On the performance of minimum redundancy array for multisource direction finding

As an application of power spectrum estimation, the multi-source direction finding has been evolved from conventional FFT method to Superresolution methods such as Multiple Signal Classification(MUSIC) algorithm. Uniform Regular Array(URA) was mainly used in all these approaches. The Minimum Redundancy array(MRA); a non-uniform thinned array which results in an input signals covariance matrix with minimum redundancy has been shown to have certain interesting properties for spectrum estimation. Only recently it was suggested to use the MRA for spatial estimation. The purpose of this research was to study the performance of this array in multi-source direction finding estimation and compare it to the result obtained with URA. Although the emphasis in this research is on using the popular MUSIC algorithm, other algorithms are also considered. Among the topics related to the MRA performance studied in the course of this research are 1. Effect of random displacement of the array element location on the performance of multi-source direction finding. 2. Performance of the MRA versus the URA using MUSIC and Minimum-Norm algorithms. 3. Performance of the MUSIC based direction finding using different covariance matrix estimates for URA and MRA. 4. The error probability of estimating the number (two in particular) of closely located sources with MRA versus URA

Signal Processing in Large Systems: a New Paradigm

For a long time, detection and parameter estimation methods for signal processing have relied on asymptotic statistics as the number $n$ of observations of a population grows large comparatively to the population size $N$, i.e. $n/N\to \infty$. Modern technological and societal advances now demand the study of sometimes extremely large populations and simultaneously require fast signal processing due to accelerated system dynamics. This results in not-so-large practical ratios $n/N$, sometimes even smaller than one. A disruptive change in classical signal processing methods has therefore been initiated in the past ten years, mostly spurred by the field of large dimensional random matrix theory. The early works in random matrix theory for signal processing applications are however scarce and highly technical. This tutorial provides an accessible methodological introduction to the modern tools of random matrix theory and to the signal processing methods derived from them, with an emphasis on simple illustrative examples

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

Toeplitz Inverse Eigenvalue Problem (ToIEP) and Random Matrix Theory (RMT) Support for the Toeplitz Covariance Matrix Estimation

"Toeplitzification" or "redundancy (spatial) averaging", the well-known routine for deriving the Toeplitz covariance matrix estimate from the standard sample covariance matrix, recently regained new attention due to the important Random Matrix Theory (RMT) findings. The asymptotic consistency in the spectral norm was proven for the Kolmogorov's asymptotics when the matrix dimension N and independent identically distributed (i.i.d.) sample volume T both tended to infinity (N->inf, T->inf, T/N->c > 0). These novel RMT results encouraged us to reassess the well-known drawback of the redundancy averaging methodology, which is the generation of the negative minimal eigenvalues for covariance matrices with big eigenvalues spread, typical for most covariance matrices of interest. We demonstrate that for this type of Toeplitz covariance matrices, convergence in the spectral norm does not prevent the generation of negative eigenvalues, even for the sample volume T that significantly exceeds the covariance matrix dimension (T >> N). We demonstrate that the ad-hoc attempts to remove the negative eigenvalues by the proper diagonal loading result in solutions with the very low likelihood. We demonstrate that attempts to exploit Newton's type iterative algorithms, designed to produce a Hermitian Toeplitz matrix with the given eigenvalues lead to the very poor likelihood of the very slowly converging solution to the desired eigenvalues. Finally, we demonstrate that the proposed algorithm for restoration of a positive definite (p.d.) Hermitian Toeplitz matrix with the specified Maximum Entropy spectrum, allows for the transformation of the (unstructured) Hermitian maximum likelihood (ML) sample matrix estimate in a p.d. Toeplitz matrix with sufficiently high likelihood