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In this paper, a new tensor-based subspace approach is proposed to estimate the direction of departure (DOD) and the direction of arrival (DOA) for bistatic multiple-input multiple-output (MIMO) radar in the presence of spatial colored noise. Firstly, the received signals can be packed into a third-order measurement tensor by exploiting the inherent structure of the matched filter. Then, the measurement tensor can be divided into two sub-tensors, and a cross-covariance tensor is formulated to eliminate the spatial colored noise. Finally, the signal subspace is constructed by utilizing the higher-order singular value decomposition (HOSVD) of the cross-covariance tensor, and the DOD and DOA can be obtained through the estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm, which are paired automatically. Since the multidimensional inherent structure and the cross-covariance tensor technique are used, the proposed method provides better angle estimation performance than Chen's method, the ESPRIT algorithm and the multi-SVD method. Simulation results confirm the effectiveness and the advantage of the proposed method.

Recently, multiple-input multiple-output (MIMO) radar [

Angle estimation is an important aspect in array signal processing and MIMO radar [

However, in the subspace methods [

The rest of the paper is organized as follows. The tensor basics and signal model are presented in Section 2. A tensor-based subspace approach for angle estimation in the presence of spatial colored noise is proposed in Section 3. The computational complexity of the method is evaluated in Section 4. In Section 5, simulation results are provided to verify the performance of the proposed algorithm. Finally, Section 6 concludes this paper.

_{i,j} and
_{i,j,k} stand for the (^{H}, (·)^{T}, (·))^{−1} and (·)^{*} denote the Hermitian transpose, transpose, inverse and complex conjugation without transposition, respectively. ⊗ and ⊙ denote the Kronecker operator and the Khatri-Rao product, respectively. diag(·) denotes the diagonalization operation, and arg(

For the readers' convenience, several tensor operations are introduced firstly, which refer to [

^{I}^{×}^{J}^{×}^{K}_{(1)} ∈ ℂ^{I}^{×}^{JK}_{(2)} ∈ ℂ^{J}^{×}^{IK}_{(3)} ∈ ℂ^{K}^{×}^{IJ}_{(1)}]_{i}_{(}_{k}_{−1)}_{J}_{+}_{j}_{i,j,k}, [[
_{(2)}]_{j}_{(}_{i}_{−1)}_{K}_{+}_{k}_{i,j,k} and [[
_{(3)}]_{k}_{(}_{j}_{−1)}_{I}_{+} _{i}_{i,j,k}, respectively.

^{I1}^{×}^{I2}^{×⋯×}^{IN} by a matrix, ^{Jn}^{×}^{In}_{n}^{I1}^{×}^{I2}^{×⋯×}^{In}_{−1}^{×}^{Jn}^{×}^{In}_{+1}^{×⋯×}^{IN}

Consider a narrowband bistatic MIMO radar system with

Both the transmit array and receive array are uniform linear arrays (UALs), and the inter-element spaces of the transmit and receive arrays are half-wavelength. At the transmit array, the transmit antennas emit the orthogonal waveforms _{1}, _{2}, ⋯, _{M}^{T} ∈ ℂ^{M}^{×}^{K}_{1}),⋯ , _{P}^{N}^{×}^{P}_{1}), ⋯ , _{P}^{M}^{×}^{P}_{p}^{jπ}^{sin} ^{θp}^{jπ}^{(}^{N}^{−1) sin} ^{θp}^{T} ∈ ℂ^{N}^{×1} and _{p}^{jπ}^{sin} ^{φp}^{jπ}^{(}^{M}^{−1)sin} ^{φp}^{T} ∈ ℂ^{m}^{×1} are the receive steering vector and transmit steering vector of the _{l}_{l}_{l}_{1}^{j}^{2}^{πf}_{d}_{1}^{lTr}_{P} e^{j}^{2}^{πf}_{dp}^{lTr}_{dP}_{r}_{l}^{N}^{×}^{K}_{l}_{m}_{m}_{1}),…, _{m}_{P}_{m}_{p}_{p}

According to

In the conventional subspace-based methods, the received signals in _{1})), vec(_{2}),…, vec(_{L}_{1} = [_{M1},_{M1}_{×(}_{M}_{−}_{M1}_{)}], _{2} = [_{M2×(}_{M−}_{M2)},_{M2}] with _{1} + _{2} = _{1} = _{1}_{2} = _{2}_{1} and
_{2}, are obtained from different matched filters, i.e,
_{1} is the output of the first _{1} matched filters and
_{2} is the output of the residual _{2} = _{1} matched filters. Thus, we have
_{21} ∈ ℂ^{N}^{×}^{M}_{2}^{×}^{N}^{×}^{M}_{1}, is formulated as:
_{2}._{1}. Since the spatial colored matrix,
_{21}, is not affected by the additive spatial colored noise. According to _{21}, is shown in

Then, the HOSVD [_{21}, yields:
_{1}, _{3} ∈ ℂ^{N}^{×}^{N}_{2} ∈ ℂ^{m2}^{×}^{m2} and _{4} ∈ ℂ^{m1}^{×}^{m1} are unitary matrices. Since
_{21} is a rank-_{s}_{21}, which can be written as:
_{is}_{i}_{s}

According to the relationship between the cross-correlation matrix and its corresponding cross-covariance tensor in _{21}, is reconstructed from
_{s}

In the subspace method [_{s}_{21}, i.e., _{21} ≈ _{s}_{s}_{s}

According to _{21}, the signal subspace, _{s}

According to _{s}_{s}^{P}^{×}^{P}_{s}_{s}

In order to estimate both the DOD and DOA, the signal subspace, _{s}_{s}_{1} = _{1}_{s}_{s}_{2} = _{2}_{s}_{s}_{3} = _{3}_{s}_{s}_{4} = _{4}_{s}_{t}^{−1}_{t}_{r}^{−1} _{r}_{t}^{jπ}^{sin} ^{φ1}, ⋯ , ^{jπ}^{sin} ^{φP}_{r}^{jπ}^{sin} ^{θ1}, ⋯ , ^{jπ}^{sin} ^{θP}_{t}_{r}_{t}_{t}_{t}_{t}_{r}_{r}_{r}^{−1}_{r}_{t}_{r}_{r}

In order to analyze the computational complexity of the proposed method, it is necessary to know the complexity of the SVD algorithm. There are a lot of methods to compute SVD, and the computational complexities of them are different. In [_{r}MNr_{r}_{s}_{21} and the truncated SVD of _{21}. The truncated HOSVD of
_{21} is equivalent to the truncated SVD of all its matrix unfolding, which needs _{r}M_{1}_{2}^{2}_{21} is _{r}M_{1}_{2}^{2}_{r}M_{1}_{2}^{2}_{r}MNLP_{21} to estimate the signal subspace, which needs _{r}M_{1}_{2}^{2}^{H} to estimate the signal subspace, which needs _{r}M^{2}

_{is}_{i}_{1}, _{2}, _{1} ≥ _{2} ≥ _{1},_{2},

In this section, some simulations are presented to evaluate the angle estimation performance of the proposed method in the presence of spatial colored noise. The multi-SVD algorithm [^{K}^{×}^{K}_{K}_{r}_{1}, _{1}) = (30°, −30°), (_{2}, _{2}) = (−40°, 10°) and (_{3}, _{3}) = (10°, 10°), and the reflection coefficients of the targets are
_{i}_{i}_{i}_{i}

_{1} = 3,

_{1} = 3,

_{1} = 3, SNR= −5dB and

In this paper, a tensor-based subspace approach is presented to DOD and DOA estimation for bistatic multiple-input multiple-output (MIMO) radar in the presence of spatial colored noise. The proposed method exploits the the multidimensional structure inherent in the received signals to construct a third-order measurement tensor. Then, two sub-tensors are obtained from the measurement tensor, which can be used to formulate a cross-covariance tensor for eliminating the influence of spatial colored noise. Finally the DOD and DOA can be estimated in conjunction with the ESPRIT method. The proposed method has better angle estimation performance than Chen's method, ESPRIT and the multi-SVD method, especially at the low SNR region. Several simulation results have verified the performance of the proposed method.

This work was supported by the New Century Excellent Talents Support Program (NCET-11-0827), innovation of science and technology talents in harbin (2013RFXXJ016) and fundamental research runs for the central universities (HEUCFX41308, HEUCFZ1110).

Bistatic multiple-input multiple-output (MIMO) radar scenario.

Root mean square error (RMSE)

Probability of successful detection

RMSE