Design of MIMO radar waveform covariance matrix for Clutter and Jamming suppression based on space time adaptive processing

Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Contents lists available at ScienceDirect

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Signal Processing

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journal homepage: www.elsevier.com/locate/sigpro

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Design of MIMO radar waveform covariance matrix for Clutter and Jamming suppression based on space time adaptive processing

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Bo Tang a,n, Jun Li b, Yu Zhang a, Jun Tang c,nn

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a

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b c

Electronic Engineering Institute, Hefei 230037, China National Key Laboratory of Radar Signal Processing, Xidian University, Xi'an 710071, China Department of Electronic Engineering, Tsinghua University, Beijing 100084, China

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a r t i c l e i n f o

abstract

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Article history: Received 15 January 2015 Received in revised form 20 October 2015 Accepted 26 October 2015

This paper studies the optimization of waveform covariance matrix (WCM) for airborne multiple-input-multiple-output (MIMO) radar systems in the presence of clutter and jamming. The goal is to enhance the target detection performance by suppressing the clutter and jamming based on space time adaptive processing (STAP). We employ the signal-to-interference-plus-noise ratio (SINR) as the figure of merit. Assuming a known target steering vector, we recast the WCM design problem into a convex optimization problem. Through a max-min approach, we also make the designed WCM robust to the target steering vector, i.e., we develop a method to design WCM that maximizes the worst-case SINR associated with an uncertainty set. We explicitly derive the target steering vector corresponding to the worst-case SINR and solve the robust design of WCM via convex optimization. Finally, we provide several numerical examples to demonstrate the superiority of the proposed algorithms over the existing methods. & 2015 Published by Elsevier B.V.

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Keywords: Multiple-input-multiple-out (MIMO) radar Space time adaptive processing (STAP) Waveform design Interference suppression Robust design

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1. Introduction

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Multiple-input-multiple-output (MIMO) radar is an emerging technology which has attracted considerable interests in recent years (see, e.g., [1–4] and the references therein). Different from traditional phased-array radar, MIMO radar system has the capability of transmitting multiple independent waveforms simultaneously. Currently, two typical configurations of MIMO radar have been proposed. The first type is called statistical MIMO radar (i.e., MIMO radar with widely separated antennas) [3]. By exploiting spatial diversity, statistical MIMO radar can improve the target detection performance and localization

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n

Principal Corresponding author. Tel.: þ86 055165927461. Corresponding author. Tel.: þ86 01062798339. E-mail addresses: [email protected] (B. Tang), [email protected] (J. Tang). nn

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accuracy [3,4]. The colocated MIMO radar is the second type providing better parameter identifiability and clutter suppression capabilities [1,5,2]. For both types of MIMO radar, we can obtain additional performance enhancement through transmitting suitable designed waveforms. Therefore, waveform optimization for MIMO radar has received significant attention (see, e.g., [6–17] and the references therein). In particular, the optimized waveform covariance matrix (WCM, or signal cross-correlation matrix) has been shown to play a central role in enhancing the performance of MIMO radar. In [6,7], the authors showed that MIMO radar could synthesize a desired transmit beampattern flexibly through the design of WCM. Considering that the steering vectors are subject to uncertainties in practice, the authors of [18,19] proposed robust designs of the WCM that had improved transmit beampattern. In [9], the authors demonstrated that the radar system could achieve better parameter estimation

http://dx.doi.org/10.1016/j.sigpro.2015.10.033 0165-1684/& 2015 Published by Elsevier B.V.

61 Please cite this article as: B. Tang, et al., Design of MIMO radar waveform covariance matrix for Clutter and Jamming suppression based on space time adaptive processing, Signal Processing (2015), http://dx.doi.org/10.1016/j. sigpro.2015.10.033i

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B. Tang et al. / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

accuracy (or lower the Cramér-Rao bound, equivalently) by designing WCM. In [8], it was shown if the WCM shared the same eigenvectors as those of the target covariance matrix, the WCM maximized the mutual information between the received signal and target scattering matrix as well as minimized the minimum mean square error (MMSE) of target parameter estimation. In [20,21], the authors designed the WCM to optimize the mutual information and the lower Chernoff bound, showing significant enhancement of the target detection performance. Detecting ground-moving targets is one important application of MIMO radar systems. Typically, such a radar system is mounted on an airborne platform. Due to the nonzero relative motion between the radar system and ground, the clutter spectrum is spread over the whole Doppler frequency and the system detection performance is limited by the strong clutter and (possible) intentional jamming. Therefore, we have to suppress the clutter and jamming simultaneously to improve the detection performance. As a well-established technique in radar society, space time adaptive processing (STAP), which refers to the processing of signals from multiple antennas and multiple pulses, has the capability of jamming mitigation and clutter cancellation (see, e.g., [22,23]). In addition, recent studies have shown that, compared with conventional singleinput-multiple-output (SIMO) STAP systems, MIMO-STAP systems (i.e., MIMO radar systems with STAP) have much sharper clutter notches and improved minimum detectable velocity (MDV) performance [1,24–26]. Other benefits of a MIMO-STAP system include lower probability of intercept (LPI), increased Doppler resolution, reduced clutter level and related hardware requirement, etc. In this paper, we focus on the WCM design problem for MIMO-STAP systems to enhance the weak-target detection performance. In [27], the authors have discussed the waveform optimization problem under a MIMO-STAP architecture. Therein, they used the output signal-tointerference-plus-noise ratio (SINR, here interference means clutter and possible jamming) as the criterion. They proposed diagonal loading of the clutter covariance matrix to formulate the waveform design problem as a convex optimization problem. However, the loading factor was chosen in a rather ad hoc way and its selection remained unsolved. In addition, the solution associated with the diagonal loading approach is suboptimal. Considering the target steering vector uncertainty, the authors in [28] studied the robust waveform design of MIMO STAP. By resorting to a max–min approach, they attempted to design waveforms which could maximize the worst-case SINR (over an uncertainty set of target steering vectors). However, the relaxed constraint (on the target steering vector) and the diagonal loading of the WCM also make the designed waveforms suboptimal. This paper employs the same signal model as that in [27] and [28], and proposes new algorithms to design WCM for MIMO-STAP systems. For the case where the target steering vector is exactly known, we obtain the optimal WCM without diagonal loading and the performance associated with the designed WCM is superior to that in [27]. For the case of uncertain target steering vectors, we derive the “worst” steering vector that leads to the

smallest SINR (over the uncertainty set of target steering vectors) and reformulate the waveform design based on the maximization of worst-case SINR into a convex optimization problem. The proposed algorithm avoids the problem of loading factor selection and enjoys better performance. The rest of paper is organized as follows. We present the signal model in Section 2. We solve the WCM design with exactly known target steering vectors in Section 3. In Section 4, we solve the robust WCM design for the case of uncertain target steering vectors. We provide several numerical examples in Section 5 to demonstrate the performance of the proposed algorithm. Finally, we conclude the paper in Section 6. Notations: Throughout this paper, matrices are denoted by bold capital letters, and vectors are denoted by bold lowercase letters. Superscript ðÞT , ðÞ and ðÞH denote transpose, complex conjugate and conjugate transpose, respectively. vecðXÞ indicates the vector which is obtained by column-wise stacking of the matrix X. trðÞ indicates the trace of a square matrix. C denotes the set of complex numbers, and Cmn are the sets of matrices of size m  n with entries from C. IM denotes an identity matrix of size M  M. For A A Cmm , A≽ð g Þ0 indicates A is positive semidefinite (definite). The symbols  and  denote Kronecker and Hadamard product, respectively. Finally, EðxÞ denotes the expectation of a random variable x.

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2. Signal model 93 Consider an airborne colocated MIMO radar with NT transmit antennas and N R receive antennas, as illustrated in Fig. 1. Assume for simplicity both of the transmit and receive arrays are equispaced, with inter-element spacings dT and dR , respectively. Then the signal reaching the ground moving target can be represented by [29,30] aT ðθt ÞS;

ð1Þ j2π f t;s

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j2π f t;s ðNT  1Þ T

; …; e  denotes the where aðθt Þ ¼ ½1; e transmit array steering vector of the target, f t;s ¼ dT sin θt =λ is the target spatial frequency, λ is the wavelength, θt is the EL target cone angle which satisfies sin θt ¼ cos θt sin ϕt , θEL and ϕ are the elevation and azimuth of the target, t t

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Fig. 1. Illustration of system layout.

Please cite this article as: B. Tang, et al., Design of MIMO radar waveform covariance matrix for Clutter and Jamming suppression based on space time adaptive processing, Signal Processing (2015), http://dx.doi.org/10.1016/j. sigpro.2015.10.033i

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B. Tang et al. / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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respectively, and S ¼ ½s1 ; s2 ; …; sNT T A CNT L is the transmit waveform matrix with sk being the transmit waveform of the k-th transmitter and L being the code length. Assume the MIMO array transmits a burst of K pulses in a coherent processing interval (CPI) at a constant pulse repetition frequency (PRF) fr, then the received target signal for the k-th pulse can be written as j2π f d;T ðk  1Þ

Y t;k ¼ αe

bðθt Þa ðθt ÞS; T

ð2Þ

where α is the complex amplitude of the target, f d;T ¼ f d =f r , fd denotes the target Doppler frequency, bðθt Þ ¼ ½1; ej2πγ f t;s ; …; ej2πγ f t;s ðNR  1Þ T is the receive array steering vector of the target and γ ¼ dR =dT . Stacking all the columns of Y t;k into a vector, we have

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yt;k ¼ vecðY t;k Þ ¼ αej2π f d;T ðk  1Þ ðST  INR Þðaðθt Þ  bðθt ÞÞ;

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where we have used the identities vecðABCÞ ¼ ðCT AÞvecðBÞ and for vectors x; y, vecðxyT Þ ¼ y  x [31]. Let yt ¼ ½yTt;1 ; …; yTt;K T , then

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yt ¼ αuðf d;T Þ  ½ðS  INR Þðaðθt Þ  bðθt ÞÞ

ð3Þ

T

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¼ α½IK  ðST  INR Þðuðf d;T Þ  aðθt Þ  bðθt ÞÞ ~ ¼ αSðuðf d;T Þ  aðθ t Þ  bðθ t ÞÞ ¼ αv t ;

c¼

ð4Þ

~ αc;k Sðuðf c;k Þ  aðθ c;k Þ  bðθ c;k ÞÞ;

ð5Þ

k¼1

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where Nc denotes the number of clutter patches in the isorange ring, αc;k , f c;k and θc;k are the amplitude, Doppler frequency and cone angle of the k-th clutter patch, respectively. Since the returns from different clutter patches are uncorrelated, then the clutter covariance matrix is given by Rc ¼ E½ccH  ¼

Nc X

~ ρc;k Sðuðf c;k Þ  aðθ c;k Þ

k¼1

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 bðθc;k ÞÞðuðf c;k Þ  aðθc;k Þ  bðθc;k ÞÞH S~ ; H

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jamming component is written as NJ X

j¼

αJ;k  sJ;k  bðθJ;k Þ;

ð7Þ

ð6Þ

where ρc;k ¼ E½αc;k αc;k  is the average power of the k-th clutter patch. We establish the model of jamming similarly to that in [22], in which the jamming signal is assumed to have a bandwidth much larger than that of the radar system, and spatially correlated with respect to the receive array while uncorrelated in fast time and from pulse to pulse, and the

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k¼1

where NJ is the number of jamming source, αJ;k A CK1 , sJ;k A CL1 , and θJ;k are the amplitude vector (in slow time), jamming signal (in fast time) and cone angle of the k-th jammer, respectively. Owing to the uncorrelation of the jammer samples in fast time and from pulse to pulse, we have E½αJ;k α

H J;k  ¼

ξk I K

ð8Þ

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and E½sJ;k sH J;k  ¼ IL ;

ð9Þ

where ξk is the average power of the k-th jammer. By using (8) and (9), we can write the jammer covariance matrix as ! NJ X H ξk bðθJ;k bðθJ;k ÞH Þ: ð10Þ RJ ¼ E½jj  ¼ IK  IL  k¼1

where uðf d;T Þ ¼ ½1; ej2π f d;T ; …; ej2π f d;T ðK  1Þ T is the temporal ~ steering vector of the target, vt ¼ Sðuðf d;T Þ  aðθ t Þ  bðθ t ÞÞ, S~ ¼ IK  ðST  INR Þ, and we have used the standard property of Kronecker product that ðA  BÞðC  DÞ ¼ ðACÞ  ðBDÞ in the second equality of (4) [31]. Next we consider the clutter model. For simplicity we neglect the clutter return from the neighborhood range cells and the range ambiguous cells (we refer to [32] for the waveform design method of MIMO radar in the presence of range ambiguous signal dependent interference). Herein, we approximate the continuous field of clutter by modeling the clutter returns with a superimposition of a number of independent clutter patches, then the clutter return in the target range cell can be written as Nc X

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Based on the established model, now we consider the following binary hypothesis testing problem for target detection 8 < H0 : z ¼ c þ j þ n ; ð11Þ f : H1 : z ¼ αvt þ c þ jþ n

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where z denotes the received signal of the cell under test, n is the receiver noise. Assume that n is dominated by the internally narrowband receiver noise, which is spatially and temporally uncorrelated, i.e., the noise covariance matrix is given by Rn ¼ E½nnH  ¼ σ 2 ID ;

ð12Þ

where σ2 denotes the power of thermal noise and D ¼ NR LK is the system dimension. In the sequel, we assume without loss of generality that σ 2 ¼ 1. The purpose of STAP is to design a D-dimensional vector w to suppress the interference and enhance the detection performance. It is well recognized in STAP society that the output SINR is closely related to the detection performance of the system, and the optimum weight maximizing the output SINR is given by [23] w¼

R u 1 vt ; H vt Ru 1 vt

ð13Þ

where Ru ¼ E½ðc þ j þnÞðc þ jþ nÞH  ¼ R c þ RJ þ Rn . As a result, when w is applied to suppress the clutter and jamming in z, the output SINR of the MIMO array for the cell under test is given by SINR ¼ jαj2

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2

jw vt j wH R u w H

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1 ¼ jαj2 vH t Ru vt :

117 ð14Þ

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Let v t ¼ uðf d;T Þ  aðθt Þ  bðθt Þ be the target space-time PN c k ¼ 1 ρc;k ðuðf c;k Þ  aðθ c;k Þ 

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bðθc;k ÞÞðuðf c;k Þ  aðθc;k Þ  bðθc;k ÞÞH , then by using (4) and

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steering vector, and let R c ¼

Please cite this article as: B. Tang, et al., Design of MIMO radar waveform covariance matrix for Clutter and Jamming suppression based on space time adaptive processing, Signal Processing (2015), http://dx.doi.org/10.1016/j. sigpro.2015.10.033i

B. Tang et al. / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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1

1=2 1=2 1=2 1=2  R~ ss R c ðR c R~ ss R c þIP Þ  1 R c R~ ss :

Table 1 Variable dimensions.

3

ð18Þ

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By using (18), the cost function of (16) can be rewritten

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Variable

Dimension

as

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vt

CNT NR K1 CNT L

1=2 H ~ 1 ~ ~H ~ ~H ~ vH Sv t ¼ v H t S ðSR c S þR v Þ t R ss v t v t R ss R c

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S S~ Rc

CNT NR KNT NR K

RJ

CLNR KLNR K

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1=2 1=2 1=2 ðR c R~ ss R c þIP Þ  1 R c R~ ss v t :

CLNR KNT NR K

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Note that

1

ð15Þ

We conclude this section by listing the dimensions of the variables used in problem modeling in Table 1. We hope it can ease the understanding of the following results and clarify the issues on the implementation difficulty of the optimization problem.

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¼ IK  Rss  R v ;

ð20Þ

where we have used ðA  BÞðC  DÞ ¼ ðACÞ  ðBDÞ in the second equality, Rss ¼ S ST is the WCM and R v ¼ INR þ R J . By using the result in (19) and introducing an auxiliary variable t, we can reformulate (16) as the following WCM design problem:

1=2 1=2 1=2 H ~  1 1=2 ~ ~ ~ R c R ss v t Z t; s:t:v H t R ss v t  v t R ss R c ðR c R ss R c þ IP Þ

3. Optimal WCM with exactly known target steering vector The aim of waveform optimization in MIMO-STAP systems is to find S (under some constraints) that maximizes the output SINR. In this study, we assume the jammer plus noise covariance matrix R J þ ID can be estimated when there is no clutter and target returns. To this end, the MIMO radar system can operate in passive mode such that the receivers collect signals only with jammer signals and noise [29]. In addition, we assume that we can predict the average power of each clutter patch from the previous scans by the sparse methods proposed in [33] or by the cognitive methods proposed in [34], such that R c is known a priori. Based on such assumptions, the results in [27] showed that after range compression the SINR only depended on the WCM and formulated the WCM design problem into a convex optimization by diagonal loading R c with an ad hoc loading factor. Next we propose a design method which does not need diagonal loading but with superior performance to the method in [27]. Considering an exactly known v t , we formulate the waveform optimization problem based on maximizing SINR as follows:  1 ~ H SR ~ c S~ H þ RJ þID ~ t s:t: S~ ¼ IK max v H Sv t S S

 ðST  INR Þ; trðSSH Þ r P 0 ;

ð16Þ

where P0 is the total available energy. PN Let Rv ¼ RJ þID ¼ IK  IL  ðINR þR J Þ, where R J ¼ k J¼ 1 H ξk bðθJ;k Þb ðθJ;k Þ. Then by using the matrix inversion lemma [31], we have

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1=2 H þ IP Þ  1 R c S~ R v 1 ;

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H ~ then where P ¼ KN T NR . Let R~ ss ¼ S~ Rv 1 S,

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~ c S~ þ RJ þ ID Þ  1 S~ ¼ R~ ss S~ ðSR H

H

1 R~ ss ¼ IK  R ss  R v ; R~ ss ≽0; R ss ≽0; trðRss Þ r P 0 ;

ð17Þ

ð21Þ

where we have used the identity trðRss Þ ¼ ½trðSSH Þ ¼ trðSSH Þ. 1=2 1=2 Since R c R~ ss R c þIP g 0, then according to the Schur complement theorem [31], the first constraint in (21) is equivalent to 2 3 1=2 ~ ~ vH vH t R ss v t  t t R ss R c 4 5≽0: ð22Þ 1=2 1=2 1=2 R c R~ ss v t R c R~ ss R c þ IP

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Therefore, we finally recast the design problem in (16) 97

as max t 2

R~ ss ;Rss ;t

~ vH t R ss v t  t s:t:4 1=2 R c R~ ss v t

1=2 ~ vH t R ss R c 1=2 1=2 R c R~ ss R c þ IP

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3 5≽0;

101 ð23Þ

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Note that the constraints in (23) are convex and hence (23) is a convex optimization problem. More precisely, the problem in (23) is a semidefinite programming (SDP) problem. Its globally optimal solution can be found by interior point method (IPM) efficiently in polynomial time [35], with public domain softwares (e.g., CVX [36]). Remark 1 (Waveform synthesis from the optimized WCM): The synthesis of transmit waveforms from the optimized WCM is an important step in the MIMO radar system. Denote the optimal solution of (23) by R opt ss . If only the energy constraint is imposed on the waveform matrix S, then the optimal waveform matrix for (16) is given by

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1 R~ ss ¼ IK  R ss  R v ; R~ ss ≽0; R ss ≽0; trðRss Þ rP 0 :

T=2 H Sopt ¼ ðRopt Q ; ss Þ

~ c S~ H þ RJ þID Þ  1 ¼ R  1 R  1 SR ~ 1=2 ðR 1=2 S~ H R  1 SR ~ 1=2 ðSR v v v c c c

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max t

R~ ss ;Rss ;t

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¼ IK  ðS ST Þ  ðINR þ R J Þ  1

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ð19Þ

R~ ss ¼ ½IK  ðS  INR Þ½IK  IL  ðINR þ R J Þ  1 ½IK  ðST  INR Þ

11 (6), we can rewrite the SINR by  1 ~ H SR ~ c S~ H þ RJ þ ID ~ t: SINR ¼ jαj2 v H Sv t S

67

ð24Þ

where Q A CLNT is an arbitrary semi-unitary matrix. In practical radar systems, waveform with constant modulus is of great interest, since it allows the radio frequency amplifier to operate at maximum efficiency and avoids unnecessary nonlinear effects in transmitters [37].

Please cite this article as: B. Tang, et al., Design of MIMO radar waveform covariance matrix for Clutter and Jamming suppression based on space time adaptive processing, Signal Processing (2015), http://dx.doi.org/10.1016/j. sigpro.2015.10.033i

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B. Tang et al. / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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If the constant modulus constraint or the finite alphabet constraint is imposed on S, we can use the methods proposed in [38,39] to synthesize waveform matrix whose covariance matrix is close to Ropt ss . However, due to the non-convexity of the constraints, the matching error between the covariance matrix of the synthesized waveforms and Ropt ss might be nonzero. Thus, the synthesized waveforms by the methods in [38,39] might be suboptimal. Besides, some applications require the waveforms to enjoy some desired properties, e.g., good range resolution, low auto-correlation and cross-correlation sidelobes. Note that the covariance matrix of such waveforms might be close to a scaled identity matrix. Hence, synthesizing a waveform matrix with the above properties and covariance matrix close to Ropt ss might be very difficult. Remark 2 (Comparison with the result in [27] and discussions): With the matrix inversion lemma, ~ c S~ H þ Rv Þ  1 can also be written as ðSR  1  1 H ~ c S~ H þR v ~ c S~ H R  1 SR ~ c þ IP ¼ Rv 1  R v 1 SR SR S~ Rv 1 : v

H ~ c S~ H þ Rv Þ  1 S~ is Then an alternative expression of S~ ðSR given by H ~ c S~ H þ Rv Þ  1 S~ ¼ R~ ss  R~ ss R c ðR~ ss R c þ IP Þ  1 R~ ss S~ ðSR ¼ ðR~ ss R c þ IP Þ  1 R~ ss :

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ð25Þ

ð26Þ

ð27Þ

It is very interesting that the expression of the SINR in (27) (without range compression by SH ðSSH Þ  1=2 ) is exactly the same as that in [27], in which the SINR was derived after range compression (i.e., matched filtering) by SH ðSSH Þ  1=2 . As a result, range compression does not change the output SINR. Thus the above observation, on the other hand, addresses the question “to compress or not to compress”, which was put forward in [9]. Nevertheless, we should also stress that after range compression by SH ðSSH Þ  1=2 , the target signal becomes

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y t ¼ αðIK  ððS ST Þ  1=2  INR ÞÞðuðf d;T Þ  aðθt Þ  bðθt ÞÞ:

43

Typically, the code length L is larger than N T . Then the dimension of the range compressed target signal, which is also the system dimension after compression, is reduced to NT NR K from LN R K. Therefore, with range compression by SH ðSSH Þ  1=2 , the computational complexity of the associated receive filter design (OððN T NR KÞ3 Þ) will be lower than that without compression (OððLN R KÞ3 Þ). In [27], the following design problem was considered

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ð28Þ

s:t:trðRss Þ r P 0 :

ð29Þ

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In order to reformulate (29) into a convex optimization, the authors of [27] diagonal loaded the clutter covariance matrix by

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R c ¼ R c þ εIP ;

55

61

63 ð31Þ

65

Next we show that the approximation in (31) is not accurate or even leads to large deviations in some situations. Since

67 69

 1  1 ~ H SðR ~ c þ εIP ÞS~ H þ Rv ~ R~ ss v t ¼ v H vH t R ss ðR c þ εIP Þ þ IP t S

71

t

t

~ t ¼ v H S~ H ðT þ Sv t

~ H 1 ~

εS~ S Þ

H 1 Zv t Sv t ; ¼ v H t Zv t  v t ZðZ þ1=εID Þ

73

ð32Þ where the first equality follows from the equivalence ~ c S~ H þ Rv , Z ¼ S~ H T  1 S, ~ and between (27) and (15), T ¼ SR the third equality is a result of the matrix inversion lemma. 1 Zv t v H If ε or P0 is very large, v H t ZðZ þ1=εID Þ t Zv t , which means the approximated cost function in the right-hand side of (31) will approach zero and any feasible solution of (29) will be nearly optimal for the approximated cost function, then the designed WCM associated with the right hand of (31) will be far from optimal.

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4. Optimal WCM with target steering vector uncertainty In practice, the prior knowledge of the target steering vector might be imprecise. Indeed, there are many nonideal factors that will lead to mismatched target steering vector, e.g., the location of target (or cone angle) and its Doppler frequency might not be exactly known, the MIMO array is not perfectly calibrated, the position of the array element might be perturbed in the flight of aircraft, and the diffuse multipath [40]. Herein, we assume that the actual target steering vector v^ t lies in an elliptical uncertainty set [41–43] given as follows: Sðv t Þ ¼ fv^ t jv^ t ¼ v t þ e; J e J C r ϵg;

ð30Þ

where ε was empirically chosen as ε ¼ λmax ðR c Þ=1000. In other words, they tried to approximate the cost function of

ð33Þ

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where p J effiffiffiffiffiffiffiffiffiffiffi J C is the C-quadratic norm [35] defined by J e J C ¼ eH Ce and C g 0. In order to make the designed WCM robust to the steering vector errors over the uncertainty set defined in (33), we consider the following design problem:

103

max t

109

S

H s:t:v^ t S~

 H

~ c S~ H þ Rv SR

1

S~ v^ t Zt; Forall v^ t A Sðv t Þ;

S~ ¼ IK  ðST  INR Þ; trðSSH Þ r P 0 :

1 ~ ~ R ss v t max v H t ðR ss R c þIP Þ R ss

53

(29) by  1  1 R~ ss v t v H R~ ss ðR c þ εIP Þ þ IP R~ ss v t : v H R~ ss R c þ IP

89

Hence, the output SINR can also be written as 1 ~ ~ SINR ¼ jαj2 v H R ss v t : t ðR ss R c þ IP Þ

5

105 107

111 ð34Þ

113

Denote the optimal value of (34) by p⋆⋆ and the optimal solution by S⋆⋆ , then for any v^ t A Sðv t Þ, the output SINR with waveform S⋆⋆ will be at least p⋆⋆ and then the performance of the designed waveforms can be guaranteed over the uncertainty set. Note that ϵ should satisfy ϵ o J v t J C (i.e., the size of uncertainty region should not be too large), otherwise 0 A Sðv t Þ and the optimal value of (34) will be zero, indicating that there will be no performance guarantee.

115

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B. Tang et al. / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

1 3

H maxmin v^ t Q v^ t S;Q

7

11 13

v^ t

s:t: trðSSH Þ rP 0 ; v^ t A Sðv t Þ; S~ ¼ IK  ðST  INR Þ;  1 H ~ c S~ H þ Rv ~ S~ SR S≽Q ; Q ≽0;

5

9

17 19 21

ð35Þ

H

23 25 27

e

29 31 33 35

ð36Þ

ð38Þ

where μ Z 0. If μ 40, then Q þ μC g 0 and Q þ μC is invertible. Thus ðv t þeÞH Q ðv t þeÞ þ μðeH Ce ϵ2 Þ

ð39Þ

Therefore,

37

1 Cv t  ϵ2 Þ; FðμÞ ¼ μðv H t Q ðQ þ μCÞ

39

and the infimum is achieved when e ¼  ðQ þ μCÞ  1 Q v t . If μ ¼ 0, we know that if e ¼  v t ,

41

1 FðμÞ ¼ 0 ¼ 0  ðv H Cvt  ϵ2 Þ: t Q ðQ þ μCÞ

43

Consequently, by summarizing the results in (40) and (41), we can obtain the Lagrange dual of (37)

45

1 Cv t  ϵ2 Þ; μ Z0: FðμÞ ¼ μðv H t Q ðQ þ μCÞ

47 49 51 53

μ

57 59 61

ð41Þ

ð42Þ

s:t: μ Z 0:

ð43Þ

Denote the optimal solution of (43) by μ⋆ . If μ⋆ 40, then the optimal solution of (37) is given by e ¼  ðQ þ μ CÞ ⋆

55

ð40Þ

In addition, since (37) is strictly feasible for any ϵ 40, Slater's condition [35] indicates that there is no gap between (37) and its duality problem given as follows: 1 max FðμÞ ¼ μðv H Cv t  ϵ2 Þ t Q ðQ þ μCÞ

⋆

1

Q vt;

ð44Þ

which illustrates that v^ t ¼ v t þe⋆ is the “worst” target steering vector with the smallest SINR. Otherwise if μ⋆ ¼ 0, then v^ t ¼ 0 is the “worst” target steering vector. By using (43), we can recast (36) into 1 max μðv H Cv t  ϵ2 Þ t Q ðQ þ μCÞ

R ss ;R~ ss ;Q ;μ

þ μCÞ  1 Q v t  μϵ2 :

67 69 71 73

2 vH t Q v t þ t  μϵ

R~ ss  Q 1=2 R c R~ ss

75 3

1=2 R~ ss R c 5≽0; Q ≽0; 1=2 1=2 R c R~ ss R c þI

77 79

1  vH Q v t Z t; t Q ðQ þ μCÞ

ð47Þ 81

1 trðRss Þ r P 0 ; μ Z 0; R~ ss ¼ IK  R ss  R v ; R~ ss ≽0; Rss ≽0:

Using the Schur complement theorem in the third constraint of (47) again, we can finally formulate (47) as max

Rss ;R~ ss ;Q ;μ;t

2

s:t:4

R~ ss  Q

1=2 R~ ss R c

1=2 R c R~ ss

1=2 1=2 R c R~ ss R c þI

83 85 87

2 vH t Q v t þ t  μϵ

3 5≽0;

"

t Q vt

vH t Q

Q þ μC

#

89 ≽0; Q ≽0;

1 trðR ss Þ rP 0 ; μ Z 0; R~ ss ¼ IK  R ss  R v ; R~ ss ≽0; R ss ≽0:

91 93

ð48Þ

¼ ðeþ ðQ þ μCÞ  1 Q v t ÞH ðQ þ μCÞðe þðQ þ μCÞ  1 Q v t Þ 1 Cv t  ϵ2 Þ: þ μðv H t Q ðQ þ μCÞ

65

1 R v ; R~ ss ≽0; Rss ≽0:

Therefore, we can further write (45) as

s:t:4

Since Q ≽0 and the constraint set in (37) is convex, the above optimization problem is convex with respect to e, and the associated Lagrange dual can be written as [35] FðμÞ ¼ inf fðv t þeÞH Q ðv t þ eÞ þ μðeH e  ϵ2 Þg;

1=2 1=2 R c R~ ss R c þ I

ð46Þ

2

ð37Þ

e

1=2 R c R~ ss

63

5≽0; Q ≽0;

H ¼ vH t Q v t  v t Q ðQ

Rss ;R~ ss ;Q ;μ;t

To proceed, we first consider the inner minimization problem of (36), which can be stated as follows: H

1=2 R~ ss R c

μðv Ht Q ðQ þ μCÞ  1 Cv t  ϵ2 Þ ¼ v Ht Q ðQ þ μCÞ  1 ðQ þ μC  Q Þv t  μϵ2

max

max min v^ t Q v^ t s:t: trðRss Þ r P 0 ; v^ t A Sðv t Þ; 2 3 1=2 R~ ss  Q R~ ss R c 4 5≽0; Q ≽0; 1=2 1=2 1=2 R c R~ ss R c R~ ss R c þ I

min v^ t Q v^ t s:t:v^ t ¼ v t þe; J e J C r ϵ:

3

R~ ss  Q

ð45Þ Note that the cost function of (45) can be written as

i.e., we try to optimize the transmit waveforms which maximize the worst-case SINR over the uncertainty set in (33). Using (18) and the Schur complement theorem together, we can obtain that (35) is also equivalent to

1 R~ ss ¼ IK  ðRss Þ  R v ; R~ ss ≽0; Rss ≽0:

s:t:4

trðR ss Þ r P 0 ; μ Z 0; R~ ss ¼ IK  R ss 

R ss ;R~ ss ;Q v^ t

15

2

Note that (34) can be reformulated as the following max-min problem

Note that the cost function of (48) is linear with respect to the optimizing variables and all the constraints are convex, thus (48) is a convex optimization problem, and the globally optimal solution can be found efficiently. Remark 3 (Comparison with the results in [28]): The authors in [28] considered a special case of (37) in which C ¼ I by the following steps: In the first step, they claimed that the spherical constraint on the steering vector error J e J ¼ J v^ t  vt J r ϵ is equivalent to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J v^ t J Z NT N R K  ϵ: ð49Þ However, it should be noted that the size of the uncertainty set in (49) is larger than that of the spherical constraint. Thus, the original optimization problem will be relaxed with the constraint in (49); In the second step of minimization, they tried to solve the relaxed optimization problem by introducing a heuristic diagonal loading of the WCM, while the problem of the selection of loading factor was not solved, either. Therefore, the solution proposed in [28] is suboptimal. Compared with [28], the advantages of the proposed algorithm are evident. First, we consider a more general uncertainty set, which can describe different kinds of steering vector error. Secondly, we explicitly derive the “worst” target steering vector which has the smallest SINR. Thirdly, without heuristic diagonal loading, we can find the globally optimal WCM which is robust to target steering vector uncertainties within polynomial time.

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B. Tang et al. / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1

5. Numerical simulations

3

In this section, we provide several numerical examples to demonstrate the performance of the proposed algorithms. The airborne MIMO radar under consideration has NT ¼ 3 antennas in the transmit array and NR ¼ 3 antennas in the receive array, with dT ¼ dR ¼ λ=2 (i.e., γ ¼ 1). The altitude of the radar platform is 9 km and the platform is moving with a speed va ¼200 m/s; Both the transmit and receive arrays are steering to broadside; The slant range of the target is Rt ¼ 12:728 km; the spatial frequency and the normalized Doppler frequency of the target are f t;s ¼ 0 and f d;T ¼ 0:3, respectively. The PRF fr is set to make the slope of the clutter ridge β ¼ 4va =λf r ¼ 1. K ¼3 pulses in a CPI are coherently collected. The clutter in the iso-range is divided into Nc ¼10,000 patches equally distributed in azimuth. The incident angle of the jammer is 151 and the jammer to noise ratio (JNR) is 60 dB. Finally, we assume the WCM Rss can be perfectly realized with the waveform matrix S.

5 7 9 11 13 15 17 19

7

First we consider the case of exactly known target steering vector. Fig. 2 shows the spatial-temporal beampatterns corresponding to the proposed algorithm, the method proposed by [27] (here we call it Wang's method), the orthogonal waveforms, and the conventional SIMOSTAP. Herein, the clutter to noise ratio (CNR) is 30 dB, the loading factor in Wang's method is set to be λmax ðR c Þ=1000 as suggested in [27], the WCM associated with SIMO-STAP ¼ P 0 =NT  aðθt ÞaH ðθt Þ, and the total transmit energy is RSIMO ss is P 0 ¼ 100. Herein, given the optimum weight in (13), the spatial-temporal beampattern is defined like that in [22], which is the response of the steering vectors as a function of spatial and Doppler frequency,

63

Pðf s ; f d Þ ¼ jwH vðf d ; f s Þj2 ;

77

ð50Þ

where vðf d ; f s Þ is the steering vector corresponding to Doppler frequency fd and spatial frequency fs. We can observe from Fig. 1 that all the four beampatterns have deep nulls in the clutter ridge and jammer

Beampattern of the proposed algorithm(dB)

23

0.5

Beampattern of Wang’s method(dB) 0.4

40 20

0.2 0.1

0

0 −20

−0.1

−40

−0.2 −0.3

35

20

0.2 0.1

0

0 −20

−0.1

−0.5 −0.5

−80 0

−0.5 −0.5

0.5

Normalized Doppler frequency

0.5

57 59 61

87 89 91 93

99 0

0.5

101

Beampattern of SIMO−STAP(dB) 0.5

50

0.4

0.4

0.3

0.3

0.2

0

0.1 0 −0.1

−50

−0.2

Spatial frequency

Spatial frequency

45

55

81

103

43

53

79

97

Normalized Doppler frequency

Beampattern of orthogonal waveforms(dB)

51

75

−60

41

49

73

−40

−0.4

39

47

71

95

−0.2 −0.3

−60

−0.4

37

40

0.3

Spatial frequency

33

Spatial frequency

0.3

31

69

85

0.5

60

0.4

25

29

67

83

21

27

65

0.2 0.1

−0.4

−100

Normalized Doppler frequency

20

109

0

111

−0.2

−0.4 0.5

107

−20

−0.1

−0.3

0

40

0

−0.3

−0.5 −0.5

105

−0.5 −0.5

−40

−60 0

0.5

Normalized Doppler frequency

Fig. 2. Spatial-temporal beampatterns of (a) the proposed algorithm; (b) Wang's method; (c) the orthogonal waveforms; (d) SIMO-STAP. Exactly known target steering vector. CNR ¼ 30 dB, P 0 ¼ 100.

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8

63

1 3 5

45

29

SINR(dB)

30 25

17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

71

25 24

73

23

The proposed method Wang’s Method Orthogonal Waveforms SIMO−STAP

15 10 5 10

69

26

15

20

25

30

35

40

45

21 50

75

The proposed method Wang’s Method Orthogonal Waveforms SIMO−STAP

22

20 10

15

20

25

ASNR(dB)

30

35

40

45

77 50

CNR(dB)

location, and very high gain in the target direction, in which the response in the target direction of the proposed method is the strongest (57.24 dB), followed by Wang's method (55.38 dB), the conventional SIMO-STAP (53.28 dB), and the orthogonal waveforms (52.96 dB). In addition, the optimum weight of the proposed method places deeper nulls in the clutter and jammer location than Wang's method. The achieved SINRs of the four waveforms are 28.62 dB, 27.69 dB, 26.64 dB, and 26.48 dB, respectively. Next we compare the SINR of the four types of waveforms against different array signal to noise ratio (ASNR) and CNR, where ASNR is defined by ASNR ¼ P 0 NT NR =σ 2 . Fig. 3a shows the output SINR of the four types of waveforms when CNR is fixed to 30 dB, and ASNR varies from 10 dB to 50 dB. It is shown both the proposed method and Wang's method outperform the orthogonal waveforms and SIMO-STAP. If ASNR is small, the performance of Wang's method is similar to that of the proposed method. For large ASNR, just as we have analyzed in Section 3 (see Remark 2), the eigenvalues of Z become larger, then the WCM obtained by Wang's method is suboptimal. Thus under this situation, our method has better performance than Wang's method. Fig. 2b presents the output SINR against CNR when ASNR is fixed to 30 dB. We can observe that, the performance of the proposed method, the orthogonal waveforms, and the SIMO-STAP is insensitive to CNR, implying that clutter can be mitigated effectively with these waveforms. In contrast, the performance of Wang's method deteriorates for a large CNR (even worse than the orthogonal waveforms for 50 dB CNR). This can be explained by that, the loading factor in Wang's method increases with CNR and makes the designed WCM deviate from optimality. As a result, the severe clutter cannot be adequately suppressed with the waveforms designed by Wang's method. Next we analyze the performance of the proposed algorithm for the case of target steering vector uncertainty. For simplicity we only consider the case in which only the

79 81

Fig. 3. Performance comparisons against ASNR and CNR. Exactly known target steering vector. (a) The output SINR against ASNR, CNR¼ 30 dB; (b) The output SINR against CNR. ASNR ¼30 dB.

83

30

85

29.5

87

29

SINR(dB)

15

67

27

20

13

65

28

35

SINR(dB)

11

30

40

7 9

50

89

28.5

The proposed method Wang’s Method Orthogonal Waveforms SIMO−STAP

28 27.5

91 93 95

27

97

26.5 26

0.28

0.285

0.29

0.295

0.3

0.305

0.31

0.315

0.32

Normalized Doppler frequency Fig. 4. The output SINR against normalized Doppler frequency. ASNR ¼ 30 dB. CNR ¼30 dB. P 0 ¼ 100. ϵ ¼ 1.

knowledge of the target Doppler frequency is uncertain. For a target with the nominal normalized Doppler frequency equal to 0.3, we can calculate that for ϵ ¼ 1 and a spherical uncertainty set (i.e., C ¼ I), the actual normalized target Doppler frequency should lie in f d;T A ½0:276; 0:324. Fig. 4 shows the output SINR of the proposed algorithm, Wang's method, and the orthogonal waveforms as well as SIMO-STAP with respect to different target Doppler frequencies in this uncertainty region, where the ASNR is 30 dB, CNR is 30 dB, and ϵ ¼ 1, the loading factor of the WCM in Wang's method is set to be P 0 =1000 as suggested in [28], and the other parameters are the same as that in Fig. 2. The result in Fig. 3 illustrates that the SINR of the proposed algorithm is larger than those of the other three types of waveforms. In Fig. 5, we show the worst-case SINR of the four types of waveforms with respect to various ASNR and CNR, where the simulation parameters in Fig. 4 are used and ϵ ¼ 1. Similar to that in Fig. 3, the performance of the

Please cite this article as: B. Tang, et al., Design of MIMO radar waveform covariance matrix for Clutter and Jamming suppression based on space time adaptive processing, Signal Processing (2015), http://dx.doi.org/10.1016/j. sigpro.2015.10.033i

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9

63

1 25

45

3 40

5

35

7

67 24

SINR(dB)

11

24.5

30

SINR(dB)

9

65

The proposed method Wang’s Method Orthogonal Waveforms SIMO−STAP

25 20 15

13 15

5

17

0 10

20

35 37 39 41

45

50

47 49 51 53 55 57 59 61

15

20

25

30

35

40

45

50

waveforms decreases with increasing size of the uncertainty set. However, the proposed algorithm always outperforms the other three types of waveforms.

SINR(dB)

−5 0.5

83 85 87

91

10

0

81

6. Conclusion

15

5

79

89

20

The proposed method Wang’s Method Orthogonal Waveforms SIMO−STAP 1

1.5

2

2.5

ε Fig. 6. The worst-case SINR against the size of uncertainty region ϵ. ASNR ¼ 30 dB. CNR¼ 30 dB.

43 45

21 10

CNR(dB)

25

27

33

40

30

25

31

35

77

Fig. 5. Performance comparisons against ASNR and CNR. Imprecisely known target steering vector, ϵ ¼ 1. (a) The worst-case SINR against ASNR, CNR ¼30 dB ; (b) The worst-case SINR against CNR. ASNR ¼30 dB.

23

29

30

75

The proposed method Wang’s Method Orthogonal Waveforms SIMO−STAP

ASNR(dB)

19 21

25

73

22.5

21.5

15

71

23

22

10

69

23.5

proposed algorithm is superior to the other three types of waveforms. Fig. 3b also shows the performance of the proposed algorithm is insensitive to the change of CNR, meaning that the clutter can be effectively suppressed. In contrast, the performance of Wang's method is only slightly better than the orthogonal waveforms for low ASNR or CNR, and deviates from the optimality at high ASNR and CNR. Comparing Fig. 3 with Fig. 2, we also observe that the performance of the orthogonal waveforms is more robust than that of SIMO-STAP; In addition, the uncertainties of the target steering vector lead to the degradation of SINR of the proposed algorithm for about 5 dB (in the worst case). Finally, we compare the performance of the four kinds of waveforms against various size of the considered uncertainty set. Fig. 6 shows their worst-case SINR versus different ϵ, where ASNR¼30 dB and CNR¼30 dB. As expected, the worst-case SINR of the four types of

In this paper, we addressed the design of WCM for an airborne MIMO radar. The aim was to improve the weaktarget detection performance of the system through simultaneously suppressing the clutter and jamming with STAP. We employed the SINR as the criterion for waveform optimization. For exactly known target steering vectors, the proposed algorithm avoided heuristic diagonal loading of the clutter covariance matrix and found the globally optimal solution. For the case where the target steering vector was uncertain, we considered a robust design of the WCM through a max–min approach. We also recast the robust design problem into a convex optimization problem without ad hoc diagonal loading the WCM. Numerical results demonstrated the effectiveness and superiority of the proposed algorithm. Possible future research track might concern the design of waveforms enjoying desired properties including low pear-to-average-power-ratio (PAR), good range resolution, and low sidelobes. In addition, the computational complexity of the proposed design methods increases dramatically with the number of array elements and the number of pulses. Thus, efficient design methods that can deal with large arrays and long integration time should be developed in the future. Moreover, algorithms robust to the mismatch of the clutter covariance matrix should also be considered.

93 95 97 99 101 103 105 107 109 111 113 115 117 119

Acknowledgments 121 The authors would like to express their gratitude to the handling editor Prof. Marius Pesavento and the

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anonymous reviewers for their constructive comments leading to improvements of this paper. This work was supported in part by the National Natural Science Foundation of China under Grant 61201379 and 61179036, and Anhui Provincial Natural Science Foundation under Grant 1208085QF103. The work of Jun Li was supported in part by National Natural Science Foundation of China under Grant 61271292. The work of Jun Tang was supported in part by National Natural Science Foundation of China under Grant 61171120, the Key National Ministry Foundation of China under Grant (9140A07020212JW0101) and the Foundation of Tsinghua University under Grant 20131089362.

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Please cite this article as: B. Tang, et al., Design of MIMO radar waveform covariance matrix for Clutter and Jamming suppression based on space time adaptive processing, Signal Processing (2015), http://dx.doi.org/10.1016/j. sigpro.2015.10.033i

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