INFORMATION THEORY AND ELECTROMAGNETISM: ARE THEY RELATED?

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1 INFORMATION THEORY AND ELECTROMAGNETISM: ARE THEY RELATED? Sergey Loyk School of Informtion Technology nd Engineering (SITE), University of Ottw 161 Louis Psteur, Ottw, Ontrio, Cnd, K1N 6N5, emil: Abstrct In this pper, we study the limittions imposed by the lws of electromgnetism on chievble MIMO chnnel cpcity in its generl form. Our pproch is three-fold one. First, we use the chnnel correltion rgument to demonstrte tht the minimum ntenn spcing under ny scttering conditions is t lest hlf wvelength. Secondly, using plnewve spectrum expnsion of generic electromgnetic wve combined with Nyquist smpling theorem in the sptil domin, we show tht the lws of electromgnetism limit the minimum ntenn spcing to hlf wvelength, λ /2, (in the cse of 1-D ntenn pertures) only symptoticlly, when the number of ntenns n. For finite number of ntenns, this limit is slightly less thn λ /2. The number of ntenns nd, consequently, the MIMO cpcity is limited for given perture size. This is scenrio-independent limit. Finlly, we study the MIMO cpcity of wveguide nd cvity chnnels. The rtionle for this is three-fold: (i) wveguide / cvity models cn be used to model corridors, tunnels nd other confined spce chnnels, (ii) this is cnonicl problem; its nlysis llows to develop pproprite techniques, which cn be further used for more complex problems, (iii) it llows to shed light on the reltion between informtion theory nd electromgnetism nd, in prticulr, to estblish the limits imposed by the lws of electromgnetism on chievble chnnel cpcity. I. INTRODUCTION It is well recognized tht the wireless propgtion chnnel hs profound impct on MIMO system performnce. In idel conditions (uncorrelted high rnk chnnel) the MIMO cpcity scles roughly linerly s the number of Tx/Rx ntenns. The effect of chnnel correltion is to decrese the cpcity nd, t some point, this is the dominnt effect. This effect is highly dependent on the scenrio considered [3]. Mny prcticlly-importnt scenrios hve been studied nd some design guidelines hve been proposed s well. In the present pper, we nlyze the effect of propgtion chnnel from completely different perspective. Electromgnetic wves re used s the primry crrier of informtion. The bsic electromgnetism lws, which control the electromgnetic field behviour, re expressed s Mxwell equtions [5]. Hence, we sk question: Wht is, if ny, the impct of Mxwell equtions on the notion of informtion in generl nd on chnnel cpcity in prticulr? In this pper, we try to nswer the second question. In other words, do the lws of electromgnetism impose ny limittions on the chievble chnnel cpcity? We re not trgeting in prticulr scenrios, rther, we re going to look t fundmentl limits tht hold in ny scenrio. Anlyzing MIMO chnnel cpcity llows one, in our opinion, to come very close to nswering this question. Our pproch is two-fold one. First, we employ the chnnel correltion rgument nd introduce the concept of n idel scttering to demonstrte tht the minimum ntenn spcing is limited to bout hlf wve length for ny chnnel (i.e., locting ntenns closer to ech other will not result in cpcity increse due to correltion). Secondly, we use the plne wve spectrum expnsion of generic electromgnetic wve nd the Nyquist smpling theorem in the sptil domin to show tht the lws of electromgnetism in its generl form (Mxwell equtions) limit the ntenn spcing to hlf wvelength (for liner ntenn rrys) only symptoticlly, when the number of ntenns n. For finite number of ntenns, this limit is slightly less thn λ /2 becuse slight oversmpling is required to reduce the trunction error when using the smpling series. In ny cse, this limits the number of ntenns nd the MIMO cpcity for given perture size. It should be emphsized tht this is scenrio independent limit. It follows directly from Mxwell equtions nd is vlid in ny scenrio. II. MIMO CHANNEL CAPACITY We employ the celebrted Foschini-Teltr formul for the MIMO chnnel cpcity [1,2], which is vlid for fixed liner n n mtrix chnnel with dditive white Gussin noise nd when the trnsmitted signl vector is composed of sttisticlly independent equl power components ech with gussin distribution nd the receiver knows the chnnel, ρ + C = log2 det I+ G G n [bits/s/hz], (1) where N is the numbers of trnsmit/receive ntenns, ρ is the verge signl-to-noise rtio, I is n n identity mtrix, G is the normlized chnnel mtrix (the entries re complex chnnel gins from ech Tx to ech Rx ntenn), tr[ GG + ] = N, which is considered to be frequency independent over the signl bndwidth, nd + denotes trnspose conjugte. In n idel cse of uncorrelted fullrnk chnnel (1) reduces to C = Nlog2 ( 1 +ρ / N), (2) i.e. the cpcity is mximum nd scles roughly linerly s the number of ntenns. III. THE LAWS OF ELECTROMAGNETISM It follows from (1) tht the MIMO chnnel cpcity crucilly depends the propgtion chnnel G. Since electromgnetic wves re used s the crrier of informtion, the lws of electromgnetism must hve n impct on the MIMO cpcity. They ultimtely determine behviour of G in different scenrios. Hence, we outline the lws of electromgnetism in MIMO system perspective. In their most generl form, they re expressed s Mxwell equtions with chrge nd current densities s the field sources [5]. Approprite boundry conditions must be pplied in order to solve them. We re interested in ppliction of Mxwell equtions to find the chnnel mtrix G in (1). Since the Rx ntenns re

2 locted t some distnce from Tx ntenns (not t the sme points in spce), we re interested in source-free region of spce, (i.e., electromgnetic wves). In this cse, Mxwell equtions simplifies to the system of two decoupled wve equtions [5]: 2 1 E 2 1 H E = 0 H = 0 (3) c t c t where E nd H re electric nd mgnetic field vectors, nd c is the speed of light. There re 6 independent field components (or polriztionl degrees of freedom ) ssocited with (3) (three for electric nd three for mgnetic fields), which cn be used for communiction in rich-scttering environment. Of course, only two of them survive in free spce ( poor scttering ). Hence, in generic scttering cse the number of polriztionl degrees of freedom vries between 2 nd 6, nd ech of them cn be used for communiction. Using the Fourier trnsform in time domin, jωt φ(, r ω ) = φ(,) r te dt (4) (3) cn be expressed s [5] φ 2 (, r ω+ω ) ( / c ) 2 φ(, r ω= ) 0 (5) where φ denotes ny of the components of E nd H, r is position vector nd ω is the frequency. For given frequency ω (i.e., nrrowbnd ssumption), (5) is second-order prtil differentil eqution in r. It determines φ (for given boundry conditions, i.e. Tx ntenn configurtion nd scttering environment) nd, ultimtely, the chnnel mtrix nd the chnnel cpcity. Note tht (5) does not require for ny significntlyrestrictive ssumptions. The source-free region ssumption seems to be quite nturl (i.e., Tx nd Rx ntenns re locted in different points in spce) nd the nrrowbnd ssumption is simplifying but not restrictive since (5) cn be solved for ny frequency nd, further, the cpcity cn be evluted using well-known techniques. Unfortuntely, the link between (5) nd the chnnel mtrix G is implicit. A convenient wy to study this link is to use the spce domin Fourier trnsform, i.e. the plne-wve spectrum expnsion, jkr φ( k, ω ) = φ( r, ω) e dr 1 (6) j( ω t kr ) φ (,) r t = φ(, ω) e d dω 4 k k (2 π) where k is the wve vector. Using (6), (5) cn be reduced to [5] k 2 ( ω/ c) 2 φ( k, ω ) = 0 (7) ( ) Hence, k =ω/ c nd the electromgnetic filed is represented in terms of its plne-wve spectrum φ( k, ω), which in turn is determined through given boundry conditions, i.e. scttering environment nd Tx ntenn configurtion. In the next sections, we discuss limittions imposed by (5)-(7) on the MIMO chnnel cpcity. IV. SPATIAL CAPACITY AND CORRELATION The chnnel cpcity is defined s the mximum mutul informtion [6], C = mx { I ( x, y )} (8) p( x ) where x, y re Tx nd Rx vectors, nd the mximum is tken over ll possible trnsmitted vectors subject to the + totl power constrint, P x = xx P t. Under some conditions, this results in (1). In order to study the impct of the electromgnetism lws on the chnnel cpcity, we definite the sptil cpcity S s the mximum mutul informtion between the Tx vector on one side nd the pir of the Rx vector nd the chnnel (ssuming perfect CSI t the Rx) on the other, the mximum being tken over both the Tx vector nd EM field distributions, + S = mx { I( x, { y, G( E) })}, const.: x x PT, p( x), E 2 (9) 2 1 E E = 0, E= E 0 { r, t} B c t where, to be specific, we ssumed tht the electric field E is used to trnsmit dt (H field cn be used in the sme wy), B is the boundry condition (due to the scttering environment), nd the lst constrint is due to the boundry condition. The first constrint is the clssicl power constrint nd the second one is due to the wve eqution. The chnnel mtrix G is function of E since the electric field is used to send dt. This mximum is difficult to find in generl since one of the constrint is prtil differentil eqution with n rbitrry boundry condition. One my consider reduced version of this problem by defining sptil MIMO cpcity s mximum of the conventionl MIMO chnnel cpcity (per unit bndwidth, i.e. in bits/s/hz) over possible propgtion chnnels (including Tx & Rx ntenn loctions nd sctterers distribution), subject to some possible constrints. In this cse, the cpcity is mximized by chnging G (within some limits), for exmple, by pproprite positioning of ntenns, S = mx { C( G) }, const.: G S ( Mxwell) (10) G where the constrint S ( Mxwell) is due to the Mxwell (wve) equtions. Unfortuntely, the explicit form of this constrint is not known. Additionl constrints my be included (due to limited perture, for exmple). Note tht this definition will give cpcity, which is, in generl, less thn tht in the first definition. Using the nlogy with the chnnel cpcity definition, one cn cll this mximum (if it exists) cpcity of given spce or sptil cpcity (since we hve to vry chnnel during this mximiztion the nme chnnel cpcity seems to be inpproprite simply becuse the chnnel is not fixed. On the other hnd, we vry chnnel within some limits, i.e. within given spce. Thus, the term cpcity of given spce, or sptil cpcity, seems to be pproprite). The question rises: wht is this mximum nd wht re the min fctors which hve n impct on it? Using the ry trcing (geometricl optics) rguments nd the recent result on the MIMO cpcity, it cn be further demonstrted tht there exists n optiml distribution of sctterers nd Tx/Rx ntenns tht provides the mximum possible cpcity in given region of spce. Hence, the MIMO cpcity per unit spce volume cn be defined in

3 fshion similr to the trditionl definition of the chnnel cpcity per unit bndwidth. Considering specific scenrio would not llow us to find fundmentl limit simply becuse the chnnel cpcity would depend on too mny specific prmeters. For exmple, in outdoor environments the Tx nd Rx ends of the system re usully locted fr wy from ech other. Hence, ny MIMO cpcity nlysis (nd optimiztion) must be crried out under the constrin tht the Tx nd Rx ntenns cnnot be locted close to ech other. However, there exists no fundmentl limittion on the minimum distnce between the Tx nd Rx ends. Thus, this mximum cpcity would not be fundmentl limit. In similr wy, prticulr ntenn design my limit the minimum distnce between the ntenn elements but it is just design constrin rther thn fundmentl limit. Similrly, the ntenn design hs n effect on the signl correltion (due to the coupling effect), but this effect is very design-specific nd, hence, is not of fundmentl nture. In other words, the link between the wve equtions (3) or (7) nd the chnnel mtrix G is very implicit since lot depends on Tx nd Rx ntenn designs nd mny other detils. We further consider reduced version of this problem. In prticulr, we investigte the cse when the Tx nd Rx ntenn elements re constrined to be locted within given Tx nd Rx ntenn pertures. We re looking for such loction of ntenn elements (within the given pertures) nd such distribution of sctterers tht the MIMO cpcity ( sptil cpcity) is mximum. While this mximum my not be chievble in prctice, it gives good indiction s to wht the potentil limits of MIMO technology re. In order to void the effect of design-specific detils, we dopt the following ssumptions. Firstly, we consider limited ntenn perture size (1-D, 2-D or 3-D) for both the Tx nd Rx ntenns. All the Tx (Rx) ntenn elements must be locted within the Tx (Rx) perture. As it is wellknown, rich scttering environment is required to order to chieve high MIMO cpcity. Thus, secondly, the rich ( idel ) scttering ssumption is dopted in its most bstrct form. Specificlly, it is ssumed tht there is infinite number of rndomly nd uniformly-locted idel sctterers (the scttering coefficient equls to unity), which form uniform scttering medium ( idel scttering) in the entire spce (including the spce region considered) nd which do not bsorb EM field. Thirdly, ntenn rry elements re considered to be idel field sensors with no size nd no coupling between the elements in the Rx (Tx) ntenn rry. Our gol is to find the mximum MIMO chnnel cpcity in such scenrio (which posses no design-specific detils) nd the limits imposed by the electromgnetism lws. It should be emphsized tht the effect of electromgnetism lws is lredy implicitly included in some of the ssumptions bove. In order to simplify nlysis further, we use the ry (geometricl) optics pproximtion (this justifies the idel scttering ssumption bove). Knowing tht the cpcity increses with the number of ntenns, we try to use s mny ntenns s possible. Is there ny limit to it? Since ntenns hve no size (by the ssumption bove), the given pertures cn ccommodte the infinite number of ntenns. However, if ntenns re locted close to ech other the chnnel correltion increses nd, consequently, the cpcity decreses. A certin minimum distnce between ntenns must be respected in order to void cpcity decrese, even in idel rich scttering [4]. This minimum distnce is bout hlf wvelength. It should be noted tht the model in [4] is two-dimensionl (2D) one. However, it cn be pplied to both orthogonl plnes nd, due to the symmetry of the problem (no preferred direction), the sme result should hold in 3D s well. We note tht, under the ssumptions bove, the ngle-of-rrivl (AOA) of multipth components is uniformly distributed over [ 0,2π ] in both plnes. Thus, the model bove cn be pplied nd the minimum distnce is bout hlf wvelength. Due to the ssumption of uniform scttering medi, ll the ntenns experience the sme multipth environment. When we increse the number of ntenns the cpcity t first increses. But t some point, due to perture limittion, we hve to decrese the distnce between djcent ntenns to ccommodte new ntenns within the given perture. When the djcent ntenn spcing decreses, the cpcity increse slows down nd finlly, when the ntenn spcing is less thn the minimum distnce, the cpcity begins to decrese. Hence, there is n optiml number of ntenns, for which the cpcity is mximum. An rgument similr to the present one hs lredy been presented erlier [8]. However, the optiml number of ntenns hs not been evluted. Using the model in [4], which results in the minimum distnce be equl to pproximtely hlf wvelength, the optiml number N opt of ntenns for given perture size L is strightforwrd to evlute (1-D perture, i.e. liner ntenn rry): Nopt 2 L/ λ+ 1 (11) where λ is the wvelength. Similr expressions cn be obtined for 2-D nd 3-D pertures s well. This is consistent with the diversity combining nlysis, where the minimum distnce is bout hlf wvelength s well [10], nd with n erlier specultion in [1]. V. SPATIAL SAMPLING AND MIMO CAPACITY In the previous section, we rgued tht the chnnel correltion limits the minimum ntenn spcing to hlf wvelength (even in the cse of idel scttering). In this section, we demonstrte tht the sme limit cn be obtined directly from the wve equtions (3) or (5), without refereeing to the chnnel correltion. Let us strt with the wve eqution (5). The field spectrum φ( k, ω) cn be computed in generl cse provided there is sufficient knowledge of the propgtion chnnel nd of the Tx ntenns (note tht we hve not mde so fr ny simplifying ssumptions regrding the propgtion chnnel). Knowing the field, which is given by the inverse Fourier trnsform in (6), nd receive ntenn properties, one my further compute the signl t the ntenn output nd, hence, the chnnel mtrix G. The result will, of course, depend on the Rx ntenn design detils. In order to find fundmentl limit, imposed by the wve equtions (5) on the chnnel cpcity (1), we hve to void ny design-specific detils.

4 Thus, s erlier, we ssume tht the receive ntenns re idel field sensors (with no size, no coupling between ntenns etc.) nd, consequently, the signl t the ntenn output is proportionl to the field (ny of the 6 field components my be used). Hence, the chnnel mtrix entries g ij must stisfy the sme wve eqution s the filed itself. In generl, different Tx ntenns will produce different plne-wve spectr round the Rx ntenns nd, hence, the wve eqution is: ( k 2 ( ω/ c) 2 ) g j ( k, ω ) = 0 (12) where g j ( k, ω) is the plne-wve spectrum produced by j-th Tx ntenn. To simplify things further, we employ the nrrowbnd ssumption: ω= const, nd, hence, k =ω/ c is constnt (the cse of frequency-selective chnnel cn be nlyzed in similr wy see below). The chnnel mtrix entries for given loctions of the Rx ntenns cn be found using the inverse Fourier trnsform in the wve vector domin: 1 j j(, ω ) = kr g r g (, ), (, ) 3 j k ω e dk gij = g j r i ω (13) (2 π) where r i is the position vector of i-th Rx ntenn, nd g j (, r ω) is the chnnel vector, i.e. propgtion fctor from j-th Tx ntenn to n Rx ntenn locted t position r. The integrtion in (13) is performed on hypersurfce k =ω/ c. As we show below, it results in very importnt consequence. Consider, for simplicity, 2-D cse (3-D cse cn be considered in similr wy). In this cse, the integrtion in (13) is performed long the line given by ( ) 2 ( ) kx + ky = ω/ c kx =± ω/ c ky (14) Assume tht the Rx ntenn is liner rry of elements locted on the OX xis, i.e. r y = 0. In this cse, (13) reduces to kmx 1 jk (, ) (, ) x r g x j x ω =, 2 g j kx ωe dkx (2 π) k (15) mx gij = g j ( xi, ω) where kmx =ω / c due to (14). We ignored the evnescent wves with k > kmx becuse they decy exponentilly nd cn be ignored t distnces more thn few λ from the source [5]. Note tht computing g ij corresponds to smpling g j ( x, ω) with smpling points being x i. Let us now pply the Nyquist smpling theorem to (15). According to it, bnd-limited signl, g j( kx, ω) in our cse (it is bnd-limited in k x -domin), cn be exctly recovered from its smples tken t rte equl t lest to twice the mximum signl frequency (Nyquist rte) [6]. In our cse, the Nyquist rte is 2k mx nd the smpling intervl is xmin = 2 π /(2 kmx ) =λ /2 (16) where λ= 2 πc / ω is the wvelength. There is no ny loss of informtion ssocited with the smpling since the originl chnnel vector g j (, r ω) (s well s the field itself) cn be recovered exctly from its smples t x = 0, ± xmin, ± 2 xmin,.... This mens tht by locting the field sensors t smpling points, which re seprted by xmin, we re ble to recover ll the informtion trnsmitted by electromgnetic wves to the receiver. Hence, chnnel cpcity is not ltered. This mens, in turn, tht the minimum spcing between ntenns is hlf wvelength: dmin = xmin =λ /2 (17) Locting ntenns more close to ech other does not provide ny dditionl informtion nd, hence, does not increse the chnnel cpcity. It should be noted tht the sme hlf-wvelength limit ws estblished in Sec. IV using the chnnel correltion rgument, i.e. locting ntenns closer will increse correltion nd, hence, cpcity will decrese. However, while the chnnel correltion rgument my produce some doubts s whether the limit is of fundmentl nture or not (correltion depends on scenrio considered), the sptil smpling rgument demonstrtes explicitly tht the limit is of fundmentl nture becuse it follows directly from Mxwell equtions (i.e., the wve eqution), without ny simplifying ssumptions s, for exmple, the geometricl optics pproximtion [7] (when evluting correltion, we hve to use it to mke ry trcing vlid). Note tht the sptil smpling rguments holds lso for brodbnd chnnel (the smllest wvelength, corresponding to the highest frequency, should be used in this cse to find x min ) nd for the cse of 2-D nd 3-D ntenn pertures. However, in the ltter two cses the minimum distnce (i.e., the smpling intervl) is different [11]. If one uses 2-D ntenn perture (i.e. 2-D smpling), the smpling intervl is x min,2 =λ / 3, (18) nd in the cse of 3-D perture, x min,3 =λ / 2. (19) While the minimum distnce in these two cses is different from the 2-D cse, xmin < xmin,2 < xmin,3 (i.e., ech dditionl dimension possesses less degrees of freedom thn the previous one), the numericl vlues re quite close to ech other. Another interprettion of the minimum distnce effect cn be mde through concept of the number of degrees of freedom. As the smpling theorem rgument shows, for ny limited region of spce (1-D, 2-D or 3-D), there is limited number of degrees of freedom possessed by the EM field itself. No ny ntenn design or their specific loction cn provide more. This is fundmentl limittion imposed by the lws of electromgnetism (Mxwell equtions) on the MIMO chnnel cpcity. An importnt note is in order on using the smpling theorem to find the minimum ntenn spcing. The smpling theorem gurntees tht the originl bndlimited function cn be recovered from its smples provided tht the infinite number of smples is used (bnd-limited function cnnot be time limited!). Hence, the hlf wvelength limit, s derived using the smpling theorem, holds true only symptoticlly, when n. When n is finite, the optiml number of ntenns my be lrger thn tht given by (11), i.e. the minimum spcing my be less thn hlf wvelength becuse slight oversmpling is required to reduce the trunction error. The mximum trunction error of the smpling series for given limited spce region (i.e., the ntenn perture in

5 our cse) decreses to zero s the number of terms in the smpling series (i.e., the number of ntenns in our cse) increses nd provided tht there is smll oversmpling [9]. In this cse, one is ble to recover lmost ll the informtion conveyed by the EM field to the ntenn perture (but not outside of the perture). Hence, one my expect tht the ctul minimum ntenn spcing is quite close to hlf wvelength for lrge number of ntenns. The chnnel correltion rgument, which roughly does not depend on n, lso confirms this. Detiled nlysis shows tht the trunction error effect cn be eliminted by pproximtely 10% increse in the number of ntenns. Fig. 1 illustrtes the effect of oversmpling by considering the MIMO cpcity versus the number of ntenns for given (fixed) perture length (liner ntenn) L = 5λ for different reliztions of n i.i.d. Ryleigh fding chnnel. Clerly, there exists mximum number of ntenns n mx ; using more ntenns does not result in higher cpcity for ny chnnel reliztion. Remrkbly, tht this mximum is slightly lrger thn tht in (11), i.e. sptil smpling nd correltion rguments gree well. Keeping this in mind, one my sy, bsed on the smpling theorem, tht the optiml number of ntenns for given perture size is given pproximtely by (11). Due to the reciprocity of (1), the sme rgument holds true for the trnsmit ntenns s well. Hence, using (2) nd (11) the mximum MIMO cpcity cn be found for given perture size. cpcity, bit/s/hz n mx 2 L + 2 λ n R Fig. 1 MIMO chnnel cpcity versus the number of ntenns for 5 L = λ. It should be noted tht, in some cses, incresing n over N opt in (11) my result in SNR increse due to ntenn gin increse nd, consequently, in logrithmic increse in cpcity. However, this increse is very slow (logrithmic) nd it does not occur if the SNR is fixed, i.e. when one fctors out the effect of the ntenn gin. Besides, the rry ntenn gin versus the number of elements for fixed perture is limited by the gin of continuous ntenn (with the sme perture). This limit is pproximtely 30% lrger thn the rry gin t d = λ /2. Keeping in mind tht the cpcity depends logrithmiclly on SNR nd, consequently, the ntenn gin, we see tht this increse in cpcity is very smll. It is interesting to note tht the MIMO cpcity nlysis of wveguide chnnels, which is bsed on rigorous electromgnetic pproch nd does not involve the usge of the smpling theorem, indictes tht the minimum ntenn spcing is bout /2 λ s well [12]. In mny prcticl cses, the minimum spcing cn be substntilly lrger thn tht in (17). For exmple, when ll the multipth components rrive within nrrow ngle spred << 1, dmin λ/(2 ) >>λ / 2 [4]. Hence, less ntenns cn be ccommodted within given perture nd, consequently, the MIMO cpcity is smller for given perture size. VI. MIMO CAPACITY OF WAVEGUIDE CHANNELS The cse of n idel wveguide MIMO chnnel is especilly interesting becuse the reltionship between informtion theory nd electromgnetics mnifests itself in the most cler form. The min ide for wveguide chnnel is to use the eigenmodes (or simply modes) s independent subchnnels since they re orthogonl (if the wveguide is lossless nd uniform) nd it is well-known tht the MIMO cpcity is mximum for independent sub-chnnels. Since ny field inside of the wveguide cn be presented s liner combintion of the modes [5], the mximum number of independent sub-chnnels equls to the number of modes nd there is no loss in cpcity if ll the modes re used. For lossy nd/or non-uniform wveguide, there exist some coupling between the modes [5] nd, hence, the cpcity is smller (due to the power loss s well s to the coupling). Thus, the cpcity of lossless wveguide will provide n upper bound for true cpcity since some loss nd non-uniformity is lwys inevitble. It should be noted tht if the coupling results in the subchnnel correltion less thn pproximtely 0.5, the cpcity decrese is not significnt [13]. We further ssume tht the wveguide is lossless nd is mtched t both ends. In this cse, the trnsverse electric fields for two different E modes, or two different H modes, or one E nd one H mode re mutully orthogonl [5] EE µ ν ds = cδµν, () S where the integrl is over the wveguide cross-sectionl re S, µ nd ν re composite mode indices, δ µν is Kronecker delt, nd c is constnt (which depends on the power trnsmitted in ech mode). () immeditely suggests the system rchitecture to chieve the mximum MIMO cpcity using the modes: t the Tx end, ll the possible modes re excited using ny of the well-known techniques nd t the Rx end the trnsverse electric field is mesured on the wveguide cross-sectionl re (proper sptil smpling my be used to reduce the number of field sensors) nd is further correlted with the distribution functions of ech mode, see Fig. 2. The signls t the correltor outputs re proportionl to the corresponding trnsmitted signls since the modes re orthogonl nd, hence, there is no cross-coupling between different Tx signls. Thus, the equivlent chnnel mtrix (i.e., Tx end-rx end-correltor outputs) is H= I N (recll tht the wveguide is ssumed to be mtched nd lossless), where I N is NxN identity mtrix, nd the cpcity chieves its mximum (2). Knowing the number of modes N, the mximum MIMO cpcity cn esily be evluted. The mximum cpcity (we cll it further simply cpcity ) of the present MIMO rchitecture described bove does not vry long the wveguide length

6 modes...α2α1 Eigenmode Eigenmode...αˆ 2αˆ1 Wveguide Modultor Demodultor (Tx) (cor. Rx) Fig. 2 MIMO system rchitecture for wveguide chnnel. nd it increses with the number of modes, s one would intuitively expect. If not ll the vilble modes re used, the cpcity decreses ccordingly. The cpcity my lso decrese if the Rx ntenns mesure the field t some specific points rther thn the field distribution long the cross-sectionl re (since the mode orthogonlity cnnot be efficiently used in this cse). In order to evlute the mximum cpcity, we further evlute the number of modes. A. Rectngulr Wveguide Cpcity Let us consider first rectngulr wveguide locted long OZ xis (see Fig. 3). The field distribution t XY plne Tx end x Rx end y O b Fig. 3 Rectngulr wveguide geometry. (cross-section) for E nd H modes is given by well-known expressions [5] nd the vrition long the OZ xis is jk given by zz e, where j is imginry unit, nd k z is the longitudinl component of the wvenumber: 2 ω πm πn kz = γmn, γ mn = c + 0 b, (21) where ω is the frequency, c 0 is the speed of light, nd m nd n designte the mode (note tht E nd H modes with the sme (m,n) pir hve the sme γ mn ). The sign of k z is chosen in such wy tht the filed propgtes long OZ xis (i.e., from the Tx end to the Rx end). The cse of γ mn >ω / c corresponds to the evnescent field, which decys exponentilly with z nd is negligible t few wvelength from the source [5]. Assuming tht the Rx end is locted is fr enough from the Tx end (i.e., t lest few wvelengths), we neglect the evnescent field. Hence, the mximum vlue of γ mn is γ mn,mx =ω / c. This limits the number of modes tht exist in the wveguide t given frequency ω. All the modes must stisfy the following inequlity, which follows from (21): m n + 4 b, (22) where = / λ, b = b/ λ nd λ is the free-spce wvelength; nd mn=, 1,2,... for E mode nd mn=, 0,1,..., m+n 0 for H mode. Using numericl procedure nd (22), the number of modes N cn be esily z evluted. A closed-from pproximte expression cn be obtined for lrge nd b by observing tht (22) is, in fct, n eqution of ellipse in terms of (m,n) nd ll the llowed (m,n) pirs re locted within the ellipse. Hence, the number of modes is given pproximtely by the rtio of res: S /4 2 e πb πs N = = w, (23) S 0 λ λ where Se = 4π b is the ellipse re, S 0 = 1 is the re round ech (m,n) pir, Sw = b is the wveguide crosssectionl re, the fctor ¼ is due to the fct tht only nonnegtive m nd n re considered, nd the fctor 2 is due to the contributions of both E nd H modes. As (23) demonstrtes, the number of modes is determined by the rtio of the wveguide cross-section re b to the wvelength squred. As we will see further, this is true for circulr wveguide s well. Hence, one my conjecture tht this is true for wveguide of rbitrry cross-section s well. This conjecture seems to be consistent with the sptil smpling rgument (2-D smpling must be considered in this cse). In fct, (23) gives the number of degrees of freedom the rectngulr wveguide is ble to support nd which cn be further used for MIMO communiction. Fig. 4 compres the exct number of modes computed numericlly using (22) nd the pproximte number (23). As one my see, (23) is quite ccurte when nd b re greter then pproximtely wvelength. Note tht the number of modes hs step-like behvior with / λ, which is consistent with (22). Using (2) nd (23), the mximum cpcity of the rectngulr wveguide chnnel cn be esily evluted. The nlysis bove ssumes tht the E-field (including both E x nd E y components) is mesured on the entire Number of modes Exct Approximte 2-D Arry 1-D Arry (OX) /λ Fig. 4. Number of modes in rectngulr wveguide for =b.

7 Cpcity, bit/s/hz D Arry 1-D Arry (OX) Exct Approximte Limit (27) /λ Fig. 5. MIMO cpcity in rectngulr wveguide for =b nd SNR= db. cross-sectionl re (or t sufficient number of points to recover it using the smpling expnsion). However, it my hppen in prctice tht only one of the components is mesured, or tht the field is mesured only long OX (or OY) xis. Apprently, it should led to the decrese of the vilble modes. This is nlysed below in detils. Let us ssume tht the E-field (both components) is mesured long the OX xis only (this corresponds to 1-D ntenn rry locted long OX). Due to this limittion, one cn compute the correltions t the Rx using the integrtion over OX xis only since the field distribution long OY xis is not known. Hence, we need to find the modes tht re orthogonl in the following sense: I = EE µ ν dx = cδµν, (24) 0 In this cse, one finds tht two different E-modes E mn 1 1 nd E mn re orthogonl provided tht m 1 m2; if these modes hve the sme m index, they re not orthogonl. The sme is true bout two H-modes nd bout one E- mode nd one H-mode. This results in substntil reduction of the number of orthogonl modes since, in the generl cse, two E-modes re orthogonl if t lest one of the indices is different, i.e. if m 1 m 2 or n1 n2. Surprisingly, if one mesures only E x component in this cse, the modes re still orthogonl provided tht m1 m2. Hence, if the receive ntenn rry is locted long OX xis, there is no need to mesured E y component it does not provide ny dditionl degrees of freedom, which cn be used for MIMO communictions (recll tht only orthogonl modes cn be used). The number of orthogonl modes cn be evluted using (22): Nx 4 / λ, (25) This corresponds to 2 / λ degrees of freedom for ech (E nd H) field. Note tht this result is similr to tht obtined using the sptil smpling rgument, i.e., independent field smples (which re, in fct, the degrees of freedom) re locted t λ /2. The similr rgument holds true when the receive rry is locted long OY xis. In this cse two modes re orthogonl provided tht n1 n2 nd there is lso no need to mesure the E x component. The number of orthogonl modes is pproximtely Ny 4 b/ λ, (26) Fig. 5 shows the MIMO cpcity of rectngulr wveguide (the sme s in Fig. 3) for SNR ρ = db. Note tht the cpcity sturtes s / λ increses. This is becuse (2) sturtes s well s N increses: lim C =ρ / ln 2 (27) N C in (2) cn be expnded s i i ρ ( 1) ρ C = ln 2 i 1 N (28) + i= 0 For lrge N, i.e. for smll ρ / N, this series converges very fst nd it cn be pproximted by first two terms: ρ ρ C 1 ln 2 2N (29) The cpcity does not chnge substntilly when the contribution of the 2 nd term is smll: ρ << 1 N > Nmx ρ (30) 2N N mx is the mximum resonble number of ntenns (modes) for given SNR (or vice vers): if N increses bove this number, the cpcity does not increse significntly. It my be considered s prcticl limit (since further increse in cpcity is very smll nd it requires for very lrge increse in complexity). Using (23) nd (25), one finds the mximum resonble size of the wveguide for the cse of 2-D nd 1-D rrys correspondingly: mx ρ (2-D rry), mx ρ (1-D OX rry), λ 2π λ 4 (31) Note tht Fig. 5 shows, in fct, the fundmentl limit of the wveguide cpcity, which is imposed jointly by the lws of informtion theory nd electromgnetism. B. Rectngulr Cvity Cpcity The nlysis of MIMO cpcity in cvities is very different from tht in wveguides in one importnt spect. Nmely, the modes of cvity exist only for some finite discrete set of frequencies (recll tht, s in the cse of wveguide, we consider lossless cvity). Hence, there my be no modes for n rbitrry frequency. To void this problem, we evlute the number of modes for given bndwidth, f [ f, f f ] 0 0+, strting t f 0. For rectngulr cvity, the wve vector must stisfy [5]: 2 πm πn πp ω k = + + = b c, (32) c0 where c is the wveguide length (long OZ xis in Fig. 1), nd p is non-negtive integer; mn, = 1,2,3,..., p= 0,1,2,... for E-modes, nd mn, = 0,2,3,..., p= 1,2,... for H-modes ( m= n = 0 is not llowed). Noting tht (32) is eqution of sphere in terms of (m,n,p), the number of modes with k [ k0, k0+ k] cn be found s the number of (m,n,p) points between two spheres with rdiuses of k 0 nd k0+ k correspondingly. Using the rtio of res pproch described bove, the number of modes is pproximtely: V /8 8 2 e πv c c f N =, (33) V 3 0 λ f0

8 2 where Ve = 4πk k is the volume between the two 3 spheres, V0 =π / Vc is the volume round ech (m,n,p) point, Vc = bc is the cvity volume; fctor 2 is due to two types of modes, nd fctor 1/8 is due to the fct tht only nonnegtive vlues of (m,n,p) re llowed. An importnt conclusion from (33) is tht the number of modes is determined by the cvity volume expressed in terms of wvelength nd by the normlized bndwidth. Detiled nlysis shows tht (33) is ccurte for lrge, b, nd c, nd if c/ λ< f0 /4 f. It should be noted tht the mode orthogonlity for cvities is expressed through the volume integrl (over the entire wveguide volume), EE µ ν dv = cδµν, (34) V c nd, hence, ll the modes re orthogonl provided tht the field is mesured long ll 3 dimensions, which, in turn, mens tht 3-D rrys must be used, which my not be fesible in prctice. If only 2-D rrys re used, then the mode orthogonlity is expressed s for wveguide, i.e. (), nd, consequently, only those modes re orthogonl tht hve different (m,n) indices. The use of 2-D rry results in significnt reductions of the number of modes for lrge c, s Fig. 6 demonstrtes. Note tht for smll c, there is no loss in the number of orthogonl modes. This is becuse different p correspond in this cse to different (m,n) pirs (this cn lso be seen from (32)). However, s c increses, different p my include the sme (m,n) pirs, which results in the number loss if 2-D rry is used. In fct, the 2-D cse with lrge c is the sme s the wveguide cse (with the sme cross-sectionl re), s it should be. The vlue of c for which the cvity hs the sme number of orthogonl modes s the corresponding wveguide cn be found from the following eqution: ct f N 0 c Nw = λ 4 f, (35) Hence, if 2-D ntenn rrys re used nd c ct, the wveguide model provides pproximtely the sme results s the cvity model does, i.e. the cross-section hs the mjor impct on the cpcity, while the effect of cvity length is negligible. The wveguide model should be used to evlute the number of orthogonl modes (nd cpcity) in this cse becuse it is more simple to del with. For exmple, long corridor cn be modelled s wveguide rther thn cvity (despite of the fct tht it is closed nd looks like cvity). Fig. 6 shows the cpcity in the cvity. While the cpcity of 2-D rry system sturtes like the wveguide cpcity, which is limited by nd b, the cpcity of 3-D system is lrger nd sturtes t the vlue given by (27). It should be noted tht (27) is the cpcity limit due to the informtion theory lws, nd (23), (25), (26), nd (33) re the cpcity limits due to the lws of electromgnetism (i.e., limited due to the number of degrees of freedom of the EM field). VII. REFERENCES [1] G.J. Foschini, M.J Gns: On Limits of Wireless Communictions in Fding Environment when Using Multiple Antenns, Wireless Personl Communictions, vol. 6, No. 3, pp , Mrch [2] I.E. Teltr, "Cpcity of Multi-Antenn Gussin Chnnels," AT&T Bell Lb. Internl Tech. Memo., June 1995 (Europen Trns. Telecom., v.10, N.6, Dec.1999). [3] D. Chizhik, G.J. Foschini, R.A. Vlenzuel, Cpcities of multielement trnsmit nd receive ntenns: Correltions nd keyholes, Electronics Letters, vol. 36, No. 13, pp , 22 nd June 00. [4] S. Loyk, G. Tsoulos, Estimting MIMO System Performnce Using the Correltion Mtrix Approch, IEEE Communiction Letters, v. 6, N. 1, pp , Jn. 02. [5] E.D. Rothwell, M.J. Cloud, Electromgnetics, CRC Press, Boc Rton, 01. [6] J.D. Gibson (Ed.), The Communictions Hndbook, CRC Press, Boc Rton, 02. [7] S.R. Sunders, Antenns nd Propgtion for Wireless Communiction Systems, Wiley, Chichester, [8] S.L. Loyk, J.R. Mosig, Sptil Chnnel Properties nd Spectrl Efficiency of BLAST Architecture, AP00 Millennium Conference on Antenns & Propgtion, Dvos, Switzerlnd, 9-14 April, 00. [9] A.J. Jerry, The Shnnon Smpling Theorem Its Vrious Extensions nd Applictions: A Tutoril Review, Proc. of IEEE, v. 65, N. 11, pp , Nov [10] Jkes, W.C. Jr.: Microwve Mobile Communictions, John Wiley nd Sons, New York, [11] D.P. Petersen, D. Middleton, Smpling nd Reconstruction of Wve-Number-Limited Functions in N-Dimensionl Eucliden Spces, Informtion nd Control, v. 5, pp , 1962 [12] S.L. Loyk, Multi-Antenn Cpcities of Wveguide nd Cvity Chnnels, IEEE CCECE 03, Montrel, My 03 [13] S.L. Loyk, Chnnel Cpcity of MIMO Architecture Using the Exponentil Correltion Mtrix, IEEE Communiction Letters, v.5, N. 9, pp , Sep 01. Number of modes Cpcity, bit/s/hz D 2-D Exct number of modes (3-D) Approximte number of modes (3-D) Exct number of modes (2-D) Wveguide number of modes (2-D) c/λ Fig. 6. Number of orthogonl modes in rectngulr cvity for = 4 λ, b= 2λ nd f / f0 = D Arry 3-D Arry Exct Approximte Limit (27) c/λ Fig. 7. Cpcity in rectngulr cvity for = 4 λ, b= 2λ nd f / f0 = 0.01.