SINR improvement based on joint design of transmit covariance matrix and receive ﬁlter design for colocated MIMO radar

This paper examines we aim to improve the performance of target detection in the presence of signal-dependent interference in multiple-input multiple-output radar through the joint design of the receiver spatial ﬁlter and the covariance matrix of waveforms. For this purpose, the signal-to-interference-plus-noise ratio is considered as the design criterion. The proposed method is a sequential algorithm that calculates the receiver ﬁlter coefﬁcients through a closed form in each step of the algorithm. The correlation of transmit waveforms is also calculated utilizing the coordinate descent method. The covariance matrix is designed in two forms. In the ﬁrst form, it is assumed that the optimal matrix is a Toeplitz matrix which leads to a simpler design. In the second form, the covariance matrix is calculated by combining orthogonal waveforms, which yields better performance for interfere reduction. Simulation results show that the proposed method has more appropriate performance in comparison with other methods.


INTRODUCTION
One of the most important features of multiple-input multipleoutput (MIMO) radar is the degree of freedom (DOF) in designing transmit waveforms compared to other radar types [1,2]. In fact, the optimal design of transmit codes has the most effect on the performance of MIMO radar. Therefore, this type of radar can be called a software radar. Accordingly, waveform design for widely separated [3] and colocated [1] MIMO radars is a major problem that has attracted many researchers in recent years [4][5][6][7][8][9][10][11][12][13][14]. The aim of this paper is to focus on the waveform design for colocated MIMO radars. Waveform design is done by direct [6,[13][14][15][16] or indirect [4,[18][19][20] design. In direct design, each transmit waveform symbols are directly calculated and the output of the algorithm will be the final waveform. In the indirect design, there are two stages; a) the transmit waveform covariance matrix R is calculated; b) the final transmit waveform symbols are generated using synthesizing the covariance matrix R. In direct design for the transmit symbols, the system computational complexity will increase by increasing the number of transmit symbols and the This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2020 The Authors. IET Communications published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology system will be more complicated. In indirect design, the computational complexity will not increase by increasing the number of transmit symbols and there are different waveforms can be generated using the designed covariance matrix [4,21]. Therefore, the aim of this paper is to design an appropriate covariance matrix.
Different criteria are used to design a MIMO radar waveform, e.g., beampattern shaping [9,22], SINR maximization [4,8,19,20], ambiguity function enhancement [23], range resolution increase [24] etc. The purpose of this paper is to design a waveform covariance matrix for improving signal to noise ratio plus interference (SINR) in presence of point like signal-dependent interferences and additive white Gaussian noise.
In collocated MIMO radars, it is usually assumed that the transmitted waveforms are narrow band and so, the point like targets and the point like scatterers as signal-dependent interferences (clutters) can be considered. It is assumed that there is a priori information about angle of target and signaldependent interferences [4, 10-14, 19, 20, 24]. In some cases, digital radio frequency memory repeat jammers can be also supposed as point like signal-dependent interferences [25,26]. It should be noted that a different model can be considered for the clutter same as models addressed in [7,27,28]. Furthermore, the subject of radar waveform design for spectrally dense environments and in the presence of communication systems has been considered in recent years [29,30]. In [30], waveform design is performed to the MIMO radar coexist with a communication system without affecting each other.
SINR is an important parameter in colocated MIMO radars and SINR improvement, directly increases the probability of target detection. In recent years, many studies have been carried out on the SINR improvement in MIMO radars [4, 8, 12-14, 18, 19, 24].
In [6,7,13,14,17], authors design waveforms directly under different constraints such as similarity, constant envelope etc. As it was mentioned before, in these methods, the complexity of the algorithm increases with the increase in the number of samples. On the other hand, with any change in target or interference locations, a new waveform should be designed that practically limits the use of this type of waveform design.
Moreover, authors design waveforms indirectly using covariance matrix under different constraints in [4,8,12,19,20]. In [19], Ahmed and Alouini aim to achieve the SINR performance of phased array radar utilizing a fixed covariance matrix for each interference angles. More DOF for interference rejection is obtained in collocated MIMO radar when the transmit signal covariance matrix is full rank. But, the covariance matrix rank in [19] is equal to 2 for different number of transmit antennas. Therefore, this method is not optimum because it cannot completely utilize the advantages of MIMO radar in comparison with the phased array radar. In [8], the authors have proposed other covariance matrices to improve the DOF of the MIMO radar addressed in [19]. But in [8], matrices are always fixed for a given antenna and are not optimal.
Some methods have been also proposed in [4,20] which are based on semi-definite programming (SDP) approach. These methods have high computational time and complexity. Therefore, the application of them is practically limited. In [4], transmit waveform covariance matrix and receive filter coefficients have been jointly optimized in two steps to prepare suitable SINR and DOF. The covariance matrix is calculated utilizing a non-convex and NP-hard optimization problem which is solved as a convex problem using Charnes-Cooper transform [4]. Haghnegahdar et al. improved the SINR by designing the proper transmit covariance matrix [20]. They proposed a sequential optimization procedure for SINR improvement, using a semi-definite programming (SDP) problem for transmit covariance matrix optimization. Their simulation results indicate better obtained SINR in comparison with phased array and correlated MIMO radars in [19].
The aim of this paper is to obtain covariance matrices that can also use the DOF of MIMO radar to eliminate the signal-dependent interferences, and to optimally adapt to target and interference changes. Therefore, we propose a joint transmit covariance matrix and filter coefficients design method. The proposed method is sequential and consists of two steps. In the first step, the filter coefficients of the receiver have been designed. In the second step, the covariance matrix has been calculated based on coordinate descent (CD) algorithm [6,31]. This algorithm is iterative and at each iteration, the algorithm determines and optimized a coordinate (element) of the covariance matrix, while fixing all other coordinates. The proposed method has been evaluated for different situations. The obtained results show the desirable performance and efficiency of the proposed method for eliminating interferences and obtaining the optimal value of SINR.
The organization of this paper is as follows: In Section 2 the problem formulation is presented for colocated MIMO radar. The covariance matrix and filter coefficient design method is proposed in Section 3. Section 4 shows the performance analysis and numerical results and conclusion is provided in Section 5.

PROBLEM FORMULATION
Assume a colocated MIMO radar with M t transmit and M r receive antenna arrays. All antenna elements are omni directional. It is supposed that both transmit and receive antenna arrays are uniform linear arrays (ULAs). Each transmit antenna element transmits narrow band signal x m (n), m = 1, … , M t , n = 1, … , N different from the other antennas, where m and n are the index of antenna elements and each sample of waveforms, respectively. Therefore, the transmit signal vector at time sample n is presented as Then the received signal in target location at will be is the steering vector of transmit antenna array. f 0 is the radar career frequency and m ( ) is the delay time for the transmit signal from antenna m to the target at an angle . For a ULA with inter-element spacing d , a t ( ) is defined as, where is the signal wavelength. Without loss generality, it is assumed that d = ∕2 here. Let us suppose that there are a point like target at angle 0 and Q signal-dependent interference sources in the same range with the point like target of interest which are located at the known angles j , j = 1, … , Q [4,8,11,12,14,19,20]. It should be noted that the presence of the target and clutters at the same range is the worst case here. These interference sources caused by point like scatterer or digital radio frequency memory repeat jammer [25,26]. Then, the received signal can be written as where 0 and i , i = 1, … , Q are complex coefficients which determine reflected power and phase of the target and interferences, respectively. These coefficients comprise path loss, transmit and receive gains and other related parameters in mono static radar equation [18]. a r ( ) is the steering vector of receive antenna array which is defined as, Vector v(n) shows the independent and identically distributed additive white Gaussian noise vector at M r receivers with zero mean and covariance matrix v 2 I. The received signal in each receive antenna is passed through a matching filter with size M t × 1. Therefore, the match filter output with size M t M r × 1 is given by where R is the correlation matrix of transmit waveforms and As the mean of transmit waveforms are usually zero in the radars, the correlation matrix is the same as covariance matrix. v c is the noise vector after passing through the match filter with zero mean and variance of 2 (I M r ⊗ R) [8,32].
As it was mentioned before, the signal detection quality and the estimation of various parameters (such as the angle of arrival estimation) in the MIMO radar is directly related to the SINR. Therefore, by increasing the SINR the detection and parameter estimation in the MIMO radar improves. The goal is to maximize the SINR at the receiver, here. Accordingly, the problem is that the received signal at the receiver is passed through a spatial filter of limited length. Therefore SINR at the filter output can be written as follows, where is interference plus noise ratio (INR) [32].
In general, the optimal value of SINR in phased array systems and MIMO radars depends on the signal and interference covariance matrices, and several studies have been conducted to estimate these covariance matrices [33][34][35][36]. Therefore, as it is shown in (7) and (8), the covariance matrix of signal dependent interferences and also the optimal SINR value is related to the covariance matrix of transmit waveform. In conventional MIMO (CM) radars, the transmit covariance matrix is an identity matrix, because transmit waveforms are uncorrelated. For the case of conventional MIMO radars, the maximum achievable SINR without interferences at the receiver can be written as follows [19], In conventional phased array (CPA) radars, all transmit array antennas send a scaled version of a waveform. Therefore all elements of the transmit covariance matrix are equal 1. The maximum achievable SINR can be expressed as (10) for phased array radars [8,37].
According to (9) and (10), CPA radars can obtain a higher SINR value in comparison with CM radars, because MIMO radars transmit the same omni-directional power, while phased array cohere the power in the target direction. Although, it can be said that MIMO radars can identify more parameters in comparison with phased array radars based on co-array concept [38], because MIMO radars utilize the waveform diversity.
In order to obtain parameters of conventional MIMO radars and achieve the SINR of conventional phased array radars, a transmit covariance matrix have been addressed in [19] as follows, cos( The authors showed that the covariance matrix has a low sidelobe level compared to phased array and MIMO radars. For this matrix [19], the maximum achievable SINR can be defined as follows, It is shown in [19] that the rank of matrix in (11) is 2 for all values of M t . The covariance matrix in (11) can be expressed as the sum of two phased array radars and therefore it does not exploit the full waveform diversity.
In [8], Authors proposed two covariance matrices (13) and (14) in order to maximize the SINR values. The proposed covariance matrices are designed based on the following criteria:(a) To be full rank matrices. to suppress more number of interferences. (b) To be able to build a constant envelope waveform to avoid destructive non-linear effect of amplifiers. (c) To have small SLL in order to be able to suppress the effect of interferences with unknown location.
The maximum SINR values which obtained from two covariance matrices addressed in [8], are expressed in (15) and (16) respectively, The elements of two covariance matrices in (13) and (14) are fixed for a specified number of transmit antennas. This problem somewhat reduces the performance of the designed MIMO radar addressed in [8]. Furthermore, covariance matrices and receive filters are not jointly designed in methods addressed in [8,19]. Therefore, these methods can not be optimal.

JOINT COVARIANCE MATRIX AND RECEIVE FILTER DESIGN
We aim to maximize the SINR with a full rank matrix in order to use more degrees of freedom and reject more interferences in comparison with conventional phased array radars. This is equivalent to the following optimization problem: Where R i, j is the corresponding element in row i and column j . The parameter Ω is the set of values that each element of the matrix R can select. To solve (17), we propose a joint transmit covariance matrix and filter coefficients design method which is a sequential method and utilizes two proposed algorithms for covariance matrix design.

Receive Filter Design
As can be seen in (8), to achieve the maximum value of SINR, it is necessary to optimize the receive filter coefficients w N M r ×1 and the proposed covariance matrix R. By considering a fixed value of R, the optimization problem can be considered as, This problem can be solved based on minimum variance distortionless response (MVDR) method [39] in terms of w and the result is equal to Then, by replacing w obtained from (19) into (17), the design of the covariance matrix R will be done according to subsection 3.2.

Covariance Matrix Design
In this subsection, two algorithms are proposed in which the covariance matrix of the transmit waveform is designed to maximize the SINR in the receiver. There are two important constraints on the covariance matrix here. The first constraint is the equal transmitted power from the antennas. Therefore to meet this constraint, the diagonal elements of R must be equal. Considering this constraint in the design of the covariance matrix causes similar RF circuits to be used in transmission, reducing the complexity of the system design, and also using the maximum gain of RF amplifiers. The second constraint is the positive semi-definite constraint for the covariance matrix. Because we should calculate a covariance matrix that is able to be synthesized to obtain the transmit waveform, otherwise the calculated matrix is not the waveform covariance matrix.
It should be noted that the SINR maximization problem in terms of the covariance matrix is a non-convex problem and almost does not have a cost-effective solution with an acceptable computational time and complexity. Therefore, we use the CD framework, also known as the alternating maximization (AM) to solve the covariance matrix design problem. This algorithm is an iterative-sequential algorithm. At each iteration, the CD algorithm determines a coordinate, which is one element of the covariance matrix here, maximizes the SINR value over the corresponding coordinate while fixing all other coordinates [40]. The algorithm is repeated for all covariance matrix elements to obtain the optimal SINR value. Accordingly, the optimization problem can be written as follows, It should be noted that the values of the covariance matrix elements cannot be independently designed and are interdependent. It means that R i, j = R * j,i . To handle the above problem, we propose two algorithms in which the design problem of the covariance matrix elements R i, j is solved by CD method.

First Algorithm
In the first algorithm, we use the TOEPLITZ structure for the covariance matrix. Considering the vector r = [1, r 12 , … , r 1M t ], the obtained covariance matrix will be as follows, It can be clearly seen that instead of element by element covariance matrix design, only elements of the vector r can be designed. As mentioned earlier, in the CD algorithm the search procedure is performed on the set of values that can be selected for each element. If it is assumed that the transmitted power is normalized, then elements of the vector r can only select values from Ω r = [−1,1]. To search on this interval, it is necessary to divide the interval into a number of points (L) with equal spacing such as follows, For a large number of points, the estimation of the desired element will be better while the computational complexity of the algorithm increases with increasing number of points. Since the covariance matrix elements can be complex values, possible phase values should be searched for R r elements in this case. Due to the large number of possible values for the amplitude and phase, the number of search repetitions for each element increases greatly and the computational complexity of the algorithm increases significantly. Now, although the covariance matrix problem has been solved by a searching method for real elements, it is possible that the covariance matrix obtained at the algorithm output is not positive semi-definite matrix. To solve the positive semidefinite problem, the eigen value decomposition is first applied to the covariance matrix, where U is the eigenvector and V is the eigenvalue of the matrix R r . Then, we force negative eigenvalues to zero, because if the eigenvalues of a TOEPLITZ matrix are all greater than or equal to zero then this matrix is positive semi-definite. Based on the above explanation, suppose that the matrixṼ is the modified matrix of eigenvalues, then the final covariance matrix of the algorithm is R r = UṼU H . In the first algorithm, it is possible that the final covariance matrix has zero eigenvalues. The rank of covariance matrix R r decreases according to the number of zero eigenvalues. As a result, the performance of the MIMO radar for interference elimination is reduced. On the other hand, since the optimization of covariance matrix elements in the first algorithm is done based on the real set, therefore the final covariance matrix is not generally optimal. The covariance matrix design based on the TOEPLITZ structure can be found in Algorithm 1.

Second Algorithm
As mentioned earlier, the covariance matrix in the first algorithm is a TOEPLITZ matrix, which results in a relative decrease in the maximum efficiency of the MIMO radar because of possible zero eigenvalues and consequently rank reduction of the covariance matrix R r . Although in the results  (19) and (20), respectively; 4. If d < L, then d = d + 1 and go to 2. 5. Find index d ⋄ corresponding to maximum SINR n d ; 6. Set r i = Ω r (d ⋄ ) and SINR n = SINR n d ⋄ ; 7. If i < M t , then i = i + 1, n = n + 1 and go to 2. 8. If i > M t , then i = 2; 9. If |SINR n − SINR n−1 | < , go to 1; otherwise, stop. 10. Set R ⋆ = R , w ⋆ = w.
section, it is shown that the first algorithm has an appropriate performance in comparison with the other methods. Therefore, the first algorithm has some limitations and the second algorithm is proposed to eliminate the limitations of the first algorithm.
In this algorithm, we aim to use the maximum covariance matrix rank and to search for a smaller set of possible values (discrete alphabets). For this purpose, it is assumed that an orthogonal set of waveforms S ∈ C K ×N is available and the transmit signal matrix X ∈ C M t ×N is obtained from their linear combination as follows, where F ∈ C M t ×K is the waveform combination matrix which is defined as follows, Thus the covariance matrix of the transmit waveforms is equal to R = E{(FS)(FS) H } = FF H . Therefore, to design the covariance matrix, the waveform matrix elements must be correctly designed.
It should be noted that the constraint of equal transmitted power from the antennas must be taken into account in waveform combination matrix design to obtain a smaller set of possible values. Furthermore in the CD-based search method, a smaller set of discrete alphabets should be used to optimize each element of the combination matrix F.
Accordingly, if the combination of the waveforms is only based on phase, i.e. the amplitudes of all elements of matrix F are fixed and only the optimal phase is designed for them, then the search for the optimal element is performed over a smaller interval (region). To use the maximum covariance matrix rank in the radar it is necessary to have K = M t , even though 1 ≤ K ≤ M t . Accordingly, the optimization problem for an ele-ALGORITHM 2 Design of covariance matrix R based on the combination of orthogonal waveforms Input: M t , M r , L, 0 , j , j = 1, … , Q, stopping threshold and Ω L Output: The optimal R ⋆ and w ⋆ . Init.: Select random phases from Ω L for F, i = 1 , j = 1 , d = 1 and n = 0. 1. Set F(i, j ) = Ω L (d ) and R = FF H ; 2. Calculate w and SINR n d using (19) and (26), respectively; 3. If d < L, then d = d + 1 and go to 1. 4. Find index d ⋄ corresponding to maximum SINR n d ; 5. Set F(i, j ) = Ω L (d ⋄ ) and SINR n = SINR n d ⋄ ; 6. If j < K , then j = j + 1, n = n + 1 and go to 1. 7. If i < M t , then i = i + 1, j = 1, n = n + 1 and go to 1. 8. If |SINR n − SINR n−1 | < , go to 1; otherwise, stop. 9. Set R ⋆ = R , w ⋆ = w. ment of matrix F can be written as follows, where Ω L = {1,̄, … ,̄L −1 }, with L indicating the alphabet size and̄= e 2 L . As the L increases, the size of Ω L increases and the elements of matrix F are more accurately designed. For L → ∞, the design problem tends to be a continuous phase. At each iteration of the algorithm, one element of matrix F is updated using a phase value that maximizes the objective function in (26). The process is performed for all elements. Finally, the desired covariance matrix using the optimal matrix F ⋆ will be R = F ⋆ F ⋆ H . Algorithm 2 illustrates the proposed method for designing the covariance matrix based on the combination of orthogonal waveforms using discrete phase alphabets.

NUMERICAL RESULTS
In this section, various simulations have been provided to evaluate the proposed algorithms. To illustrate the performance of the proposed algorithms, the simulation results are compared with CPA radar and the method addressed in [19], named as Ahmed method, for different scenarios.
It should be noted that although many methods have been proposed in recent years to maximize SINR in MIMO radars, but most of these methods are based on direct waveform design which have a lot of implementation problems (the corresponding design of the receiver with each transmitted pulse). Therefore we use some methods for comparison that can be implemented. In all the simulations, the transmit and receive antenna arrays are assumed to be uniform linear arrays (ULAs) with half wavelength spacing between elements. It is also assumed that the number of transmit and receive antennas are equal (M t = M r ) and = 10 −4 . Let us consider a MIMO radar sys- In [41], authors indicate how to shift the transmit beam pattern from angle 0 o to the desired angle. The first simulation shows the convergence of two proposed methods. There are two interferences at i = −25 • , 40 • with INR = 30 dB, here. The number of discrete points L is equal to 64 for the first algorithm. In addition, the number of distinct phase values are supposed as L = 64 (e.g. 64PSK) for the second algorithm, i.e. the interval [0, 2 ] is divided into 64 discrete phase alphabets. As can be seen in Figure 1, the algorithms behave quite upscale for different states and converge to the optimal value (SNR). The first algorithm has a higher speed to reach the final value because it only searches on three components of a vector while the second algorithm has a larger search interval. It is also seen that by increasing K the final value obtained for SINR is greater, although we have achieved good accuracy with small K as well. Increasing the value of K is more useful for the large number of interferences.
Another important problem in the covariance matrix design is its performance for different number of antennas. In the second simulation, the same assumptions are supposed as the first simulation, but the number of antennas varies from 3 to 7. As can be seen in Figure 2, with the increase in the number of antennas, the efficiency of the method is reduced in Ahmed method [19], while for the proposed methods with increasing the antenna, an improvement can be seen in SINR efficiency. Conventional phase array method also has stable behaviour.
The DOF in the MIMO radar plays an important role in eliminating interferences. We know that the conventional phase array radar performs well and has the best SNR efficiency in non-interfering conditions or in cases where the number of interferences is low compared to the number of antennas, but its performance drastically reduces with increasing the number of interferences.
In the third simulation the DOF of the proposed algorithms and the compared methods are evaluated in terms of a different number of antennas. For this simulation, the assumptions used in the first simulation are considered. However, six interferences with INR = 30 dB are supposed at Figure 3 shows obtained SINRs for the third simulation in terms of the different number of antennas. According to Figure 3, CPA radar can only eliminate the effect of six interferences when it has at least seven antennas, while the two proposed methods can easily reduce the effect of interferences with five antennas and even with four antennas (SINR value is almost 18 dB). The effect of interferences has been slightly reduced for five antennas in CPA method and Ahmed method [19], but the acquired SINR is almost 3 to 5 dB less than the proposed methods. Therefore, Ahmed method is not an optimal method for MIMO radar.
The different obtained beampatterns are compared in the fourth simulation, based on the scenario of the third simulation and using four antennas. The results obtained from Figure 4 show that the proposed methods, especially the second method, are able to detect the angle of all interferences and to provide required nulls to appropriately eliminate them, only with four antennas. According to Figure 4, Ahmed and CPA methods do not have suitable performance in accurately detecting the angle of interferences and also eliminating them, compared to the proposed methods. It should be noted that as the number of antennas increases, the depth of nulls will be greater and consequently the interferences will be better eliminated.
In the fifth simulation, we aim to evaluate the second method in terms of the number of discrete phase alphabets or the same permissible interval for selecting the matrix elements of compound F. In this simulation, the assumptions used in the first simulation are taken into account. We also assume that K = M t and the value of L is from 4 to 256 (i.e. 4PSK to 256PSK). Figure 5 indicates that SINR values obtained for different discrete phase alphabets are remarkable, even for the 4PSK case.

CONCLUSION
To design covariance matrices for increasing SINR in MIMO radar, two methods based on CD algorithm are presented in this paper. In the first method, the covariance matrix structure is based on Toeplitz form. Therefore, it has less computational complexity because only the first row elements of the covariance matrix must be optimized. In the second method, the covariance matrix is obtained by the phase combination of an