$\ell _{p}$-MUSIC: Robust Direction-of-Arrival Estimator for Impulsive Noise Environments

Abstract

A family of algorithms, named ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -MUSIC, for direction-of-arrival (DOA) estimation in impulsive noise is proposed. The ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -MUSIC estimator adopts the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -norm (1 ≤ p 2) of the residual fitting error matrix as the objective function for subspace decomposition, rather than the Frobenius norm that is used in the conventional MUSIC method. Although the matrix ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -norm minimization based subspace decomposition will lead to a nonconvex optimization problem, two iterative algorithms are designed for achieving efficient solutions. The first algorithm is the iteratively reweighted singular value decomposition (IR-SVD), where the SVD of a reweighted data matrix is performed in each iteration. The second algorithm solves the nonconvex matrix ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -norm minimization by alternating convex optimization. Two complex-valued Newton's methods with optimal step size in each iteration are devised to solve the resulting convex problem. The convergence of the iterative procedure is also proved. Numerical results verify that the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -MUSIC methodology outperforms the standard MUSIC scheme and several existing outlier-resistant DOA estimation approaches in terms of resolution capability and estimation accuracy.