# Observation Uncertainty in Gaussian Sensor Networks

### Anand D. Sarwate

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2006-3

January 23, 2006

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-3.pdf

The term "sensor network" encompasses a wide range of engineering systems with dramatically different characteristics. We consider a specific class of sensor networks whose objective is to reconstruct a source at a central terminal. Our objective in this thesis is to quantify the asymptotic error in reconstructing the source as the number of data sources, sensors, and model complexity increases. We consider three types of estimation systems -- unconstrained estimators for vector Gaussian sources that are allowed direct access to the sensor observations, estimators for discrete sources that receive information via rate constrained links from the sensors, and estimators for scalar Gaussians whose input is the output of a multiple-access channel.

We first establish bounds on the optimal estimator performance of these networks using a centralized estimator with access to all of the sensor observations. We assume the observations are noisy linear functions of the source and are thus specified by a matrix. Because the asymptotic error depends only on the spectral properties of this matrix, we can use tools from matrix analysis to give bounds on the spectrum and error in terms of the entries of the matrix for a number of different scenarios. Finally, we look at the case where the matrix is partially unknown. In some cases we can estimate the matrix directly from the data and in others we must minimize the worst mismatch distortion.

These problems can also be looked at in a more information-theoretic framework. We look at a lossless distributed source coding problem in which the joint distribution of the sources is partially unknown. Although for any finite number of sensors standard multi-terminal source codes can easily be adapted to handle the model uncertainty across time, we show a rate penalty is incurred if the number of sensors and blocklength go to $\infty$ simultaneously. This represents one kind of tradeoff between delay and complexity for the scaling behavior of these systems.

Finally, we look at the case where the sensors must communicate their observations across an additive white Gaussian noise multiple-access channel. With a known correlation structure, the optimal error converges to $0$ as $1/M$, where $M$ is the number of sensors. However, a simple feedback scheme using $K$ bits broadcast to all sensors can provide a distortion that scales to $0$ as $M^{-K/(K+2)}$. We conjecture that providing similar feedback to an optimal source code will not improve the performance beyond that of our protocol.

**Advisor:** Michael Gastpar

BibTeX citation:

@mastersthesis{Sarwate:EECS-2006-3, Author = {Sarwate, Anand D.}, Title = {Observation Uncertainty in Gaussian Sensor Networks}, School = {EECS Department, University of California, Berkeley}, Year = {2006}, Month = {Jan}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-3.html}, Number = {UCB/EECS-2006-3}, Abstract = {The term "sensor network" encompasses a wide range of engineering systems with dramatically different characteristics. We consider a specific class of sensor networks whose objective is to reconstruct a source at a central terminal. Our objective in this thesis is to quantify the asymptotic error in reconstructing the source as the number of data sources, sensors, and model complexity increases. We consider three types of estimation systems -- unconstrained estimators for vector Gaussian sources that are allowed direct access to the sensor observations, estimators for discrete sources that receive information via rate constrained links from the sensors, and estimators for scalar Gaussians whose input is the output of a multiple-access channel. We first establish bounds on the optimal estimator performance of these networks using a centralized estimator with access to all of the sensor observations. We assume the observations are noisy linear functions of the source and are thus specified by a matrix. Because the asymptotic error depends only on the spectral properties of this matrix, we can use tools from matrix analysis to give bounds on the spectrum and error in terms of the entries of the matrix for a number of different scenarios. Finally, we look at the case where the matrix is partially unknown. In some cases we can estimate the matrix directly from the data and in others we must minimize the worst mismatch distortion. These problems can also be looked at in a more information-theoretic framework. We look at a lossless distributed source coding problem in which the joint distribution of the sources is partially unknown. Although for any finite number of sensors standard multi-terminal source codes can easily be adapted to handle the model uncertainty across time, we show a rate penalty is incurred if the number of sensors and blocklength go to $\infty$ simultaneously. This represents one kind of tradeoff between delay and complexity for the scaling behavior of these systems. Finally, we look at the case where the sensors must communicate their observations across an additive white Gaussian noise multiple-access channel. With a known correlation structure, the optimal error converges to $0$ as $1/M$, where $M$ is the number of sensors. However, a simple feedback scheme using $K$ bits broadcast to all sensors can provide a distortion that scales to $0$ as $M^{-K/(K+2)}$. We conjecture that providing similar feedback to an optimal source code will not improve the performance beyond that of our protocol.} }

EndNote citation:

%0 Thesis %A Sarwate, Anand D. %T Observation Uncertainty in Gaussian Sensor Networks %I EECS Department, University of California, Berkeley %D 2006 %8 January 23 %@ UCB/EECS-2006-3 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-3.html %F Sarwate:EECS-2006-3