Statistical Properties of the Capacity of Double Nakagami-m Channels for Applications in V2V Dualhop Communication Systems

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1 9 Statstcal Propertes of the Capacty of Double Nakagam-m Channels for Applcatons n VV Dualhop Communcaton Systems Gulzab Rafq 1, Bjørn Olav Hogstad and Matthas Pätzold 3 1,3 Unversty of Agder Unversty of Navarra 1,3 Norway Span 1. Introducton The desgn and analyss of cascaded fadng models has been an actve area of research n recent years due to ts applcatons n numerous real world scenaros such as keyhole channels (Salo et al., 006; Zlatanov et al., 008), and multhop communcaton systems (Andersen, 00; Karagannds et al., 007; Talha & Pätzold, 007; Velkov et al., 009). It s shown n (Chzhk et al., 00; Ercerg et al., 1997) that n the presence of a keyhole, the fadng between each transmt and receve antenna par n a mult-nput mult-output (MIMO) system can be characterzed usng a double 1 Raylegh process. Afterwards, ths model has been extended to the double Nakagam-m fadng model n (Shn & Lee, 004). In (Salo et al., 006), the authors have lsted a few real world scenaros whch gve rse to the keyhole effect. Two such scenaros nclude dffracton through the street edges n urban mcrocellular envronments (Ercerg et al., 1997) and traversal of the propagaton paths through a narrow space for the case when the dstance between the rngs of scatterers around the transmtter and recever s large (Gesbert et al., 00). Multhop communcaton systems on the other hand fall under the category of cooperatve dversty technques (Laneman et al., 004; Sendonars et al., 003). In such systems, the wreless nodes (n a cooperatve network) assst each other by relayng the nformaton from the source moble staton (SMS) to the destnaton moble staton (DMS), hence mprovng the network coverage qute sgnfcantly. If however, the wreless nodes are assumed to be movng wth relatvely hgh speed, the concept of multhop communcaton can be appled to vehcle-to-vehcle (VV) communcaton systems, where the source vehcle (SV) or the traffc control center (TCS) communcates wth a destnaton vehcle (DV) va relay vehcles (RV). Fgure 1 depcts these two scenaros for the case of dualhop communcaton, where the nformaton from the TCS (or the SV) s receved at the DV va an RV. VV communcaton has receved a lot of attenton n recent years due to ts applcatons n traffc safety and road traffc flow (Gradnescu et al., 007). The statstcal analyss of the receved sgnal envelope under non-lne-of-sght (NLOS) propagaton condtons n an amplfy-and-forward based 1 Throughout ths chapter, we wll refer to a double process as the product of two ndependent but not necessarly dentcal processes.

2 154 Vehcular Technologes: Increasng Connectvty dualhop communcaton system can be found n (Patel et al., 006), where the overall channel between the transmtter and the recever s modeled usng a double Raylegh process. Ths model s then extended to the double Rce channel model n (Talha & Pätzold, 007), by takng the lne-of-sght propagaton condtons nto account. The statstcal propertes of the capacty of double Rce channels have been analyzed n (Rafq & Pätzold, 008). However, the Nakagam-m process s consdered to be a more general channel model as compared to the Rce and Raylegh channel models. Hence, to generalze all the aforementoned works n the regme of multhop communcaton, the authors of (Karagannds et al., 007) have presented the statstcal analyss of the N Nakagam-m model (.e., a product of N Nakagam-m processes). Moreover, second order statstcs for the double Nakagam-m process can be found n (Zlatanov et al., 008). Though a lot of papers have been publshed n the lterature employng the cascaded fadng channel model, the statstcal propertes of the capacty of double Nakagam-m channels have not been nvestgated so far. Such channels fnd applcatons both n VV communcaton systems employng dualhop communcaton, and keyhole channels (Zlatanov et al., 008). In ths chapter, we have studed the statstcal propertes of the capacty of double Nakagam-m channels. Specfcally, the nfluence of the severty of fadng on the statstcal propertes of the capacty of double Nakagam-m channels s analyzed. We have derved exact analytcal expressons for the probablty densty functon (PDF), the cumulatve dstrbuton functon (CDF), the level-crossng rate (LCR), and the average duraton of fades (ADF) of the channel capacty. Here, the LCR and ADF of the channel capacty are mportant characterstc quanttes whch provde nsght nto the temporal behavor of the channel capacty (Gorgett et al., 003), (Hogstad & Pätzold, 004). Our analyss has revealed that f the fadng severty n one or both lnks of double Nakagam-m channels decreases (.e., ncreasng the value of the severty parameter m n one or both of the cascaded Nakagam-m processes), the mean channel capacty ncreases, whle the ADF of the channel capacty decreases. Moreover, ths effect results n an ncrease n the LCR of the channel capacty at lower sgnal levels.. The double Nakagam-m channel model In ths chapter, we have consdered Scenaro n Fg. 1, where the channel between the SV and the DV va an RV s represented as a concatenaton of the SV-RV and RV-DV channels (Patel et al., 006; Talha & Pätzold, 007). Moreover, we have assumed that the fadng n the SV-RV lnk and the RV-DV lnk s characterzed by Nakagam-m processes denoted by χ 1 (t) and χ (t), respectvely. Hence, the overall fadng channel descrbng the SV-DV lnk s modelled by a double Nakagam-m process gven by (Kovacs et al., 00; Zlatanov et al., 008) Ξ(t)=A RV χ 1 (t)χ (t) (1) where A RV s a real postve constant representng the relay gan. The PDF p χ (z) of the Nakagam-m process χ (t)( = 1, ) s gven by (Nakagam, 1960) z m 1 p χ (z)= mm Γ(m )Ω m e m z Ω, z 0 () The materal n ths chapter s based on Statstcal Propertes of the Capacty of Double Nakagam-m Channels, by Gulzab Rafq, Bjørn Olav Hogstad and Matthas Pätzold whch appeared n the proceedngs of 5th IEEE Internatonal Symposum on Wreless Pervasve Computng, ISWPC 010, Modena, Italy, May IEEE.

3 Statstcal Propertes of the Capacty of Double Nakagam-m Channels for Applcatons n VV Dualhop Communcaton Systems 155 Fg. 1. The propagaton scenaros descrbng double Nakagam-m fadng channels. where Ω = E { χ (t)}, m = Ω / { Var χ (t) },andγ( ) represents the gamma functon (Gradshteyn & Ryzhk, 000). The parameter m controls the severty of the fadng. Increasng the value of m, decreases the severty of fadng and vce versa. The PDF of double Nakagam-m processes Ξ(t) s gven by (Karagannds et al., 007) p Ξ (z)= 4z m 1+m 1 ) m Γ (m ) ( K / ) m1 m (z (m1, +m Ώ )/ m Ώ z 0 (3) where Ώ 1 = A RV Ω 1, Ώ = Ω,andK n ( ) denotes the modfed Bessel functon of the second knd of order n (Gradshteyn & Ryzhk, 000, Eq. (8.43/1)). In order to derve the expressons for the PDF, CDF, LCR, and ADF of the capacty of double Nakagam-m channels, we need the jont PDF p Ξ Ξ (z, ż) of the squared process Ξ (t) and ts tme dervatve Ξ (t), aswellas

4 156 Vehcular Technologes: Increasng Connectvty the PDF p Ξ (z) of Ξ (t). The jont PDF p Ξ Ξ (z, ż) can be found by followng the procedure presented n (Zlatanov et al., 008) for the jont PDF p Ξ Ξ (z, ż) and then by usng the concept of transformaton of random varables (Papouls & Plla, 00, Eq. (7-8)), whch results n p Ξ Ξ (z, ż) = 1 4z p Ξ Ξ ( z, where and = zm 3/ π [ ż z ) m m ] m Ώ Γ (m ) 0 x m 1 m 1 e zβ1 + x x β Ω β 1 = 1 π ( ) fmax m 1 + fmax 1 zm x Ω e ż 8z( ) zβ1 + x m 1 x +x Ω β 1 dx, z 0, ż < (4) (5a) Ω β = π ( ) fmax m + fmax 3. (5b) Here, f max1, f max,and f max3 represent the maxmum Doppler frequences of the SV, RV, and DV, respectvely. The expresson for the PDF p Ξ (z) can be obtaned by ntegratng the jont PDF p Ξ Ξ (z, ż) over ż. Alternatvely, the PDF p Ξ (z) can also be found from the PDF p Ξ(z) n (3) as follows p Ξ (z) = 1 z p Ξ( z) = z m 1 +m Γ (m ) ( / ) (m1 +m Ώ )/ m K m1 m ( z ) m Ώ, z 0. (6) The expressons presented n (4) and (6) wll be used n the next secton to calculate the PDF and LCR of the channel capacty. 3. Statstcal propertes of the capacty of double Nakagam-m channels The nstantaneous capacty C(t) of double Nakagam-m channels s defned as (Nabar et al., 004) C(t)= 1 ( log 1 + γ s Ξ(t) ) = 1 ( ) log 1 + γ s Ξ (t) (bts/s/hz) (7) where γ s denotes the average receved sgnal-to-nose rato (SNR) at the DV. The factor 1 / n (7) s due to the fact that the RV n Fg. 1 operates n a half-duplex mode, and hence the sgnal transmtted from the SV s receved at the DV n two tme slots. Equaton (7) can be consdered as a mappng of a random process Ξ(t) to another random process C(t). Hence, the expressons for the statstcal propertes of the channel capacty C(t) can be found by usng the results for the statstcal propertes of the process Ξ(t) obtaned n the prevous secton. The PDF p C (r) of the channel capacty C(t) can be found n closed form wth the help of the PDF

5 Statstcal Propertes of the Capacty of Double Nakagam-m Channels for Applcatons n VV Dualhop Communcaton Systems 157 p Ξ (z) and by applyng the concept of transformaton of random varables (Papouls & Plla, 00, Eq. (7-8)) as ( ) p C (r) = ( ln() r ) 1 p γ Ξ s γ s = r+ ln() (( r 1 )/ ) (m1 +m γ )/ s ( r 1) Γ (m ) ( K / ) m1 m 1 m (m1 +m Ώ )/ γ s, m Ώ r 0. (8) The CDF F C (r) of the channel capacty C(t) can now be derved by ntegratng the PDF p C (r) and by makng the use of relatonshps n (Gradshteyn & Ryzhk, 000, Eq. (9.34/3)) and (Adamchk & Marchev, 1990, Eq. (6)) as F C (r) = = r p C (x)dx 0 1 Γ (m ) G,1 1,3 [ r 1 γ s ( m Ώ ) ] 1, r 0 (9) m 1, m,0 where G[ ] denotes the Mejer s G-functon (Gradshteyn & Ryzhk, 000, Eq. (9.301)). The LCR N C (r) of the channel capacty descrbes the average rate of up-crossngs (or down-crossngs) of the capacty through a certan threshold level r.inordertofndthelcrn C (r), we frst need to fnd the jont PDF p (z, ż) of C(t) and ts tme dervatve Ċ(t). The jont PDF p CĊ CĊ (z, ż) can be obtaned by usng the jont PDF p Ξ Ξ (z, ż) gven n (4) as p CĊ (z, ż)= ( = ) ( z+1 ln() pξ γ Ξ s z 1, γ s ( z+1 ln() ) ( z 1 ) m 3 [ ( ) ] πγs γ m m Ώ s m Γ (m ) 0 ) ż ln() / γ s z x m 1 m 1 e ( z 1)β 1 + x γ s x β ( e x m 1 for z 0and ż <. Fnally, the LCR N C (r) can be found as follows N C (r) = żp CĊ (r, ż)dż 0 = 8 π ( r 1 γ s ) m 1 [ m m ] m Ώ Γ (m ) 0 e (r 1)m γs x Ω ( z+1 ln()ż) 8γs( 1)( z ( z ) 1)β 1 γs x +x β ) + (z 1)m Ω 1 γs x Ω ( r 1)β 1 γ s x + x β x 1+m m 1 dx (10) e x m 1 Ω 1 dx (11) for r 0. The ADF T C (r) of the channel capacty C(t) denotes the average duraton of tme over whch the capacty s below a gven level r (Hogstad & Pätzold, 004; 007). The ADF

6 158 Vehcular Technologes: Increasng Connectvty T C (r) of the channel capacty can be expressed as (Hogstad & Pätzold, 007) T C (r)= F C(r) N C (r) (1) where F C (r) and N C (r) are gven by (9) and (11), respectvely. 4. Statstcal propertes of the capacty of double raylegh channels The double Raylegh channel follows as a specal case of the double Nakagam-m channel when m = 1 ( = 1, ). Hence,bylettngm = 1 ( = 1, ) n (8), (9), and (11), the PDF, CDF, and LCR of the capacty of double Raylegh channels can be expressed as and p C (r) m =1 = r ln() γ s σ σ RV ( ) K r 1 0 γ s σ, r 0 (13) σ RV F C (r) ( ) m =1 = 1 r 1 γ s σ K r 1 1 σ RV γ s σ, r 0 (14) σ RV N C (r) m =1 = r 1 ( πγs σ β + β r ) 1 1 σ RV x 4 e r 1 σ γs x e x σ RV dy, r 0 (15) γ s 0 respectvely. The ADF of the capacty C(t) of double Raylegh channels can be found usng (1), (14), and (15). In (13) (15), σ RV = A RV σ 1 and σ ( = 1, ) represent the varances of the underlyng Gaussan processes n the correspondng Raylegh processes χ (t) m =1 ( = 1, ). 5. Numercal results In ths secton, we wll dscuss the analytcal results obtaned n the prevous secton. The valdty of the theoretcal results s confrmed wth the help of smulatons. For comparson purposes, we have also shown the results for double Raylegh channels, whch represent a specal case of double Nakagam-m channels. In order to generate Nakagam-m processes χ (t), we have used the followng relatonshp (Yacoub et al., 1999) χ (t)= m l=1 μ,l (t) (16) where μ,l (t) (l = 1,,..., m ; = 1, ) are the underlyng ndependent and dentcally dstrbuted (..d.) Gaussan processes, and m s the parameter of the Nakagam-m dstrbuton assocated wth the th lnk of the dualhop communcaton systems. The Gaussan processes μ,l (t), eachwthzeromeanandvarancesσ0, were smulated usng the sum-of-snusods model (Pätzold, 00). The model parameters were computed usng the generalzed method of exact Doppler spread (GMEDS 1 ) (Pätzold et al., 009). The number of snusods for the generaton of Gaussan processes μ,l (t) was chosen to be N = 9. The parameter Ω was chosen to be equal to m σ0. Unless stated otherwse, the values of the maxmum Doppler frequences f max1, f max,and f max3 were taken to be 0, 91, and 15 Hz, respectvely. The SNR

7 Statstcal Propertes of the Capacty of Double Nakagam-m Channels for Applcatons n VV Dualhop Communcaton Systems 159 γ s was set to 15 db. The parameters A RV and σ 0 were chosen to be unty. Fnally, usng (16), (1), and (7), the smulaton results for the statstcal propertes of the channel capacty were found. The PDF and CDF of the channel capacty of double Nakagam-m channels are presented n Fgs. and 3, respectvely. Both fgures llustrate the fact that ncreasng the value of the severty parameter m (.e., a decrease n the level of the severty of fadng) n one or both lnks of the double Nakagam-m channels results n an ncrease n the mean channel capacty. Ths result s specfcally presented n Fg. 4, where the mean channel capacty s studed for dfferent values of the severty parameter m ( = 1, ). It can also be seen that double Raylegh channels (m = 1; = 1, ) have a lower mean channel capacty as compared to the mean channel capacty of double Nakagam-m channels (m = ; = 1, ). Moreover, t can also be observed from Fgs. and 3 that ncreasng the value of the severty parameter m decreases the varance of the channel capacty. Fg.. The PDF p C (r) of the capacty of double Nakagam-m channels. Fgure 5 presents the LCR N C (r) of the capacty C(t) of double Nakagam-m channels. It s observed that an ncrease n the level of severty of fadng n one or both lnks of double Nakagam-m channels ncreases the LCR N C (r) of the channel capacty at low levels r.hence, at low levels r,thelcrn C (r) of the capacty of double Raylegh channels (m = 1; = 1, ) s hgher as compared to that of double Nakagam-m channels (m = ; = 1, ). However, the converse statement s true for hgher levels r. The ADF of the capacty of double Nakagam-m channels s shown n Fg. 6. It s evdent from ths fgure that the ADF of the capacty decreases wth an ncrease n the value of the severty parameter m ( = 1, ). Fgures 7 and 8 study the nfluence of the maxmum Doppler frequences of the RV and the DV on the LCR and ADF of the channel capacty. It can clearly be observed n Fgs. 7 and 8 that the LCR and ADF are strongly dependent on the Doppler frequences of the RV and the DV. Ths means that the moblty of the RV and the DV has a sgnfcant nfluence on the LCR and ADF of the channel capacty. It s observed that ncreasng the maxmum Doppler frequences

8 160 Vehcular Technologes: Increasng Connectvty Fg. 3. The CDF F C (r) of the capacty of double Nakagam-m channels. Fg. 4. The mean channel capacty of double Nakagam-m channels for dfferent levels of fadng severty. f max and f max3 results n a sgnfcant ncrease n the LCR. However, the ADF decreases by ncreasng the maxmum Doppler frequences of the RV and the DV. 6. Concluson Ths Chapter presents the dervaton of exact analytcal expressons for the statstcal propertes of the capacty of double Nakagam-m channels, whch fnds applcatons n VV communcaton systems employng dualhop communcaton, and keyhole channels. We have studed the nfluence of the severty of fadng on the PDF, CDF, LCR, and ADF of the channel

9 Statstcal Propertes of the Capacty of Double Nakagam-m Channels for Applcatons n VV Dualhop Communcaton Systems 161 Fg. 5. The LCR N C (r) of the capacty of double Nakagam-m channels. Fg. 6. The ADF T C (r) of the capacty of double Nakagam-m channels. capacty. It s observed that an ncrease n the severty of fadng n one or both lnks of double Nakagam-m channels decreases the mean channel capacty, whle t results n an ncrease n the ADF of the channel capacty. Moreover, at lower sgnal levels, ths effects ncreases the LCR of the channel capacty. Results also show that the moblty of the RV and DV has a sgnfcant nfluence on the LCR and ADF of the channel capacty. Specfcally, an ncrease n the maxmum Doppler frequences of the RV and DV ncreases the LCR, whle t has an opposte nfluence on the ADF of the channel capacty. The presented exact results are valdated wth the help of smulatons, whereby a very good fttng s observed.

10 16 Vehcular Technologes: Increasng Connectvty Fg. 7. The LCR N C (r) of the capacty of double Nakagam-m channels. Fg. 8. The ADF T C (r) of the capacty of double Nakagam-m channels. 7. Acknowledgment The contrbuton of G. Rafq and Prof. M. Pätzold n ths chapter was partally supported by the Research Councl of Norway (NFR) through the project /S10 enttled Optmzed Heterogeneous Multuser MIMO Networks OptMO". The contrbuton of Dr. B. O. Hogstad was supported n part by the Basque Government through the MIMONET project (PC009-7B), and by the Spansh Mnstry of Scence and Innovaton through the projects COSIMA (TEC C04-0) and COMONSENS (CSD ).

11 Statstcal Propertes of the Capacty of Double Nakagam-m Channels for Applcatons n VV Dualhop Communcaton Systems References Adamchk, V. S. & Marchev, O. I. (1990). The algorthm for calculatng ntegrals of hypergeometrc type functons and ts realzaton n REDUCE system, Proc. Int. Symp. Symbolc and Algebrac Computaton, ISSAC 90, Tokyo, Japan, pp Andersen, J. B. (00). Statstcal dstrbutons n moble communcatons usng multple scatterng, Proc. 7th URSI General Assembly, Maastrcht, Netherlands. Chzhk, D., Foschn, G. J., Gans, M. J. & Valenzuela, R. A. (00). Keyholes, correlatons, and capactes of multelement transmt and receve antennas, IEEE Trans. Wreless Commun. 1(): Ercerg, V., Fortune, S. J., Lng, J., Rustako Jr., A. J. & Valenzuela, R. A. (1997). Comparson of a computer-based propagaton predcton tool wth expermental data collected n urban mcrocellular envronment, IEEE J. Select. Areas Commun. 15(4): Gesbert, D., Bölcske, H., Gore, D. A. & Paulraj, A. J. (00). Outdoor MIMO wreless channels: Models and performance predcton, IEEE Trans. Commun. 50(1): Gorgett, A., Smth, P. J., Shaf, M. & Chan, M. (003). MIMO capacty, level crossng rates and fades: The mpact of spatal/temporal channel correlaton, J. Commun. Net. 5(): Gradnescu, V., Gorgorn, C., Daconescu, R. & Crstea, V. (007). Adaptve traffc lghts usng car-to-car communcaton, Proc. 65th IEEE Semannual Vehcular Technology Conference, IEEE VTC 007-Sprng, Dubln, Ireland, pp Gradshteyn, I. S. & Ryzhk, I. M. (000). Table of Integrals, Seres, and Products, 6th edn, Academc Press. Hogstad, B. O. & Pätzold, M. (004). Capacty studes of MIMO models based on the geometrcal one-rng scatterng model, Proc. 15th IEEE Int. Symp. on Personal, Indoor and Moble Rado Communcatons, PIMRC 004, Vol. 3, Barcelona, Span, pp Hogstad, B. O. & Pätzold, M. (007). Exact closed-form expressons for the dstrbuton, level-crossng rate, and average duraton of fades of the capacty of MIMO channels, Proc. 65th Semannual Vehcular Technology Conference, IEEE VTC 007-Sprng, Dubln, Ireland, pp Karagannds, G. K., Sagas, N. C. & Mathopoulos, P. T. (007). N*Nakagam: A novel stochastc model for cascaded fadng channels, IEEE Trans. Commun. 55(8): Kovacs, I. Z., Eggers, P. C. F., Olesen, K. & Petersen, L. G. (00). Investgatons of outdoor-to-ndoor moble-to-moble rado communcaton channels, Proc. IEEE 56th Veh. Technol. Conf., VTC 0-Fall, Vancouver BC, Canada, pp Laneman, J. N., Tse, D. N. C. & Wornell, G. W. (004). Cooperatve dversty n wreless networks: Effcent protocols and outage behavor, IEEE Trans. Inform. Theory 50(1): Nabar, R. U., Bölcske, H. & Kneubühler, F. W. (004). Fadng relay channels: Performance lmts and space-tme sgnal desgn, IEEE J. Select. Areas Commun. : Nakagam, M. (1960). The m-dstrbuton: A general formula of ntensty dstrbuton of rapd fadng, n W. G. Hoffman (ed.), Statstcal Methods n Rado Wave Propagaton, Oxford, UK: Pergamon Press. Papouls, A. & Plla, S. U. (00). Probablty, Random Varables and Stochastc Processes, 4th edn, New York: McGraw-Hll.

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13 Vehcular Technologes: Increasng Connectvty Edted by Dr Mguel Almeda ISBN Hard cover, 448 pages Publsher InTech Publshed onlne 11, Aprl, 011 Publshed n prnt edton Aprl, 011 Ths book provdes an nsght on both the challenges and the technologcal solutons of several approaches, whch allow connectng vehcles between each other and wth the network. It underlnes the trends on networkng capabltes and ther ssues, further focusng on the MAC and Physcal layer challenges. Rangng from the advances on rado access technologes to ntellgent mechansms deployed to enhance cooperatve communcatons, cogntve rado and multple antenna systems have been gven partcular hghlght. How to reference In order to correctly reference ths scholarly work, feel free to copy and paste the followng: Gulzab Rafq, Bjørn Olav Hogstad and Matthas Paẗzoldt (011). Statstcal Propertes of the Capacty of Double Nakagam-m Channels for Applcatons n VV Dualhop Communcaton Systems, Vehcular Technologes: Increasng Connectvty, Dr Mguel Almeda (Ed.), ISBN: , InTech, Avalable from: InTech Europe Unversty Campus STeP R Slavka Krautzeka 83/A Rjeka, Croata Phone: +385 (51) Fax: +385 (51) InTech Chna Unt 405, Offce Block, Hotel Equatoral Shangha No.65, Yan An Road (West), Shangha, 00040, Chna Phone: Fax: