## Generalized Vandermonde Matrices for Field Estimation Using Wireless Sensor Networks

**Phil Whiting **

Macquarie University Vice-Chancellor's Innovation Fellow

## Abstract

Random Vandermonde matrices with unit complex entries and their extended versions to

multiple dimensions, d, are a natural construction with a wide range of applications in fields as diverse as finance, signal processing, wireless communications, statistical analysis, security and biology. This stems from their close relationship with the discrete Fourier transform. In this talk we will discuss the asymptotic distribution of the random singular value of such matrices taking the special case of d = 1 (Vandermonde). To elaborate we may suppose that the phases of each column are drawn uniformly on the unit circle, for some large scale version of the matrix with L columns and N rows, N going to infinity so that L/N = c > 0 and c a given constant. The existence and properties of the limiting singular value can then be understood through its limiting moments. Such analysis is relevant to estimation error in wireless sensor networks where the goal is to measure a d-dimensional vector field using noisy estimates from the sensors. As it will be shown, the mean squared error estimate of the field can be obtained as a random eigenvalue expectation. As the number of sensors L and the bandwidth of the field N grows to infinity under a suitable scaling law this expectation is well approximated by its asymptotic distribution. The results of this analysis can be used in designing networks where the sensors locations are taken random according to some underlying density.

## About Phil Whiting

Phil Whiting received his BA degree from the University of Oxford, his MSc from the University of London and his Ph. D. was in queueing theory from the University of Strathclyde. After a post-doc at the University of Cambridge with Frank Kelly, Phil concentrated on wireless research, publishing with Stephen Hanly some of the early papers on mobile networks and multi-user Information Theory. In 1993 Phil was a consultant for the Telstra trial of Qualcomm CDMA in South Eastern Australia joining the Mobile Research Centre at the University of South Australia Adelaide. From 1997 to July of this year Phil was a researcher at Bell Labs and is currently a research fellow at MacQuarie University, Sydney Australia. Phil has been a Visiting Scholar at several institutions including, Eindhoven, Melbourne and Vrij University and a Visiting Professor at the University of Korea in Seoul and Brown University. Phil's interests include wireless networks, particularly stochastic models for resource allocation and information theory. His current research includes design and performance for HetNets, analysis of scheduling in CSMA networks using fluid models and theory for random matrices.