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Digital Signal Processing 46 (2015) 41–48

Contents lists available at ScienceDirect

Digital Signal Processing

www.elsevier.com/locate/dsp

Adaptive array detection in noise and completely unknown jamming

Weijian Liu

a,b

, Jun Liu

c,d

, Libao Wang

, Keqing Duan

, Zhao Chen

, Yongliang Wang

b,∗

College of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China

Wuhan Radar Academy, Wuhan 430019, China

National Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China

Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China

Unit 95992 of PLA, Beijing, 100162, China

a r t i c l e i n f o a b s t r a c t

Article history:

Available

online 4 August 2015

Keywords:

Array

signal processing

Generalized

likelihood ratio test

Signal

detection

Unknown

jamming

Wald

test

The presence of jamming usually degrades the detection performance of a detector. Moreover, suﬃcient

information about the jamming may be diﬃcult to be obtained. To overcome the problem of adaptive

array signal detection in noise and completely unknown jamming, we temporarily assume the jamming

belongs to a subspace which is orthogonal to the signal steering vector in the stage of detector design.

Consequently, by resorting to the criteria of generalized likelihood ratio test (GLRT) and Wald test,

we propose two adaptive detectors, which can achieve signal detection and jamming suppression. It is

shown, by Monte Carlo simulations, that the two proposed adaptive detectors have improved detection

performance over existing ones.

1. Introduction

Detection of a multichannel signal in unknown disturbance is a

hot topic in the ﬁeld of array signal processing. The most pioneer-

ing

and prominent detectors are Kelly’s generalized likelihood ratio

test (KGLRT) [1], adaptive matched ﬁlter (AMF) [2,3], and adap-

tive

coherence estimator (ACE) [4]. Particularly, in [1] the signal

has a known steering vector but with an unknown amplitude, and

the noise is Gaussian distributed with an unknown covariance ma-

trix.

To estimate the covariance matrix, it is assumed that a set of

independent and identically distributed (IID) training data is avail-

able.

Consequently, the KGLRT is proposed according to the GLRT

criterion. The AMF is designed for the same detection problem

in [1], but it is obtained according to the two-step GLRT (2S-GLRT)

criterion [2,3]. The KGLRT and AMF are both conceived for the ho-

mogeneous

environment, where the training data and the test data

share a common noise covariance matrix. In contrast, the ACE is

devised in [4] based on the GLRT criterion for the partially homo-

geneous

environment, where the test data and training data share

the same noise covariance matrix only up to an unknown scaling

factor. The KGLRT, AMF, and ACE are all for the point-like target

detection, which is further investigated in [5–8] recently. More-

Corresponding author.

E-mail

addresses: liuvjian@163.com (W. Liu), junliu@xidian.edu.cn (J. Liu),

mimosar_wlb@163.com (L. Wang), duankeqing@aliyun.com (K. Duan),

tzksky@126.com (Z. Chen), ylwangkjld@163.com (Y. Wang).

over, the problem of distributed target detection is dealt with in

[9–13, and the references therein].

Note

that all the cited references above do not take into ac-

count

jamming. In practice, however, there usually exists inten-

tional

or unintentional jamming [14]. Suppression of deceptive

jamming is considered in [15], where the jamming is rejected by

multiple-input multiple-output (MIMO) radar with frequency di-

verse

array (FDA). A mainlobe jamming suppression method is

proposed in [16] based on eigen-projection and covariance matrix

reconstruction. An intrusion detection system (IDS) framework for

jamming detection and classiﬁcation is proposed in [17] for wire-

less

networks. The problem of detecting chaff centroid jamming is

addressed in [18], and it is solved with the aid of the global posi-

tioning

system (GPS) and inertial navigation system (INS). In [19]

the

jamming is deterministic and lies in a known subspace, many

GLRT-based detectors are designed. For convenience, the jamming

model in [19] is referred to as the subspace jamming, which is also

considered in [20], but it is assumed to lie in both the test and

training data, and a detector is proposed based on the method of

sieves. The problem of signal detection in subspace jamming is fur-

ther

investigated in [21–24], where the potential target is spread

in the range domain.

Remarkably,

in most of the aforementioned references involved

jamming it is assumed that some information about the jamming

is known in advance. However, in practical applications it may be

very diﬃcult to obtain suﬃcient knowledge about the jamming.

This brings a great challenge for signal detection. How to model

the completely unknown jamming and devise effective detectors is

http://dx.doi.org/10.1016/j.dsp.2015.07.006

1051-2004/

November 2015

42 W. Liu et al. / Digital Signal Processing 46 (2015) 41–48

the main motivation of this paper. Particularly, we focus on array

signal detection of a point-like target in the presence of completely

unknown jamming. An ad hoc model for the jamming is adopted

at the stage of detector design. Precisely, we temporarily assume

that it lies in a subspace orthogonal to the signal steering vec-

tor.

Subsequently, we propose two adaptive detectors according to

the GLRT and Wald test criteria. These two detectors admit cer-

tain

intuitive physical interpretations, and they can achieve signal

integration and jamming suppression simultaneously. For the per-

formance

evaluation, the cases of unknown (completely unknown

or partially unknown) jamming and no jamming are all considered.

It is shown that in the presence of unknown jamming, the two

proposed detectors exhibit improved detection performance over

the existing ones. Moreover, in the case of no jamming the pro-

posed

detector, derived according to the GLRT criterion, can still

provide slightly better detection performance than the existing de-

tectors

in some situations.

The

remainder of the paper is organized as follows. Section 2

formulates

the problem to be solved. Section 3 gives the pro-

posed

detectors and shows some important properties of them.

Numerical examples are provided in Section 4. Finally, Section 5

summarizes

the paper.

2. Problem formulation

Suppose the data are received by an N -element uniform linear

array (ULA). We want to discriminate between a binary hypothe-

sis

test, namely, hypothesis H

that a useful signal s

exists in the

data under test, which is denoted by an N × 1vector x and hy-

pothesis

that no useful signal exists in x. The useful signal s

if present, has the form s

=as, where a is the unknown nonzero

signal amplitude and s is a known normalized signal steering vec-

tor.

To sum up, the detection problem can be symbolically written



:a = 0,

:a = 0.

(1)

The normalized signal steering vector has the form

s =



1, e

j2π f

,...,e

j2π(N−1) f



√

N (2)

where f

=d cos θ

/λ, d is the interelement spacing, λ is the wave-

length,

is the angle of the target with respect to (w.r.t.) the array,

and the symbol (·)

is the transpose operation. To avoid grating

lobe, d is set to be d = λ/2. Thus f

∈[−0.5, 0.5] and f

is usually

called the normalized spatial frequency.

Besides

the possible signal, the test data x also contains distur-

bance

d, which consists of colored noise n (including clutter and

white noise) and jamming j. The noise n is modeled as a zero-

mean

complex circular Gaussian vector with an unknown covari-

ance

matrix R, which is positive deﬁnite Hermitian. The jamming

j is completely unknown. For the detector design, we temporarily

assume that j is deterministic and lies in a subspace spanned by

an N × (N − 1) matrix U

⊥

, which is a semi-unitary matrix such

that U

⊥

s =0

(N−1)×1

and U

⊥

= I

N−1

, with (·)

being the con-

jugate

transpose. Hence, j can be expressed as

j = U

⊥

α, (3)

where α is an (N −1) ×1 unknown coordinate vector. The rationale

of such a model is explained below. Note that if we deﬁne

B =[s, U

⊥

], (4)

which is an N × N unitary matrix, then B can be taken as a basis

of the entire space C

N×N

. Therefore, there exists an N × 1vector

b such that

j = Bb =a

s + U

⊥

α (5)

where b =[a

, α

]

and a

is a scalar. Equation (5) can be recast

as j = j

+ j

⊥

, where j

s and j

⊥

= U

⊥

α. Note that the com-

ponent

is the part of the jamming projected onto the signal

subspace s, with · standing for the subspace spanned by the

matrix/vector argument. Moreover, we have j

= P

j for a given

j, where P

= ss

is the orthogonal projection matrix onto the

signal subspace s. For ULAs, the signal steering vector is often

Vandermonde [25], such as (2), and the array response drops off

very quickly if the angle between the jamming and signal exceeds

the beamwidth [26]. Therefore, j

is usually small, especially for

the jamming with low or moderate power. Hence, (5) can be ap-

proximated

by (3).

customary, we also assume that a set of IID training data,

denoted by x

, l =1, 2, ..., L, is available. x

only contains noise n

which shares the same statistical property with n.

3. The proposed detectors

3.1. The Wald test approach

Let Θ be a parameter vector, partitioned as

Θ =



, Θ



(6)

where Θ

=a and Θ

=[α

, vec

(R)]

, with vec(·) being the vec-

torization

operation. Then the Wald test can be devised according

to the formula [27]

Wald

−Θ

)



−1

(

)



,Θ



−1

(

−Θ

), (7)

where

is the maximum likelihood estimate (MLE) of Θ

un-

der

, Θ

is the value of Θ

under H

, [I

−1

(

)]

,Θ

is the

(Θ

, Θ

)-part of I

−1

(Θ), evaluated at

, namely, the MLE of Θ

under H

, and

I(Θ) = E



∂

ln f

(x, X

)

∂Θ

∗

∂ ln f

(x, X

)

∂Θ



(8)

is the Fisher information matrix (FIM) for Θ [27]. The notations

E[·], ∂(·), (·)

∗

, and ln(·) stand for the statistical expectation, partial

derivative, conjugate, and natural logarithm, respectively.

The

joint PDF of x and X

 [x

, x

, ..., x

] for the problem

in (1) under H

(x, X

) = c det(R)

−(L+1)

exp



−



−1



−

−1



(9)

where c = π

−N(L+1)

, det(·) denotes the determinant of a matrix,

= x −as − U

⊥

α, and S is the sample covariance matrix (SCM)

deﬁned as

S = X

. (10)

Taking the logarithm of (9) and performing the derivative w.r.t. a

and

∗

, respectively, yield

∂ ln f

(x, X

)

∂a

= x

−1

s, (11)

∂ ln f

(x, X

)

∂a

∗

= s

−1

. (12)

Substituting (11) and (12) into (8) results in

,Θ

(Θ) = s

−1





−1

s = s

−1

s, (13)

where we have used the fact that E[x

] = R under H

. Taking

the derivative of (11) w.r.t. α

∗

or vec

∗

) and performing the

expectation operation yield the fact that I

,Θ

(Θ) is a null vector.

As a consequence, we have



−1

(Θ)



,Θ



−1



,Θ

(Θ)



= s

−1

s. (14)

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