Modeling Time-Variant Fast Fading Statistics of Mobile Peer-to-Peer Radio Channels

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1 Modeling Time-Vrint Fst Fding Sttistics of Moile Peer-to-Peer Rdio Chnnels Mingming Gn, Nicoli Czink, Polo Cstiglione, Clude Oestges 2, Fredrik Tufvesson 3, Thoms Zemen FTW Forschungszentrum Telekommuniktion Wien, Vienn, Austri 2 ICTEAM, Université ctholique de Louvin, Louvin-l-Neuve, Belgium 3 Dept. of Electricl nd Informtion Technology, Lund University, Lund, Sweden Astrct The rdio chnnels etween nodes of n indoor peer-to-peer network show specific fst fding chrcteristics. Depending on the moility nd on the scttering properties of the environment, different kinds of fding distriutions cn occur: Ricen fding etween sttic nodes, ut lso Ryleigh or even doule-ryleigh fding etween moile nodes. We investigte fst fding in indoor peer-to-peer networks sed on rdio chnnel mesurements. It turns out tht the fding sttistics chnge over time. While the predominnt fding mechnism is comintion of Ryleigh nd doule-ryleigh fding, Ricen fding lso occsionlly occurs. On top of tht, indoors, the sttistics of the fst fding chnge over time even for smll-motions of the nodes, since the propgtion environment is inhomogeneous. We comprehensively model these effects using hidden Mrkov model, prmeterized from our mesurements. The model is vlidted, reveling convincing fit etween the model nd the mesurements. Keywords peer-to-peer chnnels, fst fding, distriuted chnnel sounding, rdio chnnel modeling. I. INTRODUCTION Coopertive communictions is n emerging technology tht promises to increse the coverge, reliility nd spectrl efficiency of peer-to-peer networks [], [2]. Mny methods were introduced, where nodes help other nodes to forwrd their messge through the network. Populr schemes re relying techniques [3], distriuted spce-time coding [4], or distriuted emforming [5]. The performnce of ll of these schemes depend on the underlying rdio chnnel. Most of the works on coopertive communictions ssume Ryleigh fding etween the nodes. While this is often true in cellulr scenrios, this is not the cse in peer-to-peer chnnels. Here, we oserve strong dependence on the nodes moility nd on the scttering scenrio. For this reson, the peer-to-peer rdio chnnel must e mesured nd modeled. The second, nd most intriguing motivtion for modeling nd mesuring the peer-to-peer rdio chnnel, is tht precise chnnel chrcteriztions llows the design of coopertive protocols exploiting the specific rdio chnnel properties. For instnce, the uthors of [6] hve shown tht coopertive medium ccess control (MAC) policies for indoor-to-outdoor sttic networks cn cpitlize from the knowledge of the Ricen K-fctors. A first thorough investigtion of peer-to-peer rdio chnnels in n indoor office environment ws crried out in [7], where it turned out tht the sttistics of smll-scle fding my rnge from strongly Ricen chnnels (for sttic nodes) vi Ryleigh chnnels (when only single node of link moves) to doule- Ryleigh chnnels (when oth nodes of link move, nd rich scttering is present round oth nodes). A further prticulrity of indoor scenrios is tht the sttistics of the smll-scle fding my chnge significntly over time for moile nodes. This is due to the strongly inhomogeneous propgtion conditions resulting from the motion of other people, the interction of the ntenn nd the user holding the device, nd from the fct tht the node moility nd the size of ostcles, such s furniture, shre similr geometric scle. Similr oservtions hve een mde in [8]. Time-vrint fding sttistics my e exploited y incentive mechnisms sed on repeted gmes tht force nodes to help disdvntged ones in view of future exchnge of roles, long the lines of [9]. Contriution: In this pper, we nlyze nd model the vritions of the smll-scle fding sttistics sed on mesurements. We shortly introduce recent peer-to-peer rdio chnnel mesurement cmpign crried out in n office environment in Louvin-l-Neuve, Belgium. Bsed on these experimentl results, we evlute the smll-scle fding sttistics using the second-order scttering fding (SOSF) distriution [], which reflects ny comintion of Ricen, Ryleigh, nd doule-ryleigh fding []. We oserve sudden chnges of the fding sttistics, sometimes with the chnnel even chnging from doule-ryleigh to Ricen. To model these effects, we introduce three-step pproch: (i) for reflecting the sudden chnges of the sttistics, we define nd prmeterize hidden Mrkov model [2], (ii) the prmeters of the SOSF distriution re modeled ccording to β- distriution, fitting the mesurements, (iii) fding reliztions re generted from the SOSF distriution. After vlidting our model, we find convincing fit etween the model nd the mesurements. This pper is orgnized s follows: In Section II, we introduce the mesurements on which our work is sed on. Section III introduces the concept of estimting nd modeling time-vrint fding sttistics, nd presents the evluted model prmeters from our mesurements. Section IV provides n implementtion summry of the model. In Section V, the

2 TABLE I MEASUREMENT PARAMETERS (AFTER POST-PROCESSING)... moile Rx... moile Tx... sttic Rx... sttic Tx Prmeter Vlue Crrier frequency 3.8 GHz Bndwidth 3 MHz Recorded frequency tones 2 Recorded time smples 3 Mesurement durtion 3 s Fig.. Floorpln of peer-to-peer mesurements model is vlidted ginst mesurements. Finlly, in Section VI, we drw the conclusions from our work. A. Environment II. MEASUREMENTS Our results re sed on indoor peer-to-peer mesurements in n office environment. Figure shows floor pln of the environment. The wlls seprting the rooms were either rick wlls or plster-ord wlls. Circles indicte moile nodes, squres indicte sttic nodes, nd red nd lue colors denote receivers (Rx) nd trnsmitters (Tx), respectively. Rx re dditionlly mrked y cross. The moile nodes were either rndomly moved over smll scle (within squre of m 2 ), or over lrger scle (throughout the whole room they were in). Hence, in this environment, we cn distinguish etween single-moile links (where only one node is moving), doulemoile links (where oth nodes re moving), nd links etween sttic nodes. The sttistics of links etween sttic nodes nturlly do not chnge. Hence, this pper only considers links etween moile nodes. B. Equipment The mesurements were crried out with UCL/ULB Elektroit CS chnnel sounder [3] t crrier frequency of 3.8 GHz, using the switched-rry principle. The distriuted nodes were connected to the 8-port switches of the trnsmitter nd receiver of the chnnel sounder using long low-loss RF cles, which hd equl length. The RF cles hd excellent RF stility, even when they were slightly ent or moved during the mesurements. At the nodes, custom-mde dipole ntenns with gin of.75 db nd n lmost omnidirectionl rdition pttern were used. The chnnel sounder used long PN sequences to estimte the impulse response of the rdio chnnels etween ll comintion of Tx nd Rx nodes. Mesurements were done in urst mode, which mens tht ech mesurement of the 64 links (8 8 chnnel) ws done four times to increse the SNR y susequent verging. The mesurement prmeters re summrized in Tle I. The recorded chnnel trnsfer function is denoted y H[t,f,c], where t denotes the time index, f denotes the frequency index, nd c denotes the link index. C. Clculting Fst Fding Reliztions Following the concept of [7], the chnnel cn e seen s superposition of pth loss, sttic shdowing, dynmic shdowing, nd fst fding. To extrct the fst fding from the chnnels, we first clculte the verge power of the chnnel in its sunds nd use moving time window s P s [t,,c] = T v F su t+t v/2 t =t T v/2 F su =+( )F su H[t,f,c] 2, where T v denotes the length of the moving time window, nd ws chosen to e 2 time smples, F su is the numer of freuencies within sund, nd =...B is the sund index. Susequently, the reliztions of the fst fding re given y H[t,f,c] G[t,f,c] = Ps [t, f/f su,c]. () III. TIME-VARIANT FAST FADING STATISTICS A. The SOSF Distriution Fst fding of peer-to-peer chnnels cn rnge from eing doule-ryleigh to Ricen distriuted [7]. A distriution including ll these cses is the so-clled second-order scttering fding (SOSF) distriution []. It descries the distriution of rndom vriles generted y r = G = w e jθ +w G +w 2 G 2 G 3, (2) where G,G 2,G 3 CN(,) re complex norml distriuted. The fctors w, w, w 2, correspond to the weights of the coherent contriution, single scttering, nd doule scttering, respectively. It should e noted tht w contins ll the coherent contriutions, not necessrily the line-of-sight pth only. The proility density function of these rndom vriles is given y p SOSF (r) = r ωe w2 ω2 /4 4J (rω)j (w ω) 4+w 2 2 ω2 dω, (3) where J is the Bessel function of the first kind nd zeroth order. Normlizing the verge power s E{r 2 } =, s done in (), where E{ } denotes the expecttion opertor, we cn introduce the two prmeters of the SOSF w 2 2 = w 2, (4) +w2 +w2 2 w 2 = w 2, (5) +w2 +w2 2

3 Prmeters Prmeters Time smple () Time smple () Fig. 2. Exmples of fst fding distriution prmeters over time; () singlemoile smll-scle motion, () doule-moile, lrge-scle motion Ricen suset Mixture suset Ryleigh/DR suset.5 Empiricl pdf Fig. 3. Prmeters of the SOSF exemplry for the single-moile smll-scle motion cse. We identify four groups: the Ricen suset, the Ryleigh/doule- Ryleigh suset, the Mixture suset, nd the pure doule-ryleigh point ( =, = ). The distriution within the susets cn e well modeled y the β-distriution, s demonstrted for the Ryleigh/doule-Ryleigh suset. where nd re constrined s +, while nd. The SOSF distriution intrinsiclly includes Ricen fding (with K-Fctor K = /( ), nd = ), Ryleigh fding ( = = ), nd Doule-Ryleigh fding ( =, = ), ut lso ll comintions of these. The prmeters nd cn e estimted from experimentl fding reliztions y itertively fitting the proility density function nd/or the cumultive density function, using moment-sed estimte s strting point []. As this pper discusses the time vrition of the fst fding sttistics, our prmeters depend on time, leding to (t), (t). B. SOSF Prmeters Extrcted From Mesurements The vriility of the prmeters (t) nd (t) for n exemplry single-moile chnnel (with smll-scle motion) is investigted in Figure 2. We mostly oserve Ryleigh/doule- Ryleigh fding ( < <, = ). Only scrcely, we oserve Ricen fding or mixture of ll fding mechnisms. Looking t doule-moile chnnel (with lrge-scle motion) in Figure 2, we see tht predominntly, doule-ryleigh TABLE II EVALUATED PARAMETERS OF THE SOSF DISTRIBUTION FOR DIFFERENT MOBILITY Single moile, smll-scle motion Suset Proility Distriution Ricen. p β (3.,4.9), = RDR.78.8 p β (2.3,.5)+.9 δ( ), = Mixture.2 p β (.4,7.) Single moile, lrge-scle motion Suset Proility Distriution Ricen.7 p β (2.5,3.9), = RDR.8.7 p β (2.6,.3)+.29 δ( ), = Mixture.3 p β (2.,.4) Doule moile, smll-scle motion Suset Proility Distriution RDR p β (3.,.4)+.27 δ( ), = Mixture.5 p β (.8,9.7) Doule moile, lrge-scle motion Suset Proility Distriution RDR p β (3.,.2)+.44 δ( ), = Mixture.2 p β (.9,9.8) fding occurs. Looking t the sttistics of (t) nd (t) in Figure 3, we oserve tht their vlues cn e prtitioned into four groups: (i) the Ricen suset ( <.5), (ii) the Ryleigh/doule- Ryleigh (RDR) suset ( < <, = ), (iii) the pure doule-ryleigh cse ( =, = ), nd (iv) the mixed suset ( = ). We neglect points in the middle zone, which occur only with very low proility. It turns out tht the distriution of the prmeter for the Ricen suset cn e well pproximted y the β-distriution p β (x p,p 2 ) = Γ(p +p 2 ) Γ(p )Γ(p 2 ) xp ( x) p2, (6) where Γ( ) denotes the Gmm function. For the mixture distriution, we define = ( ) 2 + 2, denoting the points on the line =. It turns out tht lso is β-distriuted. Finlly, the distriution of the prmeter for the RDR suset cn e modeled y comintion of the β- distriution nd Dirc function, comining the mixture of Ryleigh/doule-Ryleigh fding nd pure doule Ryleigh fding, respectively. An exmple is provided in Figure 3 for the Ryleigh/doule-Ryleigh suset (excluding the doule- Ryleigh point). By tht we cn estimte (i) n overll proility of the suset, nd (ii) proility distriution of the SOSF fding prmeters within the suset. A summry of these prmeters evluted from the mesurements is provided in Tle II. Note tht for doule-moile nodes, there were too few smples in the Ricen suset such tht it ws neglected. It is noteworthy tht we oserved high proility of occurrence for pure doule-ryleigh fding, prticulrly for doule-moile scenrios. C. Hidden Mrkov Model Modeling trnsitions etween different fding distriutions cn e solved y using hidden Mrkov model (HMM) [2]. Usully in HMMs, the sttes cnnot e directly oserved (which coined the ttriute hidden ). However, in our cse,

4 TABLE III HMM STATE TRANSITION PROBABILITIES Single moile, smll-scle motion from/to Ricen RDR Mixture Ricen RDR Mixture Single moile, lrge-scle motion from/to Ricen RDR Mixture Ricen RDR Mixture Doule moile, smll-scle motion from/to RDR Mixture RDR.88.2 Mixture Doule moile, lrge-scle motion from/to RDR Mixture RDR.9. Mixture the prmetriztion is strightforwrd, since the underlying distriution cn e prtitioned into the three cses of fst fding s descried ove. Hence, we model the trnsitions etween the susets using three-stte HMM: (i) Ricen fding, (ii) Ryleigh/doule- Ryleigh fding (including pure doule-ryleigh fding), nd (iii) mixed fding. The stte trnsition proilities were estimted from the mesurements nd re presented in Tle III for the four cses. The vlidity of the Mrkov chin ws verified y testing tht ll stte trnsitions only depend on the current stte nd re uncorrelted with previous sttes. IV. MODEL SUMMARY Summrizing, we propose to model smll-scle fding with time-vrint sttistics s shown in Figure 4. We strt with rndom initil stte of the HMM. Depending on the stte, we drw the SOSF prmeters (, ) from the distriutions provided in Tle II. Using (,), we generte pre-defined numer of complex fding reliztions G using (2) nd (4),(5). Next, the stte of the HMM is updted. If the stte chnges, we drw new SOSF prmeters nd continue with generting fding reliztions. If the stte is unchnged, we skip drwing new SOSF prmeters. This simplifiction turns out to hve no noticele impct on the distriution nd helps to keep the model simple. V. VALIDATION To vlidte the model, we investigte the overll fit to mesurement dt. For tht, we generted reliztions with time-vrint sttistics using the model in Section IV. Figure 5 compres the cumultive distriution function (CDF) of the generted fst-fding smples of the singlemoile mesurements (dotted line) nd of the model (solid line), for oth lrge-scle motion (old line, lue color) nd smll-scle motion (thin line, red color). It is evident tht the distriutions fit very well. Interestingly, there is no significnt Strt with rndom initl stte Drw new, (β-distr.) Drw Yes chnnel reliztions (SOSF distr.) Drw next stte No New stte = old stte? Fig. 4. Flow digrm of the model, generting fst fding reliztion with time-vrint sttistics CDF 2 Mesured (lrge scle motion) Mesured (smll scle motion) Generted (lrge scle motion) Generted (smll scle motion) G in db Fig. 5. Fit of the model to mesurement dt: empiricl CDF of the fding reliztions evluted from the mesurement dt (single moile, smll-scle nd lrge-scle motion), nd from the generted chnnels. difference etween lrge-scle nd smll-scle motion in the overll CDF in this scenrio. Compring the mesurements with the conventionl Ryleigh, Ricen, nd doule-ryleigh distriutions, we oserve from Figure 6 tht none of these distriutions resonly fit the experimentl dt. It must e noted tht this vlidtion only tells whether the overll sttistics re reflected correctly. They do not tell, whether the trnsition etween the different fding sttistics re cptured correctly. However, this spect ws covered y vlidting the HMM s descried in Section III-C. VI. CONCLUSIONS This pper hs presented n empiricl model of timevrint fding sttistics of peer-to-peer network in n indoor office environment. The mesured dt is chrcterized y the second-order scttering fding distriution (SOSF) [], Note tht, regrdless of smll-scle or lrge-scle motion, only the fst fding distriution is evluted here, which must not e mixed with the lrge-scle fding or shdow fding.

5 CDF 2 Mesured (lrge scle motion) Pure Ryleigh Pure Ricen (K=) Pure Doule Ryleigh G in db Fig. 6. Comprison with conventionl fst-fding distriutions which is le to model ll vrieties of fst fding from doule- Ryleigh to Ricen. The prmeters of the SOSF cn e prtitioned in three groups: Ricen fding, Ryleigh/doule- Ryleigh fding, nd mixture of these. The proilities of occurrence (nd prmeters of the SOSF) in these groups depend on the scenrio, prticulrly on the moility of the nodes. Trnsitions etween the groups re descried y hidden Mrkov model. The simultions show tht our model represents the mesured distriutions correctly. Interestingly, the fst fding distriutions for lrge-scle nd smll-scle motions of the nodes re quite similr. Prticulrly, we lso oserve pure doule- Ryleigh fding which is specific to rich-scttering peer-topeer environment. [3] R. Nr, H. Boelcskei, nd F. Kneuuehler, Fding rely chnnels: performnce limits nd spce-time signl design, IEEE J. Select. Ares Commun., vol. 22, no. 6, pp. 99 9, Aug 24. [4] J. Lnemn nd G. Wornell, Distriuted spce-time-coded protocols for exploiting coopertive diversity in wireless networks, IEEE Trns. Inform. Theory, vol. 49, no., pp , Oct. 23. [5] H. Ochii, P. Mitrn, H. Poor, nd V. Trokh, Collortive emforming for distriuted wireless d hoc sensor networks, IEEE Trns. Signl Processing, vol. 53, no., pp , Nov. 25. [6] P. Cstiglione, S. Svzzi, M. Nicoli, nd T. Zemen, Impct of fding sttistics on prtner selection in indoor-to-outdoor coopertive networks, in IEEE Interntionl Conference on Communictions (ICC), Cpe Town, South Afric, My [7] C. Oestges, N. Czink, B. Bndemer, P. Cstiglione, F. Kltenerger, nd A. Pulrj, Experimentl chrcteriztion nd modeling of outdoor-toindoor nd indoor-to-indoor distriuted chnnels, Vehiculr Technology, IEEE Trnsctions on, vol. 59, no. 5, pp , jun. 2. [8] J. Kredl, A. J. Johnsson, F. Tufvesson, nd A. Molisch, A mesurement-sed fding model for wireless personl re networks, IEEE Trns. Wireless Commun., vol. 7, no., pp , 28. [9] M. Le Treust nd S. Lsulce, A repeted gme formultion of energyefficient decentrlized power control, to pper in IEEE Trnsctions on Wireless Comm., 2. [] B. Bndemer, C. Oestges, N. Czink, nd A. Pulrj, Physiclly motivted fst-fding model for indoor peer-to-peer chnnels, Electronics Letters, vol. 45, no., pp , [] C. Oestges nd B. Clerckx, MIMO Wireless Communictions. Elsevier Acdemic Press, 27. [2] Y. Ephrim nd N. Merhv, Hidden mrkov processes, Informtion Theory, IEEE Trnsctions on, vol. 48, no. 6, pp , jun. 22. [3] Elektroit EB Propsim Homepge, 28. [Online]. Aville: ACKNOWLEDGEMENTS This work ws supported y the project PUCCO funded y the Vienn Science nd Technology Fund (WWTF), prtilly supported y the Europen Commission in the frmework of the FP7 Network of Excellence in Wireless COMmunictions NEWCOM++ (contrct no. 2675), y the Austri Science Fund (FWF) through grnt NFN SISE (S6) nd y Belgin Fonds de l Recherche Scientifique FNRS (FRS-FNRS). It ws lso crried out in coopertion within the Europen COST 2 Action. FTW is supported y the Austrin Government nd y the City of Vienn within the competence center progrm COMET. The uthors re plesed to cknowledge the help of Pt Chmers, Lingfeng Liu, Frncesco Mni, Frnçois Quitin, Olivier Renudin nd Fernndo Snchez to crry out the mesurements. REFERENCES [] A. Sendonris, E. Erkip, nd B. Azhng, User coopertion diversity. Prt I: System description, IEEE Trns. Commun., vol. 5, no., pp , Nov. 23. [2], User coopertion diversity. Prt II: Implementtion spects nd performnce nlysis, IEEE Trns. Commun., vol. 5, no., pp , Nov. 23.