Parameter estimation from compressed and sparse measurements

Abstract

In this dissertation, the problem of parameter estimation from compressed and sparse noisy measurements is studied. First, fundamental estimation limits of the problem are analyzed. For that purpose, the effect of compressed sensing with random matrices on Fisher information, the Cramer-Rao Bound (CRB) and the Kullback-Leibler divergence are considered. The unknown parameters for the measurements are in the mean value function of a multivariate normal distribution. The class of random compression matrices considered in this work are those whose distribution is right-unitary invariant. The compression matrix whose elements are i.i.d. standard normal random variables is one such matrix. We show that for all such compression matrices, the Fisher information matrix has a complex matrix beta distribution. We also derive the distribution of CRB. These distributions can be used to quantify the loss in CRB as a function of the Fisher information of the non-compressed data. In our numerical examples, we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for deciding whether compression should be considered, based on the resulting loss in performance. Then, the effect of compression on performance breakdown regions of parameter estimation methods is studied. Performance breakdown may happen when either the sample size or signal-to-noise ratio (SNR) falls below a certain threshold. The main reason for this threshold effect is that in low SNR or sample size regimes, many high resolution parameter estimation methods, including subspace methods as well as maximum likelihood estimation lose their capability to resolve signal and noise subspaces. This leads to a large error in parameter estimation. This phenomenon is called a subspace swap. The probability of a subspace swap for parameter estimation from compressed data is studied. A lower bound has been derived on the probability of a subspace swap in parameter estimation from compressed noisy data. This lower bound can be used as a tool to predict breakdown for different compression schemes at different SNRs. In the last part of this work, we look at the problem of parameter estimation for p damped complex exponentials, from the observation of their weighted and damped sum. This problem arises in spectrum estimation, vibration analysis, speech processing, system identification, and direction of arrival estimation. Our results differ from standard results of modal analysis to the extent that we consider sparse and co-prime samplings in space, or equivalently sparse and co-prime samplings in time. Our main result is a characterization of the orthogonal subspace. This is the subspace that is orthogonal to the signal subspace spanned by the columns of the generalized Vandermonde matrix of modes in sparse or coprime arrays. This characterization is derived in a form that allows us to adapt modern methods of linear prediction and approximate least squares for estimating mode parameters. Several numerical examples are presented to demonstrate the performance of the proposed modal estimation methods. Our calculations of Fisher information allow us to analyze the loss in performance sustained by sparse and co-prime arrays that are compressions of uniform linear arrays.