Abstract

Ideal array responses are often desirable to a multiple-input multiple-output (MIMO) system. Unfortunately, it may not be guaranteed in practice as the mutual coupling (MC) effects always exist. Current works concerning MC in the MIMO system only account for the uniform array geometry scenario. In this paper, we generalize the issue of angle estimation and MC self-calibration in a bistatic MIMO system in the case of arbitrary sensor geometry. The MC effects corresponding to the transmit array and the receive array are modeled by two MC matrices with several distinct entities. Angle estimation is then recast to a linear constrained quadratic problem. Inspired by the MC transformation property, a multiple signal classification- (MUSIC-) like strategy is proposed, which can estimate the direction-of-departure (DOD) and direction-of-arrival (DOA) via two individual spectrum searches. Thereafter, the MC coefficients are obtained by exploiting the orthogonality between the signal subspace and the noise subspace. The proposed method is suitable for arbitrary sensor geometry. Detailed analyses with respect to computational complexity, identifiability, and Cramer-Rao bounds (CRBs) are provided. Simulation results validate the effectiveness of the proposed method.

1. Introduction

Multiple-input multiple-output (MIMO) is the technique with the most potential for the next-generation array radar system [1–3]. A MIMO system is characterized by multiple transmitting antennas and multiple receiving antennas. Unlike the traditional phased array radar, the MIMO system emits mutual orthogonal waveforms. The spatial diversity and waveform diversity enables the MIMO system to achieve a virtual aperture, which is much larger than the physical aperture of the radar system. Benefiting from the virtual aperture, the MIMO system outperforms the phased array radar with respect to clutter suppression, direction finding, and target tracking. Usually, the MIMO system can be divided into two categories in terms of its antenna configuration, namely, a distributed MIMO system and a colocated MIMO system [4, 5]. A distributed MIMO system is equipped with widely displaced transmitting and receiving antennas, and it can illuminate the area of interest from various perspectives. Therefore, it is capable of extinguishing the radar cross-section (RCS) fluctuation issue. A colocated MIMO system consists of closely spaced antennas, and it is able to provide super-resolution direction estimation. Herein, we focus on the bistatic MIMO system, which is a typical representative of the colocated MIMO system.

As a canonical issue in bistatic MIMO systems, joint direction-of-departure (DOD) and direction-of-arrival (DOA) estimation has been studied extensively in the past decades. Various methodologies have been proposed, e.g., estimation method of signal parameters via rotational invariance technique (ESPRIT) [6], multiple signal classification (MUSIC) [7], maximum likelihood [8], tensor approaches [9, 10], and sparsity-aware methods [11, 12]. A common assumption in [6–11] is that the sensors are well-calibrated. Nevertheless, owing to the radiation effects [13], sensors may be susceptible to unknown mutual coupling (MC), which would bring perturbation to the signal model and thus lead to degraded estimation performance. In general, the MC effect between sensors can be described by a MC matrix, in which the entities reveal the strength between two sensors. Although the MC matrix can be directly obtained offline via the measurement techniques such as the electromotive force [14], it may be invalid in the scenario where MC varies with time. To pursue online MC calibration, the usage of instrumental sensors or auxiliary sources have been reported [15, 16], which require additional resource. As is known to us, the MC intensity of two sensors is inversely proportional to their distance. Therefore, the MC effects between sensors that are far apart can be neglected, i.e., the associated MC coefficients can be approximated by zeros. As a result, sparse arrays, e.g., coprime arrays [17, 18], may be free from MC, but it is impossible for applications with limited space.

As mentioned earlier, the MC intensity between two sensors is inversely proportional to their interelement distance. Accordingly, the MC matrix associated with a uniform linear array (ULA) exhibits a special structure, which can be captured by a banded symmetric Toeplitz matrix. For a ULA-based MIMO system, several strategies have been proposed for MC self-calibration. In [19], a MUSIC-like estimator was presented, in which the MC coefficients are extracted into a single vector, and the DODs and DOAs are obtained via two one-dimensional (1D) peak searches. By interpreting the sensors at both ends of the arrays as instrumental sensors, the ESPRIT-like algorithms were introduced in [20, 21], where the MC effect is suppressed by using a selection matrix, and then the ESPRIT technique is followed to obtain close-form solutions for DOD and DOA estimation. In order to exploit the tensor structure of the array data, the tensor approaches were subsequently discussed in [22–24], which offer more accurate angle estimation performance than the matrix-based ones. Besides, some frameworks have investigated into the two-dimensional (2D) angle estimation issue using a polarization-sensitive array [25–28].

It should be emphasized that the above algorithms are only effective for MIMO systems with ULA configurations. In practice, the ULA-based MIMO system may occasionally encounter sensor failure [29, 30]. Besides, a more complex array manifold would be adopted in the MIMO system [31]. Therefore, the MIMO system with arbitrary Tx/Rx geometries may suffer from unknown MC, and current estimators in [19–24] will fail to work. To the best of our knowledge, only a few efforts have been devoted to robust DOA estimation for arbitrary geometry sensor array with unknown MC. In such a case, the MC matrix exhibits no special structure except Hermitian. In [32], an optimization-based framework was investigated. It formulates the DOA estimation as a block sparse inverse problem, which is solved via a joint-sparse recovery algorithm. Like the traditional counterpart, it has high complexity due to the high-dimension nonconvex problem, making it unacceptable for the MIMO system. In [33], an iterative approach was considered, in which DOA and MC coefficients are updated alternately. More recently, the MUSIC-like method was applied to robust DOA estimation in [34]. Although the approaches in [33, 34] can be directly extended to angle estimation for a bistatic MIMO system, they are inefficient due to an exhaustive grid search.

In this paper, we generalize the MC issue in a bistatic MIMO system. We consider a general scenario where the Tx/Rx arrays are in arbitrary geometries and MC effects occur in both the Tx/Rx arrays. In such a scenario, the MC matrices exhibit no special structure except symmetry. To tackle the DOD and DOA estimation problem, the eigendecomposition is firstly performed to obtain the noise subspace. Then, by exploiting the orthogonality between the noise subspace and the virtual steering vector, angle estimation is recast to a general quadratic problem. Therefore, the MC transformation property is adopted to transform the optimization problem into two different forms, in which only DOD and DOA are included. Consequently, DOD and DOA can be estimated via two individual spectrum searches. Finally, the MC coefficients can be easily obtained with the estimated DOD and DOA. The proposed method is analyzed in terms of computational complexity, identifiability, and Cramer-Rao bounds (CRBs). Numerical simulations are designed to show the effectiveness of the proposed method.

Notation, bold capital letters, e.g., , and bold lowercase letters, e.g., , denote matrices and vectors, respectively. The superscripts , , and account for transpose, Hermitian transpose, and inverse, respectively. The identity matrix is denoted by , the full one matrix is denoted by , the full zero matrix is denoted by . , , and represent, respectively, the Kronecker product, the Khatri-Rao product, and the Hadamard product; denotes the diagonalization operation; denotes rank operator; and and return the real part and the image part of a vector, respectively.

2. Problem Formulation

We consider a bistatic MIMO system with transmit sensors and receive sensors, and there are far-field targets appearing in the same range bins of the radar system. The 2D-DOA pair and the 2D-DOA of the th target are denoted by and , where and are elevation angles and and are azimuth angles. Without loss of generality, we assume that both the transmit sensors and the receive sensors are distributed in arbitrary geometries. The coordinate of the th transmit sensor is , and the coordinate of the th receive sensor is . The response vectors corresponding to the transmit array and the receive array are given bywhere λ is the carrier wavelength, and are the associated path differences with , , , and , respectively. Define and are the direction matrices corresponding to the transmit array and receive array, respectively. In the absence of the MC effect, the output from the matched filters can be written aswhere is the slow time index; is the RCS coefficient of the th targets; is the array noise, which is assumed to be Gaussian white with variance ; and . Now, we consider a scenario where both the transit sensors and the receive sensors suffer from unknown MC and the MC matrices are denoted by and , respectively, which are Hermitian. In the presence of an unknown MC, the matched output in (2) becomes

Suppose the RCS coefficients are uncorrelated, and is irrelevant with , and then, the covariance matrix of iswhere is the covariance matrix of , is the power of . With available snapshots, i.e., , can be estimated via

3. The Proposed Algorithm

3.1. DOD and DOA Estimation

Performing eigendecomposition on , one can obtain the signal subspace and the noise subspace aswhere and are two diagonal matrices that contain the dominate eigenvalues and the remind eigenvalues, respectively. and consist of the corresponding eigenvectors, which are called the signal subspace and the noise subspace, respectively. It is known that is orthogonal to , i.e.,

Since span the same subspace with the columns of , we get

The above property is similar to that in the 2D-MUSIC algorithm. However, the traditional MUSIC estimator is invalid as both and are unknown. To further process, the following conclusion in [33] is introduced.

Theorem 1. For a matrix and a vector , if there are only () distinct entities in C, thenwhere with the th column given byand is defined as

In this paper, we build on top of Theorem 1 to address the MC self-calibration challenge in the arbitrary geometry-based MIMO system. Suppose that there are distinct entities in , and distinct entities in . From Theorem 1, we havewhere and are constructed according to Theorem 1. In combination with the property , we can obtain

Inserting (13a) and (13b) into (9) establisheswhere and . To obtain the DOA, one needs to optimize

Notably, (15) is a quadratic optimization issue. To avoid the trivial solution , the above optimization problem is constrained with , i.e.,where ρ is a constant and is a vector with the first entity being one and zeros elsewhere. To solve the above issue, we can construct the following Lagrange function:where is a Lagrange multiplier. Enforcing to zeros yields

Plug (18) into (15) and remove the constant item gives

By setting a grid to contain all possible DODs, the indexes corresponding to the maximum values in (19) reveal the DODs. Similarly, we can obtain the DOA estimation via computingwhere is a vector with the first entity one and zeros elsewhere.

3.2. MC Self-Calibration

Although DOD and DOA can be separately estimated from (19) and (20), they require further pairing, since there are possible DOD-DOA pairs. Recall the properties in (12) and we can rewrite (8) aswhere . Similar to (15), to find the correct DOD-DOA pairs, we need to calculatewhich can be accomplished viawhere is a vector with the first entity one and zeros elsewhere. List all the possible DOD-DOA pairs, and the real DOD-DOA pairs can be picked out by finding the maximum values in (23).

It should be pointed out that the optimal solution of (22) is achieved ifwhere α is a constant. After the DODs and DOAs have been paired, can be estimated viawhere and denote the estimated and , respectively. Usually, we have and . After normalizing , the MC vectors can be estimated viawhere .

4. Algorithm Analysis

4.1. Computational Complexity

The computational burden (the number of complex multiplication) of the proposed is summarized as follows:

Step 1. Estimate the covariance matrix .

Step 2. Perform eigendecomposition on to obtain the noise subspace .

Step 3. Calculate the spectrum function in (19) to get the DOD estimation .

Step 4. Search the maximums of (20) to achieve the DOA estimation .

4.2. Identifiability

The proposed algorithm is effective if , , and are valid. As rank , once , is rank deficit. Similarity, if , is nonfull rank, and if , rank deficiency would occur in . As a result, a sufficient condition of the proposed estimator is that . Since and , the above condition can be simplified by . In other words, the proposed estimator can identify targets at most.

4.3. Cramer-Rao Bound (CRB)

For the sake of simplicity, (3) is rewritten aswhere Then, all the measurements are arranged into a “column” vector as . Suppose that the is deterministic but unknown. Thereafter, the mean of y is given bywhere and .

Define the unknown parameter vectors , , , , , , , and . The whole estimation parameter vector is formulated as . It is well known that the CRB matrix for is given bywhere . Next, we will concentrate on each part of . Before the detailed derivations, the following variables are defined: