SINR maximization in colocated MIMO radars using transmit covariance matrix

Highlights

• We proposed two covariance matrices in order to maximize the signal-to-interference-plusnoise ratio (SINR) and to exploit the advantages of MIMO radars.

• Two proposed matrices are full rank to suppress more interferences.

• Maximum achievable SINR is calculated analytically for both.

• Using these matrices enables us to transmit BPSK samples with the same covariance matrix.

• The second proposed matrix has a low side lobe level and therefore has a good performance in the case of unknown interference location.

Abstract

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-to-interference-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.

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 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-plus-noise 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], [7], [8], [9], [10], (b) synthesizing the transmit waveforms in order to realizes the covariance matrix R [11], [12], [13]. Several algorithms are proposed to design 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], [18], [19], [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 $\sin \left((\mathrm{\pi }/2)\mathbf{R}\right)$ 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], $\sin \left((\mathrm{\pi }/2)\mathbf{R}\right)$ 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, $\sin \left((\mathrm{\pi }/2)\mathbf{R}\right)$ 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 ${(·)}^T$, ${(·)}^H$ and ${(·)}^{-1}$, respectively and expectation operator is denoted by $E\{·\}$. The Kronecker product and the Hadamard multiplication are denoted by $\otimes$ and $\odot$, respectively. U is uniform distribution, $\det (·)$ denotes determinant of a square matrix and $|·|$ denotes the absolute value. the vectorization of a matrix denoted by vec (·).