Stochastic Signal Processing and Power Control for Wireless Communication Systems

Transcription

1 Universiy of Tennessee, Knoxville Trace: Tennessee Research and Creaive Exchange Docoral Disseraions Graduae School 2-27 Sochasic Signal Processing and Power Conrol for Wireless Communicaion Sysems Mohammed Mohsen Olama Universiy of Tennessee - Knoxville Recommended Ciaion Olama, Mohammed Mohsen, "Sochasic Signal Processing and Power Conrol for Wireless Communicaion Sysems. " PhD diss., Universiy of Tennessee, 27. hp://race.ennessee.edu/u_graddiss/255 This Disseraion is brough o you for free and open access by he Graduae School a Trace: Tennessee Research and Creaive Exchange. I has been acceped for inclusion in Docoral Disseraions by an auhorized adminisraor of Trace: Tennessee Research and Creaive Exchange. For more informaion, please conac race@u.edu.

2 To he Graduae Council: I am submiing herewih a disseraion wrien by Mohammed Mohsen Olama eniled "Sochasic Signal Processing and Power Conrol for Wireless Communicaion Sysems." I have examined he final elecronic copy of his disseraion for form and conen and recommend ha i be acceped in parial fulfillmen of he requiremens for he degree of Docor of Philosophy, wih a major in Elecrical Engineering. We have read his disseraion and recommend is accepance: Douglas Birdwell, Aly Fahy, Tsewei Wang, Jie Xiong (Original signaures are on file wih official suden records.) Seddi M. Djouadi, Major Professor Acceped for he Council: Carolyn R. Hodges Vice Provos and Dean of he Graduae School

3 To he Graduae Council: I am submiing herewih a disseraion wrien by Mohammed Mohsen Olama eniled Sochasic Signal Processing and Power Conrol for Wireless Communicaion Sysems. I have examined he final elecronic copy of his disseraion for form and conen and recommend ha i be acceped in parial fulfillmen of he requiremens for he degree of Docor of Philosophy, wih a major in Elecrical Engineering. Seddi M. Djouadi, Major Professor We have read his disseraion and recommend is accepance: Douglas Birdwell Aly Fahy Tsewei Wang Jie Xiong Acceped for he Council: Carolyn R. Hodges, Vice Provos and Dean of he Graduae School (Original signaures are on file wih official suden records.)

4 STOCHASTIC SIGNAL PROCESSING AND POWER CONTROL FOR WIRELESS COMMUNICATION SYSTEMS A Disseraion Presened for he Docor of Philosophy Degree The Universiy of Tennessee, Knoxville Mohammed Mohsen Olama December 27

5 Copyrigh 27 by Mohammed Olama. All righs reserved. ii

6 Dedicaion This disseraion is dedicaed o whom I reasure above all on his earh, and who have given meaning o my life: To my beloved moher and my grea faher, my lovely wife Suzan and my cue 9 monh-old son Omar. Wihou heir undersanding, encouragemen and suppor, his wor would no have been possible. I also dedicae his disseraion o my siser Heba, my brohers Mosafa and Muhannad, my broher in-law Sameh, my young nephew Zaid and my niece Sara. The moivaion, endurance, paience, and undersanding of my family, all crowned by heir devoion had conribued subsanially o my success. iii

7 Acnowledgmens Praise be o The Almighy Allah (God); he Lord of he universe, wihou his will and help his wor may no have come o an end. I would lie o express my profound appreciaion and grea graiude o my supervisor, Dr. Seddi M. Djouadi for his inspiring guidance, valuable suggesions, een ineres, consrucive criicism, persisen help, and friendly discussions enabled me o culminae his wor efficienly. I have learned from him a lo in research, including his solid mahemaical bacgrounds and his smar ideas. I have also learned from him how o be a good engineer, a modes person, a hard woring, and a man of principles. I can no forge his help and assisance when I had healh problems. I am sure wha I have learned from him will coninue o benefi me during he res of my life. I would lie o convey my sincere graiude o all he commiee members: Dr. Douglas Birdwell, Dr. Aly Fahy, Dr. Tsewei Wang, and Dr. Jie Xiong by which he wor conained in his disseraion was approved. I really appreciae Dr. Birdwell suggesions and commens which help me beer undersand he difficul poins in my research. I also appreciae Dr. Xiong for eaching me he conceps of probabiliy heory. Thans o Dr. Fahy for opening my mind o some imporan pracical applicaions in wireless communicaions relaed o my research. The valuable commens and suggesions of Dr. Wang on how o organize and presen my wor are very helpful. I have also learned a lo from he courses of Dr. Birdwell, Dr. Fahy, and Dr. Xiong. Special hans o Dr. Birdwell and Dr. Wang for giving me a nice space in heir lab o do my wor more efficienly. All he ime, effors, valuable suggesions and consrucive commens from all he commiee members deserve my srong appreciaion. I would lie o han Dr. Charalambos Charalambous for giving me he iniiaive and moivaion o his wor. Wihou his ousanding ideas and valuable suggesions, his wor would no have been done. I hope o mee and be in ouch wih Dr. Charalambous more closely o learn more from his nowledge and experience. iv

8 I would also lie o han Dr. Glenn Allgood and Mr. Teja Kurugani for giving me a chance o use he faciliies in Oa Ridge Naional Laboraory (ORNL), in paricular he Exreme Measuremen Communicaion Cener (EMC 2 ) laboraory ha benefis in validaing he various algorihms in my wor. I always appreciae heir confidence on me. Finally, I would lie o express my graiude and appreciaion o he many friends inside and ouside of my sudies. Thans o my close friends Anis Drira, Samir Sahyoun, Junqing Sun, Kiran Jaladhi, and Yanyan Li for he suppor and encouragemen hey gave me. Your friendships hrough boh good imes and hard imes have been a source of srengh. v

9 Absrac This disseraion is concerned wih dynamical modeling, esimaion and idenificaion of wireless channels from received signal measuremens. Opimal power conrol algorihms, mobile locaion and velociy esimaion mehods are developed based on he proposed models. The ulimae performance limis of any communicaion sysem are deermined by he channel i operaes in. In his disseraion, we propose new sochasic wireless channel models which capure boh he space and ime variaions of wireless sysems. The proposed channel models are based on sochasic differenial equaions (SDEs) driven by Brownian moions. These models are more realisic han he ime invarian models encounered in he lieraure which do no capure and rac he ime varying characerisics of he propagaion environmen. The saisics of he proposed models are shown o be ime varying, and converge in seady sae o heir saic counerpars. Cellular and ad hoc wireless channel models are developed. In urban propagaion environmen, he parameers of he channel models can be deermined from approximaing he band-limied Doppler power specral densiy (DPSD) by raional ransfer funcions. However, since he DPSD is no available on-line, a filerbased expecaion maximizaion algorihm and Kalman filer o esimae he channel parameers and saes, respecively, are proposed. The algorihm is recursive allowing he inphase and quadraure componens and parameers o be esimaed on-line from received signal measuremens. The algorihms are esed using experimenal daa, and he resuls demonsrae he mehod s viabiliy for boh cellular and ad hoc newors. Power conrol increases sysem capaciy and qualiy of communicaions, and reduces baery power consumpion. A sochasic power conrol algorihm is developed using he so-called predicable power conrol sraegies. An ieraive disribued algorihm is hen deduced using sochasic approximaions. The laer only requires each mobile o now is received signal o inerference raio a he receiver. vi

10 Several mehods for racing a user based on wave scaering models and paricle filering are presened. These algorihms cope wih nonlineariies in order o esimae he mobile locaion and velociy. They ae ino accoun non-line-of-sigh and mulipah propagaion environmens. To show ha he algorihms are robus numerical resuls are presened o evaluae heir performance of he algorihms in he presence of parameric uncerainies. vii

11 Table of Conens Inroducion.... Overview....2 Relaed Lieraure Review Wireless Channel Modeling Opimal Power Conrol Locaion Based Services Main Conribuions....4 Disseraion Ouline... 2 Sochasic Wireless Channel Modeling The General TV Wireless Channel Impulse Response Sochasic TV LTF Channel Modeling The radiional (Saic) LTF Channel Model Sochasic LTF Channel Model Spaial Correlaion of he Sochasic LTF Model Sochasic TV STF Channel Modeling The Deerminisic DPSD of Wireless Channels Approximaing he Deerminisic DPSD Sochasic STF Channel Models Soluion o he Sochasic Sae Space Model Sochasic TV Ad Hoc Channel Modeling The Deerminisic DPSD of Ad Hoc Channels Approximaing he Deerminisic Ad Hoc DPSD Sochasic Ad Hoc Channel Models Lin Performance for Cellular and Ad Hoc Channels...5 viii

12 3 Wireless Channel Esimaion and Idenificaion via he Expecaion Maximizaion Algorihm and Kalman Filering The EM Togeher wih he Kalman Filer Channel Sae Esimaion: The Kalman Filer Channel Parameer Esimaion: The EM Algorihm LTF Channel Esimaion Using he EM Togeher wih Kalman Filering STF Channel Esimaion Using he EM Togeher wih Kalman Filering Ad Hoc Channel Esimaion Using he EM Togeher wih Kalman Filering Sochasic Power Conrol Algorihms for Time Varying Wireless Newors Represenaion of TV Wireless Newors Cenralized Deerminisic PCA in TV Wireless Newors Disribued Deerminisic PCA in TV Wireless Newors Sochasic PCA in TV Wireless Newors Numerical Resuls Mobile Saion Locaion and Velociy Esimaion in Cellular Newors Sysem Model The Lognormal Propagaion Channel Model Aulin s Scaering Model The MLE Approach for MS Locaion Esimaion Recursive Nonlinear Bayesian Esimaion The EKF Design for MS Locaion and Velociy Esimaion The PF Design for MS Locaion and Velociy Esimaion The UPF Design for MS Locaion and Velociy Esimaion The SUT Mehod...2 ix

13 5.6.2 The UPF Design Numerical Resuls Conclusions and Fuure Wor...2 Bibliography...7 Via...3 x

14 Lis of Figures 2. Mean-revering power pah-loss as a funcion of and τ, for he ime varying γ (, τ ) in Example A mobile (ransmier) a a disance d from a base saion (receiver) moves wih velociy υ and in he direcion given by θ wih respec o he ransmier-receiver axis Mean-revering power pah-loss X (, τ ) for he TV LTF wireless channel model in Example Aulin s 3D mulipah channel model DPSD for β m = degrees and differen values of mobile speed DPSD, S ( f ), and is approximaion D 2 ( ω) = H( jω) via a 4 h order ransfer S funcion for mobile speed of (a) 5 m/hr and (b) 2 m/hr Inphase and quadraure componens, aenuaion coefficien, and phase angle of he STF wireless channel in Example Aenuaion coefficien in absolue unis and in db s for he STF wireless channel in Example Inpu signal, sl ( ), and he corresponding received signal, y (), for fla slow fading (op) and fla fas fading condiions (boom) Ad hoc DPSD for differen values of α 's DPSD, S( f ), and is approximaions, S ( f), using complex cepsrum algorihm for differen orders of S ( f) DPSD, S( f ), and is approximaion, S ( f ), via 4 h order funcion for differen values of α s...47 xi

15 2.3 Inphase and quadraure componens { I(), Q() }, and he aenuaion coefficien r = I + Q n() n() n() 2.4 Rayleigh aenuaion coefficien for cellular lin and wors-case mobile-o-mobile lin FER resuls for Rayleigh mobile-o-mobile lin for differen α s compared wih cellular lin Real and esimaed received signal for he LTF channel model Received signal esimaes RMSE for runs using he EM algorihm ogeher wih he exended Kalman filer The measured and esimaed inphase and quadraure componens, and received signal for 4 h order channel model using he EM algorihm ogeher wih Kalman filer Crossbow TelosB sensor node Experimenal seup: Two moving ransceivers (sensors and 2) and one fixed receiver (sensor 3) Indoor environmen considered in our experimen Oudoor environmen considered in our experimen Measured and esimaed received signal from sensor 2 by using a 4 h order ad hoc channel model Received signal esimaes RMSE for indoor and oudoor environmens using he EM algorihm ogeher wih he Kalman filer Power updae and informaion flow for uplin power conrol using predicable power conrol sraegies (PPCS) OP for he cenralized DPCA using PPCS under TV LTF models in Example xii

16 4.3 Average OP for TV LTF channel models wih δ ( ) = Average OP for TV LTF channel models wih δ ( ) = Sum of ransmied power of all mobiles for he disribued DPCA and he disribued SPCA under TV LTF channels OP under he dynamical STF channel models in Example Average OP under dynamical fla Rayleigh and Ricean STF channels Sum of ransmied power of all mobiles for he disribued DPCA and SPCA under TV STF channels Locaion and velociy esimaes of a moving MS generaed by he differen filers Locaion and velociy esimaes RMSE () generaed by he differen algorihms The UPF locaion and velociy esimaes RMSE () for imperfec nowledge of channel parameers... xiii

17 Lis of Tables 5. Performance comparison for MS locaion and velociy esimaion algorihms using he MLE, EKF, EKF/MLE, PF, and he UPF approaches...9 xiv

18 Chaper Inroducion. Overview Recen advances in wireless communicaions and elecronics, such as wireless microsensor newors, inerne echnologies and sandards, home neworing, and sensor devices in micro-scale, pose new challenges and require innovaive and mulidisciplinary soluions. These echnologies have remendous applicaions in healh, chemical, and biological monioring. On he oher hand, hese advances have enabled he developmen of new low cos, low power mulifuncional micro-sensors, which communicae over small disances []. A micro-sensing device consiss of a sensing uni, a daa processing uni, a ransceiver uni, and a power uni. These micro-sensor devices mae up micro-sensor newors comprised of hundreds or housands of ad hoc iny sensor nodes, which spread across a geographical area, and collaborae among hemselves o creae a sensing newor. These devices are preparing he ground for he emergence of new applicaions, and he developmen of new echnologies, which can provide access of informaion anyime anywhere by collecing, processing and disseminaing informaion daa. Inerne, saellie, ad hoc newors and wireless media in general, are becoming he neural sysem for a global communiy providing he means of sharing informaion colleced by micro-sensing devices. These infrasrucures-less newors consis mosly of heerogeneous wireless mobile devices wih various capabiliies, power consrains and mobiliy characerisics. Wireless communicaion newors are consrained by available bandwidh, bu he user populaion coninues o grow. Wireless sysems adap o hese pressures hrough more

19 efficien use of bandwidh; code division muliplexing (CDMA) and ransmier power conrol (PC) are wo examples. Adapive ransmission mehods including modulaion, PC, and channel coding require models of he communicaion channels and noise sources. In his disseraion we are concerned wih cellular and ad hoc wireless communicaion sysems, which deal wih large amoun of aggregae non-homogeneous raffic informaion, under very diverse operaional condiions such as differen propagaion environmens and ransmission echnologies. The operaional success of such wireless sysems depends on he correc characerizaion of he inpu-oupu response of he fading channel saisics and is realizaion hrough sae space models. The efficien uilizaion of bandwidh allocaion and in general he sysem performance relies on he deploymen of dynamic qualiy of service (QoS) measures, which are robus o he channel characerisic and he aggregae non-homogeneiy of he raffic [2, 3]. The ulimae performance limis of any communicaion sysem are deermined by he channel i operaes in [78]. Realisic channel models are hus of umos imporance for sysem design and esing. In his disseraion, new sochasic ime varying models for cellular and ad hoc newors are presened. These models capure he spaio-emporal variaions of wireless channels, which are due o he relaive mobiliy of he receiver and/or ransmier and scaerers. Unlie exising models of wireless newors ha are mainly saic in he sense ha heir saisics do no vary as funcions of ime [4-6], our wor arge he dynamic behavior of he propagaion environmen. The random variables characerizing he insananeous power in saic channel models are generalized o dynamical models including random processes wih ime varying (TV) saisics. Inphase and quadraure componens of he TV wireless channel and heir saisics are derived from he sochasic sae space models. Since hese models are based on sae space represenaions, we propose o esimae wih small finie parameer se he channel parameers and sae variables, which represen he inphase and quadraure componens direcly from received signal level measuremens. The laer are usually available or easy o obain in any wireless newor. A filer-based expecaion maximizaion (EM) algorihm [79, 8] and Kalman filer [8] are employed in he esimaion process. These filers use only he firs and second order 2

20 saisics and are recursive and herefore can be implemened on-line. Experimenal esing of he proposed models and esimaion algorihms is carried ou using received signal level measuremen daa colleced from cellular and ad hoc wireless environmens. Wireless newors are subjec o ime spread (mulipah), Doppler spread (ime variaions), pah loss, and inerference seriously degrading heir performance, power conrol is crucial o compensae for hese facors for an accepable performance in boh ad hoc and cellular sysems. The conrol of he ransmied power provides for an efficien and opimal performance of wireless sysems [2, 3, 7, 8]. I increases sysem capaciy and qualiy of communicaions, and reduces baery power consumpion. Co-channel inerference caused by frequency reuse is he mos resraining facor on he sysem capaciy. The correc usage of any power conrol algorihm (PCA) and hereby he power opimizaion of he channel models, require he use of such channel models ha capure boh emporal and spaial variaions in he channel, which exhibi more realisic behavior of wireless communicaion sysems [2, 3, 9-6]. Since few emporal or even spaioemporal dynamical models have so far been invesigaed wih he applicaion of any PCA, he suggesed dynamical models resul in new PCAs ha provide a far more realisic and efficien opimum conrol for wireless channels. The hird aspec of his disseraion is locaion based services (LBS), also nown as locaion services (LCS), and mobile or wireless locaion-based services. This is an innovaive echnology ha provides informaion or maing informaion available based on he geographical locaion of he user. The abiliy o pinpoin he locaion of an individual has an obvious and vial value in he conex of emergency services [7]. Pinpoining he locaion of people, sensor devices, and oher valuable asses also open he door o a new world of previously unimagined informaion services and m-commerce possibiliies. In his new world, faciliaions as Where is he neares ATM?, Chec raffic condiions on he highway on my roue, Find a paring lo nearby, as well as answers o Where is my advisor?, Where is my car? will be an everyday rule in our lives. Mobile locaion has also obvious applicaions in wireless micro-sensor newors, for e.g., in miliary applicaions where sensors are dropped by housands in hosile errain o probe movemen of enemy roops. Several mare sudies prediced ha mobile 3

21 locaion services will grow highly in he nex few years [8]. Some esimae ha LBS will be he nex Golden Child..2 Relaed Lieraure Review.2. Wireless Channel Modeling The wireless communicaions channel consiues he basic physical lin beween he ransmier and he receiver anennas. Is modeling has been and coninues o be a analizing issue, while being one of he mos fundamenal componens based on which ransmiers and receivers are designed and opimized. In addiion o he exponenial power pah-loss, wireless channels suffer from sochasic shor erm fading (STF) due o mulipah, and sochasic long erm fading (LTF) due o shadowing depending on he geographical area. STF corresponds o severe signal envelope flucuaions, which occur is densely build-up areas ha filled wih los of objecs lie buildings, vehicles, ec. On he oher hand, LTF corresponds o less severe mean signal envelope flucuaions, which occur in much larger sparsely populaed or suburban areas [5, 6, 9]. In general, LTF and STF are considered as superimposed and may be reaed separaely [9]. Ossanna [49] was he pioneer o characerize he saisical properies of he signal received by a mobile user, in erms of inerference of inciden and refleced waves. His model was beer suied for describing fading occurring mainly in suburban areas. The LTF model is described by he average power loss due o disance and power loss due o reflecion of signals from surfaces, which when measured in db s give rise o normal disribuions which implies ha he channel aenuaion coefficien is log-normally disribued [9]. Furhermore, in mobile communicaions, he LTF channel models are also characerized by heir special correlaion characerisics which have been repored in [97-99]. Clare [5] inroduces he firs comprehensive scaering model describing STF occurring mainly in urban areas. An easy way o simulae Clare s model using a compuer simulaion is described in [4]. This model is laer expanded o hree- 4

22 dimensional (3D) by Aulin [44]. An indoor STF is firs inroduced in [5]. Mos of hese STF models provide informaion on he frequency response of he channel, described by he Doppler power specral densiy (DPSD). Aulin [44] presened a mehodology o compue he Doppler power specrum by compuing he Fourier ransform of he auocorrelaion funcion of he channel impulse response wih respec o ime. A differen approach, leading o he same Doppler power specrum relaion was presened by Gans [6]. These STF models sugges various disribuions for he received signal ampliude such as Rayleigh, Ricean, or Naagami. The majoriy of research papers in his field use ime invarian (saic) models for he wireless channels [4-6, 9, 6]. In ime invarian models, channel parameers are random bu do no depend on ime, and remain consan hroughou he observaion and esimaion phase. This conrass wih ime varying (TV) models, where he channel dynamics become TV sochasic processes [2, 2, 46-48]. TV models ae ino accoun he relaive moion beween ransmiers and receivers and emporal variaions of he propagaing environmen such as moving scaerers. Mobile-o-mobile (or ad hoc) wireless newors comprise nodes ha freely and dynamically self-organize ino arbirary and/or emporary newor opology wihou any fixed infrasrucure suppor [9]. They require direc communicaion beween a mobile ransmier and a mobile receiver over a wireless medium. Such mobile-o-mobile communicaion sysems differ from he convenional cellular sysems, where one erminal, he base saion, is saionary, and only he mobile saion is moving. As a consequence, he saisical properies of mobile-o-mobile lins are differen from cellular ones [9], [92]. Copious ad hoc neworing research exiss on layers in he open sysem inerconnecion (OSI) model above he physical layer. However, neglecing he physical layer while modeling wireless environmen is error prone and should be considered carefully [93]. The experimenal resuls in [94] show ha he facors a he physical layer no only affec he absolue performance of a proocol, bu because heir impac on differen proocols is non-uniform, i can even change he relaive raning among 5

23 proocols for he same scenario. The imporance of he physical layer is demonsraed in [95] by evaluaing he Medium Access Conrol (MAC) performance. Mos of he research on mobile-o-mobile channel modeling, such as [52], [53], [9], [92], [96], deals mainly wih deerminisic wireless channel models. In hese models he speed of nodes are assumed o be consan and he saisical characerisics of he received signal are assumed o be fixed in ime. The Doppler power specral densiy (DPSD) is hen fixed from one observaion insan o he nex. Bu in realiy, he propagaion environmen varies coninuously due o mobiliy of he nodes a variable speeds causing newor opology o dynamically change, he angle of arrival of he wave upon he receiver can vary coninuously, and objecs or scaers move in beween he ransmier and he receiver resuling in appearance or disappearance of exising pahs from one insan o he nex. The measuremens provided in [7] are performed using boh narrow-band and wideband signals in order o obain boh LTF and STF characerisics. Measuremens using narrow-band signals provide informaion on he saisics of power loss as a funcion of disance and he Doppler spread. These measuremens confirm ha he power loss due o disance is log-normally disribued and provide values for he mean and variance. Measuremens of he DPSD are made by ransmiing a single one frequency, and measuring he flucuaions on he received signal in ime. On he oher hand, wide-band measuremens [7] are useful in deermining he number of pahs and he power-delay profile, which is a measure of he received power for differen delays. In his disseraion, new sochasic ime varying models for LTF, STF, and ad hoc environmens are presened. These models capure he spaio-emporal variaions of wireless channels, which are due o he relaive mobiliy of he receiver and/or ransmier and scaerers. The radiional models are special case of our developed models..2.2 Opimal Power Conrol Power conrol (PC) is imporan o improve performance of wireless communicaion sysems. The benefis of power minimizaion are no jus increased baery life, bu also 6

24 increased overall newor capaciy and improved call qualiy. Users only need o expand sufficien power for accepable recepion as deermined by heir QoS specificaions ha is usually characerized by he signal o inerference raio (SIR) [22]. Power conrol algorihms (PCAs) can be classified as cenralized and disribued. The cenralized PCAs require global ou-of-cell informaion available a base saions. The disribued PCAs require base saions o now only in-cell informaion, which can be easily obained by local measuremens. The power allocaion problem has been sudied exensively as an eigenvalue problem for non-negaive marices [2, 3], resuling in ieraive PCAs ha converge each user s power o he minimum power [2, 9, 4-6, 23, 24], and as opimizaion-based approaches [7]. Much of his previous wor deals wih saic ime invarian channel models. The scheme inroduced in [7, ], whereby he saisics of he received SIR are used o allocae power, raher han an insananeous SIR. Therefore, he allocaion decisions can be made on a much slower ime scale. Previous aemps a capaciy deerminaions in CDMA sysems have been based on a load balancing view of he PC problem [25]. This reflecs an essenially saic or a bes quasi-saic view of he PC problem, which largely ignores he dynamics of channel fading as well as user mobiliy. Sochasic PCAs (SPCAs) ha use noisy inerference esimaes have been inroduced in [], where convenional mached filer receivers are used. I is shown in [] ha he ieraive SPCA, which uses sochasic approximaions, converges o he opimal power vecor under cerain assumpions on he sep-size sequence. These resuls were laer exended o he cases where a nonlinear receiver or a decision feedbac receiver is used [26]. However, he channel gains are assumed o be fixed ignoring he effecs of ime variaions on he performance of he sysem. Oher resuls ha aemp o recognize he ime-correlaed naure of signals are proposed in [27], where blocing is defined via he sojourn ime of global inerference above a given level. Downlin PC for fading channels is sudied in [28] by a heavy raffic limi where averaging mehods are used. Sochasic conrol approach for uplin lognormal fading channels is sudied in [29], in which a bounded rae power adjusmen model is proposed. Recen wor on dynamic PC wih sochasic channel variaion can be found in [3-32]. 7

25 In his disseraion, various PCAs are developed based on he TV channel models. A cenralized and deerminisic PCA based on predicable power conrol sraegies (PPCS) is firs inroduced. PPCS simply means updaing he ransmied powers a discree imes and mainaining hem fixed unil he nex power updae begins. The inerference or ouage probabiliy (OP) is used as a performance measure. A disribued version of his algorihm is derived. The laer helps in allowing auonomous execuion a he node or lin level, requiring minimal usage of newor communicaion resources for conrol signaling. Subsequenly, an ieraive and disribued SPCA based on sochasic approximaions, which requires less informaion han he SPCAs described in he lieraure, is proposed. I only requires he received SIR a is inended receiver, while he received mached filer oupu (received SIR) a is inended receiver and he channel gain beween he ransmier and is inended receiver are required in he SPCA presened in [, 33]..2.3 Locaion Based Services The problem of esimaing he locaion and velociy of a mobile saion (MS) has been he subjec of much research wor over he las few years. The curren lieraure and sandards in esimaing he locaion and velociy are based mosly on ime signal informaion, such as ime difference of arrival (TDOA), enhanced observed ime differences (E-OTD), observed ime difference of arrival (OTDOA), global posiioning sysem (GPS), ec., [34-38]. However, no all of hese mehods mee he necessary requiremens imposed by specific services. In addiion, mos of hem require new hardware since localizaion is no inheren in he curren wireless sysems, for insance, GPS demands a new receiver and TDOA, E-OTD, OTDOA require addiional locaion measuremen unis in he newor [39]. Adding exra hardware means exra cos for implemenaion, which can be refleced on boh consumers and operaors. Researchers have also suggesed several MS locaion mehods based on signal power measuremens such as in [4] and [4], where a cerain minimizaion problem is solved numerically o ge an iniial esimae of he MS posiion, and hen a smoohing procedure such as linear regression [4], or he Kalman filer [4] are applied o obain a more accurae esimae. 8

26 In his disseraion, several MS racing mehods based on maximum lielihood esimaion (MLE) [72], and recursive nonlinear Bayesian esimaion (RNBE) algorihms such as he exended Kalman filer (EKF) [42], he paricle filer (PF) [43], and he unscened paricle filer (UPF) [75] are proposed. The MLE algorihm employs he average received power measuremens based on he lognormal propagaion channel model o obain an iniial MS locaion esimae [86]. These measuremens are readily available hrough newor measuremen repors or radio measuremens, in idle or acive mode, for any MS uni in 2G and 3G cellular newors. The RNBE algorihms employ he insananeous elecric field measuremens based on he 3D mulipah channel model of Aulin [44] o accoun for mulipah and non-line-of-sigh (NLOS) characerisics of he wireless channel as well as he dynamiciy of he MS. The received insananeous elecric field in his model is a nonlinear funcion of he posiion and velociy of he MS. The EKF approach is based on linearizing he nonlinear sysem model around he previous esimae, and herefore is very sensiive o he iniial sae. This moivaes he use of he ML esimae of he MS locaion as an iniial sae o he EKF. Paricle filering approaches approximae he opimal soluion numerically based on he physical model, raher han applying an opimal filer o an approximae model such as in he EKF. They provide general soluions o many problems where linearizaion and Gaussian approximaions are inracable or yield low performance. The more nonlinear he model is or he more non-gaussian he noise is, he more poenial PFs have, especially in applicaions where compuaional power is raher cheap and he sampling rae is moderae. In his disseraion, paricle filering is implemened for he generic PF and he more recen UPF. Aulin s model in [44] posulaes nowledge of he insananeous received field a he MS, which is obained hrough he circuiry of he mobile uni. The proposed RNBE algorihms ae ino accoun NLOS condiion as well as mulipah propagaion environmens. They require only one base saion (BS) o esimae he MS locaion insead of a leas hree BSs as found in he lieraure [4], [86]. However, an iniial MS locaion esimae ha requires a leas hree BSs, such as he MLE and riangulaion mehod, will improve he convergence of he RNBE filer. Paricle filering has been used 9

27 in several racing wireless applicaions [62], [87]-[89], bu he channel models used do no ae ino accoun he mulipah properies of he wireless channel. To he bes of our nowledge, he uilizaion of he PF and/or he UPF ogeher wih he classical wireless channel model o exrac he MS locaion and velociy is new. The performance of he proposed algorihms is compued numerically and in he presence of parameers uncerainy. Numerical resuls indicae ha he proposed UPF algorihm is highly accurae and superior o oher approaches..3 Main Conribuions The main conribuions of our research can be described as follows:. Developmen of dynamical wireless channel models, which capure he spacevarian and ime-varian characerisics of cellular and ad hoc wireless newors. These models are based on sae space and SDEs, and are in accordance wih he physical principles of elecromagneic wave propagaion; hey are parameric and can describe diverse propagaion environmens. They allow he ools of sysem heory, idenificaion, and esimaion o be applied o his class of problems. 2. Developmen of esimaion and idenificaion algorihms based on he EM algorihm and Kalman filering o esimae he channel model parameers and saes, respecively, from received signal measuremens for LTF, STF, and ad hoc wireless newors. These filers use only he firs and second order saisics and are recursive and herefore can be implemened on-line. The proposed models and esimaion algorihms are esed using received signal level measuremen daa colleced from cellular and ad hoc experimenal seups. 3. Developmen of PCAs based on he proposed models o compensae for pah losses, mulipah, Doppler spread, and inerferences affecing he ransmied signal. Cenralized, disribued, deerminisic and sochasic PCAs are considered. The benefis of such PCAs include: Minimize power consumpion and prolong

28 baery life of communicaing nodes, miigae inerference and increase newor capaciy, and mainain lin QoS by adaping o node movemens in ad hoc newors and random channel variaions by reducing he probabiliy of ouage. 4. Developmen of MS locaion and velociy esimaion algorihms based on he 3D mulipah scaering channel model of Aulin. The choice of his model is o accoun for he mulipah properies and NLOS of wireless newors. The insananeous elecric field is a nonlinear funcion of he posiion and velociy of he MS. The EKF, PF, and UPF are employed for he esimaion process. These esimaion algorihms are recursive and can be implemened on-line. They also suppor exising newor infrasrucure and channel signaling. 5. Developmen of experimenal and simulaion se-ups demonsraing he flexibiliy and applicabiliy of he proposed channel models in capuring he dynamics of diverse propagaion environmens, and deermining he accuracy of he proposed esimaion and idenificaion algorihms in esimaing he channel model parameers and saes. The experimenal and numerical resuls show he improvemen in performance of he developed PCAs over he radiional PCAs, and signify he high accuracy and robusness of he proposed MS locaion and velociy esimaion algorihm using paricle filering..4 Disseraion Ouline This disseraion is srucured as follows: In Chaper 2, we presen modeling of dynamical wireless channels for cellular and ad hoc newors. Lognormal shadowing or LTF channel models are discussed firs. New dynamical spaial and emporal models for he power pah loss and aenuaion coefficien of he channel are presened. These models use specific ype of SDEs whose soluion a every insan represens he correlaion properies boh in ime and space of he channel and corresponds o he saisics of he saic lognormal channel. Afer ha, analysis of STF dynamical channel models describing he received signal envelope of each mulipah

29 componen is presened. These models are based on he emporal characerisics of he channel, namely he Doppler power specral densiy. The dynamics of each mulipah componen are capured using sochasic sae-sae models. Finally, ad hoc dynamical channel models are presened in a similar way as he STF channel modeling, excep ha he deerminisic ad hoc DPSD is considered. Chaper 3 inroduces he filer-based EM algorihm combined wih he Kalman filer, o esimae he channel parameers and saes from received signal measuremens. These filers use only he firs and second order saisics. The algorihm is recursive allowing he inphase and quadraure componens and parameers o be esimaed on-line from measuremens. The proposed algorihm is esed using received signal level measuremen daa colleced from experimens for boh he cellular and ad hoc channels. Chaper 4 presens several PCAs based on he dynamical models described in Chaper 2. The cenralized deerminisic PCA is inroduced firs. The soluion of he opimal PC is obained hrough pah-wise opimizaion, which is solved by linear programming using PPCS. The PPCS are proven o be effecively applicable o such dynamical models for an opimal PC. The algorihm can be implemened using an ieraive disribued scheme. A disribued SPCA based on sochasic approximaions and which uses only measured SIR is inroduced. Numerical resuls are provided o evaluae he performance of he proposed PCAs. Chaper 5 presens MS locaion and velociy esimaion algorihms based on paricle filering. Firs, we inroduce he mahemaical models employed for he MS locaion and velociy esimaion algorihms, which are he lognormal channel model and he 3D mulipah scaering model of Aulin. The received insananeous elecric field in Aulin s model is a nonlinear funcion of he posiion and velociy of he MS. Thus, he MLE, he EKF, he PF, and he UPF are employed o esimae he MS locaion and velociy. A brief review of he heory of hese algorihms is hen inroduced. Nex, numerical resuls illusraing he accuracy and evaluaing he robusness of he proposed esimaion algorihms o uncerainies due o random variaions in he channel parameers are presened. 2

30 Finally, Chaper 6 provides concluding remars and presens a discussion on he applicabiliy of hese models and algorihms in design and analysis. I also provides an opening o emanaing fuure wor. 3

31 Chaper 2 Sochasic Wireless Channel Modeling The ulimae performance limis of any communicaion sysem are deermined by he channel i operaes in. Realisic channel models are hus of umos imporance for sysem design and esing. In his chaper, we propose new sochasic wireless channel models which capure boh he space and ime variaions of wireless sysems. The proposed channel models are based on sochasic differenial equaions (SDEs) driven by Brownian moions and represened in sochasic sae space form. Long-erm fading (LTF), shorerm fading (STF), and ad hoc wireless channel models are developed. These models are more realisic han he ime invarian ones usually encounered in he lieraure, which do no capure and rac he ime varying characerisics in he environmen. In conras wih he radiional models, he saisics of he proposed models are shown o be ime varying, bu converge in seady sae o heir saic counerpars. Thus, he radiional models are special case of our models. Pars of he resuls presened here have been published in [2, 46-48, 68, 5, 7, 9, ]. 2. The General TV Wireless Channel Impulse Response The impulse response of a wireless channel is ypically characerized by ime variaions and ime spreading [5]. Time variaions are due o he relaive moion beween he ransmier and he receiver and emporal variaions of he propagaion environmen. Time spreading is due o he fac ha he emied elecromagneic wave arrives a he receiver having undergone reflecions, diffracion and scaering from various objecs along he way, a differen delay imes. A he receiver, a random number of signal componens, copies of a single emied signal, arrive via differen pahs hus having 4

32 undergone differen aenuaion, phase shifs and ime delays, all of which are random and ime varying. This random number of signal componens add vecorially giving rise o signal flucuaions, called mulipah fading, which are responsible for he degradaion of communicaion sysem performance. The general ime varying (TV) model of wireless fading channel is ypically represened by he following mulipah low-pass equivalen impulse response [5] ( ) N ( ) ( ( ) ( )) ( ()) (2.) j n (, ) ( ) ( ) Φ τ () C ; τ = r, τ e δ τ τ = I, τ + jq, τ δ τ τ l n n n n n n= n= where C ( ; ) l ( ) N τ is he response of he channel a ime, due o an impulse applied a ime τ, N() is he random number of mulipah componens impinging on he receiver, N { } () while he se r (, τ), (, τ), τ ( ) Φ describes he random TV aenuaion, overall n n n n= N { } () phase shif, and arrival ime of he differen pahs, respecively. I (, τ), Q (, τ) N { (, ) cos (, ), (, ) sin (, )} () r τ τ r τ τ n n n n n= n n n= Φ Φ are defined as he inphase and quadraure componens of each pah. Leing s ( ) l be he low-pass equivalen represenaion of he ransmied signal, hen he low-pass equivalen represenaion of he received signal is given by [5] N jφn(, τ n() ) l () = l( ; τ) l( τ) τ = n(, τn() ) l τ n n= ( ) ( ()) y C s d r e s (2.2) and he mulipah TV band-pass impulse response is given by [5] () ( ; ) Re (, ) () n= ( n() ) jφn jωc C τ r τ e e δ τ τ N N (, τ ) = n n= ( In(, τ) cos ωc Qn(, τ) sinωc ) δ τ τn() = ( ) (2.3) where ω c is he carrier frequency, and he band-pass represenaion of he received signal is given by 5

33 ( ) N n= ( n(, τn ) cos ωc n(, τn() ) sinωc ) l( τn() ) y() = I () Q s (2.4) In general, wireless communicaion newors are subjec o ime-spread (mulipah), Doppler spread (ime variaions), pah-loss, and inerference seriously degrading heir performance. In addiion o he exponenial power pah-loss, wireless channels suffer from sochasic STF due o mulipah, and sochasic LTF due o shadowing depending on he geographical area. If he mobile happens o be in sparsely populaed area wih few buildings, vehicles, mounains ec., is signal will undergo LTF. Whereas in case he environmen is filled wih los of objecs lie buildings, vehicles ec., hen due o increase in scaering of he signal, he ype of fading will be STF [5, 6, 9]. LTF is usually modeled by lognormal disribuions and STF is modeled by Rayleigh, Ricean, or Naagami disribuions [9]. In general, LTF and STF are considered as superimposed and may be reaed separaely [9]. There exis many facors ha define he randomness and he changing condiions wih respec o ime and/or space (sochasic) wihin he wireless medium. Saic models do no ae ino accoun he ime varying behavior of he channel and herefore do no represen a realisic picure of he communicaion medium. In saic models, channel parameers are random bu do no depend on ime, and remain consan hroughou he observaion and esimaion phase. Therefore, heir saisics are ime invarian. An alernaive is o develop a new approach based on sochasic dynamical channel models o invesigae he rue behavior of he wireless communicaion newors. In TV models, he channel dynamics become TV sochasic processes [2, 2]. TV models ae ino accoun he relaive moion beween ransmiers and receivers and emporal variaions of he propagaing environmen such as moving scaerers. TV LTF, STF, and ad hoc dynamical channel models are considered in his chaper. The sochasic TV LTF channel modeling is discussed firs in he nex secion. 6

34 2.2 Sochasic TV LTF Channel Modeling 2.2. The radiional (Saic) LTF Channel Model In his secion we discuss he exising saic models and inroduce our approach on how o derive dynamical models. Before inroducing he dynamical TV LTF channel model ha capures boh space and ime variaions, we firs summarize and inerpre he radiional lognormal shadowing model, which serves as a basis in he developmen of he subsequen TV model. The radiional (ime invarian) power loss (PL) in db for a given pah is given by [9] d PL( d)[db]: = PL( d)[db] + α log + Z, d d (2.5) d where PL( d ) is he average PL in db a a reference disance d from he ransmier, he disance d corresponds o he ransmier-receiver separaion disance, α is he pah-loss exponen which depends on he propagaing medium, and Z is a zero-mean Gaussian disribued random variable, which represens he variabiliy of PL due o numerous reflecions and possibly any oher uncerainy of he propagaing environmen from one observaion insan o he nex. The average value of he PL described in (2.5) is d PL( d)[db]: = PL( d )[db] + α log, d d d (2.6) I can be seen in (2.5) and (2.6) ha he saisics of he PL do no depend on ime, and herefore hese models rea PL as saic (ime invarian). They do no ae ino consideraion he relaive moion beween he ransmier and he receiver, or variaions of he propagaing environmen due o mobiliy. Such spaial and ime variaions of he propagaing environmen are capured herein by modeling he PL and he envelope of he received signal as random processes ha are funcions of space and ime. Moreover, and perhaps more imporanly, radiional models do no ae ino consideraion he correlaion properies of he PL in space and a 7

35 differen observaion imes. In realiy, such correlaion properies exis, and one way o model hem is hrough sochasic processes, which obey specific ype of SDEs Sochasic LTF Channel Models In ransforming he saic model o a dynamical model, he random PL in (2.5) is relaxed o become a random process, denoed by { }, τ τ X (, τ ), which is a funcion of boh ime and space represened by he ime-delay τ, where τ = d/c, d is he pah lengh, c is he speed of ligh, τ = d /c and d is he reference disance. The signal aenuaion is defined by S (, ) X (, τ ) τ e, where ln() / 2 = [9]. For simpliciy, we firs inroduce he TV lognormal model for a fixed ransmier-receiver separaion disance d (or τ) ha capures he emporal variaions of he propagaing environmen. Nex, we generalize i by allowing boh and τ o vary as he ransmier and receiver, as well as scaers, are allowed o move a variable speeds. This induces spaio-emporal variaions in he propagaing environmen. When τ is fixed, he proposed model capures he dependence of { X (, τ )}, τ τ on ime. This corresponds o examining he ime variaions of he propagaing environmen X (, τ ) represens for fixed ransmier-receiver separaion disance. The process { }, τ τ how much power he signal looses a a paricular locaion as a funcion of ime. However, since for a fixed disance d, he PL should be a funcion of disance, we choose o generae { X (, )}, τ τ [45-48] τ by a mean-revering version of a general linear SDE given by ( ) dx (, τ) = β(, τ) γ(, τ) X (, τ) d + δ(, τ) dw (), X PLd db 2 (, τ) =N ( ( )[ ]; σ ) (2.7) { } where W() is he sandard Brownian moion (zero drif, uni variance) which is assumed o be independen of ( ), X τ, N ( µ ; κ ) denoes a Gaussian random variable 8

36 wih mean µ and variance κ, and PL( d)[ db ] is he average pah-loss in db. The parameer γ (, ) τ models he average ime varying PL a disance d from ransmier, which corresponds o PL( d)[ db ] a d indexed by. This model racs and converges o his value as ime progresses. The insananeous drif β (, τ) ( γ(, τ) X(, τ) ) he effec of pulling he process owards γ (, τ ), while β (, ) adjusmen owards his value. Finally, δ (, ) volailiy of he process for he insananeous drif. represens τ represens he speed of τ conrols he insananeous variance or { θ, τ } { β(, τ), γ (, τ), δ (, τ) }. If he random processes in { θ τ } Le ( ) (, ) are measurable and bounded [], hen (2.7) has a unique soluion for every X (, τ ) given by [2, 46-48] β([, ], τ) β([ u, ], τ) X(, τ) = e X(, τ) + e ( β( u, τ) γ( u, τ) du+ δ( u, τ) dw( u) ) (2.8) where β([, ], τ) β( u, τ) du. Moreover, using Io s sochasic differenial rule [] on S (, ) X(, τ ) τ = e he aenuaion coefficien obeys he following SDE 2 2 ds(, τ) = S(, τ) β(, τ)[ γ(, τ) ln S(, τ)] + δ (, τ) d + δ(, τ) dw() 2 S (, τ ) = e X (, τ ) (2.9) This model capures he emporal variaions of he propagaing environmen as he { } random parameers θ(, τ) can be used o model he TV characerisics of he channel for he paricular locaion τ. A differen locaion is characerized by a differen se of parameers { (, )} θ τ. 9

37 Now, le us consider he special case when he parameers θ (, τ ) i.e., θ ( τ ) { β ( τ ), γ ( τ), δ ( τ) } he dynamic PL X (, τ ), denoed by EX [ (, τ )] are ime invarian,. In his case we need o show ha he expeced value of, converges o he radiional average PL in (2.6). The soluion of he SDE model in (2.7) for he ime invarian case saisfies ( ) ( ) β( τ) ( ) β( τ) ( ) β( τ) u X(, τ) = e X(, τ) + γ( τ) e + δ( τ) e dw( u) (2.) where for a given se of ime invarian parameers ( ) θ τ and if he iniial X (, τ ) is Gaussian or fixed, hen he disribuion of X (, τ ) is Gaussian wih mean and variance given by [] [ ] βτ ( )( ) βτ ( )( ) ( ) βτ ( ) { } E X(, τ) = γ( τ) e + e E X(, τ) 2 Var [ X ] ( ) e Var X 2 βτ ( ) 2 ( ) e 2 βτ ( )( ) (, τ ) = δ τ + (, τ) ( ) (2.) Expression (2.) of he mean and variance shows ha he saisics of he communicaion channel vary as a funcion of boh ime and space τ. As he observaion insan,, becomes large, he random process { Xτ (, )} converges o a Gaussian random variable wih mean γτ ( ) = PL( d)[db] and variance δ 2 ( τ)/2 β( τ ). Therefore, he radiional lognormal model in (2.5) is a special case of he general TV LTF model in (2.7). Moreover, he disribuion of S (, ) X(, τ ) τ = e is lognormal wih mean and variance given by 2 [ τ ] + VarX [ τ ] 2 EX (, ) (, ) E[ S(, τ )] = exp 2 ( ) 2 exp( 2 EX [ (, τ )] + VarX [ (, τ )]) 2 [ (, τ) ] = exp 2 [ (, τ) ] + 2 [ (, τ) ] Var S E X Var X (2.2) 2

38 { } Now, les go bac o he more general case in which θ(, τ) { β(, τ), γ (, τ), δ (, τ) }. A a paricular locaion τ, he mean of he PL process EX [ (, τ )] is required o rac he ime variaions of he average PL. This can be seen in he following example. Example 2.: Le 2/ T π γ(, τ) = γm ( τ) +.5e sin T (2.3) where γ ( ) m τ is he average PL a a specific locaion τ, T is he observaion inerval, δ (, τ ) = 4 and β (, ) τ = 225 (hese parameers are deermined from experimenal measuremens as will be shown a he end of his secion), where for simpliciy δ (, τ ) and β (, τ ) are chosen o be consan, bu in general hey are funcions of boh and τ. The variaions of X (, τ ) as a funcion of disance and ime are represened in Figure 2.. The emporal variaions of he environmen are capured by a TV γ (, ) τ which flucuaes around differen average PLs γ m s, so ha each curve corresponds o a differen locaion. I is noiced in Figure 2. ha as ime progresses, he process X (, τ ) is pulled owards γ (, τ ). The speed of adjusmen owards γ (, ) choosing differen values of β (, ) τ can be conrolled by τ. In Chaper 3, we propose o recursively esimae he channel parameers direcly from received signal measuremens, using he EM algorihm combined wih he Kalman filer. Nex, he general spaio-emporal lognormal model is inroduced by generalizing he previous model o capure boh space and ime variaions, using he fac ha γ (, ) τ is a funcion of boh and τ. In his case, besides iniial disances, he moion of mobiles, i.e., heir velociies and direcions of moion wih respec o heir base saions are imporan facors o evaluae TV PLs for he lins involved. 2

39 γ(,τ) X(,τ) 9 8 X() [db] Disance (m) 5 2 Time (sec.) 3 x -4 Figure 2.: Mean-revering power pah-loss as a funcion of and τ, for he ime varying γ (, τ ) in Example

40 This can be illusraed in a simple way for he case of a single ransmier and a single receiver as follows: Consider a base saion (receiver) a an iniial disance d from a mobile (ransmier) ha moves wih a cerain consan velociy υ in a direcion defined by an arbirary consan angle θ, where θ is he angle beween he direcion of moion of he mobile and he disance vecor ha sars from he receiver owards he ransmier as shown in Figure 2.2. A ime, he disance from he ransmier o he receiver, d, () is given by d ( ) = ( d+ υ cos θ) + ( υsin θ) = d + ( υ) + 2dυcosθ (2.4) Therefore, he average PL a ha locaion is given by d () γ(, τ) = PL( d( ))[ db] = PL( d )[ db] + αlog + ξ( ), d( ) d (2.5) d where PL( d ) is he average PL in db a a reference disance d, d() is defined in (2.4), α is he pah-loss coefficien and ξ ( ) is an arbirary funcion of ime represening addiional emporal variaions in he propagaing environmen lie he appearance and disappearance of addiional scaers. Now, suppose he mobile moves wih arbirary velociy, ( vx(), vy() ), in he x-y plane, where ( x(), y() ) insananeous velociy componens in he x and y direcions, respecively. v v denoe he Rx d d() Tx θ υ Figure 2.2: A mobile (ransmier) a a disance d from a base saion (receiver) moves wih velociy υ and in he direcion given by θ wih respec o he ransmier-receiver axis. 23

41 The insananeous disance from he receiver is hus described by The parameer γ (, ) 2 2 () = ( + () x ) + ( () y ) (2.6) d d v d v d τ is used in he TV lognormal model (2.7) o obain a general spaioemporal lognormal channel model. This is illusraed in he following example. Example 2.2: Consider a mobile moving a sinusoidal velociy wih average speed 8 Km/hr, iniial disance d = 5 meers, θ = 35 degrees, and ξ () =. Figure 2.3 shows he mean revering PL X (, τ ), γ (, τ ), EX( τ ) [, ], velociy of he mobile υ and disance d() as a funcion of ime. I can be seen ha he mean of (, ) average PL γ (, ) X τ coincides wih he τ and racs he movemen of he mobile. Moreover, he variaion of X (, τ ) is due o uncerainies in he wireless channel such as movemens of objecs or obsacles beween ransmier and receiver ha are capured by he spaio-emporal lognormal model (2.7) and (2.5). Addiional ime variaions of he propagaing environmen, while he mobile is moving, can be capured by using he TV PL coefficien α () in (2.5) in addiion o he TV parameers β (, τ ) and (, ) δ τ, or simply by ξ ( ). In Chaper 3, we propose o recursively esimae he channel parameers as well as he TV PL direcly from received signal measuremens, using he EM algorihm combined wih he Kalman filer Spaial Correlaion of he Sochasic LTF Model Now, we wan o show ha he spaial correlaion of he lognormal mean-revering model of (2.7) agrees wih he experimenal spaial correlaion [97-99]. In paricular, i is repored ha he spaial correlaion for shadow fading in mobile communicaions, which compares successfully wih experimenal daa, can be modeled using an exponenially decreasing funcion muliplied by he variance of he PL process as follows 24

42 95 X(,τ) as a funcion of X(,τ) [db] 9 85 γ(,τ) 8 E[X(,τ)] X(,τ) Variable Speed v() and disance d() d() (m), v() (m/s) Time (sec.) d() v() Figure 2.3: Mean-revering power pah-loss X (, τ ) for he TV LTF wireless channel model in Example 2.2. The mobile sars moving closer o he base saion from poin 5 meers wih an angle of 35 degrees and sinusoidal speed wih average 8 m/hr (22.2 m/s). 25

43 c c Cov ( ) σ e = σ e (2.7) 2 d/ X 2 ( v/ X ) X X X where 2 σ X is he covariance of he PL process, d is he disance beween wo consecuive samples, v is he velociy of he mobile. X c is he effecive correlaion disance which is proporional o he densiy of he propagaing environmen corresponding o he disance when he normalized correlaion falls o e [99]. Here we show ha our overall spacial dynamical model capures indeed hese correlaion properies. Wihou loss of generaliy, consider he paricular case where he parameers { θ(, τ) } β( τ), γ (, τ), δ ( τ) { } =., hen we have Le X (, τ ) X (, τ) EX [ (, τ) ] dx (, τ) = β( τ) X (, τ) d + δ( τ) dw (), X N 2 (, τ) = (; σ ) (2.8) The soluion of (2.8) is given by [] X e X e dw u ( )( ) ( )( ) (, ) βτ βτ u τ = (, τ) + δ( τ) ( ) (2.9) The mean of he process X (, τ ) is zero, and is covariance is given by [] ( τ τ )( τ τ ) Cov (, ) s = E X (, ) E X (, ) X ( s, ) E X ( s, ) X 2 δ ( τ) 2 βτ ( ) βτ ( ) βτ ( )( + s) 2 βτ ( ) 2 2 ( )( s ) = e e σ + e T (2.2) where s min(, s). Leing s= +, hen we have ( ) 2 2βτ ( )( ) βτ ( ) 2 δ ( τ) 2 βτ ( )( ) Cov (, ) e σ e X + = + (2.2) 2 βτ ( ) The covariance of he overall dynamical model indicaes wha proporion of he environmen remains consan from one observaion insan or locaion o he nex, separaed by he sampling inerval. Since he mobile is in moion, i implies ha his 26

44 corresponds o a spaial covariance. If we choose he variance of he iniial condiion such 2 2 δ ( τ ) ha σ =, hen we ge 2 β ( τ ) 2 Cov ( ) ( ) 2 ( ) (, δ τ ) ( ) X X 2 ( ) e βτ e βτ + = = σ βτ Cov (2.22) Expression (2.22) indicaes ha he spaial covariance of our overall dynamical model corresponds o he repored experimenal spaial covariance given by (2.7). The comparison furher indicaes ha β(τ) is a characerisic of boh he propagaing environmen and he separaion disance of wo consecuive samples, i.e., β(τ) is inversely proporional o he densiy of he propagaing environmen, and direcly proporional o he sample separaion disance. Noe ha he spaial covariance is an imporan characerisic for our dynamical mean-revering shadow fading model since i can be clearly used in order o idenify he random parameers {β(τ), δ(τ)}. This could be accomplished by using experimenal daa of Cov ( ) in order o idenify β(τ). The laer can be used furher in conjuncion wih 2 X σ in order o idenify δ(τ). Therefore, he parameers {β(τ), δ(τ)} can be esimaed on-line from experimenal measuremens. 2 Finally, we noe ha he variance of he iniial condiion of he PL process, σ, should ineviably increase wih disance, or equivalenly δ(τ) should increase and/or β(τ) decrease. The TV LTF channel model is used o generae he lin gains for he proposed PCA which inroduced in Chaper 4. The TV STF channel model is discussed in he nex secion. 2.3 Sochasic TV STF Channel Modeling 2.3. The Deerminisic DPSD of Wireless Channels The radiional STF model is based on Ossanna [49] and laer Clare [5] and Aulin s [44] developmens. Aulin s model is shown in Figure 2.4. This model assumes ha a 27

45 Z nh mulipah componen (x, y, z ) MS O Y β n α n γ v m X Figure 2.4: Aulin s 3D mulipah channel model. 28

46 each poin beween a ransmier and a receiver, he oal received wave consiss of he superposiion of N plane waves each having raveled via a differen pah. The nh wave is characerized by is field vecor E n () given by [44] where { (), ()} n { j Φn() j ωc } E () = Re r () e e = I ()cos ω Q ()sin ω (2.23) n n n c n c I Q are he inphase and quadraure componens for he nh wave, n respecively, r = I + Q is he signal envelope, 2 2 n() n() n() Φ = is n() an ( Qn()/ In()) he phase and ω c is he carrier frequency. The oal field E ( ) can be wrien as N E() = En() = I()cos ωc Q()sin ωc (2.24) n= where { I(), Q() } are inphase and quadraure componens of he oal wave, respecively, wih N n= n I() = I () and N Q () = Q(). An applicaion of he cenral limi heorem n= n saes ha for large N, he inphase and quadraure componens have Gaussian disribuions 2 N ( x; σ ) [5]. The mean is x E { I ( )} E { Q ( )} { I } { Q } = = and he variance is 2 σ = Var ( ) = Var ( ). In he case where here is non-line-of-sigh (NLOS), hen he mean x = and he received signal ampliude has Rayleigh disribuion. In he presence of line-of-sigh (LOS) componen, x and he received signal is Ricean disribued. Also, i is assumed ha I() and Q () are uncorrelaed and hus independen since hey are Gaussian disribued [44]. Dependen on mobile speed, wavelengh, and angle of incidence, he Doppler frequency shifs on he mulipah rays give rise o a Doppler power specral densiy (DPSD). The DPSD is defined as he Fourier ransform of he auocorrelaion funcion of he channel, and represens he amoun of power a various frequencies. Define { α, β } as he direcion of he inciden wave ono he receiver as illusraed in Figure 2.4. For he case of α n is uniformly disribued and β n is fixed, he deerminisic DPSD is given by [4] n n 29

47 S ( f ) E / fm, f < f 4 2 m π f = fm, oherwise (2.25) where f is he maximum Doppler frequency, and E /2 Var { I ( )} Var { Q ( )} m = =. A more complex, bu realisic, expression for he DPSD, which assumes β n has probabiliy cos β π densiy funcion p β ( β) = for β βm, and for small angles β m, is given by 2sinβ 2 m, f > fm E S f f f f ( ) =, mcosβm m 4fmsinβm 2 2 E π 2cos βm ( f / fm) sin, f < f cos 2 m β m 4π fmsinβm 2 ( f / fm ) (2.26) Expression (2.26) is illusraed in Figure 2.5 for differen values of mobile speed. Noice ha he direcion of moion does no play a role because of he uniform scaering assumpion, and ha he DPSDs described in (2.25) and (2.26) are band limied. The DPSD is he fundamenal channel characerisic on which STF dynamical models are based on. The approach presened here is based on radiional sysem heory using he sae space approach [] while capuring he specral characerisics of he channel. The main idea in consrucing dynamical models for STF channels is o facorize he deerminisic DPSD ino an approximae nh order even ransfer funcion, and hen use a sochasic realizaion [] o obain a sae space represenaion for he inphase and quadraure componens. The wireless channel is considered as a dynamical sysem for which he inpu-oupu map is described in (2.4). In pracice, one obains from measuremens he power specral densiy of he oupu, and wih he nowledge of he power specral densiy of he inpu he power specral densiy of he ransfer funcion (wireless channel) can be deduced as 3

48 f c = 9 MHz, v = 5 m/h, 2 m/h,..., 2 m/h 5 m/h - 2 m/h Power Specrum db m/h 2 m/h Frequency Hz Figure 2.5: DPSD for β m = degrees and differen values of mobile speed. 3

49 yy 2 ( ) ( ) ( ) S f = H f S f (2.27) xx where x( ) is a random process wih power specral densiy S ( f ) represening he inpu signal o he channel, y ( ) is a random process wih power specral densiy S ( f ) represening he oupu signal of he channel, and H( f ) is he frequency response of he channel, which is he Fourier ransform of he impulse response of he channel. In general, in order o idenify he random process associaed wih S( f ) in (2.25) or (2.26) in he form of an SDE, we need o find a ransfer funcion, H( f ) whose magniude square equals S( f ), i.e. S( f ) H( f ) 2 S( s) H( s) H( s) xx =. This is equivalen o =, where s = j2π f. Tha is, we need o facorize he DPSD. This is an old problem which had been sudied by Paley and Wiener [5] and is reformulaed here as follows: Given a non-negaive inegrable funcion, S( f ), such ha he Paley-Wiener condiion ( f ) log S df < is saisfied, hen here exiss a causal, sable, minimum-phase, 2 ( + f ) 2 H( f ), such ha H( f ) S( f ) =, implying ha S( f ) is facorizable, namely, S( s) = H( s) H( s). The facor H(s) represens he frequency response of a causal, sable, and minimum-phase sysem, which, if driven by he process x(), he power specral densiy of is oupu will be given by (2.27). I can be seen ha he Paley-Wiener condiion is no saisfied when S( f ) is band limied, which is he case of wireless lins. Therefore, he deerminisic DPSD has o be firs approximaed by a raional ransfer S f, and is discussed nex. funcion, say ( ) yy Approximaing he Deerminisic DPSD A number of raional approximaion mehods [56] can be used o approximae he deerminisic DPSD, he choice of which depends on he complexiy and he required 32

50 accuracy. The order of approximaion dicaes how close he approximae curve would be o he acual one. Higher order approximaions capure higher order dynamics, and provide beer approximaions for he DPSD, however compuaions become more involved. In his secion, we consider a simple approximaing mehod which uses a 4 h order sable, minimum phase, real, raional approximae ransfer funcion. In Secion 2.4, we consider he complex cepsrum approximaion algorihm [54], which is based on he Gauss-Newon mehod for ieraive search, and is more accurae bu requires more compuaions. In he simple approximaing mehod, a 4 h order even ransfer funcion S ( s), is used o approximae he deerminisic cellular DPSD, Ss. ( ) The approximae funcion Ss () = HsH () ( s) is given by 2 K K Ss () =, Hs () s + 2 ω ( 2 ζ ) s + ω = s + 2ζω s+ ω (2.28) n n n n where Ss ( ) is he approximaion of Ss. ( ) Equaion (2.28) has hree arbirary parameers { ζ, ω, K } n, which can be adjused such ha he approximae curve coincides wih he acual curve a differen poins. The reason for presening 4 h order approximaion of he DPSD is ha we can compue explici expressions for he consans {,, K } ζ ω as funcions of specific poins on he daa-graphs of he DPSD. In fac, if he approximae densiy S ( f) coincides wih he exac densiy S( f ) a f = and f = fmax, hen he arbirary parameers {,, K } ζ ω are compued explicily as n n S() 2π f ζ = ω = = ω 2 ( ) 2ζ max 2, n, K () 2 n S S f max (2.29) Figure 2.6 shows he DPSD, S( f ), and is approximaion S( f ) via a 4 h order even funcion. 33

51 S D (f), H(jω) 2 in db S D (f) H(jω) [a] v = 5 m/h, f c = 9 MHz, β m = ; Frequency Hz. S D (f), H(jω) 2 in db S D (f) H(jω) [b] v = 2 m/h, f c = 9 MHz, β m = ; Frequency Hz. Figure 2.6: DPSD, S ( f ), and is approximaion D 2 ( ω) = H( jω) via a 4 h order ransfer S funcion for mobile speed of (a) 5 m/hr and (b) 2 m/hr. 34

52 2.3.3 Sochasic STF Channel Models A sochasic realizaion is used here o obain a sae space represenaion for he inphase and quadraure componens []. The SDE, which corresponds o H( s ) in (2.28) is given by d x( ) + 2 ζω dx( ) + ω x( ) d = KdW( ), x (), x() aregiven (2.3) 2 2 n n where { dw () } is a whie-noise process. Equaion (2.3) can be rewrien in erms of inphase and quadraure componens as d x ( ) + 2 ζω dx ( ) + ω x ( ) d = KdW ( ), x (), x () aregiven 2 2 I n I n I I I I d x ( ) + 2 ζω dx ( ) + ω x ( ) d = KdW ( ), x (), x () aregiven 2 2 Q n Q n Q Q Q Q (2.3) where { I ()} dw and { dw ()} Q whie Gaussian noises. are wo independen and idenically disribued (i.i.d.) Several sochasic realizaions [] can be used o obain a sae-space represenaion for he in-phase and quadraure componens of STF channel models. For example, he sochasic observable canonical form (OCF) realizaion [] can be used o realize (2.3) for he inphase and quadraure componens for he jh pah as () () () () () () ( ) () () () ( ) ( ) dx,, I j X I j X I I, j = + (),, 2 2 d 2 dwj 2 dx ω 2ξ ω, n n n X, K X, I j I j I j X I, j I I j() = [ ] + f (), 2 j XI, j dx, Q j X Q, j X, Q Q j = d 2 (),, 2 dx ω 2ξ ω Q, j n n n X Q, j K dwj X Q, j( ) () () X Q, j Q Qj() = [ ] + f 2 j XQ, j () ( ) (2.32) 35

53 where I j () and Qj () corresponds o he inphase and quadraure componens of he jh { j } I pah respecively, W () { j } Q and W ( ) are independen sandard Brownian moions, which correspond o he inphase and quadraure componens of he jh pah respecively, he parameers {,, K } I j () ζ ω are obained from approximaing he DPSD, n Q f and f () are arbirary funcions represening he LOS of he inphase and j quadraure componens respecively, characerizing furher dynamic variaions in he environmen. { I Q } {, } Le us denoe (), ( ) X X he sae vecors for he inphase and quadraure componens, respecively, and ( ) ( ) I Q he inphase and quadraure componens of he channel, hen (2.32) for he jh pah can be wrien in compac form as () () () () () () ( ) () () () ( ) () dx I AI X I BI dwi = d dx A + Q B Q X Q Q dwq I CI XI fi = Q C + Q XQ fq (2.33) where AI = AQ =,, 2 BI = BQ = CI = CQ = ωn 2ξ K nωn [ ] (2.34) { I (), Q() } W W are independen sandard Brownian moions which are independen of he iniial random variables X ( ) and X ( ), and ( ) ( ) I Q { I, Q ; } f s f s s are random processes represening he inphase and quadraure LOS componens, respecively. The band-pass represenaion of he received signal corresponding o he jh pah is given as: ( I I I ) ωc ( Q Q() Q() ) ω c l( τ j) () () () y = C X + f cos C X + f sin s + v( ) (2.35) 36

54 where v ( ) is he measuremen noise. As he DPSD varies from one insan o he nex, he channel parameers {,, K } ζ ω also vary in ime, and have o be esimaed on-line n from ime domain measuremens. Wihou loss of generaliy, we consider he case of fla fading, in which he mobile-o-mobile channel has purely muliplicaive effec on he signal and he mulipah componens are no resolvable, and can be considered as a single pah [5]. The frequency selecive fading case can be handled by including muliple imedelayed echoes. In his case, he delay spread has o be esimaed. A sounding device is usually dedicaed o esimae he ime delay of each discree pah such as Rae receiver [6]. Following he sae space represenaion in (2.33) and he band pass represenaion of he received signal in (2.35), he fading channel can be represened using a general sochasic sae space represenaion of he form () = ( ) ( ) + ( ) ( ) () = () () + () () dx A X d B dw y C X D v (2.36) where T AI BI X () = XI () XQ(), A() =, B() =, AQ() BQ() ( ) = cos( ω ) sin ( ω ), ( ) = cos( ω ) sin ( ω ) C c CI c C Q D c c ( ) T I Q () = () (), () = () () T I Q v v v dw dw dw ( ) (2.37) In his case, y () represens he received signal measuremens, X ( ) is he sae variable of he inphase and quadraure componens, and v( ) is he measuremen noise. Time domain simulaion of STF channels can be performed by passing wo independen whie noise processes hrough wo idenical filers, H ( s), obained from he facorizaion of he deerminisic DPSD, one for he inphase and he oher for he quadraure componen [9], and realized in heir sae-space form as described in (2.33) and (2.34). 37

55 Example 2.3: Consider a fla fading wireless channel wih he following parameers: o f = 9MHz, v = 8 m/h, β = c I Q m, and ( ) = ( ) = f f. Time domain simulaion of he inphase and quadraure componens, aenuaion coefficien, phase angle, inpu signal, and received signal are shown in Figures The inphase and quadraure componens have been produced using (2.33) and (2.34), while he received signal is reproduced using (2.35). The simulaion of he dynamical STF channel is performed using Simulin in Malab [8]. In he nex chaper, we propose o esimae he channel parameers as well as he inphase and quadraure componens direcly from received signal measuremens, using he EM algorihm ogeher wih he Kalman filer. j j Soluion o he Sochasic Sae Space Model The sochasic ime varying sae space model described in (2.36) has a soluion given by [, 2] X L( ) = Φ L(, ) XL( ) + ΦL ( u, ) BL( u) dwl( u) (2.38) where L = I or Q, Φ (, ) is he fundamenal marix, and Φ (, ) = A( ) Φ (, ) wih iniial condiion (, ) L Φ I L =, where I is he ideniy marix. A simple compuaion shows ha he mean of X L ( ) is given by [] ( ) =Φ (, ) ( ) E X E X and he covariance marix of X L ( ) is [] L L L L L L (2.39) T T T Σ L( ) = Φ L(, ) Var X L( ) + ΦL ( u, ) BL( u) BL ( u) ( ΦL ( u, ) ) du ΦL(, ) (2.4) A simple differeniaion of expression (2.4) shows ha he covariance marix Σ ( ) saisfies he Riccai equaion L 38

56 2 In-phase 2 Quadraure I() Q() sqr(i 2 ()+Q 2 ()) 2 an - [Q()/I()] ime [sec.] -.5. ime [sec.] Figure 2.7: Inphase and quadraure componens, aenuaion coefficien, and phase angle of he STF wireless channel in Example

57 3 Channel Envelop; v = 8 m/h, β m = o r() = sqr(i 2 ()+Q 2 ()) r() [db s] ime [sec.] Figure 2.8: Aenuaion coefficien in absolue unis and in db s for he STF wireless channel in Example Fla-Slow Fading s l () y() Fla-Fas Fading s l () y() ime [sec.] -.5. ime [sec.] Figure 2.9: Inpu signal, sl ( ), and he corresponding received signal, y (), for fla slow fading (op) and fla fas fading condiions (boom). 4

58 T T () A() ( ) ( ) A ( ) B( ) B ( ) Σ = Σ +Σ + (2.4) L L L For he ime invarian case, A ( ) = A and B ( ) B, simplify o L L L = expressions (2.38)-(2.4) L A ( ) A ( u) L() L( ) L L () ( ) ( ) ( ) L L X = e X + e B dw u A L = L L E X e E X T AL( ) AL AL u L() L( ) L L AL u T ( ) ( ) T ( ) Σ = e Var X e + e B B e du (2.42) I can be seen from (2.39) and (2.4) ha he mean and variance of he inphase and quadraure componens are funcions of ime. Noe ha he saisics of he inphase and quadraure componens, and herefore he saisics of he STF channel, are imes varying. Therefore, hese sochasic sae space models reflec he TV characerisics of he STF channel. As described above, he channel parameers are obained from approximaing he deerminisic DPSD. However, in realiy one can no have access o he DPSD on-line and a all imes during he esimaion process. In he nex chaper, we propose o esimae he channel parameers as well as he inphase and quadraure componens direcly from received signal measuremens, which are usually available or easy o obain in any wireless newor. The EM algorihm and Kalman filering are employed in he channel parameer and sae esimaion, respecively. This esimaion algorihm is described in Chaper 3. The TV STF channel model is used o generae he lin gains for he proposed PCA which is inroduced in Chaper 4. Following he same procedure in developing he STF channel models, he TV sochasic ad hoc channel models are developed in he nex secion. 4

59 2.4 Sochasic TV Ad Hoc Channel Modeling 2.4. The Deerminisic DPSD of Ad Hoc Channels Dependen on mobile speed, wavelengh, and angle of incidence, he Doppler frequency shifs on he mulipah rays give rise o a DPSD. The cellular DPSD for a received fading carrier of frequency f c is given by [4], f f 2 c < f S( f ) f f c = pg / π f f, oherwise (2.43) where f is he maximum Doppler frequency of he mobile, p is he average power received by an isoropic anenna, and G is he gain of he receiving anenna. For a mobile-o-mobile (or ad hoc) lin, wih f and f 2 as he sender and receiver s maximum Doppler frequencies, respecively, he degree of double mobiliy, denoed by α is defined by min ( f, f )/ max ( f, f ) α =, so α, where α = corresponds o a 2 2 full double mobiliy and α = o a single mobiliy lie he cellular lin, implying ha cellular channels are a special case of mobile-o-mobile channels. The corresponding deerminisic mobile-o-mobile DPSD is given by [52, 53, 9, 92, 96] ( pg) S ( f ) 2 2 / π f m 2 + α f f c K, f fc < ( + α ) f = 2 α ( α ) f + m α, oherwise m (2.44) where () K. is he complee ellipic inegral of he firs ind, and f max ( f, f ) m =. Figure 2. shows he deerminisic mobile-o-mobile DPSDs for differen values of α s. Thus, a generalized wo-dimensional (2D) DPSD has been found where he U-shaped specrum of cellular channels is a special case. 2 42

60 alpha = alpha =.25, f2 = 4*f alpha =.5, f2 = 2*f alpha = Figure 2.: Ad hoc DPSD for differen values of and pg = π. α 's, wih parameers fc =, f =, 43

61 Here, we follow he same procedure in deriving he sochasic STF channel models in Secion 2.3. The deerminisic ad hoc DPSD is firs facorized ino an approximae nh order even ransfer funcion, and hen use a sochasic realizaion [] o obain a sae space represenaion for inphase and quadraure componens. The complex cepsrum algorihm [54] is used o approximae he ad hoc DPSD. This algorihm is discussed nex Approximaing he Deerminisic Ad Hoc DPSD Since he ad hoc DPSD is more complicaed han he cellular one, we propose o use a more complex and accurae approximaing mehod: The complex cepsrum algorihm [54]. I uses several measured poins of he DPSD insead of jus hree poins as in he simple mehod (described in Secion 2.3.2). I can be explained briefly as follows: On a log-log scale, he magniude daa is inerpolaed linearly, wih a very fine discreizaion. Then, using he complex cepsrum algorihm [54], he phase, associaed wih a sable, minimum phase, real, raional ransfer funcion wih he same magniude as he magniude daa is generaed. Wih he new phase daa and he inpu magniude daa, a real raional ransfer funcion can be found by using he Gauss-Newon mehod for ieraive search [56], which is used o generae a sable, minimum phase, real raional ransfer funcion, denoed by H ( s), o idenify he bes model from he daa of H( f ) as where min ba, l w ( f ) ( ) ( ) 2 H f H f (2.45) = b s... bs b H ( s) = m s + a s + + as+ a (2.46) m m m m... = {,..., }, a= { a,..., a }, ( ) b b b m m w f is he weigh funcion, and l is he number of frequency poins. Several varians have been suggesed in he lieraure, where he 44

62 weighing funcion gives less aenion o high frequencies [56]. This algorihm is based on Levi [55]. Figure 2. shows he DPSD, S( f ), and is approximaion S ( f) via differen orders using complex cepsrum algorihm. Higher order of S ( f), beer approximaion obained. I can be seen ha approximaing by a 4 h order ransfer funcion gives very good resuls. Figure 2.2(a) and 2.2(b) show he DPSD, S( f ), and is approximaion S ( f) using he complex cepsrum and simple approximaion mehods, respecively, for differen values of α 's via 4 h order even funcion. I can be noiced ha he former gives beer approximaion han he laer; since i employs all measured poins of he DPSD insead of jus hree poins in he simple mehod Sochasic Ad Hoc Channel Models The same procedure as he cellular case is used o represen mobile-o-mobile channels. The sochasic OCF [] is used o realize (2.46) for he inphase and quadraure componens as I I, j() = I I, j( ) + I j( ) I () =, () + () Q Q, j() = Q Q, j() + Q j () Q () = () + () dx A X d B dw I C X f j I I j j dx A X d B dw Q C X f j Q Q, j j (2.47) where 2 m T 2 m () = (), (),, (), () = (), (),, () X I, j XI, j XI, j XI, j XQ, j XQ, j XQ, j XQ, j bm (2.48) AI = AQ =, BI = BQ =, CI = CQ = [ ] b a a a a b 2 m T 45

63 4.5 4 alpha = S(f) Original S(f) 2 Appr. wih order = 2 Appr. wih order = 4 Appr. wih order = Frequency (Hz) Figure 2.: DPSD, S( f ), and is approximaions, S ( f), using complex cepsrum algorihm for differen orders of S ( f). 46

64 Magniude (W) alpha =.5 alpha =.33 S(w) Appr. S(w) alpha =.25 alpha = Frequency (Hz) (a) Magniude (W) alpha =.5 alpha =.33 alpha =.25 S(w) Appr. S(w) alpha = Frequency (Hz) Figure 2.2: DPSD, S( f ), and is approximaion, S ( f ) (b), via 4 h order funcion for differen α s using (a) he complex cepsrum, and (b) he simple approximaion mehods. 47

65 X and X, () are sae vecors of he inphase and quadraure componens of he jh I, j() Q j pah. I j () and Qj () corresponds o he inphase and quadraure componens of he jh { j } I pah respecively, W () { j } Q and W ( ) are independen sandard Brownian moions, which correspond o he inphase and quadraure componens of he jh pah respecively, he parameers { am,..., a, bm,..., b} I Q of he ad hoc DPSD, and f ( ) and f ( ) j are obained from he approximaion j are arbirary funcions represening he LOS of he inphase and quadraure componens respecively, characerizing furher dynamic variaions in he environmen. Equaion (2.47) for he inphase and quadraure componens of he jh pah can be described as in (2.33), and he soluion of he ad hoc sae space model in (2.47) is similar o he one for STF model described in Secion The mean and variance of he ad hoc inphase and quadraure componens have he same form as he ones in he STF case in (2.39) and (2.4), which show ha he saisics are funcions of ime. The general TV sae space represenaion for he ad hoc channel model is similar o he STF sae space represenaion in (2.36) and (2.37). As he DPSD varies from one insan o he nex, he channel parameers { a,..., a, b,..., b } m m also vary in ime, and have o be esimaed on-line from ime domain measuremens. Example 2.4: Consider a mobile-o-mobile channel wih parameers v = 36 m/hr ( m/s) and v 2 = 24m/hr (6.6m/s), in which α =.66. Figure 2.3 shows ime domain simulaion of he inphase and quadraure componens, and he aenuaion coefficien. The inphase and quadraure componens have been produced using (2.47) and (2.48), while he received signal is reproduced using (2.35). In Figure 2.3 Gauss- Newon mehod is used o approximae he deerminisic DPSD wih 4 h order ransfer funcion. The simulaion is performed using Simulin in Malab [8]. 48

66 2 In-phase 2 Quadraure I() Q() Aenuaion Coefficien Aenuaion Coefficien 2 r().5.5 r() [db] Time (sec) Time (sec) Figure 2.3: Inphase and quadraure componens { I(), Q() }, and he aenuaion coefficien r = I + Q, for a mobile-o-mobile channel wih α = n() n() n() 49

67 2.5 Lin Performance for Cellular and Ad Hoc Channels Now, we wan o compare he performance of mobile-o-mobile lin wih a cellular lin. For simpliciy we will consider he case of fla fading, in which he ad hoc channel has purely muliplicaive effec on he signal and he mulipah componens are no resolvable. Thus i can be considered as a single pah [5]. We consider BPSK is he modulaion echnique and he carrier frequency is f = 9MHz. We es frames of P = bis each. We assume mobile nodes are vehicles, wih he consrain ha he average speed over he mobile nodes is 3 m/hr. This implies v + v 2 = 6m/hr, hus for a mobile-o-mobile lin wih α = we ge v = 6m/hr and v 2 =. The cellular case is defined as he scenario where a lin connecs a mobile node wih speed 3 m/hr o a permanenly saionary node, which is he base saion. Thus, here is only one mobile node, and he consrain is saisfied. We consider he NLOS case ( f = f = ), which represens an environmen wih large obsrucions. The sae space models developed in (2.32) and (2.47) are used for simulaing he inphase and quadraure componens for he cellular and ad hoc channels, respecively. The complex cepsrum approximaion mehod is used o approximae he ad hoc DPSD wih 4 h order sable, minimum phase, real, and raional ransfer funcion. The received signal is reproduced using (2.35). Figure 2.4 shows he aenuaion coefficien, 2 2 () () () r = I + Q, for boh he cellular case and he wors-case mobile-o-mobile case ( α = ). I can be observed ha a mobile-o-mobile lin suffers from faser fading by noing he higher frequency componens in he wors-case mobile-o-mobile lin. Also i can be noiced ha deep fading (envelope less han 2 db) on he mobile-o-mobile lin occurs more frequenly and less bursy (48 % of he ime for he mobile-o-mobile lin and 32 % for he cellular lin). Therefore, he increased Doppler spread due o double mobiliy ends o smear he errors ou, causing higher frame error raes. Consider he daa rae given by R = P/ T = 5 Kbps which is chosen such ha he b coherence ime T c equals he ime i aes o send exacly one frame of lengh P bis, a condiion where variaion in Doppler spread grealy impacs he frame error rae (FER). c c I Q 5

68 r() [db] r() [db] % Cellular Time (sec) 48% Ad hoc Time (sec) Figure 2.4: Rayleigh aenuaion coefficien for cellular lin and wors-case mobile-omobile lin. 5

69 Figure 2.5 shows he lin performance for frames of bis each. I is clear ha he mobile-o-mobile lin is worse han he cellular lin, bu he performance gap decreases as α. This agrees wih he main conclusion of [53], ha an increase in degree of double mobiliy miigaes fading by lowering he Doppler spread. The gain in performance is nonlinear wih α, as he majoriy of gain is from α = o α =.5. Inuiively, i maes sense ha lin performance improves as he degree of double mobiliy increases, since mobiliy in he newor becomes disribued uniformly over he nodes in a ind of equilibrium. 52

70 alpha = alpha =.5 alpha = Cellular FER E b /N (db) Figure 2.5: FER resuls for Rayleigh mobile-o-mobile lin for differen α s compared wih cellular lin. 53

71 Chaper 3 Wireless Channel Esimaion and Idenificaion via he Expecaion Maximizaion Algorihm and Kalman Filering Since he developed models are based on sae space represenaions, we propose o esimae he channel parameers and saes direcly from received signal level measuremens, which are usually available or easy o obain in any wireless newor. A filer-based expecaion maximizaion (EM) algorihm [79, 8] and Kalman filer [8] are employed in he esimaion process. These filers use only he firs and second order saisics and are recursive and herefore can be implemened on-line. The sandard EM algorihm [82] has a wide range of applicaions, for example, in he esimaion of speech signals in acousic environmens [83], in localizaion of narrowband sources [84] and in speech coding [85] o cie a few. The proposed models and esimaion algorihms are esed using received signal level measuremen daa colleced from cellular and ad hoc experimenal seups. Pars of he resuls presened here have been published in [5, 7, ]. 3. The EM Togeher wih he Kalman Filer Consider a discree-ime sae space represenaion given by x = Ax + + Bw y = C x + Dv (3.) 54

72 where n x R is a sae vecor, d y R is a measuremen vecor, m w R is a sae noise, and d v R is a measuremen noise. The noise processes o be independen zero mean and uni variance Gaussian processes. w and v are assumed The sysem parameers θ = { A, B, C, D} as well as he sysem saes x are unnown and can be esimaed hrough received signal measuremen daa, YN = { y, y2,..., yn}. The parameers are idenified using a filer-based EM algorihm and he channel saes are esimaed using he Kalman filer. The Kalman filer is inroduced nex. 3.. Channel Sae Esimaion: The Kalman Filer The Kalman filer esimaes he channel saes x for given sysem parameer θ and measuremens Y. I is described by he following equaions [8, 2] ( ) xˆ = Axˆ + P C D y C Axˆ T 2 xˆ = Axˆ, xˆ = m (3.2) where =,,2,..., N, and P is given by P = P + A B A T 2 P = C D C + B B P A B T T 2 P = AP A + B T 2 (3.3) where B = BB, and D 2 = DD T. The channel parameers θ = { A, B, C, D} are 2 T esimaed using he EM algorihm which is inroduced nex Channel Parameer Esimaion: The EM Algorihm The filer-based EM algorihm uses a ban of Kalman filers o yield a maximum lielihood (ML) parameer esimae of he Gaussian sae space model [79]. The EM algorihm is an ieraive numerical algorihm for compuing he ML esimae. Each ieraion consiss of wo seps: he expecaion and he maximizaion sep [79, 8, 82]. The filered expecaion sep only uses filers for he firs and second order saisics. The 55

73 memory coss are modes and he filers are decoupled and hence easy o be implemened in parallel on a muli-processor sysem [79]. The algorihm yields parameer esimaes wih nondecreasing values of he lielihood funcion, and converges under mild assumpions [82]. Le θ = { A, B, C, D} denoe he sysem parameers in (3.), P denoes a fixed probabiliy measure; and { P θ ; } θ Θ denoes a family of probabiliy measures induced by he sysem parameers θ. If he original model is a whie noise sequence, hen { ; } P θ θ Θ is absoluely coninuous wih respec o P [8]. Moreover, i can be shown ha under P we have : x+ w = y = v P (3.4) The EM algorihm compues he ML esimae of he sysem parameers θ, given he daa Y. Specifically, each ieraion of he EM algorihm consiss of wo seps: The expecaion sep and he maximizaion sep. The expecaion sep evaluaes he condiional expecaion of he log-lielihood funcion given he complee daa, which is described by ˆ dp θ Λ ( θ, θ) = Eˆ log Y θ dpˆ θ (3.5) where ˆ θ denoes he esimaed sysem parameers a ime sep. The maximizaion sep finds ( ˆ ) ˆ θ arg max Λ θ, θ (3.6) + θ Θ The expecaion and maximizaion seps are repeaed unil he sequence of model parameers converge o he real parameers. The EM algorihm is described by he following equaions [79, 8] 56

74 ˆ T T A = E xx Y E xx Y = = T (( )( ) ) T T T T T T T ( ( ) ( ) ( ) ( ) ) ˆ 2 B = E x Ax x Ax Y = = E xx A xx xx A A x x A Y + = ˆ T T C = E yx Y E xx Y = = T (( )( ) ) T ( ( ) ( ) ( ) ( ) ) T T T T T T ˆ 2 D = E y Cx y Cx Y = = E yy yx C C yx C xx C Y + = (3.7) where E( ) denoes he expecaion operaor. The sysem (3.7) gives he EM parameer esimaes a each ieraion for he model (3.). Furhermore, since Λ ( θ, ˆ θ ) is coninuous in boh θ and ˆ θ he EM algorihm converges o a saionary poin in he lielihood surface [79, 8, 82]. The sysem parameers { ˆ, ˆ } 2, ˆ, ˆ 2 expecaions as [79] A B C D can be compued from he condiional () T (2) T L = E xqx Y, L = E x Qx Y, = = (3) T T T (4) T T T L = E xrx + x Rx Y, L = E xsy + ysx Y = = (3.8) where Q, R and S are given by T T T T ee i j + eje i ee i j ee i l Q=, R=, S = ; i, j =,2,... n; l =,2,.. d (3.9) in which e i is he uni vecor in he Euclidean space; ha is e i = in he ih posiion, and T elsewhere. For insance, consider he case n = d = 2, hen E xx Y is = 57

75 (3) (3) T L ( R) L ( R2) E xx Y = (3) (3) (3.) = L ( R2) L ( R22) = =. The oher erms in (3.7) can be compued similarly. T where Rij { ee i j /2; i, j,2} () (2) (3) (4) The condiional expecaions {,,, } as follows [79]: L L L L are esimaed from measuremens Y ) Filer esimae of () L : L E x Qx Y Tr N P Tr N P ( ) ( ) () T () () = = = 2 2 = T () T () T () T 2 () T 2 ( 2x P r 2x P r x N x x B AP N P A B x ) = (3.) where Tr( ) denoes he marix race. In (3.), r () and recursions () N saisfy he following T T ( ) 2 ( ) r = A P C D C A r + P Qx P N P C D y C x () () r = Ar () r = m () 2 () () 2 () 2 () T 2 N = B AP N P A B 2Q () N = m m (3.2) 2) Filer esimae of (2) L : L = E x Qx Y = E x Qx Y + E x Qx Y E x Qx Y T T T T θ{ } θ θ{ } (3.3) (2) = = Therefore, (2) L can be obained from () L. 58

76 3) Filer esimae of (3) L : L = E x Rx + x R x Y = Tr N P Tr N P T T T ( ) ( ) ( ) (3) (3) (3) = 2 2 = T (3) T (3) T (3) T 2 (3) T 2 ( 2x P r 2x P r x N x x B AP N P A B x ) = (3.4) In his case, (3) r and (3) N saisfy he following recursions T T ( ) ( ) 2 T ( 2P R 2P B AP R A ) x r = A P C D C A r P N P C D y C x + + (3) (3) r = Ar (3) r = m (3) 2 (3) T 2 T 2 2 T N = B AP N P AB 2RP AB 2B AP R (3) N = m m (3) 2 (3) (3) 2 (3.5) 4) Filer esimae of (4) L : L = E x Sy + y S x Y = x P r x P r T T T T T ( ) ( ) (3.6) (4) (4) (4) = = where (4) r saisfy he following recursions r = ( A P C D C A ) r + 2P Sy (4) (4) r = Ar (4) r = m (4) T 2 (4) (3.7) Using he filers for L ( i =,2,3,4) and he Kalman filer described earlier, he () i sysem parameers { A, B, C, D} θ = can be esimaed hrough he EM algorihm described in (3.7). Numerical and experimenal resuls ha show he viabiliy of he 59

77 above algorihm in esimaing he channel parameers as well as he channel saes from measuremens are discussed in he following secions. 3.2 LTF Channel Esimaion Using he EM Togeher wih Kalman Filering The general spaio-emporal lognormal model in (2.7) can be realized by sochasic sae space represenaion of he from (, τ) = (, τ) (, τ) + (, τ) ( ) X (, τ ) () = () + () X A X B w y s e v (3.8) where A(, τ) = β(, τ ), B(, τ ) = δ (, τ) β(, τ) γ (, τ), w () = [ dw () ] T, s( ) is he informaion signal, () is he TV power loss (PL) in db, and S( ) coefficien, where = ln() / 2. v is he channel disurbance or noise a he receiver, X (, τ ) ( τ ) X, = e is he TV signal aenuaion However, for simpliciy we consider he discree-ime version of (3.8) given by x = + Ax + Bw y = se + Dv x (3.9) where n x R is a sae vecor, d y R is a measuremen vecor, m w R is a sae noise, and d v R is a measuremen noise. Noe ha he sae space model is nonlinear since he oupu equaion in (3.9) is nonlinear. We consider he general form of sae space form since he esimaion algorihm is derived for he general case. However, in our sysem model (3.8), we have n = d = and m = 2. The noise processes w and v are assumed o be independen zero mean and uni variance Gaussian processes. Furher, he noises are independen of he iniial sae x, which is assumed o be Gaussian disribued. 6

78 The unnown sysem parameers { A, B, D} θ = as well as he sysem saes x are esimaed hrough a finie se of received signal measuremen daa, Y { y, y,..., y } N =. The mehodology proposed is recursive and based on he EM algorihm combined wih he exended Kalman filer (EKF) o esimae he channel sae variables. The laer is used due o he nonlinear oupu equaion. The EKF approach is based on linearizing he nonlinear sysem model (3.9) around he previous esimae. I esimaes he channel saes x for given sysem parameer { A, B, D} θ = and measuremens 2 Y. The EKF is similar o he Kalman filer described in (3.2) and (3.3) excep we add he linearizing par o (3.2) and is described by [8] N x ( ) ( ) xˆ = Axˆ + P C D y C Axˆ xˆ T 2 = Axˆ d e C s se x = = dx x = xˆ x = xˆ (3.2) and (3.3) remains he same. The channel parameers { A, B, D} θ = are esimaed based on he EM algorihm, which is described in Secion Now, we consider he esimaion of a LTF wireless channel from received signal measuremens. In paricular, he esimaion includes he channel parameers, channel PL, and received signal. The measuremen daa are generaed by he following sysem parameers 2/ T π γ (, τ) = γm ( τ) +.5e sin, δ (, τ) = 5, β(, τ) =.2, T (3.2) where γ ( ) m τ is he average PL a a specific locaion τ and is chosen o be 25, T is he observaion inerval and is chosen o be.3 millisecond, and he variances of he sae and measuremen noises are -2 and -6, respecively. 6

79 Figure 3. shows he acual and esimaed received signal using he EM algorihm ogeher wih he exended Kalman filer for 5 sampled daa. From Figure 3., i can be noiced ha he received signal have been esimaed wih very good accuracy. Figure 3.2 shows he received signal esimaes roo mean square error (RMSE) for runs. I can be noiced ha i aes jus few ieraions (less han 5) for he filer o converge, and he seady sae performance of he proposed LTF channel esimaion algorihm using he EM ogeher wih Kalman filering is excellen. 3.3 STF Channel Esimaion Using he EM Togeher wih Kalman Filering In his secion, we consider he sochasic STF sae space models in (2.32) and (2.36), and use a se of measuremen daa provided by he Canadian communicaion research cener (CRC) o perform he EM algorihm ogeher wih he Kalman filer in order o esimae he STF channel model parameers as well as he inphase and quadraure componens respecively, which are hen compared o he ones obained from he measuremen daa. However, for simpliciy we consider he discree-ime version of (2.32) and (2.36) given by x = Ax + + Bw y = C x + Dv (3.22) where n x R is a sae vecor, d y R is a measuremen vecor, m w R is a sae noise, and d v R is a measuremen noise. We consider he general form of sae space form since he esimaion algorihm is derived for he general case. However, in our sysem model (2.32), we have n = m = 2, d = and y represens he inphase or quadraure componens corruped by noise. While in (2.36), we have n = m = 4, d = and y represens he band pass received signal. The noise processes w and v are assumed o be independen zero mean and uni variance Gaussian processes. Furher, he noises are independen of he iniial sae x, which is assumed o be Gaussian disribued. 62

80 Received Signal Esimaed.2 Real Samples Figure 3.: Real and esimaed received signal for he LTF channel model..6 RMSE Samples Figure 3.2: Received signal esimaes RMSE for runs using he EM algorihm ogeher wih he exended Kalman filer. 63

81 The unnown sysem parameers { A, B, C, D} θ = as well as he sysem saes are esimaed hrough a finie se of received signal measuremen daa, N {,,..., } Y = y y y. For he sysem model in (2.32), we use measuremen daa for he 2 N inphase componen or he quadraure componen obained separaely. While for he sysem model in (3.36), he measuremen daa correspond o a linear combinaion of he inphase componen and he quadraure componen. The measuremen daa provided by he CRC conain 98 daa files. The number of samples of he inphase and quadraure componens in each daa file is 766. The mehodology proposed is recursive and based on he EM algorihm combined wih he Kalman filer o esimae he channel sae variables. A 4 h order channel model as described in (2.36) is considered. Therefore, he sysem parameers { A, B, C, D} θ = can be represened as x b δ2 δ3 δ 4 a a2 b2 δ22 δ23 δ 24 A =, B =, δ3 δ32 b3 δ34 a a δ δ b δ ( ω ) ( ω ) [ ] C = cos c sin c, D = d d 2 (3.23) Noe ha B in (3.23) is differen from (2.36) and (2.37), since in (2.37) B( ) is bloc diagonal. In (3.23) we made B nonsingular by including some small numbers for he oher enries. As previously menioned, he esimaion of a fla fading wireless channel from received signal measuremen daa is considered. In paricular, he esimaion includes he channel parameers, inphase and quadraure componens, and he received signal, which are hen compared o he ones obained from measuremen daa. Using he measuremen daa, he sample pahs of inphase and quadraure componens are ploed ogeher, and hen compared o he ones obained by viewing he measuremens as corruped by whie noise sequences. Figure 3.3 shows he measured and esimaed inphase and quadraure componens as well as he received signal using he EM algorihm ogeher wih Kalman filer for 4 sampled daa aen from he measuremens 64

82 Inphase.5 Es Acual Quadraure Received Signal Samples Figure 3.3: The measured and esimaed inphase and quadraure componens, and received signal for 4 h order channel model using he EM algorihm ogeher wih Kalman filer. 65

83 of one channel chosen a random. From Figure 3.3, i can be noiced ha he inphase and quadraure componens of he wireless fading channel as well as he received signal have been esimaed wih very high accuracy. I can also be noiced ha he esimaion error decreases as he number of samples increases; his is because he algorihm is recursive and he channel parameers converge o he acual values as more samples are being esimaed. The sysem parameers are esimaed as ˆ A =, * B = * Cˆ = 2 cos( ωc) sin ( ω ), ˆ c D = [.9 ]. ˆ 2, (3.24) Numerical resuls indicae ha he measured daa can be generaed hrough a simple 4 h order discree-ime sochasic differenial equaion. 3.4 Ad Hoc Channel Esimaion Using he EM Togeher wih Kalman Filering In his secion, we consider he mobile-o-mobile sae space model in (2.47) and (2.36), and carry ou an experimen o measure he received signal power of moving sensors in a wireless sensor plaform. Then he EM algorihm ogeher wih Kalman filering is used o esimae he mobile-o-mobile channel parameers as well as he inphase and quadraure componens from he measured received signal. Again, we consider (3.22) as he discree-ime version of (2.47) and (2.36). The unnown sysem parameers { A, B, C, D} θ = as well as he sysem saes received signal measuremen daa, Y { y, y,..., y } N x are esimaed hrough a finie se of =. These measuremens are 2 N 66

84 colleced in he exreme measuremen communicaion cener (EMC 2 ) laboraory a Oa Ridge Naional Laboraory (ORNL) using wo moving wireless sensor nodes. The wireless sensors used in our experimen are Crossbow s TelosB sensor nodes. A single TelosB sensor node is shown in Figure 3.4. I has he following specificaions [3]: IEEE complian, daa rae is 25 bps, carrier frequency is 2.4 GHz, and has USB inerface. These sensors are implemened wih a Chipcon CC242 RF ransceiver chip which provides a buil-in received signal srengh indicaor (RSSI). This indicaor is averaged over 8 symbol periods (28 micro second). The RSSI has a dynamic range of db and is accurae o +/- 6 db. Our experimenal seup consiss of wo moving ransceivers (sensors and 2) and one passive receiver (sensor 3) conneced o a worsaion as shown in Figure 3.5. A each ime sep, sensors and 2 broadcas a pace conaining a source address and he RSSI of he mos recenly received pace from he oher sensor. Sensor 3 never ransmis; raher, i forwards paces from sensors and 2 o a worsaion for analysis. The mobile-omobile channel beween sensor and 2 is ime varying since boh sensors move wih differen (variable) velociies and direcions. Indoor and oudoor environmens are considered as well. In our experimen, he indoor environmen is he laboraory room shown in Figure 3.6, while he oudoor environmen is shown in Figure 3.7. In he esimaion and idenificaion process, a 4 h order mobile-o-mobile channel model as described in (2.47) and (2.48) is considered. Thus, he sysem parameers { A, B, C, D} θ = can be represened as in (3.23). The esimaion includes he channel parameers, inphase and quadraure componens, and he received signal, which are hen compared o he ones obained from measuremen daa. I is assumed ha he received signal measuremen daa are corruped by whie noise sequences. Figures 3.8(a) and 3.8(b) show respecively indoor and oudoor measured and esimaed received signals using he EM algorihm ogeher wih Kalman filer for 5 sampled daa aen from measuremens beween sensor and 2. A a cerain ime insan, indoor sysem parameers are esimaed as 67

85 Figure 3.4: Crossbow TelosB sensor node. Wireless Sensor Wireless Sensor 2 v 2 v Wireless Sensor 3 Wor Saion Figure 3.5: Experimenal seup: Two moving ransceivers (sensors and 2) and one fixed receiver (sensor 3). 68

86 Figure 3.6: Indoor environmen considered in our experimen. Figure 3.7: Oudoor environmen considered in our experimen. 69

87 Received Signal (db) Es. Meas Samples (a) Received Signal (db) Es. Meas Samples (b) Figure 3.8: Measured and esimaed received signal from sensor 2 by using a 4 h order ad hoc channel model for (a) indoor and (b) oudoor environmens. 7

88 ˆ A =, * B = * Cˆ = 2 cos( ωc) sin ( ω ), ˆ c D = [ ]. ˆ 2, (3.25) while oudoor sysem parameers are esimaed as ˆ A =, * B = * Cˆ = 2 cos( ωc) sin ( ω ), ˆ c D = [.725 ]. ˆ 2, (3.26) From Figures 3.8(a) and 3.8(b), i can be noiced ha he received signals from indoor and oudoor environmens have been esimaed wih very high accuracy. I aes a few ieraions (abou 5 ieraions) for he esimaion algorihm o converge. The roo mean square errors (RMSE) for indoor and oudoor environmens are shown in Figure 3.9. I can be seen ha indoor RMSE is higher han he one for oudoor because of reflecions from walls and objecs in indoor environmen. Experimenal resuls indicae ha he measured daa can be generaed hrough a simple 4 h order discree-ime sochasic differenial equaion wih excellen accuracy, and herefore demonsraing he validiy of he mehod. In he nex chaper, we inroduce one imporan applicaion, namely, sochasic PC based on he developed channel models and esimaion algorihms. 7

89 MSE 3 2 Indoor Oudoor Samples Figure 3.9: Received signal esimaes RMSE for indoor and oudoor environmens using he EM algorihm ogeher wih he Kalman filer. 72

90 Chaper 4 Sochasic Power Conrol Algorihms for Time Varying Wireless Newors The aim of he PCAs described here is o minimize he oal ransmied power of all users while mainaining accepable QoS for each user. The measure of QoS can be defined by he SIR for each lin o be larger han a arge SIR. In his chaper, differen PCAs are inroduced based on he TV channel models derived in Chaper 2. A deerminisic PCA (DPCA) is inroduced firs, and hen a sochasic PCA (SPCA) is presened. Boh cenralized and disribued PCAs are considered. Pars of he resuls presened here have been published in [46-48, 68, 4, 6, 7]. 4. Represenaion of TV Wireless Newors Consider a cellular newor wih M mobiles (users) and N base saions. The LTF spaioemporal model described in (2.7) for a cellular newor can be described as ij ( τ) = βij ( τ) ( γij ( τ) ij ( τ) ) + δij ( τ) ij ( ) 2 (, τ) = N ( ) [ ];( σ ),, dx,,, X, d, dw, ij ( ij ij ) X PL d db i j M (4.) where subscrip ij corresponds o he channel parameers beween mobile j and he base saion assigned o mobile i. The signal aenuaion coefficiens S (, ) Xij (, τ ) using he relaion S (, τ ) = e, where = ( ) ij ij τ are generaed ln / 2. Moreover, correlaion beween channels in a muli-user/muli-anenna case can be induced by leing he 73

91 T differen Brownian moions W ij s o be correlaed, i.e., E W( ) W ( ) = Q ( τ ), where ( Wij ) W() (), and Q ( τ ) is some (no necessarily diagonal) marix ha is a funcion of τ and dies ou as τ becomes large. The TV received signal of he ih mobile a is assigned base saion a ime is given by M y () = p () s () S () + n () (4.2) i j j ij i j= where p () is he ransmied power of mobile j a ime, which acs as a scaling on he j informaion signal () j of mobile i. s, and ( ) n is he channel disurbance or noise a he base saion i Similarly, following he same procedure as he LTF newor, he TV STF sae space represenaion in (2.36) for M mobiles and N base saions cellular newor is wrien as ( ) = ( ) ( ) + ( ) ( ) dx A X d B dw ij ij ij ij ij M () = () () () () + () y p s C X n i = i i (4.3) T where X () X () X (), he processes ( ) i Ii Qi X and X () are he channel saes for he inphase and quadraure componens, respecively, beween mobile j and he base saion assigned o mobile i, C( ) [ cosω sinω ] c I i, and i, j M. The TV LTF and STF channel models in (4.) and (4.3) are used o generae he lin gains of wireless newors for he PCAs proposed nex. c Q i 4.2 Cenralized Deerminisic PCA in TV Wireless Newors In his secion, we consider he uplin channel of a cellular newor and we assume ha users are already assigned o heir base saions. The cenralized PC problem for ime invarian channels for he cellular newor can be saed as follows [57] 74

92 M pg i ii min pi subjec o εi, i M M pg + η ( p,... pm ) i= j i j ij i (4.4) where p is he power of mobile i, g > is he ime invarian channel gain beween i ij mobile j and he base saion assigned o mobile i, ε i > is he arge SIR of mobile i, and η i > is he noise power level a he base saion of mobile i. Expression (4.4) for TV LTF and STF wireless newors defined in (4.) and (4.3) respecively, described using pah-wise QoS of each user over a ime inerval [,T] becomes [46] T M min pi () d, subjec o i= ( p,... pm ) T 2 2 () () () i i ii T T M i i i T p s S d () () () + () p s S d n d 2 () () () () p s C X d i i ii T T M i i i () 2 2 () () () + 2 () p s C X d n d 2 ε, for LTF i ε, for STF i (4.5) and i=,, M. A soluion o (4.5) is presened by firs inroducing he communicaion meaning of predicable power conrol sraegies (PPCS). In wireless cellular newors, i is pracical o observe and esimae channels a base saions and hen send he informaion bac o he mobiles o adjus heir power signals { p () } i i= M. Since channels experience delays, and power conrol is no feasible coninuously in ime bu only a discree-ime insans, he concep of predicable sraegies is inroduced [57]. Consider a se of discree-ime sraegies { } + p ( ), = < <... < < <... T. A ime, he base saions observe or esimae he channel informaion i M i= 75

93 { Sij si } ( ), ( ) M i, j= for LTF or { Iij Qij si } ( ), ( ), ( ) M i, j= for STF. Using he concep of predicable sraegy, he base saions deermine he conrol sraegy { pi } i= ( ) M for he nex ime insan. The laer is communicaed bac o he mobiles, which hold hese values during he ime inerval [ ),. A ime, a new se of channel informaion { S ( ), ( )} M ij si, = i j or { ij ( ), ij ( ), i ( )}, = M I Q s is observed a he base saions and he i j ime + conrol sraegies { } p ( ) M + i i= are compued and communicaed bac o he mobiles which hold hem consan during he ime inerval [, ) described in Figure 4.. Such decision sraegies are called predicable. Using he concep of PPCS over any ime inerval [, ] equivalen o +. This procedure is +, equaion (4.5) is min ( ) p + > M i= p i ( ) + subjec o ( ) ( ) (, ) (, ) ( ) + ( ) p ΓG G p η + I (4.6) where 2 2 ( ) () () + T 2 ( ) η ( ) ( ) ( ) ( ) ( ) ( ) ( ) = diag g ( ) g ( ) ( ) + g, : = s S d, i, j M for LTF, ij + j ij : = s j () C () X ij() d, i, j M for STF, + + M + i + i, :,,,,, I + + MM + p : = p,, p,, : = n d, G G, if i = j, + : =, i, j M, gij (, + ),if i j T ( η η ) Γ ( ε ε ) ( ) ( ) ( ) η, : =,,,,, : = diag,,, + + M + M (4.7) 76

94 Mobile Implemens Base Saion Calculaes -3/2 observe => calculae p m (-) - p m (-) S m (-) => p m () S( /2)p m (-) <= -/2 S( /2)p m () Send bac <=p m () p m () p m () S m (-) p m () p m () S m () => p m (+) +/2 <=p m (+) p m (+) p m (+) S m () p m (+) + Figure 4.: Power updae and informaion flow for uplin power conrol using predicable power conrol sraegies (PPCS). 77

95 and diag() denoes a diagonal marix wih is argumen as diagonal enries, and T sands for marix or vecor ranspose. The opimizaion in (4.7) is a linear programming problem in M vecor of unnowns p ( + ). Here [, ] he channel model does no change significanly, i.e., [, ] coherence ime of he channel. + is a ime inerval such ha + should be smaller han he 4.3 Disribued Deerminisic PCA in TV Wireless Newors In his secion, we consider an ieraive disribued version of he cenralized PCA in (4.6). This is convenien for on-line implemenaion since i helps auonomous execuion a he node or lin level, requiring minimal usage of newor communicaion resources for conrol signaling. The ieraive disribued PCA proposed in [2] and [4] can be used o find a disribued version o he cenralized PCA in (4.6). The consrain in (4.6) can be wrien as ( ( ) ( )) ( ) I, +, + + I (, + ) ( + ) I ΓG G p ΓG η (4.8) Defining ( ) F, ΓG (, ) G (, ) and (, ) (, ) ( ) (4.8) can be wrien as + I + + ( (, + ) ) ( + ) (, + ) u Γ G η, + I + + I F p u (4.9) If he channel gains are ime invarian, i.e., F(, + ) = F and u(, ) + = u, hen he power conrol problem is feasible if ρ F <, where ρ F is he Perron-Frobenius eigenvalue of F [2]. I is shown in [2] and [4] ha he following ieraive PCA converges o he minimal power vecor when ρ F < ( ) ( ) p + = Fp + u (4.) However, our channel gains are TV, hus a TV version of he deerminisic PCA (DPCA) 78

96 in (4.) can be defined as ( ) = (, ) ( ) + (, ) p F p u (4.) Clearly, in general he power vecor p ( ) will no converge o some deerminisic consan as in (4.). Raher, in a TV (random) propagaion environmen, i is required ha he power vecor p ( ) converges in disribuion o a well defined random variable. Since (, ) F + is a random marix-valued process, he ey convergence condiion is ha he Lyapunov exponen λ F < [58], where λ F is defined as F F( ) F( ) F ( ) (4.2) λ = lim log,, 2..., + The disribued version of (4.) can be wrien as where ( ) i ( ) ( ) εi pi( + ) = pi( ), i =,..., M (4.3) R R is he insananeous SIR defined by i R ( ) = i M j i pi( ) gii(, + ) ( ) (, ) + η (, ) p g j ij + i +, (4.4) I is shown in [33] ha he performance of he DPCA in (4.3) in erms of power consumpion is no opimal when he channel environmen is TV. Acually, he performance can be severely degraded when PCAs ha are designed for deerminisic channels are applied o TV channels [33]. Therefore, sochasic PCAs (SPCAs) mus be used in order o ensure sable opimal power consumpion. The laer is discussed in he nex secion. 79

97 4.4 Sochasic PCA in TV Wireless Newors The marix (, ) + F in (4.9) has non-negaive elemens and if he SIR arges are feasible, hen i has been shown in [33, 2] ha he power vecor which saisfies he equaliy of (4.9) minimizes he sum of he ransmied power, i.e., he power vecor ( ) + p ha saisfies ( ( + ) ) ( + ) ( + ) I F, p u, = (4.5) is he minimum power vecor. We firs briefly inroduce one basic resul on sochasic approximaion ha will be used o solve for he opimal power vecor in (4.5). Consider an unnown measurable funcion, hx ( ). A zero poin of h, x, is defined by hx ( ) =, and can be calculaed by various rapidly convergen mehods such as Newon s mehod. Assume now ha he observaion funcion is provided by h ( ) bu subjec o an addiive measuremen noise. The h measuremen is given by ( ) ( ) ( ) ( ) where y ( ) is he observaion a he h ime, and ( ) y = h x + ξ (4.6) ξ is he zero mean measuremen error a he h ime and may be dependen on x ( ). In 95, Robbins and Monro [67] suggesed a mehod for soluion of his and a more general problem, which hey called he mehod of sochasic approximaion. The purpose of using sochasic approximaion is o find a zero poin x based on he noisy observaion y ( ). Given an arbirary iniial poin x ( ) and an arbirary sequence of posiive numbers a( ) such ha a( ) =, a( ) 2 = < (4.7) = Then, i is shown in [67] ha he following approximaion sequence, 8

98 ( + ) = ( ) a( ) ( ) x x y (4.8) converges o he zero poin x of h () wih probabiliy one. Since he lin gains are random, i can be assumed ha he random par in he lef hand side of (4.5) can be represened as an addiive zero mean random noise ξ ( ), herefore, applying he sochasic approximaion algorihm in (4.8) o (4.5) we ge ( ) ( ) ( ) ( ) (, ) ( ) (, ) p + = p a + I F p u + (4.9) which can be wrien as ( ) ( ) ( ) (, ) ( ) (, ) ( ) ( ) p + = I a p + a F p + + u + (4.2) Using (4.4), he disribued version of (4.2) is ( ) ( ) ( ) ( ) ( ) ε p ( + ) = a( ) p + a p ( ) (4.2) i i i i Ri If he PC problem is feasible, he disribued SPCA in (4.2) converges o he opimal power vecor when he sep-size sequence saisfies cerain condiions. Two differen ypes of convergence resuls are shown in [59] under differen choices of he sep-size sequence. If he sep-size sequence saisfies a( ) = and a( ) 2 = = <, hen he SPCA in (4.2) converges o he opimal power vecor wih probabiliy one. However, due o he requiremen for he SPCA o rac TV environmens, he ieraion sep-size sequence is no allowed o decrease o zero. So we consider he case where he condiion a( ) 2 < is violaed. This includes he siuaion when he sep-size sequence = decreases slowly o zero, and he siuaion when he sep-size is fixed a a small consan. In he firs case when a( ) slowly, he SPCA in (4.2) converges o he opimal 8

99 power vecor in probabiliy. While in he second case he power vecor clusers around he opimal power [59]. In fac, he error beween he power vecor and he opimal value does no vanish for non-vanishing sep-size sequence; his is he price paid in order o mae he algorihm in (4.2) able o rac TV environmens. This algorihm is fully disribued in he sense ha each user ieraively updaes is power level by esimaing he received SIR of is own channel. I does no require any nowledge of he lin gains and sae informaion of oher users. I is worh menioning ha he proposed disribued SPCA in (4.2) is differen from he algorihm proposed in [33] where wo parameers, namely, he received SIRs ( ) g (, + ), are required o be nown, while only ( ) ii The selecion of an appropriae [, ] performance. For small [, ] i R and he channel gains i R are required in (4.2). + will have a significan impac on he sysem +, he power conrol updaes will be more frequen and hus convergence will be faser. However, frequen ransmission of he feedbac on he downlin channel will effecively decrease he capaciy of he sysem since more sysem resources will have o be used for power conrol. The performance of he proposed PCA is deermined numerically in he nex secion. 4.5 Numerical Resuls In his secion, we consider wo numerical examples o deermine he performance of he proposed PCAs. Example 4. and Example 4.2 consider he TV LTF and STF wireless newors, respecively. Example 4.: The LTF cellular newor has he following seup: Number of ransmiers (mobiles) is M = 24, he informaion signal ( ) i s = for i=,..., M, iniial disances of all mobiles wih respec o heir own base saions d ii are generaed as uniformly independen idenically disribued (iid) random variables (RVs) in [ ] meers, cross iniial disances of all mobiles wih respec o oher base saions dij, i j, are generaed as uniformly iid RVs in [25-55] meers, he angle θ ij beween he direcion 82

100 of moion of mobile j and he disance vecor passes hrough base saion i and mobile j are generaed as uniformly iid RVs in [ 8] degrees, he average velociies of mobiles are generaed as uniformly iid RVs in [4 ] m/hr, all mobiles move a sinusoidal variable velociies around heir average velociies such ha he pea velociy is wo imes he average speed, power pah-loss exponen is 3.5, iniial reference disance from each of he ransmiers is meers, power pah-loss a he iniial reference disance is 67 db, δij (, τ ) = 4 and βij (, ) mean and variance -2 W. τ = 225 for he SDEs, η i s are iid Gaussian RVs wih zero The performance measure is ouage probabiliy (OP). I is defined as he probabiliy ha a randomly chosen lin fails due o excessive inerference [2]. Therefore, smaller OP implies larger capaciy of he wireless newor. A lin wih a received SIR R i, less han or equal o a arge SIR i expressed as O( ε ) Prob{ R ε } ε, is considered a communicaion failure. The OP ( ) i i i O ε is =. The OP is compued using Mone-Carlo simulaions. The arges SIR, ε i, for all users are he same, and varied from 5 db o 35 db wih sep 5 db. For each value of ε i he OP is compued every 5 millisecond, i.e., [, + ] = 5 millisecond. The simulaion is performed for 5 seconds, i.e., [ ] seconds. i, T = 5 In his example, he cenralized DPCA based on PPCS in (4.6) is performed on wo differen TV LTF wireless newors; he sochasic TV channel models in (4.) and he saic models encounered in he lieraure [2]. The OP for he cenralized DPCA using PPCS based on boh sochasic and saic TV LTF channel models are shown in Figure 4.2(a) and 4.2(b), respecively. Figure 4.2 shows how he OP changes wih respec o he arge SIR, ε i, and ime. As he arge SIR increases he OP increases. This is obvious since we expec more users o fail as ε i increases. The OP also changes as a funcion of ime, since mobiles move in differen direcions and velociies. 83

101 (a) (b) Figure 4.2: OP for he cenralized DPCA using PPCS under he TV LTF models in Example 4. for (a) sochasic models, (b) saic models. 84

102 The average OP versus ε i over he whole simulaion ime (5 seconds) is shown in Figure 4.3, which shows ha he performance of PPCS using he sochasic models is on average much beer han ha of saic models. This is because he saic models do no capure he ime variaions of he channels. For example, a db arge SIR, he OP is reduced from.26 for saic models o.8 for TV sochasic ones; his represens an improvemen of over 3%. The PPCS algorihm for sochasic models ouperforms he saic ones by an order of magniude. I can be seen ha as ε i increases he performance gap beween he PPCS using sochasic and saic models decreases. This is because he effec of ε i (required QoS) is dominan. Figure 4.4 shows he average OP over he whole simulaion ime (5 seconds) for higher noise variance ( δ (, τ ) = 28). In his case he sochasic PL (, ) variaions or flucuaions around he average PL γ (, ) X τ have higher τ, since his parameer conrols he insananeous variance of he sochasic PL. The PPCS based on saic models when he acual channels have high variance gives higher OP han when he acual channels have low variance as observed in Figure 4.3 and 4.4. This is due o he fac ha channels wih high variance deviae significanly from he average (saic) channels. For example, a db arge SIR, he OP in he saic case is abou.32 while in he sochasic case, i is abou.2, an improvemen of over 37%. Therefore, he opimal ransmied power for he saic models is no longer opimal when i is used for more realisic sochasic models. Hence, sochasic models provide a far more realisic and efficien opimal conrol. Now, he performance of he disribued DPCA in (4.3) is compared wih ha of he disribued SPCA in (4.2) under sochasic TV LTF channels. Wih he same parameers as Example 4., in addiion o he arge SIRs ε i = 5 for all users and a =., he oal ransmied powers of all mobiles using he disribued DPCA in (4.3) and he SPCA in (4.2) under sochasic TV LTF channels are shown in Figure 4.5. Noe ha he power axis is logarihmic. Clearly, he disribued SPCA using sochasic approximaions provides beer power sabiliy and consumpion han ha of he disribued DPCA described in [2, 4]. 85

103 Ouage Probabiliy PPCS based on saic models PPCS based on sochasic models Targe SIR (db) Figure 4.3: Average OP for TV LTF channel models wih δ ( ) = 4. Performance comparison..5 Ouage Probabiliy PPCS based on saic models PPCS based on sochasic models Targe SIR (db) Figure 4.4: Average OP for TV LTF channel models wih δ ( ) = 28. Performance comparison. 86

104 2 Disribued DPCA Toal Power - Disribued SPCA Time (sec) Figure 4.5: Sum of ransmied power of all mobiles for he disribued DPCA and he disribued SPCA under TV LTF channels. 87

105 Example 4.2: In his example, he OP of he PPCS algorihm for boh Rayleigh and Ricean fading channel models is compued numerically and compared wih ha of no power conrol (NPC). The STF cellular newor has he following feaures: Number of mobiles M, arge SIRs ε i, sep-size sequence ( ) a, [, + ], [ ], T, average velociies of mobiles, and η i s are he same as in Example 4., carrier frequency = 9 MHz, E ii s are uniformly iid RVs in he range [4-6], E ij s (i j) are uniformly iid RVs in [25-5], angles of arrival β m s for each lin are generaed as uniformly iid RVs in [ 6] ij degrees. The OP as a funcion of arge SIR, ε i, and ime for boh PC based PPCS and NPC under a Rayleigh wireless newor are shown in Figure 4.6(a) and 4.6(b), respecively. Similar remars as in he LTF newor in Example 4. sill apply here. The average OP versus ε i over he whole simulaion ime for Rayleigh and Ricean wireless newors is shown in Figure 4.7. The performance of PPCS is compared wih he one for fixed ransmier power (i.e. NPC). Resuls show ha he PPCS algorihm ouperforms he reference algorihm. For example, a 5 db arge SIR, he ouage probabiliy of Rayleigh fla channel is reduced from.6 for NPC case o.3 for PPCS case, his represens an improvemen of 5%. Moreover, he performance of fla Ricean fading is beer han he one for fla Rayleigh fading channels. This is because he exisence of LOS componen in Ricean channels. Figure 4.8 shows he oal ransmied power of all mobiles using he disribued DPCA in (4.3) and he SPCA in (4.2) under sochasic TV STF wireless newor described in Example 4.2. The same conclusion as LTF case, he disribued SPCA using sochasic approximaions provides beer power sabiliy and consumpion han ha of he DPCA. 88

106 (a) (b) Figure 4.6: OP under he dynamical STF channel models in Example 4.2. (a) Using PPCS algorihm. (b) Using NPC. 89