同济大学：《传播信道特征估计和建模》课程教学资源（教案讲义）Chapter 09 Practices - channel modeling for modern communication systems.pdf_P6-P10

260 Practices: channel modeling for modern communication systems discrete propagation delay of the lth tap, respectively. Similar to the concept of effective scatterers in the narrowband one-ring model, the concept of an effective cluster is introduced in the new wideband multiple-ring model. From FIGURE 10.2, it is obvious that the position of the effective cluster Sl,i is identified by the intersection of the virtual ellipses and multiple rings, and the lth tap includes 2Λl effective clusters. The mean angle of the effective cluster in each tap is µl,i,r (r=1, 2) and the corresponding angle spread is ∆µl,i as illustrated in FIGURE 10.2. Therefore, the effective cluster can be completely determined by µl,i,r and ∆µl,i. Note that the angular range of µl,i,r in each tap is over [0, 2π), which means the effective cluster can be located around the MS over [0, 2π) for each tap. The setting of these two parameters follows a fixed rule. To establish this rule, firstly, we need to define the propagation delay subintervals Gl . The propagation delay interval G = [0, τ′ max] is partitioned into L mutually disjoint sub-intervals Gl . Here, we utilise the definition of subintervals as ? Gl=    0, ∆τ ′ l+1 2 , τ ′ l − ∆τ ′ l /2, τ′ l + ∆τ ′ l+1 2 , [τ ′ l − ∆τ ′ l /2, τ′ max] , l=1 l=2, 3, ..., L − 1 l=L (9.2) where ∆τ ′ l=τ ′ l−τ ′ l−1 and τ ′ max=2RL−1,ΛL−1−1 c (c is the speed of light). The propagation delay τ ′ l of the lth tap can be expressed according to the corresponding AoA φ R l,i,r (r=1, 2) as τ ′ l≈τ ′ max(1+cosφ R l,i,r)/2 ?. Solving this equation for φ R l,i,r gives φ R l,i,r=± arccos (2τ ′ l /τ ′ max − 1). According to (9.2), the expression of φ R l,i,r, and the geometrical relationship in FIGURE 10.2, the expression of the mean angle µl,i,r and the corresponding angle spread ∆µl,i of the effective cluster in the lth tap are given as µl,i,r = φ R l,i−1,r+2φ R l,i,r+φ R l,i+1,r4=± arccos 2τ ′ l−1 τ ′ max−1 +2 arccos (2τ ′ l /τ ′ max−1)+ arccos 2τ ′ l+1 τ ′ max−1 4 (9.3) ∆µl,i = φ R l,i−1,r − φ R l,i+1,r4 = arccos 2τ ′ l−1 τ ′ max−1 −arccos 2τ ′ l+1 τ ′ max−1 4. (9.4) Following the definition of the subintervals Gl and some geometrical relationship shown in FIGURE 10.2, we can determine the effective cluster in each tap according to the propagation delay τ ′ l . The time-variant tap coefficient at the carrier frequency fc of lth tap can be expressed as hl,oq (t)= lim N→∞ 1 √ N Λ Xl−1 i=0 X Rc r=1 X N n=1 exp j ψl,i,n−2πfcτl,i,oq,n + 2πfDt cos φ R l,i,r,n−γ (9.5) with τl,i,oq,n=(εl,i,on+εl,i,nq) /c. Here, τl,i.oq.n is the travel time of the wave through the link To−Sl,i,n−Rq scattered by the nth scatterer, Sl,i,n, Rc is the number of effective cluster for one ring in each tap (here Rc=2), and N is the number of effective scatterers Sl,i,n in the effective cluster Sl,i. The AoA of the wave traveling from the nth scatterer in the effective cluster Sl,i towards the MS is denoted by φ R l,i,r,n. The phases ψl,i,n are i.i.d. random variables with uniform distributions over [0, 2π) and fD is the maximum Doppler frequency. As shown in ?, the distance εl,i,on and εl,i,nq can be expressed as the function of φ R l,i,r,n as εl,i,on≈ξl,i,n−δT [cos(βT )+Θl,i sin(βT ) sin(φ R l,i,r,n)]/2 (9.6a) εl,i,nq≈Rl,i−δR cos(φ R l,i,r,n−βR)/2 (9.6b) respectively, where ξl,i,n≈D+Rl,i cos(φ R l,i,r,n). Since we assume that the number of effective scatterers in one effective cluster in this reference model tends to infinite (as shown in (9.5)), the discrete AoA φ R l,i,r,n can be replaced by the continuous expressions φ R l,i,r. In the literature, many different scatterer distributions have been proposed to characterise the AoA φ R l,i,r, such as the uniform ?, Gaussian ?, wrapped Gaussian ?, and cardioid PDFs ?. In this chapter, the von Mises PDF ? is used, which can approximate all the above mentioned PDFs. The von Mises PDF is defined as f (φ) ∆=exp [k cos (φ−µ)] /2πI0 (k) (9.7) where φ ∈ [−π, π), I0 (·) is the zeroth-order modified Bessel function of the first kind, µ ∈ [−π, π) accounts for the mean value of the angle φ, and k (k ≥ 0) is a real-valued parameter that controls the angle spread of the angle φ.

Practices: channel modeling for modern communication systems 261 For k=0 (isotropic scattering), the von Mises PDF reduces to the uniform distribution, while for k>0 (non-isotropic scattering), the von Mises PDF approximates different distributions based on the values of k ?. To better characterise the AoA in one effective cluster Sl,i, we further modify the general expression of von Mises PDF as fc φ R l,i,r =Ql,i,r exp kl,i,r cos φ R l,i,r−µl,i,r/2πI0 (kl,i,r) (9.8) where φ R l,i,r∈[µl,i,r−∆µl,i, µl,i,r+∆µl,i) and Ql,i,r is the normalisation coefficient. Here we name the PDF in (9.8) the truncated von Mises PDF. Here, truncated means that the range of AoA in this PDF is only defined within a limited interval [µl,i,r−∆µl,i, µl,i,r+∆µl,i). Therefore, the expression of the ‘tapped’ PDF of AoA in the lth tap of the proposed wideband multiple-ring channel model is given by fg φ R l,i,r = Λ Pl−1 i=0 P Rc r=1 Ql,i,r exp[kl,i,r cos(φ R l,i,r−µl,i,r)] 2πI0(kl,i,r) w φ R l,i,r, µl,i,r − ∆µl,i, µl,i,r + ∆µl,i where w (φ, φl , φu) = 1, 0, if φl < φ < φu otherwise. (9.9) Here, Ql,i,r are computed in such a way that the ’tapped’ PDF fg(φ R l,i,r) is equal to 1, i.e., R π −π fg(φ R l,i,r)dφR l,i,r=1. Note that the proposed wideband model allows one to consider the interaction of AoA, AoD, and ToA in a sensible manner. The interaction between the AoA and AoD is obtained in terms of the exact geometrical relationship, while the interaction between the AoA/AoD and ToA is calculated according to the TDL structure that allows one to investigate the correlation properties in each tap. Therefore, inspired by ?, the interaction between the AoA/AoD and ToA can be considered via setting the appropriate parameter kl,i,r for the PDF of AoA/AoD in each tap according to the PDF of ToA. 9.3.2 Generic Space-Time-Frequency Correlation Function ((10.3.2)) In this subsection, from the proposed model we first derive a new generic STF CF for wideband MIMO channels. As shown at the end of this subsection, the derived STF CF can be reduced to a compact closed-form STF CF for narrowband MIMO channels by removing the frequency-selectivity, which includes many existing CFs as special cases. • Space-Time-Frequency Correlation Function for Wideband MIMO Channels From the proposed wideband model, we derive the STF CF for each tap. The correlation properties of two arbitrary links hoq(t, τ′ ) and h ′ o′q ′ (t, τ′ ) at different frequency fc and f ′ c of a MIMO channel are completely determined by the correlation properties of hoq(t) and h ′ o ′q ′ (t) in each tap since we assume that no correlations exist between the underlying processes in different taps. Therefore, we can restrict our investigations to the following STF CF ρl,oq;l,o′q ′ (τ, χ):=E[hl,oq(t)h ′∗ l,o′q ′ (t − τ )] (9.10) where (·) ∗ denotes the complex conjugate operation and E[·] designates the statistical expectation operator. Note that the above defined CF is a function of the time separation τ, space separation δT and δR, and frequency separation χ=f ′ c−fc. As shown in Appendix 10-A, the closed-form expression of STF CF ρl,oq;l,o′q ′ (τ, χ) can be presented as ρl,oq;l,o′q ′ (τ, χ) = 2 πI0(k) Λ Pl−1 i=0 P Rc r=1 Ql,i,re jCl,i {∆µl,iI0 (Al,i,r) I0 (Bl,i,r)/2 + I0 (Bl,i,r) P∞ ℓ=1 ×Iℓ (Al,i,r) sin (ℓ∆µl,i) cos (ℓµl,i,r)/ℓ + I0 (Al,i,r) P∞ ℓ ′=1 (−1)ℓ ′ Iℓ ′ (Bl,i,r) sin (ℓ ′∆µl,i) ×cos (ℓ ′µl,i,r + ℓ ′π/2)/ℓ ′ + P∞ ℓ=1 (−1)ℓ Iℓ (Al,i,r) Iℓ (Bl,i,r) [∆µl,i cos (ℓπ/2) + sin (2ℓ∆µl,i) ×cos (2ℓµl,i,r + ℓπ/2)/2ℓ] + P∞ ℓ=1 P∞ q=1 (ℓ6=q) (−1)ℓ ′ Iℓ (Al,i,r) Iℓ ′ (Bl,i,r) [sin [(ℓ + ℓ ′ ) ∆µl,i] × cos [(ℓ+ℓ ′ )µl,i,r+ℓ ′π/2] ℓ+ℓ ′ + sin[(ℓ−ℓ ′ )∆µl,i] cos [(ℓ−ℓ ′ )µl,i,r−ℓ ′π/2] ℓ−ℓ ′

262 Practices: channel modeling for modern communication systems (9.11) where Al,i,r=al,i,r+j(XUl,i+y cos βR+x cos γ) (9.12a) Bl,i,r=bl,i,r+j(XVl,i+y sin βR+zΘl,i sin βT+x sin γ) (9.12b) Cl.i=z cos βT+XTl.i (9.12c) with x=2πfDτ, y=2πfcδR/c, z=2πfcδT /c, X=2πχ/c, al,i,r=kl,i,r cos µl,i,r, bl,i,r=kl,i,r sin µl,i,r, Tl,i=(δT /2) cos βT+D+ Rl,i, Ul,i=Rl,i+(δR/2) cos βR, and Vl,i=(δT /2)Θl,i sin βT +(δR/2) sin βR. Consequently, the STF CF between hoq(t, τ′ ) and h ′ o ′q ′ (t, τ′ ) can be shown as ρoq,o′q ′ (τ, χ) = 1 L X L l=1 ρl,oq;l,o′q ′ (τ, χ). (9.13) Note that (9.11) and (9.13) are the generic expressions which apply to the STF CF and the subsequently degenerate CFs (e.g., ST CF, frequency CF, etc.) differ only in values of Al,i,r, Bl,i,r, and Cl,i. The corresponding expressions of these three parameters for the degenerate CFs can be easily obtained by setting relevant terms (τ, δT and δT , and χ) to zero. • Space-Time-Frequency Correlation Function for Narrowband MIMO Channels A special case of the proposed model described by (9.1) is given when L=1 and Λl=1. Consequently, we have hoq (t, τ′ )=hoq (t) δ (τ ′ ), which is the complex fading envelope of the narrowband one-ring channel model. To make this evident, we express the complex fading envelop hoq (t) similarly to (9.5) by removing the subscripts (·) l , (·) i , and (·) r , i.e., hoq (t)= lim N→∞ 1 √ N X N n=1 exp j ψn−2πfcτoq,n + 2πfDt cos φ R n − γ (9.14) with τoq,n=(εon+εnq)/c, where εon and εnq can be expressed as the function of φ R n as εon ≈ξn−δT [cos(βT )+Θ sin(βT ) sin(φ R n )]/2 (9.15a) εnq≈R−δR cos(φ R n−βR)/2 (9.15b) where ξn≈D+R cos(φ R n ). In order to consistent with the proposed wideband model, the von Mises PDF (9.7) is employed to characterise the AoA φ R n of the narrowband model. In such a case, the angle spread ∆µl,i=π, which means the AoA range is over [0, 2π). As shown in Appendix 10-B, the STF CF of the narrowband one-ring model can be obtained from (9.11) after some manipulation ρoq,o′q ′ (τ, χ)=e jC I0 h A 2+B 2 1/2i.I0 (k) (9.16) where A=a+j(XU+y cos βR+x cos γ) (9.17a) B=b+j(XV+y sin βR+zΘ sin βT+x sin γ) (9.17b) C=z cos βT+XT (9.17c) with a=k cos µ, b=k sin µ, T=(δT /2) cos βT+D+R, U=R+(δR/2) cos βR, and V =(δT /2)Θ sin βT+(δR/2) sin βR. The parameters x, y, z, and X are the same as defined in (9.11). It is worth stressing that (9.16) is the generic expressions which applies to the STF CF and the subsequently degenerate CFs with the difference only in values of A, B, and C. The corresponding expressions of A, B, and C for the degenerate CFs can be easily obtained from (9.17) by setting relevant terms to zero. The derived generic STF CF (9.16) includes many existing CFs as special cases. For a SISO case, the time CF given in ? is obtained by setting δT = δR = 0 and χ = 0 in (9.16) with k 6= 0. If further setting k = 0 (isotropic scattering) in (9.16), the Clarke’s time CF in ? is obtained. For a SIMO case, the Lee’s ST CF in ? is obtained by substituting δT = 0, χ = 0, βR = 0, and k = 0 into (9.16). For a MISO case, the ST CF in ? is obtained by substituting δR = 0, χ = 0, and k = 0 into (9.16). If further substituting fD = 0 into (9.16), the space CF given in ? is obtained. For a MIMO case, the ST CF shown in ? is obtained by setting χ = 0 in (9.16) with k 6= 0.

Practices: channel modeling for modern communication systems 263 9.3.3 MIMO Simulation Models (10.3.3) In this section, based on the proposed wideband MIMO channel reference model, we propose an efficient deterministic SoS simulation model for wideband MIMO channels. The proposed wideband simulation model can be further reduced to a narrowband one by removing the frequency-selectivity. • A Deterministic Simulation Model for Wideband MIMO Channels The wideband deterministic simulation model is proposed also based on the TDL structure. The impulse response of the simulation model at the carrier frequency fc for the To−Rq link are again composed of L discrete taps according to h˜ oq (t, τ′ ) = X L l=1 h˜ l,oq (t) ×δ (τ ′ − τ ′ l ). (9.18) In (9.18), the complex fading envelope h˜ l,oq (t) is modeled by utilizing only a finite number of scatterers N and keeping all the model parameters fixed as h˜ l,oq (t)= √ 1 N Λ Pl−1 i=0 P Rc r=1 P N n=1 exp n j h ψ˜ l,i,n−2πfcτl,i,oq,n +2πfDt cos φ˜R l,i,r,n − γ io (9.19) where the phases ψ˜ l,i,n are simply the outcomes of a random generator uniformly distributed over [0, 2π), the discrete AoAs φ˜R l,i,r,n will be kept constant during simulation, and the other symbol definitions are the same as those in (9.5). Therefore, we can analyze the properties of the deterministic channel simulator by time averages instead of statistical averages. The STF CF is defined as ρ˜l,oq;l,o′q ′ (τ, χ):=D h˜ l,oq (t) ˜h ′∗ l,o′q ′ (t − τ ) E (9.20) where h·i denotes the time average operator. Substituting (9.19) into (9.20), we can get the closed-form STF CF as ρ˜l,oq;l,o′q ′ (τ, χ) = 1 N Λ Xl−1 i=0 X Rc r=1 X N n=1 e j(Cl,i+Pl,i cos φ˜R l,i,r,n+Jl,i sin φ˜R l,i,r,n) (9.21) with Pl,i=XUl,i+y cos βR+x cos γ (9.22a) Jl,i=XVl,i+y sin βR+z∆ sin βT+x sin γ (9.22b) where Cl,i, x, y, z, X, Ul,i, and Vl,i are the same as defined in (9.11). By analogy to (9.13), we can further get the STF CF between h˜ oq (t, τ′ ) and h˜′ o ′q ′ (t, τ′ ) as ρ˜oq,o′q ′ (τ, χ) = 1 L X L l=1 ρ˜l,oq;l,o′q ′ (τ, χ). (9.23) Similar to (9.11) and (9.13), (9.21) and (9.23) are the generic expressions which apply to all the CFs of the deterministic simulation model with different Cl,i, Pl,i, and Jl,i. Comparing the expressions of Al,i,r and Bl,i,r with Pl,i and Jl,i, respectively, we have Al,i,r=al,i,r+jPl,i and Bl,i,r=bl,i,r+jJl,i. From (9.21) and (9.23), it is obvious that only {φ˜R l,i,r,n} N n=1 needs to be determined for this deterministic simulation model. As addressed in Chapter 5, the MEA and MEDS have been widely used to compute the important parameters of deterministic simulation models for isotropic scattering environments. However, these two methods fail to reproduce the desired statistical properties of the reference model under the condition of non-isotropic scattering ??. Therefore, the optimization method (i.e., Lp-norm method) ? is utilized here to calculate the model parameters {φ˜R l,i,r,n} N n=1 of the deterministic simulation model based on corresponding properties of the reference model. The time CF ρl,oq;l,oq(τ)

264 Practices: channel modeling for modern communication systems frequency CF ρl,oq;l,oq(χ), and space CF ρl,oq;l,o′q ′ are identified as key properties. Then the optimization method requires the numerical minimization of the following three Lp-norms E (p) 1,l :=Z τmax 0 |ρl,oq;l,oq (τ)−ρ˜l,oq;l,oq (τ)| p dτ /τmax1/p (9.24) E (p) 2,l :=Z χmax 0 |ρl,oq;l,oq(χ)−ρ˜l,oq;l,oq(χ)| p dχ/χmax1/p (9.25) E (p) 3,l:=(Z δ max T 0 Z δ max R 0 |ρl,oq;l,o′q ′−ρ˜l,oq;l,o′q ′ | p dδT dδR/(δ max T δ max R ) ) 1/p (9.26) where p = 1, 2, ... Note that τmax, χmax, δ max T , and δ max R define the upper limits of the ranges over which the approximations ρ˜l,oq;l,oq (τ)≈ρl,oq;l,oq (τ), ρ˜l,oq;l,oq (χ)≈ρl,oq;l,oq (χ), and ρ˜l,oq;l,o′q ′≈ρl,oq;l,o′q ′ are of interest. For ρ˜l,oq;l,oq (χ) and ρ˜l,oq;l,o′q ′ , if we replace φ˜R l,i,r,n by φ˜′R l,i,r,n and φ˜′′R l,i,r,n, respectively, the three error norms E (p) 1,l , E (p) 2,l , and E (p) 3,l can be minimized independently. • A Deterministic Simulation Model for Narrowband MIMO Channels Analogous to Chapter 10.3.2.2, if we impose L=1 and Λl=1 on the wideband simulation model in (9.18), it reduces to a narrowband MIMO channel simulator. It follows that ˜hoq (t, τ′ )=h˜ oq (t) δ (τ ′ ) holds. The complex fading envelope h˜ oq (t) of the deterministic simulation model is then given by ˜hoq (t) = √ 1 N P N n=1 exp n j h ψ˜ n − 2πfcτn+2πfDt cos φ˜R n − γ io (9.27) where the phases ψ˜ n are simply the outcomes of a random generator uniformly distributed over [0, 2π), the discrete AoAs φ˜R n will be kept constant during simulation, and the other symbol definitions are the same as those in (9.14). The correlation properties of this narrowband simulation model can be obtained from (9.21) by simply neglecting the subscripts (·)l , (·)i , and (·)r in all the affected symbols. Thus ρ˜oq;o′q ′ (τ, χ) = 1 N X N n=1 e j(C+P cos φ˜R n +J sin φ˜R n ) (9.28) with P=XU+y cos βR+x cos γ (9.29a) J=XV+y sin βR+z∆ sin βT+x sin γ (9.29b) where other parameters are the same as defined in (9.16). Similar to the wideband simulation model, we have A=a+jP and B=b+jJ. From (9.28), it is clear that only {φ˜R n } N n=1 needs to be determined for this deterministic simulation model. The model parameters {φ˜R n } N n=1 can be calculated by using the same optimization method as the wideband simulation model. Therefore, by removing the subscript (·)l in (9.24)–(9.26), the model parameters can be obtained as follows E (p) 1 :=Z τmax 0 |ρoq;oq (τ)−ρ˜oq;oq (τ)| p dτ /τmax1/p (9.30) E (p) 2 :=Z χmax 0 |ρoq;oq(χ)−ρ˜oq;oq(χ)| p dχ/χmax1/p (9.31) E (p) 3 :=(Z δ max T 0 Z δ max R 0 |ρoq;o′q ′−ρ˜oq;o′q ′ | p dδT dδR/(δ max T δ max R ) ) 1/p . (9.32) Similarly, for ρ˜oq;oq (χ) and ρ˜oq;o′q ′ , if we replace φ˜R n by φ˜′R n and φ˜′′R n , respectively, the three error norms E (p) 1 , E (p) 2 , and E (p) 3 can be minimized independently