Mobile Speed Estimation Using Diversity Combining in Fading Channels

Transcription

1 Moble Speed Estmaton Usng Dversty Combnng n Fadng Channels Hong Zhang and Al Abd Dept of Elec and Comp Eng New Jersey Insttute of echnology Newar NJ 7 USA Emals: hz7@njtedu alabd@njtedu Abstract In ths paper we consder the problem of moble speed estmaton usng two common dversty schemes: selecton combnng (SC and maxmal rato combnng ( We derve three new estmators whch rely on the nphase zero crossng rate nphase rate of maxma and the nstantaneous frequency zero crossng rate of the output of SC We also propose two estmators whch wor based on the level crossng rates of the envelopes at the output of SC and he performances of all these estmators are nvestgated n realstc nosy envronments wth dfferent nd of scatterngs and dfferent number of dversty branches Our smulaton results have revealed that a two-branch envelope level crossng rate estmator provdes a performance gan In terms of the mplementaton complexty the SC-based estmator s superor to as t does not need channel estmaton I INRODUCION Accurate estmaton of the moble speed or the maxmum Doppler frequency s of mportance n wreless moble applcatons whch requre the nowledge of the rate of channel varatons ypcally these applcatons nclude handoff adaptve modulaton equalzaton power control etc [] Dversty combnng technques are often used n fadng channels and selecton combnng (SC s one of the smplest dversty methods whereas maxmal rato combnng ( s the optmal one [] However to the best of our nowledge applcaton of dversty technques to speed estmaton has not been addressed so far In ths paper we derve several new speed estmators n SC and systems n closed forms and compare the performance va Monte Carlo smulatons he organzaton of ths paper s as follows he sgnal model s dscussed n Secton II whereas fve dversty-based estmators are derved n Secton III Secton IV presents the performance comparson through extensve Monte Carlo and Secton V concludes the paper II HE SIGNA MODE Consder a nosy Raylegh frequency-flat fadng channel wth an -branch combner hen the receved lowpass complex envelope at the -th branch s z( t h( t + n( t ( where the zero-mean complex Gaussan processes h ( t and n ( t represent the channel gan (assumng a plot has been transmtted and the bandlmted addtve nose of bandwth B Hz respectvely In the Cartesan coordnates we have z( t x( t + y( t ( where j and x ( t and y ( t are nphase and quadrature components respectvely Usng the polar representaton we obtan z( t r( texp[ jθ ( t] (3 where r ( t and θ( t are the envelope and phase of z( t defned by r( t x ( t + y ( t tan θ( t y(/ t x( t ( he autocorrelaton functon of h ( t s defned by * Ch ( τ E[ h( ( ] t h t + τ We also need the n-th spectral moment of h ( t bn n gven by [3] fd n n b ( f S ( f df (5 n fd h n whch Sh ( f s the power spectral densty of h ( t Obvously b can also be wrtten n terms of C ( τ n dc ( τ (6 n h n n n jdτ τ b he results derved n the followng secton hold for a large class of correlaton functons whose power spectra are even symmetrc wth respect to the center frequency When dong the smulatons n Secton IV we use ths flexble emprcally-verfed correlaton [] whch s a natural extenson of Clar s model (also nown as Jae s model I( κ fdτ + jκcos( α fdτ Ch ( τ b (7 I( κ where α [ s the mean drecton of the angle-of-arrval (AOA κ controls the wdth of the AOA I ( s the zero-order modfed Bessel functon of the frst nd v λ vf c c s the maxmum Doppler frequency n Hz v s the moble speed λ s the wavelength f c denotes the carrer frequency and c s the speed of lght et zt ( xt ( yt ( rt ( and θ ( t denote the complex envelope nphase part quadrature part envelope and phase at the output of the SC respectvely hen based on the defnton of an -branch SC we have ξ( t ξ( t r( t r( t (8 where ξ( t {( zt xt ( yt ( rt ( θ(} t ξ ( t { z ( t x ( t y ( t r( t θ (} t whch corresponds to the -th branch On the other hand the envelope of the output of an can be wrtten as / r ( t r ( t (9 h III DIVERSIY BASED ESIMAORS he advantage of SC s that t does not requre channel estmate whereas mplementaton of needs perfect estmate/tracng of the channel In the followng secton we derve four SC-based estmators and one -based estmator Bascally one can thn of two classes of speed estmaton methods: crossng-based and covarance-based technques [5] In ths paper we concentrate on the former category Gven a statonary real random process ( t defne N ( th as the number of tmes that the process crosses the threshold level th n the postve (or negatve gong drecton over the tme nterval Also let M ( denote the

2 number of maxma of the process ( t over the tme nterval Based on [6] the expected values of N ( th and M ( can be calculated accordng to EN [ ( ] f ( d ( th th EM [ ( ] f ( d ( where f ( s the jont PDF of ( t and ( t f ( s the jont PDF of ( t and ( t and dot denotes dfferentaton wth respect to t In order to calculate the average crossng rates of nterest n dversty combners usng the above equatons we assume ndependent and dentcally dstrbuted (d branches When branches are not d one needs to use another approach [7] In what follows we frst provde closed form expressons for three crossng rates n SC dversty systems: nphase zero crossng rate ( E[ Nx( ]/ nphase rate of maxma ( E[ Mx( ]/ and the nstantaneous frequency zero crossng rate (FZCR E[ N ( ]/ hese results are beleved to be new We also θ consder the average envelope level crossng rate (ECR n SC and gven n [8] and [9] respectvely Based on these fve crossng rates we propose fve new dversty-based speed estmators hey are derved under the no nose assumpton wth sotropc scatterng e Ch ( τ b ( J fdτ where J ( s the Bessel functon of the frst nd he effect of nose and nonsotropc scatterng wll be dscussed n Secton IV o smplfy the notaton estmators are gven for f D whch s proportonal to the moble speed v A Estmator wth SC he covarance matrx of the three dmensonal Gaussan random vector V [ ] x y x at the -th branch s [3] b A b ( b where b and b do not depend on snce we have d branches Clearly the PDF ofv can be wrtten as x + y x f ( x y x exp{ ( + } (3 V 3/ / ( bb b b Another random vector of nterest s W [ ] rxx whose PDF can be determned from the PDF ofv n (3 as f ( x y x V f ( r x x W ( J ( x y x where J ( x y x y / x + y s the Jacoban [6] of the transformaton V W Based on ( we then obtan the PDF of W r r x f ( r x x exp{ ( + } (5 W 3/ / ( bb b r x b Accordng to (3 n the Appendx the jont PDF of xt ( and xt ( at the output of the -branch SC can be shown to be fxx ( x x 3/ / ( bb (6 ( r ( + r x exp{ } dr x r x b b whch fnally smplfes to ( + f xx ( xx exp{ x} b + b x exp{ } b b (7 It s nterestng to note that the nphase component of the SC complex envelope and ts dervatve at tme nstant t are ndependent Also the dstrbuton of xt ( does not depend on and stll s Gaussan By substtutng (7 nto ( we obtan the average zero crossng rate of the SC nphase component EN [ x( ] b ( b (8 + On the other hand based on (6 one gets the followng spectral moments for the Clar s model Ch( τ bj ( fdτ b bf D b 6 bf D (9 hs reduces (8 to [ ( ] ENx ( fd ( + Consequently we ntroduce the estmator as Nx( fd ( ( + When ( reduces to the well-nown zero crossng speed estmator [] B Estmator wth SC Smlar to the estmator we defne two random vectors V [ ] x x y x and W [ ] r x x θ he covarance matrx of Gaussan vector V s [3] b b b b A ( b b hen the PDF of V can be wrtten as f ( x x y x V / ( det( A (3 exp{ V AV } where det( s the determnant he Jacoban of the transformaton V W s J ( x x y x x + y r So we obtan the PDF ofw as r f ( r x x θ exp{ W / ( det( A det( A ( bbbr bb x + bbx + bbx ( 3 +bb rx cosθ b r sn θ } Based on (3 n the Appendx the jont PDF of xt ( and xt ( at the output of the -branch SC can be expressed as fxx ( x x ( (5 f ( r x x θexp( r dθdr W b After substtutng (5 nto ( usng (9 fnally we obtan the average rate of maxma of the SC nphase component EM [ ( ] x fd (

3 + / 3/ ( ( F cos θ (6 dθ 3 cos θ where F ( s the hypergeometrc functon [] Now we ntroduce the estmator as ( ( fd Mx + (7 / 3/ ( ( F cos θ dθ 3 cos θ For (7 reduces to the estmator proposed n [5] C FZCR Estmator wth SC We defne the random vector WFZCR [ r θ θ θ] whose PDF s gven by [] 3 r f ( r θ θ θ WFZCR / ( ( bbb + bb θ (8 b b r θ exp r + r θ + b b B+ bb θ where B bb b As before by substtutng (8 nto (3 we obtan the jont PDF of θ ( t and θ ( t at the output of the -branch SC f ( θθ ( θθ (9 f ( r θ θ θexp( r dθdr WFZCR b After substtutng (9 nto ( and some lengthy calculatons we obtan the average zero crossng rate of the SC nstantaneous frequency EN [ ( ] θ b b (3 b b whch nterestngly does not depend on Usng (9 (3 smplfes to EN [ ( ] θ f (3 D hs leads us to the FZCR estmator N ( θ fdfzcr (3 he above equaton agrees wth the non-dversty estmator dscussed n [] D ECR Estmators wth SC and Wthout loss of generalty let E[ h ( t ] he envelope average level crossng rate of SC dversty s nown to be [8] EN [ ( ] r ( + fd ( e (33 and of course the correspondng estmator s Nr ( fdecr (3 ( + ( e On the other hand n [9] the envelope average level crossng rate of dversty s derved as EN [ ( ] r fd (35 eγ( where Γ( s the gamma functon [] herefore the assocated estmator s eγ( N ( r f DECR (36 Although (33 and (35 have been derved prevously ther performance as speed estmators however has not been studed so far to the best of our nowledge IV PERFORMANCE ANAYSIS Snce the combned output sgnals of dversty systems are not Gaussan any more t s very dffcult to do analytc performance analyss such as those carred out n [5] herefore n ths secton we rely on Monte Carlo smulaton to compare the estmaton error of fve technques: two nphase-based estmators n ( and (7 one phase-based estmator n (3 and two envelope-based n (3 and (36 he estmaton error s defned by E[( fd fd ] Var[ fd] + ( E[ fd] fd (37 where the frst term s the varance and the second stands for the bas All the fve estmators are derved for sotropc scatterng n nosefree envronments and obvously under such condtons are unbased e E[ f D] fd In the sequel we wll loo at the effect of nonsotropc scatterng and Gaussan nose as well to study the effect of dversty combnng n more realstc envronments In each smulaton we generate ndependent realzatons of zero-mean complex Gaussan processes usng the spectral method [3] wth N complex samples per realzaton over second he spectral method s also used for generatng the complex Gaussan bandlmted nose wth a flat power spectrum over the fxed recever bandwdth of B Hz assumng that the largest possble Doppler frequency f D s Hz When applcable the sgnal-noserato (SNR at each branch s db and each branch experences the same nonsotropc scatterng wth κ α 8 deg observed n feld trals [] A Estmator wth SC he estmaton error for the estmator n ( for three dfferent cases nose-free sotropc scatterng sotropc scatterng wth nose and nose-free nonsotropc scatterng are shown n Fg versus It s demonstrated that the estmator wthout SC dversty has the best performance n all stuatons B Estmator wth SC In ths case Fg shows that we do not get any performance mprovement va SC dversty n the three dfferent propagaton envronments when usng the estmator C FZCR Estmator wth SC From Fg 3 one can see that the FZCR estmator does not gan any enhancement from the SC dversty n all three cases we mentoned above 3

4 D ECR Estmators wth SC and Fg llustrates that both the -based and SC-based ECR estmators provde the best performance wth However the SC-based estmator s preferred as t does not need any channel estmate he mprovement offered by a two-branch estmator does not seem to be sgnfcant o tae full advantage of multple observatons n a dversty recever apparently one needs to loo at more complex technques such as the maxmum lelhood estmators whch we are currently nvestgatng V CONCUSION In ths paper we have nvestgated the possblty of moble speed estmaton n cellular systems usng dversty combnng technques Several new estmators are derved for selecton combnng and maxmal rato combnng dversty methods he mpact of nose and nonsotropc scatterng are extensvely nvestgated as well as the number of dversty branches We have observed that two-branch dversty combners provde performance enhancement o acheve hgher estmaton accuracy one needs to use more complex methods whch we are worng on APPENDIX DERIVAION OF HE JOIN PDF OF ξ ( t AND ξ ( t Accordng to (8 the SC dversty system produces the sgnal ξ ( t and ts dervatve ξ ( t such that ξ( t ξ( t ξ( t ξ( t r( t r( t (38 whch means the output at tme nstant t s the output of the branch wth the largest nstantaneous envelope Of course could change as t changes We defne the event E { r max({ r } } wth the probablty P( E Based on the total probablty theorem [6] the jont dstrbuton functon of ξ ( t and ξ ( t can be obtaned as ξξ F ( ξ ξ F ( ξ ξ E P( E P( ξ < ξ ξ < ξ E (39 ξξ where F ( ξξ E P( ξ < ξξ < ξ E ξξ Wth d branches (39 smplfes to F ( ξξ P( ξ < ξξ < ξ E ξξ ( he jont PDF of ξ ( t and ξ ( t s then gven by f ( ξξ f ( r Fr ( r dr ξξ ξξ r ξ ( ξ where Fr ( represents the envelope dstrbuton functon of the frst branch In Raylegh fadng channel we have [] Fr ( r exp( r b ( herefore ( reduces to f ( ξξ ( f ( r ξξ exp( r dr ξξ rξ (3 ξ b [] G Stuber Prncples of Moble Communcaton nd ed Boston MA: Kluwer [3] A Abd On the Second Dervatve of a Gaussan Process Envelope IEEE rans Inform heory vol 8 pp 6-3 [] A Abd J A Barger and M Kaveh A parametrc model for the dstrbuton of the angle of arrval and the assocated correlaton functon and power spectrum at the moble staton IEEE rans Vehc echnol vol 5 pp 5-3 [5] A Abd H Zhang and C epedelenloglu Speed Estmaton echnques n Cellular Systems: Unfed Performance Analyss n Proc IEEE Vehc echnol Conf Orlando F 3 [6] A Papouls Probablty Random Varables and Stochastc Processes 3rd ed Sngapore: McGraw-Hll 99 [7] A Abd and M Kaveh evel crossng rate n terms of the characterstc functon: A new approach for calculatng the fadng rate n dversty systems IEEE rans Commun vol 5 pp 397- [8] X Dong and N C Beauleu Average evel Crossng Rate and Average Fade Duraton of Selecton Dversty IEEE Comm ett vol 5 No pp [9] Y C Ko A Abd M S Aloun and M Kaveh A general framewor for the calculaton of the average outage duraton of dversty systems over generalzed fadng channels IEEE rans Vehc echnol vol 5 pp [] I S Gradshteyn and I M Ryzh able of Integrals Seres and Products 5th ed A Jeffery Ed San Dego CA: Academc 99 [] SORce Statstcal propertes of a sne wave plus nose Bell Syst ech J vol 7 pp [] G Azem B Senadj and B Boashash Moble unt velocty estmaton n mcro-cellular systems usng the ZCR of the nstantaneous frequency of the receved sgnal n Proc IEEE Int Symp Sgnal Processng Applcatons Pars 3 pp 89 9 [3] K Acolatse and A Abd Effcent smulaton of space-tme correlated MIMO moble fadng channels n Proc IEEE Vehc echnol Conf Orlando F 3 REFERENCES [] A Abd K Wlls H A Barger M S Aloun and M Kaveh Comparson of the level crossng rate and average fade duraton of Raylegh Rce and Naagam fadng models wth moble channel data n Proc IEEE Vehc echnol Conf Boston MA pp

5 3 so so 3 so estmator wth SC dversty 6 FZCR estmator wth SC dversty so so 3 so nose nose 3 nose 5 5 nose nose 3 nose nonso nonso 3 nonso 5 nonso nonso 3 nonso (Hz (Hz Fg Estmaton error (n Hz of estmator wth SC dversty versus the maxmum Doppler frequency for sotropc sotropc wth nose and nonsotropc scatterng Fg 3 Estmaton error (n Hz of FZCR estmator wth SC dversty versus the maxmum Doppler frequency for sotropc sotropc wth nose and nonsotropc scatterng 3 so so 3 so estmator wth SC dversty 6 so SC so so 3 SC so 3 so ECR estmators wth SC and dversty nose nose 3 nose 5 3 nose SC nose nose 3 SC nose 3 nose nonso nonso 3 nonso 6 nonso SC nonso nonso 3 SC nonso 3 nonso (Hz (Hz Fg Estmaton error (n Hz of estmator wth SC dversty versus the maxmum Doppler frequency for sotropc sotropc wth nose and nonsotropc scatterng Fg Estmaton error (n Hz of ECR estmators wth SC and dversty versus the maxmum Doppler frequency for sotropc sotropc wth nose and nonsotropc scatterng 5