THE SPATIAL CROSS-CORRELATION OF. Mobile and Portable Radio Research Group

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1 FFCTS OF MULTIPATH AGULAR SPRAD O TH SPATIAL CROSS-CORRLATIO OF RCIVD VOLTAG VLOPS Gegy D. Dugin and Thede S. Rappapt Mbile and Ptable Radi Reseach Gup Badley Depatment f lectical and Cmpute ngineeing 43 ew ngineeing Building Viginia Plytechnic Institute and State Univesity Blacksbug, VA gdugin@vt.edu Abstact { This pape pesents a simple fmula elating multipath angula spead t small-scale fading statistics. This fmula is then applied t nd an appximate spatial css-celatin functin f eceived vltage envelpes. The analytical appach is cmpaed t 67 css-celatin functins simulated with nn-mnidiectinal multipath. The equatins in this pape pvide insight f applying spatial divesity techniques t eceives peating in the pesence f nn-mnidiectinal multipath. I. Intductin The eects f multipath angula spead n smallscale fading statistics ae imptant t cnside in the design and peatin f almst any wieless system. This pape deives fading ate and celatin statistics f Rayleigh fading envelpes based n the multipath angula spead denitin pesented in []. The key esults ae qn (6 and qn (7, which shw that the spatial css-celatin behavi f any angula distibutin f multipath pwe { egadless f spatial cmplexity{maybe chaacteized by a single angula spead paamete. In the past, the analysis f these statistics in the liteatue have been limited mstly t the idealized case f mnidiectinal multipath ppagatin []. This wk may assist the design f eceives that peate in the pesence f ealistic, nnmnidiectinal multipath. F example, the esults quantitatively shw hw a spatial divesity design is aected by multipath f abitay spatial cmplexity and andm eceive ientatin. We shw that using the classical, mnidiectinal expessins f de- This wk is spnsed by a Badley Fellwship at Viginia Tech and an SF Pesidential Faculty Fellwship. signing spatial divesity eceives will always lead t inadequate sepaatin distances f the divesity antennas wheneve a bias exists in the angle-f-aival f multipath pwe. II. Oveview f Angula Spead and Fading Statistics F typical teestial ppagatin, adi waves aive at a wieless eceive fm azimuthal diectins abut the hizn [3]. This distibutin f multipath pwe is cnveniently descibed by the functin, p(, whee is the azimuthal angle. Angula spead,, is an imptant ppagatin paamete which detemines hw spead ut multipath pwe is abut the hizn and is dened as s, jf j jfj ; F n Z p( exp(jnd ( whee F n is the nth cmplex Fuie cecient f p( []. Angula spead,, anges fm t, with denting the case f a single multipath cmpnent fm a single diectin and denting n clea bias in the angula distibutin f eceived pwe. This denitin f angula spead,, is paticulaly useful because it is diectly elated t the aveage ate at which a signal fades in a lcal aea []. Appendix A shws that the mean-squae ate-f-change f a eceived nawband vltage envelpe, R, in arayleigh fading channel is elated t the angula spead, : ( k R ( 4

2 whee k is the wavenumbe f ppagatin ( divided by wavelength,, is change in psitin, and R is the mean-squae eceived vltage f the lcal aea. qn ( states that the aveage ate f envelpe change deceases as the impinging multipath pwe becmes cncentated in a single diectin. assumes that the distance is suciently small, emaining within a lcal aea s that R is appximately wide-sense statinay (WSS []. The unit diectin vect, ^z, pints in the diectin f tavel. Mtivatin f Spatial Divesity Spatial Repesentatin f Aiving Pwe R x p( θ Angula Distibutin f Pwe θ ο P T θ, Azimuthal Angle π Figue : Angula distibutin f pwe, p(, f a sect f aiving multipath cmpnents. As just ne example, cnside a situatin whee multipath pwe is aiving unifmly ve a cntinuus ange f azimuthal angles. The functin p( f this typefchannel is The spatial css-celatin f eceive antennas at dieent psitins within a lcal aea is an imptant paamete f designing eceives that emply spatial divesity [5, 6]. The spatial css-celatin functin, (, detemines hw fa divesity antennas must be sepaated befe the fading f thei eceived vltage envelpes becmes decelated [7]. The exact calculatin f ( f a ealistic angula distibutin f multipath pwe is extemely dicult and ften nn-istpic, depending n the diectin f ^z in qn (5. p( P T : + : elsewhee (3 The angle indicates the width f the sect (in adians f aiving multipath pwe and the angle is any abitay set angle, as illustated by Figue. Fllwing qn (, the aveage envelpe fading f this sected multipath pwe is p, +cs (4 The limiting cases f qn (4 pvide deepe undestanding f this denitin f angula spead. The limiting case f a single multipath aiving fm pecisely ne diectin cespnds t, which esults in. The the limiting case f unifm illuminatin in all diectins cespnds t, which esults in the maximum angula spead f. III. Spatial Css-Celatin The spatial css-celatin functin, (, detemines the celatin between vltage envelpes sepaated in space by a distance. A denitin f nmalized spatial css-celatin is given belw [, 4]: ( fr(~ R(~ + ^zg,( frg fr g,( frg (5 whee R is the stchastic vltage envelpe as a functin f tw-dimensinal psitin vect, ~. qn (5 (a (b (c Figue : xamples f spatial divesity with andm antenna ientatins. n-istpic ( cmplicates the spacing f divesity antennas, paticulaly f eceives with andm ientatins. Figue shws seveal examples f divesity systems that may have andm azimuthal ientatin. The andm ientatin f examples (a and (b is due t mbility while the andm ientatin f example (c is due t the multiple divesity banches taken in dieent azimuthal diectins. Css-Celatin Functin Appendix B deives an appximate csscelatin functin f a diective channel, aveaged ve all pssible azimuthal ientatins: ( exp,3 R x (6 qn (6 states that the css-celatin functin will baden as the channel becmes me diective (angula spead,, deceases. Futheme, qn (6 is accuate f small, but des nt mdel the highe-de behavi f lage. Hweve, many applicatins f the spatial css-celatin functin d nt equie highe-de behavi.

3 Vltage envelpes ae cnsideed suciently decelated when ( ps belw.4, which is ughly the statistical denitin f a celatin length [5]. The celatin length, l c, f a css-celatin functin is the value that satises the fllwing equatin: (l c exp(,. Applying this denitin t qn (6, the appximate celatin length f fading envelpes pduced by any spatial distibutin f multipath pwe is l c p 3 (7 qn (7 analytically demnstates hw celatin length inceases with deceasing angula spead,. In an mnidiectinal multipath channel (, qn (7 pedicts a celatin length f l c :, which is cnmed by the classical analysis [5]. qn (6 is tested against simulated ( that ae calculated fm knwn angula distibutins f multipath pwe. The basic methd f simulatin epesents p( as a sum f multipath pwes that aive fm evenly-spaced, discete diectins in space: X p( P i, i (8 i Fm this epesentatin, it is pssible t geneate abitay ealizatins f envelpe, R, as a functin f tw-dimensinal space fm R(x; y,j exp X i x cs i p P i exp(u i (9 + y sin i whee U i is a andm vaiable unifmly distibuted ve the inteval [;. qn (9 is simply the envelpe f a supepsitin f plane waves with cnstant amplitudes, p P i. A cmpute simulatin was used t calculate ( f 67 dieent angula distibutins f pwe, p(. ach case f p( was chaacteized by qn (3 { multipath aiving ve a cntinuus, azimuthal sect f adians. The 67 cases cespnd t dieent sect widths,, that anged fm 6 t in 36 incements (3 t 36 in 5 incements. In tems f qn (8, the P i that fall within the azimuthal sect ae set t a cnstant and the P i that fall utside f the sect ae set t ze. F each simulated case f,7envelpe ealizatins f qn (9 wee geneated. ach ealizatin is a cllectin f envelpe values f a 6 by 6 aea, geneated by qn (9 using a new aw f andm phases (U i. Fm these 7 ealizatins, ( was tabulated using the denitin f qn (5 f each U i aw with a vect, ^z, that pinted in 7 azimuthal diectins (meant t simulate andm ientatin. Celatin Length (λ Celatin Lengths f Multipath Sects Result fm Methd Sect Width, (Degees Figue 3: Cmpaisn f celatin lengths using empiical and appximate analytical esults. Figue 3 pesents a plt between the appximate celatin lengths f qn (7 and the celatin lengths esulting fm simulatin. te that they ae in nea-pefect ageement except f lw values f angula spead. The slight deviatin at lwe angles is due t simulatin cuseness athe than the validity f qn (7. F 7 in qn (9, nly a few tems in the summatin f qn (8 ae nnze f an angula distibutin f pwe with a lw angula spead. Withut a lage numbe f nnze cmpnents, the st-de statistics n lnge fllw a pue Rayleigh distibutin, as nted in [8]. Gaphs f simulated vs. appximated functins f 4 f the 67 cases is shwn in Figue 4. IV. Summay The spatial css-celatin behavi f any angula distibutin f multipath, egadless f cmplexity,may be appximately chaacteized by qn (6. Futheme, Figue 3 shws that celatin lengths incease as angula spead deceases. Theefe, as angula spead deceases, eective spatial divesity at the eceive equies a lage sepaatin f dive-

4 Css celatin Functin f 5 Css celatin Functin f.5 Λ Λ Css celatin Functin f Λ Css celatin Functin f 36 Λ Figue 4: Simulated vs. analytical spatial css-celatin functins, (, f sects f incming multipath pwe f 5,,, and 36. sity antennas. If the spatial divesity wee designed using the classical, mnidiectinal analysis and the eceive was peated in ealistic envinments with lwe angula speads, then the envelpe celatin between the divesity antennas inceases and the eceive becmes vulneable t fading. Refeences ( G.D. Dugin and T.S. Rappapt, \A Basic Relatinship Between Multipath Angula Spead and awband Fading in a Wieless Channel," I lectnics Lettes, vl. 34, n. 5, pp. 43{43, Dec 998. ( W.C. Jakes (ed, Micwave Mbile Cmmunicatins, I Pess, ew Yk, 974. (3 M.J. Gans, \A Pwe-Spectal They f Ppagatin in the Mbile Radi nvinment," I Tansactins n Vehicula Technlgy, vl. VT-, n., pp. 7{38, Feb 97. (4 A.M.D. Tukmani, A.A. Awjlu, P.A. Jeffd, and C.J. Kellett, \An xpeimental valuatin f the Pefmance f Tw-Banch Space and Plaizatin Divesity Schemes at 8 MHz," I Tansactins n Vehicula Technlgy, vl. 44, n., pp. 38{36, May 995. (5 D.O. Reudink, \Ppeties f Mbile Radi Ppagatin Abve 4 MHz," I Tansactins n Vehicula Technlgy, v 974. (6 W.C. Jakes, \A Cmpaisn f Specic Space Divesity Techniques f Reductin f Fast

5 Fading in UHF Mbile Radi Systems," I Tansactins n Vehicula Technlgy, vl. VT-, n. 4, pp. 8{9, v 97. (7 R.G. Vaughn and.l. Sctt, \Clsely Spaced Mnples f Mbile Cmmunicatins," Radi Science, vl. 8, n. 6, pp. 59{66, v- Dec 993. (8 Matthias Patzld and Fank Lane, \Statistical Ppeties f Jakes' Fading Channel Simulat," in I 48th Vehicula Technlgy Cnfeence, Ottawa, CA, May 998, pp. 7{ 78. (9 S.O. Rice, \Mathematical Analysis f Randm ise," Bell System Technical Junal, vl. 3, pp. 8{33, July 944. ( H. Stak and J.W. Wds, Pbability, Randm Pcesses, and stimatin They f nginees, Pentice Hall, ew Jesey, nd editin, 994. A. Rayleigh Fading Rate as a Functin f Angula Spead Based n the pwe elatinship P R,itispssible t wite the fllwing: ( dp ( 4fP g ( which isvalid f a Rayleigh fading pcess since R and its deivative ae independent [6, 9]. In [], the auths shw that the aveage mean-squae ate f change f pwe, P, in a lcal aea is ( dp k ( fp g ( whee is angula spead, as dened in qn (. Setting qn ( equal t qn ( pduces the mean-squae fading ate esult f a Rayleigh-fading vltage envelpe in qn (. B. Css-Celatin Deivatin T develp an appximate expessin f the csscelatin f multipath elds, st expand the functin, (, int a Mclauin seies: X ( n n (n! d n ( n ( qn ( cntains nly even pwes f since any css-celatin functin will be symmetic abut. The dieentiatin f a WSS css-celatin satises the fllwing elatinship f n []: d n ( n d n n fr(~ R(~ + ^zg fr g,( frg n, d n R n (, n fr g,( frg (3 and is useful f e-expessing the Mclauin seies: P n, d n R (, n n (n! ( + n fr g,( frg, n, n h fr g,( frg i + (4 w cnside ( appximated by an abitay Gaussian functin and its Mclauin expansin: ( exp,a X (, n a n n n n!, a + (5 A Gaussian functin is chsen as a geneic appximatin t the tue css-celatin since it is a cnvenient and well-behaved celatin functin. The apppiate cnstant a is chsen by setting equal the secnd tems f qn (4 and qn (5, ensuing that the behavi f bth css-celatin functins is identical f small. The slutin f a fllws by calculating the secnd tem in qn (4 using the Rayleigh statistics fm qn ( and the fllwing elatinship: ( frg 4 R (6 w the cnstant a may be slved: a (7 4, {z } 3: Theefe, the envelpe spatial css-celatin functin in a diective channel with andm ientatin is qn (6 which nw depends nly n. qn (6 shws hw inceased clusteing f multipath abut a single diectin will baden ( by deceasing the angula spead,.