Chapter 3. GPS Signal Propagation Characteristics and its Modeling

Transcription

1 Chapter 3 GPS Sigal Propagatio Characteristics ad its Modelig 3. Itroductio GPS avigatio sigal icludes vital iformatio such as orbital parameters, clock error coefficiets etc. This received sigal is affected by several errors durig its propagatio ad thus it is extremely weak ad is of the order of 0-6 W. The oise level of this sigal is 400 times higher tha the trasmitted sigal (Kapla, 006). The situatio becomes worst particularly whe the GPS receiver is located at urba areas where the multipath effect is predomiat i the code ad carrier phase measuremets. Accurate predictio ad modelig of GPS satellite sigal propagatio characteristics is ecessary i all precise avigatio solutio applicatios such as aircraft avigatio, missile guidace ad surveyig. GPS usage is ot just limited to the aircraft e-route avigatio ad missile guidace where the user receives the satellite sigals from the ope sky. Curretly, GPS has become a essetial utility i the car avigatio, mobile phoes, surveyig ad aircraft ladig applicatios. The sigal propagatio characteristics severely affect the quality, availability ad cotiuity of the system (Braka, 99). The received sigal stregth of a GPS satellite at a give locatio o or ear the earth surface ca be predicted by aalyzig the propagatio characteristics of the chael with a appropriate propagatio model. There are two types of variatios that occur i a GPS sigal whe it propagates from the satellite to receiver o the earth s surface. They are large-scale variatios ad short-term variatios. I this chapter, large-scale ad short-term propagatio characteristics of GPS sigal are modeled ad aalysed by cosiderig various factors like carrier frequecy ad distace betwee obstacle ad receiver. The variatio of sigal quality with respect to user speed is observed usig Rayleigh ad Ricia fadig models. 3. Large-scale Variatios of a GPS sigal

2 The accuracy of the computed positio depeds o the received sigal stregth, which may degrade due to several reasos such as travellig log distaces through vacuum, dese clouds, dust particles, differet layers of the earth s atmosphere such as troposphere, ioosphere, protoosphere etc.,. I additio, the radom fluctuatios i the received sigals due to differet fadig pheomea also affect the sigal quality, system availability ad ultimately a major cause of system outages. The above metioed GPS sigal variatios ca be classified ito two types amely large-scale variatios ad short-term variatios. The large-scale variatios i a sigal are maily due to pathloss ad shadowig. The average value of the sigal stregth at ay poit depeds o distace betwee trasmitter ad receiver, carrier frequecy, type of ateas used atmospheric coditios ad so o. It may also vary because of shadowig caused by terrai ad clutter such as hills, buildigs, ad other obstacles. 3.. Models for Large-scale aalysis Large-scale variatios are very slow ad are calculated over a large area. These variatios are geerally assumed to have log ormal distributio. Empirical chael models have bee the best for aalysis of large-scale variatios. I this chapter, the Hata-Okumura model is preseted which will be used for large-scale variatio aalysis. The pathloss equatios are obtaied for urba, suburba ad rural areas Hata-Okumura Model Hata obtaied mathematical expressios by fittig the empirical curves provided by Okumura. I order to aalyze the GPS sigals i.e., the sigals trasmitted from satellite to GPS receiver, this model may ot be applicable as the distace betwee the GPS receiver ad satellite is aroud 0,00km ad hece the direct lie of sight (LOS) path is ot possible betwee satellite ad GPS receiver due to the various obstacles i the chael like hills, buildigs, trees etc. By applyig Hata-Okumura model, the propagatio characteristics of

3 GPS sigals (L ad L) are aalyzed by cosiderig the trasmitter as oe of the obstacle from where the sigal is reflected ad the height of the obstacle is assumed to be trasmitter height. The followig expressios give by Hata are used for calculatig the pathloss P Lu (db) i urba, suburba ad rural eviromets. For flat urba areas, P Lu log f 3.8logh ah logh log d (3.) c t m t where, fc = carrier frequecy (MHz) h t = height of the obstacle (meters) h m = height of the GPS receiver atea (meters) d = radio path legth (km) a(h m ) = correctio uit for GPS receiver atea height which depeds o the eviromet ad is calculated i db. Pathloss for medium-small city For a medium-small city, the correctio factor a(h m ) i db is give by (M. Hata, 980) a h.log f 0.7h.56log f m 0.8 (3.) For example, if the height of oe of the obstacles from where the GPS sigal is reflected is 30m ad the receiver atea height is.8m, the model gives the followig pathloss equatio with respect to distace at 575.4MHz (L frequecy) for a typical urba area (medium to small city). m P Lu log 0d db, d km, fc MHz (3.3)

4 It is observed that the pathloss at km from the trasmitter is 3.897dB. Similarly, the pathloss equatio for medium to small city with the same atea heights at 7.60MHz (L frequecy) is give as, P Lu log0d db, d km, fc 7. 60MHz (3.4) The pathloss i suburba ad rural areas is less tha i urba areas. Correctios for determiig pathloss i suburba ad rural areas are also determied by Hata as, For suburba area, P ( ) log f Ls PL db u (3.5) For rural area, (3.6) P Lr P Lu ( db) 4.78 log f 8.33log f Accordig to Hata model, at 575.4MHz (L frequecy), the pathloss equatios i suburba areas (P Ls ) ad rural areas (P Lr ) with respect to distace for the same example are give as, P Ls log 0 P Lr 9 35.log 0 d db d db Similarly, at 7.60MHz (L frequecy), the pathloss equatios i suburba areas (P Ls ) ad rural areas (P Lr ) with respect to distace are give as, Pathloss for large city P Ls log 0 P Lr log 0 d db d db For a large city, the correctio factors a (h m ) i db is give by h 9.9log.54 m h. a f c 00MHz m

5 h 3.log.75 m h a m f c 400MHz (3.7) For example, if the height of oe of the obstacles from where the GPS sigal is reflected is 30m ad the receiver atea height is.8m, this model gives the followig pathloss equatio with respect to distace at 575.4MHz (L frequecy) for a typical urba area (large city). P Lu log0d db, d km, fc MHz (3.8) It is observed that the pathloss at km from the trasmitter is 9.68dB. Similarly, the pathloss for a typical urba area (large city) with the same atea heights at 7.60MHz is give by P Lu log0d db, d km, fc 7. 60MHz (3.9) Accordig to Hata model, at 575.4MHz (L frequecy), the pathloss equatios i suburba areas (P Ls ) ad rural areas (P Lr ) with respect to distace are give by P Ls log 0 P Lr log 0 d db d db Similarly, at 7.60 MHz (L frequecy), the pathloss equatios i suburba areas (P Ls ) ad rural areas (P Lr ) with respect to distace are give by P Ls log 0 P Lr log 0 d db d db The model is valid for the followig rage of iput parameters: 50 h m f MHz 600, 30 ht m m 0, dkm 0 00,

6 3... Simplified Pathloss Model The complexity of sigal propagatio makes it difficult to obtai a sigle model that characterizes pathloss accurately across a rage of differet eviromets. Accurate pathloss models ca be obtaied from complex aalytical models or empirical measuremets whe tight system specificatios are met (Braka Vucetic, 99). However, for geeral trade-off aalysis of various system desigs it is sometimes best to use a simple model that captures the essece of sigal propagatio without resortig to complicated pathloss models, which are oly approximatios of the real chael ayway. Thus, the followig simplified model for pathloss as a fuctio of distace (d) is commoly used for system desig. P r P K t a d d 0 (3.0) The atteuatio i db is thus P r d log 0 d 0 db P db K ( db) 0 t a (3.) where P t ad P r are the power trasmitted ad received respectively, K a is a uit less costat that depeds o the atea characteristics ad the average chael atteuatio, d 0 is a referece distace for the atea far field, ad is the pathloss expoet. The values for K a, d 0 ad ca be obtaied to approximate either a aalytical or empirical model. Because of scatterig pheomea i the atea ear field, this model is geerally valid oly for trasmissio distaces d >d 0 where d 0 is typically assumed to be 0m idoors ad 0 00m outdoors. Whe the simplified model is used to approximate empirical measuremets, the value of K a < is sometimes set to the free-space path gai at distace d 0 assumig Omi directioal ateas

7 K a db 0log0 4d 0 (3.) ad this assumptio is supported by empirical data for free-space pathloss at a trasmissio distace of 00m. Alteratively, K a ca be determied by measuremet at distace d 0 or optimized (aloe or together with ) to miimize the mea square error (MSE) betwee the model ad the empirical measuremets. The value of depeds o the propagatio eviromet 3.. Effect of various factors o sigal propagatio I order to desig a propagatio model for the GPS, kowledge about factors cotributig to large-scale variatios is required. I the followig sectios, the effect of various factors o the sigal power is described ad appropriate chael propagatio models are desiged Effect of Distace For the evaluatio of the effect of distace o sigal power, the terrai is cosidered to be quasi-smooth where the average height of surface udulatios is 30m or less. The power received at a give distace from the obstacle varies as per the equatio (3.3) k Pr d e where, k e is a costat whose value depeds o the eviromet, lies i the rage of.5 to Effect of Frequecy The received sigal level also varies as a fuctio of the frequecy. It is iversely proportioal to the frequecy ad is give by

8 P r k f e Table 3. (3.4) where, k e is a costat which depeds o the eviromet ad varies as per Distace from trasmitter (km) Value of at 500-,000MHz bad Value of at,000-,000mhz bad Table 3. Values of expoet for the received sigal as a fuctio of the frequecy Effect of atmosphere At frequecies above 30MHz, it is foud that the troposphere has a icreasig effect o radio sigals ad radio commuicatio systems. The radio sigals are able to travel over greater distaces tha would be suggested by lie of sight calculatios. At times, coditios chage ad radio sigals may be detected over distaces of 500km or eve 000km ad more (Rao, 00). This is ormally by a form of tropospheric ehacemet, ofte called "tropo" for short. Sometimes, sigals may eve be trapped i a elevated duct i a form of radio sigal propagatio kow as tropospheric ductig. This ca disrupt may radio commuicatio liks because iterferece may be ecoutered that is ot ormally preset. The radio sigals are affected by the regio that is below a altitude of about km. As these regios are those that are greatly affected by the weather, there is a strog lik betwee weather coditios ad radio propagatio coditios ad coverage. Whe sigals are propagated over exteded distaces, as a result of ehaced tropospheric propagatio coditios, the sigals are ormally subjected to slow deep fadig. Due to this the sigals are received via a umber of differet paths. As the wids i the atmosphere move the air aroud, it meas that the differet paths will chage over a period of time. Accordigly, the sigals appearig at the receiver will fall i ad out of phase with each other due to the differet ad

9 chagig path legths ad as a result, the stregth of the overall received sigal chages. Factors like raifall storms add up to this fadig effect resultig i radom variatios i the received sigal. The Ioospheric delay due to code ad carrier is equal i magitude but opposite i sig ad is give by (3.5) 40.3TEC 40.3 I I. f f N path e ds where N e = local electro desity (electros/m 3 ) I = delay due to code I = delay due to carrier phase f = carrier frequecy TEC = Total Electro Cotet. The amout of the ioospheric delay or advace of the GPS sigal ca vary from few meters to more tha twety meters withi oe day (Guochag Xu, 003). I this thesis, carrier phase observables are used as they are more precise compared to code measuremets. The dispersive ature of ioosphere allows calculatio of the absolute TEC; if rage measuremets are available o two separate frequecies: TEC 40.3 f f ( PR PR ) (3.6) The relative TEC ca be derived from phased measuremets: TEC 40.3 f f ( ) (3.7)

10 where PR ad PR are the pseudo rage measuremets observed o L & L frequecies, ad are the phase rage measuremets observed o L & L frequecies. Equatios (3.6) ad (3.7) are the basic calculatios used to derive TEC observatios from dual frequecy GPS receivers Compariso of sigal variatios i Urba, Suburba ad Rural Areas The propagatio of radio waves i built-up areas is strogly iflueced by the ature of the eviromet, i particular by the size ad desity of buildigs (P. Osbore, 999). The sigal stregth received by a GPS receiver would deped ot oly o the satellite power, the separatio distace betwee the GPS receiver ad the obstacle, carrier frequecies but also o the terrai factors. Evirometal clutter such as buildigs, tall structures, trees, lakes or other bodies of water, the width of the streets traversed by the GPS receiver, the agle at which the sigal is icidet at the receivig atea ad the directio i which the vehicles travel with respect to the sigal propagatio also effect the sigal stregth. I propagatio studies, a qualitative descriptio of the eviromet is ofte employed usig terms such as rural, suburba, urba ad dese urba. Dese urba areas are geerally defied as beig domiated by tall buildigs, office blocks ad other commercial buildigs whereas suburba areas comprise residetial houses, gardes ad parks. The term rural defies ope farm lad with sparse buildigs, woodlad ad forests. So far, sigal variatios i urba areas are discussed. Because the effect of the evirometal clutter i suburba or rural areas is ot as severe as i urba areas, the average sigal level i these areas is comparatively better. This improvemet i the sigal levels icreases with frequecies, but does ot deped o the distace betwee obstacle ad GPS receivers. 3.3 Short-term variatios of a GPS sigal Short-term variatios are maily due to multipath reflectios ad Doppler shift which degrade the quality of received sigal particularly i urba eviromets. Sice these

11 variatios are rapid ad occur over short distaces, these are termed as short-term variatios. I urba or dese urba areas, there may ot be ay direct lie-of-sight path betwee a satellite ad a receiver atea. Istead, the sigal may arrive at a GPS receiver over a umber of differet paths after beig reflected from tall buildigs, towers, trees etc. Because the sigal received over each path has a radom amplitude ad phase, the istataeous value of the composite sigal is foud to vary radomly about a local mea (Gottapu Sasibhushaa Rao, 0) Multipath propagatio ad effects Multipath error is oe of the predomiat error sources i all GPS applicatios. Particularly, the multipath error has to be precisely estimated i the Global Navigatio Satellite Systems (GNSSs) as it is the major error source that limits the GPS receiver s performace. Wheever, a sigal is trasmitted from a GPS satellite it follows a multiple umber of propagatio paths o its way to receivig atea (Brya, 994). These multiple sigal paths are due to the fact that the sigal gets reflected back to the atea off surroudig objects, icludig the earth s surface (David L.M., 003). The GPS receiver tracks both direct ad reflected sigal compoets. Multipath is sigificat i commuicatio, sice the GPS receiver is usually located at low heights ad surrouded by obstacles. It typically uses Omi directioal (Omi azimuth) patters thus pickig up large umber of echoes. If a sigle pulse is trasmitted over a multipath chael the the received sigal will appear as a pulse trai, with each pulse i the trai correspodig to the lie of sight compoet or a distict multipath compoet associated with a distict scatterer or cluster of scatterers. The time delay spread of a multipath ca result i sigificat distortio of the received sigal. This delay spread equals the time delay betwee the arrivals of the first received sigal compoet (LOS or multipath) ad the last received sigal compoet associated with

12 a sigle trasmitted pulse. If the delay spread is small compared to the iverse of the sigal badwidth, the there is little time spreadig i the received sigal. However, if the delay spread is relatively large the there is sigificat time spreadig i the received sigal which ca lead to substatial sigal distortio. Aother characteristic of the multipath chael is its time-varyig ature. This time variatio arises because of the motio of either the trasmitter or receiver ad hece the locatio of reflectors i the trasmissio path which gives rise to variatio of multipath error. Thus, if pulses are repeatedly trasmitted from a movig trasmitter, chage i the amplitudes, delays ad umber of multipath compoets correspodig to each pulse ca be observed. Trasmitted sigal ad received sigal i a multipath eviromet is show i Figure 3.. Amplitude Amplitude time time Received sigal Trasmitted sigal Figure 3. Trasmitted sigal ad received sigal i a multipath eviromet To compare differet multipath chaels ad to quatify them, some parameters are defied. They all ca be determied from the power delay profile. These parameters ca be broadly categorized ito two types. Time dispersio parameters Frequecy dispersio parameters Time dispersio parameters These parameters iclude the mea excess delay, RMS delay spread ad excess delay spread. The mea excess delay (τ) is the first momet of the power delay profile ad is defied as

13 ak k k ak k p k k P k (3.8) where a k is the amplitude, k is the excess delay, P( k ) is power of the k th multipath sigal. The mea square excess delay spread (τ ) is defied as (3.9) k k P P k k k Sice the RMS delay spread is the square root of the secod cetral momet of the power delay profile, it ca be writte as t (3.0) where is the mea excess delay, Is the mea squared delay, is the secod momet. As a rule of thumb, for a chael to be at fadig, the followig coditio must be satisfied. t Ts 0. where t is the RMS delay spread, T S is the symbol duratio. For this case, o equalizer is required at the receiver.

14 3.3.. Frequecy dispersio parameters To characterize the chael i the frequecy domai, the followig parameters are ecessary. Coherece badwidth (B c ): This is a statistical measure of the rage of frequecies over which the chael ca be cosidered to pass all the frequecy compoets with almost equal gai ad liear phase. Such a chael is said to be flat. Practically, coherece badwidth is the miimum separatio over which the two frequecy compoets are affected differetly (Gottapu Sasibhushaa Rao, 0). If the coherece badwidth is cosidered to be the badwidth over which the frequecy correlatio fuctio is above 0.9, the it is approximated as B c 50 t where t is the RMS delay spread However, if the coherece badwidth is cosidered to be the badwidth over which the frequecy correlatio fuctio is above 0.5, the it is defied as B c 5 t The coherece badwidth describes the time dispersive ature of the chael i the local area. A more coveiet parameter to study the time variatio of the chael is the coherece time. This variatio may be due to the relative motio betwee the GPS receivers ad the satellite or the motio of the objects i the chael. Coherece time (T c ): I commuicatio system, a commuicatio chael may chage with time. This is a statistical measure of the time duratio over which the chael impulse respose is almost ivariat. Whe chael behaves like this, it is said to be slow faded. Essetially it is the miimum time duratio over which the two received sigals are affected

15 differetly. For example, if the coherece time is cosidered to be the badwidth over which the time correlatio is above 0.5, the it ca be approximated as T c 9 6f m 3.3. Doppler shift (f d ) where f m is the maximum Doppler spread ad is give by f m vr Cosider a GPS receiver movig with a costat velocity, v r, alog a path segmet havig legth d betwee the poits X ad Y, while it receives sigals from a remote source S, the differece i path legth travelled by the wave from source S to the GPS receiver at poits X ad Y is l d cos vrt cos where t is the time required for the GPS sigal to travel from X to Y, is the spatial agle betwee the directio of motio of GPS receiver ad the directio of arrival of wave, vr is assumed to be same at poits X ad Y sice the source is assumed to be very far away. The phase chage ( ) i the received sigal due to the differece i path legths ( l ) is therefore give by l vt cos (3.) ad hece the apparet chage i frequecy or Doppler shift is give by f d where f d vr cos t (3.) It is observed from Eq. (3.) that if the GPS receiver is movig towards the directio of arrival of the wave, the Doppler shift is positive (i.e., the apparet received frequecy is

16 icreased as the spatial agle is reduced) ad if the GPS receiver is movig away from the directio of arrival of the wave, the Doppler shift is egative (i.e., the apparet received frequecy is decreased as the spatial agle is icreased). Multipath compoets from a cotiuous wave sigal which arrive from differet directios cotribute to Doppler spreadig of the received sigal, thus icreasig the sigal badwidth. Figure 3. shows a Doppler power spectral desity s(v) plotted as a fuctio of Doppler-frequecy shift v. For the case of the dese-scatterers model, a vertical receiver atea with costat azimuthal gai, a uiform distributio of sigals arrivig at all arrival agles throughout the rage (0,π) ad a u-modulated CW sigal, the sigal spectrum at the atea termials is s v f d v d (3.3) where d is the Doppler shift i radias. The equality holds for frequecy shifts of that are i the rage ± d about the carrier frequecy f c, ad would be zero outside that rage. The shape of the RF Doppler spectrum described by Eq. (3.3) is classically bowlshaped, as show i Figure 3..

17 Figure 3. Power spectral desity of a wave with frequecy f c which udergoes a Doppler shift Fadig Fadig is the term used to describe the fluctuatios i the evelope of a trasmitted radio sigal. Fadig is a commo pheomeo i wireless commuicatio chaels which is caused by the superpositio of two or more versios of the trasmitted sigals which arrive at the receiver at slightly differet times. The resultat received sigal varies widely i amplitude ad phase depedig o various factors such as the relative propagatio time of the waves ad badwidth of the trasmitted sigal (Gottapu Sasibhushaa Rao, 0).. At the receiver, these multipath waves with radomly distributed amplitudes ad phases combie to give a resultat sigal that fluctuates i time ad space. Therefore, a receiver at oe locatio may receive a sigal that is much differet from the sigal received by aother receiver located at a short distace away, because of the chage i the phase relatioship amog the icomig radio waves. This causes sigificat fluctuatios i the sigal amplitude. This pheomeo of radom fluctuatios i the received sigal level is termed as fadig.

18 The most commoly kow statistical represetatios of fadig are Rayleigh, Ricia, Nakagami-m, Log Normal, ad Loo distributios. Rayleigh ad Ricia models are the commo short-term fadig models. The Nakagami distributio also falls uder this class. Log-ormal ca be used for large-scale fadig. Loo model combies both short-term fadig ad large-scale fadig. The short-term fluctuatio i the sigal amplitude caused by the local multipath is called short-term fadig ad is observed over distaces of about half a wavelegth. O the other had, log-term variatio i the mea sigal level is called largescale fadig. The latter effect is a result of movemet over distaces large eough to cause gross variatios i the overall path betwee the trasmitter ad the receiver. Large-scale fadig is also kow as shadowig. Short-term fadig ca be further classified as flat or frequecy selective, ad slow or fast (Y. Xie, 000). A received sigal is said to udergo flat fadig, if the radio chael has a costat gai ad a liear phase respose over a badwidth larger tha the badwidth of the trasmitted sigal. Uder these coditios, the received sigal has amplitude fluctuatios due to the variatios i the chael gai over time caused by multipath. Whe there is relative motio betwee the trasmitter ad the receiver, Doppler spread is itroduced i the received sigal spectrum, causig frequecy dispersio. If the Doppler spread is sigificat relative to the badwidth of the trasmitted sigal, the received sigal is said to udergo fast fadig. This form of fadig typically occurs at very low data rates. O the other had, if the Doppler spread of the chael is much less tha the badwidth of the basebad sigal, the sigal is said to udergo slow fadig. Multipath sigals are received i a terrestrial eviromet, i.e., where differet forms of propagatio are preset ad the sigals arrive at the receiver from trasmitter via a variety of paths. Therefore there would be multipath iterferece, causig multipath fadig. Addig the effect of movemet of either trasmitter or receiver or the surroudig clutter to it, the received overall sigal amplitude or phase chages over a small

19 amout of time. Maily this causes the fadig. Received sigal evelope i a typical multipath eviromet is show i Figure 3.3. Received sigal Evelope Fade width Average sigal level Fade level = - R db Fade λ/ Distace from satellite Figure 3.3 Received sigal evelope i a typical multipath eviromet The followig physical factors ifluece short-term fadig i the radio propagatio chael Multipath propagatio: Multipath is the propagatio pheomeo that results i radio sigals reachig the receivig atea by two or more paths. The effects of multipath iclude costructive ad destructive iterferece, ad phase shiftig of the sigal. Speed of the GPS receiver: The relative motio betwee the satellite ad the GPS receiver results i radom frequecy modulatio due to differet Doppler shifts o each of the multipath compoets. Speed of surroudig objects: If objects i the radio chael are i motio, they iduce a time varyig Doppler shift o multipath compoets. If the surroudig objects move at a greater rate tha the GPS receiver, the this effect domiates fadig. Trasmissio Badwidth of the sigal: If the trasmitted radio sigal badwidth is greater tha the badwidth of the multipath chael (quatified by coherece badwidth), the received sigal will be distorted.

20 As the GPS receiver moves through this sigal patter, the amplitude of the received sigal varies, goig alterately through the maxima ad miima. Whe the amplitude falls below a level give with respect to the average value, the GPS receiver is said to be goe ito a fade. The rate, N Y at which the istataeous value of the received sigal goes below the level z = E is called the level crossig rate ad is give by N Y f d E E rms e E E rms (3.4) where f d =( v / )cos is the Doppler shift due to a GPS receiver velocity v ) ad r carrier wavelegth. Average fade duratio (T Y ) of a fade at level z = E is give by ( r T Y f d E E rms e E E rms (3.5) The level crossig rate which is same as the umber of fades per secod, ad the fade duratio deped o the fade level amog may other parameters. Table 3. shows the umber of fades per secod at 575.4MHz (L) ad 7.4MHz (L). The average fade duratio for the same carrier frequecies is give i Table 3.3. Notice that deeper the fade level with respect to the average value of the sigal, the fewer the umber of fades per secod ad the shorter the fade duratio. Vehicle -0 db fades -5dB fades speed (km/h) L frequecy L frequecy L frequecy L frequecy Table 3. Number of fades per secod at L ad L frequecies

21 The type of fadig experieced by the sigal through a GPS receiver chael depeds o the relatio betwee the sigal parameters (badwidth, symbol period) ad the chael parameters (RMS delay spread ad Doppler spread). Hece there are four differet types of fadig. Two types of fadig are due to the time dispersive ature of the chael ad the other two types of fadig are based o the Doppler spread experieced by the sigal. Vehicle speed -0 db fades -5dB fades (km/h) L frequecy L frequecy L frequecy L frequecy Table 3.3 Average fade duratio (i ms) at L ad L frequecies Flat Fadig This type of fadig occurs whe the badwidth of the trasmitted sigal is less tha the coherece badwidth of the chael. Equivaletly if the symbol period of the sigal is more tha the RMS delay spread of the chael, the the fadig is called flat fadig. Thus differet compoets of the sigal will experiece the same magitude of fadig. Thus flat fadig occurs whe B S <<B C where B S is the sigal badwidth ad B C is the coherece badwidth. Ad T S >> where T S is the symbol period ad is the RMS delay spread. I such a case, GPS receiver chael has a costat gai ad liear phase respose over its badwidth Frequecy Selective Fadig Frequecy selective fadig occurs whe the sigal badwidth is more tha the coherece badwidth of the GPS receiver radio chael or equivaletly whe the symbol duratio of the sigal is less tha the RMS delay spread. This type of fadig occurs whe B S >>B C ad T S <<

22 At the receiver, multiple copies of the trasmitted sigal are obtaied, all atteuated ad delayed i time. The chael itroduces iter symbol iterferece. A rule of thumb for a chael to have fadig is T s 0. Frequecy selective fadig affects the differet spectral compoets of a radio sigal uequally. If the sigal atteuates over a portio of the badwidth of the sigal, the fadig is cosidered to be selective i frequecy domai. Frequecy selective fadig o the received sigal occurs whe a radio chael has a costat gai ad liear phase respose, but the chael badwidth is less tha that of the trasmitted sigal. Uder such coditios, the chael impulse respose has a multipath delay spread which is greater tha the reciprocal badwidth of the trasmitted sigal. The received sigal icludes multiple versios of the trasmitted sigal which are faded ad delayed i time, ad hece the received sigal is distorted. Frequecy selective fadig is due to time dispersio of the trasmitted symbols withi the chael, ad the chael iduces iter-symbol iterferece. As this effect varies with frequecy, fadig is differet at differet frequecies ad it is extremely difficult to couter its impact or compesate for the sigal loss Fast Fadig I a fast fadig chael, the chael impulse respose chages rapidly withi the symbol duratio of the sigal. Due to Doppler spreadig, sigal udergoes frequecy dispersio leadig to distortio. Therefore, a sigal udergoes fast fadig if T S >> T C, where T C is the coherece time ad B S << B D, where B D is the Doppler spread Slow Fadig

23 I a slow faded chael, the rate of chage of the chael impulse respose is much less tha the trasmitted sigal. A slow faded chael is almost costat over atleast oe symbol duratio. Hece T S << T C ad B S >> B D. It is clear that the velocity of the user plays a importat role i decidig whether the sigal experieces fast or slow fadig Fadig models of a GPS sigal Fadig chael models are used to view the effect of propagatio eviromet o radio sigals. Fadig chael models are derived by assumig various probability distributios to the sigal parameters like the agle of arrival, path delays etc. such models have lower accuracies but are easy to derive ad may aid i estimatio of chael performace Rayleigh fadig It is a statistical model for aalyzig the effect of propagatio eviromet o a radio sigal (Y. Xie, 000). Rayleigh fadig models assume that the magitude of a sigal that has passed through commuicatio chael will vary radomly or fade accordig to Rayleigh distributio which is give by (3.6) r ( t) c exp j( t cos N where c = radom path gai = agle of icomig wave d = Doppler shift i radias = iitial phase associated with th propagatio path. d )

24 It is a reasoable model for aalyzig tropospheric, ioospheric sigal propagatio ad as well as effect of heavily built up urba eviromets o radio sigals. The evelope of the chael respose will be Rayleigh distributed whe there is zero mea ad phase evely distributed betwee 0 ad radias (Berard Sklar, 997). A GPS receiver determies its positio by estimatig the distace of the receiver from each of several visible satellites. The positio of the satellites is determied from the trasmitted ephemeris. The distace is determied by estimatig the propagatio time delay of the trasmitted code. The positio of the satellite established ad the distace betwee the receiver to each satellite is estimated, the positio of the receiver ca be established by trilateratio methods. The accuracy of the positio estimated is impacted by several factors amog which are the oise i the receiver, the cofiguratio of the satellites, ad the presece of iterferece alog the trasmissio path. The propagatio path betwee the GPS satellite ad receiver is characterized by various obstacles ad reflectios. These have a large ifluece o the received sigal, whe the radio wave is propagated from the GPS satellite to receiver. The radio waves trasmitted from a satellite radiates i all directios, the received waves icludes reflected waves that are reflected off from various obstacles, diffracted waves, scatterig waves, ad the direct wave from the GPS satellite to receiver. It is show i Figure 3.4. Sice the path legths of the direct, reflected, diffracted, ad scatterig waves are differet, the time each takes to reach the GPS receiver will be differet. Direct sigals Reflected sigals

25 Figure 3.4 Priciple of multipath chael. I additio, the phase of the icomig wave varies because of reflectios. As a result, the receiver receives superpositio of several waves havig differet phase ad times of arrival. The geeric ame of a radio wave i which the time of arrival is retarded i compariso with the direct wave is called delayed wave. The receptio eviromet characterized by a superpositio of delayed waves is called a multipath propagatio eviromet. I a multipath propagatio eviromet, the received sigal is sometimes itesified or weakeed. This pheomeo is called multipath fadig ad the sigal level of the received wave chages from momet to momet. A compesatio method for this multipath fadig must be used to esure a high trasmissio performace. A GPS receiver i a typical macro cellular eviromet is usually surrouded by local scatterers so that the plae waves will arrive from so may directios without a direct LOS compoet. Two-dimesioal isotropic scatterig where the plae waves arrivig from all directios with equal probability is a very commoly used scatterig model for the forward chael i a GPS system. For this type of scatterig eviromet, the received evelope is Rayleigh distributed at ay time ad is said to exhibit Rayleigh fadig. Such chaels are modeled usig Rayleigh chael model. Cosider the mechaism by which the fadig occurs. The delayed wave with icidet agle is give by Eq. (3.7) correspodig to Figure 3.4, whe a cotiuous wave of sigle frequecy f c (Hz) is trasmitted from the satellite. where r (t ) = delayed sigal f c = carrier frequecy r jf ct t Re e t e (3.7)

26 Re () idicates the real part of a complex umber that gives the complex evelope of the icomig wave from the directio of the umber j is a complex umber. The e (t) ca be calculated by usig the propagatio path legth of the icomig waves. e t R t e L vrt cos j x t jy t (3.8) where R ad are the evelope ad phase of the th icomig wave respectively x (t) ad y (t) are the i-phase ad quadrature phase factors of e (t) v r = speed of GPS receiver i m/s = wavelegth i meters L = propagatio path legth = icidet agle. The icomig th wave shifts the carrier frequecy as vcos / (Hz) by the Doppler Effect (Hz). This is called the Doppler shift i GPS commuicatio. This Doppler shift which is described as f d has a maximum value of v / whe the icomig wave comes from the ruig directio of the receiver where cos = is the largest Doppler shift. The delayed wave that comes from the rear of the receiver has a frequecy shift of - f d (Hz). It is show by Eq. (3.9) that sice wave received by GPS, r(t) is the sythesis of the icomig waves, the icomig wave umber is made to be N s. Summatio of N s reflected sigals is give by t N s r r t

27 Re (3.9) Re N s e t exp jf t c xt jyt cos f t jsi f t c = Re[ x(t) cos π f c t + j y(t) si π f c t +j x(t) si π f c t + j y(t) cos π f c t ] = x(t) cos π f c t - y(t) si π f c t Quadrature ad i-phase represetatio of r(t) is give by r(t) xt cos f t yt si f t where, x(t) ad y (t) are radom processes give by c c c t N s x x t N s ad t y t y (3.30) where x t ad t y are ormalized radom processes, havig a average value (m r ) of zero ad dispersio of, whe N s is large eough. σ (variace) = E(x-m r ) = E(x) Eq. (3.3) gives the combiatio probability desity p(x, y), where x = x (t), y = y (t) p x, y e x y (3.3) Amplitude ad phase represetatio of the received wave is give by r (t), t Rt f t t r cos c (3.3) R t R y where R(t) is the amplitude, ad is give as ad θ(t) is the phase ad is give as t ta y / x (3.33) x

28 By usig trasformatio of variables, p(x, y) ca be coverted ito p(r, ). Rayleigh distributio of (R, ) is give by R R p R, e (3.34) By itegratig p(r, ) over from 0 to, the probability desity fuctio p(r) is obtaied as p R R e R (3.35) The probability desity fuctio p( ) is obtaied by itegratig p(r, ) over R from 0 to. p (3.36) From these equatios, it is cocluded that fluctuatio i the evelope follows a Rayleigh distributio, ad the phase fluctuatio follows a uiform distributio o the fadig i the propagatio path. I order to fid a expressio for simulatios of this Rayleigh fadig, cosider the GPS receiver that receives the radio wave as show i Figure 3.4, the arrival agle of the receivig icomig wave is uiformly distributed, ad the wave umber of the icomig waves is N. I this case, the complex fadig fluctuatio i a equivalet low pass system is r t xt jyt Cosider a u-modulated sigal S( t) cos( f t) (3.37) c Let = Agle of arrival of th ray at the receiver = Phase shift of th ray due to multipath v r = Velocity of the GPS receiver ad d = Doppler shift i radias.

29 Assume there are N S scatterers i the eviromet; the the received sigal will be of the form (3.38) N s c cos t t cos( ) R t Eq. (3.38) ca be writte as Rt Re c exp t t cos (3.39) (3.40) Reexp c N d N s s t c exp t cos c d c d Cosider the secod product term as the low pass equivalet respose of the chael ad is evaluated as (3.4) N s r( t) c exp j( t cos ) d The distiguishig feature of this type of simulator is that it cotais a low-frequecy oscillator for each Doppler shift d cos( ), i.e., made up of N S oscillators. A geeral relatio betwee the c s ad the pdf of the agles of arriva is give by c f d =,,3...N S (3.4) where f( ) is the pdf of the th agle of arrival sigal ad the c may be iterpreted as the power ratio received withi the small arc d, about the agle of arriva.

30 The first step take by Jakes is to restrict the agles of arriva from beig uiform i.e, f ) f ( )... f N ( N) / to be uiformly spaced. Ad accordig to the ( formula =,, 3 N S N (3.43) This i tur leads to the atteuatio alog the N paths beig equal, i.e., c N s From the above equatio, it is observed that sice the pdf of the agles of arrival f( ) is uiform, the power received i each arc d is same as log as the are uiformly spaced. So Eq. (3.4) becomes r( t) N s N s e j( t cos d ) (3.44) whe expaded the above equatio becomes N s N s r( t) cos( t cos( ) ) j. si( dt cos( ) ) N (3.45) x( t) jy( t) d s Ns N s cos t cos j si t cos d N s d where ad are uiformly distributed over 0 to d = Doppler frequecy shift i radias N S = umber of scatterers.

31 The Doppler frequecy shift is calculated by (3.46) vrc d cos c where v r = GPS receiver velocity = elevatio agle c = velocity of light c = carrier frequecy i radias. The level crossig rate (L Y ) ad average fade duratio (T Y ) are give by the formulae L de Y f f (3.47) T Y f f e d (3.48) Ricia fadig where = fade level f It is a stochastic model for radio propagatio aomaly caused by partial cacellatio of radio sigal by itself. If the eviromet is such that, i additio to scatterig there is a strog domiat sigal see at the receiver caused by lie of sight, the mea of the radom process is o loger zero ad varies aroud the power level of the domiat path. Such a situatio may be better modeled as Ricia fadig. Rayleigh fadig is a specialized model for Ricia fadig whe there is o lie of sight sigal. I a microcellular eviromet, the trasmitter ateas are ofte placed below the skylie of buildigs ad are surrouded by local scatterers, such that the plae waves will arrive at the base statio with a larger AOA (Agle of Arrival) spread. Furthermore, LOS path will sometimes exist betwee the satellite

32 ad receiver, while at other times LOS path does ot exist. Eve i the absece of LOS propagatio coditios, there ofte exist a domiat reflected or diffracted path betwee the satellite ad receiver. LOS or domiat reflected or diffracted path produces the specular compoet ad the multitude of weaker secodary paths cotributes to the scatter compoet of the received evelope. I this type of propagatio eviromet, the received sigal evelope still experieces fadig. However, the presece of the specular compoet chages the received evelope distributio, ad very ofte a Ricia distributed evelope is assumed. I this case the received evelope is said to exhibit Ricia fadig. Such chaels are modeled by Ricia chael model. Ricia distributio is used to model the chael whe a direct lie of sight compoet exists betwee the obstacle ad receiver i additio to the multipath compoets. It ca be expressed as a phasor sum of a costat ad a umber of scatterig poit sources. (3.49) R re j C j A e where, C = costat coheret sigal with clear LOS, A j = Amplitude of j th icomig wave, j = Iitial phase shift of j th icomig wave. Ricia probability desity fuctio is give by (3.50) (3.5) j j r C rc 0 I r p r e, r 0 r 0 p, r 0 j

33 where is the mea square value of the Rayleigh distributed compoet of r ad I 0 is the modified Bessel fuctio of order zero. The phase distributio is o loger uiform like Rayleigh distributio. The phase distributio of Ricia distributio is derived by Beckma as (3.5) p e where C cos G, 0 C G G e erf ad erf G G G e y dy 0 Clarke s simulatio model for simulatig Ricia chael is give by the chael trasfer fuctio which is give by (3.53) (3.54) Y c Y Y t Y t jy t c N s t cos t cos s N s N s t si t cos N s s d d where N s is the umber of propagatio paths, d is the maximum Doppler frequecy i radias, ad are the agle of arrival ad iitial phase of the th propagatio path respectively. Both ad are uiformly distributed over (, ) for all ad they are mutually idepedet. The chael trasfer fuctio after icludig the direct LOS compoet is give by

34 Z t Z t jz t c s Z c Y t K t cos 0 K c cos d 0 (3.55) Z s Y t K t cos 0 K s si d 0 (3.56) where K is the ratio of multipath to direct compoet, 0 ad 0 are agle of icidece ad iitial phase of direct compoet respectively. 3.4 Bit Error Rate Bit error rate is the probability of bit error. This is ofte the figure of merit for a errorcotrol code. To keep this umber small, typically less tha 0-4, differet codig techiques are used. Bit-error rate is a useful idicator of system performace o a idepedet error chael but it has little meaig o burst, or depedet error chaels. Bit error rate Number of message bits corruptedi a codeword Total umber of bits trasmitted i a codeword The error probability is thus bouded above by the probability that more tha t errors occur. Sice the bit errors i a codeword occur idepedetly o a AWGN chael, this probability is give by Pe j t j j p ( p) ( j) (3.57) where, p = the probability of error associated with trasmissio of the bits i the codeword ad = Total umber of bits trasmitted i a codeword The probability of bit error (P b ) obtaied after decodig the received codeword depeds i geeral o the specific code ad decoder ad i particular o how the bits are

35 mapped to code words, as i the bit mappig procedure associated with o biary modulatio. This bit error probability ca be approximated as Pb j p jt j j ( j) ( p) (3.58) From the above, the probability that a decoded codeword cotais a error is upper bouded by P e i t P(i,) ( j) i.e., j Pe j p ( p) jt j For a rough but reasoable performace estimate, P e is approximated as (3.59) ( t) P e P(t+,) p t This meas that a ucorrected word typically has t+ errors. O a average there will be (k/)(t+) message-bit errors per ucorrected word, the remaiig errors beig i check bits. Whe Nk bits are trasmitted i N>> words, expected total umber of erroeous message bits at the output is (k/)(t+) N P e. Hece, the Probability of bit error rate Probability of Number of message bits corruptedi a codeword Total umber of bits trasmitted i a codeword P be = ( k / )( t ) NPe Nk P be = ( t )Pe P be (3.60) j p jt j j ( j) ( p)

36 3.5 Results ad Discussio Pathloss is a importat factor i determiig the user positio i GPS. The pathloss is depedet o various factors like frequecy ad distace betwee the obstacle ad the receiver. The variatios of pathloss i urba, rural ad suburba areas are compared. The aalysis has bee used for calculatig accurate positio of GPS receiver. Medium to small city I a medium to small city, if the height of oe of the buildig from where the GPS sigal is reflected h t = 30m, the receiver atea height h m =.8m ad carrier frequecy f c = 575.4MHz (L frequecy). Accordig to Hata model, pathloss i urba (P Lu ), suburba (P Ls ) ad rural areas (P Lr ) with respect to distace are give by P Lu P Ls P Lr d, log0 db d, log 0 db d, 9 35.log 0 db where d is i km. For the same specificatios, for the carrier frequecy f c = 7.60MHz (L frequecy), pathloss i urba (P Lu ), suburba (P Ls ) ad rural areas (P Lr ) with respect to distace are give by P Lu P Ls d, log0 db d, log 0 db d, P Lr log 0 db where d is i km Distace Pathloss (db) i urba areas (km) L frequecy ( fc=575.4mhz) L frequecy ( fc=7.60mhz)

37 Pathloss (db) Table 3.4 Typical values of pathloss (db) i urba areas of a medium to small city for various distaces at L ad L frequecies. The compariso of pathloss i urba areas as a fuctio of distace at L ad L frequecies is give i Table 3.4. It is observed that pathloss obtaied i urba areas icreases with the distace betwee obstacle ad receiver. The variatio of pathloss as a fuctio of distace i urba areas at L ad L frequecies is show i the Figure 3.5. From the graph, it is observed that the pathloss obtaied at L frequecy is greater tha at L frequecy. 70 Variatio of pathloss with distace i urba areas (Small city) L L Distace (km) Figure 3.5 Variatio of pathloss (db) with distace i urba areas of a medium to small city at L ad L frequecies Distace (km) L frequecy ( fc=575.4mhz) Pathloss (db) i Suburba areas L frequecy ( fc=7.60mhz)

38 Pathloss (db) Table 3.5 Typical values of pathloss (db) i suburba areas of a medium to small city for various distaces at L ad L frequecies. Variatio of pathloss with distace i suburba areas (Small city) L L Distace (km) Figure 3.6 Variatio of pathloss (db) with distace i suburba areas of a medium to small city at L ad L frequecies. Distace Pathloss (db) i rural areas

39 Pathloss (db) (km) L frequecy ( fc=575.4mhz) L frequecy ( fc=7.60mhz) Table 3.6 Typical values of pathloss (db) i rural areas of a medium to small city for various distaces at L ad L frequecies. Variatio of pathloss with distace i rural areas (Small city) L L Distace (km) Figure 3.7 Variatio of pathloss (db) with distace i rural areas of a medium to small city at Distace (km) L ad L frequecies. Pathloss (db) for urba areas Pathloss (db) for Suburba areas Pathloss (db) for rural areas fc = fc = 7.60 fc = fc = 7.60 fc = fc = 7.60

40 Pathloss (db) MHz (L) MHz (L) MHz (L) MHz (L) MHz (L) MHz (L) Table 3.7 Pathloss i urba, suburba ad rural areas of a medium to small city as a fuctio of distace betwee obstacle ad receiver at L ad L frequecies. variatio of pathloss i urba,suburba ad rural areas with distace at L frequecy (Small city) urba sub urba rural Distace (km) Figure 3.8 Compariso of pathloss i urba, suburba ad rural areas of a medium to small city with chage i distace betwee obstacle ad receiver for L frequecy

41 Pathloss (db) variatio of pathloss i urba,suburba ad rural areas with distace at L frequecy (Small city) urba sub urba rural Distace (km) Figure 3.9 Compariso of pathloss i urba, suburba ad rural areas of a medium to small city with chage i distace betwee obstacle ad receiver for L frequecy The compariso of pathloss i suburba, rural areas as a fuctio of distace at L ad L frequecies are give i the tables 3.5 ad 3.6 respectively. From the Figures 3.6 ad 3.7, it is observed that the pathloss i suburba areas ad rural areas at L frequecy is greater tha the pathloss at L frequecy. Pathloss i urba, suburba ad rural areas as a fuctio of distace betwee obstacle ad the GPS receiver for carrier frequecies L ad L is show i Table 3.7. From the Figures 3.8 ad 3.9, it is observed that pathloss values obtaied i urba areas are more compared to the pathloss values i suburba ad rural areas ad the pathloss values obtaied at L frequecy are greater tha the values obtaied at L frequecy. Large city: For the same example, At L frequecy, pathloss equatios with respect to the distace i urba (P Lu ), suburba (P Ls ) ad rural areas (P Lr ) at L frequecy are give accordig to Hata model as,