8 research outputs found
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Structured Sub-Nyquist Sampling with Applications in Compressive Toeplitz Covariance Estimation, Super-Resolution and Phase Retrieval
Sub-Nyquist sampling has received a huge amount of interest in the past decade. In classical compressed sensing theory, if the measurement procedure satisfies a particular condition known as Restricted Isometry Property (RIP), we can achieve stable recovery of signals of low-dimensional intrinsic structures with an order-wise optimal sample size. Such low-dimensional structures include sparse and low rank for both vector and matrix cases. The main drawback of conventional compressed sensing theory is that random measurements are required to ensure the RIP property. However, in many applications such as imaging and array signal processing, applying independent random measurements may not be practical as the systems are deterministic. Moreover, random measurements based compressed sensing always exploits convex programs for signal recovery even in the noiseless case, and solving those programs is computationally intensive if the ambient dimension is large, especially in the matrix case. The main contribution of this dissertation is that we propose a deterministic sub-Nyquist sampling framework for compressing the structured signal and come up with computationally efficient algorithms. Besides widely studied sparse and low-rank structures, we particularly focus on the cases that the signals of interest are stationary or the measurements are of Fourier type. The key difference between our work from classical compressed sensing theory is that we explicitly exploit the second-order statistics of the signals, and study the equivalent quadratic measurement model in the correlation domain. The essential observation made in this dissertation is that a difference/sum coarray structure will arise from the quadratic model if the measurements are of Fourier type. With these observations, we are able to achieve a better compression rate for covariance estimation, identify more sources in array signal processing or recover the signals of larger sparsity. In this dissertation, we will first study the problem of Toeplitz covariance estimation. In particular, we will show how to achieve an order-wise optimal compression rate using the idea of sparse arrays in both general and low-rank cases. Then, an analysis framework of super-resolution with positivity constraint is established. We will present fundamental robustness guarantees, efficient algorithms and applications in practices. Next, we will study the problem of phase-retrieval for which we successfully apply the sparse array ideas by fully exploiting the quadratic measurement model. We achieve near-optimal sample complexity for both sparse and general cases with practical Fourier measurements and provide efficient and deterministic recovery algorithms. In the end, we will further elaborate on the essential role of non-negative constraint in underdetermined inverse problems. In particular, we will analyze the nonlinear co-array interpolation problem and develop a universal upper bound of the interpolation error. Bilinear problem with non-negative constraint will be considered next and the exact characterization of the ambiguous solutions will be established for the first time in literature. At last, we will show how to apply the nested array idea to solve real problems such as Kriging. Using spatial correlation information, we are able to have a stable estimate of the field of interest with fewer sensors than classic methodologies. Extensive numerical experiments are implemented to demonstrate our theoretical claims
DOA estimation for coexistence of circular and non-circular signals based on atomic norm minimization
In this paper, a gridless DOA estimation method with coexistence of non-circular and circular signals is proposed by employing an enhanced sparse nested array, whose virtual array has no holes. The virtual signals derived from both sum and difference co-arrays are constructed based on atomic norm minimization. Simulation results are provided to demonstrate the performance of the proposed method
Statistical Performance Analysis of Sparse Linear Arrays
Direction-of-arrival (DOA) estimation remains an important topic in array signal processing. With uniform linear arrays (ULAs), traditional subspace-based methods can resolve only up to M-1 sources using M sensors. On the other hand, by exploiting their so-called difference coarray model, sparse linear arrays, such as co-prime and nested arrays, can resolve up to O(M^2) sources using only O(M) sensors. Various new sparse linear array geometries were proposed and many direction-finding algorithms were developed based on sparse linear arrays. However, the statistical performance of such arrays has not been analytically conducted. In this dissertation, we (i) study the asymptotic performance of the MUtiple SIgnal Classification (MUSIC) algorithm utilizing sparse linear arrays, (ii) derive and analyze performance bounds for sparse linear arrays, and (iii) investigate the robustness of sparse linear arrays in the presence of array imperfections. Based on our analytical results, we also propose robust direction-finding algorithms for use when data are missing.
We begin by analyzing the performance of two commonly used coarray-based MUSIC direction estimators. Because the coarray model is used, classical derivations no longer apply. By using an alternative eigenvector perturbation analysis approach, we derive a closed-form expression of the asymptotic mean-squared error (MSE) of both estimators. Our expression is computationally efficient compared with the alternative of Monte Carlo simulations. Using this expression, we show that when the source number exceeds the sensor number, the MSE remains strictly positive as the signal-to-noise ratio (SNR) approaches infinity. This finding theoretically explains the unusual saturation behavior of coarray-based MUSIC estimators that had been observed in previous studies.
We next derive and analyze the Cramér-Rao bound (CRB) for general sparse linear arrays under the assumption that the sources are uncorrelated. We show that, unlike the classical stochastic CRB, our CRB is applicable even if there are more sources than the number of sensors. We also show that, in such a case, this CRB remains strictly positive definite as the SNR approaches infinity. This unusual behavior imposes a strict lower bound on the variance of unbiased DOA estimators in the underdetermined case. We establish the connection between our CRB and the classical stochastic CRB and show that they are asymptotically equal when the sources are uncorrelated and the SNR is sufficiently high. We investigate the behavior of our CRB for co-prime and nested arrays with a large number of sensors, characterizing the trade-off between the number of spatial samples and the number of temporal samples. Our analytical results on the CRB will benefit future research on optimal sparse array designs.
We further analyze the performance of sparse linear arrays by considering sensor location errors. We first introduce the deterministic error model. Based on this model, we derive a closed-form expression of the asymptotic MSE of a commonly used coarray-based MUSIC estimator, the spatial-smoothing based MUSIC (SS-MUSIC). We show that deterministic sensor location errors introduce a constant estimation bias that cannot be mitigated by only increasing the SNR. Our analytical expression also provides a sensitivity measure against sensor location errors for sparse linear arrays. We next extend our derivations to the stochastic error model and analyze the Gaussian case. We also derive the CRB for joint estimation of DOA parameters and deterministic sensor location errors. We show that this CRB is applicable even if there are more sources than the number of sensors.
Lastly, we develop robust DOA estimators for cases with missing data. By exploiting the difference coarray structure, we introduce three algorithms to construct an augmented covariance matrix with enhanced degrees of freedom. By applying MUSIC to this augmented covariance matrix, we are able to resolve more sources than sensors. Our method utilizes information from all snapshots and shows improved estimation performance over traditional DOA estimators