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Off-Grid DOA Estimation Using Sparse Bayesian Learning in MIMO Radar With Unknown Mutual Coupling

Peng Chen, Zhenxin Cao, Zhimin Chen, Xianbin Wang

arXiv:1804.04460v4eess.SP

TL;DR

The paper tackles DOA estimation in MIMO radar when antenna mutual coupling and grid-induced off-grid errors occur simultaneously. It formulates a joint sparse model and proposes the EM-based SBLMC method to estimate the relevant unknown parameters. SBLMC reportedly outperforms existing methods while maintaining acceptable computational complexity.

  • Problem

    Existing approaches did not simultaneously account for off-grid effects and unknown mutual coupling in Bayesian DOA estimation for practical MIMO radar.

  • Method

    SBLMC uses an EM-based sparse Bayesian procedure to jointly estimate noise variance, mutual coupling vectors, the off-grid vector, and scattering-coefficient variances.

  • Results

    SBLMC achieves the best reported DOA estimation performance among the compared methods under unknown mutual coupling, with acceptable computational complexity.

  • Takeaways & Limitations

    Jointly modeling off-grid effects and unknown mutual coupling supports improved DOA estimation in the studied MIMO radar setting.

Abstract

from arXiv · show

In the practical radar with multiple antennas, the antenna imperfections degrade the system performance. In this paper, the problem of estimating the direction of arrival (DOA) in multiple-input and multiple-output (MIMO) radar system with unknown mutual coupling effect between antennas is investigated. To exploit the target sparsity in the spatial domain, the compressed sensing (CS)-based methods have been proposed by discretizing the detection area and formulating the dictionary matrix, so an \emph{off-grid} gap is caused by the discretization processes. In this paper, different from the present DOA estimation methods, both the off-grid gap due to the sparse sampling and the unknown mutual coupling effect between antennas are considered at the same time, and a novel sparse system model for DOA estimation is formulated. Then, a novel sparse Bayesian learning (SBL)-based method named sparse Bayesian learning with the mutual coupling (SBLMC) is proposed, where an expectation-maximum (EM)-based method is established to estimate all the unknown parameters including the noise variance, the mutual coupling vectors, the off-grid vector and the variance vector of scattering coefficients. Additionally, the prior distributions for all the unknown parameters are theoretically derived. With regard to the DOA estimation performance, the proposed SBLMC method can outperform state-of-the-art methods in the MIMO radar with unknown mutual coupling effect, while keeping the acceptable computational complexity.

I. INTRODUCTION

The paper addresses DOA estimation in colocated MIMO radar when mutual coupling and off-grid effects occur together. It formulates a sparse model and proposes SBLMC to estimate the unknown parameters iteratively.

  • Prior methods: CS-based DOA estimation uses spatial sparsity, but discretized grids introduce an off-grid gap.Dense grids can improve performance while increasing computational complexity and dictionary mutual coherence.
  • Research gap: Mutual coupling between antennas cannot be ignored in practical MIMO radar, yet prior Bayesian methods did not jointly model coupling and off-grid effects.The paper characterizes mutual coupling through a symmetric Toeplitz matrix.
  • Contributions: The proposed sparse model jointly considers off-grid effects and unknown mutual coupling, converting DOA estimation into sparse reconstruction.The model exploits target sparsity in the spatial domain.
  • Contributions: SBLMC uses an EM-based procedure to iteratively estimate noise variance, mutual coupling vectors, the off-grid vector, and scattering-coefficient variances.The method is designed for MIMO radar with both unknown mutual coupling and off-grid effects.
  • Contributions: The paper theoretically derives prior distributions and estimation expressions for all listed unknown parameters.These include target scattering coefficients, mutual coupling vectors, the off-grid vector, and noise variance.

II. MIMO RADAR MODEL FOR DOA ESTIMATION

The model considers a colocated MIMO radar with multiple transmitting and receiving antennas, orthogonal waveforms, far-field targets, mutual coupling, and noisy pulse observations. Mutual coupling is represented through transmitter and receiver coupling matrices, with the received data reformulated for estimation.

  • Radar configuration: The colocated MIMO radar uses M transmitting antennas, N receiving antennas, orthogonal signals, and K far-field point targets in one range cell.The DOA of target k is denoted θ_k, and the system processes pulse-indexed received signals.
  • Array model: The transmitter and receiver steering vectors depend on antenna spacing and wavelength, while mutual coupling is modeled by matrices C_T and C_R.The transmitter coupling matrix C_T is a symmetric Toeplitz matrix.
  • Signal processing: Matched filters separate the orthogonal transmitted signals before the received observations are sampled and assembled into matrices or vectors.The model includes additive noise and pulse-dependent target scattering coefficients.
  • Model reformulation: A Toeplitz-matrix lemma simplifies the received-signal expression involving the coupling matrices and steering-related quantities.The construction introduces coupling vectors and auxiliary matrices for the reformulated model.
  • Estimation model: The final received-data model collects P pulses and treats the mutual coupling vector, scattering-coefficient matrix, and noise variance as unknown quantities for DOA estimation.The paper estimates DOAs from the received matrix R.

A. The Off-Grid Sparse Model

The off-grid sparse model discretizes the detection area while correcting each target’s mismatch from its nearest grid point. It represents the received data using a sparse coefficient matrix, an off-grid vector, and unknown mutual coupling vectors.

  • Grid construction: The detection area is discretized into U angle grids to construct a dictionary matrix for sparse DOA estimation.The grid size is δ = |ζ_u+1 − ζ_u|.
  • Off-grid correction: A target DOA generally does not coincide with a discretized grid, creating the off-grid mismatch addressed by the model.Each target is approximated around its nearest discretized angle using a first-order representation.
  • Sparse representation: The sparse matrix X has columns sharing a common support set, representing targets across the pulse observations.Its nonzero entries encode target scattering coefficients.
  • Sparse model: The approximate received-signal model combines the dictionary, off-grid correction, sparse coefficients, mutual coupling, and additive noise.The model is written as R ≈ [Ψ + Ξ(diag{ν} ⊗ I_MN)](X ⊗ c) + N.
  • Parameter recovery: DOAs are recovered from the support of X, scattering coefficients from its nonzero entries, and coupling matrices from c_T and c_R.The unknown parameters include X, the off-grid vector ν, and the mutual coupling vectors.

B. Sparse Bayesian Learning-Based DOA Estimation Method

The paper develops SBLMC for DOA estimation with unknown mutual coupling, modeling sparse scattering coefficients and estimating nuisance parameters through an EM-based Bayesian procedure. The method incorporates Gaussian and conjugate priors to obtain tractable posterior and parameter updates, then extracts DOAs from the recovered spatial spectrum.

  • SBLMC formulation: SBLMC estimates target DOAs in MIMO radar while accounting for unknown mutual coupling between antennas.The method is formulated as a sparse Bayesian learning approach with received signals determined by radar parameters and signals.
  • Prior modeling: Gamma priors model inverse noise variance and simplify posterior analysis through conjugacy with the Gaussian noise model.The noise precision is defined as α_n = σ^-2_n, and its Gamma prior yields a Gamma posterior.
  • Prior modeling: The sparse matrix X is assigned a Gaussian prior because it is conjugate to the Gaussian likelihood and enables closed-form estimation expressions.The paper contrasts this choice with a Laplace sparsity prior, which is not conjugate to the Gaussian likelihood.
  • Prior modeling: Gaussian distributions are assumed for mutual coupling vectors, while the off-grid parameter ν is assigned a uniform prior.Separate diagonal covariance matrices parameterize the transmitter and receiver coupling-vector distributions.
  • EM estimation: An EM procedure treats X as a hidden variable and alternates posterior estimation with MAP updates for coupling, off-grid, precision, and noise-related parameters.The unknown parameters include c_T, c_R, ν, α_n, β, ϑ_T, and ϑ_R.
  • DOA extraction: After convergence, SBLMC forms the spatial spectrum of X and estimates DOAs by selecting peaks at the K largest values.The peak-search step converts the recovered sparse matrix into DOA estimates.

IV. SIMULATION RESULTS

Simulations evaluate SBLMC for DOA estimation under unknown mutual coupling, varying SNR, coupling strength, grid size, and computational cost. SBLMC achieves stronger estimation performance than comparison methods while retaining acceptable complexity, but its resolution remains aperture-limited.

  • Simulation setup: The simulations use Matlab R2017b with a maximum of 10^3 iterations and a stopping threshold of 10^-3.Experiments were run on a 2.9 GHz Intel Core i5 PC with 8 GB of RAM.
  • Algorithm: SBLMC estimates the spatial spectrum and DOAs by locating peaks corresponding to the K largest values, using ζ + ν as the discretized angle vector.Algorithm 1 initializes received-signal, dictionary, coupling, hyperparameter, and off-grid quantities before iterative updates.
  • Overall performance: −49.31 dB is the reported DOA estimation error for SBLMC, compared with −25.20 dB for OGSBI, −26.78 dB for BCS, and −28.94 dB for MUSIC.The comparison concerns the estimated DOA vector relative to the target DOA vector in radians.
  • Overall performance: SBLMC achieves the best DOA estimation performance in the unknown-coupling scenario, but has higher computational complexity than the comparison methods.The compared methods include an L1-norm CS method, a MUSIC-like method accounting for coupling, and a Capon-like method.
  • SNR dependence: For SNR ≥5 dB, SBLMC can reduce estimation error below −50 dB and approach the CRLB when SNR > 5 dB.When SNR ≤−10 dB, all methods perform similarly poorly; above −10 dB, comparison methods remain around −28 dB.
  • Mutual coupling: With coupling between adjacent antennas from −15 dB to −2 dB, SBLMC estimates coupling vectors and can achieve errors below −50 dB when coupling is less than −2 dB.For OGSBI and MUSIC, decreasing coupling reduces error from around −25 dB to around −50 dB, while BCS ranges from −34 dB to −28 dB.
  • Grid size: Reducing grid size from 10° to 2° decreases SBLMC estimation error from −8 dB to −50 dB.The tested adjacent-angle spacing ranges from 2° to 10°; BCS and OGSBI do not improve when grid size is below 6°.
  • Computational cost: SBLMC is comparable in computational time to OGSBI, while BCS is fastest and MUSIC generally has higher complexity because of fine detection-angle discretization.MUSIC discretizes [−80°, 80°] into 1.6 × 10^6 grids in the reported comparison.

V. CONCLUSIONS

The paper addresses DOA estimation in MIMO radar with unknown mutual coupling while concurrently accounting for off-grid sparse reconstruction. It proposes SBLMC and reports superior estimation performance with acceptable computational complexity, subject to a radar-aperture separation limit.

  • Contributions: SBLMC uses expectation-maximization to estimate target DOAs while jointly considering unknown mutual coupling and off-grid effects.The method also derives prior distributions for the unknown scattering-coefficient variances, mutual coupling vectors, off-grid vector, and noise variance.
  • Results: Simulation results show that SBLMC outperforms existing DOA estimation methods under unknown mutual coupling.The reported comparison is specifically for MIMO radar with unknown mutual coupling effects.
  • Results: SBLMC retains acceptable computational complexity while improving DOA estimation performance under both off-grid and mutual coupling effects.
  • Limitation: The minimum DOA separation of SBLMC remains limited by the radar aperture, as with sparse-based super-resolution methods.

APPENDIX A THE DERIVATION OF LIKELIHOOD FUNCTION L(cT)

The appendix derives the likelihood-function derivatives needed for estimating the mutual coupling vector c_T. It proceeds from the row-vector derivative to its individual entries and final derivative expression.

  • Derivative setup: The likelihood function L(c_T) is rewritten to derive its derivative with respect to the mutual coupling vector.
  • Entrywise derivation: The derivative ∂c_T is a row vector whose m-th entry is calculated separately.
  • Entrywise derivation: The m-th entry uses an M × 1 basis vector with a one in position m and zeros elsewhere.
  • Final expression: The appendix simplifies the entrywise derivative and obtains the final derivation of G1(c_T).
  • Final expression: The coupling-related derivative components are represented using c_R ⊗ e_M vectors indexed from 0 through M−1.
  • Final expression: The complete derivative ∂c_T is given in equation (91).

APPENDIX B THE DERIVATION OF LIKELIHOOD FUNCTION L(ν)

The appendix derives the likelihood-function equation used to obtain the off-grid vector ν. It defines the derivative structure, sets it to zero, and obtains the corresponding equation.

  • Derivative setup: The likelihood function L(ν) is rewritten to derive the equation for the off-grid vector ν.
  • Entrywise derivation: The derivative with respect to ν is a 1 × U row vector whose u-th entry is considered separately.
  • Final equation: Setting the derivative with respect to ν_u to zero yields equation (92) for obtaining ν.
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