Circuits and Systems CAS-2014-00 Mekelweg 4, 2628 CD Delft The Netherlands http://ens.ewi.tudelft.nl/ M.Sc. Thesis Compressive Sensing for Near-field Source Localization Keke Hu Abstract Near-field source localization is an important aspect in many diverse areas such as acoustics, seismology, to list a few. The planar wave assumption frequently used in far-field source localization is no longer valid when the sources are in the near field. Near-field sources can be localized by solving a joint direction-of-arrival and range estimation problem. The original near-field source localization problem is a multi-dimensional non-linear optimization problem which is computationally intractable. In this thesis we study address two important questions related to near-field source localization: (i) Sparse reconstruction techniques for joint DOA and range estimation using a grid-based model. (ii) Matching the sampling grid for off-grid sources. In the first part of this thesis, we use a grid-based model and by further leveraging the sparsity, we can solve the aforementioned problem efficiently using any of the off-the-shelf ℓ1 -norm optimization solvers. When multiple snapshots are available, we can also exploit the crosscorrelations among the symmetric sensors of the array and further reduce the complexity by solving two sparse reconstruction problems of lower dimensions instead of a single sparse reconstruction problem of a higher dimension. In the second part of this thesis, we account scenarios where the true source locations are not on the grid resulting in a grid mismatch. Using the first-order Taylor approximation, we model the grid mismatch as a perturbation around the sampling grid. Based on the grid mismatch model, we propose a bounded sparse and bounded joint sparse recovery algorithms to localize near-field sources. Faculty of Electrical Engineering, Mathematics and Computer Science Compressive Sensing for Near-field Source Localization Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Telecommunications by Keke Hu born in Yichang, P. R. China This work was performed in: Circuits and Systems Group Department of Electrical Engineering Faculty of Electrical Engineering, Mathematics and Computer Science Delft University of Technology Delft University of Technology c 2014 Circuits and Systems Group Copyright ⃝ All rights reserved. Delft University of Technology Department of Electrical Engineering The undersigned hereby certify that they have read and recommend to the Faculty of Electrical Engineering, Mathematics and Computer Science for acceptance a thesis entitled “Compressive Sensing for Near-field Source Localization” by Keke Hu in partial fulfillment of the requirements for the degree of Master of Science. Dated: January 30th, 2014 Chairman: Prof. dr. ir. Geert Leus Advisor: ir. Sundeep Prabhakar Chepuri Committee Members: dr. ir. Richard Heusdens dr. ir. Radmila Pribić iv Abstract Near-field source localization is an important aspect in many diverse areas such as acoustics, seismology, to list a few. The planar wave assumption frequently used in farfield source localization is no longer valid when the sources are in the near field. Nearfield sources can be localized by solving a joint direction-of-arrival and range estimation problem. The original near-field source localization problem is a multi-dimensional non-linear optimization problem which is computationally intractable. In this thesis we study address two important questions related to near-field source localization: (i) Sparse reconstruction techniques for joint DOA and range estimation using a gridbased model. (ii) Matching the sampling grid for off-grid sources. In the first part of this thesis, we use a grid-based model and by further leveraging the sparsity, we can solve the aforementioned problem efficiently using any of the off-the-shelf ℓ1 -norm optimization solvers. When multiple snapshots are available, we can also exploit the cross-correlations among the symmetric sensors of the array and further reduce the complexity by solving two sparse reconstruction problems of lower dimensions instead of a single sparse reconstruction problem of a higher dimension. In the second part of this thesis, we account scenarios where the true source locations are not on the grid resulting in a grid mismatch. Using the first-order Taylor approximation, we model the grid mismatch as a perturbation around the sampling grid. Based on the grid mismatch model, we propose a bounded sparse and bounded joint sparse recovery algorithms to localize near-field sources. v vi Acknowledgments I would like to thank the people who have helped and supported me thoughout my project. The progress I made during my thesis project is inseparable from their help and support. First of all, I would like to express my sincere gratitude to my supervisor ir. Sundeep Prabhakar Chepuri for his continuous support of my thesis study and making me progress during the project. I trouble him a lot, and sometimes I even entangled in a minor question, but he alway provides inspired suggestions immediately with patience. From Sundeep, I learnt how to think critically and how to manage a project. Thank Sundeep, and if without his help this work can never be accomplished. I am grateful to my professor, Prof. dr. ir. Geert Leus for giving me the opportunity to work in this group and for his time, patient, encouragement, support, and invaluable guidance. Also, I would like to thank dr. ir. Richard Heusdens and dr. ir. Radmila Pribić for being my committee members. It is my honor. I would like to thank Sumeet Kumar, Shahzad Sarwar Gishkori, Dyonisius Dony Ariananda and Venkat Roy for their help and advice during my thesis project. And I would like to thank all the member of CAS group, they are enthusiasm, friendly and kindness. Finally, I would like to thank my family and friends for their support, help and encouragement. These are the source of my strength. And thank Ying He for her understanding and support. Thank my mom, if without her I won’t come to this beautiful world. And I miss her very much. Keke Hu Delft, The Netherlands January 30th, 2014 vii viii Contents Abstract v Acknowledgments vii Notations xi 1 Introduction 1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Thesis outline and contributions . . . . . . . . . . . . . . . . . . . . . . 1 1 2 3 2 Background 2.1 Compressive sensing . . . 2.2 ℓ1 -minimization algorithm 2.3 ℓ1 -SVD . . . . . . . . . . 2.4 Block sparse recovery . . . . . . . 5 5 5 6 7 3 Signal Model 3.1 Spherical wavefront model . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fresnel approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 12 4 Sparse recovery techniques for near-field source localization: based model 4.1 Grid-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Gridding on spherical wavefront model . . . . . . . . . . . 4.1.2 Gridding on Fresnel approximation model . . . . . . . . . 4.2 Two-step estimator with multiple snapshots . . . . . . . . . . . . 4.2.1 Step-1: DOA estimation . . . . . . . . . . . . . . . . . . . 4.2.2 Step-2: range estimation . . . . . . . . . . . . . . . . . . . 4.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 15 15 16 17 18 18 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grid. . . . . . . . . . . . . . . . . . . . . 5 Grid matching for near-field source localization using sparse recovery techniques 5.1 Grid mismatch and its effect . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Grid mismatch model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Sparse recovery for off-grid sources with single snapshot . . . . . . . . . 5.3.1 Bounded sparse recovery . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Bounded joint sparse recovery . . . . . . . . . . . . . . . . . . . 5.4 Sparse recovery for off-grid sources with multiple snapshots . . . . . . . 5.4.1 Step-1: Grid matching in DOA domain . . . . . . . . . . . . . . 5.4.2 Step-2: grid matching in range domain . . . . . . . . . . . . . . 5.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 23 23 24 26 26 27 28 28 30 31 6 Conclusions and future work 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Suggestion for future research . . . . . . . . . . . . . . . . . . . . . . . 37 37 38 A Performance analysis on sparse recovery A.1 The RIP and the uniqueness of sparse recovery . . . . A.2 Performance of block sparse recovery . . . . . . . . . A.3 Unbounded joint sparse recovery and its performance A.4 Performance analysis of bounded sparse recovery . . 39 39 40 40 43 x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations Single element x x Vector of element 1N (0N ) Vector of ones (zeros) X Matrix IN An identity matrix of size N ∗ (•) Complex conjugate (•)T Transposition H (•) Hermitian transposition (complex conjugate transposition) R Real number set notation C Complex number set notation E{•} Expectation operation vec(•) Vectorize a matrix diag(•) Put an vector on the main matrix ⊙ Hadamard product ⊚ Khatri-rao product ∥x∥0 The ℓ0 (-quasi) norm that indicates the number of non-zero entries of the vector x ∑ ∥x∥1 The ℓ1 norm of an N × 1 vector x equals to √ N n=1 |xn | ∑N 2 ∥x∥2 The ℓ2 norm of an N × 1 vector x equals to n=1 |xn | ∥x∥2,1 The mixed ℓ2 /ℓ1 norm of vector x, which can be divided into N sub-blocks ∑ xn , is equivalent to N n=1 ∥xn ∥2 √∑ ∑N M 2 ∥X∥F The Frobenius norm of an M ×N matrix X equals to m=1 n=1 |xmn | 2 σs,k Variance of k-th signal σw2 Variance of additive noise xi xii List of Figures 1.1 Acquisition of a micro-seismic data for seismic imaging. . . . . . . . . . 1 2.1 2.2 Multiple snapshot sparse recovery. . . . . . . . . . . . . . . . . . . . . . Comparison of block sparse recovery with traditional sparse recovery. SNR = 30 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1 A linear array receiving a signal from a near-field point source. . . . . . 11 4.1 4.2 Joint DOA and range estimation. . . . . . . . . . . . . . . . . . . . . . Two-step estimator for near-field sources localization. . . . . . . . . . . 20 21 5.1 5.2 5.3 5.4 5.5 Effect of grid mismatch in source localization . . . . . . . . . . . . . . . Near-field source localization via mismatched sensing grid. . . . . . . . DOA and range estimation via mismatched sensing grid . . . . . . . . . Multiple source localization via mismatched sensing grid. . . . . . . . . DOA and range estimation for multiple sources via mismatched sensing grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grid matching via Two-step algorithm . . . . . . . . . . . . . . . . . . 24 32 33 34 A.1 Performance of R-BJS versus SNR. . . . . . . . . . . . . . . . . . . . . A.2 Performance of R-BJS versus GoE. . . . . . . . . . . . . . . . . . . . . 47 48 5.6 xiii 9 35 36 xiv 1 Introduction In this thesis, we study the near-field source localization problem using sparse recovery techniques. This chapter provides the motivation behind this thesis, literature review, and contributions. 1.1 Literature review Source localization is an important aspect for target tracking and location-aware services, and has many applications in the field of seismology, acoustics, radar, sonar, and oceanography. In Fig. 1.1, an example of source localization applied to seismic imaging is shown. Figure 1.1: Schematic diagram showing an acquisition of micro-seismic data for seismic imaging. Image courtesy of John Logel [1]. Direction-of-arrival (DOA) estimation of narrowband signals is an extensively studied topic [2, 3]. DOA estimation can be categorized into two types, based on the distance between the source and the antenna array: (a) far-field (e.g., r ≫ 2D2 /λ), 1 and (b) near-field source localization, where r is the range between the source and the phase-reference of the array, D is the array aperture, and λ is the wavelength of the source signal. In far-field source localization, the wavefront of the signal impinging on the array is assumed to be planar [2, 4]. However, the curvature of the wavefront is no longer negligible when sources are located close to the array (i.e., in the near field or Fresnel region) like the scenario shown in Fig. 1.1. Therefore, the algorithms that depend on the planar-wave assumptions for DOA estimation are no longer valid. Such problems often arise in practical term, for example in underwater sources localization by using a vector hydrophone [5], tracking a speaker by using microphones array [6]. Unlike the far-field source localization, near-field localization is traditionally done by a joint DOA and range (distance between the source and the phase-reference of the array) estimation. Traditional approaches to the near-field localization problem extend the techniques like multiple signal classification (MUSIC) to a two-dimensional field. In [7], the MUSIC method used in a two-dimensional (DOA, range) field achieves a high resolution, but, suffers from a high computational load required for searching a twodimensional range and bearing spectrum. Moreover, the performance of the MUSIC algorithm deteriorates at low SNRs and when the sources are correlated. In [8], the wavefront is assumed to be piece-wise linear, and the uniform linear array (ULA) is divided into several subarrays. The wavefront of the signal impinging on each subarray is then assumed to be planar. By using the method proposed in [2, 4] at each subarray, the location can be estimated after gathering the DOA of each subarray. In [9, 10, 11], instead of using the piece-wise linear approximation, a quadratic approximation (the so-called Fresnel approximation) of the wavefront is made, which makes the wavefront neither planar nor spherical. The phase delay is no longer linear with the position of the antenna element (like in [2]), instead, it varies quadratically with the array position and it is characterized by the azimuth (DOA) and range of the sources (see [9] for more details). However, the array has to still satisfy the Nyquist sampling rate criterion in space, i.e., the spacing between the two adjacent antennas need to less than half a wavelength. 1.2 Motivation The previous works on near-field localization problem can provide accurate results or relative satisfying resolution but all of them are using the conventional data acquisition methods based on the Nyquist sampling rate theorem [7, 8, 9, 10, 11], which limits their performance. Since in many application the sources can be assumed as point sources and hereby the spectrum of energy versus location can be viewed as sparse with the number of sources is small. Based on this assumption, we try to localize the source locations via a perspective of sparse signal recovery. Generally the sampled measurement used for recovery is less than Nyquist sampling rate, in which case the framework is called compressive sensing (CS) [12]. In [13], the CS framework is used for DOA estimation of far-field sources. The CS framework can provide better resolution than conventional approaches [2, 3, 4] from a lesser number of samples but suffers from the computational load for multiple snapshot case. To avoid the unnecessary computational load, [14, 15] developed a tractable subspace-based ℓ1 -SVD method for multiple 2 snapshot far-field source localization which can also provide super-high resolution with low complexity. Thus, we wonder whether if the sparse recovery techniques can be applied for the near-field source localization problem also. Even though the source localization based on the sparse signal reconstruction (SSR) can perform better than the classical source localization methods, its performance is tremendously affected by the choice of the estimation grid. The grid mismatch problem is described in [16], and an error-in-variables (EIV) model is proposed in [17, 18, 19]. In [20], a joint-sparse recovery [21, 22] was used to further exploit the underlying structure with grid mismatch. Using the same concepts, we try to exploit the off-grid compressed sensing for near-field source localization. 1.3 Thesis outline and contributions In this thesis, we use the CS framework to jointly estimate the DOAs and ranges of multiple narrowband near-field sources. We address two scenarios in this thesis: in the first part, we illustrate how to localize multiple near-field sources using the SSR perspective under assumption that all the desired sources are exactly on the sampling grid; in the second part, we extend the near-field linear model to an EIV model to include the off-grid (all the desired sources need not be on the sampling grid) effect and propose a number of estimators for the same. Chapter 2: Background In this chapter, a brief introduction to CS and its applications related to our topic is provided. Firstly, we will introduce some terminologies used in CS. Then, we discuss sparse signal recovery via a ℓ1 -norm regularized least-squares algorithm. Next, we introduce ℓ1 -SVD, which we use when multiple snapshots are available. At last, we introduce the concept of block sparse recovery. Chapter 3: Signal model In this chapter, we develop the signal model that we use throughout this thesis. It includes the notation and terminologies that will be used in this thesis. We will also provide spherical wavefront model, quadratic wavefront model (based on the Fresnel approximation). Chapter 4: Sparse recovery techniques for near-field source localization: Grid-based model In this chapter, we propose a grid-based model and by further leveraging the sparsity, we solve the aforementioned problem efficiently using ℓ1 -regularized least-squares. When multiple snapshots cases are available, we exploit the cross-correlations among the symmetric sensors to further reduce the complexity by solving two sparse reconstruction problems of lower dimensions instead of a single sparse reconstruction problem of a higher dimension. 3 Chapter 5: Grid matching for near-field source localization using sparse recovery techniques In this chapter, we consider a scenario in which the true source locations are not on the sampling grid. By using the Taylor expansion, we model the grid mismatch effect. Based on the grid mismatch model, we propose a sparse recovery and a joint sparse recovery estimators to localize the near-field sources. Chapter 6: Conclusions In this chapter, we summarize this thesis and provide some suggestions for future research. 4 2 Background The compressive sensing (CS) theory is a technique to reconstruct signals at a rate that could be significantly below the Nyquist rate under the assumption that the desired signals are compressible or sparse [12]. This chapter introduces the concept of CS briefly and along with ℓ1 -norm minimization, ℓ1 -SVD, and block sparse recovery. 2.1 Compressive sensing Before we discuss the compressive sensing problem, we will introduce the concept of compressive signals. In general, we can say a signal vector x is a K-sparse vector if it has only K non-zero entries of x. Mathematically, it can be represented as ∥x∥0 ≤ K, where ∥x∥0 indicate the number of non-zero entries. We can say a signal x is compressible if it is K-sparse vector or nearly sparse (with K relatively larger coefficients compared to the rest). Given a measurement vector y ∈ CM and a sensing matrix A ∈ CM ×N , we want to recovery the original vector x ∈ CN acquired via a linear measurement process y = Ax + w, where w ∈ CM is the additive noise vector. As we known, if we want to recovery the signal x, we normally need to acquire more than N measurements, which means if M ≥ N there exists a unique solution of x. When we have fewer measurements, i.e., M ≪ N , if we know that the desired signal x is a compressive signal, with a high probability we can still recovery the signal vector x by solving an optimization problem of form x̂ = arg min x∈CN ×1 ∥x∥0 s.t. ∥y − Ax∥2 ≤ ϵ, (2.1) the above framework of reconstructing x is called CS. The wide matrix A is often referred to as the sensing matrix. The optimization problem in (2.1) is non-convex due to the ℓ0 -norm cost function and is mathematically intractable. In fact, for a general matrix A finding an approximate solution of (2.1) is NP-hard. Thus, in practical we do not propose to use this strategy to solve CS problem due to its intractability. 2.2 ℓ1 -minimization algorithm As we mentioned before, ∥x∥0 is non-convex, and the ℓ0 -norm minimization is an NPhard problem. In order to transform such a problem into a computational tractable one, it is proposed to relax the ℓ0 -norm cost with the ℓ1 -norm, and hence transforming (2.1) to a convex optimization problem of the form x̂ = arg min x∈CN ×1 ∥x∥1 s.t. ∥y − Ax∥2 ≤ ϵ. 5 (2.2) The optimization problem in (2.2) can alternatively be written in form x̂ = arg min x∈CN ×1 ∥y − Ax∥22 + µ∥x∥1 , (2.3) where parameter µ is used to control tradeoff between the sparsity and the residual norm. In this thesis, we use LASSO, a shrinkage and selection method for linear regression in [23], to solve the ℓ1 regularized optimization problem in (2.3) . In fact, it provides accurate solution of x only under the two conditions that (1) the x is sufficiently sparse and (2) the sensing matrix A satisfies an important condition named restricted isometry property (RIP) [24]. In conclusion, the ℓ1 minimization approach is one of the numerous algorithms applied for CS recovery problem and provides a powerful framework to recovering sparse signals. The advantage of ℓ1 minimization algorithm is that it provides a provably accurate solution and relaxes the CS problem to a convex optimization problem which is more tractable and can be solved by many existing efficient and accurate solvers. 2.3 ℓ1 -SVD When multiple snapshot case available, the overcomplete representation is extended and reformulated in the following form y(t) = Ax(t) + w(t), t = 1, 2, . . . , T, (2.4) where y(t) ∈ CM , x(t) ∈ CN is a K-sparse vector and the positions of non-zero entries will not change at different t, T ≫ K, and A ∈ CM ×N . One direct approach to reconstruct the sparse signal sequence is to treat each snapshot separately. We separate the problem sample by sample and solve each problem indexed by t as if each problem at index t is a single snapshot problem solved as a ℓ1 -norm minimization problem given by x̂(t) = arg min x(t)∈CN ×1 ∥y(t) − Ax(t)∥22 + µ∥x(t)∥1 , t = 1, 2, . . . , T, (2.5) then we will have a set of T solutions. This algorithm suffers too much computational load and the position of non-zero entries of the estimate may be different at different index t. In Fig. 2.1b, the result of the estimate separately (2.5) is shown. The position of non-zero entries changes for different sample due to the fact that there is no dependence between each sub-problem at different index t. The ℓ1 -SVD algorithm we will introduce in this section is an efficient regularization framework to find the related columns of A to the non-zeros entries. It is based on the singular value decomposition (SVD) of the measurement matrix and the idea is to decompose the measurement matrix into signal and noise subspaces. Stacking (2.4) into matrix Y = AX + W, where X is a row-wise sparse matrix. Consider the singular value decomposition of Y: Y = UΛVH . 6 Since only K rows of X have non-zero values, then in noiseless case the rank of Y should be K. Accordingly, the first K columns of matrix U correspond to the non-zero singular value. While with noise, the Y has full rank and the largest K singular vectors will correspond to the signal subspace and the rest are related to the noise. Therefore, we can generate a reduced dimension matrix YSV ∈ CM ×K , which only contains the most of the signal power YSV = UΛDK = YVDK , H K×T , IK is a K ×K identity matrix, and 0K is a K ×(T −K) where DK = [IK , 0H K] ∈ R matrix of all zero value. Here, we multiple both size of equation Y = AX + W with VDK , we obtain YSV = AXSV + WSV , (2.6) where XSV = XVDK and WVDK . The above equation can be expressed in the vector-form (column by column) as ysv (k) = Axsv (k) + wsv (k), for k = 1, . . . , K. Here, each column corresponds to a signal subspace singular vector. The reduced data matrix is only spatially sparse, and not in terms of the singular vector index k. In order to take into account this effect, we use a different prior obtained by computing (ℓ2 ) = the ℓ2 -norm of the singular values of a particular spatial index of xsv , i.e., xm,sv √ ∑K 2 k=1 (xm,sv (k)) , for m = 1, . . . , M . Note that, the ℓ2 -norm can be computed for all time-samples instead of only the signal subspace singular vectors, however, the former technique adds more computational complexity especially when T ≫ K. Now, we can find the range by minimizing (ℓ2 ) ∥Ysv − AXsv ∥2F + µsv ∥xsv ∥1 , (ℓ ) (ℓ ) (ℓ ) 2 2 , . . . , xM,sv ]T , and the parameter µsv controls the spatial sparsity. In where xsv2 = [x1,sv Fig 2.1c, we can see that ℓ1 -SVD performance better than the sparse recovery method based on ℓ1 -minimization separately. Since ℓ1 -SVD enforces the row-sparse structure while separate-based method only enforce the sparse elements themselves. 2.4 Block sparse recovery In many scenarios, the non-zero elements of a sparse signal appear as “clusters”, i.e., non-zero entries appear in cluster or in blocks. Estimators taking into account such underlying structures can often perform better than the conventional ℓ1 -norm regularized least-squares algorithm. We firstly introduce the concept of block sparsity. Let us consider a vector x that can be divided into N sub-blocks, x = [xT1 , xT2 , . . . , xTN ]T , out of which only K subblocks have non-zero values, then we say that vector x is a block sparse vector. Define dn is a set that contains all the indices of the entries of x that correspond to xn and define the set I = {d1 , . . . , dN }, e.g. suppose x is of 5 that can be divided into 2 sub-blocks, x1 and x2 , xT xT z }|1 { z }|2 { T x = [ x1 , x2 , x3 , x4 , x5 ] 7 l1−norm minimization sample by sample l1−SVD 20 40 40 40 60 60 60 80 100 120 Entry location of x(t) 20 Entry location of x(t) Entry location of x(t) True signal sequence 20 80 100 120 80 100 120 140 140 140 160 160 160 180 180 180 200 200 2 4 6 8 10 12 Time index, t 14 16 18 20 200 2 4 6 8 (a) 10 12 Time index, t 14 16 18 20 2 4 (b) 6 8 10 12 Time index, t 14 16 18 20 (c) Figure 2.1: Multiple snapshot sparse recovery. x(t) is a sparse vector and 40-th and 150-th elements are non-zero. SNR = 20 dB and T = 20. (a) True signal sequences. (b) ℓ1 -minimization sample by sample. (c) ℓ1 -SVD algorithm. then d1 = {1, 2} and d2 = {3, 4, 5}, and we can say x is divided via I = {d1 , d2 }. Mathematically, a K-block sparse vector x can be represented as ∥x∥0,I ≤ K where ∥x∥0,I = N ∑ I(xn ) n=1 where I(xn ) is an indicator function, i.e. I(xn ) = 1 if xn with non-zero value otherwise it is 0. An example of I(xn ) is that I(xn ) = ∥∥xn ∥2 ∥0 . The block sparse recovery is to reconstruct a K-block sparse vector through measurement vector y acquired via a linear equation y = Ax + w. One approach to block sparse recovery is to enforce the block sparsity of x via a mixed ℓ2 /ℓ1 minimization. We define the mixed ℓ2 /ℓ1 norm over the index set I = {d1 , d2 , . . . , dN } as ∥x∥2,1 = N ∑ ∥xn ∥2 . n=1 The suggested algorithm is given as x̂ = arg min x∈C ∥x∥2,1 s.t. ∥y − Ax∥2 ≤ ϵ, (2.7) which it is equivalent to x̂ = arg min x∈CN ×1 ∥y − Ax∥22 + µ∥x∥2,1 . (2.8) To discuss the uniqueness and stability of the block sparse recovery, we would like to introduce the definition of block restricted isometry property [21, 22], which can be view as a extension to the standard RIP. The definition of B-RIP indicates that there exists a unique K-block sparse solution if the sensing matrix A satisfies the BRIP of order 2K. However, we should note that a K-block sparse vector x is not 8 necessarily a K-sparse vector. It would have more than K entries with non-zero values. Note that a K-block sparse vector x is a L-sparse vector in the conventional sense, where L is the sum of the K sub-blocks element numbers (e.g. suppose each sub-block contains 3 elements, therefore, L = 3K). Fig. 2.2 shows the result of reconstruction of a block-sparse vector x by using standard sparse recovery and block sparse recovery. The original sequence x is a 5-sparse vector but also can be viewed as 1-block sparse vector. The standard sparse recovery fails to recovery x, because the prior knowledge it uses that x is 5-sparse is not sufficient to determine x ( the 5 non-zero element can be anywhere within the vector not appearing in cluster ) and σ10 > 1. While there is a unique block-sparse vector to the problem ( σ2|I < 1 ). original sequence standard sparse recovery block sparse recovery 2 amplitude 1.5 1 0.5 0 10 20 30 40 50 vector x 60 70 80 90 100 Figure 2.2: Comparison of block sparse recovery with traditional sparse recovery. SNR = 30 dB. 9 10 3 Signal Model In this chapter, the model we use thoughout this thesis will be present. In Fig. 3.1, an illustration of the geometry of source localization: sources sk (t) impinging on the uniform linear array from location (θk , rk ) producing observation outputs ym (t). sk (t) θk rm,k rk y−p y1 y0 ym yp δ Figure 3.1: A linear array receiving a signal from a near-field point source. 3.1 Spherical wavefront model Consider K narrowband sources present in the near field impinging on a array of M = 2p + 1 sensors as illustrated in Fig. 3.1. Without loss of generality, it is assumed that the phase reference of the array is at the origin, and the sensors are placed at location indices in the range [−p, p]. Denoting the spacing between two adjacent sensors as δ, the position of the m-th sensor will be mδ where m ∈ [−p, p]. Since the sources are present in the near-field region of the ULA, the curvature of the wavefront is no longer negligible. Therefore, the signal received by the m-th sensor at time t can be expressed as K ∑ 2π ym (t) = sk (t) exp(j (rm,k − rk )) + wm (t), (3.1) λ k=1 where rm,k = √ rk + m2 δ 2 − 2mδrk sin(θk ) 11 (3.2) represents the distance between the m-th sensor and the k-th source, rk is the range from the k-th source to the phase reference, sk (t) is the signal radiated by the k-th source characterized the DOA-range pair (θk , rk ), λ denotes the wavelength, and wm (t) denotes the additive noise. Stacking the measurements in y(t) = [y−p (t), . . . , yp (t)]T ∈ CM ×1 , we get y(t) = K ∑ a(θk , rk )sk (t) + w(t), for t = t1 , t2 , . . . , tT , (3.3) k=1 where tT denotes the number of snapshots, a(θk , rk ) ∈ CM ×1 is the so-called steering vector, and w(t) = [w−p (t), . . . , wp (t)]T ∈ CM ×1 is the noise vector. The non-linear measurement model in (3.3) can be concisely written as y(t) = A(θ, r)s(t) + w(t), (3.4) where A(θ, r) = [a(θ1 , r1 ), a(θ2 , r2 ), . . . , a(θK , rK )] ∈ CM ×K is the array manifold matrix, and s(t) = [s1 (t), . . . , sK (t)]T ∈ CK×1 is the source vector. 3.2 Fresnel approximation Using the Taylor series expansion of (3.2), and approximating this up to the second order, we get the so-called Fresnel approximation, which is given by rm,k ≈ rk − mδ sin θk + m2 δ 2 We can now approximate τm,k = 2π (rm,k λ cos2 θk . 2rk − rk ) as 2πδ πδ 2 sin(θk ) + m2 cos2 (θk ) λ λrk = mωk + m2 ϕk τm,k ≈ −m (3.5) where we re-parameterize the DOA and range, respectively as ωk = − 2πδ πδ 2 cos2 (θk ). sin(θk ) and ϕk = λ λrk (3.6) Using the approximation for τm,k in (3.1), we get ym (t) ≈ K ∑ 2ϕ ) k sk (t)ej(mωk +m + wm (t). (3.7) k=1 Stacking the measurements from all the M sensors, we get y(t) = K ∑ ã(ωk , ϕk )sk (t) + w(t), k=1 12 for t = t1 , t2 , . . . , tT , (3.8) where ã(ωk , ϕk ) = [e−jpωk ejp ϕk , . . . , 1, . . . , ejpωk ejp ϕk ]T ∈ CM ×1 is the modified steering vector. The output of the ULA can now be written as the following non-linear measurement model y(t) = Ã(ω, ϕ)s(t) + w(t) (3.9) 2 2 where Ã(ω, ϕ) = [ã(ω1 , ϕ1 ), ã(ω2 , ϕ2 ), . . . , ã(ωK , ϕK )] ∈ CM ×K is the array manifold, and s(t) and w(t) are the source and noise vectors, respectively. 13 14 Sparse recovery techniques for near-field source localization: Grid-based model 4 In this chapter, we localize multiple narrowband near-field sources by jointly estimating the DOA and range. Using the sparse representation framework, we form a overcomplete basis constructed using a sampling grid that is related to the possible source locations. By doing so, the original non-linear parameter estimation problem is transformed into a linear ill-posed problem. Assuming the spatial spectrum is sparse, we can localize the sources with super-resolution by solving the well-known ℓ1 -regularized leastsquares optimization problem. When multiple time samples are available, we make use of the Fresnel approximation and further assume that the source signals are mutually uncorrelated, which naturally decouples the DOA and range in the correlation domain. This allows us to significantly reduce the complexity, by solving two inverse problems of smaller dimensions one by one, instead of one inverse problem of a higher dimension. 4.1 4.1.1 Grid-based model Gridding on spherical wavefront model We provide a framework for localizing multiple near-field sources based on sparse reconstruction techniques. More specifically, we aim to jointly estimate the DOA θ = [θ1 , θ2 , . . . , θK ]T and the range r = [r1 , r2 , . . . , rK ]T . In this section, a single snapshot case is considered, with T = 1 in (3.4). The problem in (3.4) as it appears is a non-linear parameter estimation problem, where the matrix A(θ, r) depends on the unknown source locations (θ, r). In order to jointly estimate both the DOA and range using the data model in (3.4), a multi-dimensional non-linear optimization over both θ and r is required. This optimization problem is clearly computationally intractable. Suppose all possible source locations reside in the domain θk ∈ [θmin , θmax ] and rk ∈ [rmin , rmax ] for all k = 1, . . . , K. We can then cast the joint DOA-range estimation problem as a sparse reconstruction problem, where we discretize the θ-interval into Nθ and r-interval into Nr bins of resolution ∆θ and ∆r, respectively. This discretization results in an overcomplete representation of A in terms of the sampling grid (θ̄, r̄) that includes all the source locations of interest with θ̄ = [θ̄1 , θ̄2 , . . . , θ̄Nθ ]T and r̄ = [r̄1 , r̄2 , . . . , r̄Nr ]T . We construct a matrix with steering vectors corresponding to each potential source location as its columns: A(θ̄, r̄) = [a(θ̄1 , r̄1 ), a(θ̄1 , r̄2 ), . . . , a(θ̄N , r̄N )] ∈ CM ×N , where N = Nθ Nr . The matrix A is now known, and does not depend on the unknown variables (θ, r). Note that the number of potential source locations N will typically be much greater than the number of sources K or even the number of sensors M . 15 The signal is now represented by an N × 1 vector x(t), where every source can be found as a non-zero weight xn (t) = sk (t) if source k comes from (θ̄n , r̄n ) for some k and is zero otherwise, i.e., the dominant peaks in x(t) correspond to the true source locations. The discrete grid-based model for a single snapshot is given by y= N ∑ a(θ̄n , r̄n )xn + w n=1 (4.1) = A(θ̄ n , r̄n )x + w. This model allows us to transform the non-linear parameter estimation problem into a sparse recovery problem based on the central assumption that the vector x is sparse. An ideal measure for the sparsity of x is its ℓ0 (-quasi) norm ∥x∥0 , and mathematically 2 we must solve for arg min ∥x∥0 subject to ∥y − A(θ̄ n , r̄n )∥2 ≤ ε, where the parameter ε controls how much noise we wish to allow. However, this is a mathematically intractable combinatorial problem even for modestly sized problems. Hence, to simplify this problem we use an ℓ1 -norm regularization, which is the traditional best convex surrogate of the ℓ0 (-quasi) norm. The inverse problem can be solved using an ℓ1 -regularized least-squares (LS) methodology which is given by x̂ = arg min x∈CN ×1 2 ∥y − A(θ̄, r̄)x∥2 + µ∥x∥1 , (4.2) where µ is the sparsity regulating parameter. This optimization problem can be solved using any of the popular solvers available off-the-shelf (e.g., iterative thresholding, matching pursuit). 4.1.2 Gridding on Fresnel approximation model Similar as the gridding on the spherical wavefront model, we can construct an overcomplete representation also for Ã using the sampling grid (θ̄, r̄) that includes all possible source locations. This discretization results in a known matrix with steering vectors corresponding to each potential source location as its columns: Ã(ω̄, ϕ̄) = [ã(ω̄1 , ϕ̄1 ), ã(ω̄1 , ϕ̄2 ), . . . , ã(ω̄N , ϕ̄N )] ∈ CM ×N , where ω̄n = − 2πδ sin(θ̄n ) and ϕ̄n = λ grid-based model is finally given by πδ 2 λr̄n cos2 (θ̄n ) for all n ∈ {1, . . . , N }. The discrete y(t) = Ã(ω̄, ϕ̄)x(t) + w(t), (4.3) and the corresponding inverse problem for a single snapshot can be solved using an ℓ1 -regularized least-squares optimization problem as earlier. Solving the near-field localization problem using sparse regression with or without Fresnel approximation for a single snapshot incurs the same complexity. Moreover, this approximation can even deteriorate the range estimation (for more details see [9]). However, when multiple snapshots are available, the structure of the Fresnel approximated array manifold matrix allows us to significantly reduce the computational complexity. This is discussed in the next section. 16 4.2 Two-step estimator with multiple snapshots When there are multiple measurements available, we can stack (4.1) for the batch of T measurements into a matrix Y = A(θ̄, r̄)X + W (4.4) where Y = [y(t1 ), . . . , y(tT )] ∈ CM ×T , and matrices X and W are defined similarly. An important point to be noted here is that the matrix X is sparse only spatially, and is generally not sparse in time. A straightforward approach would be to use a joint sparsity promoting ℓ2 /ℓ1 -norm regularization, or an ℓ1 -SVD [14] algorithm to solve the inverse problem. In this paper, we propose to reduce the involved computational complexity of 2D-gridding by solving an inverse problem of smaller dimensions in twosteps. We do this by exploiting the spatial cross-correlation between the symmetric sensors, and the fact that the structure of the Fresnel approximated model naturally decouples the DOA and range. We now make the following assumptions1 : (a1) The source signals are mutually independent and are modeled as independent identically distributed (i.i.d.) complex circular random variables with zero mean 2 2 and covariance matrix Et {s(t)sH (t)} = diag(σs,1 , . . . , σs,K ). (a2) The noise is modeled as a zero-mean spatially white Gaussian process, and it is independent of the source signals. The noise covariance matrix is given by Et {w(t)wH (t)} = σw2 I. Under assumptions (a1) and (a2), the spatial correlation between the m-th and n-th sensor can be written as ry (m, n) = Et {ym (t)yn∗ (t)} = K ∑ 2 −n2 )ϕ 2 σs,k ej(m−n)ωk +j(m k + σw2 δ(m − n) k=1 2 where δ(.) represents the Dirac function, Et {sk (t)s∗k (t)} = σs,k denotes the signal power of the k-th source, and σw2 is the noise variance. Notice that when n = −m the spatial correlation is independent of the parameter ϕk , and we arrive at ∗ (t)} ry (−m, m) = Et {y−m (t)ym K ∑ 2 = σs,k e−2mωk j + σw2 δ(−2m). (4.5) k=1 This means that by exploiting the cross-correlation between the symmetric sensors, we can transform the original 2D (DOA and range) estimation into a 1D (DOA) estimation. Stacking (4.5) for all the symmetric sensors, we can build a virtual far-field scenario: ry = Aω (ω)rs + σw2 e, 1 These assumptions are required only for the two-step estimator discussed in Section 4.2. 17 (4.6) where ry = [ry (0, 0), ry (1, −1), . . . , ry (p, −p)]T ∈ CL×1 with L = p + 1, rs = 2 2 diag(σs,1 , . . . , σs,K ) ∈ CK×1 , and e = [1, 0Tp ]T ∈ CL×1 , and the corresponding virtual array gain pattern for the k-th source denoted by aω (ωk ) can be expressed as aω (ωk ) = [1, e−j2ωk , . . . , e−j2pωk ]T ∈ CL×1 , with the array manifold Aω (ω) = [aω (ω1 ), aω (ω2 ), . . . , aω (ωK )] ∈ CL×K . In practice, the vector ry containing the statistical correlations is approximated using the measurements from (4.4). 4.2.1 Step-1: DOA estimation As before, we can construct an overcomplete basis Aω but now with only Nθ columns corresponding to potential source directions of arrival (DOAs) using the sampling grid θ̄, i.e., Aω (ω̄) = [aω (ω̄1 ), . . . , aω (ω̄Nθ )] ∈ CM ×Nθ , sin(θ̄n ) for all n ∈ {1, . . . , Nθ } as defined earlier. The signal is where ω̄n = − 2πδ λ represented by an Nθ × 1 vector u(t), where every source can be found as a non-zero weight un (t) = sk (t) if source k comes from direction θ̄n for some k and is zero otherwise, i.e., the dominant peaks in u(t) correspond to the true source locations. The discrete grid-based model in the correlation domain is then given by ry = Aω (ω̄)u + σw2 e. (4.7) Note that the number of potential source DOAs Nθ will typically be much greater than the number of sensors M also in the correlation domain, and the model in (4.7) is still ill-posed. Hence, we solve for the unknown vector u using an ℓ1 -regularized LS minimization problem which is given by û = arg min u ∥ry − Aω (ω̄)u∥22 + µ1 ∥u∥1 , (4.8) where µ1 is the sparsity regulating parameter. Alternatively, when the noise variance σw2 is known, the unknown vector u can be obtained by solving arg min u 4.2.2 2 ∥ry − Aω (ω̄)u − σw2 e∥2 + µ1 ∥u∥1 . Step-2: range estimation Let θ̂ be the estimated DOAs from step-1, and K̂ denote the number of DOAs detected (i.e., K̂ = ∥û∥0 ). We now use the sampling grid (θ̂, r̄) to form an overcomplete basis A(θ̂, r̄) ∈ CM ×K̂Nr to arrive at Y = A(θ̂, r̄)X̃ + W, (4.9) where X̃ is obtained by removing some specific rows of the signal matrix X. In order to solve the inverse problem in (4.9) we use the ℓ1 -SVD algorithm. Note that in step-2 18 for range estimation we do not use the Fresnel approximation anymore. For the sake of completeness, the ℓ1 -SVD algorithm in [14] is briefly summarized as follows. Let Y = UΣVH be the singular value decomposition (SVD) of the data matrix. Keep a reduced M × K̂ matrix Ysv = UΣDk = YVDk , where Dk = [Ik , 0TK̂×(T −K̂) ]. The reduced data matrix contains most of the signal power, and forms the basis for the signal subspace. Similarly, let X̃sv = X̃VDk and Wsv = WVDk , to arrive at Ysv = A(θ̂, r̄)X̃sv + Wsv , (4.10) which can be expressed in vector form (column by column) as ysv (k) = A(θ̂, r̄)x̃sv (k) + wsv (k), for k = 1, . . . , K̂. Here, each column corresponds to a signal subspace singular vector. The reduced data matrix is only spatially sparse, and not in terms of the singular vector index k. In order to take this effect into account, we use a different prior obtained by computing (ℓ2 ) the √∑ℓ2 -norm of the singular values of a particular spatial index of x̃sv (k), i.e., x̃m,sv = K 2 k=1 (x̃m,sv (k)) , for m ∈ [−p, p]. Note that the ℓ2 -norm can be computed for all snapshots instead of only the signal subspace singular vectors, however, the former technique adds more computational complexity especially when T ≫ K̂ [14]. Now, we can find the range by minimizing (ℓ2 ) ∥Ysv − A(θ̂, r̄)X̃sv ∥2F + µsv ∥x̃sv ∥1 , (ℓ ) (ℓ ) (ℓ ) 2 2 T ] , and the parameter µsv controls the spatial sparsity. , . . . , x̃p,sv where x̃sv2 = [x̃−p,sv Remark 1 (Complexity reduction with multiple snapshots). Jointly estimating the DOA and range by applying the ℓ1 -SVD algorithm on the model (4.4) costs O((KNθ Nr )3 ). Using the proposed two-step estimator, the complexity is reduced significantly to O(KNr3 ) (reduction by a factor of O(Nθ3 )) with an additional complexity of solving the inverse problem in (4.8), which costs for example, O(Nθ log(Nθ )) using the iterative thresholding algorithm [25]. For a typical problem with Nr = 15 and Nθ = 180 points on the grid, the complexity reduction is significant. Remark 2 (Array geometry). Any array (uniform or non-uniform) can be used to solve for the variables (θ, r) based on the optimization problem in (4.2). For the twostep approach, any symmetric array (uniform or non-uniform) can be used. 4.3 Simulations We consider a symmetric ULA with M = 15 sensors placed such that the inter-sensor spacing δ = λ/4 (To avoid aliasing when the virtual far-field model (4.7) in is used), where λ represents the wavelength of the narrowband source signals. For this array, the far-field distance is beyond 2D2 = 24.5λ, and any sources within the range of 24.5λ from the array will be in the near field. We compare the proposed algorithms with matched-filter beamforming [3] for a single time sample scenario. For the multiple time 19 Beamforming −50 DOA (Degrees) DOA (Degrees) True source locations 0 50 −50 0 50 4 6 8 10 range in λ 12 14 4 6 12 14 with Fresnel approx. −50 DOA (Degrees) DOA (Degrees) w/o Fresnel approx. 8 10 range in λ 0 50 −50 0 50 4 6 8 10 range in λ 12 14 4 6 8 10 range in λ 12 14 Figure 4.1: Joint DOA and range estimation. Two near-field sources with DOAs: 0◦ and 10◦ , and ranges: 5λ and 10λ. SNR = 20 dB, and T = 1. The sampling grid has a resolution of ∆θ = 10◦ and ∆r = 1λ. sample scenario we compare the performance with the beamforming method [3] and the 2D-MUSIC algorithm [7]. The optimization problems in the proposed algorithms are solved using CVX [26]. The regularization parameter is chosen via cross-validation. In Fig. 4.1, we illustrate the joint DOA and range estimation obtained by solving the sparse regression problem for a single time sample scenario. We consider two sources at locations (0◦ , 5λ) and (10◦ , 10λ). The SNR is 20 dB with T = 1. As can be seen from the plots, high resolution can be achieved by using the sparse modeling framework as compared to the conventional beamforming technique. The performance loss with and without (w/o) Fresnel approximation for the considered scenario is negligible. In Fig. 4.2, we show the proposed two-step estimator for near-field source localization. We consider four sources are locations (0◦ , 5λ), (0◦ , 17λ) and (50◦ , 8λ). The simulations are provided for SNRs of 10 dB and 30 dB with T = 200 snapshots.In the two-step estimator, we solve for the DOAs in the correlation domain in which the range parameters are naturally decoupled from the DOAs (due to the Fresnel approximation). 20 −50 0 50 10 15 range in λ 2D−MUSIC (SNR = 10 dB) −50 0 50 10 15 range in λ 2D−MUSIC (SNR = 30 dB) −50 0 50 5 10 15 range in λ 0 50 20 5 10 15 20 range in λ Two−step estimator (SNR = 10 dB) 5 10 15 20 range in λ Two−step estimator (SNR = 30 dB) −50 0 50 20 DOA (Degrees) 5 −50 20 DOA (Degrees) DOA (Degrees) 5 DOA (Degrees) Beamforming (SNR = 30 dB) DOA (Degrees) DOA (Degrees) True source locations −50 0 50 5 10 15 range in λ 20 Figure 4.2: Two-step estimator. Three near-field sources at locations: (0◦ , 5λ), (0◦ , 17λ), and (50◦ , 8λ), and T = 200. The sampling grid has a resolution of ∆θ = 1◦ and ∆r = 0.5λ. In the second step, we solve for the range using the ℓ1 -SVD algorithm where we use the number of sources as K̂ from the step-1. Even though, 2D-MUSIC can achieve high resolution in the DOA domain, its performance deteriorates along the range comain especially when two sources shares the same DOA. The two-step approach would further allow to increase the gridding resolution because of the involved smaller overcomplete dictionary as compared to the dictionary obtained from a 2D grid (cf. (4.1) ) 21 22 Grid matching for near-field source localization using sparse recovery techniques 5 In this chapter, we discuss the grid mismatch effect associated with near-field source localization using sparse recovery techniques. The grid based model presented is the previous chapter is extended to an error-in-variables (EIV) model to account for cases when the true source locations are not on the assumed sampling grid. We propose a number of estimators for correcting such perturbations when both single and multiple time snapshots are available. 5.1 Grid mismatch and its effect Source localization based on the sparse signal reconstruction (SSR) can provide a high resolution compared with the classical source localization methods, but it should be noted that its performance is tremendously affected by the choice of the sampling grid. In Fig. 5.1, we illustrate a grid mismatch scenario and show how it influences the estimation result, , where the estimator is based on the sparse recovery technique discussed in Chapter 4. Consider K narrowband sources present in near field impinging on a array comprising of M = 2p + 1 sensors. We recall the signal model described in Chapter 3 y(t) = A(θ, r)s(t) + w(t), where (θ, r) = [(θ1 , r1 ), (θ2 , r2 ), . . . , (θK , rK )] indicates the unknown source location. Suppose all the possible source location reside in the domain θk ∈ [θmin , θmax ] and rk ∈ [rmin , rmax ] for all k = 1, . . . , K. We showed in Chapter 4 that we can discretize the DOA-range domain into N = Nθ Nr bins to obtain the grid-based model (4.1) y= N ∑ a(θ̄n , r̄n )xn + w n=1 = A(θ̄, r̄)x + w, where (θ̄, r̄) = [(θ̄1 , r̄1 ), . . . , (θ̄N , r̄N )] denotes the sampling grid. In the grid-based estimation, we assume that the desired sources lie on top of the sampling grid. However, in practice there is no reason to believe that the true source locations lie on the sampling grid. When the source location (θk , rk ) for any k does not lie on the considered sampling grid, the discrete model in (4.1) leads to grid mismatch. Note that the grid mismatch leads to less sparse (or maybe even non-sparse) solutions due to the energy leakage of the desired signals over the adjacent grid points. To illustrate this effect, consider a unit amplitude source present within the near-field region at (10◦ , 10λ) and we try to localize the near-field source by using the classical 23 DOA (Degrees) True sources location 5 6 7 8 9 10 11 12 13 14 15 4 5 6 7 8 9 10 range in λ 11 12 13 14 15 13.5 14.5 DOA (Degrees) Mismatched grid 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 range in λ 11.5 12.5 Figure 5.1: Source located at (10◦ , 10λ). The sampling grid has a resolution of ∆θ = 1◦ and ∆r = 1λ within the domain [5.5◦ , 14.5◦ ] and [4.5λ, 14.5λ]. grid-based approach, where the sampling grid has the resolution of ∆θ = 1◦ and ∆r = 1λ within the domain [5.5◦ , 14.5◦ ] and [4.5λ, 14.5λ] such that the true source locations does not lie on the sampling grid. As seen in Fig. 5.1, the non-zero coefficients that correspond to the true source locations are spilled over the adjacent grid points. On the other hand, increasing the gridding resolution would lead to the columns of A that are highly correlated (i.e., A would have high coherence) which makes the sparse recovery difficult. The classical sparse regression does not account for such perturbations due to the grid mismatch. 5.2 Grid mismatch model The EIV model mentioned in [17, 18, 19] treats the grid mismatch effect as an additive error matrix E ∈ CM ×N . Taking the perturbation into account, we can rewrite the signal model in (4.1) as y(t) = Âx(t) + w, where Â = A + E. (5.1) If a source location (θk , rk ) is not on the sampling grid, then there exists a pair 24 of unknown error parameters (eθ̄n , er̄n ) that accounts for the perturbation, (θk , rk ) = (θ̄n + eθ̄n , r̄n + er̄n ). Using a first-order Taylor expansion of the steering vector a(θ̄n + eθ̄n , r̄n + er̄n ), we can approximate it up to the second order as follows ∂a(θ, r) ∂a(θ, r) a(θ̄n + eθ̄n , r̄n + er̄n ) ≈ a(θ̄n , r̄n ) + e + er̄ . (5.2) ∂θ θ=θ̄n θ̄n ∂r r=r̄n n The misaligned grid can be corrected by estimating the perturbation (eθ̄n , er̄n ). In other words, near-field localization accounting for the grid mismatch is a joint estimation problem in which we jointly estimate (θ̄n , r̄n ) as well as (eθ̄n , er̄n ). Using the approximation in (5.2), we can write the measurement vector y(t) for T = 1 as: ) N ( ∑ ∂a(θ, r) ∂a(θ, r) y(t) = eθ̄n + er̄n xn (t) + w a(θ̄n , r̄n ) + ∂θ ∂r θ= θ̄ r=r̄ n n n=1 N N ∑ ∑ ∂a(θ, r) = a(θ̄n , r̄n )xn (t) + eθ̄n ∂θ θ= θ̄ n n=1 i=1 N ∑ ∂a(θ, r) er̄ + w, + xn (t) ∂r r=r̄n n n=1 and can be concisely written as y = Ax + Aθ [eθ ⊙ x] + Ar [er ⊙ x] + w, (5.3) [ ]T [ ]T where eθ = eθ̄1 , . . . , eθ̄N ∈ RN[ ×1 and er = eθ̄1 , . . . , eθ̄N] ∈ RN ×1 are the perturbation vectors, and Aθ = ∂a(θ,r) , . . . , ∂a(θ,r) ∈ CM ×N , matrices ∂θ ∂θ θ=θ̄1 θ=θ̄N [ ] ∂a(θ,r) ∂a(θ,r) ∈ CM ×N contains the first-order derivatives. Ar = , . . . , ∂r ∂r r=r̄1 =r̄N ( Using matrix-vector properties, we have A [e ⊙ x] + A [e ⊙ x] = A θ Eθ + θ θ r r ) Ar Er x, where Eθ = diag(eθ ) and Er = diag(er ). Recall that the error matrix E in (5.1) is given by E = Aθ Eθ + Ar Er ∈ CM ×N . Now the problem is briefly stated as follows: given the wide matrix Φ = [A|Aθ |Ar ] ∈ CM ×3N and the measurement vector y, estimate the unknown parameters x, eθ , and er . Remark 3: (1) Since the perturbations are related to the DOA and range, both eθ and er are real parameters. (2) When xn = 0 for a certain n, then eθ̄n and er̄n can take any value without any contributions to the observation vector y. Hence, there is no meaning to recovery such eθ̄n and er̄n . Hence, through out this paper we focus only on recovering the entries of eθ and er related to non-zero entries of x. 25 Algorithm 1 Alternating minimization 1. Input: A, Aθ , Ar , ϵ, imax , η 2. Output: êθ , êr , x̂ (0) (0) 3. Initial: eθ = 0, er = 0 4. for i = 0 to imax (i) (i) 5. x̂(i) = arg min ∥x∥1 s.t. ∥Ax + Aθ [eθ ⊙ x] + Ar [er ⊙ x] − y∥2 ≤ ϵ 6. update x̂ (i) 7. if x̂n ≤ η (i) 8. x̂n = 0 9. end 10. l = |x̂(i) |./ max(|x̂(i) |) (i+1) (i+1) 11. {eθ , er } = arg mineθ ,er ∥Ax + Aθ [eθ ⊙ x̂(i) ] + Ar [er ⊙ x̂(i) ] − y∥2 s.t. −0.5∆θl < eθ < 0.5∆θl −0.5∆rl < er < 0.5∆rl 12. end 13. I is the set of the support of the non-zero elements of x̂(imax ) 14. θ̂k = θ̄k + eθ̄k ; r̂k = r̄k + er̄k , where k ∈ I Table 5.1: Alternation minimization algorithm. 5.3 5.3.1 Sparse recovery for off-grid sources with single snapshot Bounded sparse recovery A pair of unknown perturbations (eθ̄n , er̄n ) indicates the error between a source location (θk , rk ) and the nearest grid point (θ̄n , r̄n ) and is always bounded by the size of grid cell by definition. With bounded perturbations eθ̄n and er̄n , we can reconstruct the sparse signal x in the mismatch model (5.3) by solving an optimization problem of the form [20] [x̂, êθ , êr ] = s.t. 1 µ1 ∥x∥1 + ∥ (A + Aθ Eθ + Ar Er ) x − y∥22 2 x∈CN ,eθ ∈RN ,er ∈RN Eθ = diag(eθ ), Er = diag(er ) (5.4) −0.5∆θ < eθ̄n < 0.5∆θ; for n = 1, 2, . . . , N. −0.5∆r < er̄n < 0.5∆r; arg min The above optimization is non-convex and generally difficult to solve. In [27], an alternating minimization algorithm to solve such non-convex optimization problems was proposed. We adapt the algorithm to suit our problem, and this is summarized in the Table. 5.1. Typically, the threshold η in Algorithm 1 is chosen depending on the noise level, i.e., η ≥ σw , where σw is the standard deviation of the additive noise. 26 5.3.2 Bounded joint sparse recovery The model in (5.3) can be expressed as q z }| { Φ x [z }| {] y = A Aθ Ar eθ ⊙ x +w er ⊙ x = Φq + w, (5.5) where Φ ∈ CM ×3N , and q ∈ C3N ×1 is a 3K-sparse vector. Recalling remark 3, the vector q can be grouped into N sub-blocks and each block contain 3 elements, i.e. the j-th sub-block will have elements qj = [xj , eθ̄j xj , er̄j xj ]T for j = 1, 2, . . . , N . Then the vector q can be viewed as K-block sparse vector since only K sub-blocks have non-zero values. In Chapter 3, we discussed the sparse recovery techniques involving block sparse recovery. We can enforce block sparsity on q via a mixed ℓ2 /ℓ1 -norm minimization to recover vector q instead of enforcing the sparsity of q via ℓ1 -norm minimization [21, 22]. Define the mixed ℓ2 /ℓ1 -norm of q as ∥q∥2,1 = N √ ∑ x2n + (eθ̄n xn )2 + (er̄n xn )2 . n=1 The optimization problem taking this structure into account can be formulated as [x̂, êθ , êr ] = s.t. 1 µ2 ∥q∥2,1 + ∥Φq − y∥22 , 2 −0.5∆θ|x| < qθ < 0.5∆θ|x|, −0.5∆r|x| < qr < 0.5∆r|x|, arg x∈CN ,e min N N θ ∈R ,er ∈R (5.6) where |x| denotes the entry-wise absolute value of vector x, qθ = eθ ⊙ x and qr = er ⊙x. The optimization problem in (5.6) is non-convex, and can be relaxed to a convex optimization problem using separating technique, where we assume x = x+ − x− . The relaxed optimization problem is referred to as the relaxed bounded joint sparse (R-BJS) estimator [27], and is given by 1 ∥A(x+ − x− ) + Aθ qθ + Ar qr − x∥22 + γ∥q∥2,1 x ,x ,q 2 s.t. x+ ≥ 0, x− ≥ 0 − 0.5∆θ(x+ + x− ) < qθ < 0.5∆θ(x+ + x− ) − 0.5∆r(x+ + x− ) < qr < 0.5∆r(x+ + x− ) [x̂, êθ , êr ] = arg +min − (5.7) The above convex problem provides an accurate solution only when qθ ⊙qr = 0. During simulations it was observed that this non-convex constraint is always satisfied. The RBJS method is much more efficient than the alternating algorithm. The performance analysis of R-BJS is given in appendix A.4. 27 Remark 4 The result of R-BJS is an approximate K-block sparse vector, then we choose K columns of matrix Φ related to the K sub-blocks with largest ℓ2 -norm value to generated a new sensing matrix Φ̃, and substitute into (5.7) to refine the R-BJS results. For example, in single source case, the estimated result is related to the j-th sub-block, where j = 1, 2, . . . , N , hence we pick up the related columns of Φ to generate sensing matrix Φ̃, then run optimization (5.7) again to find the optimal perturbation parameters. 5.4 Sparse recovery for off-grid sources with multiple snapshots When multiple snapshots are available, we can stack (3.4) for the batch of T measurements into a matrix Y = A(θ, r)S + W (5.8) where Y = [y(1), . . . , y(T )] ∈ CM ×T , and matrices S and W are defined similarly. 5.4.1 Step-1: Grid matching in DOA domain In Chapter 4, we introduced a method to transform the original 2D (DOA and range) estimation into 1D (DOA) estimation via exploiting the cross-correlation between the symmetric sensors of the array based on the Fresnel approximation. The signal model based on the Fresnel approximation is given by ( cf.(3.8) ) y(t) = K ∑ ã(ωk , ϕk )sk (t) + w(t), for t = 1, 2, . . . , T, k=1 where ωk = − πδ 2 2πδ sin(θk ) and ϕk = cos2 (θk ) λ λrk The structure of the array manifold based on Fresnel approximation naturally decouples the DOA and range. As in Chapter 4, we have a virtual far-field model (4.6) depending only on DOA, and it is given by ry = Aω (ω)rs + σw2 e with the array manifold Aω (ω) = [aω (ω1 ), aω (ω2 ), . . . , aω (ωK )] ∈ CM ×K , where ry = [ry (−p, p), . . . , ry (0, 0), ry (1, −1), . . . , ry (p, −p)]T ∈ CM ×1 with M = 2p + 1, 2 2 , . . . , σs,K ]T ∈ CK×1 , and e = [1, 0Tp ]T ∈ CL×1 . rs = [σs,1 Based on the virtual far-field model, we can construct an overcomplete basis Aω as we did in Chapter 4 with only Nθ columns corresponding to the potential source DOAs using the sampling grid θ̄, i.e., Aω (ω̄) = [aω (ω̄1 ), . . . , aω (ω̄Nθ )] ∈ CM ×Nθ , 28 where ω̄n = − 2πδ sin(θ̄n ) for all n ∈ {1, . . . , Nθ } is as defined earlier. If an ωk is not on λ the sampling grid, then there exists an error parameter eω̄n that accounts perturbation, ωk = ω̄n + eω̄n . ω e , we can Using a first-order Taylor expansion aω (ω̄n + eω̄n ) = aω (ω̄n ) + ∂a ∂ω ω=ω̄n ω̄n rewrite (4.6) as ( ) ry = Aω (ω̄) + Ãω (ω̄)Eω u + σw2 e, (5.9) where [ ∂aω ∂ω Eω ∈ = diag(eω̄1 , eω̄2 , . . . , eω̄Nθ ) ] ∂aω , . . . , ∂ω ω=ω̄ ∈ CM ×Nθ . ω=ω̄1 RNθ ×N θ and Ãω (ω̄) = Nθ Choosing a proper constraint box [−α, α] limiting perturbation parameter eω̄n , the optimization problem for (4.6) is formulate as 1 ∥Aω u + Ãω pω − ry ∥22 + µ3 ∥z∥2,1 u,pω 2 s.t. u ≥ 0, z = [uT , pTω ]T , and − αu ≤ pω ≤ αu, ∑ θ √ 2 un + p2ω,n . where pω = eω ⊙ u, and ∥z∥2,1 = N n=1 [û, p̂ω ] = arg min 5.4.1.1 (5.10) Realigning the DOA estimate In the first step the matching model we build is under the Fresnel approximation, and the estimated DOA result suffers from the Fresnel approximation. The perturbation generated by the approximation in Fresnel zone influences less in the classical gridbased estimation, and we can still obtain the accurate result, which can be seen in the results presented in Chapter 4. However, for the grid mismatch, the error we estimate is the perturbation on the virtual far-field grid based on the Fresnel approximation. For example, assume we have a near-field source located at (50◦ , 5λ), the estimated DOA based on the Fresnel approximation via algorithms proposed in Chapter 4 (note that the Fresnel zone is related to the aperture of ULA, thus different ULA has different Fresnel zone), will be 48.8067◦ . This motivates us to realign the estimated result in the Fresnel model to the the estimated result in spherical wavefront model. Consider a ULA with M = 2p + 1 element as illustrated in Fig. 3.1. The distance to each sensor is depicted in (3.2), √ rm,k = rk + m2 δ 2 − 2mδrk sin(θk ). In Fresnel approximation, we assume that rm,k ≈ rk − mδ sin θk + m2 δ 2 cos2 θk . 2rk The range difference dm,k = rm,k − rk is given by dm,k ≈ −mδ sin(θk ) + m2 = mδck,1 + m2 29 δ2 cos2 (θk ) 2rk δ2 ck,2 . 2 Define r(k) = [r−p,k , . . . , rp,k ]T as the distance vector of k-th source to each sensor of the ULA. Stack dm,k , we can obtain r(k) − rk 1 = d̂k , k = 1, 2, . . . , K. We can now build linear equation of the form r(k) − rk 1 = Bck . where −δp 0.5δ 2 (−p)2 .. ∈ RM ×2 B = ... . δp 0.5δ 2 (p)2 is a tall matrix and ck = [ck,1 , ck,2 ]T . We can then find Fresnel based location (θ̂, r̂) via the true delay vector by using least-squares method. Then we can build a table for us to realign the Fresnel approximated one to the one based on spherical model. 5.4.2 Step-2: grid matching in range domain The measurement covariance matrix is given by Ryy = A(θ, r)Rss A(θ, r)H + σw2 IM . (5.11) By using the Khatri-rao product, (5.11) can be written as p = [A(θ, r)∗ ⊚ A(θ, r)]rs + σw2 ê = Ψ(θ, r)rs + σw2 ê, where p represents the vec(Ryy ) ∈ CM 2 ×1 , ê = vec(IM ) ∈ CM Ψ(θ, r) = [ψ(θ1 , r1 ), . . . , ψ(θK , rK )] ∈ CM 2 ×1 , and 2 ×K . Let θ̂ be the DOA from step-1, and K̂ (notice that K̂ ≤ K, e.g. two sources may share the same DOA but at different range) denotes the number of different source DOAs deˆ tected (i.e., K̂ = ∥û∥0 ). We now firstly realign θ̂ to θ̂ at different range, and then using ˆ sampling grid (θ̂, r̄) (in Chapter 3, we have K̂Nr points {(θ̂1 , r̄1 ), (θ̂1 , r̄2 ), . . .}, and here ˆ ˆ the nodes number is not changed but {(θ̂1 , r̄1 ), (θ̂2 , r̄2 ), . . .}) to form an overcomplete 2 2 ˆ ˆ basis Ψ(θ̂, r̄) ∈ CM ×K̂Nr as well as the partial derivative matrix Ψr (θ̂, r̄) ∈ CM ×K̂Nr ] [ ˆ ˆ ˆ ˆ ∂ψ( θ̂ , r) ∂ψ( θ̂ , r) ∂ψ( θ̂ , r) ∂ψ( θ̂ , r) ˆ 1 e 1 K̂ ,..., , ,..., . Ψr (θ̂, r̄) = ∂r ∂r ∂r ∂r r=r̄1 r=r̄N r=r̄1 r=r̄N r And then we generate the grid matching model on range domain ) ( ˆ ˆ p = Ψ(θ̂, r̄) + Ψr (θ̂, r̄)Ẽr r̃s + σw2 ê, 30 r (5.12) where r̃s ∈ CK̂Nr ×1 is a K-sparse vector containing the signal power spectrum. Rewrite (5.12) as Φr z }| [z ]{ [ ˆ ˆ p = Ψ(θ̂, r̄), Ψr (θ̂, r̄) u }|r ]{ r̃s +σw2 ê er ⊙ r̃s = Φr ur + σw2 ê (5.13) where Φr ∈ CM ×2K̂Nr , and ur ∈ C2K̂Nr ×1 is a 2K-sparse vector, which can be divided into K̂Nr sub-blocks and be viewed as K-block sparse vector. Letting q̃r = ẽr ⊙r̃s , then we can reconstruct the K-block sparse vector r̃s via the mixed ℓ2 /ℓ1 -norm minimization optimization 2 1 ∥p − Φr ur ∥22 + µ4 ∥ur ∥2,1 r̃s ,q̃r 2 s.t. r̃s ≥ 0, and − 0.5∆rr̃s < q̃r < r̃s 0.5∆r, arg min where ∥ur ∥2,1 = 5.5 (5.14) ∑K̂Nr √ 2 2 . rs,n + qr,n n=1 Simulations We consider a symmetric ULA with M = 15 sensors and an inter-sensor spacing δ = λ4 where λ represents the wavelength of the narrowband source signals. The near-field region for this array is within the distance 2D2 = 24.5λ. We compare the results of the proposed agorithms to the grid matching problem and the optimization problems we solved in such estimations is done by CVX [26]. In Fig. 5.2, we show the results of grid matching for near-field source localization when single snapshot is available. We consider a unit amplitude single source at location (0.3◦ , 1.28λ). The sampling grid has a resolution of ∆θ = 1◦ and ∆r = 0.1λ, resulting in a maximum DOA and range misalignment of 0.5◦ and 0.05λ, respectively. The SNR is 40 dB. The threshold of alternating algorithm is 0.4. Alternating algorithm performed better than R-BJS form the plot, but R-BJS is much faster and suffer less computational load. In Fig. 5.3, we put all the proposed algorithms together and the performance of the proposed algorithm can be visually compared. Also, it can be seen that refined R-BJS improves the performance of R-BJS. In Fig. 5.4, we show the results for multiple near-field sources localization via mismatched sensing grid for single snapshot scenario. We consider two sources with a unit amplitude locating at (0.3◦ , 1.28λ) and (−3.2◦ , 1.52λ), using the same sampling grid that was used in Fig 5.3. The SNR is 40 dB. As can be seen that refining the result of R-BJS and improve the performance and provide a much more accurate estimation result. The improvement can be seen much more visually in Fig. 5.5. In Fig. 5.6, we show the proposed two-step approach to grid matching for near-field source localization when multiple snapshots are available. We consider three near-field sources at locations (−20.82◦ , 6.1λ) and (10.12◦ , 10.9λ). The simulations are provided for SNRs of 0 dB with T = 200 snapshots. In the two-step grid matching algorithm, we 31 R−BJS −4 −3 −3 −2 −2 DOA (degrees) DOA (degrees) True source location −4 −1 0 −1 0 1 1 2 2 3 3 1 1.1 1.2 1.3 range in λ 1.4 1.5 1.6 1.7 1 1.1 1.2 (a) range in λ 1.4 1.5 1.6 1.7 1.5 1.6 1.7 (b) Alternating algorithm Refined R−BJS −4 −4 −3 −3 −2 −2 DOA (degrees) DOA (degrees) 1.3 −1 0 −1 0 1 1 2 2 3 3 1 1.1 1.2 1.3 range in λ 1.4 1.5 1.6 1.7 1 (c) 1.1 1.2 1.3 range in λ 1.4 (d) Figure 5.2: Near-field source localization via mismatched sensing grid. Source located at (0.3◦ , 1.28λ). SNR = 40 dB, and T=1. The sampling grid has a resolution of ∆θ = 1◦ and 0.1λ within the domain [−4◦ , 3◦ ] and [1λ, 1.7λ]. (a) True source location. (b) R-BJS. (c) Alternating minimization algorithm with threshold η = 0.4. (d) Refined R-BJS. solve for the Fresnel approximated DOAs in the correlation domain in which the range parameters are naturally decoupled from the DOAs (due to the Fresnel approximation). Due to the Fresnel approximation, in the next step, we check the Fresnel error at ˆ different range to the corresponding Fresnel estimated DOAs and realign to θ̂ at different range. By solving the optimization problem proposed in grid matching in range domain, we locate the near-field sources. The results have been zoomed in for visual comparison. 32 1 0.9 original R−BJS Refined R−BJS Alternating algorithm 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.5 0 0.5 1 DOA (degrees) (a) DOA estimation result. 1 0.9 original R−BJS Refined R−BJS Alternating algorithm 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1.15 1.2 1.25 range in λ 1.3 1.35 (b) Range estimation result. Figure 5.3: DOA and range estimation via mismatched sensing grid. Source located at (0.3◦ , 1.28λ). SNR = 40 dB, and T=1. The sampling grid has a resolution of ∆θ = 1◦ and 0.1λ within the domain [−4◦ , 3◦ ] and [1λ, 1.7λ]. 33 True source location DOA (degrees) −4 −2 0 2 1 1.1 1.2 1.3 range in λ 1.4 1.5 1.6 1.7 1.4 1.5 1.6 1.7 1.5 1.6 1.7 R−BJS DOA (degrees) −4 −2 0 2 1 1.1 1.2 1.3 range in λ Refined R−BJS DOA (degrees) −4 −2 0 2 1 1.1 1.2 1.3 range in λ 1.4 Figure 5.4: Multiple source localization via mismatched sensing grid. Near-field sources with DOAs: 0.3◦ and −3.2◦ , and ranges: 1.28λ and 1.52λ. SNR = 40 dB, and T = 1. The sampling grid has a resolution of ∆θ = 1◦ and 0.1λ within the domain [−4◦ , 3◦ ] and [1λ, 1.7λ]. 34 1 original R−BJS Refined R−BJS 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −3.5 −3 −2.5 −2 −1.5 −1 DOA (degrees) −0.5 0 0.5 1 (a) DOA estimation result. 1 0.9 original R−BJS Refined R−BJS 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1.15 1.2 1.25 1.3 1.35 range in λ 1.4 1.45 1.5 (b) Range estimation result. Figure 5.5: DOA and range estimation for multiple sources via mismatched sensing grid. Near-field sources with DOAs: 0.3◦ and −3.2◦ , and ranges: 1.28λ and 1.52λ. SNR = 40 dB, and T = 1. The sampling grid has a resolution of ∆θ = 1◦ and 0.1λ within the domain [−4◦ , 3◦ ] and [1λ, 1.7λ]. 35 Grid matching in DoA estimation; SNR=0 1 true Alternating BJS 0.9 0.8 0.7 Power 0.6 0.5 0.4 0.3 0.2 0.1 0 −25 −20 −15 −10 −5 DOA (Degrees) 0 5 10 15 (a) DOA estimation Grid matching in DoA estimation; SNR=0 1 true Alternating BJS 0.9 0.8 0.7 Power 0.6 0.5 0.4 0.3 0.2 0.1 0 4 6 8 10 12 range in λ 14 16 18 20 (b) range estimation Figure 5.6: Grid matching via Two-step algorithm. Two near-field source at locations: (−20.82◦ , 6.1λ) and (10.12◦ , 10.9λ). SNR = 0 dB and T = 200. The sampling grid has a resolution of ∆θ = 1◦ and ∆r = 1λ within domain [−90◦ , 90◦ ] and [4λ, 20λ]. 36 Conclusions and future work 6 In this chapter, we summarize this thesis and provide some suggestions for the future research. 6.1 Conclusions In this thesis, we have considered the problem of localization of near-field point sources. The classical near-field source localization problem is a non-linear (joint DOA and range) parameter estimation problem. The frequently used planar-wave assumption is no more valid for near-field sources as the wavefronts are spherical. Using the sparse representation framework, we transform the original non-linear problem into a linear ill-posed inverse problem. Based on the assumption that the spatial spectrum (i.e., the number of point sources) is sparse, we can localize the sources with a high resolution by solving an ℓ1 -regularized sparse regression. In this thesis, we address two closely related problems in near-field source localization via sparse recovery: 1. Localize near-field sources with perfect sampling grid. 2. Localize near-field sources with mismatched sampling grid. For the first situation, sparse recovery can localize near-field sources with single snapshot and provide a relatively accurate result. Additionally, when multiple snapshots are available and the sources are uncorrelated, the DOA and range parameters are naturally decoupled in the correlation domain. This enables us to solve two smaller dimension sparse regression problems instead of one higher dimension sparse regression problem which leads to a significant complexity reduction. Compared with 2D-MUSIC, the two-step estimator via sparse recovery method can even work in low-SNR situation regimes with high-resolution. Grid mismatch can affect the estimation result tremendously. In this part, extended the grid-based model into EIV model to account for the grid mismatch. We have proposed two estimators based on sparse recovery and block sparse recovery for grid matching. When multiple snapshots are available, we extend the two-step estimator that we proposed in Chapter 3 to account for perturbations also. In the two-step estimator for grid matching, since in the step-1 DOA estimation we use the Fresnel approximation, the estimation result has errors both from the Fresnel approximation as well as perturbation error. Thus, we realign the DOA so as to overcome the effect of Fresnel approximation. The realigned DOA is now used in step-2 in which the range is estimated. 37 6.2 Suggestion for future research Sparse recovery is now widely used in source localization but most of them are on farfield model. So we can do lots of work on this aspect. And here is some direction that can be possible to extend in the application of CS in near-field source localization: • Automatic method to generate the induced parameter µ in Lasso. As we did in this thesis, the recovery result also relies on the choice of regularization parameter, and till now we haven’t find a way to automatic generate the suitable regularization parameter, and have to find manually. Find a algorithm that can automatically generate suitable inducing parameter will improve the near-field source localization efficiently and in other field also. • Wideband near-field source localization. Find an algorithm by using sparse recovery method to localize the wideband near-field sources with relative high performance and computational advantage. • Using circle array to localize near-field sources. ULA is the simplest array used in array processing, and it can only localize source within domain [−90◦ , 90◦]. Circle array can estimate omni-direction. • In this research, we discretize the searching domain to localize the near-field source. E. J. Candes provides a super high resolution algorithm using sparse recovery but no need to discretize the searching domain. And we can avoid the effect of grid mismatch. We test this method and it could work in near-field localization but it need to find a way to achieve a super high resolution as it works in far-field domain. 38 Performance analysis on sparse recovery A.1 A The RIP and the uniqueness of sparse recovery RIP is an important condition that provide guarantee to recovery approximated sparse vector. Definition 1. A matrix A satisfies the restricted isometry property (RIP) of order K if there exists a σK [28] ∈ (0, 1) such that (1 − σK )∥x∥22 ≤ ∥Ax∥22 ≤ (1 + σK )∥x∥22 (A.1) holds for all x ∈ ΣK . Denotes that ΣK = {x : ∥x∥0 ≤ K} as the set of all K-sparse vector. The definition of RIP indicates that there exists a unique solution for the optimization problem in (2.3) if the sensing matrix A satisfies the RIP of order 2K. And this can be proved as follow: Proof. If a matrix A satisfies the RIP condition of order 2K, it means ∃ σ2K ∈ (0, 1) such that (1 − σK )∥h∥22 ≤ ∥Ah∥22 ≤ (1 + σK )∥h∥22 holds for all h ∈ Σ2K . ⇒ ∥Ah∥22 > 0. Assume ∃ x1 , x2 ∈ ΣK satisfying bf y = Ax + w and x1 ̸= x2 , thus we obtain A(x1 − x2 ) = 0. Let h = x1 − x2 and h ∈ Σ2K due to x1 , x2 ∈ ΣK , which contradicts the fact that ∥Ah∥22 > 0. Therefore, x1 = x2 and the uniqueness has been proven. Additionally, [24] mentioned√that the solution of (2.2) is that of (2.1) if the RIP constants of A satisfies σ2K < 2 − 1. More details and applications on RIP can be seen in [24, 28, 29, 30, 31]. The induced parameter µ is unknown in general. The general approach to choosing µ, cross-validation, is mentioned in [32, 33, 34] as well as other approaches. 39 A.2 Performance of block sparse recovery Suppose that a measurement of a ULA are corrupted by bounded noise so that y = Ax + w where ∥w∥2 ≤ ϵ. Using the matlab tool CVX [26] to recovery x via the optimization problem (2.7) x̂ = arg min ∥x∥2,1 s.t. ∥y − Ax∥2 ≤ ϵ. (A.2) x∈C The B-RIP can be defined as Definition 2. A matrix A satisfies the block restricted isometry property (B-RIP) of order K if there exists a constant σK|I ∈ (0, 1) such that (1 − σK|I )∥x∥22 ≤ ∥Ax∥22 ≤ (1 + σK|I )∥x∥22 (A.3) holds for all K-block sparse vectors x which can be divided into N sub-blocks, x = [x1 , . . . , xN ], via I = {d1 , . . . , dN }, where dn = {i : xi ∈ xn } for n = 1, 2, . . . , N and i is the entry index of x. Then the performance bound of (2.7) can be derived in the following theorem. Theorem 1.√[21] Assume that a matrix Φ ∈ CM ×3N satisfies the B-RIP of order 2K with σ2K < 2 − 1 and let y = Φq + w where ∥w∥2 ≤ ϵ. Then the solution q̂ of optimization (2.7) obeys c0 ∥q − q(K) ∥2,1 c1 √ ∥q̂ − q∥2 ≤ √ +√ ϵ (A.4) 3 3 K where q(K) indicates the best K-block sparse approximation to q, such that q(K) is Kblock sparse and minimizes ∥q − d∥2,1 over all K-block sparse vectors d and both c0 and c1 will be given in lemma 1. The proof of above theorem can be found in [21]. A.3 Unbounded joint sparse recovery and its performance In order to recovery the sparse signal in the mismatch model (5.3), it can be refer to the given optimization problem with unbounded perturbation error eθ̄n and er̄n , the optimization problem [20] can be formulated as arg min x∈CN ,eθ ∈RN ,er ∈RN µ5 ∥x∥1 + ∥ (A + Aθ Eθ + Ar Er ) x − y∥22 (A.5) with Eθ = diag(eθ ) and Er = diag(er ). As we known, the model in (5.3) can be expressed as q z }| { Φ x [z }| {] y = A Aθ Ar eθ ⊙ x +w er ⊙ x = Φq + w, 40 (A.6) where Φ ∈ CM ×3N , and q ∈ C3N ×1 is a 3K-sparse vector. Then the optimization (A.5) can be written in a relaxed joint sparse recovery approach arg min x∈CN ,eθ ∈RN ,er ∈RN where ∥q∥2,1 = N √ ∑ µ6 ∥q∥2,1 + ∥y − Φq∥22 , (A.7) x2n + (eθ̄n xn )2 + (er̄n xn )2 . n=1 The performance bound for the optimization (A.7) can be derived based on the theorem 1 in the following lemma and the proof will be provided later. M ×3N Lemma satisfies the B-RIP of order 2K with √ 1. Assume that a matrix Φ ∈ C σ2K < 2 − 1 and let y = Φq + w where ∥w∥2 ≤ ϵ. Then the optimal solution q̂ of the optimization (A.7), where q̂ = [x̂; êθ ⊙ x̂; êr ⊙ x̂], satisfies ∥x̂ − x∥2 ≤ C1 ∥q − q(K) ∥2,1 + C2 ϵ ∥(êθ − eθ ) ⊙ x̂∥2 ≤ C3 ∥q − q(K) ∥2,1 + C4 ϵ ∥(êr − er ) ⊙ x̂∥2 ≤ C5 ∥q − q(K) ∥2,1 + C6 ϵ (A.8) (A.9) (A.10) where q(K) indicates the best K-block sparse approximation to q and √ c0 c1 (1 − σ2K ) + 2 2 C1 = (1 − σ2K ); C2 = c2 c2 √ 2 1 1 − σ2K −1 ) ( + )c3 C3 = (1 + ∥Ar ∥2 ∥Ar ∥2 1 − σ2K √ 2 1 − σ2K −1 1 C4 = (1 + ) ( + )(2 + c3 ) ∥Ar ∥2 ∥Ar ∥2 1 − σ2K √ 1 2 1 − σ2K −1 ) ( + )c3 C5 = (1 + ∥Aθ ∥2 ∥Aθ ∥2 1 − σ2K √ 1 − σ2K −1 1 2 C6 = (1 + ) ( + )(2 + c3 ) ∥Aθ ∥2 ∥Aθ ∥2 1 − σ2K √ 1 1 − σ2K √ c0 = 2 3 k− 2 1 − (1 + 2)σ2K √ √ 1 + σ2K √ c1 = 4 3 1 − (1 + 2)σ2K √ c2 = (1 − ∥∆θ ∥2 − ∥∆r ∥2 )(1 − σ2K ) − 2∥Φ∥2 (1 + ∥∆θ ∥2 + ∥∆r ∥2 ) c3 = 2 + ∥Φ∥2 (1 + ∥∆θ ∥2 + ∥∆r ∥2 ) with ∆θ = diag(eθ ) and ∆r = diag(er ). Proof. In theorem 1, we have c0 ∥q − q(K) ∥2,1 c1 √ ∥q̂ − q∥2 ≤ √ + √ ϵ. 3 3 K 41 (A.11) According to Cauchy-Schwarz inequalities, we obtain x − x̂ ∥q̂ − q∥2 = eθ ⊙ x − êθ ⊙ x̂ er ⊙ x − êr ⊙ x̂ 2 ) 1 ( ≥ √ ∥q̂ − q∥2 + ∥eθ ⊙ x − êθ ⊙ x̂∥2 + ∥er ⊙ x − êr ⊙ x̂∥2 3 And we know that ∥eθ ⊙ x − êθ ⊙ x̂∥2 = ∥eθ ⊙ (x − x̂) − x̂ ⊙ (eθ − êθ )∥2 ∥er ⊙ x − êr ⊙ x̂∥2 = ∥er ⊙ (x − x̂) − x̂ ⊙ (er − êr )∥2 . Defining dp = x − x̂, dθ = eθ − êθ , dr = er − êr and ϵ0 = ∥q − q(K) ∥2,1 . Using the inequality in (A.11) ∥dp ∥2 + ∥eθ ⊙ dp + x̂ ⊙ dθ ∥2 + ∥er ⊙ dp + x̂ ⊙ dr ∥2 ≤ c0 ϵ0 + c1 ϵ ⇒ ∥dp ∥2 (1 − ∥Θ∥2 − ∥R∥2 ) ≤ c0 ϵ0 + c1 ϵ + ∥x̂ ⊙ dθ ∥2 + ∥x̂ ⊙ dr ∥2 [ ] √ x̂ ⊙ dθ ≤ c0 ϵ0 + c1 ϵ + 2 x̂ ⊙ dr (A.12) 2 Using B-RIP on [Aθ , Ar ] [ ] [ ] √ x̂ ⊙ dθ x̂ ⊙ d θ 1 − σK ([Aθ , Ar ]) x̂ ⊙ dr ≤ [Aθ , Atheta ] x̂ ⊙ dr 2 2 0 x̂ ⊙ dθ ≤ Φ x̂ ⊙ dr 2 d p ≤ ∥Φq∥2 − Φ eθ ⊙ d p er ⊙ dp 2 ≤ 2ϵ + ∥Φ∥2 ∥d̂p ∥2 (1 + ∥Θ∥2 + ∥R∥2 ) (A.13) It is known that σK ([Aθ , Ar ]) < σ2K and from (A.12) and (A.13) we obtain ∥x̂ − x∥2 ≤ C1 ∥q − q(K) ∥2,1 + C2 ϵ. (A.14) Using B-RIP on Aθ 0 √ x̂ ⊙ dθ Φ 1 − σk (Aθ )∥x̂ ⊙ dθ ∥2 ≤ 0 2 ≤ 2ϵ + ∥Φ∥2 ∥d̂p ∥2 (1 + ∥Θ∥2 + ∥R∥2 ) +∥Φ∥2 ∥x̂ ⊙ dr ∥2 . (A.15) Combining (A.12), (A.13) and (A.15), we obtain ∥(êθ − eθ ) ⊙ x̂∥2 ≤ C3 ∥q − q(K) ∥2,1 + C4 ϵ 42 (A.16) In the same way we can get ∥(êr − er ) ⊙ x̂∥2 ≤ C5 ∥q − q(K) ∥2,1 + C6 ϵ (A.17) Proof has been done. A.4 Performance analysis of bounded sparse recovery In order to analyze the performance of the the constrained block sparse recovery, we recall the definition of B-RIP mentioned in Chapter 2. The sensing matrix Φ ∈ CM ×3N is to said obey B-RIP if there exist a constant σK 1 such that for every K-block sparse vector q ∈ C3N (A.18) (1 − σK )∥q∥22 ≤ ∥Φq∥22 ≤ (1 + σK )∥q∥22 . Using this definition, the performance bound for a constrained sparse recovery can be obtained by using the techniques mentioned in [21, 29, 27]. The performance bound for (5.4) is given in the following lemma. (√ )−1 Lemma 2. Assume that the B-RIP constant σ2K < 2c4 + 1 and ∥w∥2 ≤ ϵ. Then the optimal solution (x̂, êθ , êr ) to the optimization problem (5.4) satisfies ∥x̂ − x∥2 ≤ C7 ∥x − x(K) ∥1 + C8 ϵ ∥(êθ − eθ ) ⊙ x̂∥2 ≤ C9 ∥x − x(K) ∥1 + C10 ϵ ∥(êr − er ) ⊙ x̂∥2 ≤ C11 ∥x − x(K) ∥1 + C12 ϵ (A.19) (A.20) (A.21) where x(K) is the best approximation to x with K non-zeros entries, such that x(K) is K-sparse vector and minimizes ∥x − d∥1 over all K-sparse vectors d, and { √ [√ ] √ } c4 2(1 − σ2K ) 2 k 2 √ √ + 0.5∆θ + 0.5∆r + √ C7 = 1 − ( 2c4 + 1)σ2K 1 − σ2K k k √ 2(1 − σ2K ) 2 1 + σ2K √ C8 = 1 − ( 2c4 + 1)σ2K 1 − σ2K ) √ √ ( )−1 ( √ 1 − σ2K 1 2 +√ c4 ∥Φ∥2 C7 C9 = 1+ ∥Ar ∥2 ∥Ar ∥2 1 − σ2K ) √ √ )−1 ( ( √ 1 1 − σ2K 2 +√ C10 = 1+ (2 + c4 ∥Φ∥2 C8 ) ∥Ar ∥2 ∥Ar ∥2 1 − σ2K ) ( √ √ ( )−1 √ 1 − σ2K 1 2 C11 = 1+ +√ c4 ∥Φ∥2 C7 ∥Aθ ∥2 ∥Aθ ∥2 1 − σ2K ) √ √ )−1 ( ( √ 1 − σ2K 1 2 +√ C12 = 1+ (2 + c4 ∥Φ∥2 C8 ) ∥Aθ ∥2 ∥Aθ ∥2 1 − σ2K c4 = 1 + 0.25∆θ2 + 0.25∆r2 1 In the rest thesis, we use σK for the block-RIP constant instead of σK|I used in Chapter 2 unless we declare 43 In the above lemma, the performance bound for the box-constrained sparse recovery relies on the noise level and resolution of the sampling grid. The proof is given below: Proof. As we known, the optimization (5.4) is equivalent to optimization min x∈CN ,eθ ∈RN ,er ∈RN ∥x∥1 s.t. ∥y − Φq∥ ≤ ϵ. (A.22) Denote by x̂ = x + h the solution to (5.4) or (A.22). And we decompose h as h = ∑l−1 i=0 hIi , where hIi is the restriction of h to the set Ii which consists of K entries. We define I0 as the set of indices for which x is the K largest entries (since the noise x is not K-sparse but approximately). Chosen such that the norm of hI1 over I1 is the largest, the norm over I2 is the second largest and so on. To find the bound of h, we need use the inequalities below ∥h∥2 = ∥hI0 ∪I1 + h(I0 ∪I1 )c ∥2 ≤ ∥hI0 ∪I1 ∥2 + ∥h(I0 ∪I1 )c ∥2 . (A.23) • Bound on ∥h(I0 ∪I1 )c ∥2 According to Cauchy inequalities, we develop that l−1 1 1 ∑ ∥h(I0 ∪I1 )c ∥2 ≤ √ ∥hIi ∥1 = √ ∥hI0c ∥1 K i=1 K √ ∥hI0c ∥1 ≤ ∥hI0 ∥1 ≤ K∥hI0 ∥2 (A.24) (A.25) And according to the optimization (A.22) and Ii ∩ Ij = ∅ where i ̸= j, we obtained ∥x∥1 ≥ ≥ ∥xI0 + xI0c ∥1 ≥ ⇒ ∥hI0c ∥1 ≤ ∥x̂∥1 = ∥xI0 + hI0 ∥1 + ∥xI0c + hI0c ∥1 ∥xI0 ∥1 − ∥hI0 ∥1 + ∥hI0c ∥1 − ∥xI0c ∥1 ∥xI0 ∥1 − ∥hI0 ∥1 + ∥hI0c ∥1 − ∥xI0c ∥1 2∥xI0c ∥1 + ∥hI0 ∥1 Substitute into (A.24) and invoke the the fact that x = xI0c + xI0 and (A.25), we conclude that 2 2 ∥h(I0 ∪I1 )c ∥2 ≤ √ ∥x − x(K) ∥1 + ∥hI0 ∥2 ≤ √ e0 + ∥h(I0 ∪I1 )c ∥2 K K (A.26) where e0 = ∥x − x(K) ∥1 . • Bound on ∥hI0 ∪I1 ∥2 To bound ∥hI0 ∪I1 ∥2 , we invoke the relationship with ∥fI0 ∪I1 ∥2 . Define f = q̂ − q, and fIi indicates the K blocks sparse vector. According to the fact that ΦfI0 ∪I1 = ∑ Φf − l−1 i=2 ΦfIi , we obtain that ∥ΦfI0 ∪I1 ∥22 = ⟨ΦfI0 ∪I1 , Φf ⟩ − l−1 ∑ i=2 44 ⟨ΦfI0 ∪I1 , ΦfIi ⟩ (A.27) According to the definition of B-RIP, we obtained that √ |⟨ΦfI0 ∪I1 , Φf ⟩| ≤ ∥ΦfI0 ∪I1 ∥2 ∥Φf ∥2 ≤ 1 + σ2K ∥fI0 ∪I1 ∥2 ∥Φf ∥2 (A.28) Since f = q̂ − q and both q̂ and q are feasible ∥Φf ∥2 = ∥Φ(q̂ − q)∥ ≤ ∥Φq̂ − x∥2 + ∥Φq − x∥2 ≤ 2ϵ (A.29) √ |⟨ΦfI0 ∪I1 , Φf ⟩| ≤ 2ϵ 1 + σ2K ∥fI0 ∪I1 ∥2 (A.30) Then, According to the Lemma 2.1 in [29], we have l−1 l−1 ∑ ∑ | ⟨ΦfI0 ∪I1 , ΦfIi ⟩| ≤ σ2K (∥fI0 ∥2 + ∥fI1 ∥2 ) ∥fIi ∥2 i=2 i=2 ≤ √ 2σ2K ∥fI0 ∪I1 ∥2 l−1 ∑ ∥fIi ∥2 (A.31) i=2 Notice that f = ∥fIi ∥2 = where c4 Meanwhile l−1 ∑ i=2 ≤ = h eθ ⊙ h + x̂ ⊙ dθ er ⊙ h + x̂ ⊙ dr 0 h I i eθ ⊙ hIi + x̂Ii ⊙ dθ er ⊙ hIi x̂Ii ⊙ dr 2 √ c4 ∥hIi ∥2 + 2(0.5∆θ + 0.5∆r)∥x̂Ii ∥2 1 + 0.25∆θ2 + 0.25∆r2 . ∥x̂Ii ∥2 ≤ l−1 ∑ ∥x̂Ii ∥1 = ∥x̂(I0 ∪I1 )c ∥1 ≤ e0 (A.32) (A.33) i=2 Substitue (A.32) and (A.33) into (A.31), we obtain | l−1 ∑ ⟨ΦfI0 ∪I1 , ΦfIi ⟩| ≤ √ √ 2σ2K ∥fI0 ∪I1 ∥2 ( c4 ∥h(I0 ∪I1 )c ∥2 i=2 +2(0.5∆θ + 0.5∆r)∥e0 ) (A.34) Substitute (A.30) and (A.34) into (A.27), and invoke the B-RIP we obtain (1 − σ2K )∥fI0 ∪I1 ∥22 ≤ ∥ΦfI0 ∪I1 ∥22 { √ √ ≤ ∥fI0 ∪I1 ∥2 2ϵ 1 + σ2K + σ2K 2c4 ∥hI0 ∪I1 ∥2 [ ]} √ √c4 +2 2 √ + 0.5∆θ + 0.5∆r (A.35) K 45 and as we known that ∥hI0 ∪I1 ∥2 ≤ ∥fI0 ∪I1 ∥2 , we obtain the bound of ∥hI0 ∪I1 ∥2 as below { √ 2(1 − σ2K ) 2 1 + σ2K √ ∥hI0 ∪I1 ∥2 ≤ ϵ 1 − σ2K 1 − ( 2c4 + 1)σ2K √ [√ ] } c4 2 K √ + 0.5∆θ + 0.5∆r e0 + (A.36) 1 − σ2K K • Bound on ∥h∥2 Substitute the bound of ∥hI0 ∪I1 ∥2 and ∥h(I0 ∪I1 )c ∥2 into (A.23), we got ∥h∥2 = ∥x̂ − x∥2 ≤ C7 ∥x − x(K) ∥1 + C8 ϵ (A.37) • Bound on ∥(êθ − eθ ) ⊙ x̂∥2 and ∥(êr − er ) ⊙ x̂∥2 By using the similar method we used in Appendix A.2, we could get ∥(êθ − eθ ) ⊙ x̂∥2 ≤ C9 ∥x − x(K) ∥1 + C10 ϵ ∥(êr − er ) ⊙ x̂∥2 ≤ C11 ∥x − x(K) ∥1 + C12 ϵ The last, to make sure all the constant will be larger than zero, (√ )−1 larger than zero, thus, σ2K ≤ 2c4 + 1 . 2(1−σ2K ) √ 1−( 2c4 +1)σ2K (A.38) (A.39) should be According to lemma 2, SNR and grid offset error (GoE) are the two important factor that influence the performance of the sparse recovery algorithm in the mismatched situation. Here, we analyze performance of RBJS in different SNR and GoE, respectively by invoking Monte-Carlo simulation for 100 runs for each parameter that is used for comparison. The normalized-root-mean-squared-error (NRMSE) of the estimate is used as a measure of performance and is given by √ Eθ = E{|θ − θ̂|2 }/∆θ × 100% (A.40) √ Er = E{|r − r̂|2 }/∆r × 100% (A.41) where θ and r are the true location parameters of the sources and θ̂ and r̂ are the estimated results. We calculate the NRMSE for different signal-to-noise ratios (SNRs) and grid offset error (GoE), which is the ratio of perturbation to the size of grid cell in percentage. The SNR is ranging from 20 to 65 dB. The perturbation parameters eθ ranges from 0.1◦ to 0.45◦ and er ranges from −0.045λ to −0.005λ. Define the grid offset error in DOA domain GoEθ as GoEθ = eθ /∆θ × 100% and the grid offset error in range domain GoEr as GoEr = er /∆r × 100%, e.g., if the size of grid cell is 1◦ ×0.1λ, then the GoEθ and GoEr of a pair of perturbation, eθ = 0.1◦ and er = 0.005λ, are 10% and 5% respectively. In Fig.A.1, the source is 46 located at fixed location (0.3◦ , 1.28λ), and shows the performance of the R-BJS for different SNR. In the R-BJS algorithm, the NRMSE is decreasing as SNR increasing. In Fig.A.2, the SNR is fixed in 40 dB and shows the performance with different GoEθ and GoEr . The performance of R-BJS becomes worse as the GoE increasing and the results agree with lemma 2. NRMSE in DOA domain versus SNR 50 Eθ in % 40 30 20 10 0 20 25 30 35 40 45 SNR in dB 50 55 60 65 55 60 65 NRMSE in range domain versus SNR 40 Er in % 30 20 10 0 20 25 30 35 40 45 SNR in dB 50 Figure A.1: Performance of R-BJS versus SNR. A fixed single unit source located at (0.3◦ , 1.28λ) with different SNR. SNR ranges from 20 dB to 65 dB. The sampling grid has a resolution of ∆θ = 1◦ and 0.1λ within the domain [−4◦ , 3◦ ] and [1λ, 1.7λ]. 47 NRMSE in DOA domain versus GoE 50 GoEr = 5% Eθ in % 40 30 GoEr = 20% GoEr = 30% GoEr = 45% 20 10 0 10 15 20 25 30 35 40 45 35 40 45 GoEθ in % NRMSE in range domain versus GoE 20 GoE = 5% r 15 GoEr = 20% GoE = 30% Er in % r 10 GoEr = 45% 5 0 10 15 20 25 30 GoEθ in % Figure A.2: Performance of R-BJS versus GoE. A unit amplitude single source located close to the grid cell (0◦ , 1.3λ) with different perturbation parameters, eθ ranges from 0.1◦ to 0.45◦ and er ranges from −0.045λ to −0.005λ. SNR is 40 dB and T = 1. The sampling grid has a resolution of ∆θ = 1◦ and 0.1λ within the domain [−4◦ , 3◦ ] and [1λ, 1.7λ]. 48 Bibliography [1] Christopher R. 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