Waveform covariance matrix design for robust signal-dependent interference suppression in colocated MIMO radars

Accepted Manuscript

Waveform Covariance Matrix Design for Robust Signal-Dependent Interference Suppression in Colocated MIMO Radars Mostafa Bolhasani, Esfandiar Mehrshahi, Seyed Ali Ghorashi PII: DOI: Reference:

S0165-1684(18)30204-4 10.1016/j.sigpro.2018.06.007 SIGPRO 6844

To appear in:

Signal Processing

Received date: Accepted date:

10 March 2018 4 June 2018

Please cite this article as: Mostafa Bolhasani, Esfandiar Mehrshahi, Seyed Ali Ghorashi, Waveform Covariance Matrix Design for Robust Signal-Dependent Interference Suppression in Colocated MIMO Radars, Signal Processing (2018), doi: 10.1016/j.sigpro.2018.06.007

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Waveform Covariance Matrix Design for Robust Signal-Dependent Interference Suppression in Colocated MIMO Radars

CR IP T

Mostafa Bolhasania , Esfandiar Mehrshahia , Seyed Ali Ghorashia,b,∗ a Cognitive

Telecommunication Research Group, Department of Telecommunications, Faculty of Electrical Engineering, Shahid Beheshti University, G.C., Tehran, Iran b Cyberspace Research Institute, Shahid Beheshti University, G. C, Tehran, Iran.

AN US

Abstract

In multiple-input multiple-output (MIMO) radars, interference can be properly suppressed by designing transmit waveforms. However, having an error in initial estimation of interferers’ locations can result in degradation of signal-tointerference-plus-noise ratio (SINR), significantly. In this paper we consider the

M

problem of robust interference suppression and SINR increment in collocated MIMO radars, in the presence of signal-dependent interference resources. We propose an efficient method to generate transmit waveform covariance matrix

ED

with the following properties: a) robustness in interference suppression, b) high SINR at the receiver, c) using the same power level for all transmit antennas and d) suitable for generating BPSK waveforms in a closed-form. We solve

PT

the resulted optimization problems with numerical approaches in a short period of time. The proposed covariance matrix, by exploiting the correlated MIMO

CE

radar concept and a technique we use in the transmitter, yields to a better SINR level compared to conventional MIMO radars, phased array method and the recent introduced schemes. Simulation results validate our claims and show

AC

the advantages of our work. Keywords: multiple-input multiple-output (MIMO) radar, interference ∗ Corresponding

author Email addresses: [email protected] (Mostafa Bolhasani), [email protected] (Esfandiar Mehrshahi), [email protected] (Seyed Ali Ghorashi)

Preprint submitted to Journal of LATEX Templates

June 12, 2018

ACCEPTED MANUSCRIPT

suppression, convex optimization, covariance matrix design.

1. Introduction

CR IP T

Using the similar idea of multiple-input multiple-output (MIMO) commu-

nications, recently a new type of radar is introduced, called MIMO radar [1].

Based on the antennas configurations, MIMO radars can be classified into two 5

categories: collocated radars [2] and widely distributed [3] ones. In collocated

radars, which is the focus of this paper, the transmitting antennas are spaced

AN US

close enough, so that all of the antennas see the same target cross section.

Therefore, the collocated configuration with Mt transmit and Mr receive antennas cannot provide spatial diversity. However, it can lead to an Mt Mr length 10

virtual steering vector, which can increase the spatial resolution [4]. In widely distributed MIMO radars, transmit antennas of MIMO radar are widely separated in order to overcome the fluctuations of target amplitude on the radar

M

display, as each of the antennas sees a different aspect of target. The waveform diversity in this topology can increase the spatial diversity of the system [5, 15

6] similar to the multi-path diversity concept in wireless communications over

ED

fading channels [7, 8]. Unlike to phased array radars in which all of the transmit antennas emit a scaled version of the same waveform, MIMO radars allow

PT

the transmit antennas to transmit independent waveforms. This provides more degrees of freedom (DOF) compared to the phased array counterpart and can 20

be exploited to improve the radar performance [5]. Therefore, one of the most

CE

important issues in MIMO radars is the waveform design and recently, there have been several research studies focusing on this issue [9-24].

AC

Waveform design is usually done in two ways: a) direct design of waveform

symbols [19- 22], b) designing ”waveform covariance matrix” and then gener-

25

ating the waveforms, based on this matrix [23-25]. In the direct method, LMt symbols must be designed (L is the number of symbols transmit from each

antennas) while in the covariance matrix based method, only Mt (Mt − 1)/2 elements of the waveform covariance matrix should be designed, because co-

2

ACCEPTED MANUSCRIPT

variance matrix is a Hermitian matrix. Therefore, waveform design using the 30

covariance matrix can reduce the complexity [26]. The waveforms can be designed by both of the above mentioned methods, analytically or numerically.

CR IP T

In some works such as a method proposed in [13], authors have showed that if the waveform covariance matrix satisfies some specific constraints, then the BPSK waveforms that realize this matrix can easily be generated in a closed35

form solution. In [21] for uniform linear arrays (ULAs), a closed-form solution is proposed in which, symbol design is based on discrete Fourier transform (DFT)

coefficients. Compared to the iterative algorithm introduced in [15], this method

AN US

has a lower computational complexity. In [19] we proposed a method to design the transmitting waveforms and receive combining filter with a quasi-convex 40

based method, however, its complexity order is high. Generally, the time required to use closed-form waveform design methods is much lower than that of iterative counterparts [26].

There are two key points in collocated MIMO radar design; the first is im-

45

M

proving SINR in various scenarios. Having high output SINR at the receiver, increases the probability of target detection, directly. In detection of a target

ED

at a certain direction, the reflections from all the other directions should be suppressed as much as possible. In collocated MIMO radars, it is usually assumed that the transmitted waveforms are narrowband and so, the point like

50

PT

targets and the point like scatterers (signal-dependent interferences) are considered. Also it is assumed that there is a priori information about directions

CE

of target and interferences [15-26]. This information comes from initial target/interference direction estimation methods [27, 28]. for example, in a radar network that systems work in a same transmitted frequency or when there are

AC

multiple strong clutters (e.g. surrounding mountains) in the environment, we

55

should have the priori information about the interferers directions. These recent proposed methods [16-24] by using this priori information, can be used to track the targets or to confirm the initial estimation results [29, 30]. Therefore, this type of radar can be used parallel to the other types of radar to build a hybrid radar system. At the receiver, optimal filter coefficients can be used by 3

ACCEPTED MANUSCRIPT

60

minimum variance distortionless response (MVDR) beamformer [31] in order to put nulls in the initial estimated directions of interferences. The second key point in collocated MIMO radars is the covariance matrix

CR IP T

rank, which represents the number of distinct transmitted waveforms. According to the concept of co-array [17], using waveform diversity (which is equiv65

alent to a covariance matrix with two or more eigenvalues) higher SINR can

be achieved compared to the phased array method (which is equivalent to a covariance matrix with rank of one). This means that if the number of interfer-

ence directions and their powers is proportional to the number of phased array

70

AN US

antennas, the phased array method achieves higher SINR compared to the conventional MIMO radar (MIMO radar which transmit orthogonal waveforms). However, by increasing the number of interference directions or their powers, the SINR level of the phased array method decreases, dramatically [24]. In the interference free scenarios, although in the conventional MIMO radar, the parameters can be identified better [4], it has lower signal-to-noise ratio (SNR) performance [23] compared to the phased array radar, due to orthogonal trans-

M

75

mit waveforms and non-coherent processing in MIMO radars. Therefore, to ex-

ED

ploit the benefits of both conventional MIMO radars and phased array method, partially correlated waveform design techniques are provided, that leads to the concept of so called ”correlated MIMO radars” [23-25]. In [23] in order to achieve SNRs close to that of the phased array method, authors proposed a

PT

80

covariance matrix whose elements are the cross-correlation of two orthogonal

CE

waveforms. In [24], we proposed two full rank waveform covariance matrices in order to enhance SINR as well as the interference suppression capability. In [25] a method called phased-MIMO scheme is introduced as a trade-off between phased array and conventional MIMO. In all of these methods [23-25] it is as-

AC

85

sumed that initial estimation of interference directions is available without any error or uncertainty, which cannot be a practical assumption and motivate us to overcome this issue. Also, in [22] an algorithm is introduced that designs only the phases of the transmit signals, according to the interference directions in the

90

transmitter. This method has good properties in the radar ambiguity function 4

ACCEPTED MANUSCRIPT

in order to improve the range resolution. Considering the auto/cross correlation of these waveforms, we can see that they are approximately orthogonal, similar to conventional MIMO radars.

95

CR IP T

Any change in the cross-correlation of the waveforms may change in the power levels of the resultant waveforms. Radio frequency amplifiers (RFAs) cannot have their maximum power efficiency at all power levels, because of

their non-linearity properties. Hence, according to the different power levels, we require different RFAs at different transmit antennas and therefore, by changing the transmit waveform power, system hardware should be modified. In order to

avoid this hardware modification and to make it possible to use multi-function

AN US

100

software radar [26], identical RFAs that all work at the same power level, should be used. Also constant envelope and equal power waveforms have to be used. Hence, the transmission of equal amounts of power from all antennas is an important practical constraint in waveform design techniques.

In this paper, we propose an efficient method to generate transmit wave-

105

M

form covariance matrix in MIMO radars, with the following properties: a) robustness in interference suppression, i.e. it considers the possible existence of

ED

error in the initial estimation of interference directions. b) high SINR at the collocated MIMO radar receiver and c) using the same power level for all trans110

mit antennas. This means that in our proposed waveform design method, the

PT

optimization problem is solved with the following constraints: cohere power transmission in the targets’ directions, deep nulls around the initial estimated

CE

directions of interference, transmission of the same power levels from all antennas and a positive semi-definite covariance matrix. Next, to exploit the benefits

115

of BPSK waveforms, we propose a sequential optimization algorithm, based on

AC

the mentioned method, which generates BPSK waveforms in a closed-form. The key benefits of using the waveforms of the proposed covariance matrix

in the transmitter part of a MIMO radar are as follows:

120

• The transmit beam-pattern has much deeper nulls in the directions of interference, compared to the recent method in [22]. This is especially vital when we do not know the interference directions, exactly. Note that having 5

ACCEPTED MANUSCRIPT

several nulls in each possible interference direction, results in decreasing the depth of the nulls [22]. If we have some errors in initial estimation of interference directions, the proposed method leads to a significant improvement of output SINR compared to the methods in [22-25], due to its

CR IP T

125

interference-robust design and using correlated MIMO radar concept.

• When the number of interferers (or similarly their power) increases, our

proposed method has lower loss in the output SINR and achieves higher SINR, compared to the performance of methods in [22-25].

• Similar to methods in [22-24] and unlike to [25], our proposed method

130

AN US

satisfies constant envelope practical constraint in order to avoid any hard-

ware modification (exploit multifunction software radar concept) in radar system and avoids the destructive effect of amplifier non-linear behavior. • The proposed method of designing BPSK waveform covariance matrix, leads to an innovative algorithm that can be implemented for radar an-

135

tenna waveform design in real-time due to its low computational complex-

M

ity.

The organization of the rest of this paper is as follows: In section 2 the

140

ED

signal model of collocated MIMO radars is introduced. In section 3 the proposed waveform design method is introduced. Section 4 shows the numerical results and conclusion is provided in section 5.

PT

Notaion: Lower case letters, x, and bold upper case letters, X denote vectors and matrices, respectively. (.)T and (.)H denote transposition and conjugate

CE

transposition, respectively, and inverse of a matrix is denoted by (.)−1 . Expec145

tation operator is denoted by E (.). The k times hadamard product of matrix X is denoted by Xk . The Kronecker product is denoted by ⊗ and |.| denotes

AC

the modulus of a complex number.

2. Signal Model Consider a collocated MIMO radar system with a transmit array of Mt and

150

a receive array of Mr isotropic antennas. Each antenna array element sends

6

ACCEPTED MANUSCRIPT

xm (n), m = 1, ... , Mt , n = 1, ... , L which can be different from each other due to cross correlation value, n is the time index and L is the number of symbols T

in each waveform. Assume that x(n) = [x1 (n), x2 (n), ..., xMt (n)]

is a Mt × 1

155

CR IP T

vector of transmit waveforms in time index n. Then, the signal at target location θ would be at (θ)T x(n), n = 1, ..., L. We assume that the transmit/receive array is a ULA with d = λ/2 inter-element spacing, thus, at (θ) is given by: at (θ) = [1 , e−j2π(d/λ) sin(θ) , ..., e−j2π(d/λ)(Mt −1) sin(θ) ]T = [1 , e−jπ sin(θ) , ..., e−jπ(Mt −1) sin(θ) ]T .

(1)

AN US

Consider a point like target at direction θ0 and Q signal-dependent interference sources (point like scatterers or digital radio frequency memory repeat

jammer [36]) at directions θi , i = 1, .., Q. Therefore, the baseband signal at the 160

receiver can be written as:

T

y(n) = βt ar (θ0 )at (θ0 ) x(n) Q P T + βi ar (θi )at (θi ) x(n) + v(n), i=1

(2)

M

n = 1, ..., L

where βt and βi are the independent complex amplitudes proportional to RCSs

ED

of target and interference sources, respectively, ar (θ) represents the Mr × 1 received steering vector and it can be expressed as follows:

PT

ar (θ) = [1 , e−jπ sin(θ) , ..., e−jπ(Mt −1) sin(θ) ]T

(3)

v (n) is Mr × 1 distributed additive white Gaussian noise vector at Mr receivers with covariance matrix of σv2 IMr ×Mr and zero mean.

CE

165

Also the transmit beam-pattern of the waveform covariance matrix R in

AC

specific direction θ can be written as follows [14]:   H H H p(θ) = E (at (θ) X)(at (θ) X)
H = at (θ) E XXH at (θ)

(4)

H

= at (θ) Rat (θ).

Where X is Mt × L transmitted waveforms matrix. Eq. (4) shows that the transmit beam-pattern can be controlled by designing the transmit covariance 7

ACCEPTED MANUSCRIPT

170

R. The received signal at each element of the receive array is passed through Mt matched filters. Then, the output of matched filters for Mr receive antennas

y = βt ar (θ0 ) ⊗ Rat (θ0 ) Q P + βi ar (θi ) ⊗ Rat (θi ) + vc i=1

CR IP T

can be written as a Mt Mr × 1 vector as follows: (5)

where vc is the noise vector with zero mean and variance of σ 2 (IMr ⊗ R) [24].

Finally, the output of matched filters pass through FIR filter of length LMr × 1 and SINR at the receiver is written as follows: 2 ρ wH ar (θ0 ) ⊗ Rat (θ0 ) SIN R = wH Rin w Rin =

Q X i=1

AN US

175

2

ηi |ar (θi ) ⊗ Rat (θi )| + (IMr ⊗ R)

(6)

(7)

n o. n o. 2 2 where ρ = E |βt | σ 2 is SNR and ηi = E |βi | σ 2 is interference plus

H

R−1 in ar (θ0 ) ⊗ Rat (θ0 ) H

ar (θ0 ) ⊗ at (θ0 ) RH R−1 in ar (θ0 ) ⊗ Rat (θ0 )

ED

w=

M

noise ratio (INR). The optimum value of wLMr ×1 using MVDR [31] is as follows: (8)

by replacing (8) in (6), the output SINR for a specific R is: H

H

(9)

PT

SINR = ρar (θ0 ) ⊗ at (θ0 ) RH R−1 in ar (θ0 ) ⊗ Rat (θ0 ).

As can be seen in (9), SINR at the receiver is a function of waveform co180

variance matrix R. Maximum achievable SINR will be obtained when there is

CE

only noise (interferers free case). Therefore, the design of waveform covariance matrix can directly affects the output SINR and consequently, the probability of

AC

target direction. There are various methods in the literature in order to increase SINR by designing the waveform covariance matrix [23-25].

185

3. Proposed Method for Waveform Covariance Matrix Design In this section, by exploiting the correlated MIMO radar concept, we propose a covariance matrix whose transmit beam-pattern has enough specific nulls 8

ACCEPTED MANUSCRIPT

with a proper depth in interference directions. We try to cohere the transmit power in the targets’ directions and so, this method achieves higher SINR com190

pared to the recent methods introduced in [22-25]. In contrast to phased array

CR IP T

method, conventional MIMO method and methods in [23-25] which the interference directions do not affect the transmit beam-pattern, in our design we consider the interference directions in the transmitter and do not have any constraint on waveform orthogonality. In order to suppress the interferences, the 195

most important factor is the covariance matrix rank. This rank is the number of

non-zero eigenvalues and these values correspond to the number of independent

AN US

(not necessarily orthogonal) columns/rows of the matrix. Therefore, we relax the waveforms orthogonality constraint in our design. While the methods in

[23-25] and the phased array method can cohere the transmit power in only one 200

target direction, our proposed method can cohere the transmit power in several targets’ directions. For this purpose, consider J targets in θj , j = 1, ..., J and Q interference resources (includes initial estimated interference directions and

M

virtual interference directions around these directions based on uncertainty region width) in θi , i = 1, ..., Q. We assume that the transmit power in targets’ directions is k and try to minimize the transmit power value in interference

ED

205

directions. According to the transmit power in (4), we have to solve this optimization problem:

CE

PT


H  min aH at (θi ), i = 1, ..., Q  t (θi )E XX   X 
  s.t. aH (θ )E XXH a (θ ) = k, j = 1, ..., J j t t j p1 2  E  |xm,n | = LMt t , m = 1, ..., Mt , n = 1, ..., L     

(10)

AC

where Et is the total transmit power and the last constraint in p1 guarantees that the waveforms have constant envelope and so, the same amount of power

210

will transmit from all transmit antennas. This is a very important practical constraint and is an advantage of each method that satisfies it. The optimization problem of p1 is a nonconvex quadratically constrained quadratic program (QCQP) and NP-hard [33]. Therefore, we use a semidefinite relaxation (SDR) 9

ACCEPTED MANUSCRIPT

Mt

CR IP T

215

method to convert this optimization problem to a convex one. By introducing
a new variable R = E XXH , we have    min aH t (θi )Rat (θi ), i = 1, ..., Q   R    s.t. aH t (θj )Rat (θj ) = k, j = 1, ..., J p2 (11)   R  0      R(m, m) = Et , m = 1, ..., M t

the constraint R  0 results in having a positive semi-definite matrix, because

it is a covariance matrix. Also, due to the fact that R(m, m) represents the transmit power of the mth antenna, the last constraint in p2 guarantees that the

220

AN US

same amount of power will transmit from all transmit antennas. In p2, all of the

objective functions and constraints are linear functions of R and therefore, the optimization problem of p2 is a multi-objective linear programming (MOLP). Now, to solve this problem, we convert the problem p2 to equivalent standard epigraph form of SDP [36] which is single objective multi-constraint optimization

optimization problem of p2 converts to the following form:   min η   R      j   s.t. RB − k tr ≤ η, j = 1, ..., J  1     µ = εη p3
  tr RBi2 ≤ µ, i = 1, ..., Q       R0      Et R(m, m) = M , m = 1, ..., Mt t

(12)

PT

ED

225

M

problem. Indeed, we consider two auxiliary parameters µ and η. Then, the

CE

H i where Bj1 = at (θj )aH t (θj ), B2 = at (θi )at (θi ) and ε is a small positive quantity.

This type of optimization problem can be solved effectively, by numerical ap-

AC

proaches such as interior-point algorithm [15, 33]. SeDuMi [34] is an algorithm that used as a core solver in CVX [35] and can speed up the solution process of

230

solving the SDP. Therefore, p3 can be effectively solved (e.g., by using convex optimization toolbox CVX [35]) in a short period of time. Finally, after designing the desired covariance matrix, generating the waveforms that realize this covariance matrix can be performed. The infinite alpha10

ACCEPTED MANUSCRIPT

bet waveforms can be generated by the iterative method in [14]. It should be 235

noted that in [22] that waveforms are designed directly, their symbols’ phases come from an infinite alphabet such as the method in [14]. Therefore, if we use

CR IP T

the effective simulated annealing based method in [22] or the simulated annealing based method in [37], by a personal computer with 4 GHz CPU, 8 GB of RAM and for waveforms with 128 symbols, the design process needs about 3 240

and 9 minutes, respectively. These values, approximately are the same as the

time that the iterative methods need to generate the waveforms realizing the covariance matrix [26]. Therefore, the computational time needed to generate our

AN US

proposed covariance matrix is comparable with that of the method introduced in [22].

Next, if BPSK waveforms are needed specifically, which is focused recently

245

in the literature [13, 23, 24] due to their implementation simplicity, we propose an algorithm to achieve BPSK waveforms in a closed-form. In [13], authors have showed that if the waveform covariance matrix satisfies following constraints,

closed-form solution:

   R0   Et R(m, m) = M , m = 1, ..., Mt t  
  sin π R  0

PT

ED

250

M

then the BPSK waveforms that realize this matrix can easily be generated in a

2

the last constraint represents that sin

π 2R




(13)

must be positive semi-definite.

CE

Therefore, we add this constraint to our porposed optimization problem as fol-

AC

low:

  min η   R      j   s.t. RB − k tr ≤ η, j = 1, ..., J  1      µ = εη  
p4 tr RBi2 ≤ µ, i = 1, ..., Q      R0      Et  , m = 1, ..., Mt R(m, m) = M   t  
 π sin 2 R  0 11

(14)

ACCEPTED MANUSCRIPT

The optimization problem in p4 due to the new added constraint, is not a convex optimization problem and we have to relax this constraint to a convex
one. Using a Taylor series, sin π2 R can be written as
3
5
7
1 π π π π π  R R R R 2 2 2 sin R = 2 − + − +··· (15) 2 1! 3! 5! 7!

CR IP T

255

Now, according to the fact that the covariance matrix elements are equal to one

or less than one, we consider only the first two terms of Eq. (16) and rewrite this equation as follows:

260

2



R ' 1


1 π 2R

1! 3 π R ( 2 )

−


3 π 2R 3!

(16)

AN US

sin

π

( π R) by this approximation, if 2 1! − 3! will be positive semi-definite, then
sin π2 R is positive semi-definite with high probability, however, this is not

guarantee, because we ignore the remaining small value terms (which containt

negative values) in the taylor series. Therefore, we rewrite the problem of p4 as    min η   R      j  s.t. RB − k  tr ≤ η, j = 1, ..., J 1       µ = εη  
p5 tr RBi2 ≤ µ, i = 1, ..., Q      R0     Et   R(m, m) = M , m = 1, ..., Mt  t   3 1 π  π R) (  R ( ) 2 2  − 3!  0 1!

the optimization problem in p5 is still non-convex and therefore, we propose an

CE

265

(17)

PT

ED

M

follows:

iterative algorithm to relax the last constraint and solve this problem, effectively. Real waveforms such as BPSK can only be used to generate symmetric beam-

AC

patterns, because the corresponding covariance matrix has real value elements. if we have several interferences in θi , i = 1, ..., Q and a target in θt (θt 6= θi ), we

270

can assume several interferences in −θi , i = 1, ..., Q in order to have symmetric transmit beam-pattern. In this way, the corresponding covariance matrix is a real matrix and therefore, BPSK waveforms can be generated in a closed-form by an algorithm which is demonstrated in Algorithm 1. 12

ACCEPTED MANUSCRIPT

In our proposed algorithm, we set an initial value for R. We set the or275

thogonal MIMO covariance matrix as R0 which satisfies positive semi-definity 3 1 ( π2 R) ( π R) of 2 1! − 3! . Also, consider a target in θt and Q interference resources

CR IP T

in θi , i = 1, ..., Q. In each iteration of the proposed algorithm, since the last constraint is now a linear function of R, the optimization problem is convex and can be effectively solved in a short period of time. In step 6, we set a state 280

to control the convergence of the algorithm and δ is a small value which is set by user. These iterations give us an covariance matrix which is candidate to

AN US

285

be a BPSK waveforms covariance matrix with high probability. However, this
may leads that sin π2 R has some small negative eigenvalues which violates the
positive semi-definity of sin π2 R . Therefore, we first compute the eigenvalues
and eigenvectors matrices corresponding to C = sin π2 R by solving CΣ = λΣ (this equation give us eigenvectors and eigenvalues of the matrix C ) as follows: 

0

ED

0 .. .

PT

Σ=

0

h



0

0

···

0

λ2

0

0 .. .

λ3 .. .

··· .. . .. .

0 .. .

0

···

0

λMt

M

     Λ=    

λ1

Σ1

Σ2

···

0

ΣMt

i

         

(18)

(19)

where λi , i = 1, · · · , Mt are eigenvalues of C and Σi , i = 1, · · · , Mt are eigen-

CE

vectors of C. Then, if Λ has some negative diagonal elements, we force these values to zero. According to the fact that C is a real symmetric matrix, it is

diagonalizable and Σ is an invertible matrix. Therefore, we can rewrite C as:

AC

290

C = ΣΛΣ−1

(20)

This matrix is certainly positive semi-definite, because it is a toeplitz matrix and has not any negative eigenvalues. Then, we compute our proposed covariance matrix as follows: RBP SK =

  2 sin−1 (C) π 13

(21)

ACCEPTED MANUSCRIPT

we must proof that this matrix is positive semi-definite, because this is a covariance matrix. Using Taylor series, sin−1 (C) can be written as: 3

−1

sin

5

7

(C) 3 (C) 5 (C) (C) = C + + + +··· 6 40 112

(22)

CR IP T

295

Considering the fact that for two arbitrary matrices V and U, Y = V + U is

positive semi-definite when both when V and U are positive semi-definite [38]. p

300

Also if V is positive semi-definite, then (V) will be positive semi-definite where
p is a natural number [38]. Therefore, RBP SK = π2 sin−1 (C) is always pos-

itive semi-definite. Finally, the BPSK waveforms which realise this covariance

AN US

matrix are generated as follows [13]:

 1  X = sign ψΛ /2 ΣH

(23)

where X is the transmit waveforms matrix and ψ is a matrix including zero

AC

CE

PT

ED

M

mean and unit variance Gaussian random variables.

14

ACCEPTED MANUSCRIPT

Algorithm 1 Proposed Sequential Optimization Algorithm Q

Input: θt , {θi }i=1 and R0 = ROrthogonal−M IM O Output: RBP SK and BPSK waveforms in closed-form

Set n = 0, R = R0 ,

CR IP T

Initialisation:

if (Inteferences directions is not symmetric) then assume −θi , i = 1, ..., Q as additional interference directions. (n)

Use (4) to compute p1 (θ);

2:

Use the following optimization problem to compute R(n) :    min η   R       s.t. tr RBj1 − k ≤ η, j = 1, ..., J        µ = εη  
tr RBi2 ≤ µ, i = 1, ..., Q      R0     Et   R(m, m) = M , m = 1, ..., Mt  t   3 1 π  π ( 2 R)  ( 2 R)  − 3!  0 1!

(24)

M

AN US

1:

ED

(n)

3:

Use (4) to compute p2 (θ);

4:

Set n = n + 1, R0 = R;

Repeat step 1 to step 4;  2 90 P (k) (k) 1 6: Until M SE = 181 p2 (θi ) − p1 (θi ) ≤ δ;
θi =−90 7: Set C = sin π R ; 2

PT

5:

Use (19) and (20) to compute eigenvalues and eigenvectors matrices of C;

9:

Force negative eigenvalues of Λ to zero;

CE 8:

AC

10: 11:

305

Rewrite C = ΣΛΣ−1 ; Output RBP SK   1 sign ψΛ /2 ΣH .

=

2 π




sin−1 (C) and waveforms matrix X

=

Based on the computational analysis at each step, the overall computational

complexity of Algorithm 1 for p iterations is in order of O pMt3.5 + O Mt3 , 15

ACCEPTED MANUSCRIPT


which mainly comes from the steps 2 and 8 which is equal to O Mt3.5 [20]
and O Mt3 [39], respectively. Numerical simulations show that Algorithm 1

CR IP T

requires about p ≥ 3 iterations to converge a good solution. 4. Simulation Results

In this section, several numerical simulations are considered in order to eval-

310

uate the efficiency of the proposed covariance matrix design. At first, we eval-

uate the null generating efficiency of the proposed method compared to the method in [22]. Then the SINR performance of the proposed method compared

315

AN US

to the methods in [22-25] is evaluated, when there are two interferers in the

environment. Next, by increasing the number of interferences from 2 to 10, we investigate SINR behavior for all methods as a function of INR. Then, we consider the scenario in which the initial estimation of the interference directions has some errors, and we show the SINR performance of all methods in this case.

320

M

In all of the simulations, we assume that transmit/receive array is ULA with half-wavelength inter-element spacing.

In the first simulation, we consider a target at θ0 = 0◦ and an interference at

ED

θi = 50◦ . The goal is to have one peak in transmit beam-pattern in the target direction and to have one null in the interference direction as deep as possible.

325

PT

The number of transmit antennas is 16 and the total power is normalized to one. Fig. 1 shows the transmit beam-pattern using the proposed method and the method in [22]. As can be seen, our proposed method places the peak power in

CE

the target direction and creates one null with -124 dB depth in the interference direction, while the method in [22] generates a null with -75 dB depth [22]. Although -75 dB null depth is enough for most applications, the importance

of having deeper nulls appears when we want to make the radar robust against

AC 330

the initial estimation error of interference directions. Increasing the number of nulls in the transmit beam-pattern with fixed number of transmit antennas, decreases the nulls depths [22]. Therefore, the method that creates deeper nulls has a better performance.

16

CR IP T

ACCEPTED MANUSCRIPT

(a)

(b)

AN US

Figure 1: Transmit beam-pattern to create three nulls in the interference directions using (a) the proposed method and (b) the method in [22].

In the next simulation, we consider a target at θ0 = −30◦ and three interfer-

335

ence resources at θi = [15◦ , 30◦ , 60◦ ]. The number of transmit antennas is 16 and

M

total power is normalized to one. Fig. 2 shows the transmit beam-pattern using the proposed method and the method in [22]. As can be seen, because of increasing the number of nulls from one to three, the performance of the method in [22] is degraded about 15 dB [22]. However, the performance of our proposed method

ED

340

is not degraded. If there are only three interferences at [a◦ , b◦ , c◦ ] and the initial estimation can have one degree of error, we need to create 9 nulls in the following

PT

directions: [(a − 1)◦ , a◦ , (a + 1)◦ , (b − 1)◦ , b◦ , (b + 1)◦ , (c − 1)◦ , c◦ , (c + 1)◦ ]. This shows the importance of using a method which can create several nulls as deep as possible in the transmit beam-pattern in order to hold its perfor-

CE

345

mance, when the number of nulls increases or we do not know exactly the

AC

interference directions. In the next simulation, we assume a target located in θ0 = 0◦ and the ini-

tial estimation shows three interference directions at θi = [−40◦ , 25◦ , 60◦ ] with

350

1◦ uncertainty. The number of transmit antennas is 20. The goal is to cohere the transmit power in target direction and put 9 nulls in θi = [−41◦ , −40◦ , −39◦ , 24◦ , 25◦ , 26◦ , 59◦ , 60◦ , 61◦ ]. Fig. 3 shows the transmit beam-pattern using the proposed method in order

17

CR IP T

ACCEPTED MANUSCRIPT

(a)

(b)

AN US

Figure 2: Transmit beam-pattern to create a null in the interference direction using (a) the proposed method and (b) the method in [22].

0

M

Proposed Method

ED

-40

PT

Pt( ) (dB)

-20

CE

-60

AC

-80

-80

-40

0

40

80

(Degrees)

Figure 3: Transmit beam-pattern using the proposed method in order to make it robust against the uncertainty in interference directions.

18

ACCEPTED MANUSCRIPT

to make it robust against the initial estimation error of interference directions. As can be seen, the proposed method puts the peak transmit power in the 355

target direction and creates 9 nulls about -80 dB proper depth in the uncertainty

CR IP T

region of the interference directions. Obviously, according to the scenario and the uncertainty region width, the number of nulls in specific directions can be adjusted.

Now, in order to evaluate the SINR performance of the methods in [22360

25] and our proposed method, we consider several simulations. In the next simulation, we assume a target located at θ0 = 0◦ and two interferers located

AN US

at θi = [−10◦ , 30◦ ]. To consider the effect of various number of antennas, we

swipe the number of transmit and receive antennas from 8 to 20 with the step of 4 and get a mean from the results. Also we set the INR to 50 dB. Fig. 4 365

shows the SINR behavior as a function of SNR. According to the mentioned co-array concept, the phased array method achieves higher SINR performance compared to the recently methods in [22-25]. However, our proposed method

M

due to the used technique in the transmitter, has a bit higher SINR than that of phased array method, even in the known interference direction scenario. The performance difference of our proposed method compared to the phased array

ED

370

method, method in [22], method in [23], the first and the second methods in [24], phased-MIMO method and conventional MIMO method is 0.23 dB, 9.8 dB,

PT

3.49 dB, 1.17 dB, 4.08 dB, 2.93 dB and 10.57 dB, respectively. In the next simulation, we would like to evaluate the effect of increasing the contribution of transmit antennas among total antennas. We assume that the

CE

375

number of receive antennas is 8 and swipe the number of transmit antennas from 12 to 30 with the step of 6 and get a mean from the results. All other

AC

parameters are the same as the previous simulation. Increasing the number of transmit antennas leads to the reduction of difference between the obtained

380

transmit beam-pattern and the desired one. Fig. 5 shows the SINR behavior as a function of SNR for this case. As can be seen, by increasing the contribution of transmit antennas, the performance difference between our proposed method and the other counterparts is higher than that of previous simulation values, as 19

ACCEPTED MANUSCRIPT

it was expected. Compared to the phased array method and the second method 385

in [24], the difference is 1.2 dB and 1.49 dB, respectively.

CR IP T

45 40

SINR (dB)

35

AN US

30 25

Method in [23] Conventional MIMO Method Phased Array Method First Method in [24] Second Method in [24] Proposed Method Phased-MIMO in [25] Method in [22]

20

10 -10

-5

M

15

0 SNR (dB)

5

10

ED

Figure 4: SINR comparison between the proposed method, methods in [22-25], phased array and conventional MIMO methods.

PT

In the next simulation, we increase the number of interferences to 10 and then, by increasing the INR, we compare the SINR performance of all methods. Consider 10 interferers located at −10◦ , 30◦ , 45◦ , −40◦ , 50◦ , 55◦ , −70◦ , 80◦ , −85◦

CE

and 15◦ . Also the SNR is set to 20 dB and we swipe the number of transmit

390

and receive antennas from 18 to 30 with the step of 2 and get a mean from the results. Fig. 6 shows the SINR behavior as a function of INR. At first, when

AC

the INR is low, due to co-array concept, the phased array method outperforms all other methods. However, by increasing the INR, the performance of this method degrades significantly because of not using waveform diversity. On the

395

other hand, our proposed method, similar to the conventional MIMO and the methods in [22, 24, 25] holds its performance and suppresses the interferences

20

ACCEPTED MANUSCRIPT

45

CR IP T

40

SINR (dB)

35 30 25

Method in [23] Conventional MIMO Method Phased Array Method First Method in [24] Second Method in [24] Proposed Method Phased-MIMO in [25] Method in [22]

AN US

20 15 10 -10

-5

0 SNR (dB)

5

10

M

Figure 5: SINR comparison between the proposed method, methods in [22-25], phased array and conventional MIMO methods in case of increasing the transmit antennas

ED

contribution.

effectively and achieves higher SINR compared to its counterparts. If the INR increasing continues, since the number of antennas is fixed, the performance of

PT

all methods will be degraded.

In the next simulation, similar to the simulation corresponding to Fig. 5,

400

CE

we evaluate the effect of increasing transmit antennas. We assume 5 receive antennas and swipe the number of transmit antennas from 18 to 30 with the step of 2. All other parameters are the same as the previous simulation. As

AC

can be seen in Fig. 7, the SINR performance difference between our proposed

405

method and that of counterparts is increased due to increasing DOF in the transmitter. Also, because of decreasing the number of total antennas compared to the previous simulation, the performance of the method in [23] outperforms phased array counterpart. The rank of covariance matrix in [23] and phased

21

ACCEPTED MANUSCRIPT

70 60

CR IP T

50

30 20 10 0

-10 -20 -10

Proposed Method Phased Array Method Second Method in [24] Method in [23] First Method in [24] Method in [22] Conventional MIMO Method Phased-MIMO in [25]

0

10

AN US

SINR (dB)

40

20 30 INR (dB)

40

50

60

M

Figure 6: Comparison the SINR behavior as a function of INR for the proposed method, methods in [22-25], phased array and conventional MIMO methods.

410

ED

array method are 2 and 1, respectively and therefore, the method in [23] has more interference suppression capability. In the next simulation, we assume the scenario with 2 interferers located

PT

at θi = [−10◦ , 30◦ ], while due to an initial estimation error at the receiver, the interference directions are estimated at θi = [−11◦ , 29◦ ]. Assume that INR

CE

is 30 dB and we swipe the number of transmit and receive antennas from 12

415

to 20 with the step of 2 and get a mean from the results. Fig. 8 shows the SINR performance as a function of SNR for this case. As can be seen, even one

AC

degree of error in the initial estimation of only two interferers’ directions, causes significant SINR degradation for all methods except our proposed method and method in [22]. The reason is in both of these methods, in the transmitter, we

420

can create 6 nulls in θi = [−11◦ , −10◦ , −9◦ , 29◦ , 30◦ , 31◦ ]. However, unlike the method in [22] which uses orthogonal waveforms, our proposed method achieves

22

ACCEPTED MANUSCRIPT

60 50

CR IP T

40

20 10 0

-10 -20 -30 -10

Proposed Method Phased Array Method Second Method in [24] Method in [23] First Method in [24] Method in [22] Conventional MIMO Method Phased-MIMO in [25]

0

10

AN US

SINR (dB)

30

20 30 INR (dB)

40

50

60

M

Figure 7: Comparison the SINR behavior as a function of INR for the proposed method, the methods in [22-25], the phased array and the conventional MIMO methods

ED

in case of increasing the transmit antennas contribution.

much higher SINR compared to the method in [22], because of using correlated waveforms. The SINR difference of our proposed method compared to that

425

PT

of phased array method, method in [22], method in [23], the first and second methods in [24] and conventional MIMO method is 16.96 dB, 7.63 dB, 16.21 dB,

CE

21.18 dB, 17.9 dB and 22.46 dB, respectively. It should be noted that in this scenario, the SINR performance of the method in [23] and the second method in [24] is improved compared to the other methods. The reason is that the SLLs of

AC

the receive beam-pattern for these two methods is much lower than that of the

430

other methods. Therefore, these methods reduce the sensitivity to interferers location. We showed that sin

π 2 RBP SK




is positive semidefinite, which enables us to

realize RBP SK with BPSK waveforms in a closed-form. Now, we want to eval-

23

ACCEPTED MANUSCRIPT

50

CR IP T

40

SINR (dB)

30

20

Method in [23] Conventional MIMO method Phased Array Method First Method in [24] Second Method in [24] Proposed Method Method in [22]

AN US

10

0

-10 -10

-5

0 SNR (dB)

5

10

M

Figure 8: Comparison the SINR behavior as a function of SNR for the proposed method, methods in [22-24], phased array and conventional MIMO method in the

ED

scenario which the initial estimation of the interference directions has some errors.

uate SINR performance difference between infinite alphabet and BPSK wave435

forms which realize our proposed covariance matrix. We assume a target located

PT

at θ0 = 0◦ and two interferences located at θi = [−15◦ , 40◦ ]. We swipe the number of transmit and receive antennas from 12 to 20 with the step of 4 and get a

CE

mean from the results. Also we set the INR to 50 dB and assume 128 symbols for each waveform. Fig. 9 shows the SINR level for infinite alphabet and BPSK

440

waveforms which realize R and RBP SK , respectively. As can be seen, SINR

AC

performance of BPSK waveforms is lower than infinite alphabet waveforms (as expected) and the difference is about 1.5 dB. However, BPSK waveforms and infinite alphabet waveforms with 128 symbols can be generated in 10−4 seconds and 3 minutes, respectively [26]. Therefore, in addition to implementation sim-

445

plicity of BPSK waveforms, they can be generated in much fewer times than

24

ACCEPTED MANUSCRIPT

that of counterpart and they are suitable for real-time applications. It is worth noting if we increase the number of waveform symbols, the difference between the infinite alphabet waveforms covariance matrix and covariance

CR IP T

matrix which is realized by BPSK waveforms will be decrease.

50

Infinite alphabet waveforms BPSK waveforms

45

SINR (dB)

AN US

40

35

30

-5

ED

20 -10

M

25

0 SNR (dB)

5

10

Figure 9: Comparison of SINR obtained by infinite alphabet and BPSK waveforms

5. Conclusion

CE

450

PT

which realize R and RBP SK , respectively.

In this work, we proposed a waveform covariance matrix design method and

AC

an innovative algorithm that makes this covariance matrix a good candidate for generating BPSK waveforms in a closed-form. By exploiting the correlated MIMO radar concept, this proposed method increases the achieved SINR com-

455

pared to that of conventional MIMO radar, phased array method and recent proposed methods. Also due to the technique we used in the transmitter and

25

ACCEPTED MANUSCRIPT

creating several nulls with enough depth in the uncertainty region of the interferers’ locations, we reduced the sensitivity to have some errors in the initial estimation of the interference directions. Therefore, when there are some errors in the initial estimation of interferers, our proposed method achieves much

CR IP T

460

higher SINR compared to the recent methods. The proposed optimization prob-

lem to design the waveform covariance matrix and our proposed algorithm can effectively be solved by numerical approaches in a short period of time due to their low computational complexity. Unlike the phased-MIMO method, our pro465

posed method satisfies the constant envelope practical constraint. In contrast

AN US

to the most recent methods which proposed fixed covariance matrices that co-

here the transmit power only in one target direction, our proposed method can cohere the transmit power in several targets’ directions. Simulation results are inline with our analytical results.

6. Refrences

M

470

References

USA, 2009.

ED

[1] J. Li, P. Stoica, MIMO Radar Signal Processing, Wiley, New York, NY,

PT

[2] P. Stoica, J. Li, and X. Zhu, ”MIMO radar with co-located antenna: Review of some recent work,” IEEE Signal Process. Mag., vol. 24, no.5, pp. 106-114,

475

CE

Sep. 2007.

[3] A. M. Haimovich, R. S. Blum, and L. J. Cimini, ”MIMO radar with widely separated antennas,” IEEE Signal Process. Mag., vol. 25, no. 1, pp. 116-129,

AC

Jan. 2008.

480

[4] J. Li, P. Stoica, L. Xu and W. Roberts, ”On Parameter Identifiability of MIMO Radar,” in IEEE Signal Processing Letters, vol. 14, no. 12, pp. 968971, Dec. 2007.

26

ACCEPTED MANUSCRIPT

[5] Fishler, E., Haimovich, A., Blum, R. S., Cimini, L. J., Chizhik, D., and Valenzuela, R. A. Spatial diversity in radars - Models and detection performance. IEEE Transactions on Signal Processing, 54, 3 (Mar.2006), 823-838.

485

CR IP T

[6] Yang, Y., and Blum, R. S. MIMO radar waveform design based on mutual

information and minimum mean-square error estimation. IEEE Transactions on Aerospace and Electronic Systems, 43, 1 (Jan.2007), 330-343.

[7] S. M. Alamouti, A simple transmitter diversity scheme for wireless communications, IEEE J. Sel. Areas Commun., vol. 16, pp. 1451-1458, Oct. 1998.

490

AN US

[8] D. Tse and P.Viswanath, Fundamentals of Wireless Communication. Cambridge, U.K.: Cambridge Univ. Press, 2005.

[9] D. R. Fuhrmann and J. S. Antonio, ”Transmit beamforming for MIMO radar systems using signal cross-correlation,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, no. 1, pp. 171-185, Jan. 2008.

495

M

[10] S. Imani, M. M. Nayebi and S. Ghorashi, ”Transmit Signal Design in Colocated MIMO Radar Without Covariance Matrix Optimization,” in IEEE

ED

Transactions on Aerospace and Electronic Systems , vol.PP, no.99, pp.1-1. [11] Jian Li, Petre Stoica and Xumin Zhu, ”MIMO radar waveform synthesis,” 2008 IEEE Radar Conference, Rome, 2008, pp. 1-6.

PT

500

[12] S. Jardak, S. Ahmed and M. S. Alouini, ”Generation of Correlated Finite

CE

Alphabet Waveforms Using Gaussian Random Variables,” in IEEE Transactions on Signal Processing, vol. 62, no. 17, pp. 4587-4596, Sept.1, 2014.

AC

[13] S. Ahmed, J. S. Thompson, Y. R. Petillot and B. Mulgrew, ”Finite Al-

505

phabet Constant-Envelope Waveform Design for MIMO Radar,” in IEEE Transactions on Signal Processing, vol. 59, no. 11, pp. 5326-5337, Nov. 2011.

[14] P. Stoica, J. Li, and X. Zhu, ”Waveform synthesis for diversity-based transmit beampattern design,” IEEE Trans. Signal Process., vol. 56, no. 6, pp. 2593-2598, June 2008. 27

ACCEPTED MANUSCRIPT

510

[15] P. Stoica, J. Li and Y. Xie, ”On Probing Signal Design For MIMO Radar,” in IEEE Transactions on Signal Processing, vol. 55, no. 8, pp. 4151-4161, Aug. 2007.

CR IP T

[16] M. Haghnegahdar, S. Imani, S. A. Ghorashi and E. Mehrshahi, ”SINR Enhancement in Colocated MIMO Radar Using Transmit Covariance Matrix Optimization,” in IEEE Signal Processing Letters, vol. 24, no. 3, pp. 339-

515

343, March 2017.

[17] J. Liu, H. Li and B. Himed, ”Joint Optimization of Transmit and Receive

no. 1, pp. 39-42, Jan. 2014. 520

AN US

Beamforming in Active Arrays,” in IEEE Signal Processing Letters, vol. 21,

[18] S. Imani and S. Ghorashi, ”Transmit signal and receive filter design in colocated mimo radar using a transmit weighting matrix,” Signal Processing Letters, IEEE, vol. 22, no. 10, pp. 1521-1524, Oct 2015.

M

[19] S. Imani and S. A.Ghorashi, ”Sequential quasi-convex-based algorithm for waveform design in colocated multiple-input multiple-output radars,” IET Signal Processing, vol. 10, no. 3, pp. 309-317, May 2016.

ED

525

[20] G. Cui, H. Li, and M. Rangaswamy, ”Mimo radar waveform design with constant modulus and similarity constraints,” IEEE Trans. Signal Process.,

PT

vol. 62, no. 2, pp. 343-353, Jan 2014. [21] J. Lipor, S. Ahmed, M.-S. Alouini, Fourier-based transmit beampattern design usingMIMO radar, IEEE Trans. Signal Process. 62 (9) (2014) 2226-

CE

530

2235.

AC

[22] H. Deng, Z. Geng and B. Himed, ”Mimo radar waveform design for transmit

535

beamforming and orthogonality,” in IEEE Transactions on Aerospace and Electronic Systems, vol. 52, no. 3, pp. 1421-1433, June 2016.

[23] S. Ahmed and M.-S. Alouini, ”Mimo-radar waveform covariance matrix for high sinr and low side-lobe levels,” IEEE Trans. Signal Process., vol. 62, no. 8, pp. 2056-2065, April 2014. 28

ACCEPTED MANUSCRIPT

[24] S. Imani, S. A.Ghorashi and M. Bolhasani ”SINR maximization in colocated MIMO radars using transmit covariance matrix,” Signal Processing, vol. 119, pp. 128-135, Feb 2016.

540

CR IP T

[25] A. Hassanien and S. A. Vorobyov, ”Phased-MIMO radar: A tradeoff be-

tween phased-array and MIMO radars,” IEEE Trans. Signal Process., vol. 58, no. 6, pp. 3137-1351, Jun. 2010.

[26] S. Ahmed and M. Alouini, ”A survey of correlated waveform design for

multifunction software radar,” in IEEE Aerospace and Electronic Systems

545

AN US

Magazine, vol. 31, no. 3, pp. 19-31, March 2016.

[27] A. Aubry, A. DeMaio, A. Farina and M. Wicks, ”Knowledge-Aided (Potentially Cognitive) Transmit Signal and Receive Filter Design in SignalDependent Clutter,” in IEEE Transactions on Aerospace and Electronic Systems, vol. 49, no. 1, pp. 93-117, Jan. 2013.

550

M

[28] A. Duly, D. Love, and J. Krogmeier, ”Time-division beamforming for MIMO radar waveform design,” IEEE Trans. Aerosp. Electron. Syst., vol.

ED

49, no. 2, pp. 12101223, Apr. 2013.

[29] A. De Maio, Y. Huang, and M. Piezzo, ”A Doppler robust max-min approach to radar code design,” IEEE Trans. Signal Process., vol. 58, no. 9,

555

PT

pp. 49434947, Sep. 2010.

[30] W. Zhu and J. Tang, ”Robust Design of Transmit Waveform and Receive

CE

Filter For Colocated MIMO Radar,” in IEEE Signal Processing Letters, vol. 22, no. 11, pp. 2112-2116, Nov. 2015.

[31] H. L. V. Trees, Optimum array processing Part IV of detection, estimation,

AC

560

and modulation theory. John Wiley, 2002.

[32] M. Greco, F. Gini and A. Farina, ”Radar Detection and Classification of Jamming Signals Belonging to a Cone Class,” in IEEE Transactions on Signal Processing, vol. 56, no. 5, pp. 1984-1993, May 2008.

29

ACCEPTED MANUSCRIPT

565

[33] Z. q. Luo, W. k. Ma, A. M. c. So, Y. Ye and S. Zhang, ”Semidefinite Relaxation of Quadratic Optimization Problems,” in IEEE Signal Processing Magazine, vol. 27, no. 3, pp. 20-34, May 2010.

CR IP T

[34] J. F. Sturm, ”Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optim. Meth. Softw., vol. 11-12, pp. 625653, 1999. 570

[35] M. Grant and S. Boyd, ”CVX: Matlab software for disciplined convex programming, version 2.1,” http://cvxr.com/cvx, Mar. 2014.

[36] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge: U.K.:

AN US

Cambridge University Press, 2004.

[37] Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P. Optimization by simulated annealing. Science, 220 (May 1983), 671-680.

575

[38] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cam-

M

bridge Univ. Press, 1985.

[39] J. Demmel, I. Dumitriu, and O. Holtz, Fast linear algebra is stable, Numer.

AC

CE

PT

ED

Math. 108 (2007), no. 1, 59-91.

30