1050 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 2007

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1 5 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 7 Stochastc Modelng and Smulaton of Frequency-Correlated Wdeband Fadng Channels Cheng-Xang Wang, Member, IEEE, Matthas Pätzold, Senor Member, IEEE, andqyao Abstract For the smulaton of practcal frequency-dversty wreless communcaton systems, such as frequency-hoppng systems, multcarrer code-dvson multple-access systems, and orthogonal frequency-dvson multplexng systems, t s often desrable to produce multple Raylegh fadng processes wth gven frequency correlaton propertes. In ths paper, a novel stochastc wde-sense statonary sum-of-snusods channel smulator s proposed to emulate frequency-correlated wdeband fadng channels, where the frequency correlaton propertes are controlled by only adjustng the constant phases. Closed-form expressons are provded for all the parameters of the smulaton model. Ths enables us to nvestgate analytcally the overall correlaton propertes not only the correlaton coeffcents of the smulated processes wth respect to both tme separaton and frequency separaton. It s shown that the wdeband channel smulator wll be reduced to a narrowband Raylegh fadng-channel smulator by removng the frequency selectvty. Furthermore, the COST 7 typcal-urban and rural-area channels are appled to evaluate the performance of the resultng wdeband and narrowband channel smulators, respectvely. The correlaton propertes of the smulaton models approach the desred ones of the underlyng reference models as the number of exponental functons tends to nfnty, whle very good approxmatons are acheved wth the chosen lmted number of exponental functons. Index Terms Frequency correlaton, Raylegh fadng channels, statstcs, stochastc sum-of-snusods channel smulators, wdeband fadng channels. I. INTRODUCTION MULTIPLE correlated Raylegh fadng sgnals are commonly encountered n frequency-dversty wrelesscommuncaton systems, such as frequency-hoppng FH systems, multcarrer code-dvson multple-access MC-CDMA systems, and orthogonal frequency-dvson multplexng OFDM systems. For convenence, researchers nvestgatng these systems have typcally assumed that the frequency separaton of two sgnals at dfferent carrer frequences s suffcently large compared wth the coherence bandwdth of the channel. Ths assumpton mples that the fadng characterstcs of df- Manuscrpt receved December, 3; revsed November 4, 4, January 3, 6, Aprl, 6, and May 9, 6. The revew of ths paper was coordnated by Prof. A. Abd. C.-X. Wang s wth Jont Research Insttute of Sgnal and Image Processng, School of Engneerng and Physcal Scences, Herot Watt Unversty, EH4 4AS Ednburgh, U.K. e-mal: cheng-xang.wang@hw.ac.uk. M. Pätzold and Q. Yao are wth the Faculty of Engneerng and Scence, Agder Unversty College, 4876 Grmstad, Norway e-mal: matthas. paetzold@ha.no; qyao@hotmal.com. Dgtal Object Identfer.9/TVT ferent frequency-separated channels are uncorrelated ], and therefore, the underlyng channel smulator can be mplemented by usng uncorrelated fadng processes. However, n practcal frequency-dversty systems, the frequency separaton of dfferent channels s often smaller than the coherence bandwdth due to spectrum and frequency plannng constrants ] 8]. In ths case, the frequency correlaton wll exst among dfferent channels, resultng n the so-called frequency-correlated channels. Snce the assumpton of uncorrelated fadng does not take the frequency correlaton nto account, postvely based performance results are often obtaned for the nvestgated systems. Therefore, accurate theoretcal analyses and smulatons of multple frequency-correlated Raylegh fadng channels are of great mportance n nvestgatng the nfluence of the frequency correlaton on the performance of real frequency-dversty systems. In the lterature, dfferent algorthms have been presented for the generaton of two 4] or any number 5] 8] of frequencycorrelated Raylegh fadng processes. The authors of 4] 8] have all employed the method of generatng correlated processes by usng a lnear transformaton of uncorrelated processes. To ths end, multple uncorrelated processes must frst be produced, and the Cholesky decomposton technque has to be used to factorze the gven correlaton coeffcent matrx 4] 8]. Ths sgnfcantly ncreases the complexty and restrct the applcaton of the algorthms 4] 8] to real systems, partcularly when the requred number of correlated processes s large. Furthermore, the papers n 4] 8] have only studed the correlaton coeffcents of the generated processes and not the overall correlaton propertes wth respect to both tme separaton and frequency separaton. The sum-of-snusods approach 9], ] s well known to be an effcent and flexble method for the smulaton of moble fadng channels. Its applcaton to the desgn of fadng-channel smulators ncludes the modelng of narrowband channels ], ], wdeband channels ] 4], spatal-temporal channels 5], 6], and dgtal channels generatve models 7], 8]. For the purpose of smulatng space dversty channels and multple-nput multple-output channels, the sum-of-snusods method has also been wdely employed to generate multple uncorrelated 9] 3] and correlated 4] Raylegh fadng processes. However, the channel smulators descrbed n the study n 9] 4] cannot be drectly appled to the smulaton of frequency-correlated Raylegh fadng channels. Based on the same mathematcal reference model, as descrbed n the /$5. 7 IEEE

2 WANG et al.: STOCHASTIC MODELING AND SIMULATION OF WIDEBAND FADING CHANNELS 5 study n ], two determnstc sum-of-snusods channel smulators 5], 6] have been developed to model FH Raylegh fadng channels wth gven frequency correlaton propertes. Although not explctly ndcated n the study n 5] and 6], these two channel smulators can also be used to smulate other frequency-dversty channels, such as MC-CDMA and OFDM channels. Unfortunately, there stll exst relatvely large devatons between some of the statstcal propertes of both channel smulators 5], 6] and those of the underlyng reference model ]. Moreover, these two channel smulators are lmted to model only narrowband channels, where the transmtted symbol rate s much smaller than the channel coherence bandwdth. Modern frequency-dversty wrelesscommuncaton systems are often desgned for hgh symbol rates. If the symbol rate s so hgh as n the order of the coherence bandwdth, the channel becomes wdeband or frequency selectve. The am of ths paper s to develop a stochastc sum-of-snusods channel smulator that can smulate multple frequency-correlated wdeband fadng channels wth accurate statstcal propertes. For our purpose, a mathematcal reference model based on the tapped-delay-lne structure 7], 8] s frst presented n modelng frequency-correlated wdeband fadng channels. In order to be consstent wth the study n ], we have assumed a -D sotropc scatterng propagaton envronment wth omndrectonal antennas at recevers 9] and a truncated negatve exponental power delay profle PDP. It s mportant to menton that a dfferent profle, e.g., the Gaussan functon 4], can also be deployed as the PDP. The applcaton of the present modelng procedure to dfferent PDPs s straghtforward. The overall correlaton propertes among dfferent frequency-separated channels are nvestgated n detal wth respect to both tme separaton and frequency separaton. By removng the frequency selectvty from the proposed wdeband reference model, t s shown that the derved correlaton functons wll be reduced to the results presented n, p. 5] for narrowband Raylegh fadng channels. Hence, the narrowband reference model n, p. 48] s only a specal case of the proposed wdeband reference model. Then, we propose a novel stochastc sum-ofsnusods smulaton model, whch can emulate accurately all of the correlaton propertes of the derved wdeband reference model. Smlarly, a narrowband Raylegh fadng channel smulator can be obtaned by removng the frequency selectvty from the wdeband smulaton model. Rather than frst generatng uncorrelated processes, as n 4] 8] and 4], the proposed channel smulator can drectly smulate multple frequencycorrelated processes by only adjustng the constant phases of the employed harmonc functons. The statonarty propertes of stochastc sum-of-snusods channel smulators have recently been addressed n several papers, e.g., n 3] 3]. We wll show that the proposed stochastc channel smulator s wdesense statonary WSS. The remander of ths paper s organzed as follows. In Secton II, a reference model for frequency-correlated wdeband fadng channels s proposed. It s shown that the wdeband reference model wll be reduced to the narrowband reference model n ] by removng the frequency selectvty. A novel stochastc sum-of-snusods-based wdeband channel smulator s descrbed n Secton III. In ths secton, we also show how the model parameters can be determned. Secton IV deals wth the performance evaluaton and verfcaton of the proposed smulaton model. Fnally, the conclusons are drawn n Secton V. II. MATHEMATICAL REFERENCE MODEL A. Reference Model for Frequency-Correlated Wdeband Fadng Channels Bello s WSS uncorrelated-scatterng WSSUS model 33] has wdely been accepted n modelng wdeband multpath fadng channels. Regardng the desgn of hardware or software smulaton models for wdeband channels, a dscretzaton of the propagaton delays has to be performed. A sutable and often used model s the so-called dscrete tapped-delay-lne model 7], 8]. It s mportant to menton that a dscrete tappeddelay-lne model s a WSSUS model f the tme-varant tap coeffcents are uncorrelated Gaussan random processes. The complex baseband mpulse responses at two dfferent carrer frequences f c and f c can be expressed as L hτ,t: µ l tδτ τ l, at f c a l L h τ,t: µ l tδτ τ l, at f c b l where L s the number of taps, and µ l tµ l t and τ l ndcate the complex tme-varant tap coeffcent and the dscrete propagaton delay of the lth l,,...,l tap, respectvely. Accordng to the authors of 4] and 34], a proper PDP pτ for a wdeband channel s gven by the followng truncated negatve exponental profle: pτ b exp τ, τ τ max where represents the rms delay spread, and τ max denotes the maxmum propagaton delay. The normalzaton factor b / e τ max/ has been ntroduced here to fulfll the condton τ max pτ dτ. Ths means that the average power of the wdeband fadng channel s normalzed to unty, and hence, the PDP n can be consdered as the probablty densty functon pdf of the propagaton delays. An example of s gven by µs and τ max 7µs. In ths case, the PDP of the wdeband typcal-urban TU channel for global system for moble communcatons GSM s obtaned 34]. The frequency correlaton functon FCF Rχ s related to the PDP through the Fourer transform 33],.e., Rχ b τ max pτ exp jπτ χdτ { exp τ max ]} + jπχ. 3 +jπχ

3 5 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 7 For a dscrete tapped-delay-lne model, each tap s consdered as the combnaton of a number of multpath components. The propagaton delay nterval I,τ max] s parttoned nto L mutually dsjont subntervals I l,.e., I L l I l and I l I λ { } for every l and λ ll, λ,,...,l 3], 4]. Each subnterval I l corresponds to the range wthn whch a number of multpath components are combned to the lth tap. For smplcty, the subntervals I l can be defned as follows: I l { τ l τ l /,τ l + τ l+ /, l,,...,l τ l τ l /,τ l + τ l+ /], l L where τ τ, τ l τ l τ l l,,...,l, and τ L τ max τ L. The above expresson can be rewrtten as, τ l+ /, l I l τ l τ l /,τ l + τ l+ /, l,,...,l. τ l τ l /,τ max], l L 5 Note that the tme-varant tap coeffcent µ l t correspondng to the dscrete propagaton delay τ l actually stands for the contrbuton of the channel mpulse response hτ,t n the nterval τ I l,.e., µ l t τ I l hτ,tdτ. By analogy, µ l t τ I l h τ,tdτ holds. It s mportant to stress here that the delay spacng between the taps should be chosen to avod a regular spacng,.e., τ l τ l+. Ths s to ensure the FCF of the rado channel has a suffcently large perod at least larger than twce the system bandwdth, whch s partcularly mportant for FH systems. Under the WSSUS condton, the tme-varant tap coeffcents µ l t and µ l t are zero-mean complex Gaussan random processes. It follows that the ampltude processes µ l t and µ l t are Raylegh dstrbuted. The Central-Lmt Theorem 36] justfes that a Gaussan random process can be modeled by a superposton of an nfnte number of properly weghted snusods. Therefore, we propose to model µ l t and µ l t as µ l t µ,l t+jµ,l t lm N l M l n m C n,m,l 4 exp j πf max t cos β n,l Φ n,m,l ˆθ ] n,m,l 6a µ l t µ,l t+jµ,l t lm N l M l n m C n,m,l exp j πf max t cos β n,l Φ n,m,l ˆθ ] n,m,l 6b wth Φ n,m,l πf c ϕ n,m,l and Φ n,m,l πf cϕ n,m,l. Here, f max s the maxmum Doppler frequency, and C n,m,l, β n,l, and ϕ n,m,l are ndependent random gans, random angles of arrval, and random propagaton delays of the lth tap, respectvely. The phases ˆθ n,m,l are ndependent random varables unformly dstrbuted over, π. Equatons 6a and 6b can be nterpreted as follows. For the dffuse components µ l tµ l t of the lth tap, the nth wave at the angle of arrval β n,l s also composed of M l waves wth ampltudes C n,m,l and propagaton delays ϕ n,m,l I l. All of these M l waves experence the same Doppler shft f max cos β n,l. Adoptng Clarke s -D sotropc scatterng model 9], the angles of arrval β n,l are unformly dstrbuted over, π,.e., pβ /π, β<π. The average fractonal power of the lth tap as a functon of propagaton delays ϕ n,m,l stll follows the truncated negatve exponental profle ntroduced n,.e., pϕ b exp ϕ, ϕ I l. 7 Clearly, τ max pτ dτ L l ϕ I l pϕdϕ holds. The random gans C n,m,l can be related to the power assocated wth each ndvdual wave of the lth tap as follows: Cn,m,l pβ n,l,ϕ n,m,l dβdϕ ]. Note that pβ,ϕdβdϕ descrbes the fracton of the ncomng power wthn the nfntesmal ntervals dβ and dϕ. From 6, t follows that the correlaton propertes of µ l t and µ λ tl, λ,,...,l are completely determned by the underlyng real Gaussan nose processes µ t and µ j,λ t, j,. Therefore, we can restrct our nvestgatons to the followng autocorrelaton functons ACFs and cross-correlaton functons CCFs: r µ µ j,λ τ :E {µ tµ j,λ t + τ} } r µ µ {µ j,λτ,χ:e tµ j,λt + τ 8a 8b where E{ } denotes the statstcal average wth respect to β n,l, ϕ n,m,l, and ˆθ n,m,l. It should be observed that 8b s a functon of both tme separaton τ t t and frequency separaton χ f c f c. Usng 6, we can obtan the followng correlaton functons of the wdeband reference model: r µ µ τ r µ τ µ ba l B l J πf max τ 9a r µ,l µ,l τ r µ τ,l µ,l r µ,l µ,l τ r µ τ,l µ,l 9b r µ µ j,λ τ r µ τ µ j,λ 9c

4 WANG et al.: STOCHASTIC MODELING AND SIMULATION OF WIDEBAND FADING CHANNELS 53 r µ µ τ,χ bj πf max τ +πχ ] exp ϕ πχ snπχϕ cosπχϕ] ϕτ l + τ l+ / ϕτ l τ l / r µ,l µ τ,χ r µ,l µ,l,lτ,χ bj πf max τ +πχ ] exp ϕ snπχϕ+πχ cosπχϕ] ϕτ l + τ l+ / ϕτ l τ l / 9d 9e r µ µ j,λτ,χ 9f for, j, and l, λ,,...,l wth l λ, where J denotes the zeroth-order Bessel functon of the frst knd, A l exp τ l τ l //], and B l exp τ l + τ l+ //]. In 9d and 9e, fx xx xx s an abbrevaton for fx fx. Snce the dervatons of 9a 9f are smlar, only the dervaton of 9e s gven n Appendx A, whle others are omtted here for brevty. The delay power Ω l assgned to the lth tap can be obtaned from 9a accordng to Ω l r µ µ ba l B l. The total delay power s, therefore, Ω p L l Ω l. The expresson 9b clearly ndcates the uncorrelatedness between the n-phase and quadrature components of µ l t µ l t. The expressons 9c and 9f state that there are no cross correlatons between the underlyng processes n dfferent taps due to the uncorrelated-scatterng US condton mposed on the model. From 9d and 9e, t s clear that r µ µ and τ,χ r µ,l µ are separable nto two parts one depends,lτ,χ on the tme-separaton varable τ and the other one on the frequency-separaton varable χ. Therefore, we can wrte r µ µ τ,χr µ µ τr µ µ χ and r µ,l µ,lτ,χ r µ µ τr µ,l µ χ, where r µ µ τ L l r µ µ τ,l J πf max τ. We can easly calculate R µ µ and χ R µ,l µ usng 9d and 9e, respectvely. It can be shown,lχ further that R µ µ χ and R µ,l µ are related to the,lχ FCF Rχ n 3 as follows: L ] Rχ R µ µ χ+jr µ,l µ.,lχ l B. Reference Model for Frequency-Correlated Narrowband Raylegh Fadng Channels A specal case of the wdeband channel model descrbed by s gven when L. Consequently, hτ,tµtδτ and h τ,tµ tδτ hold. These are actually the mpulse responses of a narrowband-channel model. To make ths evdent, we express the stochastc processes µt and µ t smlarly to 6a and 6b by removng the subscrpt l,.e., µt µ t+jµ t lm N N M C n,m M n m exp j πf max t cos β n Φ n,m ˆθ ] n,m µ t µ t+jµ t N M lm C n,m N M n m exp j πf max t cos β n Φ n,m ˆθ ] n,m a b wth Φ n,m πf c ϕ n,m and Φ n,m πf cϕ n,m. In order to be consstent wth ], we employ for the narrowband-channel model a negatve exponental PDP,.e., pϕ exp ϕ, ϕ whch s a specal case of the truncated exponental PDP n for τ max. Agan, the PDP n can actually be regarded as the pdf of the propagaton delays, snce the average power of the narrowband fadng channel s normalzed to be one,.e., pϕdϕ holds. Note that the value of n must be much less than the symbol duraton for a narrowband channel. An example s the rural-area RA channel model 34] developed for GSM, where.86 µs, and the symbol duraton s about 3.7 µs. By takng the Fourer transform of the PDP n, the FCF s gven by Rχ +jπχ. 3 The correlaton propertes of the above narrowband-channel model can be easly obtaned from 9 by neglectng the subscrpt l. In ths case, τ τ /, and τ + τ / τ max,.e., A, B, and b hold. The resultng correlaton functons are gven by r µ µ τ r µ τ µ J πf max τ 4a r µ µ τ r µ τ µ r µ µ τ r µ τ µ 4b r µ µ τ,χ J πf max τ +πχ 4c ] r µ µ τ,χ r µ µ τ,χ πχr µ µ τ,χ. 4d As nvestgated n the study n 4] 8] and 4], the correlaton coeffcents of the correspondng stochastc processes can easly be ganed by evaluatng the correlaton functons n 4 at τ. The separablty property of r µ µ τ,χ s clear from

5 54 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 7 the followng expresson: r µ µ τ,χr µ µ τr µ µ χ, where R µ µ χ / + πχ ]. Smlarly, r µ µ τ,χ r µ µ τr µ µ χ holds, where R µ µ πχ/ + χ πχ ]. A close comparson of R µ µ χ, R µ µ and χ, the FCF Rχ n 3 gves the followng relaton: Rχ R µ µ χ+jr µ µ χ. Note that the frequency-correlated narrowband Raylegh fadng channel model descrbed by s essentally the same as the equvalent baseband-channel model gven n, p. 48]. As a result, the correlaton functons n 4 are n agreement wth those presented n, p. 5], as well as n 5] and 6]. Therefore, the proposed wdeband-channel model descrbed by and 6 ncludes the narrowband-channel model n, p. 48] as a specal case f the truncated negatve exponental PDP s employed. However, the proposed modelng procedure can also be appled to other forms of PDP, e.g., as shown n 4]. It should be clarfed at ths pont that the mathematcal model descrbed by and 6, as well as ts specal case descrbed by, stands for a nonrealzable stochastc reference model, snce the employed numbers of exponental functons tend to nfnty. Nevertheless, the correlaton functons shown n 9a 9f and 4a 4d provde the bass for the performance evaluaton of the realzable smulaton model descrbed n the next secton. III. NOVEL STOCHASTIC SUM-OF-SINUSOIDS SIMULATION MODEL A. Smulaton Model for Frequency-Correlated Wdeband Fadng Channels The stochastc smulaton model proposed here s also based on the dscrete tapped-delay-lne structure as n the reference model, but the manner how the complex random processes are modeled s completely dfferent. The mpulse responses of the smulaton model are agan composed of L dscrete taps accordng to L ĥτ,t: ˆµ l tδ τ τ l, at f c 5a l L ĥ τ,t: ˆµ l tδ τ τ l, at f c 5b l where the dscrete propagaton delays τ l are dentcal to those of the reference model. Concernng the smulaton model, t s mportant to menton that τ l must be an nteger multple of the samplng nterval T s,.e., τ l q l T s, wth q l denotng a non-negatve nteger. Gven τ l, T s can be determned by T s gcd{τ l }L l, where gcd{ } represents the greatest common dvsor ]. In case that the orgnal smulaton samplng nterval s so large that the dscrete propagaton delays τ l are not multples of the samplng nterval, t s necessary to ncrease the samplng rate up-samplng to accommodate all the dfferent dscrete delays. Ths wll reduce the smulaton speed and result n computatonally neffcent smulatons. To solve ths problem, several converson methods were dscussed n the study n 35] to use an alternatve model from the orgnal one havng all dscrete propagaton delays matchng the samplng nterval wthout usng up-samplng technques. In 5a and 5b, the stochastc processes ˆµ l t and ˆµ l t are modeled by a fnte double sum of properly weghted exponental functons ˆµ l t ˆµ,l t+j ˆµ,l t n N l + m ˆµ l t ˆµ,l t+j ˆµ,l t n N l + m c n,m,l exp j πf n,l t θ m,l ˆθ ] n,m,l 6a c n,m,l exp j πf n,l t θ m,l ˆθ ] n,m,l 6b wth θ m,l πf c φ m,l, and θ m,l πf cφ m,l. Here, N l M l defnes the fnte number of exponental functons of the lth tap, manly determnng the realzaton expendture and the performance of the resultng channel smulator. The gans c n,m,l, the dscrete Doppler frequences f n,l, and the dscrete delays φ m,l wll be determned durng the smulaton-setup phase and kept constant durng smulaton. The phases ˆθ n,m,l are ndependent random varables unformly dstrbuted n the nterval, π. The expressons 6a and 6b provde us wth further observatons: When dfferent carrer frequences f c and f c f c + χ are employed n the physcal channel model, we only have to adopt the correspondng sets of constant phases θ m,l and θ m,l θ m,l +πχφ m,l m,,...,m l n the smulaton model, whle mantanng all other parameters unchanged. Therefore, ths channel smulator can drectly smulate frequency-correlated processes by smply adjustng the constant phases, whch s completely dfferent from the method of generatng correlated processes by usng a lnear transformaton of uncorrelated processes n 4] 8] and 4]. Followng 5a, Fg. a shows the dscrete tapped-delaylne structure of the resultng stochastc smulaton model correspondng to the carrer frequency f c. The nput and output sgnals of the channel are represented here by xt and yt, respectvely. The structure of the underlyng complex stochastc process ˆµ l t see 6a] of the lth l,,...,l tap s llustrated n Fg. b. By substtutng n Fg. b the constant phases θ m,l by θ m,l, whch performs the same functon as we substtute n Fg. a ˆµ l t by ˆµ l t, we wll mmedately obtan a model correspondng to another carrer frequency f c see 5b and 6b]. The correlaton propertes of the stochastc processes ˆµ t and ˆµ j,λ t, j, and l, λ,,...,l are calculated accordng to the followng defntons: ˆr µ µ j,λ τ :E {ˆµ tˆµ j,λ t + τ} 7a } ˆr µ µ {ˆµ j,λτ,χ:e tˆµ j,λt + τ. 7b

6 WANG et al.: STOCHASTIC MODELING AND SIMULATION OF WIDEBAND FADING CHANNELS 55 Fg.. Stochastc sum-of-snusods smulaton model for wdeband channels. a Dscrete tapped-delay-lne structure. b Underlyng complex stochastc process ˆµ l t. Here, E{ } ndcates the statstcal average wth respect to the random phases ˆθ n,m,l. The substtuton of 6 nto 7 results n the followng relatons: ˆr µ µ τ ˆr µ τ µ n N l + m ˆr µ,l µ,l τ ˆr µ τ,l µ,l ˆr µ,l µ,l τ ˆr µ τ,l µ,l n N l + m c n,m,l c n,m,l cosπf n,l τ snπf n,l τ 8a 8b ˆr µ µ j,λ τ ˆr µ τ µ j,λ 8c ˆr µ µ τ,χ n N l + m c n,m,l cosπf n,l τ πφ m,l χ 8d ˆr µ,l µ τ,χ ˆr µ,l µ,l,lτ,χ n N l + m c n,m,l snπf n,l τ πφ m,l χ 8e ˆr µ µ j,λτ,χ 8f

7 56 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 7 for all, j, and l, λ,,...,l wth l λ. Equaton 8a clearly states that the ACFs of ˆµ t and ˆµ t depend only on the tme separaton τ t t. Moreover, t can be easly shown that the mean values of ˆµ t and ˆµ t are equal to zero. Accordng to the study n 36], a stochastc process s sad to be WSS f ts mean value s constant and ts ACF depends only on the tme separaton τ. Therefore, ˆµ t and ˆµ t are WSS processes. Consequently, the resultng stochastc channel smulator s WSS. In the sequel, our am s to fnd proper smulaton model parameters n such a way that all of the correlaton propertes of the smulaton model descrbed by 8a 8f wll approxmate as close as possble those of the gven reference model descrbed by 9a 9f. The gans c n,m,l and the dscrete Doppler frequences f n,l are derved by usng the method of exact Doppler spread ], whch results n the followng closed-form expressons: c n,m,l ba l B l N l M l π f n,l f max sn n ] N l 9a 9b for n N l +, N l +,...,N l, m,,...,m l, and l,,...,l. After substtutng 9a nto 8a and 8b, we realze that M l has no nfluence on ˆr µ µ τ and ˆr µ,l µ,l τ. Moreover, the receved power ˆΩ l of the lth tap can be obtaned from 8a, accordng to ˆΩ l ˆr µ µ ba l B l, whch equals Ω l for the reference model. The substtuton of 9a and 9b nto 8b makes clear that ˆr µ,l µ,l τ r µ,l µ,l τ always holds for any gven numbers of N l and M l, snce the f n,l s fulfll the followng symmetry condton: f Nl +,l f Nl,l,...,f,l f,l. The symmetry property of f n,l s evdently llustrated n Fg. a, where f max 9Hz and N l were used as an example. Equaton 9b also states that the US condton see 8c and 8f], whch requres f n,l ±f k,λ for l λ,s always fulflled n the case where N l /N λ n /k holds, where n,,...,n l and k,,...,n λ 3]. Concernng the computaton of φ m,l appearng n 8d and 8e, the new method proposed n the study n 3] s appled n Appendx B, where the followng closed-form expresson s derved: ] φ m,l ln A l m. M l A l B l Note that the dscrete delays φ m,l are located n the nterval τ l τ l /,τ l + τ l+ /], n whch τ l τ l / and τ l + τ l+ / correspond to φ,l and φ Ml,l, respectvely. An example of φ m,l wth µs, l τ τ, τ. µs, and M l s plotted n Fg. b. In ths case, φ m,l,.] µs holds. Fg.. a Dscrete Doppler frequences f n,l wth f max 9 Hz and N l. b Dscrete delays φ m,l wth µs, l τ τ, τ. µs, and M l. B. Smulaton Model for Frequency-Correlated Narrowband Raylegh Fadng Channels Analogous to Secton II-B, f we mpose L on the wdeband smulaton model n 5, t reduces to a narrowband Raylegh fadng channel smulator. It follows that ĥτ,t ˆµtδτ and ĥ τ,tˆµ tδτ hold. The stochastc processes ˆµt and ˆµ t are then gven by ˆµt ˆµ t+jˆµ t N M c n,m exp j πf n t θ m ˆθ ] n,m n N+ m n N+ m a ˆµ t ˆµ t+jˆµ t N M c n,m exp j πf n t θ m ˆθ ] n,m b wth θ m πf c φ m and θ m πf cφ m. The correlaton propertes of ths narrowband smulaton model can be obtaned

8 WANG et al.: STOCHASTIC MODELING AND SIMULATION OF WIDEBAND FADING CHANNELS 57 from 8a 8f by smply neglectng the subscrpt l n all the affected symbols. Thus ˆr µ µ τ ˆr µ τ µ N M n N+ m ˆr µ µ τ ˆr µ τ µ ˆr µ µ τ ˆr µ τ µ N ˆr µ µ τ,χ M n N+ m N M n N+ m ˆr µ µ τ,χ ˆr µ µ τ,χ N M n N+ m c n,m cosπf nτ a c n,m snπf nτ b c n,m cosπf nτ πφ m χ c c n,m snπf nτ πφ m χ. d If we remove the subscrpt l n 9 and and further take nto account A, B, and b, all the parameters for the narrowband channel smulator are obtaned as follows: c n,m NM π f n f max sn N φ m ln n ] 3a 3b 4 m / M for n N +, N +,...,N, and m,,...,m.as shown n 4, we have substtuted on the rght-hand sde the quantty m by m / n order to avod φ m when m M. It s mportant to pont out that the derved correlaton functons, as shown n 8a 8f and a d, are always vald for the gven smulaton model see 6 and ], whle ndependent of any gven reference model. However, the derved smulaton-model parameters, as shown n 9,, 3, and 4, are only vald for the gven reference model, where -D sotropc-scatterng condtons and a truncated negatve exponental PDP have been assumed. If the channel condtons of the reference model are dfferent, the smulaton model parameters have to be rederved. IV. PERFORMANCE EVALUATION The fnal acceptablty of a novel channel smulator s usually establshed by analytcal and numercal performance evaluatons. The extent to whch the correlaton propertes of the proposed channel smulator and reference model agree wll govern the degree of confdence we place n the modelng approach. In ths secton, we wll valdate the frequency-correlated wdeband and narrowband Raylegh fadng channel smulators from both the analytcal and smulaton ponts of vew. A. Comparson of Correlaton Functons for Wdeband Fadng Channels The correspondng correlaton functons of the wdeband smulaton model see 8a 8f] wll be compared wth those of the wdeband reference model see 9a 9f]. As aforementoned n Secton III-A, 8b, 8c, and 8f are dentcal to 9b, 9c, and 9f, respectvely. Moreover, by substtutng 9a and 9b nto 8a, we can show that the ACF ˆr µ µ τ of the smulaton model approaches the desred one r µ µ τ f the number of exponental functons tends to nfnty,.e., ˆr µ µ τ r µ µ τ as N l. By analogy, the substtuton of 9a, 9b, and nto 8d and 8e results for N l and M l n ˆr µ µ τ,χ r µ µ τ,χ and ˆr µ,l µ,lτ,χ r µ,l µ respectvely. For brevty of presentaton, only,lτ,χ, the proof of ˆr µ,l µ τ,χ r µ,l µ for N,l,lτ,χ l and M l s shown n Appendx C. Other proofs are omtted here. Next, the dscrete sx-tap COST 7 TU channel model 34] wth µs and τ max 7µs wll be appled to nvestgate the degradaton effects of the smulaton model wth a practcal lmted number of exponental functons. The dscrete propagaton delays τ l of the reference model and the smulaton model are then drectly equated wth the values specfed n 34]: {τ l }5 l {,.,.5,.6,.3, 5} µs. Note that the sets {τ l }L l can also be determned by some other computaton methods, e.g., n 4] and 37]. In order to provde a gude on how to choose the number of exponental functons, t s necessary to defne some approprate measures for the errors between the approxmate correlaton functons and the desred ones. Close observatons of 8a, 8d, and 8e tell us that M l has no nfluence on ˆr µ µ τ ˆr µ µ whle N τ,, l has no nfluence on ˆr µ µ,χ and ˆr µ,l µ A deep nsght nto the performance and complexty of the wdeband-channel smulator,l,χ. can be ganed by evaluatng the followng three performance crtera. The frst performance crteron s the mean-square error MSE ɛ of the ACF ˆr µ µ τ defned by ɛ τ max τ max rµµ τ ˆr µ µ τ] dτ 5 where τ max denotes an approprate tme nterval,τ max ] over whch the approxmaton of r µ µ τ s of nterest. As suggested n the study n ], the value τ max N l /f max has turned out to be sutable. The relatonshp between ɛ

9 58 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 7 Fg. 3. a MSE ɛ of the ACF ˆr µ µ τ wth dfferent N l.bmsesɛ of ˆr µ µ,χ and ɛ 3 of ˆr µ,l µ wth dfferent M l for the COST,l,χ 7 TU channel f max 9Hz, µs, τ max 7µs, and l. and N l for l s gven n Fg. 3a. It clearly llustrates the convergence behavor of ˆr µ µ τ r µ µ τ for ncreasng N l. It s apparent that a very good approxmaton of ˆr µ µ τ r µ µ τ has already been acheved f N l 5. The MSEs of the CCFs ˆr µ µ and,χ ˆr µ,l µ whch are defned by,l,χ, ɛ χ max ɛ 3 χ max χ max χ max r µ µ,χ ˆr µ µ dχ 6a,χ] r µ,l µ,χ ˆr µ,l µ dχ 6b,l,l,χ] respectvely, are another two mportant performance crtera. In 6, χ max represents the maxmum frequency separaton to whch the approxmaton of r µ µ,χ and r µ,l µ,l,χ s stll of nterest. Needless to say, χ max should be larger than or equal to the coherence bandwdth of the channel. It was mentoned n the study n 3] that the coherence bandwdth of a channel n practcal cellular systems s on the order of MHz. Ths nctes us to choose χ max 5MHz. Both of Fg. 4. ACFs r µ µ τ of the reference model and ˆr µ µ τ of the smulaton model for the COST 7 TU channel f max 9Hz, µs, and τ max 7µs. a l, N l 5,andM l.bl 3, N l, and M l. the resultng MSEs ɛ and ɛ 3, n terms of M l,areshownn Fg. 3b. Ths fgure demonstrates that ɛ 3 s always smaller than ɛ for the same M l. However, n both cases, M l 5 can always result n very good fttngs. Accordng to the analyses above and takng account of the requred condton N l /N λ n /k for n,,...,n l and k,,...,n λ, M l l,,...,5, N 5, N 6, N 8, N 3, N 4 4, and N 5 3 were selected for the wdeband smulaton model as a good compromse between the model s precson and complexty. Fg. 4a and b mpressvely shows the excellent accordance between r µ µ τ and ˆr µ µ τ wthn the tme nterval τ,n l /f max ] for l and l 3, respectvely. Wth the gven quanttes f max 9Hz, N 5, and N 3, N /f max.84 s and N 3 /f max.99 s hold. In case that τ>n l /f max, ˆr µ µ τ and r µ µ τ wll gradually dverge and never converge agan ]. In order to check the valdty of the analytcal expressons, the smulaton results obtaned by computng the statstcal average of 5 random realzatons are also llustrated n these fgures. To demonstrate the excellent approxmaton qualty of ˆr µ µ τ,χ r µ µ the numercal τ,χ,

10 WANG et al.: STOCHASTIC MODELING AND SIMULATION OF WIDEBAND FADING CHANNELS 59 Fg. 5. CCFs for the COST 7 TU channel f max 9 Hz, µs, and τ max 7 µs. a r µ µ reference model, l. τ,χ b ˆr µ µ τ,χ smulaton model, l, N l 5,andM l. Fg. 6. CCFs for the COST 7 TU channel f max 9 Hz, µs, and τ max 7 µs. a r µ,l µ reference model, l.,lτ,χ b ˆr µ,l µ smulaton model, l, N l 5,andM l.,lτ,χ results of 9d and 8d for l are plotted n Fg. 5a and b, respectvely. A good agreement between r µ,l µ,lτ,χ and ˆr µ,l µ s also observed, as shown n Fg. 6,lτ,χ for l. B. Comparson of Correlaton Functons for Narrowband Raylegh Fadng Channels The correspondng correlaton functons of the narrowband smulaton model see a d] wll be compared wth those of the narrowband reference model see 4a 4d]. Smlarly, we can conclude that ˆr µ µ τ r µ µ τ always holds. Also, t can easly be shown that ˆr µ µ τ r µ µ τ, ˆr µ µ τ,χ r µ µ τ,χ, and ˆr µ µ τ,χ r µ µ as N and M. τ,χ As a good tradeoff between complexty and performance of the narrowband-channel smulator, N and M were selected. Fg. 7 demonstrates the excellent approxmaton result for ˆr µ µ τ r µ µ τ over the nterval,n/f max ]. The smulaton result by averagng 5 realzatons s also plotted n ths fgure for the purpose of valdaton. On the bass of the COST 7 RA channel 34], the numercal results of 4c and c wth the gven moderate values of N and M are shown n Fg. 8a and b, respectvely. It s clear that ˆr µ µ τ,χ matches r µ µ τ,χ very closely. Fg. 7. ACFs r µ µ τ of the reference model and ˆr µ µ τ of the smulaton model wth N and M for f max 9Hz. Fg. 9a and b ndcates that the approxmaton error between r µ µ τ,χ and ˆr µ µ wth the chosen values of N and M τ,χ s small. Compared wth the two determnstc sum-of-snusods channel smulators developed n 5] and 6], the proposed stochastc narrowband Raylegh fadng channel smulator enables better fttngs to the underlyng reference model, whle t has to pay hgher computatonal efforts due to the stochastc property and the used double sum of exponental functons. It should

11 6 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 7 Fg. 8. CCFs for the COST 7 RA channel f max 9 Hz and.86 µs. a r µ µ τ,χ of the reference model. b ˆr µ µ τ,χ of the smulaton model wth N and M. also be stressed here that ths stochastc narrowband-channel smulator has the smlar computatonal complexty to that of the stochastc sum-of-snusods channel smulators n ], 3], and 38]; however, t has the addtonal capablty of drectly smulatng multple frequency-correlated Raylegh fadng processes wth gven temporal and frequency correlaton propertes. V. C ONCLUSION In ths paper, we have proposed a novel stochastc WSS sum-of-snusods channel smulator, whch s capable of smulatng multple frequency-correlated wdeband and narrowband Raylegh fadng channels. Analytcal expressons have been derved for all the parameters of the smulaton model. The performance of the proposed channel smulator has been nvestgated wth respect to the ACFs and CCFs of the smulated processes. Three performance crtera have been employed n order to provde a gude on how to choose the number of exponental functons n the channel smulator. The numercal and smulaton results show that the correlaton propertes of the smulaton model wth the chosen numbers of exponental functons match very closely those of the underlyng reference model. The resultng stochastc channel smulator s very useful for the smulaton of practcal frequency-dversty fadng channels, such as FH, MC-CDMA, and OFDM channels. Fg. 9. CCFs for the COST 7 RA channel f max 9 Hz, and.86 µs. a r µ µ τ,χ of the reference model. b ˆr µ µ of the τ,χ smulaton model wth N and M. APPENDIX A. Dervaton of 9e In ths Appendx, we derve the CCF r µ,l µ of µ,lτ,χ,l t and µ,l t accordng to 8b: } r µ,l µ {µ,lτ,χe,l tµ,lt + τ lm N l M l M l n m p q E { C n,m,l C p,q,l cosπf max t cos β n,l πf c ϕ n,m,l ˆθ n,m,l sn πf max t + τcosβ p,l πf cϕ p,q,l ˆθ ] } p,q,l lm Cn,m,l N l M l n m snπf max τ cos β n,l πχϕ n,m,l lm pβ n,l,ϕ n,m,l N l M l n m snπf max τ cos β n,l πχϕ n,m,l dβdϕ

12 WANG et al.: STOCHASTIC MODELING AND SIMULATION OF WIDEBAND FADING CHANNELS 6 τ l + τ l+ / π τ l τ l / pβ,ϕ snπf max τ cos β πχϕdβdϕ τ l + τ l+ / π τ l τ l / π b exp ϕ cosπf max τ cos β snπχϕdβdϕ bj πf max τ τ l + τ l+ / τ l τ l / exp ϕ snπχϕdϕ. 7 The ndefnte ntegral n the rght-hand sde of the above equaton can be solved by usng 39, eq..663.] as follows: exp ϕ snπχϕdϕ exp ϕ snπχϕ πχ cosπχϕ] +πχ exp ϕ snπχϕ+πχ cosπχϕ]. 8 +πχ The substtuton of 8 to 7 leads to an expresson of r µ,l µ,lτ,χ shown n 9e. B. Dervaton of In ths Appendx, we apply the parameter-computaton method proposed n the study n 3] to derve a closed-form expresson for the dscrete delays φ m,l shown n. Close observatons of the expressons 8a 8f show us that the dscrete delays φ m,l appear only n 8d and 8e. Therefore, we wll calculate φ m,l n such a way that for a lmted number of exponental functons, ˆr µ µ,χ and ˆr µ,l µ can ap-,l,χ proxmate as well as possble r µ µ,χ and r µ,l µ,l,χ, respectvely. Begnnng wth the frst par ˆr µ µ and,χ r µ µ,χ, we obtan ther Fourer transforms as follows: Ŝ µ µ φ b A l B l 4M l m δφ φ m,l +δφ + φ m,l ] 9a S µ µ φ b 4 exp φ. 9b By comparng 9b wth 7, t s clear that S µ µ s related to the PDP, as s Ŝµ µ φ Let us ntroduce the ntervals φ. I m φ m,l,φ m,l ] wth φ,l τ l τ l /. We demand that the average delay power of the reference model s equal to the average delay power of the smulaton model wthn the nterval I m,.e., φ I m S µµ φdφ φ I m Ŝ µµ φdφ 3 for m,,...,m l. To solve ths problem, we defne an auxlary functon accordng to Gφ m,l : Applyng 9b, we can wrte Gφ m,l b 4 φ m,l S µ µ φdφ. 3 exp φ ] m,l. 3 If we keep 3 n mnd and use 9a and 3, another form of Gφ m,l can be obtaned: Gφ m,l τ l τ l / b 4 A l+ S µ µ φdφ + m k φ I k m k φ I k Ŝ µµ φdφ S µµ φdφ b 4 A l+ bm 4M l A l B l. 33 The comparson between 3 and 33 leads to an analytcal expresson for the dscrete delays φ m,l, as gven n. Applyng the above proposed computaton procedure, the same result for φ m,l can be acheved from another relaton par ˆr µ,l µ,l,χ and r µ,l µ,l,χ. C. Proof of ˆr µ,l µ τ,χ,,r µ,l µ for N,l,lτ,χ l and M l Performng the substtuton of 9a and 9b and nto 8e gves, for N l, and M l, the followng desred result: lm ˆr N l µ,l µ,lτ,χ M l lm N l M l { sn n N l + m π πf max τ sn N l πχ ln ba l B l 4N l M l n ] A l m M l A l B l ]}

13 6 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 7 lm N l lm N l n N l + ba l B l 4N l { π sn πf max τ sn n ] N l πχ ln n N l + b 4N l { sn πf max τ sn b π π π ln B l ln A l A l xa l B l ln B l ln A l π N l exp z snπf max τ sn y πχzdzdy b π π π ln B l ln A l exp z ]} dx n ] } πχz dz cosπf max τ sn ydy exp z snπχzdz bj πf max τ +πχ ] exp z snπχz+πχ cosπχz] zτ l + τ l+ / zτ l τ l / r µ,l µ,lτ,χ. 34 Thus, ˆr µ,l µ τ,χ converges to r µ,l µ f N,l,lτ,χ l and M l. ACKNOWLEDGMENT The authors would lke to thank the anonymous revewers for ther helpful and constructve comments. REFERENCES ] W. C. Jakes, Ed., Mcrowave Moble Communcatons. Pscataway, NJ: IEEE Press, 994. ] T. C. Wu, C. C. Chao, and K. C. Chen, Capacty of synchronous coded DS SFH and FFH spread-spectrum multple-access for wreless local communcatons, IEEE Trans. Commun., vol. 45, no., pp., Feb ] Z. Kostc, I. Marc, and X. D. Wang, Fundamentals of dynamc FH n cellular systems, IEEE J. Sel. Areas Commun.,vol.9,no.,pp.54 66, Nov.. 4] N. C. Beauleu, Generaton of correlated Raylegh fadng envelopes, IEEE Commun. Lett., vol. 3, no. 6, pp. 7 74, Jun ] B. Natarajan, C. R. Nassar, and V. Chandrasekhar, Generaton of correlated Raylegh fadng envelopes for spread spectrum applcatons, IEEE Commun. Lett., vol. 4, no., pp. 9, Jan.. 6] S. Sorooshyar and D. G. Daut, Generaton of correlated Raylegh fadng envelopes for accurate performance analyss of dversty systems, n Proc. IEEE PIMRC, Bejng, Chna, Sep. 3, pp ] T. Klngenbrunn and P. Mogensen, Modelng frequency correlaton of fast fadng n frequency hoppng GSM lnk smulatons, n Proc. IEEE VTC Fall, Amsterdam, The Netherlands, Sep. 999, pp ] N. C. Beauleu and M. L. Meran, Effcent smulaton of correlated dversty channels, n Proc. IEEE WCNC, Chcago, IL, Sep., pp. 7. 9] S. O. Rce, Mathematcal analyss of random nose, Bell Syst. Tech. J., vol. 3, pp. 8 33, Jul ] S. O. Rce, Mathematcal analyss of random nose, Bell Syst. Tech. J., vol. 4, pp , Jan ] M. Pätzold, Moble Fadng Channels. New York: Wley,. ] P. Höher, A statstcal dscrete-tme model for the WSSUS multpath channel, IEEE Trans. Veh. Technol., vol. 4, no. 4, pp , Nov ] K. W. Yp and T. S. Ng, Karhunen-Loève expanson of the WSSUS channel output and ts applcaton to effcent smulaton, IEEE J. Sel. Areas Commun., vol. 5, no. 4, pp , May ] E. Chavaccn and G. M. Vtetta, GQR models for multpath Raylegh fadng channels, IEEE J. Sel. Areas Commun., vol. 9, no. 6, pp. 9 8, Jun.. 5] M. Pätzold and N. Youssef, Modelng and smulaton of drectonselectve and frequency-selectve moble rado channels, Int. J. Electr. Commun., vol. AEÜ-55, no. 6, pp , Nov.. 6] Q. Yao and M. Pätzold, Down-lnk channel modelng for moble communcatons wth smart antennas at the base staton, n Proc. IEEE VTC Sprng, Jeju, Korea, Apr. 3, pp ] C. X. Wang and M. Pätzold, A generatve determnstc model for dgtal moble fadng channels, IEEE Commun. Lett., vol.8,no.4,pp.3 5, Apr. 4. 8] C. X. Wang and M. Pätzold, A new determnstc process based generatve model for characterzng bursty error sequences, n Proc. IEEE PIMRC, Barcelona, Span, Sep. 4, pp ] P. Dent, G. E. Bottomley, and T. Croft, Jakes fadng model revsted, Electron. Lett., vol. 9, no. 3, pp. 6 63, Jun ] Y. B. L and Y. L. Guan, Modfed Jakes model for smulatng multple uncorrelated fadng waveforms, n Proc. IEEE ICC, New Orleans, LA, Jun., pp ] Y. R. Zheng and C. Xao, Improved models for the generaton of multple uncorrelated Raylegh fadng waveforms, IEEE Commun. Lett., vol. 6, no. 6, pp , Jun.. ] Y. L and X. Huang, The smulaton of ndependent Raylegh faders, IEEE Trans. Commun., vol. 5, no. 9, pp , Sep.. 3] C. X. Wang and M. Pätzold, Methods of generatng multple uncorrelated Raylegh fadng processes, n Proc. IEEE VTC Sprng, Jeju, Korea, Apr. 3, pp ] C. X. Wang and M. Pätzold, Effcent smulaton of multple crosscorrelated Raylegh fadng channels, n Proc. IEEE PIMRC, Bejng, Chna, Sep. 3, pp ] M. Pätzold, F. Laue, and U. Kllat, A frequency hoppng Raylegh fadng channel smulator wth gven correlaton propertes, n Proc. IEEE ISPACS, Kuala Lumpur, Malaysa, Nov. 997, pp. S8.. S ] C. X. Wang, M. Pätzold, and B. Itsaracha, A determnstc frequency hoppng Raylegh fadng channel smulator desgned by usng optmzaton technques, n Proc. IEEE PIMRC, Lsbon, Portugal, Sep., pp ] J. D. Parsons, The Moble Rado Propagaton Channel, nd ed. New York: Wley,. 8] G. L. Stüber, Prncples of Moble Communcatons, nd ed. Boston, MA: Kluwer,. 9] R. H. Clarke, A statstcal theory of moble-rado recepton, Bell Syst. Tech. J., vol. 47, no. 6, pp. 957, Jul./Aug ] M. F. Pop and N. C. Beauleu, Lmtatons of sum-of-snusods fadng channel smulators, IEEE Trans. Commun., vol. 49, no. 4, pp , Apr.. 3] M. F. Pop and N. C. Beauleu, Desgn of wde-sense statonary sumof-snusods fadng channel smulators, n Proc. IEEE ICC, New York, Apr. 8 May, pp ] M. Pätzold, On the statonarty and ergodcty of fadng channel smulators basng on Rce s sum-of-snusods, n Proc. IEEE PIMRC, Bejng, Chna, Sep. 3, pp

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M. Ryzhk, Tables of Integrals, Seres, and Products, 6th ed. Boston, MA: Academc,. Cheng-Xang Wang S M 5 receved the B.Sc. and M.Eng. degrees n communcaton and nformaton systems from Shandong Unversty, Shandong, Chna, n 997 and, respectvely, and the Ph.D. degree n wreless communcatons from Aalborg Unversty, Aalborg, Denmark, n 4. From to, he was a Research Assstant wth the Department of Communcaton Networks, Techncal Unversty of Hamburg Harburg, Hamburg, Germany. From to 5, he was a Research Fellow of moble communcatons wth Agder Unversty College, Grmstad, Norway. From January to Aprl 4, he was a Vstng Researcher at the Baseband Algorthms and Standardzaton Laboratory, Semens AG- Moble Phones, Munch, Germany, conductng research and development of error models for Enhanced General Packet Rado Servce EGPRS systems wthn the framework of the 3GPP GERAN System Concept R&D Project. Snce 5, he has been a Lecturer n Moble Communcatons and Networks at Herot Watt Unversty, Ednburgh, U.K. He s also an honorary fellow of the Unversty of Ednburgh and has been an Adjunct Professor wth Guln Unversty of Electronc Technology, Guln, Chna, snce June 6. Hs current research nterests nclude moble propagaton channel modelng, error models, smart antennas and vrtual multple-nput multple-output systems, orthogonal frequency-dvson multplexng, ultrawdeband, cogntve rado, space-tme codng, cross-layer desgn of wreless networks, moble ad hoc and wreless-sensor networks, and 3GPP long-term evoluton. He has publshed about 7 research papers n journals and conference proceedngs. Dr. Wang was the recpent of the 999 Excellent Paper Award for two of hs papers from the Sxth Natonal Youth Communcaton Conference of Chna, Bejng, Chna. He serves as an Edtoral Board Member of the Wreless Communcatons and Moble Computng WCMC and as a Techncal Program Commttee Member for 4 major nternatonal conferences, ncludng IEEE VTC 5-Fall, ICIC 6, Globecom 6, WCNC 7, ICC 7, and ICIC 7. He also served as Sesson Char for ICCCAS 6. He s a member of the Insttute of Engneerng and Technology IET. Matthas Pätzold M 94 SM 98 receved the Dpl.-Ing. and Dr.-Ing. degrees n electrcal engneerng from Ruhr-Unversty Bochum, Bochum, Germany, n 985 and 989, respectvely, and the Habltaton degree n communcatons engneerng from the Techncal Unversty of Hamburg Harburg, Hamburg, Germany, n 998. From 99 to 99, he was wth ANT Nachrchtentechnk GmbH, Backnang, Germany, where he was engaged n dgtal satellte communcatons. From 99 to, he was wth the Department of Dgtal Networks at the Techncal Unversty Hamburg Harburg. Snce, he has been a Full Professor of moble communcatons wth Agder Unversty College, Grmstad, Norway. He s the Author of the books Moble Rado Channels Modellng, Analyss, and Smulaton n German Veweg, 999 and Moble Fadng Channels Wley & Sons,. Hs current research nterests nclude moble-rado communcatons, partcularly multpath fadngchannel modelng, multple-nput multple-output systems, channel-parameter estmaton, and coded-modulaton technques for fadng channels. Prof. Pätzold was the recpent of the 998 and Neal Shepherd Memoral Best Propagaton Paper Award from the IEEE Vehcular Technology Socety. He was the recpent of the 3 Excellent Paper Award from the IEEE Internatonal Symposum on Personal, Indoor, and Moble Rado Communcatons PIMRC 3 n Bejng, Chna, as well as of the Best Paper Award from the eghth Internatonal Symposum on Wreless Personal Multmeda Communcatons WPMC 5 n Aalborg, Denmark. He was Local Organzer of the conference Kommunkaton n Vertelten Systemen KVS and Organzer of the second Internatonal Workshop on Research Drectons n Moble Communcatons and Servces. He served as a member of the Techncal Program Commttee for IST 5, VTC 5-Fall, and WPMC 5. He also served as Sesson Char for several reputed nternatonal conferences VTC 4-Sprng, NRS 4, PIMRC 4, IST 5, and WPMC 5. Q Yao receved the B.Sc. and M.Eng. degrees n communcaton and nformaton systems from Shandong Unversty, Shandong, Chna, n 997 and, respectvely. She s currently workng toward the Ph.D. degree at Aalborg Unversty, Aalborg, Denmark. In, she worked as a Student Research Assstant wth the Department of Communcaton Networks, Techncal Unversty Hamburg Harburg, Hamburg, Germany. Snce October, she has been wth Agder Unversty College, Grmstad, Norway. Her current research nterests nclude moble communcatons, adaptve antenna systems, moble fadng-channel modelng, and channel-parameter estmaton.