Sparse source localization using perturbed arrays via bi-affine modeling
Introduction
Directions-of-arrival (DOA) estimation of energy emitting sources using sensor arrays finds important application in problems ranging from target localization in radar system to speech enhancement using microphone arrays. In recent times, new sparse array geometries, such as coprime [1] and nested arrays [2], have been proposed that are capable of identifying sources using just M sensors, exploiting the enhanced degrees of freedom offered by their difference co-arrays (or, virtual arrays) [3]. In order to exploit the enhanced degrees of freedom, so far, two main approaches for DOA estimation have been proposed: 1) Subspace methods, 2) Sparsity based methods. In the former approach, which is based on the MUSIC algorithm, the subspace properties of the spatially smoothed co-array manifold is used to estimate the DOAs [4]. However, in the latter approach, the range of all possible directions is discretized into a grid, and then the DOA estimation problem is reformulated as a sparse representation problem [5], [6], [7], [8]. We review this approach in more details in Section 2.
It is well known that array imperfections such as gain and/or phase error, perturbations in sensor locations, and mutual coupling, can significantly degrade the performance of DOA estimation algorithms [9], [10]. This is mainly due to the strong dependence of these algorithms on the accurate knowledge of the underlying array manifold. In this paper, we consider the sensor location error as the only imperfection associated with the physical array, i.e., we assume that the sensor locations are perturbed from their nominal positions. The problem of DOA estimation using such perturbed arrays has been well studied for more than two decades. Existing approaches mostly treat the perturbations as unknown but deterministic parameters, and then estimate these parameters jointly with the DOAs. Classical methods such as [10], [11], [12], [13], resolve array uncertainties using eigenstructure-based methods, or variants of the maximum-likelihood approach. Recently, [14] proposed a unified framework for different kind of array imperfections, and proposed a Bayesian approach for array calibration and DOA estimation. However, these approaches mostly work for an overdetermined signal model (fewer sources than sensors), primarily because many of them consider a uniform linear array.
In recent times, the problem of blind gain and phase calibration (BGPC) has been formulated as a bilinear problem [15], which in turn, can be recast as a convex optimization problem, using the idea of “lifting” [16], [17], [18]. However, such a formulation does not consider the concept of co-array, and, hence their guarantees are not applicable for an underdetermined signal model where the number of sources can possibly be .
In contrast, the authors in [19], studied the effect of co-array geometry on the BGPC problem and proposed a new self-calibration algorithm for nested arrays in presence of gain/phase errors. Their approach builds on and extends the method in [12], which was originally proposed for a ULA. However, in this paper, we consider perturbations in sensor locations, which gives rise to a signal model, which is distinctly different from that considered in [19]. In BGPC problems, the gain and/or phase of the sensors are unknown, and the goal is to resolve both unknown gain and/or phase and the DOAs. In our case, we assume that the phase and gain of the signals received from the sensors are ideal, but the location of the sensors are perturbed. We will compare the signal model defined for gain/phase error, which has been studied in [19], against sensor location error in Section 2.2, and establish important differences between them.
Since the self calibration algorithm developed in [19] cannot be directly applied to our case, we follow a different approach in this paper. We assume that the perturbations are small, so that we can approximate the coarray manifold using its first order Taylor series expansion. This formulation leads to a “bi-affine” model, which is linear in source powers, and affine in the perturbation variable. We show that it is possible to recover the DOAs even in presence of the nuisance perturbation variables, via a clever elimination of variables. By exploiting the pattern of repeating elements, it is possible to reduce the said bi-affine problem to a linear underdetermined (sparse) problem in source powers, which can be efficiently solved using minimization. We establish precise conditions under which such reduction is possible, for both ULA and a robust version of coprime arrays.
The paper is organized as follows. In Sec. 2 we compare and contrast different kinds of array imperfections (gain/phase error, sensor location perturbation) in terms of their effects on the difference co-array. In Sec. 3, we introduce the bi-affine model for DOA estimation with perturbed sensors. Sec. 4, establishes a transformation under which we can write the bi-affine problem as a linear problem in source powers, via elimination of the unknown perturbation variable. The specific details of this transformation depend on the array geometry. In Sec. 5, we review an iterative algorithm proposed in [20] to jointly solve for DOAs and source powers when we only have an estimate of the covariance matrix. Numerical simulations are conducted in Sec. 6. Sec. 7 concludes the paper.
Notation: Throughout this paper, matrices are represented by upper case bold letters, and vectors by lower case bold letters. The symbol represents the ith entry of a vector x. The symbol ȷ denotes the imaginary unit . The symbols stand for the conjugate, transpose, and hermitian, respectively. The symbols ∘, ⊙, ⊗ represent the Hadamard product, Khatri–Rao product, and Kronecker product, respectively. The symbol denotes the matrix Frobenius norm and represents the vectorized form of a matrix.
Section snippets
Signal model for gain/phase error vs location errors
Consider a linear array of M antennas impinged by K narrow-band sources with unknown directions of arrival (DOA) , . Let be the vector of signals received by the M antennas, represent the emitted signals from K sources, and be the additive noise (all corresponding to the lth time snapshot). The source signals are assumed to be zero mean, and pairwise uncorrelated, and the noise vector is zero mean, i.i.d. with variance , and uncorrelated from the
The bi-affine model
In this section, we derive a bi-affine model from the covariance matrix (5) corresponding to a sensor array with perturbed locations. Our main assumption is that δ is small enough so that we can approximate the coarray manifold using the first order Taylor series expansion as follows: which can be also written in the matrix form as where denotes the unperturbed co-array manifold, , and is
Source localization: bi-affine to linear transformation
Under the grid-based model, the DOAs can be estimated from the support of the sparse vector p that is a solution to the bi-affine system of equations (13). In general, (13) can admit multiple solutions in the variables . While the column rank of a matrix describing a linear system of equations determines if it admits a unique solution, to the best of our knowledge, no such general condition exists for a bi-affine (or even bi-linear) system which can be used as a test for existence of
Iterative algorithm for finite snapshots and noise
While our main results show that it is fundamentally possible to eliminate the nuisance variable δ and solve for DOAs, they are derived under the assumption that the ideal covariance matrix is available. In practice however, we can only estimate using a finite number of snapshots. For the estimated covariance matrix, the technique for variable elimination in the proofs of Theorem 1, Theorem 2 may not be robust (although it works perfectly for the ideal covariance matrix). This prompts us
Simulations
In this section, we conduct three different sets of numerical experiments to validate our theoretical claims. In all the simulations, we assume points on the grid. The DOAs are chosen uniformly between and , and assigned to the closest point on the grid. The perturbations are assumed to be . (Notice that, following the model given in Section 2, the sensor locations and the perturbations are normalized with respect to half of the wavelength .) In the first and second sets
Conclusion
In this paper, we investigated the robustness of coprime arrays to unknown perturbations on the locations of sensors. We assumed that the perturbations are small and developed a bi-affine model in terms of the unknown perturbations and the source powers. We used the redundancies of the difference coarray to eliminate the nuisance variables, and reduce the bi-affine problem to a linear underdetermined (sparse) problem in source powers, which can be solved using minimization. We derived this
Acknowledgements
This work was supported in parts by the University of Maryland, College Park, and the Department of Defense.
Ali Koochakzadeh was born in Tehran, Iran, on November 1, 1990. He received the B.S. degree in electrical engineering from Sharif University of Technology, in 2014. He is currently pursuing the M.S. degree in the field of digital signal processing at University of Maryland, College Park. His research interests include digital signal processing and applications in sensor and array signal processing, and compressive sensing.
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Ali Koochakzadeh was born in Tehran, Iran, on November 1, 1990. He received the B.S. degree in electrical engineering from Sharif University of Technology, in 2014. He is currently pursuing the M.S. degree in the field of digital signal processing at University of Maryland, College Park. His research interests include digital signal processing and applications in sensor and array signal processing, and compressive sensing.
Piya Pal is an Assistant Professor of Electrical and Computer Engineering at the University of Maryland, College Park. She received her Bachelors in Technology degree in Electronics and Electrical Communication Engineering from Indian Institute of Technology, Kharagpur, India in 2007, and her Ph.D. in Electrical Engineering from Caltech in 2013, where her Ph.D. thesis won the Charles and Ellen Wilts Prize for Outstanding Doctoral Thesis in Electrical Engineering. In January 2014, she joined the Department of Electrical and Computer Engineering at the University of Maryland, College Park, where she is also affiliated with the Institute for Systems Research. Her research interests span compressive and structured sampling, high dimensional statistical signal processing with applications in radar and sensor array processing high resolution imaging, tensor methods, and statistical learning. She is an elected member of the IEEE SAM Technical Committee of the IEEE Signal Processing Society, and Sensory Systems Technical Committee of the IEEE Circuits and Systems Society. She is a recipient of the 2016 NSF CAREER Award.