SOON AFTER the publication of Claude Shannon s fundamental

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1 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 electromagnetic theory and information theory, identifying some unavoidable limitations imposed by the laws of electromagnetism to communication 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, antennas, channel capacity, multiple-input multiple-output (MIMO) systems, number of degrees of freedom (NDF). I. INTRODUCTION SOON AFTER the publication of Claude Shannon s fundamental 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, computation, 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 electromagnetism 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 understanding 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 informational quantities: the amount of information associated to a spatial 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 complexity of a set of functions, e.g., the infimum of the number Manuscript received June 25, 2007; revised February 15, Current version published October 3, The author is with the Microwave Laboratory of the University of Cassino, Via Di Biasio 43, Cassino, Italy ( mdmiglio@unicas.it). Digital Object Identifier /TAP 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 completely 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 includes classic antennas, adaptive antennas and multiple-input multiple-output (MIMO) antennas. The starting point to reach this goal is to consider the physical system consisting of the field radiated by an electromagnetic source (usually the antenna, but more generally a complex ensemble of antennas and scattering objects) and observed on a given manifold. Like any physical system, the field can be represented 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 practical 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 surface, observed on an observation manifold external to, can be evaluated as [14] (1) X/$ IEEE

2 MIGLIORE: ON ELECTROMAGNETICS AND INFORMATION THEORY 3189 Fig. 3. geometrical interpretation of the sphere-covering approximation of the range of the radiation operator; the empty circles are the elements of an -net. Fig. 1. Geometry of the problem; D is the domain containing the sources, is the observation manifold. Fig. 2. Radiating system model; A models the radiation operator; M models the measurement operator, u the uncertainty level affecting the measured data z. where is the source current density in the space, is the dyadic Green s function [14], and the dot denotes the matrixvector product. We can discuss the above problem in a more abstract way, using classical tools of functional analysis, in which the function is an element in a metric space, and the function is an element in a metric space. In the following we restrict our attention to Hilbert space equipped with the usual norm on. In this space many concepts can be explained using a simple geometrical approach, helping to clarify some otherwise quite abstract concepts. The main results showed in this Section regarding approximation and information content are valid in much more general metric spaces. The interested reader can find more details in [15] [18]. In Fig. 2 an abstract model of the radiating system is drawn, wherein is the integral operator in (1), while the role of the measurement operator will be discussed in the following. Regarding the domain of the operator, we suppose that is a bounded set, e.g.,. Geometrically, the set of the source currents is contained in a sphere having finite radius. The square radius of the sphere represents the energy (radiated and stored) of the source. Since the observation manifold is outside the domain containing the sources, the kernel of the radiation operator is analytic. Consequently is a compact (or completely continuous) operator [14], and can be expanded using the Hilbert-Schmidt decomposition (or singular value decomposition) obtaining wherein,, are the left singular functions, the right singular functions and the singular values of the operator respectively, and denotes the inner product in. For more (2) details about the Hilbert- Schmidt decomposition the reader can refer to [19]. In our context it is useful to recall that the singular values are an infinite countable set of real, positive numbers, and due to the compactness of the operator [15], [19]. From (2) we can note that a compact operator maps a ball of radius in an ellipsoid having the th semi-axis (Fig. 3). It is useful to note that the semi-axis of the ellipsoid tends to become very short for increasing -values due to the behavior of the singular values. Broadly speaking, a compact operator transforms a sphere having infinite dimensions in an almost finite dimensional ellipsoid, provided that the radius of the sphere is finite. This is a geometrical illustration of the definition of a compact operator, e.g., an operator that maps an arbitrary bounded set into precompact set (e.g., a set whose closure is compact) [15]. Now, let us introduce a measure of the amount of information associated to the functional set, using the approach developed by Kolmogorov in [11]. The basic idea is that in practical instances the observable quantity is not directly, but the output quantity, let be, of a measurement process. This process is modelled in Fig. 2 as a linear operator wherein is a Hilbert space equipped with the usual norm on. The operator depends on the details of the measurement process, for example the kind, the number and the position of the elements of the receiving antenna, the noise level of the receivers and so on. Paralleling the approach following in [5] we will skip the details of the measurement system supposing that the measurement operator is the identity operator plus a quantity that will be called uncertainty, which is an element in a bounded set of space, let, wherein will be called uncertainty level. The presence of this quantity models an unavoidable physical problem: in any practical instance any element of can be measured only within a finite accuracy due to the presence of noise and measurement uncertainty. Accordingly, it is impossible to distinguish two elements whose distance is, let us say, less than [13]. A relevant consequence is that all the elements placed at a distance less than from an element can be substituted by without any loss of information. Repeating the process, we can identify a number of elements allowing to represent any element belonging to within an uncertainty level. From an informational point of view, if we consider as an information source, the extraction of such a set of elements is equivalent to a source coding process. In particular, the number of elements of the smallest of such sets (e.g., after eliminated all the redundancies) represents the amount of information associated to at an degree of accuracy. The evaluation of such quantity can be obtained by means of an -covering of, according to the approach outlined by Kolmogorov in [11].

3 3190 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 10, OCTOBER 2008 We recall that a system of sets is called an -covering of the set if the diameter of an arbitrary does not exceed and [11], [14]. A set is called an -net for the set if every element of the set is at a distance not exceeding from some element of [11]. With reference to Fig. 3, we can image an -covering of as a family of open balls with centres in and radius whose union includes. The minimal number of -balls required to cover, expressed in bits, is called the Kolmogorov -entropy (called also metric entropy, in contradistinction to the Shannon probabilistic entropy) [11], [16]. Any -net having a number of elements equal to the Kolmogorov -entropy allows to approximate at level using the smallest subset, e.g., in the most economical way, and will be called an optimal -net. With reference to Fig. 3, if we suppose that the -covering is minimal, the set of elements on which the balls are centred gives an optimal -net. For a more rigorous discussion regarding the relationship between -covering and -net in metric spaces the reader can refer to [11] and [16]. Since is precompact the -entropy turns out to be finite for any [11], [13]. In our specific context, it means that any pattern belonging to can be approximated by a pattern belonging to a finite ( -entropy) number of patterns for any accuracy. Note that any optimal -net represents a minimum length codebook able to represent any element of Y within a level of accuracy. Accordingly, the -entropy is the amount of information, measured in bits, associated to at an degree of accuracy [13]. It is interesting to note that an increasing of (e.g., ) increases the length of the semi-axes of the ellipsoid in Fig. 3. Consequently, also semi-axes associated to small singular values give a not negligible contribution, increasing the volume of the ellipsoid, and consequently the -entropy. In particular, if we leave the domain of unbounded we can obtain an arbitrarily large -entropy. This happens, for example, imposing a constraint only on the total radiated energy (associated to the radiated power), allowing an arbitrarily high level of reactive energy. Accordingly, superdirective sources [20] are able to give, at least theoretically, fields whose -entropy can be arbitrarily high. Starting from the sphere-covering approach it is possible to identify a strict connection between antenna synthesis and the Shannon rate distortion theory (see [21, Sec. 13.5], for a geometrical interpretation of the distortion theory using sphere-covering approach). However, in Section III we will focus our attention toward the classic problem of information theory, consisting of the reliable identification of a codeword belonging to a finite codebook from data observed at the output of a noisy channel. B. The Amount of Information Reliably Conveyed by the Electromagnetic Field in Presence of an -Level of Uncertainty Basically, in a communication system information is associated to different waveforms. Consequently, roughly speaking, the maximum amount of information conveyed by a communication channel is equal to the maximum number of different Fig. 4. Geometrical interpretation of the receiving process as search in the sphere-packed range of the radiation operator;z is the uncertainty-corrupted observed data, y the noiseless data, x the codeword. waveforms which are distinguishable also when the observed data are affected by uncertainty. In order to be distinguishable, the waveforms must be enough different, or equivalently their distance, evaluated in a suitable norm, must be larger than a quantity that depends on the uncertainty level. The mathematical tool that will be used to quantify the above qualitative observations was introduced by Kolmogorov in [11] and is based on the -packing of. An -packing of is defined as a family of open balls, with centers in and radius whose pairwise intersections are all empty [11], [15]. Geometrically, the problem consists of filling the set by means of not intersecting open balls having radius, as shown in Fig. 4. In our context, this means that the distance among the patterns associated to the centres of the open balls is not smaller than. Since is precompact the supremum (e.g., the least upper bound) of the number of open spheres requires to fill is finite for any. This number, expressed in bits, is the Kolmogorov -capacity of [11]. This result allows to quantify the maximum amount of information transmissible through the operator in presence of an uncertainty level on the observable quantity. In fact, let us suppose that the presence of noise and measurement uncertainty makes possible to distinguish only patterns whose distance is not smaller than. Since it is possible to encode a different information to each of such patterns, the number of bits that we can reliably recover by observing the (uncertainty corrupted) pattern is equal to the Kolmogorov -capacity, while the codebook is given by all the for which is the center of a sphere. Again, if we fix only the radiated energy, we leave the domain of the operator unbounded, obtaining an arbitrary large -capacity. However, in this case the codebook includes superdirective sources, e.g., sources having very high reactive energy compared to the radiated energy. According to the above approach, the operator acts like a communication channel, and in the following will be called channel operator. Summarizing, the electromagnetic field can convey an amount of information equal to the -capacity of the set of functions belonging to the range of the radiation operator, or in other words equal to the number of -separated patterns, wherein is fixed by noise and measurement uncertainty. Consequently the construction of the optimal codebook of a spatial communication system can be seen as the synthesis of different -separated patterns. However, it is useful to stress again that not all the current density distributions associated to -separated patterns can be physically realizable, since we can have superdirective sources. This point will be further discussed in Sections II-C and D.

4 MIGLIORE: ON ELECTROMAGNETICS AND INFORMATION THEORY 3191 Upper bound and lower bound of the -entropy and -capacity of the channel in space can be obtained directly from the geometrical interpretation in terms of sphere-packing and spherecovering, considering ellipsoids having respectively dimensions and dimensions, wherein denotes the number of singular values. Taking into account that the th semiaxis of the ellisposids is long, and supposing, we have [18] wherein is the -entropy (related to uncertainty level). The concepts of -entropy and -capacity are extremely useful to understand the theoretical limitations in the amount of information contained and transmissible by electromagnetic systems, but their application in practical electromagnetic problems is cumbersome. In fact, in electromagnetic theory the field is usually represented by superposition of suitably basis functions. This naturally suggests a different approach which is closer to classic tools used in electromagnetic theory for the approximation of, e.g., the identification of a basis with the minimum number of elements which allows a linear approximation of any element of within an error. Such a basis is called an optimal basis. In order to clarify the importance of such a basis, let us suppose that the dimension of an optimal basis is. Any element of is identified (at an -level of approximation) by the coefficients of the linear expansion. Furthermore, since the basis of the expansion is optimal, is also the minimum number of parameters required to identify any element of y within an approximation level using a linear approximation, and consequently fixes the number of variables to identify the state of the field radiated on (e.g., its spatial distribution) at a level of approximation. Accordingly, the dimension of an optimal basis is the number of degrees of freedom of at the -level of approximation (called in the following) [13]. The problem of identifying an optimal space, and hence the, can be rigorously solved by means of the Kolmogorov -width (or -diameter) of the set. The Kolmogorov - width of in is the infimum of the distance between and any -dimensional subspace of [17]. Note that the minimum such that the -width is not greater than fixes also the minimum dimension of linear subspaces approximating the elements of at level of accuracy, and hence the of. In particular, with reference to the Hilbert Schmidt decomposition in (2), the -width of is, while an optimal subspace is given by the [17]. Accordingly. the the of is equal to the number of singular values greater than, equal to e.g., In many cases the singular values of the compact operators have a step-like behavior, with a rapid decrease after the knee. In this caseallthesingularvalueswhose indexislargerthanaquantity, let be, are negligible for any practically reasonable uncertainty level. Consequently, the is scarcely dependent on (3). The relevant consequence is that in this case we can make referenceto the Number of Degrees of Freedom without explicitly indicating the approximation level, introducing the concept of [13]. Furthermore, with reference to (3), we can note that in this case and the -entropy number of elements at the center of the -spheres in the spherecovering as well as the -capacity number of elements at the center of the -spheres in the sphere-packing belong to an dimensional space. Now, let us consider the probabilistic Shannon theory, in which signal and noise are stochastic quantities. The connections between Kolmogorov and Shannon approach are discussed in [11]. For our purpose, we can note that Fig. 4 suggests a straightforward parallelism between the Kolmorogov -capacity approach and the approach used in Information Theory for bandlimited channels corrupted by additive white Gaussian noise (see for example [21, Sec. 10.1] for a sphere-packing interpretation of the Shannon capacity). Paralleling the classic Information Theory approach, we can use the Hilbert-Schmidt decomposition of the channel operator to decompose the channel operator in parallel Gaussian channels. In particular in the Appendix at the end of this paper it is shown that in a statistical approach each singular function associated to the first NDF singular values of the channel operator is potentially able to convey statistically independent information along the communication channel. As last observation, it is important to stress again the paramount importance of the concept of the. From a physical point of view, the is the minimum number of parameters that allows to identify any element of, e.g., any configuration of the electromagnetic field on the observation manifold, within a level of approximation suitable for practical applications, using a linear approximation. Since the spatial distribution of the field is uniquely identified by parameters, we can associate independent information to each of these parameters. Accordingly, while the -capacity measures the maximum number of distinguishable signals, the gives the effective number of dimensions of the space in which these signals lie, e.g., the dimension of the signal space. Consequently, the NDF has a fundamental role both in pattern approximation and in the information conveyed by the field suggesting a practical way to identify the connection between these two problems. This way will be followed in Section III. An intuitive discussion on the role of the in pattern approximation and information transmission is also reported in [22]. C. Some Examples of Spatial Channel Operators In order to discuss the properties of the spatial channel operator, let us consider a 2D scalar example regarding a circular domain D limited by a circumference having radius, and a circular observation curve having the same center of (inset of Fig. 5). In this case the singular system of the channel operator can be explicitly evaluated, and the singular functions and singular values are [23] (4)

5 3192 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 10, OCTOBER 2008 Fig NDF as a function of the Signal/Noise Ratio (SNR) normalized to n for different values of the distance d= of the observation circle from the circle including the sources; inset: geometry of the problem;. wherein is the Bessel function of order, is the nth order Hankel function of the second kind, is the free space wavenumber, and the index ranges from to. The representation in (2) (that considers a positive index and not increasing singular values for increasing index) can be straightforwardly obtained by rearranging the index. The singular values have a step like behavior with a knee after the first singular values. However, the decreasing rate of the singular values after depends on the distance between the observation points and the sources. In fact, even if the kernel of the integral operator is continuous, the presence of very close (in terms of wavelengths) singularities of the field (associated with the sources) to the observation domain causes fast variations in the kernel. Consequently, the field on the observation curve has peaks, whose reconstruction requires high order exponential functions, i.e., left singular functions associated to large indexes. Accordingly, the number of relevant singular values can be much larger than in case of close proximity of the sources [23]. In such a case the decreasing rate of the singular values is slow, and the and consequently the -capacity is strongly dependent on the ratio. Consequently the number of dimensions significantly increases as regards the, and the -capacity of the channel can grow significantly faster than the usual logarithmic law. In the following the ratio will be indicated simply as Signal/Noise Ratio (SNR), but it is understood that it is related to both active (radiated) and reactive power. If the distance between the field singularities and the observation curve is at least a couple of wavelengths, the kernel has a sufficiently smooth variation, and the number of singular functions required to reconstruct the field on turns out to be only (5) (6) slightly larger than and scarcely dependent on the SNR. Consequently, it is possible to define the of the space-domain signal [13]. In the following this quantity will be called (Space-domain signal ). The dimension of the space to be filled by open -spheres is equal to the SNDF and the -capacity of the channel can be estimated from (3). In order to clarify the dependence of the on the distance from the sources, we consider a simple example, regarding sources included in a circumference having radius 50 and an observation circumference at distance from ranging from 0.01 to. The SNR ranges from 1 db to 60 db. The normalized to is plotted in Fig. 5. The figure shows that when the observation distance is larger than the is a very slightly increasing function. In this case the can be considered constant over all the SNR range shown in the figure, and only slightly larger than. Note however that if the SNR is extremely large, the can be significantly greater than. This can occur when the uncertainty is extremely small, or when is extremely high (as happens for example in superdirective sources, in which the stored reactive energy is very high [24]). In the case of a very short distance on the contrary the increases rapidly. It is interesting to note that the distance required to obtain such an increase in the corresponds to the zone wherein the reactive field is preponderant compared to the radiated field. Clearly the increase of information is related to the reactive energy, and this could give some doubts about the possibility of recovering such information. However, the very basic concept in information theory is the possibility to distinguish different spatial configurations even if they are corrupted by noise. In this special case information is encoded in the electric and magnetic energy stored in the close proximity of the electromagnetic source, so that a proper detector must be sensible to variation of these quantities. For example, in microwave microscopy resonant cavities are used to restore information conveyed by the reactive near field of the object under measurement, breaking the Rayleigh resolution limit of [25]. The analysis of in reactive zone could be also useful for RFID applications. Finally, the Lorentz reciprocity theorem [27] permits the exchange of the role of source and observation point. Consequently, the in Fig. 5 is also equal to the in the case of sources on and observation curve, e.g., in the case of sources surrounding the observation curve. Let us consider now the full vector case. The definition of the is, of course, independent of the nature (scalar or vector) of the function. Indeed, the definition of the for the electromagnetic field in [13] is introduced considering directly the vector case. Since in practical cases the observable quantities are scalar components of the vector field, it is of interest to investigate how the of the field are distributed among the components, or equivalently how many components of the electric and magnetic field must be measured to reconstruct the vector field. Firstly, in the electrodynamics the currents and the charges are related by the equation of continuity, and consequently electric and magnetic fields are related each to the other. Further-

6 MIGLIORE: ON ELECTROMAGNETICS AND INFORMATION THEORY 3193 more, the field outside a surface including all the sources is uniquely defined by the two tangential components of the electric (or magnetic) field on the surface due to the uniqueness theorem. Consequently, in this case it is possible to estimate the of the (vector) electromagnetic field by evaluating the of two (tangential) components of the electric field on the observation surface. Let us consider a simple example consisting of electromagnetic sources spatially limited by a spherical surface having radius, and a spherical observation manifold having radius concentric to. The field outside can be expanded in spherical harmonics [27] that represent an optimum basis in. Only a finite number of spherical harmonics are required to represent the field on within a finite approximation. Such a number of harmonics is the of the field at the required level of approximation [28]. The number of spherical harmonics required to represent the field on, and hence the, for a not superdirective source tends to for and for an observation surface at least a couple of wavelengths from [27]. In particular, the field on is represented by means of radial TE and radial TM modes, that can be reconstructed from the knowledge of the two components of the electric (or magnetic) field tangent to, each of them characterized by number of degrees of freedom. The of the vector field is consequently obtained by the sum of the number of degrees of freedom of these two scalar components. Note that when the radiating system becomes electrically small the does not tend to zero, and the formula cannot be applied. This is due to the fact that any electrically small antenna acts like a superdirective source [27], increasing the reactive energy compared to the radiated energy when. Even if theoretically no upper bound exists for the, the fast increasing of the Q factor limits in practice the potentially useful modes to the lowest six ones [29]. As last observation, the field radiated by an electromagnetic source having finite size is an analytic function on the observation domain [26], and hence has an infinite number of degrees of freedom. However in practical instances we can observe only the portion of this function falling on the observation domain. Furthermore, also if we extend the observation domain to the whole space, we do not have access to the analytic continuation of the observed field. In practice, we are able to observe a noise-corrupted analytic function only on an observation domain having finite extension. The consequence of this lack of knowledge is that an upper-bound exists for the number of degrees of freedom associated to the spatial domain even if we extend the observation domain to the whole space [5]. For a not superdirective radiating system, such an upper-bound is of the order of [31], wherein is the area of the smallest convex surface including the source, and is the wavelength. The existence of an upper-bound for the spatial is an important difference between space-domain and time-domain communication systems. In fact, since it is possible to observe a (temporal) signal for an arbitrarily long temporal interval, no in time domain communica- upper-bound exists for the tion systems. D. An Example of Space-Time Channel Operator In Section II-A the theory was explained with special reference to the radiation operator, e.g., considering a pure spatial channel [5], in which information are encoded in spatial variation of the electromagnetic field on the observation surface. Of course the theory is general, and other channel operators can be considered. Operators working on time-domain functions give the classic time-domain communication channels, while operators working on signals defined both in time domain and space domain give the space-time communication channels used in MIMO communication system analysis. However, Hilbert-Shmidt decomposition of the operators, and in particular operators working on functions defined in the space-time domain, is analytically cumbersome. In this Section a simple upper bound for the channel capacity of space-time communication systems is presented, using band limitation approximation of the space-time received signal. The use of band-limited approximation is advantageous in our context since there is a large and well established literature on the optimal basis and on the for this class of functions [33]. In particular, the results showed in this section are a straightforward extension of the results regarding the well-known time-domain bandlimited channel corrupted by a stochastic process modelled as additive white Gaussian noise (AWGN) [21]. In the following the TNDF, SNDF and STNDF will denote respectively the of the time-domain signal, of the space-domain signal (e.g., the spatial distribution of the electromagnetic field on the observation surface) and of the space-time-domain signal. Let us consider a time-domain bandlimited signal having (time-domain) bandwidth (wherein and are the minimum and maximum frequency of the signal), radiated by not superdirective sources placed inside a convex surface, and observed on a curve, on which we introduce a proper parameterization, wherein is the curvilinear abscissa along. As discussed in [30], [31], for each frequency of the temporal signal we can identify a spatial bandwidth for the field observed on. For the sake of simplicity, as a first step in this section we suppose that the spatial bandwidth can be considered flat in the time bandwidth, e.g.,. Accordingly, the field can be expanded in the Whittaker-Kotel nikov-shannon (WKS) sampling series [32] wherein is the (time-domain) field observed at time in the point of coordinate along the observation curve, is related to a suitable phase function subtracted from the field [32], is the free space light velocity, and the number of (spatial) samples in the representation is practically equal to the. The formula in (7) gives a simple explanation regarding the limits of space-time communication systems. The value of the field (and consequently also of any signal carried by the electromagnetic field) in a point of the observation manifold can be obtained by a linear combination of the field in other points (7)

7 3194 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 10, OCTOBER 2008 Fig. 6. The space-bandlimited time-bandlimited AWGN channel. of the observation manifold provided that a proper time shift is taken into account. The number of positions that carry independent information is finite, and equal to the for non superdirective sources, wherein is the length of the observation curve. In practice the is an upperbound for the number of SISO subchannels that can be obtained for any configuration of TX antennas spatially limited by and RX antennas placed on [5]. Let us suppose that the received signal is corrupted by AWGN noise having power (Fig. 6). The observable quantity is consequently: wherein is the (space-time) bandlimited AWGN affecting the observed data. Under the hypothesis that the time-signals are bandlimited with bandwidth and observation interval, the received signal can be represented by a double WKS series, in which the number of temporal samples is essentially [33] and the number of spatial samples is essentially [13]. The space-time number of degrees of freedom (STNDF) of the flat spatial-bandwidth space-time channel is. The channel capacity (bits/s) of the space-time communication system can be obtained paralleling the classic approach regarding the time-domain bandlimited channel [12], obtaining wherein is the received average power and has the same role of the noise spectral density defined in the classic time-domain signal channel Finally, we can note that the supremum of the amount of information that can be transmitted along the channel in a space interval and a time interval is not greater than wherein is the energy of the signal. As discussed in Section II-C, it is theoretically possible to increase the in a spatial interval beyond the standard value of by means of superdirective radiating systems [4]. Since superdirective sources have small bandwidth, if we increase the SNDF significantly above, we also obtain a decrease of the TNDF, so that the most important quantity in a communication system is the STNDF. Let us consider now the case of a space-time channel having (time) bandwidth with an average power constraint. Following the classic approach of Information Theory, the (8) (9) channel is divided into a large number of narrow sub-channels having bandwidth, in which the spatial bandwidth can be considered practically constant. The spatial bandwidth is a linear function of the frequency, and will be expressed as, wherein the constant is the spatial bandwidth for unit frequency, while, and do not depend on the frequency [31]. The channel capacity can be obtained by maximizing the expression with the constraint (10) (11) Using the method of Lagrange multipliers we obtain that the channel capacity is (12) wherein and. Comparing (12) multiplied by and (9), we obtain the value of the. Again, the STNDF is the product of two quantities, the TNDF, equal to 2BT, and a quantity,, that represents the SNDF of the wide-band communication system. III. APPLICATION TO ANTENNAS A. A Unified Approach to Classic, Adaptive and MIMO Antennas As discussed in Section II-B, there is a strong connection between the degree of complexity of a set of functions, e.g., the defined as the infimum of the number 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. Broadly speaking, the is a money that can be spent in two different ways: to obtain a function that concentrates the energy in some desired intervals, or to send statistically independent information. In the first case the degrees of freedom are used in the approximation theory sense, while in the second case they are used in the communication theory sense. This is indeed exactly what happens in antenna synthesis. In order to clarify the double role that the of the field can play, let us consider an antenna consisting in electromagnetic currents having a finite spatial extension as in Fig. 1. As first step, we suppose that the source radiates in the free space, and only one state of polarization is used to transmit information. In the classic synthesis approach the receiving antenna is in the far-field region at a distance such that it is seen as point-like from the radiating system. In presence of only AWG noise, under

8 MIGLIORE: ON ELECTROMAGNETICS AND INFORMATION THEORY 3195 the hypothesis that the receiving antenna aperture is unitary, the channel capacity is (13) wherein is the power at the input of the transmitting antenna, is the directivity (supposed equal to the antenna gain) of the transmitting antenna, is the path loss, the noise power, and the bandwidth of the time-domain signal. In order to show the role played by the in the channel capacity, let us consider a sphere centered in the transmitting antenna whose radius tends to infinity as observation surface. By expanding the radiation operator (1) as in (2) we identify the of the field. Note that the choice of the observation surface maximizes the available. In fact, any further degree of freedom is associated to the analytical continuation of the field, e.g., to singular values of the radiation operator that are negligible in the case of not superdirective source. This will be called in the following. The goal of the antenna designer is to set the coefficients of the currents whose distributions are given by the left singular functions, in order to maximize the channel capacity. This goal can be reached by increasing the directivity of the antenna. Indeed, there is a strict relationship between the directivity and the number of singular functions used in the synthesis problem [24]. For example, the maximum directivity of a source enclosed in a sphere having radius is proportional to the number of singular values (in the specific case the spherical harmonics) used to synthesize the pattern [34]. The practical limitation of the directivity of antennas is due to the fact that only a finite number of singular functions can be effectively used in the synthesis [20], or equivalently that the effective is finite. For example, in the case of spherical source discussed at the end of Section II-B, the directivity of the radiating system is practically limited to almost [34]. By using singular functions associated to singular values after the knee we can increase the directivity at any desired value. However, in this case we have a superdirective source. Among the many well known problems regarding the superdirective sources, we can note that the bandwidth of the system decreases. Since an increase of the directivity gives only a logarithmical increase of the channel capacity, while a decrease of the bandwidth gives a linear decrease of the channel capacity, generally we would obtain a decrease of the channel capacity using superdirective sources even if we were able to keep the antenna losses at a negligible level. Summarizing, in the above example all the available are used to obtain a pattern that maximizes the directivity. Let us suppose that the we must synthesize a receiving antenna in presence of a number of interference signals modelled as AWGN. In this case it is advantageous to filter out the interference signals in order to increase the Signal/Noise ratio. In order to reach this goal, a number of available degrees of freedom is used to synthesize nulls toward the directions of the interference signals. Again, the available limits the performance of the system, both in terms of maximum number of interference signals that can be filtered out, and in terms of tradeoff between the directivity and the number of nulls of the pattern. Clearly, the maximization of the channel capacity is obtained by maximizing the term inside the logarithm function of the channel capacity expression. This is exactly the goal of optimal beamforming algorithms, like Widrow or Howells-Applebaum algorithms [35], used in the adaptive antennas. It is interesting to note that adaptive antennas are used in a double role. On a side, they are used to synthesize a pattern with nulls toward the angle of arrivals (AoA) of the interferences, on the other side they are used to identify the AoA of the signals, e.g., to extract information from the environment. A well known result is that an array of radiating elements can identify the AoA of up to uncorrelated signals. However, the identification of the AoA is related to the possibility to distinguish the subspace of the received signals correlation matrix associated to the signal (the signal space) from the subspace associated to the noise (the noise space) [35]. If we increase above the, the spreading of the singular values associated to the signal space tends to fast increase. In this case the presence of noise tends to cover some signal space eigenvalues, making impossible to divide the signal space from the noise space, and consequently also the identification of the AoA of the signals. Accordingly, theoretically we do not have limitation regarding the number of signals that we can identify, but when the number of (uncorrelated) signals becomes significantly larger than the a situation similar to the superdirectivity occurs, so that in practice the number of signals is limited by the electrical dimension of the antenna. In practice we have basically the same limitations when the antenna is used to synthesize a pattern, and when it is used to extract information from the environment. Let us suppose now that we have to synthesize a transmitter antenna in the case of not point-like receiving antenna. By expanding the radiation operator relating the source currents and the observation curve wherein the receiving antenna is placed, we can evaluate the available. Due to the extremal properties of the singular values, the first singular function assures the maximum concentration of the power in the area covered by the receiving antenna, and consequently the maximization of the argument of the logarithm in the channel capacity expression, obtaining the following bit rate: (14) wherein is the first singular value. This solution is known as MIMO beamforming. However, we can follow a different strategy, associating statistically independent information at each singular function. For sake of simplicity, let us suppose that all the relevant singular values of the radiating operator are constant. In this case we have that (15) that is larger than the bit rate in (14) in the case of significantly greater than zero (note that in the case of not constant singular values the maximization of the capacity requires to distribute the available average power among the different spatial

9 3196 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 10, OCTOBER 2008 subchannels associated to the singular values; the solution of the problem is given by the so called water-filling algorithm [38]). This strategy is followed in spatial multiplexing MIMO systems. Accordingly, in practice a MIMO antenna is a radiating system that conveys statistical independent information on more than one spatial degree of freedom. Summarizing, if the NDF of the field incident on the receiving antenna is only one, the maximization of the received power (or of the SINR in presence of not cooperative interference sources) gives also the maximization of the bit rate, since it is not possible to distinguish different radiated field spatial configuration, and information can be encoded only by means of temporal variations. In this case, classic antenna synthesis based on an energetic approach allows also the maximization of the bit rate. However, if the incident field has more than one degree of freedom, the maximization of the received power does not assure maximization of the bit rate. In this case it is more advantageous to use the available degrees of freedom of the incident signal to transmit statistically independent information, e.g., to encode information also by modulating the spatial distribution of the field. Since the distance between the transmitting and receiving antennas is very large in terms of wavelengths, the local incident field on the receiving antenna is usually a plane wave. Furthermore, information is usually associated to only one of the two independent polarizations of the vector field. In this case the incident field has only one degree of freedom and the classic synthesis approach, based on an energetic point of view, gives the same results of an antenna synthesis based on an informational point of view (e.g., whose goal is not to control the distribution of the radiated power in the space, but only to maximize the bit rate). Note however that if we are able to use both the polarizations of the incident plane wave, we have two degrees of freedom of the field also in the classic free-space far-field link. Accordingly, the double-polarized antennas are a particular case of MIMO antennas [22]. A further important case regards antennas operating in complex environments, e.g., environments containing a large number of scattering objects. In fact, in case of dense scattering environment the incident field is the superposition of a large number of plane waves, and hence has more than one degree of freedom. From a mathematical point of view, the channel operator is an integral operator like (1), wherein the Green s function must be calculated in presence of scattering objects. The integral operator is compact provided that the observation curve does not intersect sources of the field, and the general discussion regarding the does not change. However, due to the presence of the secondary sources associated to the scattering objects, the on a given observation manifold is usually higher than the on the same manifold in the case of absence of scatterers. Consequently, the presence of scattering objects allows to have a value of higher than one also on observation domains having small (only some wavelengths) length, as happens in practical MIMO applications. Of course, the in absence of scatterers is an upper bound for the of the field, provided that the scattering properties of the objects do not change. In fact, the possibility to control the scattering properties of the objects gives further degrees of freedom to modify the field configuration. A practical application of controllable scattering objects is represented by MIMO parasitic antennas [36], [37]. In these antennas parasitic elements terminated on electronically controllable loads are used as low cost solution to increase the of the field compared to the of the radiating system consisting of the sole active antennas. B. Application of the Double Role of the NDF in Antenna Synthesis According to the above observations, in complex environment it is advantageous to modify classical approaches in order to use the available NDF of the received field to send statistically independent information. Let us consider an example of such an approach. One of the techniques proposed in the framework of personal communication system is the spatial filtering for interference reduction (SFIR), consisting in imposing the maximum of the radiation pattern of the antenna toward the subscriber of interest, and imposing nulls in the direction of interference sources using adaptive antennas. In terms of degrees of freedom, SFIR uses some degrees of freedom to impose nulls of the field pattern, and the remaining ones to maximize the power received by the subscriber of interest. Let us suppose that the system works in a dense scattering environment, as often happens. In this case we can use part of the degrees of freedom to impose nulls of the field on the elements of the antenna that we do not want to illuminate, let us call it antenna D, and use the remaining degrees of freedom to send statistically independent information to the antenna of interest, let us call it antenna B, obtaining a MIMO system with null constraint. The transmitting antenna will be called antenna A. We consider a preliminary step, in which A and D cooperate to identify the channel matrix from A to D. In the classic MIMO matrix notation [38], [39], we have, wherein is the vector collecting the signals at the input of the transmitting array A, having radiating elements, is the vector collecting the signals at the input of the receiving array D, having radiating elements, and is the channel matrix. In order to null the field on D, we must impose. The projector onto the nullspace of is, wherein and are the right singular vectors of the singular value decomposition (SVD) of associated to singular values of the matrix having zero value [40]. Then the transmission toward the antenna B having M elements starts. Using the standard MIMO notation, in absence of null constraint we have, wherein is the vector collecting the signals at the input of the transmitting array is the vector collecting the signals at the input of the receiving array B, and is the channel matrix. In order to assure the null toward D, we consider the subset of lying in the null of. Consequently, the input of the array A is the vector, and the input-output relationship becomes. Accordingly, in presence of null constraint we have an equivalent channel matrix. Using classic MIMO approach [38], [39] we are able to maximize the bit rate toward B, without interfering with D.

10 MIGLIORE: ON ELECTROMAGNETICS AND INFORMATION THEORY 3197 Fig. 7. Channel capacity in the case of MIMO system with null constraint (solid line), MIMO system without null constraint (dotted line) and MIMO beamforming with null constraint (dashed line); in the inset a scheme of the geometry of the problem is drawn; the large filled circles represent the scattering objects, randomly placed. In the following a simple 2-D example is reported. The scenario consists in 40 2-D point-like scatterers placed in an area, and an array (antenna A) consisting of elements placed along the axis, with the central element placed in (see inset of Fig. 7). The antenna D is a linear array of elements, parallel to antenna A with the central element placed in. Then the average channel capacity considering an array (antenna B) parallel to the antenna A and having elements, whose central one is placed in with ranging between 4 and 45, is evaluated. The average channel capacity is evaluated considering 100 different random scenarios. The contribution of the line of sight (LOS) between the transmitting and the receiving array is subtracted, and only the scattered field is considered in the numerical simulations. The channel capacity, evaluated in the case of uniform power distribution between the elements of the transmitting antennas (e.g., no waterfilling solution), is plotted in Fig. 7 as solid line. In the same figure the channel capacity in the case of MIMO system without null constraint is plotted as dotted line, while the MIMO beamforming case, that maximizes the received energy with the null constraint, is plotted as dashed line. The plots confirm that the hybrid energetic/informational solution allows to assure the null constraint, with better performance compared to the pure energetic solution (e.g., the dashed line). C. Application of the Double Role of the NDF in Antenna Characterization A further simple example of the use of the double role of the NDF is in the MIMO antenna characterization. As preliminary step, let us recall that a well established method for characterization of classic antennas is based on near-field measurements. In practice, this technique allows to identify the of the antenna under test from measurements of radiated power density [41]. Now let us consider the characterization of a MIMO antenna. From an informational point of view, the quantity of interest is the amount of statistically independent information that we can obtain at the output of the MIMO antennas. This goal can be reached by analyzing the eigenstructure of the covariance matrix of the output signals, under the classic hypothesis of Gaussian distribution of the signals. Unfortunately, such a quantity depends not only on our antenna, but on the whole communication system, e.g., also on the transmitting antenna and on the environment. In order to identify a quantity that allows to characterize the antenna itself from an informational point of view, we can use the double role of the SNDF. In fact, according to the observations outlined in this paper, a MIMO antenna is basically a (usually linear) operator that transforms a signal belonging to a theoretically infinite dimensional space (the incident electromagnetic field) into a signal belonging to a finite dimensional space (the signals at the output of the elements of the antenna). The number of dimensions of the output space (e.g., the received signal space) is equal to the number of elements of the MIMO antenna, let be, while the number of (effective) dimensions of the input space is equal to the SNDF. This number cannot be larger than the of the field radiated by the antenna in the whole space, that has been denoted as. Accordingly, for a fixed frequency the antenna operator is completely characterized (at a fixed degree of accuracy) by a table of elements. This table, that can be obtained from standard near-field measurements [41], is an intrinsic property of the antenna, and allows to characterize the information content associated to the received signal space for any communication channel configuration. A simple example of application of the above observations regards a MIMO antenna having elements, and spatially limited by a spherical surface having radius. The field radiated by each element of the antenna, with the other elements terminated on the working loads, is expanded in spherical harmonics, as discussed in Section II-C. The table of coefficients allows to evaluate the covariance matrix of the received signals, and consequently also the channel capacity, after fixing the environmental model. This approach was followed by Gustafsson and Nordebo in the case of Rayleigh environment (e.g., the so called rich scattering environment) [10]. Note that the possibility of characterizing the antenna from an informational point of view using methods (e.g., near-field measurements) currently used to characterize the antenna from an energetic point of view is a practical application of the double role of the degrees of freedom. This observation allows to extend the well established literature on the characterization of classic antennas to MIMO antennas. As an example, a widely adopted method is to sample the field radiated by the antenna using near-field measurement systems. In particular, according to [31], [41], the field radiated by each element of the MIMO antenna can be represented by a proper sampling representation whose number of terms is only slightly larger than the. This suggests to sample the field in these points using classic near-field measurement systems. Then the field is interpolated in a large number of directions, and the correlation matrix is numerically evaluated by considering uncorrelated waves impinging from these directions.

11 3198 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 10, OCTOBER 2008 Fig. 8. Eigenvalues of the correlation matrix for a MIMO cube antenna in the case of uncorrelated signals impinging from an half-space. Fig. 9. Eigenvalues of the correlation matrix in the case of a TX cube antenna and a RX cube antenna placed in a corridor; in the inset the plan of the corridor, with the position of the TX antenna (filled circle) and the positions of the RX antennas (thick line), is shown. A practical problem regarding MIMO antennas is that they rarely work in a true dense scattering environment. Consequently, it is of interest to study the loss of performance when the antenna works in different propagation conditions. The advantage of the sampling approach is that it is possible to simulate different environmental conditions in a simple and very natural way. For example, let us suppose that the waves impinge only from a hemispherical region, as happens for example if we put the antenna on a wall. In this case, we need only to simulate a large number of waves impinging from this semispherical region. In the following an examples regarding the MIMO cube antenna considered also in [10], consisting of 12 electric dipoles placed along the sides of a cube, is reported. The cube antenna is characterized by numerical simulations of near-field measurements of the field radiated by the antenna. In Fig. 8 the eigenfunctions of the received signal covariance matrix obtained by the sampling approach above outlined considering waves impinging from an hemispherical region are plotted as a function of the antenna dimension (e.g., the radius of the minimum sphere enclosing the radiating elements of the antenna). We can take also into account a specific environment simulated by ray-tracing techniques or full-wave methods. For example, in Fig. 9 a specific environment, consisting in a corridor of the Faculty of Engineering in Cassino, is simulated using 3D ray tracing. A working frequency of 2.4 GHz, typical of Wi-Fi applications, is chosen, corresponding to a free space wavelength equal to m. A TX cube antenna with is positioned in a fixed position (filled circle in the inset of Fig. 9) while an RX cube antenna with is placed in 1280 different positions from 3 m to 43 m from the TX antenna (thick line in the inset of Fig. 9). The two antennas are 1.5 m above the floor and 2.5 m below the roof. Finally, it is interesting to note that the method is not limited to antennas having spherical-like shape, neither to spherical near-field measurement surfaces, but allows to handle much more general antenna shapes and scanning geometries [41]. IV. CONCLUSION Given a spatial communication channel, the amount of information that can be reliably conveyed by the channel is limited under the hypothesis that the energy of the set of the input signals is bounded. In particular, an important role in the limitation of the amount of transmissible information is played by the singular values of the channel operator above the noise level. This number is equal to the. Each singular function associated to a singular value above the noise level can be associated to an equivalent informational subchannel characterized by an input-output coefficient equal to the singular value. This results can be applied to any communication channel, including the space-time continuous channel. In particular the space-time channel can be decomposed in a number of parallel channels equal to the of the space-time channel operator. The Number of Degrees of Freedom of the field is a key quantity to understand the physical limitations of radiating systems both from an energetic and informational point of view. Different classes of antennas use the available spatial degrees of freedom in different ways. In particular, in MIMO antennas they are used to convey statistically independent information, allowing to encode information also by means of spatial variation of the electromagnetic field, while in classic antennas they are used to control the spatial distribution of the radiated energy. Accordingly, MIMO antennas are classic antennas that use the available degrees of freedom in a different way [22]. The above observations suggest that a number of methods developed for classic antennas could be useful also for MIMO antennas provided that we take into account the different role played by the degrees of freedom of the field. An example of the extension of classic approaches to MIMO systems is discussed with reference to antenna synthesis and antenna characterization. As last observation, it is useful to note that the role of antennas in MIMO communication systems is often considered of secondary importance compared to space-time processing algorithms. According to the observations outlined in this paper, a

12 MIGLIORE: ON ELECTROMAGNETICS AND INFORMATION THEORY 3199 MIMO antenna is basically a (usually linear) operator that transforms a signal belonging to a space whose effective dimension is equal to (the incident electromagnetic field) into a signal belonging to a finite dimensional space (the signals at the output of the elements of the antenna). In the mapping information associated to the null space of the operator is missed, and no mathematical trick can restore such information. In order to maximize the amount of information transmitted in the wireless channel, the MIMO system must be degrees of freedom matched, in the sense that the dimension of the received signal space must be equal to the essential dimension of the electromagnetic field. Indeed, the antenna itself can be considered a communication channel [42] that limits the throughput of the communication system. Apart from the specific interest in space-time communication systems, the approach outlined in this paper can be used also in other fields in which space-time data processing are used. In fact, the approach can be extended to include also detection problems, enabling the understanding of the intrinsic limitations of a number of emerging techniques like for example microwave tomography and MIMO RADAR [43]. APPENDIX With reference to Fig. 2, let us suppose that the input signal is a zero mean white Gaussian stochastic process with unit variance. Furthermore, we suppose that the observable quantity, let be, is the signal corrupted by zero mean additive white Gaussian noise with variance, e.g.,. With reference to the Hilbert-Shmidt expansion of, we can obtain the classic model of parallel Gaussian channels used in Information Theory [21], [44] using the Karhunen-Loève theorem and expanding the input signal on the basis represented by the singular functions, obtaining the coefficients, and the output signal and the noise on the basis represented by the singular functions, obtaining the coefficients and. The coefficients of the input signal expansion and of the noise expansion are i.i.d. Gaussian processes, having variance equal to one, and having variance equal to. Furthermore, since (wherein is the expectation operator, is the th singular value and is the Kroneker symbol, equal to zero if and equal to 1 if ) also the coefficients of the (noiseless) output signals (being Gaussian and uncorrelated) are independent. The communication channel can be decomposed in a number of parallel Gaussian channels, each of them able to transmit statistically independent information. In particular the th channel is able to convey bits [21]. In case of step-like behavior of the singular values, with a knee at the th singular value, becomes negligible for, and consequently for reasonable SNR only the first parallel channels are able to convey a significant amount of information. In more general situations, the exact number of parallel Gaussian channels that maximizes the total mutual information is given by the water-filling algorithm. ACKNOWLEDGMENT The author wish to thank the anonymous referees, whose suggestions helped to greatly improve the clarity and readability of the paper. REFERENCES [1] L. Brillouin, Science and Information Theory. New York: Academic Press, [2] J. W. Wallace and M. A. Jensen, Intrinsic capacity of the MIMO wireless channel, in Proc. IEEE Vehicular Tech. Conf., Vancouver, CA, Sep. 2002, pp [3] A. S. Y. Poon, R. W. Brodersen, and D. N. C. Tse, Degrees of freedom in multiple-antenna channels: A signal space approach, IEEE Trans. Info. 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13 3200 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 10, OCTOBER 2008 [26] E. Wolf and F. Nieto-Vesperinas, Analyticity of the angular spectrum amplitude of scattering fields and some of its consequences, J. Opt. Soc. Am. A, vol. 2, pp , Jun [27] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw Hill, [28] M. D. Migliore, An intuitive electromagnetic approach to MIMO communication systems, IEEE Antennas and Propag. Mag., vol. 48, no. 3, pp , Jun [29] M. Gustafsson and S. Norbedo, On the spectral efficiency of a sphere, in Prog. Electromagn. Res., 2007, pp [30] O. M. Bucci and G. Franceschetti, On the spatial bandwidth of scattered field, IEEE Trans. Antennas Propag., vol. AP-36, pp , Dec [31] O. M. Bucci, C. Gennarelli, and C. Savarese, Representation of electromagnetic fields over arbitrary surfaces by a finite and non redundant number of samples, IEEE Trans. Antennas Propag., vol. AP-46, pp , Mar [32] O. M. Bucci, G. D Elia, and M. D. Migliore, Optimal time domain field interpolation from plane-polar samples, IEEE Trans. Antennas Propag., vol. AP-45, no. 6, pp , Jun [33] H. J. Landau and H. O. Pollak, Prolate spheroidal wave function, Fourier analysis and uncertainty III: The dimension of the space of essentially time- and band-limited signals, The Bell Syst. Tech. J., pp , Jul [34] R. F. Harrington, On the gain and beamwidth of directional antennas, IRE Trans. Antennas Propag., vol. 6, no. 3, pp , Jul [35] H. L. Van Trees, Optimum Array Processing. New York: Wiley, [36] M. Wennstrom and T. Svantesson, An antenna solution for MIMO channels: The switched parasitic antenna, in IEEE Proc. 12th Inter. Symp. on Personal, Indoor and Mobile Radio Communications, Oct. 2001, vol. 1, pp. A-159 A-163. [37] M. D. Migliore, D. Pinchera, and F. Schettino, Improving the channel capacity using adaptive MIMO antennas, IEEE Trans. Antennas Propag., vol. AP-54, no. 11, pp , Nov [38] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, [39] M. A. Jensen and J. Wallace, A review of antennas and propagation for MIMO wireless communications, IEEE Trans. Antennas Propag., vol. 52, no. 11, pp , Nov [40] G. H. Golub and C. F. Van Loan, Matrix Computation. Baltimore, MD: The Johns Hopkins University Press, [41] O. M. Bucci and M. D. Migliore, A new method to avoid the truncation error in near-field antennas measurements, IEEE Trans. Antennas Propag., vol. AP-54, no. 10, pp , Oct [42] M. D. Migliore, The MIMO antenna as a communication channel, in Proc. Antennas and Propagation Symp., Honolulu, HI, Jun [43] M. D. Migliore, Some physical limitations in the performance of statistical MIMO RADARs, IET Microw., Antennas Propag., to be published. [44] A. D. Wyner, The capacity of the band-limited Gaussian channel, Bell Syst. Tech. J., no. 45, pp , Mar Marco Donald Migliore (M 04) received the Laurea degree (honors) in electronic engineering and the Ph.D. degree in electronics and computer science from the University of Napoli Federico II, Naples, Italy, in 1990 and 1994, respectively. He is currently an Associate Professor at University of Cassino, Cassino, Italy, where he teaches adaptive antennas, radio propagation in urban area and electromagnetic fields. He has also been appointed Professor at the University of Napoli Federico II, where he teaches microwaves. In the past he taught antennas and propagation at the University of Cassino and microwave measurements at the University of Napoli Federico II. He is also a Consultant to industries in the field of advanced antenna measurement systems. His main research interests are antenna measurement techniques, adaptive antennas, MIMO antennas and propagation, and medical and industrial applications of microwaves. Dr. Migliore is a Member of the Antenna Measurements Techniques Association (AMTA), the Italian Electromagnetic Society (SIEM), the National Inter-University Consortium for Telecommunication (CNIT) and the Electromagnetics Academy. He is listed in Marquis Who s Who in the World, Who s Who in Science and Engineering, and in Who s Who in Electromagnetics.