Deterministic Model of Multipath Fading for Circular and Parabolic Reflector Patterns

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Winfred Austin
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1 To appear i the Proceedigs of the 5 IEEE outheastco, (Ft. Lauderdale, FL), April 5 Determiistic Model of Multipath Fadig for Circular ad Parabolic Reflector Patters Dwight K. Hutcheso dhutche@clemso.edu Daiel L. Noeaker dao@ces.clemso.edu Milto W. Holcombe Departmet of Electrical ad Computer Egieerig Clemso Uiversity Abstract Traditioally, a probabilistic model has bee used to aalyze a wireless commuicatio system s susceptibility to multipath fadig. However, the characteristics of fadig i a actual wireless commuicatio lik deped o the particular determiistic geometry of reflectors that the sigal ecouters. I this paper, we preset a determiistic model of multipath propagatio, ad we apply the model to two reflector geometries: a circular patter of reflectors ad a parabolic patter of reflectors. Numerical results for each case are icluded, ad it is show that the geometry of the reflectors that the sigal ecouters has a substatial effect o the rate of chage i sigal stregth observed by a mobile receiver. 1. Itroductio I a wireless mobile commuicatio system, the received sigal is made up of compoets travelig over multiple propagatio paths of differig legth, ofte with o lie-of-sight compoet. The sigal compoets do ot arrive at the receiver i phase with oe aother, i geeral, ad the relative phase agles ad amplitudes of the compoets at the receiver determie the overall stregth of the received sigal. If the receiver chages locatios, the phase relatioship amog the reflected sigal compoets chages, ad hece, the stregth of the received sigal varies. This pheomeo is referred to as multipath fadig, ad it results i a sigal stregth at the receiver that is a fuctio of its positio. Positiodepedet sigal stregth results i time-depedet sigal stregth if the trasmitter, receiver, or reflectors are i motio. Aalyses of commuicatio-system performace i multipath fadig typically employ a statistical model of the fadig observed at the receiver. The most commoly used models fall withi the class of Gaussia wide-sese statioary ucorrelated scatterig (GWU) chaels itroduced i [1]. Oe popular GWU chael is the so-called Jakes-Clarke model [] which exhibits the log time-correlatio tails that are ofte observed i field measuremets of cellular commuicatio systems. Aother GWU chael with a expoetial time-correlatio fuctio [3] is equivalet to a Gauss-Markov chael, ad it is also widely used i system performace aalysis. These statistical models are favored for use i Mote Carlo simulatio of commuicatio systems because the models are ameable to efficiet software implemetatio ad because they provide commo bechmarks for system desigs. (The Gauss-Markov chael has a particularly simple algorithmic implemetatio.) The relatioship of the statistical models to actual propagatio eviromets that are ecoutered i practice is problematic, however. The Jakes-Clarke model is based o a circular array of ifiitesimal reflectors ad idepedet radom phase shifts of the reflected sigal compoets. May other GWU models, such as the Gauss-Markov model, are ot easily related to ay physical cofiguratio of reflectors. I this paper we examie the effect that the physical cofiguratio of reflectors has o the empirical fadig characteristics observed at the receiver. From this we gai isight ito the extet to which the properties of the chael may deviate from those predicted by the use of a sigle statistical model. I this paper, a model is preseted for the stregth of a sigle-frequecy sigal at the receiver as a fuctio of the receiver s positio i a determiistic multipath eviromet. The model is applied to two reflector geometries: a circular reflector patter ad a parabolic reflector patter. The fadig characteristics are cosidered for the two reflector geometries for a choice of model parameters that allow for a fair compariso betwee them. It is show that there are substatial differeces i the fadig that results from the two reflector geometries. pecifically, there is a greater fluctuatio i the sigal stregth over the same area with a circular reflector patter tha with a parabolic reflector patter of similar dimesios. Thus the fadig characteristics of a mobile receiver of a give velocity is show to deped sigificatly o the particular reflector geometry. 1

2 . Developmet of the Model I this model, a give reflector patter is cosidered ad used to determie the stregth of a sigle-frequecy sigal as a fuctio of receiver positio. Let c be the frequecy of the sigal i radias per secod. Because this is the oly frequecy at which eergy is preset i the sigal, the time-domai expressio for the sigal received at a poit,, i the x-y plae, ca be writte i the form: ε t) E cos( ω t + φ ). (1) ( c Alteratively, it ca be writte the form: jωct ε ( t) Re{ E e }, () where E Ee jφ (3) is the basebad-equivalet complex represetatio of the sigal. The sigal received at ca also be writte as a sum of the compoets travelig alog each reflective path from the trasmitter to the receiver at : E E + E1 + + E 1 + E + (4) uppose the multipath compoet with complex amplitude E is chose as the referece compoet. The the complex amplitude of ay other multipath compoet ca be expressed i terms of E. Let kπfd/c deote the free-space phase costat ad deote the differece i the path propagatio distaces of compoets E ad E. The the sigal compoet E ca be writte as jk E E e. (5) I geeral, a expressio for the sigal at the receiver is obtaied by summig all of the sigal compoets expressed i the form of equatio (5)..1. Circular Reflector Patter The sceario for the circular reflector patter cosists of a trasmitter at a poit Q i the x-y plae o the egative x-axis, which is at a distace q > from the origi, O. The referece reflector is positioed at P, which has a distace r > from the origi ad is o the egative x-axis. Aother reflector is positioed at P, which also has a distace r > from the origi, but is o a lie at a agle with the egative x-axis. The receiver is located at, which is o a lie at a agle with the positive x-axis, at a distace of ρ from the origi. Figure 1. Circular reflector patter I order for the sigal compoet E to be expressed i the form show i equatio (5), the phase shift of E relative to E must be foud. It is assumed that the distace from the trasmitter to the origi is much larger tha the distace from ay reflector to the origi (q>>r). It is also assumed that the distace betwee ay reflector ad the origi is much larger tha the distace betwee the receiver ad the origi (r>>ρ). Uder these assumptios, accurate first-order approximatios result i r ( 1 cosθ ) + ρ cos( θ + η). (6) Therefore, if N reflectors are uiformly distributed aroud the circle cetered at the origi with radius r, the sigal received at ca be writte as jk ( r (1 cosθ ) + ρ cos( θ + η)) E E e. N 1 If equatio (7) is ormalized by N ad N is allowed to approach ifiity we obtai: E E π π e jk ( r(1 cosθ ) + ρ cos( θ + η )) dθ (7). (8) By maipulatig terms ad usig a expressio for the Bessel fuctio [4], equatio (8) ca be simplified oce more to obtai the model equatio for the received sigal at poit due to a ifiitely populated circular patter of reflectors: E ( k r + ρ ρ cosη ) jkr Ee J + r where J is the Bessel fuctio of the first kid of zero order... Parabolic Reflector Patter The sceario for the parabolic reflector patter cosists of a trasmitter at a poit Q i the x-y plae o the egative x-axis, which is at a distace q > from the origi, O. The reflectors i this sceario populate a portio of a parabola described by the equatios x g( ay h (a ad h are positive costats) ad y mi y (9)

3 y max. The referece reflector is located at the vertex of the parabola, ad this poit is deoted P. Aother reflector is located at poit P with coordiates (g(y ), y ). The receiver is located at poit, which has coordiates ( x, y ). Agai we cosider the case i which the distace from the trasmitter to the origi is much greater tha the distace from ay reflector to the origi. Figure. Parabolic reflector patter I the case of the circular reflector patter, the expressio for uiform placemet of reflectors aroud the circle permits a aalysis for the limitig case of a cotiuum of reflectors with uiform reflected sigal desity aroud the circle. A similar approach could be used for the parabolic reflector patter, but it results i aalytical difficulties. Istead a expressio is developed for reflectors that are placed o the parabola at fixed itervals with respect to the y-axis coordiate. The received sigal at reflector P is weighted by the arc legth of the parabola betwee P -1 ad P. This correspods to a uiform reflected sigal desity alog the parabola i the limit as the umber of reflectors approaches ifiity. Aother key differece i the developmet of the model for the parabolic reflector patter versus that of the circular reflector patter is the cosideratio of sigal stregth atteuatio with distace. Because the receiver is ear the ceter of the circle i the circular reflector patter case, all multipath compoets arrivig at the receiver have approximately the same stregth. This is ot true i the case of the parabolic reflector patter. Therefore, a multiplicative path-loss term must be itroduced i the complex amplitude of each received sigal compoet to accout for its positio-depedet atteuatio. This term 1 is of the form where r( is the distace from { r( } the reflector ad is the expoetial path-loss parameter. Because a two-dimesioal spatial model is cosidered i this developmet, a atural choice for this parameter is 1/. Let y ) represet the sigal compoet reflected at P. For closely spaced reflectors, the arc legth alog the parabola betwee P -1 ad P is closely approximated by the distace betwee P -1 ad P. By the mea-value theorem there exists y *, y -1 y * y, such that the distace betwee P ad P -1 ca be writte as P P + g'( y *) ( y y ). (1) Thus for large N the received sigal is give by: N y*) 1+ g' ( y*) E ( y y 1). (11) 1 { r( y*)} As the umber of reflectors approaches ifiity, the received sigal, ormalized by N, becomes a defiite itegral, give by y max 1+ g' ( E dy. (1) { r( } y mi The expressio for the sigal compoet is developed i a similar maer as the expressio for E is developed i the circular reflector patter case by fidig the extra distace that compoet travels i relatio to the distace traveled by a referece sigal, E ). ome first-order approximatios are used i the developmet to simplify the expressio. The model for the received sigal for the parabolic reflector patter case is determied to be where ad y max 1+ (a E dy (13) { r( } y mi E e ( ay + h) + + x y y jk ( ay h ( ay h) y ) + ( ay h) + y (14) r( ( ay h ) + ( y ). (15) 3. Numerical Results x y Numerical values are chose for the parameters i the circular ad parabolic reflector patter models to permit a fair compariso of their fadig characteristics. A compariso of the fadig characteristics for oe set of parameters is detailed below. This set of parameters provides a fair compariso betwee the two reflector patters, because the distace from the receiver to the earest reflector is similar for each case. For both models, the uits of distace ad the x ad y coordiates o the plots are give i terms of the sigal s wavelegth. 3

I order for the distace from the receiver to the earest reflector to be similar for both reflector patters, the parameters a ad h are chose such that the parabolic reflector patter will cross the x,

4 Hece, the free-space costat used is kπ/λπ. Also, the complex costat, E, is chose to equal 1 i both models. For the circular reflector patter, the radius of the circular patter is chose to be te wavelegths. Recall that the parabolic reflector patter is described by the equatio, x ay h. I order for the distace from the receiver to the earest reflector to be similar for both reflector patters, the parameters a ad h are chose such that the parabolic reflector patter will cross the x, +y, ad y axes at the same poits as the circular reflector patter. Therefore these parameters are chose as a.1 ad h1. The parabola s trucatio is specified by y max -y mi. ice approximatios are used i the developmet of the models that the distace from the origi to the receiver is much smaller tha the distace from the origi to ay reflector, sigal stregths are calculated for receiver locatios at x ad y. Figure 3 shows both reflector patters ad the area over which receiver locatios are cosidered. plot from a top view. Darker shadig correspods to a smaller sigal stregth. Figures 5(a) ad 5(b) show similar plots for the case of the parabolic reflector patter. Figure 4(a). igal stregth for circular reflector patter model Figure 4(b). igal stregth for circular reflector patter model (top view) Figure 3. Reflector patters for umerical aalysis igal stregths are calculated for each reflector patter for uiformly distributed receiver locatios withi the chose area. All figures show results for 441 sample locatios except Figures 6 ad 7, which show results for 1, sample locatios withi the same area. Figure 4 (a) shows a plot geerated from the data from the circular model. The magitude of the received sigal at a receiver locatio (x,, is displayed as the correspodig z- coordiate (axis labeled E ). Figure 4 (b) shows the same 4

Figure 5(a). igal stregth for parabolic reflector patter model Figure 6.

5(b). igal stregth for parabolic reflector patter model (top view) The bar graphs of Figures 6

respectively, of the sigal magitudes for the case of the circular reflector.

of the sigal magitudes for the parabolic reflector case. Figure 7.

5 Figure 5(a). igal stregth for parabolic reflector patter model Figure 6. Probability distributio fuctio of sigal stregth for circular reflector patter model Figure 5(b). igal stregth for parabolic reflector patter model (top view) The bar graphs of Figures 6 ad 7 show the empirical probability distributio fuctio ad probability desity fuctio, respectively, of the sigal magitudes for the case of the circular reflector. Figures 8 ad 9 show the empirical probability distributio fuctio ad probability desity fuctio of the sigal magitudes for the parabolic reflector case. Figure 7. Probability desity fuctio of sigal stregth for circular reflector patter model Figure 8. Probability distributio fuctio of sigal stregth for parabolic reflector patter model 5

desity fuctio i equatio (17). Note that both asymptotic aalytical results yield close agreemet with the empirical results of our example. 4. Coclusio Figure 9.

6 desity fuctio i equatio (17). Note that both asymptotic aalytical results yield close agreemet with the empirical results of our example. 4. Coclusio Figure 9. Probability desity fuctio of sigal stregth for parabolic reflector patter model For the circular reflector patter, the miimum ad maximum sigal magitudes for receiver locatios over the chose area are ad.1116, respectively. Therefore, the sigal stregth fluctuates by 99.47% of its maximum value. For the parabolic reflector patter, i cotrast, the miimum ad maximum sigal magitudes are.985 ad 1.175, respectively. This meas the sigal stregth oly fluctuates by 13.4% of its maximum value. Thus the receiver observes much less spatial variatio i the sigal stregth (ad much less severe fadig) as it moves ear the origi if the reflector patter is parabolic rather tha circular. The probability distributio fuctio ad probability desity fuctio for sigal magitudes at the receiver ca also be derived aalytically for the case of the circular reflector patter uder the coditios we cosider. Let the radom variable Z represet the received sigal stregth at a radomly selected locatio ear the origi. It ca be show that i the limit as q>>r, ad r>>ρ, the probability distributio fuctio ad probability desity fuctio of Z are z F Z z) arcsi π α 1 ad f Z z < z α (, (16) ( z) απ z 1 α z > α 1 z < z α z > α, (17) I this paper we preset a model for the stregth of a sigle-frequecy sigal at a receiver i the presece of multipath propagatio. A determiistic reflector patter is modeled, ad the model is used to determie the stregth of the sigal at the receiver as a fuctio of the receiver s positio. igal-stregth expressios are developed for geeral circular ad parabolic reflector patters. Examples of both reflector patters are cosidered, usig parameters for each which permit a fair compariso of their fadig characteristics. The parabolic reflector patter is foud to produce substatially less variatio i the sigal stregth as a fuctio of receiver positio tha the circular reflector patter. This illustrates that the fadig characteristics deped sigificatly o the physical cofiguratio of the reflectors, ad it highlights the drawbacks associated with usig a sigle statistical model to predict fadig chael properties. 5. Ackowledgemets This work was supported i part by the Natioal ciece Foudatio uder grat EEC ad i part by the U.. Army Research Laboratory ad the U.. Army Research Office uder grat DAAD Refereces [1] P. A. Bello, "Characterizatio of radomly time variat liear chaels," IEEE Tras. Commu. yst., vol C-11, pp , Dec [] R. H. Clarke, "A statistical theory of mobile-radio receptio," Bell yst. Tech. J., vol. 47, o. 4, pp , July-Aug [3] P. A. Bello ad B. D. Neli, "The ifluece of fadig spectrum o the biary error probabilities of icoheret ad differetially coheret match filter receivers," IRE Tras. Commu. yst., vol C-1, pp , Jue 196. [4] F. Bowma, Itroductio to Bessel Fuctios, Dover Publicatios, March 1968 respectively, where represets the greatest sigal stregth observed. The probability distributio fuctio i equatio (16) is plotted i Figure 6 as the solid black lie. A similar plot is show i Figure 7 for the probability 6

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