EFFECTS OF MULTIPATH ANGULAR SPREAD ON THE SPATIAL CROss -correlation OF RECEIVED VOLTAGE ENVELOPES

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1 IEEE Vehcular Technology Conference, Houston,TX, May 16-19, 1999 pp EFFECTS OF MULTIPATH ANGULAR SPREAD ON THE SPATIAL CROss -correlation OF RECEIVED VOLTAGE ENVELOPES Gregory D. Durgn and Theodore S. Rappaport Moble and Portable Rado Research Group Bradley Department of Electrcal and Computer Engneerng 432 New Engneerng Buldng Vrgna Polytechnc Insttute and State Unversty. Blacksburg, VA e-mal: gdurgncvt. edu Abstract - Ths paper presents a smple formula relatng multpath angular spread to small-scale fadng statstcs. Ths formula s then appled to fnd an approxmate spatal cross-correlaton functon for receved voltage envelopes. The analytcal approach s compared to 67 cross-correlaton functons smulated wth non-omndrectonal multpath. The equatons n ths paper provde nsght for applyng spatal d. versty technques to recevers operatng n the presence of non-omndrectonal multpath. I. Introducton The effects of multpath angular spread on smallscale fadng statstcs are mportant to consder n the desgn and operaton of almost any wreless system. Ths paper derves fadng rate and correlaton statstcs for Raylegh fadng envelopes based on the mult path angular spread defnton presented n [1]. The key results are Eqn (6) and Eqn (7), whch show that the spatal cross-correlaton behavor of any angular dstrbuton of mult path power - regardless of spatal complexty- may be characterzed by a sngle angular spread parameter. In the past, the analyss of these statstcs n the lterature have been lmted mostly to the dealzed case of omndrectonal multpath propagaton (2]. Ths work may assst the desgn of recevers that operate n the presence of realstc, nonomndrectonal multpath. For example, the results quanttatvely show how a spatal dversty desgn s affected by multpath of arbtrary spatal complexty and random recever orentaton. We show that usng the classcal, omndrectonal expressons for de- sgnng spatal dversty recevers wll always lead to nadequate separaton dstances for the dversty antennas whenever a bas exsts n the angle-of-arrval of multpath power. II. Overvew of Angular Spread and Fadng Statstcs For typcal terrestral propagaton, rado waves arrve at a wreless recever from azmuthal drectons about the horzon (3]. Ths dstrbuton of multpath power s convenently descrbed by the functon, p( ), where s the azmuthal angle. Angular spread, A, s an mportant propagaton parameter whch determnes how spread out multpath power s about the horzon and s defned as A= Fn = j p(o) exp(jnj)dj (1) where Fn s the nth complex Fourer coeffcent of p(o) [1]. Angular spread, A, ranges from to 1, wth denotng the case of a sngle multpath component from a sngle drecton and 1 denotng no clear bas n the angular dstrbuton of receved power. Ths defnton of angular spread, A, s partcularly useful because t s drectly related to the average rate at whch a sgnal fades n a local area [1]. Appendx A shows that the mean-square rate-of-change of a receved narrowband voltage envelope, R, n a Raylegh fadng channel s related to the angular spread, A: 27r Ths work s sponsored by a Bradley Fellowshp a.t Vrgna. Tech and an NSF Presdental Faculty Fellowshp.

2 where k s the wavenumber of propagaton (27r dvded by wavelength,.-), r s change n poston, and E { R 2 } s the mean-square receved voltage of the local area. Eqn (2) states that the average rate of envelope change decreases as the mpngng multpath power becomes concentrated n a sngle drecton. Spatal Representaton of Arrvng Power Rx a -c::c -Q. Angular Dstrbuton of Power 9, Azmuthal Angle 21t Fgure 1: Angular dstrbuton of power, p(o), for a sector of arrvng multpath components. As just one example, consder a stuaton where multpath power s arrvng unformly over a contnuous range of azmuthal angles. The functon p( ) for ths type of channel s () = { : Oo =5S Bo+a p : elsewhere (3) The angle a ndcates the wdth of the sector (n radans) of arrvng mult path power and the angle (J s any arbtrary offset angle, as llustrated by Fgure 1. Followng Eqn (2), the average envelope fadng for ths sectored multpath power s A = V a co a 2et (4) The lmtng cases of Eqn ( 4) provde deeper understandng of ths defnton of angular spread. The lmtng case of a sngle multpath arrvng from precsely one drecton corresponds to a=, whch results n A =. The other lmtng case of unform llumnaton n all drectons corresponds to a = 271', whch results n the maxmum angular spread of 1. III. Spatal Cross-Correlaton The spatal cross-correlaton functon, p( r), determnes the correlaton between voltage envelopes separated n space by a dstance r. A defnton for normalzed spatal cross-correlaton s gven below [2, 4]: (r) = E {R(ro)R(ro + rz)}- (E {R} ) 2 p E {R}- (E {R}) 2 (5) where R s the stochastc voltage envelope as a functon of two-dmensonal poston vector, r Eqn (5) assumes that the dstance r s suffcently small, re manng wthn a local area so that R s approxmately wde-sense statonary (WSS) [2]. The unt drecton vector, z, ponts n the drecton of travel. Motvaton for Spatal Dversty The spatal cross-correlaton for recever antennas at dfferent postons wthn a local area s an mportant parameter for desgnng recevers that employ spatal dversty [5, 6].. The spatal cross-correlaton functon, p(r), determnes how far dversty antennas must be separated before the fadng of ther receved voltage envelopes becomes decorrelated [7]. The exact calculaton of p(r) for a realstc angular dstrbuton ofmultpath power s extremely dffcult and often non-sotropc, dependng on the drecton of z n Eqn (5). (a) (b) (c) Fgure 2: Examples of spatal dversty wth random antenna orentatons. Non-sotropc p(r) complcates the spacng of dversty antennas, partcularly for recevers wth random orentatons. Fgure 2 shows several examples of dversty systems that may have random azmuthal orentaton. The random orentaton of examples (a) and (b) s due to moblty whle the random orentaton of example (c) s due to the multple dversty branches taken n dfferent azmuthal drectons. Cross-Correlaton Functon Appendx B derves an approxmate crosscorrelaton functon for a drectve channel, averaged over all possble azmuthal orentatons: p(r) exp [-23A 2 Gr] (6) Eqn ( 6) states that the cross-correlaton functon wll broaden as the channel becomes more drectve (angular spread, A, decreases). Furthermore, Eqn (6) s accurate for small r, but does not model the hgher-order behavor for larger r. However, many applcatons of the spatal cross-correlaton functon do not requre hgher-order behavor.

3 Voltage envelopes are consdered suffcently decorrelated when p( r) drops below.4, whch s roughly the statstcal defnton of a correlaton length [5]. The correlaton length, le, of a cross-correlaton functon s the value that satsfes the followng equaton: p(/e) = exp(-1). Applyng ths defnton to Eqn (6), the approxmate correlaton length for fadng envelopes produced by any spatal dstrbuton of mult path power s lc = A.y23 (7) " Eqn (7) analytcally demonstrates how correlaton length ncreases wth decreasng angular spread, A. In an omndrectonal multpath channel (A = 1), Eqn (7) predcts a correlaton length of lc =.21A, whch s confrmed by the classcal analyss (5]. Smulaton Eqn (6) s tested aganst smulated p(r) that are calculated from known angular dstrbutons of multpath power. The basc method of smulaton represents p( fj) as a sum of N mult path powers that ar ve from evenly-spaced, dscrete drectons n space: p(o) = L: P; 6 -..!!!_ =t N ( 2 ') N (8) From ths representaton, t s possble to generate arbtrary realzatons of envelope, R, as a functon of two-dmensonal space from N R(:c, y) = L: exp(21tu) x (9) =l exp [- 2 1r (cos + ysn )] I where U s a random varable unformly dstrbuted over the nterval [, 1). Eqn (9) s smply the envelope of a superposton of N plane waves wth constant ampltudes, Vf{. A computer smulaton was used to calculate p( r) for 67 dfferent angular dstrbutons of power, p(o). Each case of p(o) was characterzed by Eqn (3) - mult path arrvng over a contnuous, azmuthal sector of a radans. The 67 cases correspond to dfferent sector wdths, a, that ranged fron f to 21T n :S ncrements (3 to 36 n 5 ncrements). In terms of Eqn {8), the P that fall wthn the azmuthal sector a are set to a constant and the P that fall outsde of the sector are set to zero. For each smulate case of a, 72 envelope realzatons of Eqn (9) wer generated. Each realzaton s a collecton of envelope values for a 6, by 6.. area, generated by Eqn (9) usng a new draw of random phases (21TU) From these 72 realzatons, p(r) was tabulated usng the defnton of Eqn (5) for each U draw wth a vector, z, that ponted n 72 azmuthal drectons (meant to smulate random orentaton) !!1 o.a.!!! o.a.4.2 Correlaton Lengths tor Multpatt Sectors Result trom Smulaton - - Analytcal Method,, Sector Wdth, a (Degrees) Fgure 3: Comparson of correlatn lengths usng emprcal and approxmate analytcal results. Fgure 3 presents a plot between the approxmate correlaton lengths of Eqn (7) and the correlaton lengths resultng from smulaton. Note that they are n near-perfect agreement except for low values of angular spread. The slght devaton at lower angles s due to smulaton courseness rather than the valdty of Eqn (7). For N = 72 n Eqn (9), only a few terms n the summaton of Eqn (8) are nonzero for an angular dstrbuton of power wth a low angular spread. Wthout a large number of nonzero components, the frst-order statstcs no longer follow a pure Raylegh dstrbuton, as noted n [8]. Graphs of smulated vs. approxmated functons for 4 of the 67 cases s shown n Fgure 4. IV. Summary The spatal cross-correlaton behavor of any angular dstrbuton of multpath, regardless of complexty, may be approxmately characterzed by Eqn (6). Furthermore, Fgure 3 shows that correlaton lengths ncrease as angular spread decreases. Therefore, as angular spread decreases, effectve spatal dversty at the recever requres a larger separaton of dver-

4 Cross-correlaton Functon tor a = 5 Cross-correlaton Functon for a = 1 s Q. c.5.g.!!! e o Smulaton Analytcal p=e-1 A= "'""'"'"'""''''''"'''''"'"'"''"'"'"'""'''''''''"''''''''''''''""' 'C' : Smulaton Analytcal p=e-1 A=.4785 r:..5 '"'"'"'"'"'"''"'''""'"''"''''"''""'"'""'"'""''''''""'''''''''"''"'"'"' 8 I ' Dstance (rfa) 6 2 Dstance (r/'j..) 3 4 Cross-correlaton Functon for a = 2 Cross-correlaton Functon for a = 36 'C' a: g-.5 Smulaton Analytlcat p-e-1 A=.8255 f..... e o --..._,, ' s Q. Smulaton Analytcal p-e-1.5 A=1 '1a... e o.5 1 Dstance (rfa) Dstance (r/'j..) Fgure 4: Smulated vs. analytc spatal cross-correlaton functons, p(r), for sectors of ncomng multpath power for a = 5, 1, 2, and 36. sty antennas. If the spatal dversty were desgned usng the classcal, omndrectonal analyss and the recever was operated n realstc envronments wth lower angular spreads, then the envelope correlaton between the dversty antennas ncreases and the recever becomes vulnerable to fadng. References (1) G.D. Durgn and T.S. Rappaport, "A Basc Relatonshp Between Multpath Angular Spread and Narrowband Fadng n a Wreless Channel," lee Electroncs Letter-s, vol. 34, no. 25, pp , 1 Dec (2) W.C. Jakes (ed), Mcrowave Moble Communcatons, IEEE Press, New York, (3) M.J. Gans, "A Power-Spectral Theory of Propagaton n the Moble Rado Envronment," IEEE Transactons on Vehcular Technology, vol. VT-21, no. 1, pp , Feb ( 4) A.M.D. Turkman, A.A. Arowojolu, P.A. Jefford, and C.J. Kellett, "An Expermental Evaluaton of the Performance of Two-Branch Space and Polarzaton Dversty Schemes at 18 MHz," IEEE Transactons on Vehcular Technology, vol. 44, no. 2, pp , May (5) D.O. Reudnk, "Propertes of Moble Rado Propagaton Above 4 MHz," IEEE Transactons on Vehcular Technology, Nov (6) W.C. Jakes, "A Comparson of Specfc Space Dversty Technques for Reducton of Fast

5 Fadng n UHF Moble Rado Systems," IEEE Transactons on Vehcular Technology, vol. VT-2, no. 4, pp , Nov (7) R.G. Vaughn and N.1. Scott, "Closely Spaced Monopoles for Moble Communcatons," Rado Scence, vol. 28, no. 6, pp , Nov Dec (8) Matthas Patzold and Frank Lane, "Statstcal Propertes of Jakes' Fadng Channel Smulator," n IEEE 48th Vehcular Technology Conference, Ottawa, CA, May 1998, pp (9) S.O. Rce, "Mathematcal Analyss of Random Nose," Bell System Techncal Journal, vol. 23, pp , July (1) H. Stark and J.W. Woods, Probablty, Random Processes, and Estmaton Theory for Engneers, Prentce Hall, New Jersey, 2nd edton, A. Raylegh Fadng Rate as a Functon of Angular Spread Based on the power relatonshp P = R 2, t s possble to wrte the followng: whch s vald for a Raylegh fadng process snce R and ts dervatve are ndependent [6, 9]. In [1], the authors show that the average mean-square rate of change of power, P, n a local area s where A s angular spread, as defned n Eqn (1). Settng Eqn (1) equal to Eqn (11) produces the mean-square fadng rate result for a Raylegh-fadng voltage envelope n Eqn (2). B. Cross-Correlaton Dervaton To develop an approxmate expresson for the crosscorrelaton of mult path felds, frst expand the functon, p(r), nto a Mclaurn seres: - r2n d2.np( r') I p( r) - L...J (2n)! dr'2n (12) n=o r 1 = Eqn. ( 12) contans only even powers of r snce any cross-correlaton functon wll be symmetrc about r =. The dfferentaton of a WSS cross-correlaton satsfes the followng relatonshp for n ;:=: 1 (1]: d2np(r') I = j.;.e{r(f.)r(r. + r'z)}l-o dr'2n r'=o E {R2}- (E {R})2 E{(d")2} ( l)n dr = - E {R2}- (E {R})2 (13) and s useful for re-expressng the Mclaurn seres: p(r) Now consder p(r) approxmated by an arbtrary Gaussan functon and ts Mclaurn expanson: p(r) exp [-a Gr] f: (-l)an ut n=o n. A 1-a(x)2+... (15) A Gaussan functon s chosen as a generc approxmaton to the true cross-correlaton snce t s a convenent and well-behaved correlaton functon. The approprate constant a s chosen by settng equal the second terms of Eqn (14) and Eqn (15), ensurng that the behavor of both cross-correlaton functo;j.s s dentcal for small r. The soluton for a follows by calculatng the second term n Eqn (14) usng the Raylegh statstcs from Eqn (2) and the followng relatonshp: (E {R}) 2 = E { R 2 } (16) Now the constant a may be solved: a= Ul:21r] A2 (17) "-...--' 23. Therefore, the envelope spatal cross-correlaton functon n a drectve channel wth random oren taton s Eqn ( 6) whch now depends only on A. Eqn (6) shows how ncreased clusterng of multpath about a sngle drecton wll broaden p( r) by decreasng the angular spread, A.