SINR maximization in colocated MIMO radars using transmit covariance matrix

Signal Processing ] (]]]]) ]]]–]]]

1

Contents lists available at ScienceDirect

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Signal Processing

5

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

7 9 11 13

SINR maximization in colocated MIMO radars using transmit covariance matrix

15 Q1

Sadjad Imani a,n, Seyed Ali Ghorashi a,b, Mostafa Bolhasani a

17

a Cognitive Telecommunication Research Group, Department of Telecommunications, Faculty of Electrical Engineering, Shahid Beheshti University, G.C., Tehran, Iran b Cyber Research Centre, Shahid Beheshti University, G.C., Tehran, Iran

19 21 23 25 27 29 31

a r t i c l e i n f o

abstract

Article history: Received 6 March 2015 Received in revised form 8 July 2015 Accepted 15 July 2015

The main waveform design features in multiple-input multiple-output (MIMO) radars include (a) signal transmission with full rank covariance matrix in order to use the maximum waveform diversity and to suppress more number of interferers, (b) constant envelope in order to have simplicity in deployment and to reduce the destructive effect of nonlinear amplifiers and (c) small side lobe level (SLL) in order to reduce the effect of interferers with unknown location. Therefore, in order to maximize the signal-tointerference-plus-noise ratio (SINR) and to exploit the advantages of MIMO radars, in this paper we have proposed two full rank transmit covariance matrices and maximum achievable SINR is calculated analytically for both. We have shown that the two proposed covariance matrices can be used to generate BPSK waveforms which satisfy constant modulus constraint. Simulation results show that when the angle location of interferences is known, the first proposed matrix achieves a higher level of SINR compared to the second one, while the second proposed matrix has a lower SLL compared with the first one. Also we have shown that the two proposed covariance matrices can handle more interferences compared to phased-array and the recently proposed methods in MIMO radars. & 2015 Elsevier Ltd All rights reserved.

Keywords: Multiple-Input Multiple-Output radar Covariance matrix Signal-to-interference-plus-noise ratio Waveform design

33 35 37 39 41 43 45 47 49 51 53

63

1. Introduction Using a similar idea of MIMO communications, recently a new type of radar that is known as MIMO radar is introduced [1]. MIMO radar emits different waveforms through its antennas, while phased array radar sends phase-shifted versions of a single waveform. Transmitting different waveforms in MIMO radar provides more degrees of freedom (DOF) and therefore has many benefits compared with traditional phased array radars [2,3]. MIMO radar can be classified into two main groups: widely separated [4] and colocated

55 57 59

n

Corresponding address. E-mail address: [email protected] (S. Imani).

antennas [5]. In radars with widely separated antennas, transmit antennas are far enough from each other and therefore the target radar cross sections (RCS) for different transmitting paths are independent random variables and spatial diversity of target improves. Co-located MIMO radar utilizes the antenna configuration of phased array, but sends different waveforms and create virtual arrays that provides more flexibility in beampattern matching design. The purpose of waveform design in radars is to transmit power in certain directions in order to enhance the signal-to-interference-plusnoise ratio (SINR) at the receiver. The process of waveform design can be divided into two parts; (a) designing the transmit waveform covariance matrix R [6–10], (b) synthesizing the transmit waveforms in order to realizes the covariance matrix R [11–13]. Several algorithms are proposed to design

http://dx.doi.org/10.1016/j.sigpro.2015.07.011 0165-1684/& 2015 Elsevier Ltd All rights reserved.

61 Please cite this article as: S. Imani, et al., SINR maximization in colocated MIMO radars using transmit covariance matrix, Signal Processing (2015), http://dx.doi.org/10.1016/j.sigpro.2015.07.011i

65 67 69 71 73 75 77 79 81

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S. Imani et al. / Signal Processing ] (]]]]) ]]]–]]]

transmit waveform with different constraints such as low peak-to-average power ratio (PAPR) and proper ambiguity function [14]. A sequential algorithm is presented in [14] to jointly design the transmit waveform and receive combining filter in order to maximize the SINR for a point-like target in the presence of multiple interferences based on convex optimization. This algorithm is very slow because of its high computational complexity. Also in [15] we design the transmit signal and receive combining filter, in order to maximize the SINR with a quasi-convex based method. But the order of complexity of our method is high. Recently, authors in [16] proposed a transmit covariance matrix to achieve the SINR of phased array radars and also reduce the side lobe level (SLL) in the receive beampattern. In [15] and [16], it is assumed that the target and interference locations are known (the direction of target and interference can be estimated using [17–20]). However, generally there are someinterference with unknown locations around the MIMO radar. Therefore, having lower SLL enables MIMO radars to suppress unknown interferences. It is shown in [16] that the rank of proposed covariance matrix for all number of transmit antennas is always two. This means that the method in [16] does not exploit the full waveform diversity in MIMO radars. Also in [13] it is shown that for a
 given covariance matrix, R, if sin ðπ=2ÞR is positive semidefinite, binary phase shift keying (BPSK) waveforms can be designed in closed form in order to realize R.
However,  for the proposed covariance matrix in [16], sin ðπ=2ÞR is not positive semidefinite. In this paper we propose two toeplitz covariance matrices in order to maximize the SINR at the receiver. The proposed covariance matrices are designed based on the following criteria: (a) to have full rank. Notice that when covariance matrix has full rank, then by the same number of antennas it would be possible to suppress more number of interferers compared with matrices with lower rank order. (b) To be able to built a constant envelope waveform based on it, because to have a constant envelope is one of the important features in waveform design which yields to avoid destructive nonlinear effect of amplifiers. (c) To have small SLL in order to be able to combat the effect of interferers with unknown location. It is shown mathematically for the first proposed covariance matrix that the maximum achievable SINR is higher than that of the proposed one in [16], but it has a higher SLL. Therefore, when the angle location of target and interferences are known, our proposed method is more efficient. The second covariance matrix is proposed in order to maximize SINR and reduce the SLL similar to [16]. The important contribution of this paper is that both of our proposed covariance matrices are full rank and therefore they can suppress more number of interferences based on co-array concept [21]. Also, we have proved that for both of
our proposed covariance matrices, sin ðπ=2ÞR is positive semidefinite, which means that BPSK waveforms can be used to realize both of our covariance matrices. BPSK is a signaling that despite of its simple structure has a constant envelope. This means that our proposed covariance matrices can be easily implemented. Simulation results show that the two proposed covariance matrices are able to suppress more interferences compared with phased array radars and other existing correlated MIMO radars. The organization of this paper is as follows: Section 2 introduces the problem

formulation for a single point-like target in the present of multiple interferences and SINR maximization problem. The proposed covariance matrices are introduced in Section 3. Section 4 shows the numerical results and conclusion is provided in Section 5. Notation: Bold upper case letters, X, and lower case letters, x, denote matrices and vectors, respectively. Transposition, conjugate transposition and inverse of a matrix are denoted by ðÞT , ðÞH and ðÞ  1 , respectively and expectation operator is denoted by Efg. The Kronecker product and the Hadamard multiplication are denoted by  and  , respectively. U is uniform distribution, detðÞ denotes determinant of a square matrix and j  j denotes the absolute value. the vectorization of a matrix denoted by vec (  ).

Consider a co-located MIMO radar system equipped with a transmit array of M t and a receive array of M r isotropic antennas. Each transmit antenna emits sm ðnÞ; m ¼ 1; …; M t ; n ¼ 1; …; L which is different from other transmitted waveforms. L is the sample number in each waveform. Also, assume that sðnÞ ¼ ½s1 ðnÞ; s2 ðnÞ; …; sMt ðnÞT is an M t  1 vector of transmit waveforms in time sample n. Then, the signal at target location θ is at ðθÞT sðnÞ; n ¼ 1; …; L: For a ULA with half-wavelength inter-elements spacing, at ðθÞ is given by sin ðθ Þ

65 67 69 71 73 75 77 79

2. Problem formulation

at ðθÞ ¼ ½1; e  jπ

63

; …; e  jπ ðMt  1Þ sin ðθÞ T :

yðnÞ ¼ α0 ar ðθ0 Þat ðθ0 ÞT sðnÞ þ

83 85 87 89 91

ð1Þ

Assume that there is a point-like target at the location θ0 as well as Q signal-dependent interference sources at locations θj ; j ¼ 1; ‥; Q . Therefore, the baseband signal at the receiver can be expressed as Q X

81

αj ar ðθj Þat ðθj ÞT sðnÞ þ νðnÞ

93 95 97 99

j¼1

ð2Þ where α0 and αj are the RCSs of the target and interference sources respectively, and ar ðθÞ represents the Mr  1 received steering vector. νðnÞ describes the independent and identically distributed additive white Gaussian noise vector at M r receivers with covariance matrix σ 2v I. Also assume that αj 's are mutually uncorrelated with zero mean and variance σ 2j . The power of the transmitted signal at target location can be expressed as  2 H  ð3Þ pðθÞ ¼ EaH t ðθ Þs ¼ at ðθ ÞRat ðθ Þ where R ¼ EðssH Þ is the covariance matrix of the transmitted signal. As can be seen in (3), transmit beampattern can be controlled by designing the transmit covariance matrix R. This matrix must be positive semidefinite because it is a covariance matrix. In waveform design, the transmit power of all antennas are equal. The received signal at the elements of receive array is passed through M t matched filters and will be stored in an M r  Mt matrix Z as Z ¼ α0 ar ðθ0 Þat ðθ0 ÞT R þ

Q X

101 103 105 107 109 111 113 115 117 119 121

αj ar ðθj Þat ðθj ÞT R þ νsH

ð4Þ

j¼1

Please cite this article as: S. Imani, et al., SINR maximization in colocated MIMO radars using transmit covariance matrix, Signal Processing (2015), http://dx.doi.org/10.1016/j.sigpro.2015.07.011i

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S. Imani et al. / Signal Processing ] (]]]]) ]]]–]]]

1 3 5

  using vecðAXBÞ ¼ BT  A vecðX Þ we can write     y ¼ vec ZT ¼ α0 vec Rat ðθ0 Þar ðθ0 ÞT þ

Q X





αj vec Rat ðθj Þar ðθj ÞT þ vc

j¼1

7 ¼ α0 ar ðθ0 Þ  Rat ðθ0 Þ þ 9 11 13 15

Q X

αj ar ðθj Þ  Rat ðθj Þ þ vc

ð5Þ

j¼1

where vc is the noise vector with zero mean (Eðvc Þ ¼ 0) and   


 T  T H vec νsH σ 2vc ¼ E vc vHc ¼ E vec νsH  

 ¼ E ððI  sÞvecðνÞÞððI  sÞvecðνÞÞH ¼ σ 2 E ðI  sÞ I  sH

17 19 21

using ðA  BÞðC  DÞ ¼ ðAC  BDÞ, the variance of vc can be expressed as


 σ 2vc ¼ σ 2 E I  ssH ¼ σ 2 I  E ssH ¼ σ 2 ðI  R Þ ð6Þ

radars the maximum achievable SINR can be written as [22]

63

SINRphased ¼ ρM r M2t :

65

27 29 31 33 35 37 39 41 43

2.1. Maximizing the SINR Since a high SINR level improves the parameter estimation and signal detection in the presence of Gaussian interference, the goal is to maximize the SINR at a receiver by designing the transmit covariance matrix and receive beamforming filter. The output of matched filters pass through receive beamforming filter wNMr 1 and therefore SINR is as follows:   ρwH ar ðθ0 Þ  Rat ðθ0 Þ2 ð7Þ SINR ¼ wH R in w 

P where ρ ¼ E jα0 j2 =σ 2 and Rin ¼ Qi¼ 1 ηi ar ðθi Þ  Rat ðθi Þj2 n  o 2 þðInR  RÞ and ηj ¼ E αj  =σ 2 . As can be seen from (7), maximizing the SINR needs the knowledge of target locations and interferences. However, designing the optimum receive filter w is independent of transmit covariance matrix R. Using the MVDR method, the receive combining filter can be calculated as 1 Rin ar ðθ0 Þ  Rat ðθ0 Þ

45

w¼

47

by inserting (8) in (7) the optimal SINR is

49 51 53 55

1 ar ðθ0 ÞH  at ðθ0 ÞH R H Rin ar ðθ0 Þ  Rat ðθ0 Þ

1 ar ðθ0 Þ  Rat ðθ0 Þ: SINR ¼ ρar ðθ0 ÞH  at ðθ0 ÞH RH Rin

ð8Þ

67 69 71 73 75 77 79 81 83

ð12Þ

85

They showed that this matrix has a low side lobe level compared to phased array and MIMO radars. For the proposed matrix in [16], maximum achievable SINR is ! 1
 : SINRRx ¼ 2ρMr ð13Þ 1  cos π =M t

87

For large number of transmit antennas, the maximum SINR obtained by Rx is SINRRx C 0:4SINRphased . It is shown in [16] that the rank of matrix in (12) is 2 for all values of Mt . The covariance matrix in (12) can be expressed as the sum of two phased array radars and therefore it does not exploit the full waveform diversity. Our goal is to maximize the SINR with a full rank matrix in order to use more degrees of freedom in MIMO radars. This DOF leads to reject more interferences compared to phased array radars. In addition, it is shown in
 [16] that for a given covariance matrix, R, if sin ðπ =2ÞR is positive semidefinite, in order to realize R, binary phase shift keying (BPSK) waveforms can be designed in closed form. However, the covariance matrix in [16] is not positive semidefinite.

89 91 93 95 97 99 101 103 105 107 109

ð9Þ

As can be seen from (9), the optimal SINR is a function of transmit covariance matrix. In conventional MIMO radars, transmit waveforms are uncorrelated and therefore the transmit covariance matrix is an identity matrix. When there is no interference, the maximum achievable SINR in the case of conventional MIMO radars at the receiver can be written as

57

SINRMIMO ¼ ρM r M t :

59

In phased array radars, all transmit array antennas send a scaled version of a waveform, and therefore all elements of the transmit covariance matrix are 1. For phased array

61

ð11Þ

As can be seen in (10) and (11), phased array radars can achieve a higher level of SINR compared to conventional MIMO radars (MIMO radars transmit the same power in all directions, while phased arrays cohere the power in the direction of target). However, based on co-array concept [21] we know that MIMO radars can identify more parameters compared to phased arrays (because MIMO radars use the waveform diversity). In order to exploit the performance of MIMO radars and achieve the SINR of phased array radars, authors in [16] have proposed a transmit covariance matrix as follows:    3 2 1 cos Mπt ⋯ cos π ðMMt t 1Þ 6 7   6 7 6 cos π 7 1 ⋱ ⋮ Mt 6 7 6 7:   Rx ¼ 6 7 ⋮ ⋱ ⋱ cos Mπt 6 7 6 7     4 5 1Þ π ⋯ cos 1 cos π ðMMt  M t t

23 25

3

ð10Þ

3. Proposed covariance matrices 111 Here we propose two covariance matrices. The first one has a higher level of SINR compared to the second one as well as the covariance matrix introduced in [16], but its SLL is higher than them. The SINR of the second one is close to the SINR in [16] and has the same SLL as covariance matrix introduced in [16]. However, both proposed covariance matrices are full rank and therefore they use the full diversity which leads to reject more interferences compared to covariance matrix in [16] and phased array radars.
 Also in both proposed covariance matrices sin ðπ =2ÞR is positive semidefinite, which enables us to realize R with BPSK waveforms in a closed form.

Please cite this article as: S. Imani, et al., SINR maximization in colocated MIMO radars using transmit covariance matrix, Signal Processing (2015), http://dx.doi.org/10.1016/j.sigpro.2015.07.011i

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S. Imani et al. / Signal Processing ] (]]]]) ]]]–]]]

4

1

14

3

12

M =5 M =10 M =15 M =20

10

7

8

9

4

13

2

15

Fig. 1. Transmit beampattern using the first proposed covariance matrix for different number of transmit antennas. The total transmit power is normalized to one.

 Rp1 at ðθ0 Þ    ðaÞ 1 1 ¼ ρ ar ðθ0 ÞH  at ðθ0 ÞH RH p1  IM r  R p1 ar ðθ 0 Þ   Rp1 at ðθ0 Þ   ðbÞ ¼ ρ ar ðθ0 ÞH  at ðθ0 ÞH RH p1 ar ðθ 0 Þ  at ðθ 0 Þ

3.1. The first proposed covariance matrix

¼ ρMr at ðθ0 ÞH RH p1 at ðθ 0 Þ

-60

-40

-20

17 19

0

20

40

60

θ (Degrees)

21

ðcÞ

23 25 27 29 31

As mentioned previously, we can cohere the power by proper R in the region of interest (ROI). Therefore, the power of back scattered signals from target at the receiver increases. In order to increase the SINR level and get close as much as possible to the SINR level of phased arrays as an upper limit, and at the same time in order to exploit the full waveform diversity in MIMO radars, we propose the covariance matrix as follows: 2

33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

R p1

1 6 6 Mt  1 6 Mt ¼6 6 ⋮ 6 4

Mt  1 Mt

⋯

1

⋯

⋱

⋱

⋯

Mt  1 Mt

1 Mt

1 Mt

3

7 ⋮ 7 7 7 Mt  1 7 Mt 7 5 1

since the proposed matrix is real and symmetric, the

Q3 transmit beampattern P t ðθ Þ is symmetric (Fig. 1). In Fig. 1

it is depicted that side lobes in transmitted beampattern reduce without oscillation and the maximum power is focused at the beampattern center where the given target is assumed to be located. Assume that our desired matrix is R, which is built from BPSK samples z's, where z ¼ signðyÞ and y's are Gaussian samples with covariance matrix Ry . It is shown in [13] that R y ðm; nÞ ¼ sin ððπ =2ÞRðm; nÞÞ or briefly R y ¼ sin ððπ =2ÞRÞ, where Rðm; nÞ is the element at mth row and nth column of matrix R. Since, R y is positive semidefinite, the transmit covariance matrix sin ððπ =2ÞRÞ has to be positive semidefinite. Hence, R y is obtained from R and then the matrix of Gaussian random variable y's that can realize R y can be easily determined with a de-whitening transformation.
 Now we prove that for Rp1 , sin ðπ =2ÞRp1 is positive semidefinite because a symmetric matrix A is positive semidefinite if all eigenvalues are nonnegative [16]. In order to find eigenvalues, we write 

det sin

π 2

R p1



  λI ¼ 0

ð14Þ

nðn þ 1Þ i¼ 2 i¼1

ð19Þ

¼ ρM r

MX t p

p¼1

i¼0

Mt X

p1 Mt  i X Mt  i þ 1 Mt Mt i¼0

Mt  p þ 1 

p¼1

¼ ρM r

Mt X

i¼0

Mt 

p¼1

¼ ρM r

¼ ρM r

MX t p

1 Mt

MX t p i¼0

iþ

Mt 

p¼1

¼ ρMr M 2t 

85 87 89 91 93 95 97 99 101 103

! 107

p 1 X i i þp 1 Mt M t i¼0 p 1 X

!

109 111

!! 113

i

i¼0

  Mt  X 1 pðp 1Þ ðM t  p þ 1ÞðM t pÞ þ Mt  Mt 2 2 p¼1 Mt X

83

105

the maximum SINR can be written as Mt X

75

81 ð17Þ

n X

n3 n2 n i2 ¼ þ þ ; 3 2 6 i¼1

73

79

where p represents the row number. According to (17), (18) and n X

71

77

where (a) comes from ðA  BÞ  1 ¼ A  1  B  1 . Eqs. (b) and (c) both come from ðA  BÞðC  DÞ ¼ AC  BD. In order to more simplify the maximum SINR, we assume that target is located at θ0 ¼ 01, hence at ðθ0 Þ ¼ ½1 1 ⋯ 1T and 2 3 MX 0 t 1 M t i X M t i 6 þ 1 7 6 7 Mt i¼0 6 i ¼ 0 Mt 7 6 7 ⋮ 6 7 6 M p 7 p  1 t 6 7
 6 X M t  i X M t i 7 þ  1 R p1 at θt ¼ 6 ð18Þ 7 Mt 6 i ¼ 0 Mt 7 i¼0 6 7 6 7 ⋮ 6 7 6 7 MX t 1 6 7 M t i 4 5 M t i¼0

SINRRp1 ¼ ρM r

67 69

1 SINR ¼ ρar ðθ0 ÞH  at ðθ0 ÞH R H ar ðθ0 Þ p1  ðIM r  R p1 Þ

0

65

ð16Þ


 Therefore, sin ðπ =2ÞR p1 is positive semidefinite. The maximum SINR will be achieved when there is no interference (except noise component) [16]. Therefore, R in ¼ ðIMr  RÞ, and the maximum achievable SINR for R p1 can be written as

6

11

ð15Þ


 Eq. (15) shows that sin ðπ =2ÞRp1 has ðM t  2Þ zero eigenvalues and two other eigenvalues are

λ1  0:179Mt λ2  0:821Mt :

Pt(θ)

5

63

which yields to   λMt  2 λ2  Mt λ þ0:147M2t ¼ 0

1 M2 Mt p2 p þ t þ  pMt Mt 2 2

!!

!  X Mt  2 Mt M 2t Mt X p p   p þ Mt 2 2 p¼1 p¼1

Please cite this article as: S. Imani, et al., SINR maximization in colocated MIMO radars using transmit covariance matrix, Signal Processing (2015), http://dx.doi.org/10.1016/j.sigpro.2015.07.011i

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S. Imani et al. / Signal Processing ] (]]]]) ]]]–]]]

¼ ρM r M2t þ

3 5 7 9 11

¼ ρM r ¼

ρMr  3

Mt Mt 1 X 1 X p p2 Mt p ¼ 1 Mt p ¼ 1

Mt 1 M2t þ þ  2

2

1 Mt

M 3t

!

M 2t

Mt þ þ 3 2 6

M =5



2M 2t þ 1 :

15

M =10

8

!!

M =20

67 6

ð20Þ

As can be seen from (20) when M t increases, the maximum SINR increases too and for large values of M t the maximum SINR obtained by Rp1 would be SINRRp1 C 23SINRRphased which is SINRMIMO , and for M t 42, SINRRp1 4SINRMIMO .

69

5 4

71

3

73

2

75 1

3.2. The second proposed covariance matrix

77

0 -60

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

In order to efficiently suppress the interference in multiantenna systems, it is necessary that the location of target and interferences is known. However, in some cases the location of interferences is unknown. In order to improve the efficiency of system in the presence of unknown interferences, we have to design the transmit and receive beampatterns with low side-lobe level to reduce the power of back scattered signal from unknown interferences. With this consideration, the second covariance matrix, Rp2 , is proposed as follows: 2 3 M t  2 Mt  4  Mt þ 2 1 ⋯ Mt Mt Mt 6 7 Mt  2 6 Mt  2 1 ⋱ ⋮ 7 6 Mt 7 Mt 6 7 Mt  4 7 6 Mt  4 Mt  2 1 ⋱ R p2 ¼ 6 M t 7: Mt Mt 6 7  Mt  2 7 6 ⋮ ⋱ ⋱ ⋱ 6 7 Mt 4 5  Mt þ 2 Mt  4 M t  2 ⋯ 1 Mt Mt Mt Similar to Rp1 , the proposed matrix Rp2 is real and symmetric. Therefore, the transmit beampattern is symmetric, too. The transmit beampattern for the proposed matrix is plotted for different number of transmitters, Mt , in Fig. 2. It can be seen from Fig. 2 that the second proposed matrix similar to the proposed matrix of [16] concentrates the power around beampattern center (and not exactly on its center). It is also demonstrated in Fig. 2 that side lobes are oscillating. Similar to (20), the maximum SINR obtained by Rp2 is 1 ar ðθ0 Þ SINRRp2 ¼ ρar ðθ0 ÞH  at ðθ0 ÞH RH p2  ðIM r  R p2 Þ

R p2 at ðθ0 Þ

¼ ρM r at ðθ0 ÞH R H p2 at ðθ 0 Þ:

65

M =15

7

greater than the obtained SINR by Rx . For M t ¼ 2, SINRRp1 ¼ 13

63

9

Pt(θ)

1

5

ð21Þ

Again assume that the target is located at θ0 ¼ 01, hence at ðθ0 Þ ¼ ½1 1 ⋯ 1T and 2 3 MX 0 t 1 M t  2i X Mt  2i 6 þ 1 7 6 7 Mt i¼0 6 i ¼ 0 Mt 7 6 7 ⋮ 6 7 6 M p 7 p 1 t 6 7 X X
 6 Mt  2i Mt  2i þ 1 7 Rp2 at θt ¼ 6 ð22Þ 7 M M 6 i¼0 7 t t i¼0 6 7 6 7 ⋮ 6 7 6 7 MX t 1 6 7 M  2i t 4 5 Mt i¼0

-40

-20

0

20

40

60

79

θ (Degrees)

Fig. 2. Transmit beampattern using the second proposed covariance matrix for different number of transmit antennas. The total transmit power is normalized to one.

81 83

using (21), (22) and (19), the maximum SINR can be written as !   pX  MX 1  Mt t p X Mt  2i M t 2i þ 1 SINRRp2 ¼ ρMr Mt Mt p¼1 i¼0 i¼0 Mt X

¼ ρM r

  MX t p 2i

M t  p þ1 

p¼1

¼ ρM r

i¼0

Mt X

Mt 

p¼1

¼ ρM r

E¼0

Mt X

Mt 

p¼1

¼ ρMr

Mt  X p¼1

¼ ρM r

2 Mt

ρMr  3

i¼0

2i 1 Mt

MX t p i¼0

iþ

p 1 X

91

95

!!

97

i

i¼0

! 

Mt Mt 1 X 1 X ðMt þ1Þ þ 2pðMt þ1Þ  2p2 M M t t p¼1 p¼1 p¼1 Mt X

 M2t þ2 :

89

93

 pX ! 1  2i 2i  Mt Mt i¼0

Mt  1 X 2p2 2p þ M2t þM t 2pM t Mt p ¼ 1

¼ ρM r M 2t  M t ðM t þ1Þþ

¼

þp 

87

!



  2 pðp  1Þ ðMt p þ1ÞðMt pÞ þ Mt  Mt 2 2

¼ ρMr M2t  M 2t 

 MX t p

Mt

pX 1 

85

ðM t þ1Þ2 M t 2 M 3t M 2t M t þ þ  Mt 3 Mt 2 6

99 101 103

!

!!

105 107 109 111

ð23Þ

As can be seen from (23), SINRRp2 is always greater than SINRMIMO . For large number of transmit antennas, the maximum SINR obtained by Rp2 is SINRRp2 CSINRRphased =3. Although the SINR obtained by Rp2 is lower than SINR obtained by R x (interference free case), but for our proposed
 covariance matrix, sin ðπ =2ÞRp1 is always positive semidefinite. In order to obtain eigenvalues of SINRRp2 , we write π    ð24Þ det sin Rp2  λI ¼ 0 2

Please cite this article as: S. Imani, et al., SINR maximization in colocated MIMO radars using transmit covariance matrix, Signal Processing (2015), http://dx.doi.org/10.1016/j.sigpro.2015.07.011i

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S. Imani et al. / Signal Processing ] (]]]]) ]]]–]]]

6

3 5 7 9 11 13 15

then we have   λMt  2 λ2  Mt λ þ0:25M2t ¼ 0

Rp

ð25Þ

0.9

Rp


 Equation (25) shows that sin ðπ =2ÞRp1 has ðM t  2Þ zero eigenvalues. Two other eigenvalues are

0.8

Rx in [16] Phased-array

λ1 ¼ 0:5Mt λ2 ¼ 0:5Mt :

0.6

ð26Þ


 Therefore, sin ðπ =2ÞRp1 is positive semidefinite. Our proposed covariance matrices are designed to detect the target in location θ0 ¼ 01. To detect the target at the location θ, the covariance matrix can be easily modified as follows [8]: R~ ¼ R  at ðθÞaH t ðθ Þ

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In this section, several simulations are considered in order to evaluate the efficiency of the proposed covariance matrices. In simulations, the efficiency of the proposed covariance matrices is compared with conventional MIMO radars, phased array radars and proposed covariance matrix, Rx , in [16]. Also in all of the following simulations we assume that transmit and receive arrays are ULA with half-wavelength inter-element spacing. The receive beampattern using combining beamformer at the receiver can be calculated as follows:  2 P r ðθÞ ¼ wH ar ðθÞ  Rat ðθÞ : ð28Þ where w is the MVDR beamformer [23], which is calculated in [16]. In the first simulation, we assume that target is located at θ0 ¼ 01 and we are going to cohere the power in this direction. We also assume that there are 10 transmit and receive antennas and total transmit power is normalized to one. Fig. 3 shows transmit beampattern for all methods. As can be seen in this figure, the phased array technique and the proposed covariance matrix R p1 make higher level of power at the target location compared to other methods. It should be considered that the covariance matrix Rp2 as well as the proposed covariance matrix R x in [16] cannot make the peak power at the target location and therefore, these methods have a degradation in the transmit power, which decreases the SINR performance. As mentioned before, the proposed covariance matrices Rp2 and Rx in [16] have low side lobe levels compared to the two other methods, which is useful in the unknown interferences cases. The second simulation shows the SINR behavior as a function of SNR for all methods. In this simulation, both the number of transmit and receive antennas are 9. A point-like target is located at θ0 ¼ 01 and two signal dependent interferences with SINR¼30 dB are located at θ1 ¼  101 and θ2 ¼ 301. Fig. 4 shows the SINR behavior as a function of SNR. As can be seen in this figure, the achieved SINR with R p1 has the closest value to the SINR of phased array. The difference between the performance of phased array and using Rp1 , R p2 and Rx are 1.45 dB, 4.3 dB and 3.6 dB, respectively. Although the SINR obtained by Rp2 is 0.7 dB lower than SINR obtained by R x , the correlated MIMO radars with R p2 can use the full waveform diversity which enables us to reject

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Fig. 3. Comparison between transmit beampattern of phased array, two proposed covariance matrices and covariance matrix proposed in [16].

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Fig. 4. Comparison of obtained SINR using two proposed covariance matrices, omnidirectional MIMO-radar, phased-array, and correlated MIMO-radars in [16].

more interferences compared to the correlated MIMO radars
with Rx in [16]. Also sin ðπ =2ÞRp2 is positive semidefinite and can be used to generate the BPSK waveforms in closed form. Fig. 5, shows the receive beampattern for all methods with simulation parameters in Fig. 4. As can be seen in Fig. 5, the receive beampattern for correlated MIMO radar with Rp2 is the same as Rx and the SLL of these two methods is lower than other two methods. Also it can be seen that the SLL of Rp1 is lower than SLL of phased array. As mentioned before, a method with lower SLL can improve SINR level in the presence of unknown interference locations. To show the effect of low SLL in SINR level, consider the simulation parameters in Fig. 4. Also we assume that there is two other interferences with unknown locations (θ1 Uð30; 90Þ and θ2 Uð 10;  90Þ). The interference-tonoise ratio for these two unknown interferences is equal to 20 dB. This simulation is performed for 200 different values of θ3 and θ4 and the average SINR level is shown in Fig. 6. As

Please cite this article as: S. Imani, et al., SINR maximization in colocated MIMO radars using transmit covariance matrix, Signal Processing (2015), http://dx.doi.org/10.1016/j.sigpro.2015.07.011i

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Fig. 5. Receive beampattern of phased array, two proposed covariance matrices and covariance matrix proposed in [16].

can be seen in this figure, the low SLL properties are very effective in the performance of SINR. In this simulation, the best SINR performance belongs to correlated MIMO radar with Rp2 and R x . Also we can see the SINR level of correlated MIMO radar with Rp1 is better than that in phased array. It should be considered that, maximum SINR level is achieved when there is no interference in the environment. However, Fig. 6 is drawn when there is unknown interference in the environment, and therefore, we cannot expect that the SINR level is maximized in this figure. On the other hand, our first proposed matrix despite that it has a beam-pattern with larger SLLs with respect to what is proposed in [16], but in the case of known interference location, the efficiency of our first proposed matrix is considerably more that of proposed in [16]. Also in the case of unknown interference location which needs low SLLs, we have proposed our second matrix whose efficiency is very close to that of the proposed in [16]. This second proposed matrix in spite of the proposed matrix in [16] is full rank (to use the maximum waveform diversity), is positive definite and can be easily implemented. One of the most important advantages of MIMO radar is waveform diversity which provides more degrees of freedom compared with phased-array ones. Based on the concept of sum co-array, the DOF of phased array equals to M t and therefore it can suppress at most Mt  1 interferences. For a full rank covariance matrix, the DOF of MIMO radar equals to 2Mt  1 and can suppress at most 2M t 2 interferences. As Rx is not full rank, it's DOF is lower than MIMO radar with full rank covariance matrix. A simulation is performed in order to show the DOF properties. In this simulation for the target located at θ0 ¼ 01, SNR¼20 dB and 10 interferences are located at  101; 301; 451; 401; 551; 501;  701; 801;  851 and 151. Fig. 7 shows the SINR level for first Q interferences from the above list. The interference-to-noise ratio (INR) is 30 dB. As can be seen, as the two proposed covariance matrices are full rank, they can suppress 2M t 2 ¼ 8 interferences effectively while the phased array can only suppress M t  1 ¼ 4 interferences. Also we can see that the proposed method in [16] can suppress 6 interferences which shows that this method has not used the full

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Fig. 6. Comparison of obtained SINR using two proposed covariance matrices, omnidirectional MIMO-radar, phased-array, and correlated MIMO-radar proposed in [16] for 2 interferences with unknown locations. 40

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waveform diversity of MIMO radars, however, both of our proposed methods use full waveform diversity. Fig. 8 shows the SINR behavior for all methods in the presence of 9 first interferences from the above list versus the interference-to-noise ratio. As can be seen in this figure, when INR increases, the reduction of SINR level for both of our proposed covariance matrices is very low, while the performance of phased array and correlated MIMO radar with Rx degrades dramatically in high INR region. For both covariance matrices, we showed that
 sin ðπ =2ÞR is positive semidefinite, which enables us to realize R, with BPSK waveforms in a closed form. Fig. 9 shows the SINR level for both of our proposed covariance matrices with BPSK waveforms (for more details, see [13]). Simulation parameters are similar to Fig. 4. This figure shows that the difference between the SINR obtained by two R p1 and Rp2 and SINR obtained by BPSK waveforms (with 50 samples) which realized R p1 and Rp2 is about 0.5 dB. This result shows that our proposed

Please cite this article as: S. Imani, et al., SINR maximization in colocated MIMO radars using transmit covariance matrix, Signal Processing (2015), http://dx.doi.org/10.1016/j.sigpro.2015.07.011i

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using the second proposed covariance matrix is lower than other methods with the same SLL, but it is full rank. Therefore, it can handle more interferences compared to existing correlated MIMO radars which is an important advantage.

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covariance matrices can reduce the effect of non-linear properties of radio amplifiers in MIMO radar systems.

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5. Conclusion

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In this paper two transmit waveform covariance matrices are proposed. They are full rank and therefore
 full waveform diversity is achievable. Also sin ðπ=2ÞR for both proposed covariance matrices is positive semidefinite, thus, they can be used to generate BPSK waveforms in a closed form. Simulation results show that the first proposed covariance matrix achieves more level of SINR compared with existing correlated MIMO radars when the target and interference angle locations are known. Also this matrix has a lower SLL than that of phased array. Also numerical results indicate that although SINR obtained

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