Elsevier

Digital Signal Processing

Volume 61, February 2017, Pages 15-25
Digital Signal Processing

Sparse source localization using perturbed arrays via bi-affine modeling

https://doi.org/10.1016/j.dsp.2016.06.004Get rights and content

Highlights

  • A co-array model with perturbed sensor locations is considered.

  • A novel self-calibration approach is proposed for underdetermined DOA estimation.

  • Assuming small perturbations, and on-grid DOAs leads to a bi-affine model.

  • Using coarray redundancies, the bi-affine model is reduced to a linear sparse model.

  • Reductions are derived for both ULA, and a robust version of coprime arrays.

Abstract

Non-uniform spatial sampling geometries, such as nested and coprime arrays, are provably capable of localizing O(M2) sources using only M sensors. However, such guarantees require the physical locations of the sensors to satisfy certain constraints, as dictated by the corresponding array geometries. In this paper, we consider the scenario when these constraints may be violated, leading to unknown perturbations on the locations of sensors. Such perturbations can have detrimental effect on the performance of virtual array based direction-of-arrival (DOA) estimation algorithms, since the perturbed virtual array will no longer be a uniform linear array (ULA). We propose a novel self-calibration approach for underdetermined DOA estimation with such arrays, that makes extensive use of the redundancies (or repeated elements) in the virtual array. Assuming small perturbations, and a sparse grid-based model for the DOAs, we extract a novel “bi-affine” model (affine in the perturbation variable, and linear in the source powers) from the covariance matrix of the received signals. The redundancies in the co-array are then exploited to eliminate the nuisance perturbation variable, and reduce the bi-affine problem to a linear underdetermined (sparse) problem in source powers, from which the DOAs can be exactly recovered under suitable conditions. This reduction is derived for both ULA and a newly-introduced robust version of coprime arrays, when the covariance matrix of the received signals is exactly known. Our approach is compared and contrasted with recently developed algorithms for blind gain and phase calibration (BGPC), whose signal model is fundamentally different from ours. We also provide an iterative algorithm to jointly solve for the DOAs and perturbation values when we can only estimate the covariance matrix using a finite number of snapshots.

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 O(M2) 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 O(M2).

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 1 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 xi represents the ith entry of a vector x. The symbol ȷ denotes the imaginary unit 1. The symbols (.),(.)T,(.)H stand for the conjugate, transpose, and hermitian, respectively. The symbols ∘, ⊙, ⊗ represent the Hadamard product, Khatri–Rao product, and Kronecker product, respectively. The symbol .F denotes the matrix Frobenius norm and vec(.) 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) θRK, θ=[θ1,θ2,,θK]T. Let y[l]CM be the vector of signals received by the M antennas, x[l]CK represent the emitted signals from K sources, and w[l] 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 σw2, 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 Aca(δ) using the first order Taylor series expansion as follows:(Aca(δ))(m1)M+m,ieȷ(dmdm)ωi(1+(δmδm)ȷωi), which can be also written in the matrix form asAca(δ)Aca,0+ΔAca,0ϒ where Aca,0 denotes the unperturbed co-array manifold, ϒ=ȷdiag(ω1,,ωNθ), and ΔRM2 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 (δ,p). 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 RY is available. In practice however, we can only estimate RY 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 Nθ=200 points on the grid. The DOAs are chosen uniformly between 60 and 60, and assigned to the closest point on the grid. The perturbations are assumed to be δ=αδ0. (Notice that, following the model given in Section 2, the sensor locations and the perturbations are normalized with respect to half of the wavelength λ/2.) 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 1 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.

References (20)

  • P.P. Vaidyanathan et al.

    Sparse sensing with co-prime samplers and arrays

    IEEE Trans. Signal Process.

    (2011)
  • P. Pal et al.

    Nested arrays: a novel approach to array processing with enhanced degrees freedom

    IEEE Trans. Signal Process.

    (2010)
  • R.T. Hoctor et al.

    The unifying role of the coarray in aperture synthesis for coherent and incoherent imaging

    Proc. IEEE

    (1990)
  • P. Pal et al.

    Coprime sampling and the music algorithm

  • Y.D. Zhang et al.

    Sparsity-based DOA estimation using co-prime arrays

  • P. Pal et al.

    Correlation-aware techniques for sparse support recovery

  • P. Pal et al.

    On application of lasso for sparse support recovery with imperfect correlation awareness

  • P. Pal et al.

    Correlation-aware sparse support recovery: Gaussian sources

  • L. Seymour et al.

    Bearing estimation in the presence of sensor positioning errors

  • Y.-M. Chen et al.

    Bearing estimation without calibration for randomly perturbed arrays

    IEEE Trans. Signal Process.

    (1991)
There are more references available in the full text version of this article.

Cited by (16)

  • Calibrating nested sensor arrays for DOA estimation utilizing continuous multiplication operator

    2020, Signal Processing
    Citation Excerpt :

    On the other hand, they also lack a theoretical analysis on their performances. Another method for calibrated nonuniform linear arrays is investigated in [25], which mainly consider the case that the location of the sensors are perturbed, whereas the gain and phase of the signals received from the sensors are ideal. In this paper, we also address the problem of calibrating nested arrays with model errors, where only gain-phase errors are considered, and the other array uncertainties are assumed to be ideal.

  • Joint direction of arrival estimation and array calibration for coprime MIMO radar

    2019, Digital Signal Processing: A Review Journal
    Citation Excerpt :

    The DOA estimation performance will degrade seriously in the presence of mutual coupling [23,24] or array gain-phase errors [25]. With small perturbations or few unknown parameters, the parameters can be obtained via bi-affine modeling [48], sparse total least squares (STLS) [49], non-convex optimization [50] or minimal solver [51], but it is shown that there are no sufficient equations to recover all the unknown gain-phase errors when the perturbations are large [49]. Therefore, using additional well-calibrated antennas is an effective way to obtain joint DOA and gain-phase error estimation [52,53], and it has also been adopted in MIMO radar [54,55].

  • Array auto-calibration using a generalized least-squares method

    2019, AEU - International Journal of Electronics and Communications
    Citation Excerpt :

    Therefore, it can be repeated easily and periodically. A large number of auto-calibration algorithms have been proposed for GPM calibration and MC calibration [11–17]. In MC calibration, Most of these methods are designed for only one kind of array, e.g. [18] for uniform linear arrays (ULA), and the RARE family algorithms [19,20] for uniform circular arrays (UCA).

  • Sparsity based off-grid blind sensor calibration

    2019, Digital Signal Processing: A Review Journal
    Citation Excerpt :

    In [34], compressive sensing based array self-calibration algorithms are proposed for direction finding in which the signal parameters and unknown complex sensor gains are estimated in an alternating manner. In [35], a sparse DOA estimation approach is proposed using arrays with unknown perturbations on sensor locations. DOA and perturbation values are solved iteratively.

View all citing articles on Scopus

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.

View full text