Space-time duality in multiple antenna channels

Transcription

1 Spae-time duality in multiple antenna hannels Massimo Franeshetti, Kaushik Chakraborty 1 Abstrat The onept of information transmission in a multiple antenna hannel with sattering objets is studied from first physial priniples. The amount of information that an be transported by eletromagneti radiation is related to the spae-wavenumber and the time-frequeny spetra of the system omposed by the transmitting antennas and the sattering objets. The spatial information ontent is quantified in a similar fashion than its temporal ounterpart, by reduing the inverse problem of field reonstrution to a ommuniation problem in spae, and determining the relevant ommuniation modes of the hannel by rigorously applying the sampling theorem on the field s vetor spae. One onsequene for narrow-band frequeny transmission is that spae and time an be deoupled, leading to a spae-time information duality priniple in the omputation of the apaity of the radiating system. Interestingly, in the ase of wide-band frequeny transmission, it is shown that time and spae annot be deoupled and they jointly haraterize the wave s information ontent. I. INTRODUCTION The basi onsideration that is the leitmotiv of this paper an be stated as follows: in propagation of eletromagneti EM) waves, spae and frequeny are separate but intimately linked objets that pose fundamental limits on the amount of information arried by sattered fields. Resolving the amount of information that an be ommuniated through EM radiation is a venerable subjet that has been treated by a great number of authors in different fields, often leading to redisovering of basi fats time and again. Our attempt here is to present a unified approah based on the physial notions of spae-wavenumber and time-frequeny spetra of the propagating field. We onsider information sent in the form of waves, hene as a physial proess: waves propagate in spae through line of sight, refletion, sattering, and diffration. Like any other physial phenomenon, this is governed by the laws of nature. These laws determine not only the proess itself, but also the Researh supported in part by the National Siene Foundation CAREER award CNS and by the California Institute of Teleommuniation and Information Tehnologies CALIT2). The authors are with the California Institute for Teleommuniations and Information Tehnology, Department of Eletrial and Computer Engineering, University of California at San Diego, La Jolla, CA 92037, massimo@ee.usd.edu, khakrab@usd.edu August 23, 2007 DRAFT

2 amount of diversity that a wave arries along its path. One kind of diversity is due to the wave being a funtion of time, hene being haraterized by a frequeny spetrum. Another kind of diversity is due to the wave propagating through spae. In this ase the wave interats with the propagation environment and this interation introdues spatial diversity that an be haraterized in terms of spatial bandwidth. Both kinds of diversities influene the amount of information that an be ommuniated through wave propagation. The number of hannels available for ommuniation and the amount of information that an be transported over these hannels depend on both the spatial and the frequeny bandwidth of the radiating system. This paper starts by onsidering the spatial bandwidth of a radiating system and its information ontent, and then onsiders the interation between spatial and frequeny bandwidths in terms of information apaity. The broad onlusions that an be drawn from our analysis are as follows. Spae an be viewed as a apaity bearing objet: the amount of information ommuniated by a sattering system inreases with its size and with the size of the reeiving antenna array. This is shown rigorously from physis, by applying the sampling theorem in the spae domain. Furthermore, it turns out that in the narrow-band frequeny ase, time and spae an be deoupled, and the information rate measured in bits per unit time and summed over all spatial information hannels available for transmission, is equivalent to measuring information rate in bits per unit spae and summing over all frequeny information hannels available for transmission, thus establishing a spae-time information duality priniple. On the other hand, in the ase of wide-band frequeny transmission, time and spae are strongly oupled and they jointly haraterize the information ontent of the radiating system. Next, we wish to spend few words on the organization of the paper. Our analysis starts with a preliminary setion where we identify the physial diversity limits of frequeny narrow-band) sattered fields, that were first introdued by Bui and Franeshetti in the late nineteen eighties [2], [3], using a funtional analysis approah. We then ast these results in a ommuniation theoreti setting, viewing the inverse problem of field reonstrution through sampling interpolation as a ommuniation problem. This provides the link between the notion of degrees of freedom of sattered fields and the orresponding information theoreti definition. That is, between the minimum number of independent parameters suffiient to ompletely reonstrut the radiated EM field up to arbitrary preision ɛ, and the number of eigenvalues of the MIMO operator sattering matrix) that are above an arbitrary level ɛ. This link was reently pointed out by Migliore [11]. Following these preliminaries, in Setion 3 we present a geometri orollary to Bui and Franeshetti s work to determine the relationship between the minimum non-redundant antenna spaing at the reeiver, the wavelength, and the angular aperture of 2

3 the radiating system and we relate this orollary to to the works of Poon et. al. [15] and Miller [13]. In Setion IV, we examine the spatial information ontent of the sattered field in more detail, deriving a purely spatial additive white gaussian noise AWGN) hannel and its orresponding spae-time version, and we point out an information duality priniple between spae and time, arising from the analysis of suh hannels. In Setion V we extend the treatment to frequeny wide-band transmission, showing the mutual interation of spae and time in the resolution of the information ontent of the propagating wave. Finally, in Setion VI, we draw onlusions and outline diretions for future work. Throughout the paper, we point out the pratial impliations of our analysis and we provide few illustrative examples. II. PRELIMINARIES A. Physis bakground: Bui-Franeshetti diversity limits Any real measurement system is invariably affeted by a degree of unertainty. This an stem from several fators, e.g., bakground noise, sensitivity and preision of the measuring system, dynamis of the measurement apparatus, and errors due to numerial approximation. Hene, it is both reasonable and neessary to onsider two eletromagneti field vetors E 1 and E 2 indistinguishable if their differene, in a given norm, is below resolution ɛ, that is E 1 E 2 < ɛ. 1) Let us now onsider a sattering system onsisting of an arbitrary number of transmitters soures) and satterers, whih are enlosed within a ball B of finite radius a. The reeivers are loated on an observation domain lying on an analytial manifold M, whih is assumed to be some wavelengths away from B. For the ase of a one-dimensional open) observation domain, M is an ar spanning the range [ S, S] in an arbitrary urvilinear oordinate system s. The arlength is normalized with respet to r m, whih is the minimum distane of M from the enter of B. All results desribed here are valid for one-dimensional observation domains and two-dimensional radiating system, but an be easily extended to higher dimensions. A shemati diagram of the sattering model is depited in Figure 1. The primary soures are assumed to be uniformly bounded, so that the urrent density inluding the polarization urrents) indued on the satterers satisfies the onstraint Jr ) dr η, 2) B where η is a onstant. In this setion we shall onsider narrowband soures, i.e., Jr ) onsists of two impulses in the temporal) spetral domain at angular frequenies ω and ω. 3

4 The field indued by the sattering system B on the observation manifold M an be expressed as E 0 r) = G 0 r, r ) Jr ) dr, 3) B where G 0 is the free-spae dyadi Green s funtion f., e.g., [6]), G 0 r, r ) = jωµ I + ) 4π β 2 exp jβr), 4) R I being the identity matrix; β is the wavenumber satisfying β = ω ɛµ = ω, where ɛ and µ are the permeability and the permittivity of the medium, and is the veloity of propagation of the wave; and R = r r ) r r ). It is onvenient at this point to extrat the propagation fator exp jβr) from E 0 r), and onsider the redued radiated field Er) = E 0 r) expjβr). 5) Notie that the redued field E is still represented by 3), provided that one substitutes G 0 with G = G 0 expjβr), thus obtaining Er) = B Gr, r ) Jr ) dr. 6) In [2], [3], the following results were presented. It was shown that the redued sattered field an be well approximated by spae-bandlimited funtions. The approximation error dereases with the bandwidth, undergoing a sharp transition in the viinity of a ritial value of bandwidth see [3], Figure 3), whih is heneforth referred to as the effetive bandwidth of the field, W 0 = βa. 7) The transition is very sharp for large sattering systems i.e., for systems with a λ). The redued field an then be represented in terms of linear ombinations of onvenient basis elements. By appropriate hoie of the basis, the oeffiients of interpolation an be the sampled values of the field on the observation manifold [3]. Given a fixed preision level ɛ, the minimum number of basis funtions neessary to obtain a satisfatory representation of the field elements is defined to be the degrees of freedom N 0. For large sattering systems, the number of degrees of freedom of the sattered fields an be identified with the Nyquist number N 0 = 2SW 0 π = 2Sβa π, 8) evaluated in terms of the effetive bandwidth W 0. Note that 2S is the length of the detetor surfae M, normalized with respet to r m, see Figure 1. For large sattering systems, the number of degrees of freedom is pratially insensitive to the error level ɛ, i.e., a well-defined quantity [3]. For small sattering 4

5 systems, the dependene of the degrees of freedom on the error level ɛ was investigated numerially in [22], but this ase is not onsidered here. In the examples below, we onsider some pratial senarios and illustrate sample values of the spatial bandwidth and of the number of degrees of freedom for various operating frequenies. Notie that the spatial bandwidth W 0 is normalized to the wavelength λ and hene is a pure number. We shall ome bak to these examples several times throughout the paper. Example 2.1: Consider a radiating sphere of radius a = 1 Km enlosing all the transmitters and the satterers. Suppose the observation manifold is loated at a distane of r m = 20 Km, and the reeiving antenna array is of absolute) length 2S = 200 m. This situation an orrespond to a base station loated on a hilltop serving mobiles loated in a distant valley. In this ase, a/r m = 0.05, and 2S/r m In Table I, the spatial bandwidth W 0 ) and the number of degrees of freedom N 0 ) are omputed for several arrier frequenies. Example 2.2: Consider a radiating sphere of radius a = 100 m enlosing all the transmitters and satterers. Suppose the observation manifold is a irle enlosing the radiating sphere, with radius r m = 250 m. This is an idealized urban senario, where a number of reeiving antennas are plaed around the mobiles and the satterers. In this ase, a/r m = 0.4, and 2S/r m = 2π. In Table II, W 0 and N 0 are omputed for several arrier frequenies. B. From physis to ommuniation Let us now formally desribe the ommuniation problem following from the physis outlined in the previous setion. This set up is analogous, with some minor variations, to the ones in Miller [13], Migliore [11], and Poon et. al. [15]. Let us onsider the orthonormal basis expansion of the field measured over a finite spatial interval [ S, S] on M, Er) = α i Ψ i r), 9) i=0 where {α i } i=1 are the oeffiients of expansion, and {Ψ ir)} i=1 omprise a omplete orthonormal vetor) basis set. An arbitrary soure urrent density Jr ) inside B an also be desribed by a omplete orthonormal vetor) basis set {Φ l r )} l=1 as Jr ) = β l Φ l r ), r B, 10) l=1 where {β l } l=1 are the oeffiients of expansion. Next, substituting 9) and 10) into 6), we have α k = h kl β l, k = 1, 2,, 11) l=1 5

6 where h kl = M B Ψ k r) Gr, r ) Φ l r ) drdr. The omponents h kl an be thought as the oupling oeffiients between the transmission mode l in the ball B and the reeption mode k in the manifold M. In matrix form, 11) is equivalent to where α = {α k } k=1 and β = {β l} l=1 α = H β, 12) are the vetors of oeffiients and the elements of the matrix H are the omponents {h kl }. The matrix H an be viewed as the ommuniation operator viz. the sattering matrix) between the radiating ball and the reeiving manifold. The parameter R = rankh) defines the number of degrees of freedom of the ommuniation system, as it determines the number of parallel independent ommuniation modes. For large sattering systems, reall that the number of basis funtions {Ψ k } required to represent the reeived field intensity within a given preision tends to N 0, the number of degrees of freedom of the sattered field. Therefore, the linear subspae spanning the rows of H is at most of dimension N 0, so that R N 0. This is the main observation of [11]. Note that the soures and the satterers also play an important role in the determination of the exat value of R, sine the rank of H is also dependent on the size of the linear subspae spanned by the olumns of H, whih is ditated by the number of orthonormal basis funtions {φ l } required to represent the soure urrent density. In other words, the EM degrees of freedom represent an upper bound on the degrees of freedom of the MIMO ommuniation hannel obtained by onsidering all possible onfigurations of soures and satterers that an our inside the radiating ball. Due to the ompatness of the radiation operator, it is possible to perform an eigenvalue deomposition of the infinite dimensional matrix H, where only a finite number of diagonal elements are essentially nonzero. The resulting eigenfuntions yield the ommuniation modes. Formally, we an write H = U Λ V, 13) where U and V are infinite-dimensional) unitary matries, and Λ is a diagonal matrix with R nonzero omponents {l 1,..., l R }. Without loss of generality, we assume that Λ is normalized, i.e., R i=1 l2 i = 1. Then 12) is transformed into α = Λ β, 14) where the omponents of the linear vetors α = U α and β = V β onstitute the ommuniation modes. This eigenvalue deomposition allows to deompose the ontinuous spae MIMO hannel into a finite number of orthogonal SISO subhannels. The omponents {l 1,..., l R } determine the oupling strengths of the respetive subhannels, and are ritial in the EM haraterization of the apaity of the spae-time MIMO hannel, as we shall see in Setion IV-B. 6

7 III. MINIMUM NON REDUNDANT ANTENNA SPACING We now derive a geometri orollary to Bui and Franeshetti s work whih provides an elegant interpretation of the number of degrees of freedom, and leads to the determination of the minimum nonredundant antennas spaing in terms of wavelength and angular aperture of the system. We expliitly note that this minimum spaing is not heuristi, but it follows diretly from physis and from appliation of the sampling theorem in the spae domain, in light of the onsiderations made in the previous setion. For the purpose of this disussion, we expliitly indiate the normalization fator r m and assume that the non-normalized length of the observation domain is 2S. It follows from 8), if the manifold is at approximately onstant distane from the radiating system, then ) ) ) 2S 2a 2a N 0 = Ω r, 15) λ λ r m where Ω r is the angle subtended by the reeiving manifold on the transmitting volume see Figure 2a)). Thus, the number of degrees of freedom is approximately equal to the produt of the angle subtended by the reeiving manifold on the transmitting sphere times the size of the transmitter normalized with respet to the arrier wavelength. Similarly, 2a N 0 = r m ) ) 2S λ ) 2S Ω t, 16) λ where Ω t is the angle subtended by the transmitting volume on the reeiving manifold see Figure 2b)). Thus, the number of degrees of freedom is approximately equal to the produt of the angle subtended by the transmitting volume on the reeiving manifold times the size of the reeiver normalized with respet to the arrier wavelength. Above formulas show that for fixed size of the radiating system, the number of degrees of freedom inreases with the angular resolution of the reeiver; while for fixed size of the reeiver it inreases with the angular spread of the transmitter. This interpretation reovers the results in [15]. These authors onsidered a slightly different, but essentially equivalent model in whih lusters of satterers are loated in the far field with respet to both the soures and the detetors. The number of degrees of freedom for this model is reported to be the minimum of two produt terms, eah of whih are of the form 15) and 16). Furthermore, in a previous work [13] again the same kind of formula appears. In fat, this size times angular resolution formula has been known, at least non-rigorously, sine the nineteen fifties, dating bak at at least to the florentine shool of Toraldo di Frania [19] [21]. The formal onnetion between the number of degrees of freedom and the sampling rate, a diret onsequene of Bui and Franeshetti s band-limitation property, onstitutes the derivation provided above. 7

8 It has also been reported in [15] and [17] that, for linear transmitting and reeiving antenna arrays, the minimum antenna spaing to obtain a non-redundant representation of the field at the reeiver is suh that the antenna elements are separated by a distane λ/2. We now show that this minimum spaing also depends on the angle subtended by the transmitting ball at the reeiver. From 15) or 16) we have that 2S a = λ N 0 r m 2. 17) Sine 2S/N 0 is the Nyquist sampling interval of the spae bandlimited field, it follows from 17) that the minimum non-redundant antenna spaing is given by s = 2S N 0 = r mλ 2a. 18) Therefore, half a wavelength is the minimum non-redundant antenna spaing only when r m a and the manifold embraes the sphere, while in general for r m a the appropriate non-redundant spaing is given by 18). Notie that s inreases with r m. The reason for this should be lear: onsider for example different values of r m orresponding to different onentri ars subtended by the same angle θ at the enter of the ball B. The lengths of these ars are different: the one losest to the radiating ball is of the shortest absolute) length. However, the normalized lengths with respet to their distane from the enter of B are the same. Therefore, the number of samples needed for reonstrution in eah of these setions are the same, whih implies that the spaing between the adjaent samples is larger for the ones loated at larger distane from the enter. This agrees with pratial design priniples of wireless ommuniation systems, where the antennas at the ellular base station are plaed several wavelengths apart when the base station is far from the radiating system, as for example in a hilltop setting, while they are tightly paked together in miroell base stations, where the base station is lose to the radiating system, as for example in an urban setting f. [5] and the following examples). Example 3.1: Reall the hilltop example f. Example 2.1) with a = 1 Km, r m = 20 Km, and 2S = 200 m. In this ase, with f = 900 MHz i.e., λ = 0.33 m), the minimum antenna spaing is given by s = 2S N 0 = 200 s 60 = 3.33m = 10λ. This an also be seen from 18) sine λ = 1 Ω t = r m 2a = 10. Hene, the antennas should be widely separated. Consider now the urban example f. Example 2.2) with a = 100 m, r m = 250 m, and 2S = 2πr m. In this ase, with f = 900 MHz, the minimum antenna spaing is given by s = 2S N 0 = 2π = 0.417m = 1.25λ. This is again onsistent with 18), sine s λ antennas an be tightly paked together. = r m 2a = Hene, in this ase, the Both of the ases above are in agreement with pratial design priniples of ellular systems, see [5]. 8

9 IV. INFORMATION DUALITY PRINCIPLE We are now ready to give an information theoreti haraterization of the radiated field. This is the natural extension of the results in [11]. We first introdue a pure spatial and then a spatio-temporal MIMO ommuniation hannel model, based on interpolation through sampled field points; then we present a spae-time information duality priniple that arises from the analysis of these two hannels. A. A spatial AWGN ommuniation hannel We start with a very simple model of additive white Gaussian noise AWGN) spatial ommuniation hannel that is progressively refined in the next setions. The hannel we onsider here is a purely spatial hannel, in the sense that information is onveyed from the radiating ball to the observation domain in a single time step, using the spatial ommuniation modes. Furthermore, for the sake of simpliity, this hannel is onstruted at the reeiver under the assumption that the soure is apable of generating a given field pattern on the manifold M this assumption will be removed in the next setions). The odewords are the eletri field expansion oeffiients measured on the manifold, orresponding to a omplete orthonormal basis funtion set {ψ k }, and the bakground noise due to measurement errors and approximation errors is umulatively modeled as independent Gaussian random variables that are added to eah expansion oeffiient. Consider a odeword of n symbols α 1, α 2,..., α n ) that we wish to ommuniate. We assoiate these symbols to the expansion oeffiients of the field on the manifold M and assume that the eletri field Er) on the whole manifold an be reovered from these symbols within a given preision level by using non-redundant sampling interpolation in spae. We first derive a bound on the average spatial power measured in the interval [ S, S]. This an be obtained from the onstraint on the soure urrent density 2) and the redued field equation 6) as follows. Er) = Gr, r ) Jr )dr sup Gr, r ) r B B Jr ) dr sup r Gr, r ) η, 19) where the last inequality follows from 2). Squaring and integrating both sides of 19), we have ) 1 2 Er) 2 dr η2 sup Gr, r ) dr = P, 20) 2S 2S r M M where P is a onstant beause the integrand is bounded, sine the manifold M is outside the ball B so that r r ; and the integral is over a bounded support 2S. Notie that P is but for a normalizing resistane) the average spatial power over the manifold. Inequality 20) learly holds also for the o-polar omponent of the field, Er), whih we onsider next. 9

10 Assuming an orthornormal basis expansion of the field, by 20) it follows that we have the odeword onstraint 1 n n α i 2 = 1 n i=1 S S Er) 2 dr 2SP n. 21) Notie that 2S/n stays onstant provided that the sampling rate over the manifold has been fixed and the spatial odeword length S grows with n. We now introdue the bakground noise due to measurement and approximation errors at the reeiver. Let a Gaussian random variable N 0, σ 2 ) be added independently to eah transmitted symbol α i, so that for a transmitted odeword α 1,..., α n ), the orresponding reeived odeword γ 1,..., γ n ) is obtained by γ i = α i + z i, i = 1,..., n, 22) where the z i s are realizations of i.i.d. N 0, σ 2 ) random variables. We remark here that the index i is a spae index. We an now use Shannon s formula to state that, Theorem 1: The information apaity of the spatial additive Gaussian noise hannel desribed above and subjet to the onstraint 21) is given by C = 1 2 log 1 + P σ 2 ) bits per spatial symbol. 23) It is appropriate at this point to summarize the assumptions we need for above apaity formula to hold. First of all, we need a non-redundant sampling representation of the field. This ensures that the oeffiients in the odeword, being essentially the same number as the degrees of freedom N 0 of the field, are independent. Sine the field is effetively bandlimited [2], this an be ensured by hoosing appropriate basis funtions. The seond point to notie is that we need the soure to be apable of generating the desired pattern of field oeffiients {α i } on the manifold M. This might not be true in general and depends on the atual onfiguration of the soures and the satterers inside the ball, whih determines the hannel ommuniation modes. For example, a portion of the manifold might be shaded by absorbing obstales present inside the ball. In this ase, the oeffiients {α i } on this portion of the manifold annot be exited. In fat, reall from Setion II-B that the number of degrees of freedom of the field measured on the manifold M is an upper bound on the number of degrees of freedom of the ommuniation system. This upper bound an be ahieved onsidering all possible onfigurations of soures and satterers inside the ball. In the next setions we shall make the dependene on the soures and the ommuniation modes expliit. Finally, above apaity formula is only valid in the limit of large 10

11 odeword size, that is as n. This requires the manifold size to diverge as well, so that 2SP/n remains onstant. In pratie, we are onstrained to a finite number of observation points plaed on a manifold of finite size. We shall address this last problem in detail at the end of this setion. We now perform one more step to obtain the apaity per unit normalized spae over the hannel. This follows by introduing the additional physial onstraint of the effetive spatial bandwidth W 0. This limits the amount of detetable variation over the manifold of any transmitted signal. By 8), at most N 0 = 2SW 0 /π different symbols an be transmitted over a spatial region of normalized length 2S. Thus, the apaity per unit normalized spae is given by C = N S 2 log + Pσ ) 2 = W 0 1 2π log + Pσ ) 2 bits per normalized length. 24) The noise is also related to the spatial bandwidth W 0. The average power of the spatial) white Gaussian noise proess is proportional to the spatial bandwidth, so that σ 2 = N s W s, where W s = W0 2π and N s is the onstant spatial) noise power spetral density. Thus, we reover the spatial version of the Shannon- Hartley formula for ommuniation over a bandlimited hannel subjet to additive white Gaussian noise, Theorem 2: The information apaity of the spatial MIMO radiating system depited in Figure 1, where transmitters are narrowband with wavelength λ, and the reeiving array is subjet to additive white Gaussian noise is given by where W s = W 0 2π = a λ. C = W s log 1 + P ) bits per normalized length, 25) N s W s Some observations are now in order. First of all, we note that spatial apaity is measured in bits per unit of normalized) spae, whih orrespond to the bits per unit time in the usual temporal formula. Furthermore, the spatial apaity is assoiated to an observation interval of normalized) length 2S; similarly, the temporal apaity is assoiated to a time interval of T seonds. In fat, the number of bits that an be transmitted over the manifold sales with its normalized) size. In the temporal analogy this orresponds to the trivial observation that the amount of information transmitted over a time interval T = 2T orresponds to twie the amount of information sent over a time interval T. In the spatial domain, the inrease is due to a longer reeiving array whih enhanes the angular resolution, i.e., the detetion apability of the reeiver. In the temporal domain this orresponds to transmitting more symbols over the hannel using a longer time window. We also note that the spatial apaity sales with the spatial bandwidth W 0 = βa, i.e., with the wavelength-normalized) size of the radiating system in a same fashion as the temporal apaity sales with the frequeny bandwidth. Sine W 0 is proportional to the size of the 11

12 radiating system, this means that a larger radiating system inreases the information apaity by working as a lens enhaning its fousing apability. Furthermore, W 0 is inversely proportional to the wavelength, whih means that transmitting at higher frequenies i.e. shorter wavelengths) an lead, in priniple, to higher information rates. Next, we turn to the problem that in the usual temporal formulation the apaity formula is derived by letting T, where T is the time interval of transmission and reeption of the ontinuous funtion of time representing the odeword symbols. Instead, in the spatial formulation the normalized spatial interval is bounded by the total angular spread 2π in the ase of a one-dimensional observation domain). This invariably implies bounded spatial odeword lengths, and hene that the apaity formulas derived above are not ahievable, but rather provide upper bounds on the ahievable information rates of the spatial hannel under onsideration. We emphasize that this is true even if we streth the observation domain to infinity, sine the number of degrees of freedom of the eletri field indued on the unbounded) observation manifold by soures and satterers enlosed in a bounded volume is still finite, and bounded by the total angular spread, as it was disussed at the end of Setion III. In this ontext, it is therefore useful to onsider the notion of error exponents arising beause the normalized spatial interval is bounded. The probability of an error event, whih ours when the deteted odeword is different from the transmitted odeword, deays exponentially with n, the number of symbols in the odeword. The exponent of the error probability indiates the rate of deay of the error event, and is a useful hannel parameter. For the AWGN hannel, upper bounds on the error exponent are well known f., e.g., [7], Theorem 7.4.4). In the following example we apply suh bounds showing that, with a larger N 0, information rates loser to apaity an be ahieved. A large value of N 0 an be ahieved by using a higher arrier frequeny, and/or by using a large sattering system, and/or by using a large observation domain size. It should be noted that these estimates are based on the upper bounds on error probability, and the atual ahievable rates an in fat be higher. Example 4.1: Reall the hilltop example Example 2.1) with a = 1 Km, r m = 20 Km, and 2S = 200 m, and the urban example Example 2.2) with a = 100 m, r m = 250 m, and 2S = 2πr m. In Table III, the ratio of the maximum ahievable information rate to the hannel apaity R max /C) orresponding to P e 10 6 for SNR = 0 db are omputed for several arrier frequenies for all of these senarios. B. A spae-time AWGN ommuniation hannel In this setion, the soure geometry is taken into aount so that an expliit relation between input represented by the urrent density inside the ball) and output represented by the observed EM field on 12

13 the manifold) is obtained. Furthermore, we onsider transmission over both spae and time. There has been a plethora of reent publiations that provide a spae-time MIMO model based on the physis of propagation [1], [8] [13], [15] [17], [23]. The ommon denominator of all of these works follows the outline given in Setion II-B, where the MIMO hannel is deomposed into a set of R parallel SISO subhannels, as given by equation 14). As in the ase of the purely spatial hannel model, we an introdue the effets of bakground noise via a simple AWGN model. From 14) it follows that the measured ommuniation modes are given by γ i = α i + z i = l i β i + z i, i = 1,, R, 26) where {z i } are realizations of i.i.d. N 0, σ 2 ) random variables whih are added to eah spae omponent at a given instant of time; {l i, i = 1,, R} are the nonzero diagonal elements of Λ; and the input oeffiients are subjet to a power onstraint 1 R R β l 2 P, 27) l=1 where P an be immediately related to η. Reall from Setion II-B that R is the number of independent parallel SISO subhannels, as determined by the singular value deomposition of the hannel matrix H; furthermore, R N 0, where N 0 is the number of degrees of freedom of the sattered field, i.e., the number of orthogonal basis funtions required to represent the field within a given preision level. We now let the R spatial hannels defined by equation 26) evolve over time, assuming independent additive Gaussian noise added at eah measurement over time as well. We thus obtain a parallel Gaussian hannel, formed by the R spatial ommuniation modes, whose apaity is well known f. [18]) to be C = R i=1 W t log 1 + P i ) l2 i W t N t bits per unit time, 28) where W t is the temporal frequeny bandwidth, N t is the onstant) power spetral density of the additive white Gaussian noise in the time evolution, W t N t = σ 2, and {Pi } are the waterfilling power alloations [4]. If the ommuniation modes are roughly of the same strength, i.e., l 2 i 1 R, then the spae-time apaity expression 28) simplifies to the sum of R idential parallel subhannels, given by C = RW t log 1 + P ) RW t N t where the waterfilling power alloation alloates equal power P to all hannels. 13 bits per unit time, 29)

14 C. A spae-time information duality priniple Assuming the soures to be able to exite all N 0 degrees of freedom of the field, we an perform the a substitution R = N 0 = 2Ω r λ = W s2ω r, obtaining P ) C = 2Ω r W s W t log 1 + 2Ω r W s W t N t bits per unit time. 30) Above formula shows the spae-time apaity in bits per unit time when the spae-time hannel is used over time and this hannel also exploits the spae diversity given by N 0 = 2Ω r W s spatial degrees of freedom. Similarly, we an obtain a dual formula for the spae-time apaity in bits per normalized length, when the spae-time hannel is used in spae as defined by Theorem 2, and this hannel also exploits the time diversity of W t 2T degrees of freedom given by a frequeny bandwidth W t. In this ase, we need to modify 25) to relate the oeffiients {α i } to the ommuniation modes. By assuming as before all the ommuniation modes to be of the same strength, we let γ i = α i + z i = lβ i + z i, with l 2 = 1 W t 2T. The waterfilling power alloations alloate equal power P to all frequeny hannels and we have C = 2T W t W s log 1 + P 2T W t W s N s ) bits per unit length. 31) By omparing equations 30) and 31), we an now state the following spae-time information duality priniple. Information duality priniple. The physial spae-time AWGN hannel an be operated in time, by summing all the relevant spatial information hannels that are related to the spatial) degrees of freedom of the sattered field through appliation of the sampling theorem in spae. In a dual way the same hannel an be operated, in priniple, in spae given aeptable error exponents due to the angular resolution limit), by summing all the relevant frequeny information hannels that are related to the temporal) degrees of freedom of the sattered field through appliation of the sampling theorem in time. It follows that there are two equivalent ways of measuring information apaity: in the first ase this is measured in bits per unit time, while in the seond ase it is measured in bits per units of normalized spae. The two information measures are equivalent representations of the same onept of information rate through spae and time, in a MIMO system, projeted along eah of these dimensions. Notie that while the time dimension is present in the first information measure, the seond information measure is 14

15 a pure number. This is beause the bandwidth W t is measured in Hz, while the spatial bandwidth W s, being normalized to the wavelength, is a pure number. It follows that the two equations 30) and 31) annot be diretly ompared. However, we an make both representations independent of dimension by normalizing the bandwidth W t to the arrier frequeny f 0 = 1/τ 0. In this ase we measure 30) in bits per time normalized to τ 0, and 31) in bits per length normalized to λ. A quantitative omparison is now possible, and we see that the two equations oinide, provided that normalized variables Ω r = 2S/r m and T/τ 0 are equal. This means that in our ideal dual-operation of the spae-time hannel, the odeword spatial length orresponding to the observation domain 2S, normalized to r m, must be equal to the odeword temporal length T, normalized to the period of the temporal arrier frequeny τ 0. V. WIDE-BAND FREQUENCY SPECTRUM The information theoreti haraterization of the radiated field presented in the previous setion was based on narrow-band frequeny signals. We have assumed that the spatial bandwidth W 0 remains onstant over the narrow) frequeny band W t used for transmission. In this way, the information ontents along the spae and time dimensions ould be treated independently and led to the information duality priniple. We now want to make a first step in the diretion of understanding the properties of the signal reeived on the manifold M when the narrow-band assumption is relaxed. In this ase, we need to aount for the dependene between the spatial band and the frequeny band of the radiated wave, generalizing the single frequeny treatment of Bui and Franeshetti [2], [3]. As we shall see in the following, this leads to a variable spae-time sampling rate defining the information ontent of the sattered wave. Let us start writing the spae-time domain representation of the field over the manifold, whih is a real funtion, et, s) = 1 2π + ωa/ dω e iωt du e ius Eω, u), 32) ωa/ where Eω, u) is the frequeny-wavenumber representation of the reeived signal, s is the urvilinear oordinate over the manifold, a is the size of the radiating ball, is the speed of light, and ωa/ = W 0 is the spatial bandwidth of the radiating system at angular frequeny ω. Note that in 32) the real) spae-time field representation is obtained by transforming Eω, u) with respet to the wavenumber and then inverse transforming with respet to the angular frequeny. Letting the inner integral be Fω, s), we note that et, s) = 1 2π + dω e iωt Fω, s) 33) 15

16 is real if F ω, s) = F ω, s). Hene, we an write { et, s) = 1 } ωa/ π Re dω e iωt du e ius Eω, u). 34) 0 ωa/ We now assume the funtion Eω, u) to be frequeny band-limited to the interval [ω 0 ω/2, ω 0 + ω/2]. It follows that { et, s) = 1 ω0+ ω/2 } ωa/ π Re dω e iωt du e ius Eω, u). 35) ω 0 ω/2 ωa/ Above equation is the starting point for the analysis of the two ases of narrow-band and wide-band spetra. A. Case 1. Narrow-band Let us onsider first radiating a pure sinusoidal signal at angular frequeny ω 0. That is, a time domain signal of the type osω 0 t). In this ase, for ω > 0, we have Eω, u) = πδω ω 0 ). 36) When no spatial modulation has been enfored on Eω, u), we have from 35), { ω0 + ω/2 et, s) = Re dω e iωt δω ω 0 ) 2ωa ωas ) } sin ω 0 ω/2 { 2ω0 a ω0 as ) } = Re sin e iω0t = 2ω 0a ω0 as ) osω 0 t) sin = 2W 0 osω 0 t) sin W 0 s), 37) where we have adopted the onvention sinx) = sin x)/x. Above equation shows that the spatial signal is onentrated around s = 0, as expeted, beause no spatial modulation has been enfored on Eω, u). Clearly, we an move the maximum of the signal in spae by appropriate spatial modulation, thus obtaining a different spatial illumination of the type sin[ ω0a s s 0)]. In this way, two suffiiently distint sin ) patterns an be singled out provided that the spaing s between them orresponds to at least half a lobe of the sin ) funtion, that is ω0a s = 2πa s > π, i.e., λ s > λ 2a = π W 0. 38) This time-domain analysis is onsistent with the orresponding frequeny domain analysis of Setion III, leading to 18), and with antenna theory, whih sets a limitation on the attainable resolution of antennas of dimension 2a. While the frequeny domain analysis leading to 18) is based on the appliation of the 16

17 sampling theorem in spae, the time-domain viewpoint leading to the equivalent 38) sets the resolution limit on the manifold as the angular separation needed to obtain spots over the manifold where eah reeiving antenna an detet essentially a single signal. Notie that while in 38) s is normalized to r m, this is not the ase in 18) where the normalizing fator appears expliitly. It is interesting now to onsider radiating a pure sinusoidal signal of spatial wavenumber u 0, i.e., a spae domain signal of the type osu 0 s). In this ase, for u > 0, we have The inner integral in 35) an then be evaluated to yield ωa/ du e ius πe iu0s, if ωa Eω, u) = ωa/ Eω, u) = πδu u 0 ). 39) 0, otherwise. < u 0 < ωa, Clearly, in order to generate a sinusoid at an arbitrary spatial frequeny, we need to let the radiating system dimension a. In this ase, 35) an be evaluated as follows, { } ω0 + ω/2 et, s) = Re dω e iωt e iu 0s ω 0 ω/2 { = Re ω sin ) } e iω 0t u 0 s) ωt 2 ωt = ω osω 0 t u 0 s) sin 2 40) ). 41) By analogy with 37) we see that now the time signal is onentrated around t = 0 and that two distint time signals an be singled out, by performing appropriate frequeny modulation, provided that the time spaing t between them satisfies the Nyquist resolution limit, t > 2π ω. 42) We onlude that in the narrow-spae and narrow-frequeny ase, spae and time signals are deoupled and the appropriate sampling rate is determined by the frequeny and spae bandwidths respetively. Next, we onsider the ase of wide band signals, where it turns out that spae and time signals annot be deoupled. Sine the spatial bandwidth is a funtion of the radiated frequeny, the amount of diversity available in spae depends on a speifi position in frequeny, and similarly the amount of diversity available in time depends on a speifi position in spae. B. Case 2. Wide-band We now want to onsider a radiated waveform of frequeny spetrum ω and spae spetrum W 0 ω) that are not negligible. For onveniene, we assume the frequeny-wavenumber representation of the 17

18 reeived signal Eω, u) to be a onstant E 0 over both the frequeny and the wavenumber intervals of interest. The inner integral of 35) an then be written as follows, Fω, s) = E 0 ωa/ ωa/ Computing the outer integral we have that ω0+ ω/2 ω 0 ω/2 du e ius = E 0 is dω e iωt Fω, s) = E 0 ω is e iωas/ e iωas/). 43) { [ ω e iω0t+as/) sin t + as ) ] 2 [ ω t as ) ]}. 44) 2 e iω0t as/) sin To obtain et, s) we finally multiply by 1/π and take the real part, obtaining et, s) = E { 0 ω sin [ω 0 t + as )] [ ω sin t + as ) ] πs 2 sin [ω 0 t as )] [ ω sin t as ) ]}. 45) 2 We now ompare 45) with 37) and 41). It is lear that while time and spae are deoupled in the narrowband ase, a more omplex relationship arises in the wideband ase. In the frequeny narrowband ase 37) the spatial resolution is ditated by the amount of shift of a single sin ) funtion translated in spae; similarly in the spae narrowband ase 41) the time resolution is ditated by the amount of shift of a single sin ) funtion translated in time. Instead, time and spae appear now linked to eah other in 45), whih is a onsequene of the fat that the spatial band W 0 ω) = ωa/ now varies with the frequeny ω over the interval [ω 0 ω/2, ω 0 + ω/2]. A somewhat learer piture arises if we elaborate on 45) and rewrite it as follows, where F t, s) = ωa F + t, s) = ω 0a et, s) = E 0 ω π ω0 as sinω 0 t) os osω ω0 as 0t) sin [ F t, s) + F + t, s) ], 46) ) { sin [ ω 2 ) { [ ω sin 2 )] [ t + as sin ω 2 t + as ω as t as ) ] [ ω + sin 2 )]} t as, 47) ) ]}. 48) These equations learly show that the signal is the superposition of two quadrature omponents and allow omparison of the orresponding envelopes. We notie that the frequeny narrowband ase expressed by 37) is immediately reovered by letting ω 0 while keeping E 0 ω onstant, and notiing that lim F ωa ω0 as ) { ) } ωt t, s) = lim sinω 0 t) os sin = 0. 49) ω 0 ω 0 2 The spatial narrowband ase annot be diretly reovered by letting a, beause the limit annot be ωt 2 arried inside the integral 35), while the proper steps to be followed are the ones leading to 37). 18

19 VI. CONCLUSION In MIMO systems ommuniation is performed through EM waves and the onept of information transmission is related to the amount of diversity EM waves an arry. Suh diversity lies in two different dimensions: time and spae. The lassial view of Shannon s information theory typially onsiders only the time dimension along with its transformed ounterpart: the frequeny spetrum. We have shown that Shannon s theory an also be applied to the spae dimension whih, analogous to time, beomes a apaity bearing objet. The onept of transformed spae domain and spae bandwidth has been introdued and an information duality priniple has shown that spae and time an naturally be treated as the dual of one another. In the wide-band transmission regime a muh more omplex senario arises. In this ase, time and spae are linked together, as the spae bandwidth depends on the radiated frequeny. Our analysis appears to be open to several refinements. In partiular, haraterization of the apaity in the frequeny wide-band regime and showing that although spae and time are not deoupled, information is onserved in spae-time are interesting open problems. VII. ACKNOWLEDGEMENT The authors wish to aknowledge enlightening onversations they had on the subjet with Prof. Giorgio Franeshetti, to whom this work is dediated. REFERENCES [1] T. D. Abhayapala, T. S. Pollok and R. A. Kennedy, Spatial deomposition of MIMO wireless hannels, in Pro. Seventh International Symposium on Signal Proessing and its Appliations, ISSPA 2003, Paris, pp , Vol. 1, July [2] O. M. Bui and G. Franeshetti, On the spatial bandwidth of sattered fields, IEEE Trans. Antennas Propagat., 3512): , De [3] O. M. Bui and G. Franeshetti, On the degrees of freedom of sattered fields, IEEE Trans. Antennas Propagat., 377): , July [4] T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, New York, [5] M. P. Fitz, C. Shen, and M. Samuel, The wireless ommuniations physial layer, in G. Franeshetti and S. Stornelli Eds., Wireless Networks: From the Physial Layer to Communiation, Computing, Sensing, and Control, Chapter 2, Elsevier In., [6] G. Franeshetti, Eletromagnetis: Theory, Tehniques, and Engineering Paradigms, Kluwer Aademi, [7] R. G. Gallager, Information Theory and Reliable Communiation, John Wiley & Sons, New York, [8] L. W. Hanlen and M. Fu, Wireless ommuniation systems with spatial diversity: a volumetri approah, Proeedings of the ICC 2003, , vol. 4, May [9] L. W. Hanlen and M. Fu, Capaity of MIMO hannels: a volumetri approah, Proeedings of the ICC 2003, , vol. 5, May

20 [10] R. A. Kennedy and T. D. Abhayapala, Spatial onentration of wave-fields: towards spatial information ontent in arbitrary multipath sattering, in Pro. Fourth Australian Communiations Theory Workshop, AusCTW2003, pp 3845, Melbourne, Australia, Feb 5-7, [11] M. D. Migliore, On the role of the number of degrees of freedom of the field in MIMO hannels, IEEE Trans. Antennas Propagat., 542): , Feb [12] M. D. Migliore, Restoring the symmetry between spae domain and time domain in the hannel apaity of MIMO ommuniation systems, in Pro IEEE Antennas & Propagation International Symposium, pp , July [13] D. A. B. Miller, Communiating with waves between volumes: evaluating orthogonal spatial hannels and limits on oupling strengths, Appl. Optis, 3911): , Apr [14] T. S. Pollok, T. D. Abhayapala and R. A. Kennedy, Spatial Limits to MIMO Capaity in General Sattering Environments, in Pro. 7th International Symposium on DSP for Communiation Systems DSPCS03), pp 49-54, De [15] A. S. Y. Poon, R. W. Brodersen, and D. N. C. Tse, Degrees of freedom in multiple-antenna hannels: a signal spae approah, IEEE Trans. Inform. Theory, 512): , Feb [16] A. S. Y. Poon, R. W. Brodersen, and D. N. C. Tse, Impat of sattering on the apaity, diversity, and propagation range of multiple-antenna hannels, IEEE Trans. Inform. Theory, 523): , Mar [17] A. M. Sayeed, Deonstruting multiantenna fading hannels, IEEE Trans. Signal Proessing, 5010): , Ot [18] I. E. Telatar. Capaity of multi-antenna Gaussian hannels, European Transations on Teleommuniations, 106), , [19] G. Toraldo di Frania, Resolving power and information, J. Optial Soiety Ameria, 457): , July [20] G. Toraldo di Frania. Diretivity, super-gain and information, IRE Trans. Antennas Propagat., AP-43): , July [21] G. Toraldo di Frania, Degrees of freedom of an image, J. Optial Soiety Ameria, 597): , July [22] J. Xu and R. Janaswamy, Eletromagneti degrees of freedom in 2-D sattering environments, IEEE Trans. Antennas Propagat., vol. 54, no. 12, pp , De [23] J. W. Wallae and M. A. Jensen, Intrinsi apaity of the MIMO wireless hannel, in Pro. IEEE Vehiular Tehnology Conf., vol. 2, pp , Vanouver, CA, Sep

21 Fig. 1. Geometry of the sattering problem. The transmitters denoted by rosses) and the satterers denoted by ovals) are assumed to be enlosed within a ball B of radius a, and the observation manifold M is loated in the far field r m a). The manifold is measured in urvilinear oordinates of variable s. Carrier frequeny f in MHz) W 0 N π π π Table I: Spatial bandwidth and number of degrees of freedom for Example

22 S S Transmitting ball B Transmitting ball B Ω r 0 Ω t 0 2a 2a -S Reeiving manifold M -S Reeiving manifold M a) b) Fig. 2. Geometri interpretation of the number of degrees of freedom of the sattered field. Carrier frequeny f in MHz) W 0 N π π π Table II: Spatial bandwidth and number of degrees of freedom for Example 2.2. Carrier frequeny R max /C f in MHz) Example 2.1 Example Table III: Table of values for Example