3188 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 10, OCTOBER 2008 On Electromagnetics and Information Theory Marco Donald Migliore, Member, IEEE Abstract—Some connections are described between electromag- netic theory and information theory, identifying some unavoidable limitations imposed by the laws of electromagnetism to communi- cation systems. Starting from this result, the role of the degrees of freedom of the field in radiating systems is investigated. Different classes of antennas use the available degrees of freedom in different ways. In particular, a multiple-input multiple-output antenna is a radiating system conveying statistically independent information on more than one degree of freedom of the field. Applications of the theory to antenna synthesis and antenna characterization in complex environments are shown. Index Terms—Antenna measurement, antenna synthesis, an- tennas, channel capacity, multiple-input multiple-output (MIMO) systems, number of degrees of freedom (NDF). I. INTRODUCTION S OON AFTER the publication of Claude Shannon’s fun- damental papers on Information theory, a great deal of research has been devoted to the application of this theory to physics [1]. This effort has shown interesting connections between information theory and a large number of different fields of physics, including thermodynamics, optics, computa- tion, quantum theory and astrophysics. Instead, the connection between electromagnetic theory and information theory has been object of little research in the past, and only recently there is an interest toward the connections between these two theories [2]–[10]. As a consequence, a clear understanding of the unavoidable limitations imposed by the laws of electro- magnetism to our ability to communicate is not available yet. This understanding is fundamental since it enables the Shannon “mathematical theory of communication” to be connected to the real world. This paper represents a further contribution toward the un- derstanding of the relationship between the information theory and the electromagnetic theory, and how to use this relationship for practical applications. A key point to reach this goal is the use of Kolmogorov approach [11] besides the classic Shannon probabilistic approach [12] to introduce two important informa- tional quantities: the amount of information associated to a spa- tial distribution of the electromagnetic field and the amount of information reliably conveyed by the electromagnetic field. The advantage of the Kolmogorov approach in our context is that it is an operatorial-based communication theory, and consequently it naturally matches the classic electromagnetic theory approach. In particular, a strong connection between the “degree of com- plexity” of a set of functions, e.g., the infimum of the number Manuscript received June 25, 2007; revised February 15, 2008. Current ver- sion published October 3, 2008. The author is with the Microwave Laboratory of the University of Cassino, Via Di Biasio 43, 03043 Cassino, Italy (e-mail: mdmiglio@unicas.it). Digital Object Identifier 10.1109/TAP.2008.929444 of functions required to represent the set of functions within a given accuracy, and the “amount of information” that the set of functions can convey in presence of noise is shown. This result, that regards completely general communication systems, when applied to antennas naturally suggests a com- pletely new point of view, in which antennas are characterized from their ability to transmit information. A straightforward consequence is a novel unified approach to antennas that in- cludes classic antennas, adaptive antennas and multiple-input multiple-output (MIMO) antennas. The starting point to reach this goal is to consider the phys- ical system consisting of the field radiated by an electromagnetic source (usually the antenna, but more generally a complex en- semble of antennas and scattering objects) and observed on a given manifold. Like any physical system, the field can be rep- resented by means of a number of state variables. The minimum number of state variables allowing to represent the field on the observation manifold within a desired accuracy is the number of degrees of freedom (NDF) of the field [13]. A key result of this paper is that an electromagnetic source can use the available NDF of the field basically in two different ways, to approximate the field to a desired field distribution, or to send or receive statistically independent information. Classic antennas use the NDF in the first way, while MIMO antennas use them in the latter way. The “information based” approach to antennas proposed in this paper is not only conceptually interesting, but has prac- tical applications. Examples regarding antenna synthesis and antenna characterization in complex environments are shown in the paper. II. SOME LIMITATIONS IN THE AMOUNT OF INFORMATION ASSOCIATED TO THE ELECTROMAGNETIC FIELD A. The Amount of Information Associated to the Spatial Distribution of the Electromagnetic Field at an -Level of Uncertainty As preliminary step, let us recall some analytical properties of the field radiated by an harmonic source having finite spatial extension. In the following the time dependence ( being the frequency of the signal) will be understood and dropped. With reference to Fig. 1, the field radiated by an harmonic electromagnetic source placed in a domain D limited by a sur- face , observed on an observation manifold external to , can be evaluated as [14] (1) 0018-926X/$25.00 © 2008 IEEE