Pooria Pakrooh

Ph.D. FinalApr 09, 2015, 2:00 pm - 4:00 pm

LSC376

PARAMETER ESTIMATION FROM SPARSE AND COMPRESSED 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-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

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 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.

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

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 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

Co-Adviser: Louis L. Scharf

Non-ECE Member: Chris Peterson

Member 3: Edwin K. P. Chong

Addional Members: J. Rockey Luo

Publications:

P. Pakrooh, A. Pezeshki, L. L. Scharf, D. Cochran, and S. D. Howard, â€œAnalysis of Fisher Information and the Cramer-Rao Bound for Nonlinear Parameter Estimation after Compressed Sensingâ€, Submitted to IEEE Trans. on Signal Processing.

P. Pakrooh, A. Amini, and F. Marvasti, â€œOFDM Pilot Allocation for Sparse Channel Estimationâ€, EURASIP Journal on Advances in Signal Processing, vol. 59, March 2012.

P. Pakrooh, A. Pezeshki, and L. L. Scharf, â€œModal Analysis Using Sparse and Co-prime Arraysâ€, in preparation.

P. Pakrooh, A. Pezeshki, and L. L. Scharf, â€œThreshold Effects in Parameter Estimation from Compressed Dataâ€, 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, â€œCharacterization of Orthogonal Subspaces for Alias-Free Reconstruction of Damped Complex Exponential Modes in Sparse Arraysâ€, 48th Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, Nov. 2-5, 2014 (invited paper).

P. Pakrooh, A. Pezeshki, L. L. Scharf, D. Cochran, and S. D. Howard, â€œAnalysis of Fisher Information and the Cramer-Rao Bound for Nonlinear Parameter Estimation after Compressed Sensingâ€, Submitted to IEEE Trans. on Signal Processing.

P. Pakrooh, A. Amini, and F. Marvasti, â€œOFDM Pilot Allocation for Sparse Channel Estimationâ€, EURASIP Journal on Advances in Signal Processing, vol. 59, March 2012.

P. Pakrooh, A. Pezeshki, and L. L. Scharf, â€œModal Analysis Using Sparse and Co-prime Arraysâ€, in preparation.

P. Pakrooh, A. Pezeshki, and L. L. Scharf, â€œThreshold Effects in Parameter Estimation from Compressed Dataâ€, 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, â€œCharacterization of Orthogonal Subspaces for Alias-Free Reconstruction of Damped Complex Exponential Modes in Sparse Arraysâ€, 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

N/A

ECE 514

ECE 516

ECE 614

ECE 651

ECE 652

MATH 519

STAT 530

N/A