Advances in Radio Science

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1 Advances in Radio Science, 3, , 25 SRef-ID: /as/ Copenicus GmbH 25 Advances in Radio Science Radiation Patten Analysis of Antenna Systems fo MIMO Divesity Configuations O. Klemp H. Eul Univesity of Hannove, Depatment of High Fequency Technology Radio Systems, Hannove, Gemany Abstact. Multiple-input multiple-output MIMO) antenna systems antenna configuations fo wideb multimode divesity ank among the emeging key technologies in net geneation wieless communication systems. The analysis of such tansmission systems usually neglects the influences of eal antenna adiation chaacteistics as well as the influences of mutual coupling in a multielement antenna aangement. Nevetheless, to achieve a detailed desciption of divesity gain channel capacity by using seveal tansmit- eceive antennas in a wieless link, it is essential to take all those effects into account. The epansion of the adiation fields in tems of spheical eigenmodes allows an analytical desciption of the antenna adiation chaacteistics accounts fo all the coupling effects in multielement antenna configuations. Theefoe the adiation patten analysis by spheical eigenmode epansion povides an efficient altenative to establish an analytical appoach in the calculation of envelope coelation o channel capacity. 1 Intoduction The spheical mode epansion SME) is a well known technique in tems of the chaacteiation of a vaious numbe of applications of classical electomagnetic scatteing adiation poblems. Numeous authos such as Statton 1941) Wene Mitta 2) have pesented a genealied theoy employing spheical mode analysis adopted this method to study a multiplicity of wave popagation poblems. Applications of the spheical mode epansion have been applied e.g. to the numeical analysis of Cassegainianfed paaboloids in Potte 1967), the investigation of multiple scatteing of electomagnetic waves in Buning Lo 1971), the computation of antenna adiation pattens of abitay wie antennas in Chen Simpson 1991b,a) the modeling of base station antenna equipment in Adane et al. 22). Futhemoe the application of spheical mode epansion to pefom a tansfomation between nea- fafield elated field quantities as descibed in Ludwig 1971) is a fundamental basis fo todays techniques of spheical antenna nea field measuement. Coespondence to: O. Klemp klemp@hft.uni-hannove.de) In this aticle the analysis of the adiation behavio using spheical eigenmodes will be applied to the chaacteiation of the divesity pefomance of multielement antenna configuations in tems of inteelement coelation. Theefoe we pesent a modified technique of spheical mode epansion as in Chen et al. 1992) fo the analysis of abitay antennas multielement antennas. Based upon a suitable statistical model, e.g. povided in Waldschmidt Wiesbeck 24) to lude the spatial popeties of a multipath popagation scenaio, a closed fom epesentation fo antenna coelation will be deived. The aticle is oganied as follows, Sect. 2 eviews the fundamental popeties of the spheical mode epansion technique with espect to the electic- magnetic vecto potential fo a given elementay souce distibution. Section 3 deives a genealied epesentation of antenna adiation patten due to the Faunhofe appoimation of the Geen s adiation function. Subsequently the modal epesentation of antenna adiation pattens is used in Sect. 4 to specify the powe coelation between inteactive antenna elements in a divesity configuation to establish a measue fo thei divesity pefomance. The pesented analysis is adopted to an aay of esonant λ/2 dipoles given in Sects a configuation of multi-pot wideb logaithmically peiodic antennas in Sect. 6. The latte analysis will be completed by esults deived fom spheical nea-field measuement. Section 7 concludes this aticle. 2 Modal Solutions of Vecto Potentials The spheical mode epansion as given in this section will be used to accomplish the tansfomation between a known distibution of electic magnetic cuents e.g. on the metaliation sufaces of an antenna stuctue) the esulting electomagnetic field distibution in the field egion, which is consideed to be souce-fee. In Ludwig 1971) this tansfomation technique was applied to deive a genealied mapping ule between field quantities in diffeent coodinate systems. Theefoe in SME analysis a sepaation of coodinate systems fo the given souce-egion the equested fieldegion is equied. Fo a given volume V containing electic magnetic cuent densities J J m as shown in Fig. 1 the magnetic- electic vecto potentials A F can be

2 Figues 158 O. Klemp H. Eul: Radiation Patten Analysis fo MIMO Divesity Antennas ' J') J m ') V - ' y P n n m)! Fig. Fig Distibution Distibution of of electic electic magnetic magnetic cuent cuent densities densities J J J m inpvolume n cos ξ) V. = ɛ m n + m)! m= J m in volume V. Pn m cos ϑ) P n m cos ϑ ) cos m ϕ ϕ )) antenna i epesented using Chen et al. 1992) as follows: Eq. 5) gives a genealied solution to the magnetic vecto k ant. j ij i P, ) A) = 1 J ) potential. Pn m cos ϑ) denotes the associated Legende function of degee n ode m, ɛ m specifies Neumann s num- 4π V ) j e jk dv 1) be 1 fo m=, 2 fo m>). F ) = 1 J - m ) 4π y V ) e jk dv. A ) = 5) n In Eq. 1) k=2π jk a m n ɛµf denotes the wave numbe of the electomagnetic field, whee ɛ epesents the pemittivity of fee N n m h2) n k) Pn m cos ϑ) cos mϕ) n= m= Fig. 2. Two-element antenna configuation in fee space with antennas i j. n space µ the espective pemeability. Hamonic time dependence will be assumed, so that f is equivalent to the ope- n= m= jk b m n N n m h2) n k) Pn m cos ϑ) sin mϕ). ational fequency of any elementay souce consideed in the In Eq. 5) the esponse to the given electic cuent distibution is sepaated with espect to souce point- field point volume V. The vecto epesents the distance P, ) between the oigin the pointk of obsevation gives the displacement of the souce point elation d dependent quantities using the nomaliation N to the oigin. Theefoe n m given in Eq. 6). A F epesent a solution to the vecto Helmholt equation of the total dipole electic 2 field E magnetic field H. Subsequently, the analysis - will be esticted to the y distibution Nn m of electic cuent densities, whee the total adiation field is = 2n + 1) n m)! ɛ m 6) 4π n + m)! eclusively given in tems of the dipole magnetic 1 vecto potential. Using the theoem of dual fields in electodynamics an abitay distibution of the magnetic cuent density J m can be systems appoves a sepaation of functional quantities that Othogonality between the aes of the spheical coodinate Fig. elated 3. Configuation to an electic of two cuent paallel density oiented J, λ/2 fodipoles this pupose with a elative 1) displacement ae solely elated of d along to athe single -ais. coodinate diection. Theefoe epesents a complete solution of the adiation poblem. Povided the field solution of the souce distibution is considnates) may be analyed independently fom the espective the souces of the adiation fields given in pimed coodieed in tems of the spheical coodinate system, ϑ, ϕ), the electomagnetic fields at the field-points. Any souce dependence is given in tems of the vecto epansion coefficients Geen s function may be epesented as in Chen Simpson 1991a) using the eo-ode spheical Hankel function a m n b m n of the field epansion as given in Eqs. 7) h 2) of the second kind as given in Eq. 2) with R= : 8). e jkr R = jk h2) kr). 2) Fo the validity of Eq. 2), the case whee > has to be consideed, whee the antennas will be analyed in the tansmit case. Applying the piple of ecipocity in antenna analysis, the eceive case is implicitly given in Eq. 2). To deive a modal epesentation of the electomagnetic 17 field, fist of all Eq. 2) can be e-fomulated in tems of an infinite seies as given in Eq. 3). Using n.-ode spheical Bessel functions j n k ) in tems of the souce-point dependence, n.-ode spheical Hankel functions h 2) n k) of the second kind fo the dependence of the obsevation-point Legende polynomials P n cos ξ) of degee n, Eq. 2) is the fist step to deive a sepaation of souce- field egion. h 2) kr) = n= 2n + 1) j n k ) h 2) n k) P n cos ξ). 3) Using the addition theoem fo Legende polynomials as in Abamovit Stegun 1972), a m n b m n = = V ) V ) 4) N m n J ) j n k ) 7) P m n cos ϑ ) cos mϕ ) dv N m n J ) j n k ) 8) P m n cos ϑ ) sin mϕ ) dv Theefoe, Eqs. 7) 8) denote a genealied desciption of all consideed field souces in volume V in tems of elementay spheical wave contibutions.

3 O. Klemp H. Eul: Radiation Patten Analysis fo MIMO Divesity Antennas Antenna Radiation patten analysis Consideing the adiation behavio of antenna elements usually the fa-field appoimation by Faunhofe is applied. This appoimation given in Eq. 9) neglects diffeences in amplitude of all oming elementay waves at a cetain fa field distance so that the esponse to all field souces leads to a wave popagation with plana amplitude distibution. Thus effects of intefeence as a consequence of spatial sepaation between the field souces lead to a maimum phase deviation of π/8 fo all field points that adhee to the fa field conditions. > 8 a2 λ. 9) In Eq. 9) a denotes the adius of a sphee that completely encloses the egion of distibuted souce points. As a consequence of the Faunhofe appoimation, Eq. 3) can be simplified with espect to the spheical Hankel function: lim >8 a2 λ h 2) n k) = j n+1 e jk. 1) k In spheical coodinates ϑ, ϕ), the vecto potentials A o F may be diectly mapped to the espective field quantities E o H. Unde the tems of an eclusive distibution of electic cuents J in the consideed souce egion a field point adheing to the fa field condition by Faunhofe, Eq. 11) epesents the complete solution of the electic field in tems of the spheical components E ϑ E ϕ. [ ] [ ] Eϑ Aϑ E ϑ, ϕ) = = jkz. 11) E ϕ A ϕ In Eq. 11) Z denotes the value of chaacteistic fee-space impedance given by Z=12π. E ϑ, ϕ) = e jk n N m { n a m n cos mϕ) n= m= + b m n sin mϕ)} j n P m n cos ϑ). 12) Theefoe, Eq. 12) epesents the fa-field solution of the electomagnetic field fo a tansmit antenna in tems of a seies epesentation of spheical hamonics. The espective souce distibution is fomulated in tems of an electic cuent distibution in accodance to Fig. 1 Eq. 1). Due to the limitation of fa field conditions, the spatial oientation of tansvese electomagnetic field components becomes othogonal in the fa field plane, that again ehibits a nomal oientation to the adial diection of wave popagation at any point of obsevation. Theefoe the espective dependence of electical field E magnetic vecto potential A is specified by Eq. 11) fo the special case of spheical coodinates. Applying Eq. 12) a genealied desciption of antenna adiation patten is deived by means of an infinite seies epansion implying spheical hamonics. Fo this eason Eq. 13) ludes the spatial popeties of antenna adiation in tems of the vecto antenna adiation patten C ϑ, ϕ). E ϑ, ϕ) = e jk C ϑ, ϕ). 13) The vecto epansion coefficients of the field appoach, a m n b m n in tems of the coesponding antenna adiation patten, ae elated to the epansion coefficients of the magnetic vecto potential a m m n b n employing Eqs. 9) 11) as follows: { a m } { } n,ϑ a m a m = jkz n,ϑ, 14) n,ϕ a m n,ϕ { b m n,ϑ b m n,ϕ } = jkz { } b m n,ϑ. 15) b m n,ϕ The nomalied eigensolutions to the vecto Helmholt equation ae given in tems of the nomalied spheical hamonics Ynm e ϑ, ϕ) Y nm o ϑ, ϕ) as given in Statton 1941) due to: { } { } Y e nm ϑ, ϕ) Ynm o ϑ, ϕ) = Nn m P n m cos mϕ) cos ϑ). 16) sin mϕ) With espect to the othogonality popety between spheical hamonics of diffeent degee n o ode m as shown in Wene Mitta 2), the field epansion as given in Eq. 12) epesents a complete solution to the adiation behavio of an abitay souce configuation epesented in tems of vecto epansion coefficients a m n b m n. Fo pactical consideations the infinite uppe summation bound of Eq. 12) has to be limited to a finite value N. Impaiments esulting fom the limitation of the infinite summation can be isolated with espect to the equiement of appoimative field convegence e.g. given by Naasimhan et al. 1985). Theefoe modal indices geate N min =ka ae egaded to have little significance to the field composition, whee a denotes the minimum adius of a sphee that completely encloses the souce antenna. A numeical study of the convegence behavio of the field epansion is given e.g. in Chen Simpson 1991a) Chen et al. 1992). The vecto coefficients of the field epansion can be detemined accoding to Witte et al. 23) by methods of numeical point matching as given in Hafne 199). As descibed in Ludwig 1971) the matching pocedue will be applied on the suface on which the fields ae known. Subsequently the fa field sphee is egaded to be the suface whee the field quantities ae given in tems of the fa field components of antenna adiation patten. In ode to detemine the equested modal weights, an equiangula gid is placed upon the fa field sphee poviding constant angula steps of ϕ=2π/p ϑ=π/q. P Q coespond to the numbe of field points in the diections of aimuth elevation, espectively. Antenna adiation pattens ae deived fom numeical analysis employing the FEM. 4 Computation of Multielement Antenna Coelation Antenna divesity eception is an effective technique fo suppessing multipath fading in a mobile antenna system. Divesity eception attains effective fading mitigation by the synthesis, selection, switching of seveal eceived signals

4 Fig. 1. Distibution of electic magnetic cuent densities J J m in volume V. 16 O. Klemp H. Eul: Radiation Patten Analysis fo MIMO Divesity Antennas k - ant. j j ij antenna i i P, ) ρ Pi,j 2 ρi,j 2 = R i,j σi 2σ j 2. 17) 17 Subsequently, the elation between powe coelation coefficient ρ Pi,j comple coelation coefficient ρ i,j as given in Eq. 17) between antenna elements i j will be consideed. The powe coelation theefoe depends on the covaiance R i,j between the two antenna elements the vaiances σi 2 σj 2 that ae detected at the espective divesity banches. Figue 2 depicts the coesponding configuation of a two-element antenna in eceive mode. The distance vecto ij accounts fo the spatial sepaation of antennas i j. As shown in Leife 22) the spatial sepaation between the antenna elements esults in a eceive gain due to spatial divesity. The diection of plane waves impinging at a diection ϑ ϕ ae epesented in tems of k. Povided an antenna fa field epesentation that may be decomposed into a ϑ- ϕ dependent faction which ae given y by C ϑ,i ϑ, ϕ) C ϕ,i ϑ, ϕ) in tems of antenna element i a 2-D polaiation invaiant pobability distibution p ϑ,ϕ ϑ, ϕ) of the AOA with the appopiate nomaliation given in Waldschmidt Wiesbeck 24), Eq. 18) denotes antenna covaiance. 2π π [ R i,j = K XPR C ϑ,i ϑ, ϕ) Cϑ,j ϑ, ϕ) 18) The atio of the mean ident powe in tems of a vetical Fig. Fig. 2. Two-element 2. Two-element antenna antenna configuation configuation in fee in space fee space with with antennas i j. antennas i j. polaied eceive signal to the mean ident powe of a hoiontal polaied eceive signal is denoted by the facto of coss-polaiation atio XP R. In Eq. 18), means conjugate opeation, K is a popotionality constant. that ae coelated to the smallest possible etent. In the case P, ) of antenna divesity, k eduction of the coelation coefficient 2π π [ between the eceiving antenna dbanches leads to an enhanced σi 2 = K XPR Cϑ,i ϑ, ϕ) 2 divesity effect. Coelation analysis may be pefomed using a epesentation of tansmission line paametes like input impedances dipole 2 + C ϕ,i ϑ, ϕ) 2] p ϑ,ϕ ϑ, ϕ) sin ϑdϑdϕ. 19) -scatteing paametes at the individual divesity banches as demonstated in Kildal y Rosengen As the spatial popeties of the consideed tansmit channel 23) Deneyd Kistensson 24). The deived stongly influence to the popeties of antenna coelation, coelation coefficients do not account dipole 1fo the spatial popeties of the tansmit channel, whee the eceive antenna aay cal fomulation of antenna adiation pattens theefoe sim- they have to be inevitably taken into account. An analyti- is applied, but inheently assume a unifom distibution of plifies the computation of coelation coefficients fo divesitydisplacement analysis. Inof Leife d along 22) the -ais. a simplified multipole com- Fig. the 3. Configuation angle of aival ofaoa). two paallel Essentially, oientedinteelement λ/2 dipolescoela- tion coefficients in a multielement antenna configuation ae putational model was published in tems of paallel oiented with a elative stongly affected by the popeties of the consideed multipath popagation envionment as will be shown in the fol- vecto antenna adiation pattens C i,j ϑ, ϕ) of antennas i dipole antennas. It is theefoe staightfowad to epess the lowing. In Jakes 1974) the spatial popeties of a multielement divesity antenna configuation may be acquied by ap- Povided the seies epansion fo the adiation fields as in j in tems of a spheical mode epansion as given in Sect. 3. plying two-dimensional pobability distibutions pϑ, ϕ) in Eq. 12), the epessions fo antenna covaiance vaiance become a function of modal epansion coefficients a m n tems of the AOA distibution of the oming electomagnetic field in elevation- aimuth diection. Eamples fo b m n. Equation 2) accounts fo the seies epesentation coelation analysis in multielement antennas may be found of antenna adiation pattens of antennas i j by means of in Tsunekawa Kagoshima 199), Svantesson 22) spheical eigenmodes epesents a genealied desciption of antenna covaiance in tems of a spheical mode e- Waldschmidt Wiesbeck 24). pansion. + C ϕ,i ϑ, ϕ) C ϕ,j ϑ, ϕ) ] e jk ij p ϑ,ϕ ϑ, ϕ) sin ϑdϑdϕ R i,j = K n i n j n i = m i = n j = m j = ) XPRa ni m i,ϑan j m j,ϑ + a n i m i,ϕan j m j,ϕ ) + XPRb ni m i,ϑbn j m j,ϑ + b n i m i,ϕbn j m j,ϕ ) + XPRa ni m i,ϑbn j m j,ϑ + a n i m i,ϕbn j m j,ϕ + XPRb ni m i,ϑan j m j,ϑ + b n i m i,ϕan j m j,ϕ { 2) I ee n i m i n j m j I oo n i m i n j m j I eo n i m i n j m j ) I oe n i m i n j m j }. As can be seen in 2), covaiance computation equies a e,o e,o foufold summation of 2-D coupling integals In i m i n j m j that ae weighed by the espective epansion coefficients. Theefoe an evaluation of Eq. 2) might become quite time consuming with the advantage to deive an analytical fomulation of the coelation popeties in multielement antenna configuations. The coupling integals ae given in tems of a poduct of two nomalied spheical hamonics Ynm e,o ϑ, ϕ)

5 Fig. 2. Two-element antenna configuation in fee space with antennas i j. O. Klemp H. Eul: Radiation Patten Analysis fo MIMO Divesity Antennas 161 dipole 2 k - d dipole 1 P, ) y Fig. 3. Configuation of two paallel oiented λ/2 dipoles with a elative displacement of d along the -ais. Fig. 3. Configuation of two paallel oiented λ/2 dipoles with a elative displacement of d along the -ais. with the angula distibution of oming waves, p ϑ,ϕ ϑ, ϕ) nom. distance d/ as shown in Eq. 21). 2π π Fig. 4. Powe coelation Fig. of 4. isolated- Powe coelation coupled of isolated- dipole antennas coupled fo ipole angula antennas spead of σ ϕ = 1 e,o e,o In i m i n j m j = Yn e,o i m i ϑ, ϕ) Yn e,o j m j ϑ, ϕ) 21) fo an angula spead of σ ϕ =1. e jk ij p ϑ,ϕ ϑ, ϕ) sin ϑdϑdϕ. Using the othogonality elation as in Wene Mitta 2) fo spheical hamonics, Eq. 2) may be simplified with espect to a cetain symmety of the AOA distibution p 17 ϑ,ϕ ϑ, ϕ) the antenna aay topology. Applying the spheical wave epansion to the geneal fomulation of vaiances as in Eq. 19) fo antenna element ν, ν=1,2 yields: The coelation analysis of an aangement of two paallel oiented, esonant λ/2 dipoles in fee space depicted in Fig. 3 will be accomplished in this section. In ode to compae the esults fom the customay computation of antenna coelation by an evaluation of the twofold Riemann sum in Eqs. 18) 19) with the method based upon a spheical mode epansion of antenna adiation pattens as given in Sect. 4 the pattens ae evaluated in tems of a spheical mode epansion delimited by N. In tems of the coelation analysis of the two-element dipole configuation two diffeent cases have to be consideed. The fist case assumes both dipole antennas isolated fom each othe, so that the effect of electomagnetic inteaction is neglected by definition. The assumption justifies a modification of the field epansion applying the aay facto of the antenna configuation into the coelation analysis. The esults of this study ae summaied in Sect In the second appoach electomagnetic coupling between the dipole antennas is taken into account. powe coelation coefficient P12 powe coelation coefficient P12 1,,8,6,4,2 no inteelement coupling no coup., SME N=2 mutual coupling,,,25,5,75 1, 1,25 1,5 Section 5.2 povides the esults fo the coupling-case of the dipole1, aay.,8 5.1 Analysis of isolated Dipoles no inteelement coupling no coup., SME N=2 mutual coupling In tems,6 of a configuation of identical isolated antenna elements, the epesentation fo the total adiation patten can be simplified with espect to the appopiate aay facto 2π { π σ 2 ν = K n ν Gϑ,,4 ϕ) of the antenna configuation. Theefoe the computation of antenna coelation simplifying equies the sphe- XPR a nν m ν,ϑyn e ν m ν 22) n ν = m ν = ical mode,2 epansion of the adiation fields of the isolated +b nν m ν,ϑyn o n ν antenna element that is identical fo all elements within the 2 ν m ν + a nν m ν,ϕyn e ν m ν aangement. The consideation of an aay of identical isolated antenna, elements,25,5 theefoe,75 leads 1, to an effective 1,25 educ- 1,5 n ν = m ν, = +b nν m ν,ϕyn o ν m ν 2} p ϑ,ϕ ϑ, ϕ) sin ϑdϑdϕ. tion of the computationalnom. esouces distance fo coelation d/ analysis. Fo convenience, the AOA distibution will be esticted to Fig. 5. Powe coelation the of y-plane isolated- hence coupled be dipole given in antennas tems of in atems Laplacian of equally dis-distibutetibution as in Svantesson 22) that is popely descibed A 5 Analysis of Two-Element Dipole Aay hoiontal plane. by a mean value m ϕ the espective spead σ ϕ. All simulations ae pefomed using a mean value of m ϕ =9 using the coodinate system depicted in Fig. 3. Thee diffeent values of angula spead σ ϕ ae consideed. Figue 4 depicts antenna powe coelation coefficient with espect to the nomalied inteelement distance d/λ a naow angula spead σ ϕ =1. In ode to addess to the influences of electomagnetic coupling on antenna coelation, the dashed lines in Fig. 4 epesent the coelation coefficient in tems of coupled dipole antennas deived fom the customay evaluation of Eqs. 18) 19). The esults deived fom a spheical wave epansion of the adiation patten of the isolated dipole antennas ae given as cicles. The seies epesentation in Eq. 12) is esticted with espect to N=2 leading to a maimum of 12 comple vecto coefficients in the field epansion. Compaing the pocesses of antenna coelation fo the uncoupled aangement the case of an 18 electomagnetically coupled antenna configuation, inteelement coupling leads to a decease of antenna coelation fo

6 162 O. Klemp H. Eul: Radiation Patten Analysis fo MIMO Divesity Antennas Fig. 5. Powe coelation of isolated- coupled dipole antennas in tems of equally distibuted AOAs in the hoiontal plane. Fig. 6. Powe coelation of coupled dipole antennas fo an angula spead of σ ϕ =1 diffeent degees of SME. all inteelement coupling distances. This obsevation is in accodance to Tsunekawa Kagoshima 199) as well as Leife 22). The degadation of antenna coelation is due to the effect of patten divesity that can only be contibuted of an aangement of electomagnetically coupled antenna elements. Antenna coelation of the electomagnetically decoupled dipole antennas is esticted to the occuence of spatial divesity due to antenna sepaation ij. The esults fo the coelation analysis based upon the spheical mode epansion of the adiation field matches vey well with the esults deived fom customay coelation analysis, unless only 12 vecto coefficients ae involved in the field epansion. This may be eplained in tems of the convegence citeion given by Naasimhan et al. 1985) that claims modal indices N min =ka, with a=λ/2 with espect to the configuation of λ/2-esonant, uncoupled dipole antennas. Theefoe the convegence citeion equies intege indices of N min =2. It has to be claified, that the pocesses of antenna coelation deived fom the customay numeical analysis of Eq. 17) using Eqs. 18) 19) may only be consideed to be appoimative solutions. As a esult of numeical simulation applying the FEM, antenna fa field epesentation may not be given in tems of an analytical fomulation. The esults wee obtained by applying an equiangula gid on the fa field sphee, whee the adiation fields wee calculated. Thoughout this pape equal gid distances in the diections of aimuth elevation, ϕ=3 ϑ=3 wee applied. Figue 5 consides the case of an equal angle of aival distibution in the y-plane. Neglecting the effects of inteelement coupling between the dipole antennas, the obseved antenna coelation coesponds to the well known esult fo the powe coelation of Jakes 1974) between antennas i j using the Bessel function of. ode, with ρ Pi,j =J 2 k d). As obseved fom Fig. 4 mutual coupling between adjacent antenna elements also leads to a decease of powe coelation e.g. Fig. 5) due to patten divesity gain, especially fo small antenna displacements. The esults of coelation analysis with espect to the SME of the adiation fields fo N=2 again eveals a good ageement to the customay computation technique. 5.2 Analysis of coupled Dipoles Fom Sect. 5.1 it can be concluded, that mutual coupling between adjacent elements within a multielement antenna aangement may esult in an impovement of divesity pefomance. This ease of divesity gain is a esult of a degadation of antenna adiation pattens losing thei aial symmety. Theefoe the convegence of antenna adiation patten analysis is influenced by the appeaance of mutual coupling causing highe ode modes in the field epansion. As a esult, a lage numbe of vecto epansion coefficients has to be taken into account to guaantee an accuate appoimation of antenna adiation behavio. This effect can be deived fom Fig. 6 that shows the esults of antenna coelation fo an angula spead of σ ϕ =1. The computation of inteelement antenna coelation is based upon field epansions dealing with summation limits of N =2 N =5 in accodance with Eq. 12). The diffeences in simulation esults between the customay computation technique by sampling the fa field sphee the seies epansion in tems of spheical hamonics ae minimied with espect to an ease of N. This is due to the violation of the convegence citeion N min =ka fo low degees of the field epansion. With an ease of the nomalied inteelement distance d/λ of the coupled dipole configuation, the effective antenna apetue eases. Theefoe an enhanced numbe of spheical modes has to be taken into account, to ensue sufficient field accuacy. E.g. fo a given inteelement spacing of d/λ=1.5, modal indices of degee N min =4 should be consideed in the field appoach. Figue 7 epesents the esults of antenna coelation fo an equally distibuted AOA in the hoiontal plane. As epected the esults show the same bias. Due to the ease

7 O. Klemp H. Eul: Radiation Patten Analysis fo MIMO Divesity Antennas 163 Fig. 8. Tapeoidal antenna geometies fo the ealiation of fequency-independent divesity antennas. Fig. 7. Powe coelation of coupled dipole antennas in tems of equally distibuted AOAs in the hoiontal plane diffeent degees of SME. of effective antenna apetue the degadation of antenna adiation behavio fom the aial symmety of the isolated dipole antenna, a lage seies limit N is equied to account fo a sufficient appoimation of antenna adiation fields. With an ease of the numbe of vecto epansion coefficients in Eq. 12) the cuves of antenna coelation tend to convege. 6 Analysis of Fequency-Independent Plana Antennas Compact multimode antennas as the biconical antenna that has been epoted by Svantesson 22) o the plana multiteminal spial- sinuous antennas epoted by Waldschmidt Wiesbeck 24) seem to be pomising fo the integation in MIMO tansmission systems with a combination of polaiation-, patten- modal divesity. In this section the fou-am log.-pe. tapeoid antenna, that was epoted by Klemp et al. 24a) a miniatuied deivative of this antenna type, the adial modulated fou-am log.-pe. tapeoid antenna epoted by Klemp et al. 24b) will be analyed with espect to thei popety of polaiation divesity. The antennas adhee to the piple of self-complement theefoe povide lage opeational bwidths. Both antenna types ae depicted in Fig. 8. The ability of a good polaiation divesity behavio is based on the degeneation of two spatially othogonal eigenfield distibutions, that ae supplied by the symmetical fou-am antennas in Fig. 8. In the ideal case the fou ams of the log.-pe. antenna stuctues can be decomposed into two othogonal am pais that epesent two decoupled tansmit channels with othogonal linea polaiation leading to uncoelated output voltages at the feeding bidge of the antenna stuctue. Fo the evaluation of polaiation divesity pefomance a 2-D pobability distibution of the angle of aival is pesumed. In analogy to the eceive divesity case of the dipole antenna aay, the aimuthal dependence of AOA is modeled using a Laplacian distibution. In tems of the elevational dependence, a Gauss distibution is assumed in accodance with Waldschmidt Wiesbeck 24). Mean value angula spead in the elevation ae given as m ϑ =45, σ ϑ =15. The Laplace distibution is detemined by a mean value of m ϕ =1 a spead of σ ϕ =2. Figue 9 shows the esult of powe coelation coefficient ρ Py in tems of polaiation divesity pefomance with equal mean banch powe in each polaiation component -polaiation o y-pol.) in the specified fequency ange fom 1 GH to 6 GH. Due to the high degee of coss polaiation discimination of each pai of opposite antenna ams, a low degee of powe coelation can be obseved in the entie fequency ange of opeation. ρ P always emains below.12. The esults of an odinay computation of powe coelation ae compaed to the esults of a spheical mode epansion N =5) fo the tapeoid antenna leading to a good ageement. The low degee of coelation can be justified by using the esults deived fom spheical antenna nea field measuement at the distt fequency positions f 1 =1.8 GH, f 2 =2.45 GH f 3 =5.85 GH. Based upon the measuement esults with an angula esolution of 1 in ϑ- ϕ-diection the powe coelation coefficient was computed using the twofold Riemann appoimation of Eqs. 17) to 19). As fo the miniatuied deivative of the fou-am tapeoid antenna depicted in Figs. 8b, 1 epesents the esults of powe coelation against opeating fequency. Results deived fom an appoach of Eq. 17) by taking the equivalent Riemann sums of Eqs. 18) 19) as well as the evaluation of powe coelation by spheical mode epansion of the individual adiation pattens yield a good ageement. Numeical data of powe coelation can be veified vey well by esults fom spheical nea field measuement of the miniatuied tapeoid antennas at the espective fequency positions f 1 to f 3 as shown in Fig Conclusions In this pape, a genealied appoach fo the computation of powe coelation in multielement antenna aangements was

8 164 O. Klemp H. Eul: Radiation Patten Analysis fo MIMO Divesity Antennas Fig. 9. Powe coelation fo log.-pe. tapeoid antenna. Simulation measuement esults. Fig. 1. Powe coelation fo miniatuied log.-pe. tapeoid antenna. Simulation measuement esults. pesented, which is based upon the epansion of antenna adiation fields in tems of spheical hamonics. Theefoe antenna adiation behavio was consideed in eceive-mode. The spatial popeties of a multipath popagation scenaio wee modeled using 2-D pobability density distibutions. In tems of a finite seies epesentation of antenna adiation fields, an analytical investigation of antenna coelation popeties can be accomplished. An analytical fomulation of antenna powe coelation was pesented involving elementay 2-D coupling integals ove a poduct of spheical hamonics, that ae weighed by the vecto seies epansion coefficients of the coesponding antenna adiation pattens. Due to the fact that the contibutions of elementay coupling integals emain fied fo a given aay topology popagation envionment, the amount of antenna coelation only depends on the epansion coefficients of thei adiation patten theefoe fom the cuent distibution on the coesponding metaliation sufaces. Hence the pesented analysis povides a majo simplification in ode to deive the espective coelation popeties of individual antenna elements in a multipath popagation scenaio. Futhemoe the analysis may seve as an initial point fo the optimiation of antenna adiation popeties with espect to low inteelement coelations by adjusting the modal content of the espective adiation fields. The pesented coelation analysis was applied on a configuation involving two paallel esonant λ/2 dipole antennas in eceive mode diffeentiating between isolated electomagnetically coupled antennas. The esults of customay coelation analysis SME based computation of antenna coelation agee vey well. Likewise the two diffeent appoaches in coelation analysis wee adopted to the chaacteiation of two wideb log.-pe. plana antennas in polaiation divese eceive mode. The esults also eveal a vey good ageement could be veified vey well by esults deived fom spheical antenna nea field measuement. The self-complementay log.-pe. fou-am tapeoid antennas may be consideed fo an implementation in a tansmission system with wideb polaiation divesity eception seem pomising fo the integation in MIMO tansmission systems with multimode opeation due to the multi-pot antenna concept low coelations between the diffeent opeational modes. Refeences Abamovit, M. Stegun, I. A.: Hbook of Mathematical Functions, Dove Publications, New Yok, Adane, Y., Gati, A., Wong, M.-F., Dale, C., Wiat, J., Hanna, V. F.: Optimal Modeling of Real Radio Base Station Antennas fo Human Eposue Assessment Using Spheical-Mode Decomposition, IEEE Antennas Wieless Popagation Lettes, 1, , 22. Buning, J. H. Lo, Y. T.: Multiple Scatteing of EM Waves by Sphees Pat I-Multipole Epansion Ray-Optical Solutions, IEEE Tansactions on Antennas Popagation, 19, , Chen, Y. Simpson, T.: Radiation Patten Analysis of Abitay Wie Antennas Using Spheical Mode Epansions with Vecto Coefficients, IEEE Tansactions on Antennas Popagation, 39, , 1991a. Chen, Y. Simpson, T. L.: Compact Repesentation of Radiation Pattens using Vecto Spheical Mode Epansions, Antennas Popagation Society Intenational Symposium, June 1991, 3, , 1991b. Chen, Y., Simpson, T. L., Ho, T. Q.: Highly efficient technique fo solving adiation scatteing poblems, IEE Poceedings- H, 139, 7 1, Deneyd, A. Kistensson, G.: Antenna signal coelation its elation to the impedance mati, Electonics Lettes, 4, 41 42, 24. Hafne, C.: The Genealied Multipole Technique fo Computational Electomagnetics, Atech House, Inc., Boston, London, 199.

9 O. Klemp H. Eul: Radiation Patten Analysis fo MIMO Divesity Antennas 165 Jakes, W. C.: Micowave Mobile Communications, IEEE Pess, Inc., New Yok, Kildal, P.-S. Rosengen, K.: Electomagnetic Analysis of Effective Appaent Divesity Gain of Two Paallel Dipoles, IEEE Antennas Wieless Popagation Lettes, 2, 9 13, 23. Klemp, O., Schult, M., Eul, H.: A Plana Boadb Antenna fo Applications in WLAN 3G, Intenational Symposium on Signals, Systems Electonics ISSSE, Lin, Austia, August 1 13, 24a. Klemp, O., Schult, M., Eul, H.: Miniatuiation Techniques Fo Logaithmically-Peiodic Plana Antennas, IEEE Intenational Symposium on Pesonal Indoo Mobile Communications PIMRC, Bacelona, Spain, Septembe 5 9, 24b. Leife, M. C.: Signal Coelations in Coupled Cell MIMO antennas, Antennas Popagation Society Intenational Symposium, June 22, 3, , 22. Ludwig, A. C.: Nea-Field Fa-Field Tansfomations Using Spheical-Wave Epansions, IEEE Tansactions on Antennas Popagation, 19, , Naasimhan, M. S., Chistophe, S., Vaadaangan, K.: Modal Behavio of Spheical Waves fom a Souce of EM Radiation with Application to Spheical Scanning, IEEE Tansactions on Antennas Popagation, 33, , Potte, P. D.: Application of Spheical Wave Theoy to Cassegainian-Fed Paaboloids, IEEE Tansactions on Antennas Popagation, 15, , Statton, J. A.: Electomagnetic Theoy, McGaw Hill, New Yok, Svantesson, T.: Coelation Channel Capacity of MIMO Systems Employing Multimode Antennas, IEEE Tansactions on Vehicula Technology, 51, , 22. Tsunekawa, K. Kagoshima, K.: Analysis of a Coelation Coefficient of Built-In Divesity Antennas fo a Potable Telephone, Antennas Popagation Society Intenational Symposium, 7 11 May 199, 1, , 199. Waldschmidt, C. Wiesbeck, W.: Compact Wide-B Multimode Antennas fo MIMO Divesity, IEEE Tansactions on Antennas Popagation, 52, , 24. Wene, D. H. Mitta, R.: Fonties in Electomagnetics, IEEE Pess Seies on Micowave Technology RF, New Yok, 2. Witte, E. D., Giffiths, H. D., Bennan, P. V.: Phase mode pocessing fo spheical antenna aays, Electonics Lettes, 39, , 23.