Internatonal Journal of Eletrons and Eletral Engneerng Vol. 4, No. 5, Otober 06 Improvng the Performane of Fadng Channel Smulators Usng New Parameterzaton Method Omar Alzoub and Moheldn Wanakh Department of Communaton Engneerng, Hgh Insttute of Appled Senes and Tehnology, Damasus, Syra Emal: omar_alzoub@hotmal.om, wanakh@ss-net.org Abstrat The modelng and smulaton of moble fadng hannels an be done effently by usng the fnte sum of weghted snusods wth equally dstrbuted phases known as the onept of Re method. In ths paper, we evaluate the statstal propertes of Webull-lognormal fadng hannel model, suh as Level Crossng Rate (LCR), Average Duraton of Fades (ADF), and Probablty Densty Funton (PDF). Several results are obtaned by usng dfferent methods to desgn the determnst smulaton model parameters. New omputaton method of determnst smulaton model parameters s presented. Ths proedure s a ombnaton of two methods, Method of Equal Areas and Method of Exat Doppler Spread. It s alled a ombnaton of MEA and MEDS. Comparsons between Autoorrelaton Funtons (ACFs) of both referene and smulaton models are ntrodued for dfferent methods. It s demonstrated by several smulaton results that statstal propertes of smulaton and referene models wll be muh loser aordng to the new method than MEDS and MEA. Fnally, the results ndate the superorty of the new method over the MEDS and MEA wth respet to LCR and ADF of Webull-lognormal fadng hannel model. Index Terms level rossng rate, average duraton of fades, autoorrelaton funtons I. INTRODUCTION The modelng of fadng hannels has a great mportane n the desgn, test and mprovement the performane of ellular rado ommunaton systems. The hannel smulator must be effent, flexble and aurate. In addton, the statstal propertes of the hannel smulator should be very lose to the statstal behavor of the desred referene model []. Ths wll depend on the desgn method of hannel smulator. Dependng upon the rado propagaton envronment, varous multpath fadng models are avalable n lterature []. Both of the advanes on lassal fadng models and a bref summary of some new fadng models were presented n []. Moble rado hannels are lassfed nto two man ategores, namely frequeny-nonseletve and frequeny-seletve hannels. The frst type s modeled by usng an approprate stohastal models, suh as Raylegh, Re and Suzuk proesses [4], whereas frequeny-seletve hannels an be modeled by usng (npath) tap delay lne model [5], whh requres n Manusrpt reeved Aprl, 05; revsed January, 06. 06 Int. J. Eletron. Eletr. Eng. do: 0.878/jeee.4.5.44-448 44 oloured Gaussan proesses. Therefore, omputer smulaton models an be mplemented by means of the Re method [6], whh depends on approxmaton of the oloured Gaussan proesses by fnte sum of weghted snusods wth phases unformly dstrbuted. Fndng proper desgn method for omputng parameters of smulaton models provdes determnst proesses at the output of hannel smulator wth a statstal propertes losed to those of orrespondng stohast proesses, espeally statsts of seond order suh as LCR and ADF. Analytal expressons for these quanttes have been derved for Raylegh and Re [7], [8]. In ths researh we are nterested n modellng frequenynonseletve hannels, where Suzuk proess [9] s onsdered to be a more sutable model for non-seletve frequeny ellular rado hannels n many ases. Suzuk proess s obtaned by multplyng Raylegh and Lognormal proesses wth eah other. The reader an fnd detaled nformaton on the extended and modfed stohast versons of Suzuk models n [9]. It s worth mentonng that these stohast models are not apable of modelng non unform satterng. In order to ombne nhomogeneous dffuse satterng wth shadow fadng, the Webull-lognormal proess appears as an approprate omposte model [0]. The Webull-lognormal model, onssts of three stohast Gaussan proesses, should be desgned by a proper method. There are many methods to alulate parameters of smulaton model (doppler oeffents and dsrete doppler frequenes), for example the Method of Equal Areas (MEA) [], [9], whh provdes a satsfed approxmaton of the desred statsts for Jakes Doppler power spetral densty even for a small number of snusods, but t fals and requres large number of snusods for other types of Doppler power spetral denstes suh as those wth Gaussan shapes []. Furthermore, t s found that MEA does not result n a perod ACF due to unequal dstanes between dsrete Doppler frequenes []. There s another method alled Method of Exat Doppler Spread (MEDS), whh ompute parameters n suh way the Doppler spread s the same for both referene and smulaton models []. In ths paper, new optmal desgn method of smulaton model parameters s presented. Ths method s named a ombnaton of MEA and MEDS. Therefore, a bref revew of referene models for Raylegh and Webull-lognormal hannel models and the statstal propertes for them s gven n Setons II, III respetvely. After that n Seton IV t s shown how the
Internatonal Journal of Eletrons and Eletral Engneerng Vol. 4, No. 5, Otober 06 LCR and ADF of Raylegh Proesses (t ), whh belong to the statstal propertes of the seond degree, are very mportant n assessng the performane of hannel smulators. LCR s defned as the rate (rossngs per seond) at whh the envelope (t ) rosses a gven level r n the postve (or negatve) gong dreton. LCR of Raylegh proess an be represented by [], [9]: determnst smulaton model for Webull-lognormal s obtaned by usng the onept of Re s sum of snusods. For ths purpose three omputaton methods of smulaton model parameters s presented n Seton V. In Seton VI, the performane of the three methods s evaluated by omparng statstal propertes of referene and smulaton models over Webull-lognormal hannel model, where t s observed that new method mproved statstal propertes performane of smulaton models more than other methods, whh wll be refleted on the performane of the smulator. Fnally the onluson s gven n Seton VII. II. N ( r ) (5) p ( r ) It s obvous from (5) that LCR of Raylegh proess s proportonal to ts PDF by the onstant whh s the RAYLEIGH CHANNEL MODEL reverse urve of ACF r (0),,. In the ase of Raylegh and Re hannels are the most mportant hannel models n moble ommunatons. Usually, Raylegh and Re proesses are preferred for modellng fast-term fadng, whereas slow-term fadng s modelled by a lognormal proess. Re proess s proposed an approprate stohast model for desrbng the envelope of reeved sgnal n rural areas, where the Lne of Sght (LOS) s taken nto onsderaton, whereas Raylegh proess s onsdered sutable for urban regons, where LOS s not exst [], [9]. A Raylegh proess (t ) s obtaned by takng the absolute value of the zero-mean omplex Gaussan proess (t ) (t ) j (t ),.e.: (t ) (t ) sotrop satterng, where the ACF r ( ) s gven by (), the quantty may be wrtten as: ( 0 fmax ) (6) On the other hand, ADF T ( r ) s the mean value of the length of all tme ntervals over whh the envelope (t ) remans below a gven level r. In general, the ADF s defned by [], [9]: F ( r ) (7) T ( r ) N (r ) where F ( r ) the Cumulatve Dstrbuton Funton () where (t ) represents sattered omponent n the reeved sgnal wth unorrelated real and magnary parts, and varanes var{ (t )} 0,,. (CDF) of (t ) defned by F (r ) pr [ (t ) r ]. III. WEIBULL-LOGNORMAL CHANNEL MODEL Ths model s used to smulate non unform satterng wth shadow fadng, where the frst one models the possble satterng non unformtes of the hannel, whereas the seond aounts for the slow term varatons of the loal mean due to shadowng. The Webulllognormal proess WL(t) s obtaned by multplyng a Webull proess W(t) by a lognormal one L(t). The Webull proess results from a Raylegh one R(t) as A typal shape for the Doppler Power Spetral Densty (PSD) of the omplex Gaussan proesses s gven by the Jakes PSD [7]: o, f f ) S ( f ) S ( f ) max ( f max 0 where f max f f max,, f f max / W (t ) R(t ) and a parameter expressng the fadng severty [0]. As we have seen n the Seton II, the Raylegh proess an be obtaned by (), whereas the lognormal proess s generated by a real valued Gaussan proess (t ) wth zero mean and unt varane as () denotes the maxmum Doppler frequeny. Takng the nverse Fourer transform of the Jakes PSD results n the followng ACF: r ( ) r ( ) o J 0 f max () L(t ) exp[ s (t ) m], where s and m determne the where J 0. s the zeroth-order Bessel funton of the type of shadowng envronment. The stohast Webulllognormal proess wll be [0]: frst knd. The PDF of Raylegh proess (t ), whh desrbes the statstal sgnal varatons, s gven by the Raylegh dstrbuton: x e, x 0 P ( x ) o 0, x 0 x WL(t ) a [ (t ) (t )] / where a exp( m) 0 exp[ s (t )] (8). The ampltude PDF PWL ( ) of WL(t) s gven by [0]: o 06 Int. J. Eletron. Eletr. Eng. (4) PWL ( x) 444 x sa exp( 0 x z a ) exp( ln z s )dz (9)
Internatonal Journal of Eletrons and Eletral Engneerng Vol. 4, No. 5, Otober 06 We note from (9) that the PDF of Webull-lognormal proess depends on three parameters ( s, a, ), and then ths model shows hgh flexblty and nludes Suzuk, Webull and Raylegh models as speal ases. The LCR of Webull-lognormal proess was approxmated n [0] from the assumpton of a slowly tme varyng lognormal proess ompared to the Webull one. So LCR s defned by the equaton [0]: N r ( x) f max x s a / / exp( x z a 0 orrespondng dsrete Doppler frequenes f, n. The Doppler phases, n, (,, ), are realzatons of a random varable unformly dstrbuted wthn the nterval (0, ] [], [9]. The proedures wll be named by Method of Equal Areas (MEA), Method of Exat Doppler Spread (MEDS), and the new one s named by Combnaton of MEDS and MEA method. Here, we wll not present n detal the smulaton modelng employng sum of snusods, but for the nterested reader we refer to [9] for detaled and well-presented analyss of the man methods used n the sum of snusods smulaton sheme. ) exp( ln z s ) dz (0) Now ADF easly an be found by usng (0) wth CDF gven by [0]: Fr ( x) s u exp( a 0 IV. x z A. Method of Equal Areas (MEA) The Doppler oeffents, n have been desgned n ) exp( ln z s ) dz terms of fulfllng the power onstrant 0 0. The () frequenes f, n an be found by parttonng the Doppler power spetral densty of (t ) nto N setons of equal DETERMINISTIC SIMULATION MODEL power and usng the upper frequeny lmts, related to these areas. The Doppler oeffents, n and frequenes An effent smulator for Webull-lognormal fadng hannels s obtaned by usng the onept of Re s sum of snusods [6]. Aordng to that prnple, we replae the stohast Gaussan proesses (t ), (t ), and f, n are omputed by [], [9], []:, n 0 (t ) of the referene model by the followng (t ), n os( f, n t, n ),, f, n f max sn( () where N denotes the number of snusods. The oeffents, Doppler frequenes and Doppler phases respetvely. These parameters have to be omputed durng the smulaton setup phase, e.g., by one of the methods desrbed n the followng setons. From the fat that all parameters are known quanttes, t follows that (t ) an be onsdered as a determnst proess. Applyng the MEDS results the frequenes f, n n the followng relaton: f, n f max sn[ In order the determnst proess (t ),,, to be exp[ s (t )] (7) wth a smple modfaton of MEDS method by means of the relatons [], [9]: envelope of Webull-lognormal fadng hannels an be modeled aordng to: r (t ) a [ (t ) (t )] ( n )] N The dsrete frequenes f,n of (t ) are alulated obtan the three determnst Gaussan proesses (t ),,,. By analogy wth (8), the reeved / respetvely for n,,..., N,,. unorrelated, we defne N N [], [9], thus we (6) ) N B. Method of Exat Doppler Spread (MEDS) The MEDS s doumented n [], [9]. For the omputaton of the gans, n, the same s vald as MEA. f, n, and, n are alled Doppler n respetvely for n,,..., N,,. n parameters, n, (5) N determnst proesses: N n f, n erf ( N () In general, the ACF of (t ) and (t ) are gven by f, N ) 0, n,,.., N (8) N N f, n n [], [9]: N r ( ) n V. As (t ) has a Gaussan PSD S ( f ) defned as [],,n os( f, n ),, (4) S ( f ) COMPUTATION METHODS FOR THE MODEL PARAMETERS exp( f ) (9) where a parameter related to the db-ut-off Ths seton presents three dfferent methods for the determnaton of the Doppler oeffents, n and the 06 Int. J. Eletron. Eletr. Eng. [9]: frequeny f aordng to f ln, whereas f s 445
Internatonal Journal of Eletrons and Eletral Engneerng Vol. 4, No. 5, Otober 06 of both referene and smulaton models are muh loser aordng to the new method than MEA and MEDS, and of ourse ths wll affets the statstal propertes later when LCR and ADF are studed for Webull-lognormal model. From now the number of snusods was assumed N N 5 and N 6 for Webull-lognormal muh smaller than the maxmum Doppler frequeny f max, f s.e. f f max, so the frequeny rato K max f muh greater than one,.e. K. In real worlds hannels K 0 []. fadng hannel. The Fg. 4 shows the PDF of the reeved envelope of Webull-lognormal fadng hannel as a funton of K, wth a parameter set defned as C. Combnaton of MEDS and MEA The new method reles on the applaton of both methods MEDS, MEA, but we must apply the method N MEDS on the frst half number of snusods. Therefore,, n and f, n are gven as follows: =.8, s 0.5, m, 0 0.8, fmax 9Hz. From Fg. 4 an exellent onformty between the referene and smulaton PDF s s revealed, no matter what the value of K s. Beause PDF of Webull-lognormal proess s, n 0 f, n f max sn[ ndependent of tme seletvty of the lognormal proess. In Fg. 5, Fg. 6, and Fg. 7, the normalzed LCR of determnst Webull-lognormal proesses for all ntrodued methods wth K, K, and K s (0) N ( n )] N () shown, respetvely. It an be observed learly that LCR of the smulaton model s very lose to that of the referene model aordng to the ombnaton of MEDS and MEA n omparson wth MEA and MEDs. The orrespondng graphs for the normalzed ADF are presented n Fg. 8, Fg. 9, and Fg. 0. In addton nreasng K mproves the performane of LCR and where n,,..., N,,, then MEA method s appled on the seond half of snusods number:, n 0 f, n f max sn[ () N ( n )] N ADF of the smulaton model. Fnally, the results doumented n the Fg. 5-Fg. 0 ndate a superorty of the ombnaton of MEDS and MEA over the MEDS and MEA wth respet to LCR and ADF. () where n ( N ), ( N ),..., N,,. Fnally the formulas for, n and f, n aordng to ombnaton of MEDS and MEA are gven by, n [, n,, n ] and f, n [ f, n, f, n ] respetvely. VI. COMPARISON OF STATISTICAL PROPERTIES BETWEEN REFERENCE AND SIMULATION MODELS The statstal propertes of the referene model for Webull-lognormal fadng hannel are ompared wth the orrespondng smulaton results. Assumng that smulaton model parameters have been found n one of the prevously desrbed methods, In ths ase, the parameters n () are known quanttes and the ACF r ( ) of the smulaton model an be alulated for Fgure. ACFs of referene and smulaton models usng MEA, by means of (4), whereas the ACF r ( ) of the referene model s obtaned from (). Both of ACFs r ( ), r ( ) are ompared wth N 5,, through Fg., Fg., and Fg., where the omputaton of the smulaton model parameters was based on MEDS, MEA, and ombnaton of MEDS and MEA, respetvely. It s observed that fttng between ACFs s exellent at least up to the N th zero-rossng of r ( ) for all methods, but the Fg. shows the error has beome less between the referene and smulaton models, espeally n the last three snusodal harmons of the ACF n omparson wth Fg. and Fg.. Ths means that ACFs 06 Int. J. Eletron. Eletr. Eng. Fgure. ACFs of referene and smulaton models usng MEDS 446
Internatonal Journal of Eletrons and Eletral Engneerng Vol. 4, No. 5, Otober 06 Fgure. ACFs of referene and smulaton models usng ombnaton of MEDS and MEA Fgure 7. The LCR of determnst Webull-lognormal proesses usng MEDS, MEA, and ombnaton of MEDS and MEA wth K Fgure 4. The PDF of determnst Webull-lognormal proesses wth varaton of K Fgure 8. The ADF of determnst Webull-lognormal proesses usng MEDS, MEA, and ombnaton of MEDS and MEA wth K Fgure 5. The LCR of determnst Webull-lognormal proesses usng MEDS, MEA, ombnaton of MEDS and MEA wth K Fgure 9. The ADF of determnst Webull-lognormal proesses usng MEDS, MEA, and ombnaton of MEDS and MEA wth K Fgure 6. The LCR of determnst Webull-lognormal proesses usng MEDS, MEA, ombnaton of MEDS and MEA wth K Fgure 0. The ADF of determnst Webull-lognormal proesses usng MEDS, MEA, and ombnaton of MEDS and MEA wth K 06 Int. J. Eletron. Eletr. Eng. 447
Internatonal Journal of Eletrons and Eletral Engneerng Vol. 4, No. 5, Otober 06 VII. CONCLUSION The onept of Re s sum of snusods s used to desgn an effent determnst smulaton models for Webull-Lognormal fadng hannels. A study of the statsts of suh types of smulaton models was the top of the present paper, espeally for the PDF, LCR, and ADF. New omputaton method of determnst smulaton model parameters s presented, ths method s alled a ombnaton of MEA and MEDS. The performane of dfferent parameter omputaton methods s dsussed and evaluated by omparng ACFs of both referene and smulaton models. It s observed that ACFs of referene and smulaton models are muh loser aordng to the new method than MEA and MEDS. In addton, the new method gave us an exellent results orrespondng wth PDF, LCR, and ADF of determnst smulaton model. Therefore, the determnst smulaton model, based on the ombnaton of MEDS and MEA, wll be very lose n ts statstal propertes to those of the referene model. Fnally, the mproved performane of the statstal propertes of determnst smulaton fadng hannel models lead to get a hgh auray and effeny fadng hannel smulator. REFERENCES [] M. Pätzold, Moble Rado Channels, John Wley & Sons, 0. [] P. M. Shankar, Fadng and Shadowng n Wreless Systems, New York: Sprnger, 0. [] J. F. Pars, Advanes n the statstal haraterzaton of fadng: From 005 to present, Internatonal Journal of Antennas and Propagaton, vol. 04, no. 6, pp. -5, 04. [4] J. Proaks, Dgtal Communatons, 4th ed., New York: MGraw- Hll, 00. [5] J. D. Parsons, The Moble Rado Propagaton Channel, nd ed., Chhester, England: John Wley & Sons, 00. [6] S. O. Re, Mathematal analyss of random nose, Bell Syd. Teh. J., vol., pp. 8-, July 944 and vol. 4, pp. 46-56, Jan. 945. [7] W. C. Jakes, Mrowave Moble Communatons, New York: IEEE Press, 99. [8] S. O. Re, Dstrbuton of the duraton of fades n rado transmsson: Gaussan nose model, Bell Syst. Teh. J., vol. 7, pp. 58-65, May 958. [9] M. Pätzold, Moble Fadng Channels, Chhester: John Wley & Sons, 00. [0] P. Karadmas and S. A. Kotsopoulos, A generalzed modfed suzuk model wth setored and nhomogeneous dffuse satterng omponent, Journal Wreless Personal Communatons, vol. 47, no. 4, pp. 449-469, 008. [] M. Pätzold, U. Kllat, F. Laue, and Y. L, On the statstal propertes of determnst smulaton models for moble fadng hannels, IEEE Trans. Veh. Tehnol., vol. 47, no., pp. 54-69, Feb. 998. [] M. Pätzold, U. Kllat, and F. Laue, A determnst dgtal smulaton model for Suzuk proesses wth applaton to a shadowed Raylegh land moble rado hannel, IEEE Trans. Veh. Tehnol., vol. 45, no., pp. 8-, May 996. [] M. Pätzold and F. Laue, Level-Crossng rate and average duraton of fades of determnst smulaton models for re fadng hannels, IEEE Trans. Veh. Tehnol., vol. 48, no. 4, pp. -9, July 999. Omar Alzoub was born on September 8, 978, n Homs, Syra. He reeved hs Master degree n Communatons Engneerng n 009 from Albaath Unversty, Homs, Syra. And he s urrently a Ph.D. student n Communaton Engneerng department n Hgh Insttute of Appled Senes and Tehnology n Damasus, Syra. Moheldn Wanakh was born n January 7, 948, n Konatra, Syra. He reeved hs PhD degree n Cybernets & Informaton Theory n 980 from Polytehn Kev-USSR. Currently he s a head of Communaton Networks lab n Hgh Insttute of Appled Senes and Tehnology n Damasus, Syra. Hs urrent researh area nludes dgtal ommunaton, statsts, and nformaton theory. 06 Int. J. Eletron. Eletr. Eng. 448