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Graduate Exam Abstract


Pooria Pakrooh

Ph.D. Preliminary

May 19, 2014, 8:00 AM -10:00 AM

ECE Conference Room (C101B)

Parameter Estimation from Compressed and Sparse Measurements


Abstract: In this dissertation, the problem of parameter estimation from compressed and sparse noisy measurements is studied. In the first part, 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 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 here are those that satisfy a version of the Johnson-Lindenstrauss lemma. Analytical lower and upper bounds on the CRB are derived for estimating parameters from randomly compressed data. These bounds quantify the potential loss in CRB as a function of Fisher information of the non-compressed data. Also, the effect of compression on performance breakdown regions for subspace 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, subspace methods 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 second part, we look at the problem of parameter estimation for p damped complex exponentials, from the observation of sparse and coprime samples of their weighted and damped sum. This problem arises in many areas such as modal analysis, speech processing, system identification and direction of arrival estimation. We are interested in the estimation of the mode parameters through characterization of the orthogonal subspace of the generalized Vandermonde matrix associated with the signal component of the sensor measurements. This characterization becomes useful when we are interested in maximum likelihood or least squares estimation of the modes from noisy measurements. Here, we present characterizations of the orthogonal subspaces for certain sparse and coprime arrays. After estimating the parameters representing the orthogonal subspace using Iterative Quadratic Maximum Likelihood (IQML) method, we find the roots of two polynomials associated with these coefficients and match up the roots. We show that for coprime geometries, matching up the roots removes aliasing and yields the actual modes in the noise-free case. Naturally, all of our developments in this part also apply to estimation of complex exponential modes from time series data.

Adviser: Ali Pezeshki
Co-Adviser: Louis Scharf
Non-ECE Member: Chris Peterson
Member 3: Edwin Chong
Addional Members: Jie Luo

Publications:
P. Pakrooh, L. L. Scharf, A. Pezeshki, and Y. Chi, "Analysis of Fisher information and the Cramer-Rao bound for nonlinear parameter estimation after compressed sensing", in Proc. 2013 IEEE Int. Conf. on Acoust., Speech and Signal Process. (ICASSP), Vancouver, BC, May 26-31, 2013, pp. 6630--6634.

P. Pakrooh, A. Pezeshki, and L. L. Scharf, "Threshold effects in parameter estimation from compressed data", in Proc. 1st IEEE Global Conference on Signal and Information Processing, Austin, TX, Dec. 3-5, 2013 (invited paper).


Program of Study:
ECE 514
ECE 516
ECE 614
ECE 651
ECE 652
MATH 519
STAT 530
N/A