270 Practices:channel modeling for modern communication systems D=2f
270 Practices: channel modeling for modern communication systems pn1 H )( 2 n s )( 1 n s X R XT )(n3 s To Toc Rq Rqc pn2 1qn H H 2qn H n1 [ n2 [ pn3 H 3qn H )(n3 [ T )( 3 n [ R nn 21 H D 2 f RT RR n1 I T n2 I T n3 IT n1 I R n2 I R n3 I R R J T J E R E T 2a G T G R OT OR LoS Rq I H pq H Fig. 9.7 A generic channel model combining a two-ring model and an ellipse model with LoS components, single- and doublebounced rays for a MIMO V2V channel (MT = MR = 2)
Practices:channel modeling for modern communication systems 271 where 0=V不+ Kagog e-j2xfeTmci2frm.t.com+)+2 n.tcom() (9.34a) -学-V2-=立衣 xdf2=rtm(ego-n）+2sn-tm(o0-nj】 (9.34b) 0=V思 xe2frma.t cos(g)-r)+2 co-(og》-7nj月 (9.34c) ntp广光出波高 andspecify how much the single-and dou ounced rays contribute to the total s varables with unifor distributions over anthe imum Dopeie tthedpivNthatthe oDd ar independent for douboue nd a (scenano)can be expresed s Eog-kgoR cos(5-BR) (9.35) Eon RT -koor cos(of)-Br) (9.36) en1g≈En,-kaog cos(6g）-BR) (9.37刀 eoa≈6m-kaor cos(pg}-r） 9.38) g≈RR-koon cos(-g)-n）】 (9.39) eoma≈al-kadr cos(-r） (9.40) ng≈a)-koon cos(©g-R) (9.41) D-RT cos+R co (9.42) whee%≈，D-kras所，=(D2+碍-29,D+R陨+2 DRo,= h2+fP+2 af cos）/k+f cos）,Lb吼a+fcos）,k。=(ur-2o+1)/2,andk,=(Mr-2g+l/2.Hereb
Practices: channel modeling for modern communication systems 271 where h LoS oq (t) = s KoqΩoq Koq + 1 e −j2πfcτoq e j h 2πfTmax t cos π−φ LoS Rq +γT +2πfRmax t cos φ LoS Rq −γR i (9.34a) h SB oq (t) = X I i=1 h SBi oq (t) = X I i=1 s ηSBiΩoq Koq + 1 lim Ni→∞ X Ni ni=1 1 √ Ni e j(ψni−2πfcτoq,ni ) ×e j h 2πfTmax t cos φ (ni ) T −γT +2πfRmax t cos φ (ni ) R −γR i (9.34b) h DB oq (t) = s ηDBΩoq Koq + 1 lim N1,N2→∞ N X1,N2 n1,n2=1 1 √ N1N2 e j(ψn1,n2−2πfcτoq,n1,n2 ) ×e j h 2πfTmax t cos φ (n1) T −γT +2πfRmax t cos φ (n2) R −γR i . (9.34c) In (9.34), τoq=εoq/c, τoq,ni=(εoni+εniq)/c, and τoq,n1,n2=(εon1+εn1n2+εn2q)/c are the travel times of the waves through the link To−Rq, To−s (ni)−Rq, and To−s (n1)−s (n2)−Rq, respectively. Here, c is the speed of light and I=3. The symbols Koq and Ωoq designate the Ricean factor and the total power of the To−Rq link, respectively. Parameters ηSBi and ηDB specify how much the single- and double-bounced rays contribute to the total scattered power Ωoq/(Koq+1). Note that these energy-related parameters satisfy PI i=1 ηSBi+ηDB=1. The phases ψni and ψn1,n2 are i.i.d. random variables with uniform distributions over [−π, π), fTmax and fRmax are the maximum Doppler frequencies with respect to the Tx and Rx, respectively. Note that the AoD φ (ni) T and AoA φ (ni) R are independent for double-bounced rays, while they are interdependent for single-bounced rays. From FIGURE 10.7 and based on the application of the law of cosines in appropriate triangles, the distances εoq, εoni , εniq, and εn1n2 in (9.34) for any scenario (macro-cell (D≥1000 m), micro-cell (300<D<1000 m), or pico-cell (D≤300 m) scenario) can be expressed as εoq ≈ ε − kqδR cos(φ LoS Rq − βR) (9.35) εon1 ≈ RT − koδT cos(φ (n1) T − βT ) (9.36) εn1q ≈ ξn1 − kqδR cos(φ (n1) R − βR) (9.37) εon2 ≈ ξn2 − koδT cos φ (n2) T − βT (9.38) εn2q ≈ RR − kqδR cos φ (n2) R − βR (9.39) εon3 ≈ ξ (n3) T − koδT cos φ (n3) T − βT (9.40) εn3q ≈ ξ (n3) R − kqδR cos φ (n3) R − βR (9.41) εn1n2 ≈ D − RT cos φ (n1) T + RR cos φ (n2) R (9.42) where φ LoS Rq ≈π, ε≈D−koδT cos βT , ξn1= D2+R2 T−2D ×RT cos φ (n1) T −1/2 , ξn2= q D2+R2 R+2DRR cos φ (n2) R , ξ (n3) T = a 2+f 2+2af cos φ (n3) R / a+f cos φ (n3) R , ξ (n3) R =b 2/(a+ f cos φ (n3) R , ko=(MT−2o+1) /2, and kq=(MR−2q+1) /2. Here b denotes the semi-minor axis of the ellipse and the equality a 2=b 2+f 2 holds. As shown in Appendix 10-C, based on the newly proposed general method to derive the exact relationship between the AoA and AoD for any shape of the
272 Practices:channel modeling for modern communication systems scattering region,we have sin)=Rr sin+D2-2RTD cosof (9.43) Cos)-(D-Rr com+D3-2R D cond) (9.44) sin )=RRsin+D2+2RRDcos) (9.45) cos)D+RRc0s/V+D2+2RnD cos (9.46) sima-Psin/(a2+P+2a时cosg） (9.47刀 cos9-(2af+(a2+fP)cos）/(d2+P+2 af cos） (9.48) cieptenlandsaeoroaiocenaro。 h8物 can further reduce to the widely used raeo and T-AT/D ficanth d b in?and(2，(28)，and(32)in7. d on pure such as (A1-(a3》 thedan be replaced by the comtinuousprsoandctivel characterise AoD and AoAwe use the von Mises PDE given in (9.7). related to the single-bounce m ode than single-bounced rays due to the large distance D(larger distance D results in the independence of the AoD and channel statis cs,wenedtoditin8nihtswecnetc.To ng cars around the Tx and Rx and the stationary roadsi els in mi and pico-cell scen However,to th of t aut knowledge,this is the ver is mainl from waves reflected by the stationary roadsi bounced.This indicates that ne: .he is smaller than combined two-ring model and ellipse model with a LoS component. In this section,based on the proposed channel model in (9.33),we will derive the STF CF and the corresponding SDF PSD for a non-isotropic scattering envir
272 Practices: channel modeling for modern communication systems scattering region, we have sin φ (n1) R = RT sin φ (n1) T / q R2 T + D2 − 2RT D cos φ (n1) T (9.43) cos φ (n1) R = − D − RT cos φ (n1) T / q R2 T + D2 − 2RT D cos φ (n1) T (9.44) sin φ (n2) T = RR sin φ (n2) R / q R2 R + D2 + 2RRD cos φ (n2) R (9.45) cos φ (n2) T = D + RR cos φ (n2) R / q R2 R + D2 + 2RRD cos φ (n2) R (9.46) sin φ (n3) T = b 2 sin φ (n3) R / a 2 + f 2 + 2af cos φ (n3) R (9.47) cos φ (n3) T = 2af + (a 2 + f 2 ) cos φ (n3) R / a 2 + f 2 + 2af cos φ (n3) R . (9.48) Note that the above derived expressions in (9.35)–(9.48) are sufficiently general and suitable for various scenarios. For macro- and micro-cell scenarios, the assumption D≫max{RT , RR}, which is invalid for pico-cell scenarios, is fulfilled. Then, the general expressions of ξn1 and ξn2 can further reduce to the widely used approximate expressions as ξn1≈D−RT cos φ (n1) T and ξn2≈D+RR cos φ (n2) R . In addition, the general expressions (9.43)–(9.46) for the two-ring model can further reduce to the widely used approximate expressions as φ (n1) R ≈π−∆T sin φ (n1) T and φ (n2) T ≈∆R sin φ (n2) R with ∆T≈RT /D and ∆R≈RR/D. Moreover, the relationships (9.47) and (9.48) for the ellipse model obtained by using our method significantly simplify the relationships derived based on pure ellipse properties, such as (A1)–(A3) in ? and (27), (28), and (32) in ?. Since the number of effective scatterers are assumed to be infinite, i.e., Ni→∞, the proposed model is actually a mathematical reference model and results in the Ricean PDF. For our reference model, the discrete expressions of the AoA, φ (ni) R , and AoD, φ (ni) T , can be replaced by the continuous expressions φ (SBi) R and φ (SBi) T , respectively. To characterise AoD φ (SBi) T and AoA φ (SBi) R we use the von Mises PDF given in (9.7). Note that the proposed model is adaptable to a wide variety of V2V propagation environments by adjusting model parameters. It turns out that these important model parameters are the energy-related parameters ηSBi and ηDB, and the Ricean factor Koq. For a macro-cell scenario, the Ricean factor Koq and the energy parameter ηSB3 related to the single-bounce ellipse model are very small or even close to zero. The received signal power mainly comes from singleand double-bounced rays of the two-ring model, in which we assume that double-bounced rays bear more energy than single-bounced rays due to the large distance D (larger distance D results in the independence of the AoD and AoA), i.e., ηDB>max{ηSB1 , ηSB2 }≫ηSB3 . This means that a macro-cell scenario can be well characterised by using a two-ring model with a negligible LoS component. In contrast to macro-cell scenarios, in micro- and pico-cell scenarios, the VTD significantly affects the channel characteristics as presented in ?. To consider the impact of the VTD on channel statistics, we need to distinguish between the moving cars around the Tx and Rx and the stationary roadside environments (e.g., buildings, trees, parked cars, etc.). Therefore, we use a two-ring model to mimic the moving cars and an ellipse model to depict the stationary roadside environments. Note that ellipse models have been widely used to model F2M channels in micro- and pico-cell scenarios ?, ?. However, to the best of the authors’ knowledge, this is the first time that an ellipse model is used to mimic V2V channels. For a low VTD, the value of Koq is large since the LoS component can bear a significant amount of power. Also, the received scattered power is mainly from waves reflected by the stationary roadside environments described by the scatterers located on the ellipse. The moving cars represented by the scatterers located on the two rings are sparse and thus more likely to be single-bounced, rather than doublebounced. This indicates that ηSB3>max{ηSB1 , ηSB2 }>ηDB holds. For a high VTD, the value of Koq is smaller than that in the low VTD scenario. Also, due to the large amount of moving cars, the double-bounced rays of the two-ring model bear more energy than single-bounced rays of two-ring and ellipse models, i.e, ηDB>max{ηSB1 , ηSB2 , ηSB3 }. Therefore, micro-cell and pico-cell scenarios with consideration of the VTD can be well characterised by utilising a combined two-ring model and ellipse model with a LoS component. • Generic Space-Time-Frequency Correlation Function and Space-Doppler-Frequency Power Spectral Density In this section, based on the proposed channel model in (9.33), we will derive the STF CF and the corresponding SDF PSD for a non-isotropic scattering environment. 1) New Generic Space-Time-Frequency Correlation Function
Practices:channel modeling for modemn communication systems 273 ssumption.This m ans that we study the V2V channel over a short distance,when the WsS condition is fulfilled X)= E[hm()h哈gt-r】 VSoQle'y (9.49 where (.)denotes the c plex conjugate operation,E is the statistical expectation operator,(12... and.).It should be ob e tion definition widely used in other referencese?? ki (x)=E [hog (t)hig (t+)/Soog. (9.50) The CF definition in (9.49)is actually the correct one following the CF definition given in Stochastic Processes(see when pi(x)is a rea -V+可4- Xej2(Ocos Br-Q cos BR)(fTcosT-fRco7R) (9.51) where0=(d-o)r/入，Q=(d-g)6r/入，=(Mr-2d+1)/2,and=(Mr-2g+1)/2. Applying the von Mise PDF to the two-ring model,we for the AoD and f-exp cos/2()]for the AoA Substituting (9.34b) and (9.36)-(9.39)into (9.49),we can express the STF CF of the single-bounce two-ring model as P(= 好一(-3路 2π(k）VK+Kg+可J x2rh("r)+m(m-）l。2rocm(。m-)*0m(om-m】 xe学n女(m-(om-刃w (9.52) ng clos g=wV+( √K+Kog+o(》 (9.53) where A)cos()Cos TT(R) +j2r0(Q)cos Br(R)-j2rxXArm)/c (9.54a)
Practices: channel modeling for modern communication systems 273 As mentioned in Chapter 3, WSS channels have fading statistics that remain constant over short periods of time or distance (e.g., in the order of tens of wavelengths). In the developed channel model, we have used the WSS assumption. This means that we study the V2V channel over a short distance, when the WSS condition is fulfilled. Under the WSS condition, the normalised STF CF between any two complex fading envelopes hoq (t) and h ′ o ′q ′ (t) with different carrier frequencies fc and f ′ c , respectively, is defined as ? ρhoqh ′ o′q′ (τ, χ)= E hoq (t) h ′∗ o′q ′(t − τ) p ΩoqΩo′q ′ =ρhLoS oq h ′LoS o′q′ (τ, χ)+X I i=1 ρ h SBi oq h ′SBi o′q′ (τ, χ)+ρhDB oq h ′DB o′q′ (τ, χ) (9.49) where (·) ∗ denotes the complex conjugate operation, E [·] is the statistical expectation operator, o, o′ ∈ {1, 2, ..., MT }, and q, q′ ∈ {1, 2, ..., MR}. It should be observed that (9.49) is a function of time separation τ, space separation δT and δR, and frequency separation χ = f ′ c − fc. Note that the CF definition in (9.49) is different from the following definition widely used in other references, e.g., ???? ρ˜hoqh ′ o′q′ (τ, χ) = E hoq (t) h ′∗ o′q ′ (t + τ) / p ΩoqΩo′q ′ . (9.50) The CF definition in (9.49) is actually the correct one following the CF definition given in Stochastic Processes (see Equation (9-51) in ?). It can easily be shown that the expression (9.50) equals the complex conjugate of the correct CF in (9.49), i.e., ρ˜hoqh′ o′q′ (τ, χ) = ρ ∗ hoqh ′ o′q′ (τ, χ), and thus is an incorrect definition. Only when ρ ∗ hoqh ′ o′q′ (τ, χ) is a real function (no imaginary part), ρ˜hoqh ′ o′q′ (τ, χ) = ρhoqh ′ o′q′ (τ, χ) holds. Substituting (9.34a) and (9.35) into (9.49), we can obtain the STF CF of the LoS component as ρhLoS oq h ′LoS o′q′ (τ, χ) = s KoqKo′q ′ (Koq + 1) (Ko′q ′ + 1)e j 2πχ c (D−ko′ δT cos βT +kq′ δR cos βR) ×e j2π(O cos βT −Q cos βR) e j2πτ(fTmax cos γT −fRmax cos γR) (9.51) where O = (o ′ − o) δT /λ, Q = (q ′ − q) δR/λ, k ′ o = (MT − 2o ′ + 1) /2, and k ′ q = (MR − 2q ′ + 1) /2. Applying the von Mise PDF to the two-ring model, we obtain f φ SB1 T =exp h k T R T cos φ SB1 T −µ T R T / 2πI0 k T R T for the AoD φ SB1 T and f φ SB2 R =exp h k T R R cos φ SB2 R −µ T R R i / 2πI0 k T R R for the AoA φ SB2 R . Substituting (9.34b) and (9.36)–(9.39) into (9.49), we can express the STF CF of the single-bounce two-ring model as ρ h SB1(2) oq h ′SB1(2) o′q′ (τ, χ) = ηSB1(2) 2πI0 k T R T(R) p (Koq + 1) (Ko′q ′ + 1) Zπ −π e k T R T (R) cos φ SB1(2) T (R) −µ T R T (R) ×e j2πτ fTmax cos φ SB1(2) T −γT +fRmax cos φ SB1(2) R −γR e j2π O cos φ SB1(2) T −βT +Q cos φ SB1(2) R −βR ×e j2πχ c RT (R)+ξn1(2) −ko′ δT cos φ SB1(2) T −βT −kq′ δR cos φ SB1(2) R −βR dφSB1(2) T(R) (9.52) where the parameters sin φ SB1 R , cos φ SB1 R , sin φ SB2 T , and cos φ SB2 T follow the expressions in (9.43)–(9.46), respectively. For the macro- and micro-cell scenarios, (9.52) can be further simplified as the following closed-form expression ρ h SB1(2) oq h ′SB1(2) o′q′ (τ, χ) = ηSB1(2) e jC SB1(2) T (R) I0 (r A SB1(2) T(R) 2 + B SB1(2) T(R) 2 ) p (Koq + 1) (Ko′q ′ + 1)I0 k T R T(R) (9.53) where A SB1(2) T(R) =k T R T(R) cos µ T R T(R)+j2πτfT(R)max cos γT(R) +j2πO(Q) cos βT(R)−j2πχXAT (R) /c (9.54a)
274 Practices:channel modeling for modern communication systems B2=k张sin&+j2xr(fr(R)sin TT(R+frma△r(R)sin 7R(T）) +j2πO(Q)sin Br+Q(O)△TuR)Sin BR(TY-XXBr/c) (9.54b) C=2fR(T)COS TR(T)2Q(O)cos BR(T)+/c (9.54c) kd on sin BR- D+krcos Applying the von Mises PDF to the ellipse model,we get f=expcos -】/[2x(馈] iion of (46),()an(1)int )we can obrain the sT CF of the singl-bou xe-))3cfo)) e)(e-a) (9.55) bounce two-ring model g学动=nV+9可{Vg9+可 VK+(Kog+6(得)6) (9.56) where Acos(.cosTj(Q)cos r(R) 2mX(Rr干ka)cosr)/c (9.57a) B2限+sin+i2 TfT()典T(a +j2O(Q)sin Br(+j2xk()sin Br(/c (9.57b) CDB=2xx(Rr+RR+D)/c (9.57c sic hedef)mila,ony the brief ouline of the derivaion of) isemanyspecial cases.If wethenode since the CF definition(0)is used n.Consequently the derived STC(includes otherd in as special cases,whenis replaced byIf we conider thefor aF2M channel (includes the CChaper and.subs can be obtained from (9.49).=x=nsB,=nD=0. .Purtheedncan also be obtainedom) Applying the Fourier transform to theSTFCFin()n terms of we can obtain the corresponding SDF PSD as k (o)F(ph()e-Terdr Sh(fD,X)+∑& (9.58)
274 Practices: channel modeling for modern communication systems B SB1(2) T(R) =k T R T(R) sin µ T R T(R)+j2πτ(fT(R)max sin γT(R)+fR(T)max∆T(R) sin γR(T)) +j2π(O(Q) sin βT(R)+Q(O)∆T(R) sin βR(T)−χXBT (R) /c) (9.54b) C SB1 T(R)=∓2πτfR(T)max cos γR(T)∓2πQ(O) cos βR(T)+2πχXCT(R) /c (9.54c) with XAT =RT −ko ′δT cos βT , XBT =−ko ′δT sin βT −kq ′δR∆T sin βR, XCT =RT +D−kq ′δR cos βR, XAR =−RR− kq ′ δR cos βR, XBR =−kq ′δR sin βR−ko′δT ∆R sin βT , and XCR =RR +D+ko′δT cos βT . Applying the von Mises PDF to the ellipse model, we get f φ SB3 R = exp h k EL R cos φ SB3 R −µ EL R / 2πI0 k EL R . Performing the substitution of (9.34b), (9.40), and (9.41) into (9.49), we can obtain the STF CF of the single-bounce ellipse model as ρ h SB3 oq h SB3′ o′q′ (τ, χ) = ηSB3 2πI0 k EL R p (Koq + 1) (Ko′q ′ + 1) Zπ −π e k EL R cos φ SB3 R −µ EL R ×e j2πτh fTmax cos φ SB3 T −γT +fRmax cos φ SB3 R −γR i e j2π h O cos φ SB3 T −βT +Q cos φ SB3 R −βR i ×e j2πχ c h 2a−ko′ δT cos φ SB3 T −βT −kq′ δR cos φ SB3 R −βR i dφSB3 R (9.55) where the parameters sin φ SB3 T and cos φ SB3 T follow the expressions in (9.47) and (9.48), respectively. The substitution of (9.34c), (9.36), (9.39), and (9.42) into (9.49) results in the following STF CF for the doublebounce two-ring model ρhDB oq h ′DB o′q′ (τ, χ) = ηDBe jCDB I0 q ADB T 2 + BDB T 2 I0 q ADB R 2 + BDB R 2 p (Koq + 1) (Ko ′q ′ + 1)I0 k T R T I0 k T R R (9.56) where A DB T(R) =k T R T(R) cos µ T R T(R)+j2πτfT(R)max cos γT(R)+j2πO(Q) cos βT(R) ∓j2πχ RT(R)∓ko ′(q ′) cos βT(R) /c (9.57a) B DB T(R)=k T R T(R) sin µ T R T(R)+j2πτfT(R)max sin γT(R) +j2πO(Q) sin βT(R)+j2πχko′(q ′) sin βT(R)/c (9.57b) C DB = 2πχ (RT +RR+D) /c. (9.57c) Since the derivations of (9.51)–(9.53), (9.55), and (9.56) are similar, only the brief outline of the derivation of (9.53) is given in Appendix 10-D, while others are omitted for brevity. The derived STF CF in (9.49) includes many existing CFs as special cases. If we only consider the two-ring model (ηSB3=0) for a V2V channel in a macro- or micro-cell scenario (D≫max{RT , RR}) with the frequency separation χ=0, then the CF in (9.49) will be reduced to the CF in (18) of ?, where the time separation τ should be replaced by −τ since the CF definition (9.50) is used in ?. Consequently, the derived STF CF in (9.49) also includes other CFs listed in ? as special cases, when τ is replaced by −τ. If we consider the one-ring model only around the Rx for a F2M channel in a macro-cell scenario (ηSB1=ηSB3=ηDB=fTmax =0) with non-LoS (NLoS) condition (Koq=0), the derived STF CF in (9.49) includes the CF (9.16) in Chapter 10.3 and, subsequently, other CFs listed in Chapter 10.3 as special cases, when τ is replaced by −τ. Furthermore, the CF (7) in ? can be obtained from (9.49) with Koq=fTmax =χ=ηSB3=ηDB=0. Consequently, other CFs listed in ? can also be obtained from (9.49). 2) New Generic Space-Doppler-Frequency Power Spectral Density Applying the Fourier transform to the STF CF in (9.49) in terms of τ, we can obtain the corresponding SDF PSD as Shoqh ′ o′q′ (fD, χ) = F n ρhoqh ′ o′q′ (τ, χ) o = Z∞ −∞ ρhoqh ′ o′q′ (τ, χ) e −j2πfDτ dτ = ShLoS oq h ′LoS o′q′ (fD, χ) +X I i=1 Sh SBi oq h SBi ′ o′q′ (fD, χ) + ShDB oq h ′DB o′q′ (fD, χ) (9.58)