0 ⎢ r1 ⎢

  k = E zk zkH = ⎢ . ⎣ ..

2. Problem formulation



√

where τ0 and τk are positive and possibly random numbers, namely, the texture component, determining the local scattering power. g is the speckle component and can be modeled as an N −dimensional circularly symmetric zero-mean vector with positive definite covariance matrix





r N −1

r¯1 r0

..

. ···

⎤ · · · r¯ N −1 · · · r¯ N −2 ⎥ ⎥ .. ⎥ , .. . . ⎦ r1

(5)

r0

r j = E zi z¯ i + j , 0 ≤ j ≤ N − 1, 1 ≤ i ≤ N ,   where r j = E zi z¯ i + j is called the correlation coefficient and r¯ denotes the complex conjugate of r. k is a Toeplitz HPD matrix. According to the ergodicity of a wide sense stationary, the correlation coefficient of data zk can be calculated by averaging over time instead of its statistical expectation,

rˆ j =

1 N− j

1− j N−  n =1

zk (n) z¯ k (n + j ) ,

0 ≤ j ≤ N − 1,

(6)

where zk (n) is the nth element of zk . As to the considered distances, we focus on the Euclid, PowerEuclid (PE) and Log-Euclid (LogE) distances. The considered three

Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

distances and the corresponding geometric mean estimators formally are defined below. • Euclid distance



dE ( A , B ) =

tr ( A − B ) ( A − B )

H



(7)

,

where tr (·) stands for the trace operator and the corresponding geometric mean is

¯ E = arg min 1  K

0

K 



ˆk d2E , 



k =1

(8)

K 2 1   ˆ k = arg min  −   .

K

0

¯E= 

K 1 

K

ˆ k. 

(9)

ˆ k = zk z H , but in It is noted that (9) is the NSCM estimator when  k ˆ k is HPD. this paper we use (5) to ensure  • Power-Euclid distance

   tr

K

0

= arg min 0

ˆ k0 = αE 0 + (1 − αE )  ˆ k , 0 ≤ αE ≤ 1, 

Aβ − B β



Aβ − B β

H  ,

(10)

K 



ˆk d2P , 



k =1

K  1   β ˆ β 2  − k  . K

K   ˆ E = arg min 1 ˆ k0  d2E , 



0,αE >0 K

First, we take the derivative of (16) with respect to  and set it equal to zero, obtaining the KA-E estimator (see Appendix A for detailed derivations) K K    ˆE= 1 ˆ k0 = 1   αE 0 + (1 − αE ) ˆ k .

Second, we rewrite (18) to obtain an approximate result before taking the derivative with respect to αE , namely, (see Appendix B.1 for detailed derivations) K  K  2 2α  2 2  E  ˆ ˆ E = arg min αE ˆ k ¯ E  0 −   − k −   + cE ,

where cE =

(12)

(13)

where log A = U logU H , log = diag (log λ1 , log λ2 , · · · , log λ N ). The corresponding geometric mean is K   ¯ L = arg min 1 ˆk  d2L ,  k =1

K 2 1   ˆ k = arg min log − log  .

K

(14)

¯ L = exp 

1 K

k =1

2 K   ¯ ˆ k E −   is a constant. Nulling the derivative

k=1

αE , we obtain

2 K   ˆ ¯ E k −  

αˆ E = k=K 1 

2 .   ˆ  −   0 k

(20)



ˆk . log

Equation (20) can be further written as

ηE

αˆ E = 

 , 0 −  ¯ E 2

where

ηE = K12

to ensure

¯E= 

1 K

K 

(21)

2 K   ˆ ¯ E k −   . According to [33], we use min(1, αˆ E )

k=1

α less than 1. It is noted that when ˆ k = c k c kH and

k=1

c k c kH , namely, the SCM, (21) is the (20) in [33].

ˆ β = αP β + (1 − αP )  ˆ β , 0 ≤ αP ≤ 1.  0 k0 k

(22)

Then, we can obtain the similar result of the Euclid case.

k =1

K 

1 K

Similarly, the KA-PE covariance estimator of each cell can be modeled as

Similarly, taking the derivative of (14) with respect to  and we can obtain the LogE mean estimator



k =1

k =1

   dL ( A , B ) = tr (log A − logB ) (log A − logB )H ,

0

K

k =1

of (19) with respect to

k =1

0

(18)

i =1

(19)

When β = 1, β = −1 and β = 0.5, (12) is the Euclid mean, rightsided KL mean [38] and root-Euclid mean [35], respectively. In this paper we choose β = −1. • Log-Euclid distance

K

K

i =1

(11)

1/β .

(17)

k =1

0,αE >0 K

k =1

K 1  β ˆ  k K

k =1

K 2 1   ˆ k = arg min  − αE 0 − (1 − αE )   .

Taking the derivative of (10) with respect to  and we can obtain the PE mean estimator

¯P= 

(16)

where αE is the weighting factor, 0 is the prior covariance matrix of the CUT, which can be obtained from the SAR imagery [28], physics-based models [29], DEM [30] and history or simulation data [31,32]. Then substituting (16) into (8) leads to

K

where A = U U H , A β = U β U H ,  = diag (λ1 , λ2 , · · · , λ N ), λi (i = 1, · · · , N ), is the eigenvalues of A. The corresponding geometric mean is

¯ P = arg min 1 

In this section we derive our covariance estimators. The KA-E estimator of each cell can be modeled as [33]

0,αE >0 K

k =1

dP ( A , B ) =

3. Proposed estimators

k =1

Then, taking the derivative of (8) with respect to , and we can obtain the Euclid mean estimator

3

(15)

K   ˆ P == arg min 1 ˆ k0  d2P , 

0,αP >0 K

k =1

K  2 1   β β ˆβ  . = arg min  − αP 0 + (1 − αP )  k 

0,αP >0 K

k =1

(23)

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Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

Nulling the derivative of (23) with respect to β leads to (see Appendix A for detailed derivations)

 ˆP= 

K



1/β

K 1 

ˆ k0 

=

i =1

K 1 

K

1/β

αP 0 + (1 − αP ) ˆ kβ β



.

i =1

(24) Equation (23) (see Appendix B.2 for detailed derivations) can be recast as K  αP2  

ˆ P = arg min 

2 β ˆ β 0 −  k

0,αP >0 K

−

  ˆ β ¯ β 2 k − P  + cP ,

2αP K

where cP =

k =1

K  

(25)

 K    ¯ β ˆ β 2 P − k  is a constant. Nulling derivative of

1 K

k=1

αP , we obtain

ηP

 ,  β ¯ β 2 0 − P 

where

ηP =

 , log0 − log ¯ L 2

where

ηL =

1 K2

(31)

2 K      ˆ ¯ L logk − log  , and we also use min 1, αˆ L

k=1

to ensure α less than 1. So far, we obtain three new KA estimators. The proposed estimators are summarized in Table 1. There are two main characteristics of proposed covariance estimation. First, the estimators consist of the linear or nonlinear combination of prior covariance and estimated covariance. Second, compared with existing KA estimator, i.e., CC estimator, the proposed estimators use the geometric estimators instead of SCM estimator as the estimated covariance to improve the detection performance in non-Gaussian clutter [36, 38]. In next section, we conduct numerical experiments to demonstrate the advantage of our proposed estimator.

k =1

(25) with respect to

αˆ P = 

ηL

αˆ L = 

1 K2

(26)

 K      ˆ β ¯ β 2 k − P  . Then, using min 1, αˆ P we obtain

k=1

the KA-PE estimator. When it comes to the KA-LogE estimator, we model the estimation as

ˆ k0 = αL log0 + (1 − αL ) log ˆ k, log

0 ≤ αL ≤ 1.

(27)

4. Performance analysis In this section, we analyze the estimation and detection performance of the proposed estimators with the existing estimators. We compare the KA-E, KA-PE, and KA-LogE estimators with their natural competitors, namely, Euclid, PE and Log-E estimators. Also, we compare them with other regularized estimators, i.e., the CC [33] and SFP [43] estimators. First, we compare estimation performance of the proposed estimators with the existing estimators. Second, we use the adaptive normalized matched filter (ANMF) detector [44] with each estimator to further illustrate the detection performance of the proposed estimators. Finally, we analyze the detection performance via measured sea clutter data. The covariance matrix of speckle component can be modeled as

Then, we obtain

ˆ L = arg min 

K 1 

0,αL >0 K

d2E



ˆ k0 , 

 = ρ |i − j | exp ((i − j ) f c ) , 1 ≤ i , j ≤ N



where ρ is the one-lag correlation coefficient, N is the DOF of the system and f c = 0.05 is the normalized frequency of clutter. Therefore, the covariance matrix of clutter of each cell can be modeled as

k =1

K 2 1   ˆ k log − αL log0 − (1 − αL ) log  , 0,αL >0 K k =1

= arg min

(28) and null the derivative of (28) with respect to log , we have (see Appendix A for detailed derivations)

 ˆ L = exp 

K 1 

K

 = exp

 ˆ k0 

i =1 K  

1 K



.

(29)

i =1

Equation (28) (see Appendix B.3 for detailed derivations) can be recast as

ˆ L = arg min 

K  αL2  

2 ˆ k log0 − log 

0,αL >0 K

−

2αL K

where cL =

k =1

K  

2  ˆ ¯ L logk − log  + cL ,

(30)

2 K    ¯ ˆ k logL − log   is a constant. Then we take

k=1

derivative of (30) with respect to obtain the weighting factor

αL and set it equal to zero to

(33)

where τk follows the inverse Gamma distribution with shape parameter λ and scale parameter μ = 1, namely

1

μλ (λ)



τk −(λ+1) exp −

1

μτk

 ,

(34)

where λ denotes the shape parameter which represents the nonGaussianity of clutter and μ denotes the scale parameter, λ and μ are positive parameters. The prior covariance matrix 0 is taken as a perturbed matrix of the actual covariance matrix  and can be generated in the following way [34]

0 =   tt T ,

k =1 1 K

R k = τk , k = 0, 1, . . . , K ,

f (τk ) =



αL log0 + (1 − αL ) logˆ k

(32)

(35)

where  denotes the Hadamard matrix product, and t = [t 1 , t 2 , · · · , t N ]T is a vector of which each element follows the Gaussian distribution with mean 1 and variance σ 2 (t i ∼ N (1, σ 2 ), i = 1, 2, · · · , N). The perturbed vector t determines the effectiveness of the prior covariance matrix 0 . When the variance σ 2 is big, 0 can be regarded as invalid prior covariance matrix and the weighting factor α will become small. When the variance σ 2 is

Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

5

Table 1 Proposed estimator. Distance

Corresponding KA estimator K  

Euclid

ˆE = 

i =1

K 

¯E = 

K  

i =1 ˆP =⎢  ⎣

⎡ Log-Euclid

Basic estimator



K

⎡ Power-Euclid

αE 0 +(1−αE )ˆ k

αP β0 +(1−αP )ˆ kβ K

K  

i =1 ˆ L = exp ⎢  ⎣

Fig. 1. NFN of error matrix, N = 8, K = 8, f c = 0.05, λ = 3,

⎤1/β

K

,

⎤

⎛

⎥ ⎦

⎞1/β



⎟ ⎠

K 

αˆ P = min 1, 

ˆk log

k=1 ¯ L = exp ⎜  ⎝ K

⎞ ⎟ ⎠

η



P  ¯ P 2 0 −



αˆ L = min 1, 



 ¯ E 2 0 −

ηL



 ¯ L 2 0 −

4.2. Detection performance In this subsection, we address the problem of detecting a known complex signal vector s in received data, which can be formulated as a binary hypotheses test

In this subsection, we use the simulation data to measure the estimation performance of our estimators. We calculate the normalized Frobenius norm (NFN) of the error matrix [45] to display the estimation accuracy of the estimators. The NFN of the error matrix, defined as [45]



ˆβ  k

ηE

ρ = 0.9. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

4.1. Estimation performance

NFN 

K 

k=1 ¯P =⎜  ⎝ K

αL log0 +(1−αL ) textlog ˆ k



αˆ E = min 1, 

K

⎛

⎥ ⎦

small, 0 can be regarded as an effective prior covariance matrix and the weighting factor α will become big.

# "  ˆ E  − 

k=1

Color loading factor

ˆk 

(36)

ˆ is the covariance estimation. where  We calculate the mean of 10,000 Monte Carlo experiments of the NFN to obtain the NFN estimation values of each estimator, shown in Fig. 1. According to the NFN results, we can find that when σ 2 is small the value of NFN of the three proposed KA estimators are small and the KA-E has the smallest value, followed by KA-PE and KA-LogE. The CC estimator has the worst NFN result, though the prior covariance is accurate. On the other hand, when σ 2 becomes big, as shown in Fig. 1(b), the estimation accuracy of the KA-E and KA-PE estimators are close to their basic estimators, the NFN of KA-LogE is smaller than its basic estimator namely the LogE. For the CC estimator, the NFN result becomes much bad. The NFN value can suggest that the proposed estimators, especially the KA-E, have better estimation and detection performance than existing estimators.



H0 :

y0 = c0,

y k = c k , k = 1... K

H1 :

y 0 = as + c 0 ,

y k = c k , k = 1... K

where

,

(37)

s = √1 [1, exp ( j2π f d ) , exp ( j2π 2 f d ) , · · · exp( j2π ( N − N

1)) f d )]T , f d is the normalized Doppler frequency of target. a is an unknown deterministic parameter, which accounts for both the target and the channel effects. We use the ANMF detector [46] with the KA-E, KA-PE, KA-LogE, Euclid, PE, LogE, CC and SFP estimators, respectively, to construct our detectors. The structure of ANMF detector is

$ $ $ H ˆ $2 $s  y 0 $ H1   ≷ ηANMF , ˆ −1 s y H  ˆ −1 y 0 H0 sH  0

(38)

where ηANMF is the threshold of ANMF. In addition, when the covariance  of clutter is known, we call equation (38) the normalized matched filter (NMF) detector which provides the upper bound on the detection performance for the ANMF detector. 4.2.1. CFAR property This subsection illustrates the constant false alarm ratio (CFAR) property of proposed detectors. The standard Monte Carlo tech-

6

Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

Fig. 2. Threshold against λ, N = 8, K = 8, f c = 0.05, f d = 0.2, ρ = 0.9.

Fig. 3. PD against

ρ , N = 8, K = 8, f c = 0.05, f d = 0.2, λ = 3.

nique based on 100/PFA independent trails is employed to obtain the detection thresholds for a given probability of false alarm (PFA). The PFA is set as 10−4 to alleviate the computational burden, namely, we use 106 independent trails to obtain the detection thresholds. We change the shape parameter λ and the one-lag correlation coefficient ρ to obtain the corresponding thresholds. The results are shown as Fig. 2 and Fig. 3. It is shown that the detection threshold of the proposed detectors do not dramatically alter with the change of λ or ρ . In other words, the proposed detectors have the CFAR property. However, the threshold of ANMF with CC changes responding to the two parameters, therefore the ANMF with CC do not maintain CFAR property. 4.2.2. Detection property The detection probabilities (PD) of detectors are computed via 104 Monte Carlo experiments. In the compound-Gaussian case, the signal-to-clutter ratio (SCR) is defined as [47]

SCR =

E{a2 } E{τ }

(39)

, μ

where E{τ } = (λ−1) for λ > 1. In all simulations, we consider a coherent pulse train of N = 8 and set the normalized frequency of the target f d = 0.2.

Fig. 4 depicts the probability of detection (PD) with the σ 2 = 0.1 and σ 2 = 0.9 when the number of secondary data K = 1N. The SCR varies from -5 to 25 dB with an interval of 1 dB. Comparing Fig. 4 (a) with Fig. 4 (b), it can be noted that the ANMF detector with the proposed estimators have better detection performance than other detectors. When the prior knowledge is accurate, the ANMF detectors with the KA-E and KA-PE are close to the optimal performance, namely, the NMF detector. The KA-E has slight advantage with the KA-PE. When the prior knowledge becomes imprecise, the KA-E and KA-PE are still better than other estimators but the KA-LogE falls behind the SFP due to the bad basic estimator the LogE. The performance of the CC estimator has a relatively bad performance because of the sample starvation and non-Gaussianity clutter, which can be reflected in Fig. 4. The SFP estimator has similar performance with the Euclid and PE estimators. We further reduce λ and ρ to generate more spiky clutter. As shown in Fig. 5, overall the detection performances are declined but the proposed detectors are still better than existing detectors. Moreover, for the chosen environment parameters, the prior covariance matrix has little effect on detection performance of the KA-E and KA-PE. At the case of imprecise prior knowledge (Fig. 5(b)), just the KA-LogE and CC have roughly 1 dB and 3 dB decline when PD = 0.8.

Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

7

Fig. 4. PD against SCR, N = 8, K = 8, f c = 0.05, f d = 0.2, λ = 3, ρ = 0.9.

Fig. 5. PD against SCR, N = 8, K = 8, f c = 0.05, f d = 0.2, λ = 1.5, ρ = 0.5.

To further verify the superior performance of our proposed estimators, we use the receiver operating characteristic (ROC) curve shown in Fig. 6 to reflect the PD against PFA under SCR=10 dB and 0 dB. The results of ROC curve indicate that the proposed estimators perform better than the SFP and CC estimator in different PFAs. The KA-E has the best performance and the CC and LogE are not work well. To some extent, the threshold can reflect the detection performance of detectors. Therefore, we calculate the threshold of each detector to further analyze the performance of each estimator. Fig. 7 shows the threshold of the ANMF detectors under difference PFAs. In general, the threshold goes down as the PFA rise. From Fig. 7 we can find that the KA-E, KA-PE and their basic estimators have similar threshold and smaller than other detectors. The LogE has the largest value of threshold so its detection result is the worst. Moreover, the prior knowledge has little impact on the threshold of KA-E and KA-PE, which can infer that the proposed detector need not the knowledge to keep the CFAR property. Fig. 8 analyzes the effect of target speed on the PD. The SCR is set to be 10 dB, the normalized Doppler of the target varies from -0.5 to 0.5 with an interval of 0.05. The PD of KA-E and KA-PE estimators have relatively narrower notch than the CC, SFP and their basic estimators. Therefore, the proposed estimators have better detection performance when target Doppler near the clutter frequency. In addition, all the detectors tend to similar detection

performance when the target frequency is away from the clutter frequency, except for the CC and LogE. When the knowledge becomes bad, the ANMF with KA-E and KA-PE estimators are slight better than the ANMF with the SFP and the ANMF with CC estimator degrades seriously. The similar results can be found in Fig. 9, which we fix the target Doppler as 0.2 and change the Doppler center of clutter to obtain corresponding PDs. 4.3. Measured sea clutter data In this subsection, we apply the ANMF detector with proposed estimators to one set of measured data, the McMaster IPIX radar in Grimsby, which is the typical case of the sea clutter. Moreover, a detailed statistical analysis of the adopted data has been conducted in [31,48]. As for the IPIX Radar, we use the 19980223_170435_ANTSTEP.CDF data [48] to analyze the proposed detector. The data consist of 60,000 pulses and 34 range cells. The IPIX data is shown in Fig. 10, the strongest clutter appears at the 25th range cell and about the 40,000th to 60,000th pulses. Therefore, we choose the 25th% range cell as CUT, exploiting K = 1N surrounding cells [21, 24] [26, 29] as training data. We apply the statistical tests to the N = 8 temporal samples. Using the (50, 000 − N + 1) th to 60,000th available pulses via standard Monte Carlo technique to obtain the detection threshold for a given PFA=10−2 since measured data is not enough to calculate

8

Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

Fig. 6. ROC curve, N = 8, K = 8, f c = 0.05, f d = 0.2, λ = 3, ρ = 0.9.

Fig. 7. Threshold against PFA, N = 8, K = 8, f c = 0.05, f d = 0.2, λ = 3, ρ = 0.9.

Fig. 8. PD against f d , N = 8, K = 8, f c = 0.05, λ = 3, ρ = 0.9, SCR=10 dB.

Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

Fig. 9. PD against f c , N = 8, K = 8, f d = 0.2, λ = 3, ρ = 0.9, SCR=10 dB.

Fig. 10. IPIX radar clutter data.

the threshold with less value of PFA. The PDs of each detector are also obtained via the same data. We add a point target in the 25th range cell for different SCRs and set the normalized Doppler of the target as f d = 0.2. According to [49], we use the MLE to estimate the mean parameter values of the (50, 000 − N + 1) th to 60,000th pulses and the parameters are λ = 1.5383 and μ = 1.7745. According to [28,32], we use the previous N t = 107 temporal samples of current CUT as the historical data to estimate 100 covariance matrices. Then, we obtain the prior covariance matrix 0 of current CUT via calculating the mean of the 100 covariance matrices 0i . Precisely, as shown in the Fig. 11, we slide the window to obtain the ith covariance matrix estimation 0i , namely

0i =

N y 0i y 0i H

H

y 0i y 0i

(40)

,

where y 0i is the ith sliding vector in the CUT. Then, we average all of 0i to obtain the prior knowledge covariance matrix, namely

0 =

1 Nt − N + 1

Nt − N +1 i =1

0i .

(41)

Fig. 11. Procedure for estimating 0 .

9

10

Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

5. Conclusions In this paper, we have proposed three estimators, namely, the KA-E, KA-PE and KA-LogE estimators, for adaptive signal detection in non-Gaussian clutter under the small number of secondary data. The KA-E KA-PE and KA-LogE can achieve improved signal detection performance under small number of secondary data. From experimental results, it is found that the quality of prior knowledge has the important effect on improving the detection performance. When the precision of prior covariance is unknown, among three proposed estimators, the KA-E has the best performance. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Fig. 12. PD against SCR for IPIX Radar data, N = 8, K = 8, f d = 0.2.

Appendix A. Derivation of (18), (24) and (29) For the geometry mean

¯ = arg min A 0

K 1 

K

 A − A k 2 ,

(A.1)

k =1

first we prove that the problem (A.1) is convexity then we obtain the geometry mean. Proof. For the function f ( A ) =

1 K

K  k=1

 A − Ak 2 we have

K  ∂ A − Ak 2

∂A

k =1

Fig. 13. PD against f d for IPIX Radar data, N = 8, K = 8, SCR=15 dB.

As shown in Fig. 12, the detection results of all the detectors are worse than the simulation results in Fig. 4, because of the nonGaussianity of measured data is more serious than the simulation data. From Fig. 12 we can find that the proposed estimators are better than the existing estimators and the ANMF with KA-E estimator has the best performance, the PDs can raise rapidly. In contrast, the PDs of KA-LogE raise slowly and its basic estimator has a worse detection performance. The performance of KA-PE is fall in between KA-E and KA-LogE. Among three basic geometric mean estimators, the Euclid and Power-Euclid mean estimators are better than the SFP. Therefore, in the worst condition, it is believed that the KA-E and KA-PE estimators can better than the SFP. In summary, the KA-E estimator has the best property followed by the KA-PE and KA-LogE. Fig. 13 depicts the PD versus normalized Doppler of target. From the graph we can find the clutter frequency range approximately from -0.1 to 0.1 and near the clutter frequency the detection performance drops rapidly. The KA-E has the narrowest notch while the CC estimator has the widest notch. The ANMF with SFP estimator only better than the ANMF with the LogE and CC estimators. Summarizing the simulation and measured data results, we can find that, the three proposed KA estimators are better than their natural competitors and the existing regularized methods. Among proposed estimators, the KA-E has the best performance both in estimation and detection.

  K  ∂ tr ( A − A k ) ( A − Ak )H = ∂A k =1   K  ∂ tr A A H − A AkH − Ak A H + Ak A kH = ∂A k =1         K  ∂ tr A A H − ∂ tr A AkH − ∂ tr A H Ak + ∂ tr Ak AkH = . ∂A k =1

(A.2) When A and A k are HPD, we can obtain A = A and Then we arrive at H

A kH

= Ak .

        K  ∂ tr A A H − ∂ tr A AkH − ∂ tr A H Ak + ∂ tr Ak AkH ∂A k =1

=

K  ∂ tr ( A A ) − 2∂ tr ( A Ak ) + ∂ tr ( Ak Ak )

=2

K 

(A.3)

∂A

k =1

( A − Ak ).

k =1

Hence, we obtain

∂ 2 f ( A) ∂ A2

= 2K I > 0.

(A.4)

Therefore, the optimization problems are convexity. Furthermore, the solution of (A.1) is unique because of when A → +∞.

1 K

K 

k=1

 A − Ak 2 → +∞

Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

Therefore, the geometry mean can be obtained via nulling (A.3), namely

A=

K 1 

K

Ak .

(A.5)

k =1

For the KA-E, KA-PE and KA-LogE, A is , β and log , reˆ k , αP β + (1 − αP )  ˆ β and spectively. And A k is αE 0 + (1 − αE )  0 k

αL log0 + (1 − αL ) logˆ k , respectively. Then we can obtain the results (18), (24) and (29).

lim

K 1 

K →∞

K

11

ˆ k = . 

(B.3)

k =1

¯ E , to replace  since the true covariance maWe use (9), namely  trix  is unknown and we have K 2αE 

K

 

¯H ˆH ¯ ˆH ¯ ¯H Re tr 0  E − 0 k + E k − E E



= 0.

(B.4)

k =1

Then we obtain the approximate (17) given in (19). B.2. Derivation of (25)

Appendix B. Derivation of (19), (25) and (30) Similarly the (23) can be recasted as B.1. Derivation of (19)

ˆ E is obtained in this ApThe detail formula derivation of  pendix. Let recast (17) as

 2 β ˆ P = arg min 1  ˆ β  β − αP 0 − (1 − αP )  k 0,αP >0 K

= arg min

 2 ˆ E = arg min 1  ˆ k   − αE 0 − (1 − αE )  

0,αP >0 K

0,αE >0 K

K 1 

= arg min

0,αE >0 K

−

K 1 

K

2 

k =1

 

K 1 

K 1 

K



2 E 0

α

K

" 

2αE Re tr



= arg min

K 1 

0,αE >0 K

−

1

K 

ˆ H +  ˆ H − H 0  − 0  k k

−

#

k =1

+

 2  ˆ  2αE Re  −   k  

ˆ H +  ˆ H − H 2αE Re tr 0 H − 0  k k



− −

According to (B.1), we obtain K  αE2  

0,αE >0 K

−

K

2 ˆ k 0 −  

K

β

H

 β ˆβ H − 0  k

k =1

K  1   β ˆ β 2  − k  . K

K 

K 1 K

+

k =1

 2  ˆ  2αE Re k −  + cE

1

K  k =1 K 

(B.2)

 

 ˆ H +  ˆ H − H . Re tr 0 H − 0  k k

k =1

¯ E and  are closest in Euclid distance when K According to (9),  tends to infinity, namely



k =1

 

2  

ˆ − β  2αP Re  k β

+ cP

&   H β β ˆβ H 2αP Re tr 0 β − 0  k

k =1

β

k =1

2αE

& 

2αP Re tr 0 β

0,αP >0 K

(B.1)

−

 2  ˆβ β 2αP Re k −  

K  2  ˆ P = arg min 1  αP2 β0 − ˆ kβ 

k =1

K 1 

(B.5)

k =1

k =1

ˆ E = arg min 

K

2

k =1

k =1

K 1 



αP2 β0 − ˆ kβ 

Thus, we obtain

K 2 1   ˆ k +  −   .

K

K

k =1

'  H   ˆ β − β β H + β  k

 2 αE2 0 − ˆ k 

k =1

K 1 

K

−



K  1   β ˆ β 2  − k  K

 H ' ˆβ ˆβ  + k k

K 1 

0,αP >0 K K 1 

2

k =1

k =1

= arg min

H

ˆ k ˆH− ˆ H + H ˆ k H −  +  k k

K



i =1

i =1

K



αP2 β0 − ˆ kβ  +

&   H β β ˆβ H 2αP Re tr 0 β − 0  k

− k 

K  2 1  2  ˆ k ˆ k −  +  −  

k =1

K 1 

  ˆβ β H

k =1

ˆH− ˆH ˆ k H +  ˆ k 2αE Re tr 0 H − 0  k k

0,α >0 K

−

−

K 2 1   ˆ k  −   K

k =1

= arg min −

 

αE2 0 − ˆ k  +

K 1 

ˆβ  k

H



 β H

−  β

(B.6)

' .

¯ P , to replace  since  ¯ P and  Similarly, we use (12), namely  are closest in Power Euclid distance and we have K 2αP 

K

& 



β H

¯ Re tr 0  P β

 β ˆβ H − 0  k

k =1

 H  H ' ˆβ − ¯β ¯β  ¯β  = 0, + P P P k and we obtain the approximate (23) given in (25).

(B.7)

12

Z. Shang et al. / Digital Signal Processing 97 (2020) 102616

B.3. Derivation of (30) Recasting (27) as

 2 ˆ L = arg min 1  ˆ k  log  − αL log 0 − (1 − αL ) log   0,αL >0 K

= arg min

K 1 

0,αL >0 K

+ −

1 K

2 

k =1

K  

2  ˆ k log  − log  

k =1

K 1 

K

 

αL2 log 0 − log ˆ k 

 

ˆH 2αL Re tr log0 logH − log0 log k

k =1

ˆ k logH + log ˆ k log ˆH − log k = arg min

K 1 

0,αL >0 K

− −

K 1 

K



2 ˆ k − log  

 

2  

2 L log 0

α

ˆ k − log  2αL Re log   

ˆH 2αL Re tr log 0 log H − log 0 log  k

k =1

ˆ H − log  log H + log  log  k +

(B.8)

k =1

K 1 

K

k =1





K 2 1   ˆ k log  − log   , K k =1

¯ L , and leading to replacing  by (15), namely  K 2αL 

K

 

ˆ 0 log  ¯H ˆH Re tr log  L − log 0 log k

i =k

¯ L log H − log  ¯ L log  ¯H + log  L k



(B.9)

= 0.

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Zheran Shang received the B.S. and M.S. degrees from Wuhan Radar Academy, Wuhan, in 2014 and 2016, respectively. He is currently working toward the Ph.D. degree at National University of Defense Technology, Changsha, China. His current research interests include statistical signal processing and target detection. Kai Huo was born in Hubei, China in 1983. He received the B.S. degree in communication engineering in 2005 and Ph.D. degree in electronic science and technology in 2011 from National University of Defense Technology (NUDT), China. Currently he is a lecturer at NUDT. His research interests include radar signal processing and feature extraction.

13

Weijian Liu received the B.S. degree in information engineering and M.S. degree in signal and information processing, both from Wuhan Radar Academy, Wuhan, China, and the Ph.D. degree in information and communication engineering from National University of Defense Technology, Changsha, China, in 2006, 2009, and 2014, respectively. Now he is a lecturer at Wuhan Radar Academy. His current research interests include multichannel signal detection and statistical and array signal processing. Sun Yang was born in 1992. He received his B.S. degree and M.S. degree in electronic engineering from Information Engineering University in 2014 and 2017 respectively. Now he is Ph.D. candidate in information and communication engineering of National University of Defense Technology. His research interests are signal processing and image processing. Yong-Liang Wang received his Ph.D. degrees in electrical engineering from Xidian University, Xian, China, in 1994. From June 1994 to December 1996, he was a postdoctoral fellow with the Department of Electronic Engineering, Tsinghua University, Beijing, China. He has been a full professor since 1996, and he was the director of the Key Research Laboratory, Wuhan Radar Academy, Wuhan, China, from 1997 to 2005. 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. He has authored or coauthored three books and more than 200 papers. His recent research interests include radar systems, space-time adaptive processing, and array signal processing. Dr. Wang is a member of the Chinese Academy of Sciences and also a Fellow of the Chinese Institute of Electronics.