Handbook on Array Processing and Sensor Networks

CHAPTER 1

Wavefields

Alfred Hanssen

Department of Physics and Technology, University of Troms, Norway

1.1 INTRODUCTION

The theory of univariate stochastic processes [1, 2] forms the backbone of the analysis and processing of single sensor data. Stochastic processes X(t) are parameterized by a single free variable t, which is a timelike variable. By performing some appropriate processing on samples from realizations of the process X(t), one hopes to be able to infer physical properties about the source of the signal, about the transmission medium, or both. To aid in the development of processing techniques, it is customary to seek simplifying assumptions. A standard assumption is that of stationarity [3], which allows for drastic simplifications of moment functions and related quantities. Needless to say, the standard assumptions that we apply are often questionable for real-world situations.

However, our problems of interest in the physical world are often parameterized in four-dimensional space–time coordinates rather than in time alone. Examples of such are abundant; just think about any wave or fluctuation phenomena in physical space [4–7]. For engineering applications, the propagation and space–time distribution of radio waves is an important example [8–11], as is the propagation ...