Joint Angles and Mutual Coupling Estimation Algorithm for Bistatic MIMO Radar

Abstract

We study the problem of angle estimation for a bistatic multiple-input multiple-output (MIMO) radar with unknown mutual coupling and proposed a joint algorithm for angles and mutual coupling estimation with the characteristics of uniform linear arrays and subspaces exploitation. We primarily obtain an initial estimate of DOA and DOD, then employ the local one-dimensional searching to estimate exactly DOA and DOD, and finally evaluate the parameters of mutual coupling coefficients via the estimated angles. Exploiting twice of the one-dimensional local searching, our method has much lower computational cost than the algorithm in (Liu and Liao (2012)), and automatically obtains the paired two-dimensional angle estimation. Slightly better performance for angle estimation has been achieved via our scheme in contrast to (Liu and Liao (2012)), while the two methods indicate very close performance of mutual coupling estimation. The simulation results verify the algorithmic effectiveness of our scheme.

1. Introduction

Since multiple-input multiple-output (MIMO) radars exploit multiple antennas to manipulate parallel processing for transmitting diverse waveforms and receiving the reflected signals, several potential advantages over conventional phased-array radars such as angular diversity, flexible beampattern, and more degrees of freedom have been demonstrated in [1–4]. In contrast to other radar systems with a single transmit antenna, the additional degrees of freedom gained from MIMO radar has the capability of overcoming fading effect, enhancing spatial resolution, strengthening parameter identifiability, and improving target detection performance [5–8]. Several established angle estimation algorithms for MIMO radar comprise of the signal parameters estimation through the algorithms reliable on rotational invariance techniques (ESPRIT) [9], Capon [10], multiple signal classification (MUSIC) [11], parallel factor (PARAFAC) analysis [12], and adaptive PARAFAC [13], to name a few. However, most of the algorithms mentioned above take little concern for the problem of mutual coupling, which always exists in practical situations. The algorithm in the presence of unknown mutual coupling [14] for this practical situation achieves well estimation of angles and mutual coupling coefficients, while it is compulsory to perform twice of one-dimensional peak searching and additional pairing. Actually, the algorithmic complexity in [14] can be further reduced through the schemes of multidimensional data processing.

In our study, we present a low-complexity joint algorithm for the angles and mutual coupling coefficients estimation in bistatic MIMO radars. By employing the transformation of special matrix, we combine the signal subspace and noise subspace of the received data together to achieve the joint estimation of DOD and DOA. The angles can be automatically obtained via two local one-dimensional searching without the knowledge of mutual coupling, whose coefficients can be easily estimated after the angle estimation.

The remainder of this paper is structured as follows. Section 2 develops the data model for a MIMO radar system with unknown mutual coupling; Section 3 establishes the joint algorithm for joint angles and mutual coupling estimation in addition with the complexity analysis and Cramér-Rao Bound (CRB). In Section 4, simulation results are presented to verify improvement for the proposed algorithm, while the final conclusions are summarized in Section 5.

Notation
Bold lower (upper) case symbols denote column vectors or matrices; (⋅)^𝑇,(⋅)^∗, (⋅)^𝐻, (⋅)⁻¹, and (⋅)^† denote the operators of transpose, conjugate, conjugate-transpose, inverse, and pseudoinverse, respectively, diag(𝐯) stands for diagonal matrix whose diagonal element is a vector v; 𝐷_𝑛(⋅)denotes the operation of taking the 𝑛th row of the matrix to construct a diagonal matrix; 𝐈_𝐾 is a 𝐾×𝐾 identity matrix; ⊗ and ⊙ represent the Kronecker product and Hadamard product, respectively; the real part of complex numbers is denoted by Re(⋅) and the minimum elements of an array are denoted by min(⋅); 𝐸[⋅] stands for the expectation operator; det(⋅) denotes the determinant of a matrix, vec(⋅) denotes an operator stacking the columns of a matrix on top of each other.

2. Data Model

We consider a MIMO radar system equipped with 𝑀 transmit antennas and 𝑁 receive antennas, both of the antenna arrays are permuted as the uniform linear array (ULA), where the arrays of transmitter and receiver have been collocated in the 𝑦-axis with the interelement distance being half-wavelength [12]. The transmitted 𝑀-encoded pulse signals are orthogonal to each other; denote the mth signal as 𝐬_𝑚=[𝑠_𝑚(1),𝑠_𝑚(2),…,𝑠_𝑚(𝐿)]^𝑇, where 𝑚=1,2,…,𝑀, and 𝐿 is the length of the signal. We assume that there are 𝐾 uncorrelated targets, 𝜃_𝑘,𝜙_𝑘 are direction of departure (DOD) and direction of arrival (DOA) of the kth target with respect to the transmit array normal and the receive array normal, respectively. Assume the transmit and receive steering vectors are 𝐚_𝑡(𝜃_𝑘)and𝐚_𝑟(𝜙_𝑘), and hence, the signal arrive at the kth target is written as 𝐚^𝑇_𝑡(𝜃_𝑘)𝐒, 𝑘=1,2,…,𝐾, where 𝐒=[𝐬₁,𝐬₂,…,𝐬_𝑀]^𝑇∈ℂ^𝑀×𝐿. Assume also that the reflection coefficients 𝛽_𝑘(𝑡) and the phase shift 𝑓_𝑘 are constant between pulse intervals, and the reflected signals of different targets are independent.

The receive data of the 𝑁 receive antennas of the tth pulse can be denoted as𝐘(𝑡)=^𝐾_𝑘=1𝛽_𝑘(𝑡)𝑒^{𝑗2𝜋𝑓𝑘𝑡}𝐚_𝑟𝜙_𝑘𝐚^𝑇_𝑡𝜃_𝑘𝐒+𝐖(𝑡), (1a) where 𝐘(𝑡)∈ℂ^𝑁×𝐿 and W stand for the 𝑁×𝐿 receive data matrix and the noise matrix with identical determinants, respectively.

When each signal received by the 𝑁 antennas has been processed by a number of 𝑀 match filters [15], we obtain an 𝑀×𝑁 data matrix, which is equivalent to [𝐘(𝑡)𝐒^𝐻]^𝑇. After iterative vectorization for the input data, we finally construct the output of the tth pulse as𝐱(𝑡)=vec𝐘(𝑡)𝐒^𝐻𝑇. (1b) Due to the orthogonality of the transmit signals towards each other, 𝐒𝐒^𝐻=𝐈 is satisfied for the unitary matrix 𝐒. Hence, we have𝐱(𝑡)=vec^𝐾_𝑘=1𝛽_𝑘(𝑡)𝑒^{𝑗2𝜋𝑓𝑘𝑡}𝐚_𝑡𝜃_𝑘𝐚^𝑇_𝑟𝜙_𝑘+vec𝐖𝐒^𝐻𝑇=^𝐾_𝑘=1𝛽_𝑘(𝑡)𝑒^{𝑗2𝜋𝑓𝑘𝑡}vec𝐚_𝑡𝜃_𝑘𝐚^𝑇_𝑟𝜙_𝑘(𝑡)=𝐚+𝐧_𝑟𝜙₁⊗𝐚_𝑡𝜃₁,…,𝐚_𝑟𝜙_𝐾⊗𝐚_𝑡𝜃_𝐾𝐛(𝑡)+𝐧(𝑡), (1c)where 𝐛(𝑡)=[𝑏₁(𝑡),𝑏₂(𝑡),…,𝑏_𝐾(𝑡)]^𝑇∈ℂ^𝐾×1, 𝑏_𝑘(𝑡)=𝛽_𝑘(𝑡)𝑒^{𝑗2𝜋𝑓𝑘𝑡}; 𝐧(𝑡) is an 𝑀𝑁×1 Gaussian white noise vector; 𝐚_𝑟(𝜙_𝑘)=𝐂_𝑟𝐚_𝑟(𝜙_𝑘); 𝐚_𝑡(𝜃_𝑘)=𝐂_𝑡𝐚_𝑡(𝜃_𝑘). It follows that 𝐂_𝑟∈ℂ^𝑁×𝑁 and 𝐂_𝑡∈ℂ^𝑀×𝑀 represent the matrices containing the mutual coupling coefficients. According to the characteristics of ULA, if the elements have the same separation distance, the mutual coupling coefficients will be the same, so this will lead to two Toeplitz matrices as shown below.𝐂_𝑟=𝑐^𝑟₁⋯𝑐^𝑟_𝑛𝑐0⋯0_𝑟2𝑐^𝑟₁⋯𝑐^𝑟_𝑛⋮𝑐0⋯0_𝑟𝑛⋯𝑐^𝑟₁⋯𝑐^𝑟_𝑛0⋯00𝑐^𝑟_𝑛⋯𝑐^𝑟₁⋯𝑐^𝑟_𝑛⋮0⋯00⋯0𝑐^𝑟_𝑛⋯𝑐^𝑟₁,𝐂_𝑡=𝑐^𝑡₁⋯𝑐^𝑡_𝑚𝑐0⋯0_𝑡2𝑐^𝑡₁⋯𝑐^𝑡_𝑚⋮𝑐0⋯0_𝑡𝑚⋯𝑐^𝑡₁⋯𝑐^𝑡_𝑚0⋯00𝑐^𝑡_𝑚⋯𝑐^𝑡₁⋯𝑐^𝑡_𝑚⋮0⋯00⋯0𝑐^𝑡_𝑚⋯𝑐^𝑡₁, (2) where 𝑐^𝑟_𝑝(𝑝=1⋯𝑛) and 𝑐^𝑡_𝑞(𝑞=1⋯𝑚) stand for the mutual coupling coefficients. 𝐚_𝑟(𝜙_𝑘) and 𝐚_𝑡(𝜃_𝑘) represent the receive and the transmit steering vectors for the kth target, respectively. It is rewritten that𝐚_𝑟𝜙_𝑘=1,exp−𝑗𝜋sin𝜙_𝑘,…,exp−𝑗(𝑁−1)𝜋sin𝜙_𝑘^𝑇,𝐚_𝑡𝜃_𝑘=1,exp−𝑗𝜋sin𝜃_𝑘,…,exp−𝑗(𝑀−1)𝜋sin𝜃_𝑘^𝑇. (3) Define 𝐀=[𝐚_𝑟(𝜙₁)⊗𝐚_𝑡(𝜃₁),…,𝐚_𝑟(𝜙_𝐾)⊗𝐚_𝑡(𝜃_𝐾)], which can be denoted by 𝐀𝐀=^𝑇𝐷₁𝐀_𝑅𝐀_𝑇𝐷₂𝐀_𝑅⋮𝐀_𝑇𝐷_𝑁𝐀_𝑅, (4) where 𝐀_𝑇=[𝐚_𝑡(𝜃₁),…,𝐚_𝑡(𝜃_𝐾)]∈ℂ^𝑀×𝐾, 𝐀_𝑅=[𝐚_𝑟(𝜙₁),…,𝐚_𝑟(𝜙_𝐾)]∈ℂ^𝑁×𝐾. There exists a transformation matrix H corresponding to the finite number of row interchanged operations such that𝐅=𝐇𝐀=𝐚_𝑡𝜃₁⊗𝐚_𝑟𝜙₁,…,𝐚_𝑡𝜃_𝐾⊗𝐚_𝑟𝜙_𝐾. (5) And the matrix 𝐅 can be denoted by𝐀𝐅=^𝑅𝐷₁𝐀_𝑇𝐀_𝑅𝐷₂𝐀_𝑇⋮𝐀_𝑅𝐷_𝑀𝐀_𝑇. (6) By collecting 𝐽 samples and representing the received data as 𝐗=[𝐱(1),𝐱(2),…,𝐱(𝐽)], the transmitted signal matrix can now be generalized as𝐗=𝐀𝐁+𝐍, (7) where 𝐍∈ℂ^{𝑀𝑁×𝐽}stands for the noise matrix, 𝐁=[𝐛(1),𝐛(2),…,𝐛(𝐽)]∈ℂ^𝐾×𝐽.

The covariance matrix of the received signal 𝐑=𝐸[𝐱(𝑡)𝐱^𝐻(𝑡)] can be decomposed as𝐑=𝐄_𝑠𝐃_𝑠𝐄^𝐻_𝑠+𝐄_𝑛𝐃_𝑛𝐄^𝐻_𝑛, (8) where 𝐃_𝑠 denotes a 𝐾×𝐾 diagonal matrix formed by 𝐾 largest eigenvalues and 𝐃_𝑛 denotes a diagonal matrix formed by the rest 𝑀𝑁-𝐾 smaller eigenvalues. 𝐄_𝑠 and 𝐄_𝑛 represent the signal subspace and noise subspace, respectively, of which 𝐄_𝑠 stands for the eigenvectors corresponding to the 𝐾 largest eigenvalues 𝐄_𝑛 consists of the rest eigenvectors. In case of noise absence, the equation between 𝐄_𝑠 and 𝐀 can be formulated as𝐄_𝑠=𝐀𝐓, (9) where 𝐓 represents a nonsingular 𝐾×𝐾 matrix.

3. Joint Angles and Mutual Coupling Estimation Algorithm

3.1. Angles Estimation

According to the MUSIC method, the joint estimation of DOD and DOA can be achieved by the peak searching function as follows:𝐹𝜃,𝜙,𝐂_𝑟,𝐂_𝑡=1𝐐𝜃,𝜙,𝐂_𝑟,𝐂_𝑡=1𝐚_𝑟𝜙⊗𝐚_𝑡(𝜃)^𝐻𝐄_𝑛𝐄^𝐻_𝑛𝐚_𝑟𝜙⊗𝐚_𝑡(𝜃.) (10) However, the method does not work if we cannot obtain 𝐂_𝑟 and 𝐂_𝑡. Even if the mutual coupling coefficients is already known, a two-dimensional searching is still required resulting into a considerably heavy computation burden.

With respect to the (4) and (9), the signal subspace matrix is represented by𝐄_𝑠=𝐀_𝑇𝐷₁𝐀_𝑅𝐀_𝑇𝐷₂𝐀_𝑅⋮𝐀_𝑇𝐷_𝑁𝐀_𝑅𝐓. (11a) For which, it is partitioned as𝐄_𝑠=𝐄_𝑠1𝐄_𝑠2⋮𝐄_𝑠𝑁, (11b)where 𝐄_𝑠𝑙∈ℂ^𝑀×𝐾, 𝑙=1,…,𝑁.

Taking the 𝑛th and (𝑛+1)th part into account, now we have𝐄_𝑠𝑛=𝐀_𝑇diag𝐂^𝑟_𝑛𝐀_𝑅𝐄𝐓,_𝑠(𝑛+1)=𝐀_𝑇diag𝐂^𝑟_𝑛+1𝐀_𝑅𝐓, (12a)(12b)where 𝐂^𝑟_𝑛 is the 𝑛th row of 𝐂_𝑟. When 𝑛 satisfies 𝑛≤𝑁/2, it is found that 𝐂^𝑟_𝑛+1 is the right shift of 𝐂^𝑟_𝑛, and hence,diag𝐂^𝑟_𝑛+1𝐀_𝑅=diag𝐂^𝑟_𝑛𝐀_𝑅Φ_𝑟, (13) where Φ_𝑟=diag(exp[−𝑗𝜋sin𝜙₁],…,exp[−𝑗𝜋sin𝜙_𝐾]). Equation (12b) can be rewritten as𝐄_𝑠(𝑛+1)=𝐄_𝑠𝑛𝐓⁻¹Φ_𝑟𝐓. (14)

The diagonal elements of Φ_𝑟 are the eigenvalues of 𝐄^†_𝑠𝑛𝐄_𝑠(𝑛+1), and the corresponding eigenvectors are displayed as 𝐓⁻¹. In case of noise absence, 𝐓⁻¹=𝐓⁻¹𝚷, where 𝚷 is a column permutation matrix with 𝚷⁻¹=𝚷^𝑇. The estimation of Φ_𝑟 is Φ_𝑟=𝚷^𝑇Φ_𝑟𝚷, and the DOA estimation can be obtained via𝜙_𝑘^𝑖𝑛𝑖=sin⁻¹−angle𝑢_𝑘𝜋, (15) where 𝑢_𝑘 is the kth diagonal element of Φ_𝑟.

The following matrix 𝐄_𝑐 stands for the column transformation of the direction matrix 𝐅:𝐄_𝑐=𝐇𝐄_𝑠𝐓⁻¹=𝐅𝚷. (16a) This matrix 𝐄_𝑐 can now be similarly partitioned as𝐄_𝑐=𝐄_𝑐1𝐄_𝑐2⋮𝐄_𝑐𝑀, (16b) where 𝐄_𝑐𝑙∈ℂ^𝑁×𝐾, 𝑙=1,…,𝑀. Considering the mth and (𝑚+1)th part of 𝐄_𝑐, by the similar method in equations from (11a) and (11b) to (13), the iterative matrix updates for 𝐄_𝑐𝑚 now become𝐄_𝑐(𝑚+1)=𝐄_𝑐𝑚𝚷^𝑇Φ_𝑡𝚷, (16c) where Φ_𝑡=diag(exp[−𝑗𝜋sin𝜃₁],…,exp[−𝑗𝜋sin𝜃_𝐾]). We assume that 𝑣_𝑘 is the kth diagonal element of 𝐄^†_𝑐𝑚𝐄_𝑐(𝑚+1), then the angle estimates can be achieved by𝜃_𝑘^𝑖𝑛𝑖=sin⁻¹−angle𝑣_𝑘𝜋. (16d)Therefore, the initial estimations of DOA and DOD have been obtained with the characteristics of automatic pairing.

However, the estimation obtained by (15) and (16d) has low accuracy and bias, for it just employs partial information of the antennas, the more accurate estimation of the angles will be obtained below.

Referring to [14, Lemma 1], the estimation for the transmitted signals and their paired response can be obtained by 𝐚_𝑡(𝜃)=𝐂_𝑡𝐚_𝑡(𝜃)=𝐓_𝑡(𝜃)𝐜_𝑡,𝐚_𝑟𝜙=𝐂_𝑟𝐚_𝑟𝜙=𝐓_𝑟𝜙𝐜_𝑟, (17) where𝐜_𝑡=[𝑐^𝑡₁,…,𝑐^𝑡_𝑚], 𝐜_𝑟=[𝑐^𝑟₁,…,𝑐^𝑟_𝑛],𝐓_𝑡(𝜃)=𝐓_𝑡1(𝜃)+𝐓_𝑡2(𝜃), and the nonzero elements calculated from 𝐓_𝑡1(𝜃)∈ℂ^𝑀×𝑚 and 𝐓_𝑡2(𝜃)∈ℂ^𝑀×𝑚 can now be written as𝐓_{𝑡1𝑖𝑗}=𝐚_𝑡(𝜃)_{𝑖+𝑗−1}𝐓,𝑖+𝑗≤𝑀+1,_{𝑡2𝑖𝑗}=𝐚_𝑡(𝜃)_{𝑖−𝑗+1},𝑖≥𝑗≥2. (18) It is shown for 𝐐(𝜃,𝜙,𝐂_𝑟,𝐂_𝑡)that𝐐𝜃,𝜙,𝐂_𝑟,𝐂_𝑡=𝐜_𝑟⊗𝐜_𝑡^𝐻𝐐_𝑀𝜃,𝜙𝐜_𝑟⊗𝐜_𝑡, (19) where 𝐐_𝑀(𝜃,𝜙)=(𝐓_𝑟(𝜙)⊗𝐓_𝑡(𝜃))^𝐻𝐄_𝑛𝐄^𝐻_𝑛(𝐓_𝑟(𝜙)⊗𝐓_𝑡(𝜃)).

Since 𝐜_𝑟⊗𝐜_𝑡≠𝟎, and when 𝑀𝑁−𝐾≥𝑚𝑛, we construct the function to search for the estimations of DOA and DOD [16] as follows:𝜃,𝜙=argmax_𝜃,𝜙1det𝐐_𝑀𝜃,𝜙. (20) Equation (20) requires a two-dimensional searching, which has a high complexity. Considering the initial estimation of DOD and DOA obtained in (15) and (16a), (16b), (16c), and (16d), we can use two one-dimensional searching to obtain more accurate estimation of DOD and DOA𝜃_𝑘=argmax_𝜃∈𝜃_𝑘^𝑖𝑛𝑖_−Δ𝜃,𝜃_𝑘^𝑖𝑛𝑖_+Δ𝜃1det𝐐_𝑀𝜃,𝜙_𝑘^𝑖𝑛𝑖. (21) Throughout the local searching for 𝜃 within [𝜃_𝑘^𝑖𝑛𝑖−Δ𝜃,𝜃_𝑘^𝑖𝑛𝑖+Δ𝜃], where Δ𝜃 is a small value, we obtain more accurate estimation of the DOD 𝜃_𝑘. Similarly, 𝜙_𝑘 can be also estimated from 𝜙 within [𝜙_𝑘^𝑖𝑛𝑖−Δ𝜙,𝜙_𝑘^𝑖𝑛𝑖+Δ𝜙], where Δ𝜙 is a small value. Hence we have𝜙_𝑘=argmax_𝜙∈𝜙_𝑘^𝑖𝑛𝑖_−Δ𝜙,𝜙_𝑘^𝑖𝑛𝑖_+Δ𝜙1det𝐐_𝑀𝜃_𝑘,𝜙, (22) where 𝜃_𝑘 is estimation of 𝜃_𝑘 via (21).

3.2. Mutual Coupling Estimation

According to (19), the estimates of 𝐜_𝑟𝑡=𝐜_𝑟⊗𝐜_𝑡 will be obtained from the minimum eigenvector of (1/𝐾)∑^𝐾_𝑘=1𝐐_𝑀(𝜃_𝑘,𝜙_𝑘). After normalizing 𝐜_𝑟𝑡(𝐜_𝑟𝑡(1)=1), and reshaping it to an 𝑚×𝑛 matrix 𝐂_𝑟𝑡, we have𝐂_𝑟𝑡=𝐜_𝑡𝐜^𝑇_𝑟. (23) Then we obtain the estimation of 𝐜_𝑟 and 𝐜_𝑡 from the singular value decomposition of 𝐂_𝑟𝑡. Let us assume 𝐂_𝑟𝑡=𝐔𝐒𝐕^𝐻 and the largest left singular vector is 𝐮_𝑡, while the largest right singular vector is 𝐯_𝑟, it is easy to construct that𝐜̂_𝑡=𝐮_𝑡𝐮_𝑡(1),𝐜̂_𝑟=𝐯_𝑟𝐯_𝑟(1)^∗. (24) Till now, we have achieved the proposal of our scheme for the joint angle and mutual coupling estimation in MIMO-radar. We show the major algorithmic steps as follows.(1)Estimate the covariance matrix of the received data through 𝐑=∑^𝐽_𝑡=1𝐱(𝑡)𝐱^𝐻(𝑡).(2)Perform eigenvalue decomposition of 𝐑 to get 𝐄_𝑠 and 𝐄_𝑁, and partition the matrix 𝐄_𝑠 to get 𝐄_𝑠1,𝐄_𝑠2,…,𝐄_𝑠𝑁. It follows to obtain 𝜙_𝑘^𝑖𝑛𝑖 from the eigenvalues of 𝐄^†_𝑠𝑛𝐄_𝑠(𝑛+1) and 𝐓 from the corresponding eigenvectors.(3)Compute 𝐄_𝑐, and get the initial estimation of the DOD 𝜃_𝑘^𝑖𝑛𝑖.(4)Obtain the estimation of DOD 𝜃_𝑘 through one-dimensional searching via (21) while keeping 𝜙_𝑘^𝑖𝑛𝑖 fixed.(5)Obtain the estimation of DOA 𝜙_𝑘 through one-dimensional searching via (22) while keeping 𝜃_𝑘 fixed.(6)Obtain the estimation of 𝐂_𝑟𝑡 from the minimum eigenvector of (1/𝐾)∑^𝐾_𝑘=1𝐐_𝑀(𝜃_𝑘,𝜙_𝑘), and step further to get the estimation of 𝐜_𝑟 and 𝐜_𝑡 from the singular value decomposition of 𝐂_𝑟𝑡 via (24).

3.3. Complexity Analysis and Cramer-Rao Bound (CRB)

The proposed algorithm has lower complexity than the algorithm in [14]. Our scheme costs 𝑂{𝑀²𝑁²𝐽+𝑀³𝑁³+2𝐾²𝑀+3𝐾³+2𝐾²𝑁+𝐾(2𝑁+2) + 2𝑛₁𝐾[𝑀𝑁𝑚𝑛+𝑀𝑁𝑚𝑛(𝑀𝑁−𝐾) + (𝑚𝑛)! + (𝑀𝑁−𝐾)𝑚²𝑛²]+𝑚³𝑛³+min(𝑛²𝑚,𝑚²𝑛)}, where 𝑛₁ is the number of steps within the local searching range, while the algorithm in [14] requires 𝑂{𝑀²𝑁²𝐽+𝑀³𝑁³+𝑛₂[𝑀²𝑁𝑛(𝑀𝑁−𝐾) + (𝑀𝑁−𝐾)𝑀²𝑛²+2(𝑀𝑛)!+𝑁²𝑀𝑚(𝑀𝑁−𝐾) + 𝑁²𝑚²(𝑀𝑁−𝐾)]+𝐾²[𝑀𝑁𝑚𝑛+𝑀𝑁𝑚𝑛(𝑀𝑁−𝐾) + (𝑚𝑛)!+(𝑀𝑁−𝐾)𝑚²𝑛²]+𝑚³𝑛³+min(𝑛²𝑚,𝑚²𝑛)}, where 𝑛₂ is the number of steps within the global searching range, and 𝑛₂≫𝑛₁.

According to [17], we derive the CRB for the joint angles estimation and mutual coupling algorithm in bistatic MIMO radar as follows:𝜎CRB=²2Re𝐃^𝐻𝚷^⟂_𝐆𝐃⁻¹, (25) where 𝜎²stands for the covariance of the noise. 𝐃=𝜕𝐮𝜕𝜃₁,…,𝜕𝐮𝜕𝜃_𝐾,𝜕𝐮𝜕𝜙₁,…,𝜕𝐮𝜕𝜙_𝐾,𝜕𝐮𝜕𝑐^𝑡₂,…,𝜕𝐮𝜕𝑐^𝑡_𝑚,𝜕𝐮𝜕𝑐^𝑟₂…,𝜕𝐮𝜕𝑐^𝑟_𝑛 (26) with (1)⋮(𝐽)𝐮=𝐀𝐛𝐀𝐛,𝚷^⟂_𝐆=𝚷^⟂_𝐀0⋱0𝚷^⟂_𝐀, (27) with 𝚷^⟂_𝐀=𝐈_𝑀𝑁−𝐀𝐀^𝐻𝐀⁻¹𝐀^𝐻. (28)

3.4. Advantages of the Proposed Algorithm

By compares and contrasts to [14], the algorithmic advantage of our approach can be briefly summarized as follows:(1)Our approach can achieve the estimation for automatically paired two-dimensional angles, while the algorithm in [14] requires additional pairing.(2)The proposed algorithm only requires the local one-dimensional searching and hence has much lower complexity, while the algorithm in [14] urges two global one-dimensional searching.(3)For the performance of angle and mutual coupling estimation, slightly improvement can be verified in our method than the one in [14].(4)By fully exploiting both the signal subspace and the noise subspace, our method displays more validity and effectiveness, while the algorithm in [14] just employs the noise subspace.

4. Simulation Results

The Root mean square error (RMSE) of the angle and the RMSE of the mutual coupling coefficients [12] can be defined asRMSE_𝑎=1𝐾^𝐾𝑘=111000^1000𝑙=1𝜙_𝑘,𝑙−𝜙_𝑘²+𝜃_𝑘,𝑙−𝜃_𝑘²,RMSE_𝑐=11000^1000𝑙=1𝐜̂_𝑟,𝑙−𝐜_𝑟²_𝐹+𝐜̂_𝑡,𝑙−𝐜_𝑡²_𝐹, (29) where 𝜃_𝑘,𝑙 is the estimate of DOD, 𝜃_𝑘of the lth Monte Carlo trial, 𝜙_𝑘,𝑙 is the estimate of DOA 𝜙_𝑘 of the lth Monte Carlo trial, 𝐜̂_𝑟,𝑙is the estimate of 𝐜_𝑟 of the lth Monte Carlo trial, and 𝐜̂_𝑡,𝑙is the estimate of 𝐜_𝑡 of the lth Monte Carlo trial. We assume that there are 𝐾=3 targets with their angles correspondingly distributed as (𝜃₁,𝜙₁)=(50^∘,−20^∘), (𝜃₂,𝜙₂)=(−50^∘,20^∘), (𝜃₁,𝜙₁)=(10^∘,40^∘). 𝐜_𝑡=[1,0.1174+0.0577𝑗], and 𝐜_𝑟=[1,−0.0121−0.1029𝑗].

Figure 1 depicts the estimation result of the proposed algorithm for all three targets with 𝑀=8, 𝑁=10, 𝐽=200, and SNR = 10 dB. It is explicit to show that the angles can be clearly observed.

Figure 1 

Angle estimation performance of the proposed algorithm with SNR = 10 dB.

We compare the proposed algorithm against the algorithm in [14] and CRB. Figures 2 and 3 present the comparison of the algorithms. From Figures 2 and 3, we can find that the angle estimation performance of the proposed algorithm is slightly better than the algorithm in [14], while our approach has very close performance for mutual coupling estimation to that of [14]. The reasons are (1) the method in [14] estimates the DOA and DOD separately, while the proposed considers the two angle’s information simultaneously. (2) The proposed obtains an initial estimation firstly, which provides more information of the angles in estimation.

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Figure 2 

Angle and mutual coupling estimation performance comparison (𝑀=8,𝑁=10,𝐽=200,𝐾=3).

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Figure 3 

Angle and mutual coupling estimation performance comparison (𝑀=8,𝑁=6,𝐽=150,𝐾=2).

Figures 4 and 5 present the estimation performance of the proposed algorithm with 𝐽=150, 𝐾=2, and variable proportion of 𝑀 to 𝑁. From Figures 4 and 5, the angle estimation performance of the proposed algorithm is improved with the increment of receive/transmit antenna numbers. Due to the diversity gain, multiple receive/transmit antennas improve angle estimation performance [12]. Meanwhile, it can be verified that the mutual coupling estimation performance of our method is improved with the decrement of receive/transmit antenna numbers. The dimensions of the mutual coupling matrix enlarge in condition that the number of receive/transmit antennas is increasing.

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Figure 4 

Angle estimation performance with 𝑁=6 and different values of 𝑀.

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Figure 5 

Angle estimation performance with 𝑀=8 and different values of 𝑁.

Figure 6 presents angle estimation performance of our scheme with 𝑀=8, 𝑁=6, 𝐾=2 and different values of 𝐽. It is indicated in Figure 6 that the angle estimation and mutual coupling estimation performance of the proposed algorithm is improved with 𝐽 increasing. When 𝐽 increases, we obtain a more accurate covariance matrix, and thereafter gain a more accurate subspace, which leads to more accurate angle estimation.

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Figure 6 

Angle estimation performance with different values of𝐽.

5. Conclusion

In this paper, we have presented a low-complexity algorithm for joint angles and mutual coupling estimation in bistatic MIMO radar with uniform linear arrays (ULAs). This method first obtains an initial estimate of DOA and DOD, then employs the local one-dimensional searching to achieve exact estimates for DOA and DOD, and finally estimates mutual coupling coefficients via the estimated angles. Automatically paired two-dimensional angle estimation can be also performed by our proposed algorithm. We have reduced the algorithmic complexity in comparison with that of [14] and established better performance of angle estimation.

Acknowledgments

This work is supported by China NSF Grants (60801052) and the Fundamental Research Funds for the Central Universities (NZ2012010, kfjj120115, kfjj20110215).