Polarization Degrees of Freedom

Polarizatio Degrees of Freedom Ada Poo Electrical ad Computer Egieerig Uiversity of Illiois, Urbaa-Champaig poo@uiucedu David Tse Electrical Egieerig ad Computer Sciece Uiversity of Califoria, Berkeley dtse@eecsberkeleyedu Table I MULTIPLICATIVE GAIN IN DEGREES OF FREEDOM FROM POLARIZATION FOR DIFFERENT ARRAY GEOMETRIES AND SCATTERING CONDITIONS k^ E φ Scatterig Array Geometries Coditio Poit Liear Plaar Spherical E θ Fully scattered 6 6 4 Azimuth oly 6 4 z ~ θ Abstract This paper uifies the differet coclusios o the polarizatio degrees of freedom i multiple-atea chaels It shows that the multiplicative gai i the degrees of freedom from polarizatio depeds o the array geometry ad the scatterig coditio Figure Coordiates for a electric dipole orieted alog the z-axis I MAIN RESULTS The umber of degrees of freedom supported by a commuicatio chael is a simple but fudametal performace measure I multiple-atea chaels, there are differet coclusios o the multiplicative gai i the degrees of freedom from polarizatio Adrews et al [ demostrated that there are 6 degrees of freedom from co-located ateas orthogoal curret loops plus orthogoal electric dipoles Marzetta [ stated that there are 4-fold icrease i the degrees of freedom from usig polarimetric atea elemets I our earlier paper [, we claimed that there are oly -fold icrease ad therefore, scatterers would ot icrease the degrees of freedom from usig polarimetric atea elemets This paper uifies the differet perspectives o the polarizatio degrees of freedom While all the reported results hold, they are true oly uder certai geometry of the atea arrays ad scatterig coditio With referece to the umber of degrees of freedom from sigle polarizatio arrays derived i [, Table I summarizes the multiplicative gai from polarizatio for differet array geometries ad scatterig coditios The 6-fold multiplicative gai oly holds for a poit source with co-located orthogoal ateas ad a liear array of these poit sources i a fully scattered eviromet Usig higher dimesioal atea arrays reduces the multiplicative gai II VECTOR POINT SOURCES We will first uderstad the 6 degrees of freedom from a pair of trasmit ad receive polarimetric ateas Cosider the radiatio patter from oe of the six compoets made up the polarimetric ateas the electric dipole orieted alog the z-axis I the far field, the radiated field E is perpedicular to the directio of propagatio ˆk = θ, φ Alog ˆk, the radiated field lies o the plae cosistig of the dipole ad ˆk ad therefore, the radiated field cosists of oly the ˆθ compoet Its magitude is proportioal to the projectio of the z-axis atea orietatio to ˆθ field directio Its radiatio patter ca the be writte as a 0 θ, φ = ˆθ si θ The factor of is to ormalize a 0θ, φ to uity The radiatio patters for the electric field orieted alog the x- axis ad the y-axis are rotated versio of a 0 θ, φ, ad are give by a θ, φ = a θ, φ = ˆθ cos θ cos φ + ˆφ si φ ˆθ cos θ si φ ˆφ cos φ The radiatio patters for the orthogoal curret loops are the dual of the electric dipoles ad are give by a θ, φ = ˆφ si θ a 4 θ, φ = ˆθ cos φ + ˆφ cos θ si φ a 5 θ, φ = ˆθ si φ + ˆφ cos θ cos φ Now the sigal that is radiated by the mth compoet of the trasmit polarimetric ateas ad itercepted by the th For ease of expositio, we do ot iclude the factor expressios e i kr i the

compoet at the receiver, has a chael gai of H m = a ϑ, ϕ Hϑ, ϕ, θ, φ a m θ, φ dω t dω r for, m = 0,, 5, where ˆκ = ϑ, ϕ is the directio of the icidet field at the receiver, dω t = si θ dθ dφ, ad dω r = si ϑ dϑ dϕ The chael respose Hˆκ, ˆk is a complex itegral kerel It models the chael gai, ad the chage i polarizatio due to scatterers Now we cosider two scatterig coditios: multipaths spread over the etire θ, φ propagatio space ad multipaths oly spread over the azimuth directios, φ I the first case, the scatterig respose satisfies E [ H ij ˆκ, ˆkH i j ˆκ, ˆk = δ ii,jj δˆκ ˆκ, ˆk ˆk i, i, j, j = 0, I the secod case, we assume that scatterers are lyig o the xy-plae ad therefore the scatterig respose is o-zero oly at θ = π/ The respose satisfies E [ H ij π, ϕ, π, φh i j π, ϕ, π, φ = δ ii,jj δϕ ϕ, φ φ i, i, j, j = 0, I both cases, we model H ij ˆκ, ˆk s as zero-mea white Gaussia radom processes A Fully Scattered i θ ad φ Now we look ito the correlatio amog etries of the chael matrix H i with property give i As a θ, φ s are orthoormal over θ, φ, H m s are iid complex Gaussia radom variables Thus, the chael is equivalet to the iid Rayleigh fadig MIMO chael Furthermore, H is a 6 6 radom matrix, the umber of degrees of freedom is 6 B Scattered i φ Oly The radiatio patters { a π, φ, = 0,, 5} are orthogoal over φ Deotig the orm of a π r, φ by σ, they are give by σ 0 = σ = 4 σ = σ = σ 4 = σ 5 = 8 Therefore, elemets of the chael matrix H are idepedet with variaces give by varh m = σ σ m The ratio of the largest to the smallest variaces is 4 The degrees of freedom should be close to that of iid Rayleigh fadig chael Thus, the umber of degrees of freedom is 6 III LINEAR ARRAYS Now we cosider liear arrays of vector poit sources A Fully Scattered i θ ad φ Suppose the liear arrays are orieted alog the z-axis such that the array resposes ivolve θ oly This choice of coordiates is for coveiece Suppose the elemet spacig o the trasmit array is t, ad the umber of vector poit sources is t, ad therefore, the legth of the trasmit array is L t = t t All legth quatities are ormalized to a wavelegth Similar defiitios apply to r, r, ad L r The chael matrix is give by H = e r cos ϑ B ϑ, ϕ Hϑ, ϕ, θ, φ e t α = t e tcos θ Bθ, φ dω t dω r 4 ad is of dimesio 6 r 6 t, where the trasmit ad the receive array resposes are e iπ tα e iπ rα e iπ tα e rα = respectively, the elemet respose is r Bθ, φ = [ a 0 θ, φ a 5 θ, φ T θφ ad the trasformatio matrix is 0 0 0 0 0 T θφ = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e iπ rα The arrowest beam from the trasmit array is give by f t α := e t0 e t α = e iπt tα siπl tα t siπ t α It attais its mai lobe at α = 0, has zeros at multiples of /L t, ad is periodic with period / t As α spas over [,, the umber of atea elemets is optimized whe the period equals to, that is, t = / ad t = L t Same coclusio ca be draw by lookig at the array resolutio over [, which is /L t Hece, the umber of atea elemets is //L t = L t Uder the optimal samplig, the trasmit array respose ca be approximated by e t α t/ p= t/ f t α p L t et p L t 5 as L t 6 The f t cos θ p L t s are the samplig fuctios over θ Similarly, we defie f r α As e t p/l t, p = t,, t ad e r q/l r, q = r,, r are idividually orthoormal, the statistical properties of H ca be approximated by ˆ H

where its elemets are give by Ĥ 6q+r/+,6p+ t/+m q = f r cos ϑ b L cos q, ϕ Hϑ, ϕ, θ, φ r L r b m cos p, φ f p t cos θ dωt dω r 7 L t L t ad b θ, φ is the th colum of Bθ, φ The use of a ew set of fudametal modes give by the trasformatio by T θφ makes sure that b m cos p L t, φft cos θ p L t, m, p ad b cos q L r, φfr cos θ q L r,, q are idividually orthogoal over θ, φ Cosequetly, elemets of ˆ H are idepedet with variaces give by where varĥ6q+ r/+,6p+ t/+m = ς q,σ p,m σ p,0 = σ p, = π p L t + p L t σp, = σp,5 = p + σp, = σp, = p π L t π L t Similar defiitios apply to the ς q, s The variaces are close to 0 at the corers of ˆ H Whe Lt, L r, the umber of degrees of freedom of ˆ H should be close to that of iid Rayleigh fadig chael, ad it is also close to that of H Therefore, the asymptotic umber of degrees of freedom is mi{6 t, 6 r } which is 6 times of the scalar chael B Scattered i φ Oly We cosider liear arrays orieted alog the x-axis The array resposes are e t si θ cos φ ad e r si θ cos φ At θ = π/, they become e t cos φ ad e r cos φ which are the same as those orieted alog the z-axis except that φ spas betwee 0 ad π while θ spas betwee 0 ad π At cos φ = p L t, it is seemigly that the 6 elemet respose Bθ, φ is a rak costat matrix However, si φ i the respose takes differet values over 0 φ < π ad π φ < π Therefore, the elemet respose is ot a costat matrix but is a fuctio of φ To derive the dimesio of the fuctioal space spaed by the colums of Bθ, φ, we use a differet trasformatio matrix: T φ = I Now the elemet respose is [ B π, φ = 0 0 0 cos φ si φ 0 si φ cos φ 0 0 At cos φ = p L t, the st ad the 5th colums spa the same space Similarly, the rd ad the 4th colums spa the same space but differ from that of the st colum The d, rd, 5th, ad 6th colums are orthogoal Cosequetly, the asymptotic umber of degrees of freedom is mi{4 t, 4 r } which is 4 times of the scalar chael Liear arrays ca oly resolve either θ or φ betwee 0 ad π I each of the resolved directios, we could obtai degrees of freedom from polarizatio Whe the agular spread of scatterers is over both θ ad φ, the extra -fold icrease i the degrees of freedom is due to the use of vector poit sources to resolve the remaiig agular directio, ot resolved by the arrays Whe the agular spread is over φ oly, there is o eed to resolve the θ by the vector poit sources Istead, the vector sources are used to distiguish betwee 0 φ < π ad π φ < π, which yields the extra -fold icrease i the degrees of freedom IV PLANAR ARRAYS We cosider disk-like plaar arrays lyig o the xy-plae Suppose x p, y p is the coordiates of the pth vector source ormalized to a wavelegth The its respose i the far field is e iπxp si θ cos φ+yp si θ si φ Bθ, φ It is more coveiet to use the polar coordiates where the respose becomes e iπρp si θ cosφ φp Bθ, φ ad ρ p, φ p is the correspodig coordiates of the pth vector source We arrage the vector sources o the array i cocetric circles: ρ p = ρ t, =,,, t,ρ 8a φ p = m φ t, m = 0,,, π 8b The trasmit array respose is e t,π +m si θ, φ = t e iπ ρt si θ cosφ m φt A Fully Scattered i θ ad φ As the array respose ivolves both θ ad φ, the samplig fuctios are derived i two steps The resolutio over θ is derived from f t,θ si θ := e t0, 0 e t si θ, φ = t,ρ t = R t = m=0 Rt π πrt 0 Rt 0 e iπ ρt si θ cosφ m φt 0 e iπρ si θ cosφ φ ρ dφ dρ J 0 πρ si θρ dρ = J πr t si θ πr t si θ = jic si θ 9 as ρ t, φ t 0, where R t is the radius of the trasmit array, J is the th order Bessel fuctio of the first kid, ad jic is the jic fuctio The jic fuctio is similar to the sic fuctio: jicx is maximized at x = 0 ad jic0 = ; ad asymptotically, jicx has zeros at multiples of Thus, the resolutio over si θ is approximately equal to

At si θ = p θ, the arrowest beam over φ is give by f t,φ p θ, φ := e t p θ, 0 e t p θ, φ jic si φ 0 The resolutio over φ is approximately equal to si Puttig together, the samplig fuctios over θ, φ are f t,θ si θ p θ f t,φ p θ, φ p φ si for p θ =,, ad p φ = 0,, π si Whe p θ, the resolutio ca be further approximated by p θ ad p φ = 0,, πp θ Comparig the rages of p θ, p φ to those of, m i 8, we obtai ρ t = / ad φ t = at optimal samplig p θ At si θ = ad φ = p φ si, φ is uiquely defied, but cos θ i Bθ, φ takes differet values over 0 θ < π/ ad π/ θ < π Therefore, the elemet respose Bθ, φ is a fuctio of θ, ad the dimesio of the fuctioal space spaed by the colums of Bθ, φ is 4 The asymptotic umber of degrees of freedom is mi{4 t, 4 r } which is 4 times of the scalar chael B Scattered i φ Oly At θ = π/ ad φ = p φ si, Bθ, φ is a 6 costat matrix which is of rak oly Therefore, the asymptotic umber of degrees of freedom is mi{ t, r } which is times of the scalar chael Ulike liear arrays, plaar arrays ca resolve both θ ad φ I each resolved directios, we obtai degrees of freedom from polarizatio Whe the agular spread of scatterers is over both θ ad φ, vector poit sources are used to distiguish betwee 0 θ < π/ ad π/ θ < π, which yields the extra -fold icrease i the degrees of freedom However, whe the agular spread is over φ oly, there is o extra icrease i the degrees of freedom V SPHERICAL ARRAYS We ca cotiue usig the samplig represetatio ad derive the degrees of freedom for spherical arrays However, it will ivolve more additioal approximatios We therefore switch gear to a physical approach that ca apply to all field regios the ear ad the far fields ad all sizes of atea arrays Suppose the radius of the trasmit array is R t Huyges priciple states that kowledge of the wave field o a surface eclosig the source currets is sufficiet to determie the wave field outside the surface We therefore observe the wave field o a sphere eclosig the trasmit array ad of radius R s The mappig from curret sources o the source sphere to wave field o the observatio sphere is obtaied from the free-space dyadic Gree s fuctio: Ḡr, r = I + e ik 0 r r k0 4π r r This yields the trasmit array respose: Ā t ˆk, p = ḠR sˆk, R tˆp, ˆk, ˆp S S Now we are lookig for the decompositio similar to 6 which decomposes the array respose ito a sequece of dyads That is, we are lookig for two sets of orthoormal fuctios, {u i ˆk} ad {vi t ˆp} such that ḠR sˆk, Rtˆp = ik 0 σ u ˆkv t ˆp The set of sigular values {σ i } i descedig order yields the spectrum of the array respose which determies the umber of degrees of freedom A Spectrum of the Array Resposes We first make use of the multipole expasio of the Gree s fuctio i spherical coordiates [4, Ch 7: Ḡr, r [ = ik 0 Mm r + ˆM mr,m + N m r ˆN mr where M m ad N m are the magetic ad the electric multipole fields respectively, ad ˆM m ad ˆN m are { the correspodig source fuctios As stated i [5, Mm Rˆr, N m Rˆr } ad { ˆMm Rˆr, ˆN m Rˆr } are idividually orthogoal over the uit sphere S Next, we fid the orm of these fuctios From defiitio, M m r = [ rh Y m ˆr = i + h X m ˆr where h is the spherical Hakel fuctio of the first kid, Y m is the spherical harmoic fuctio, ad X m ˆr is defied as the vector spherical harmoic fuctio i [6, Sec 97 Usig 90 i [6, we obtai M mrˆrm m Rˆr dω = + h k 0 R δ,mm To fid the orm of N m Rˆr s, it requires some work that is ot i stadard electromagetic textbooks From defiitio, N m r = k 0 [ rh Y m ˆr Usig c i [7, we obtai = k 0 [ h r Y m ˆr N m r = + h ˆrY m ˆr + d [ rh r Y m ˆr dr = + h ˆrY m ˆr [ + h k 0r h r Y m ˆr

Furthermore, we have ˆrY m ˆr ˆrY m ˆr dω = δ δ mm ˆrY m ˆr r Y m ˆr dω = 0 Thus, the orm of N m Rˆr is give by [ + N mrˆrn m Rˆr dω = h + k 0R + + h + + k 0R δ δ mm Replacig h with j yields the orm of ˆM m Rˆr ad ˆN m Rˆr } where j is the spherical Bessel fuctio As a whole, the sigular value decompositio of ḠR sˆk, Rtˆp is [ ḠR sˆk, Rtˆp = σm u m ˆkv t m ˆp = m= + σ m u m ˆkv t ˆp m where the sigular values are give by 4 σm = h k 0 R s jk 0 R t 5a [ + σm = h + k 0R s + h + + k 0R s [ + + j k 0 R t + + j +k 0 R t 5b ad the correspodig orthoormal fuctios are u m ˆk = M mr sˆk u m ˆk = N mr sˆk M m R sˆk N m R sˆk v m ˆp = ˆM m R tˆp ˆM v m ˆp = ˆN m R tˆp m R tˆp ˆM m R tˆp Similar defiitios apply to ς ml as sigular values ad ˆq as source orthoormal fuctios of the receive array v r ml B Fully Scattered i θ ad φ Followig, a discrete chael model is obtaied: H r + r+m r+l r, t +t+mt+lt = v r rm rl r q Cq, pv t tm tl t p dpdq =ς rm rl r σ tm tl t u rm rl r ˆκ Hˆκ, ˆku tm tl t ˆk dˆkdˆκ As u ml ˆk s are orthoormal, etries of the chael matrix H are idepedet with variaces give by varh r + r+m r+l r, t +t+mt+lt = ς rm rl r σ tm tl t The umber of degrees of freedom is determied by the miimum of the umber of sigificat ς ml s ad that of σ ml s By studyig the distributios summarized i 5, we ca obtai this umber i ay field regio with arbitrary array sizes Now let us study the distributio i the far field with large arrays I the far field, k 0 R s The spherical Hakel fuctio ca be approximated by [ h eik0rs + k 0 R s = + O k 0 R s k 0 R s ad the magitude of the asymptotic term is idepedet of ad equals to k 0R s The sigular values become σm = j kr t kr s σ m = [ + kr s + j k 0 R t + 6a + j +k 0 R t 6b Whe k 0 R t, j kr t 0 for > k 0 R t Thus, both σ m ad σ m vaish whe > k 0 R t The umbers of degrees of freedom cotributed by the magetic multipoles ad the electric multipoles are asymptotically the same, ad equal to k 0 R t = A t where A t = π k0rt π is defied as the effective aperture of the trasmit array i our earlier paper [ Cosequetly, the asymptotic umber of degrees of freedom is mi{a t, A r } which is times of the scalar chael derived i [ C Scattered i φ Oly As scatterers do ot icrease the degrees of freedom i the fully scattered case, they will ot icrease them either i the curret case At θ = π/, u ml ˆk s are orthogoal for differet m ad l oly That is, for the same m, l, the dimesio of the fuctioal space spaed by u ml ˆk s is If the magetic electric multipoles vaish at = N = N, the maximum m is N + N + I the far field, the asymptotic umber of degrees of freedom is mi{4k 0 R t, 4k 0 R r } = mi{8 πa t, 8 πa r } Fially, the vector poit sources a ˆk i Sectio II are the lowest order modes u ml ˆk i the far field Thus i this sectio, we cosider large array i the far field while the smallarray couterpart has bee cosidered i Sectio II REFERENCES [ M R Adrews, P P Mitra, ad R decarvalho, Triplig the capacity of wireless commuicatios usig electromagetic polarizatio, Nature, vol 409, pp 6 8, Ja 00 [ T L Marzetta, Fudametal limitatios o the capacity of wireless liks that use polarimetric atea arrays, i Proc IEEE ISIT, July 00, p 5 [ A S Y Poo, R W Broderse, ad D N C Tse, Degrees of freedom i multiple-atea chaels: a sigal space approach, IEEE Tras Iform Theory, vol 5, o, pp 5 56, Feb 005 [4 W C Chew, Waves ad Fields i Ihomogeeous Media IEEE Press, 995 [5 A J Devaey ad E Wolf, Multipole expasios ad plae wave represetatios of the electromagetic field, J Math Phys, vol 5, o, pp 4 44, Feb 974 [6 J D Jackso, Classical Electrodyamics, rd ed Wiley, 998 [7 R G Barrera, G A Estévez, ad J Giraldo, Vector spherical harmoics ad their applicatio to magetostatics, Eur J Phys, vol 6, pp 87 94, 985