Walter Scott, Jr. College of Engineering

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