Walter Scott, Jr. College of Engineering

Graduate Exam Abstract

Pooria Pakrooh
Ph.D. Final
Apr 09, 2015, 2:00 pm - 4:00 pm
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-orthogonally 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 normalized Fisher information
matrix after compression 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
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 co-prime
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.
Adviser: Ali Pezeshki
Co-Adviser: Louis L. Scharf
Non-ECE Member: Chris Peterson
Member 3: Edwin K. P. Chong
Addional Members: J. Rockey Luo
P. Pakrooh, A. Pezeshki, L. L. Scharf, D. Cochran, and S. D. Howard, “Analysis of Fisher Information and the Cramer-Rao Bound for Nonlinear Parameter Estimation after Compressed Sensing”, Submitted to IEEE Trans. on Signal Processing.

P. Pakrooh, A. Amini, and F. Marvasti, “OFDM Pilot Allocation for Sparse Channel Estimation”, EURASIP Journal on Advances in Signal Processing, vol. 59, March 2012.

P. Pakrooh, A. Pezeshki, and L. L. Scharf, “Modal Analysis Using Sparse and Co-prime Arrays”, in preparation.

P. Pakrooh, A. Pezeshki, and L. L. Scharf, “Threshold Effects in Parameter Estimation from Compressed Data”, in preparation.

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).

P. Pakrooh, A. Pezeshki, and L. L. Scharf, “Characterization of Orthogonal Subspaces for Alias-Free Reconstruction of Damped Complex Exponential Modes in Sparse Arrays”, 48th Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, Nov. 2-5, 2014 (invited paper).

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