50 Generic channel models 3.3 Specular-path model az nuth and the poua the wa tua (2005):and Cox (2002)dersen and Mogense (1):Vaughan ()Yin et(003). in the on domain algorithms for extracting the polarization characteristics of the propagation channel. zed anter ennas can not transmit or receive in one polarization only To differentiate the two dominant direction of al fold on,specity wave introduced,which is composed of the com pah The model th of thehwave t=em62 c.(aa）a(a】aia-ea0）aa u(t-n) (3.10) (s)den 11 pi,(pi 1,2)denotes the polarization index. 0-)。 Q) t-) cxpj2U,t） A uv.(1-T)c Figure 3.8 Contribution of the th wave to the received signal in a MIMO system incorporating dual-antenna arrays
50 Generic channel models 3.3 Specular-path model In the specular-path model, the electromagnetic waves are considered to propagate along multiple specular paths. For parameter estimation, the parameters describing a path may include its delay, direction of arrival (i.e. azimuth and elevation of arrival), direction of departure (i.e. azimuth and elevation of arrival), Doppler frequency and polarization matrix. Other parameters may be also included, e.g. the time-variability of these parameters in the time-variant case. In the following, the bidirection-delay-Doppler-Dual-polarization specular path-model is introduced. Dual polarization is specifically considered as the polarization of an electromagnetic wave, especially for the TEM waves, can be projected into two orthogonal directions. From the parameter estimation point of view, by considering dual polarizations, more samples of the channel observation are obtained, and consequently, the estimation accuracy of the path parameters is improved compared with the signal-polarization. Furthermore, since using dual polarized antenna array for boosting the capacity of MIMO system has been considered in advanced wireless communication standards Deng et al. (2005); Kyritsi and Cox (2002); Pedersen and Mogensen (1999); Vaughan (1990); Yin et al. (2003), modeling the channels in the dual-polarization domain becomes popular Acosta-Marum et al. (2010); Degli-Esposti et al. (2007); Hämäläinen et al. (2005); Kwon and Stüber (2010). Therefore, it is necessary to design estimation algorithms for extracting the polarization characteristics of the propagation channel. As shown in Figure 6.4, Rx and Tx antenna arrays with dual-polarized antenna configuration considered. Each of the dual antenna transmits/receives signal within two polarizations at the same time. This assumption is based on the realistic experience that antennas can not transmit or receive signals in one polarization only. To differentiate the two polarizations in the underlying model, one of them is called main polarization, specifying the dominant direction of the signal field pattern. The other is correspondingly referred to the complementary polarization. In order to describe the wave polarization, a polarization matrix Aℓ is introduced, which is composed of the complex weights for the attenuations along the propagation paths. The signal model describing the contribution of the ℓth wave to the output of the MIMO system reads s(t; θℓ) = exp(j2πνℓt) c2,1(Ω2,ℓ) c2,2(Ω2,ℓ) αℓ,1,1 αℓ,1,2 αℓ,2,1 αℓ,2,2 · c1,1(Ω1,ℓ) c1,2(Ω1,ℓ) T · u(t − τℓ), (3.10) where ci,pi (Ω) denotes the steering vector of the transmitter array (i = 1) with totally M1 entries or receiver array (i = 2) with totally M2 entries, where M1 and M2 are the amount of antennas in the Tx and Rx respectively. Here pi ,(pi = 1, 2) denotes the polarization index. Polarization Matrix of Path " " A (ȍ ) ,11,1,1 " c ( ) 1 " tu W ( ) 1 " u t W M ( ) " u t W (ȍ ) 1,1,2 ,2 " c ),( 1 ș" ts );( ș" exp( j2 t) s t SX " (ȍ ) ,12,1,1 " c (ȍ ) M1 2,,1 ,1 " c ),( 2 ș" s t M (ȍ ) 2,1,2 ,2 " c (ȍ ) M1 ,11,,1 " c (ȍ ) M2 1,,2 ,2 " c (ȍ ) M2 2,,2 ,2 " c Figure 3.8 Contribution of the ℓth wave to the received signal in a MIMO system incorporating dual-antenna arrays
Generic channel models 51 Cyele 1 Cyele2 T 2 Mi-1 M Tx 2…网图☐2…网a··· 12.M-可M▣12·.M-可M间· Rx Figure3.9 Timing structure of sounding and sensing windows Written in matrix form,(6.12)is reformulated as s(t:0)=exp(j2rvet)C2(D2.)ACf(S)u(t-Te) (3.11) with C2(2d=[c2.12.d）c2.22.tj (3.12) C(1.0=[c,11.）c1,2(01, (3.13) =ai8a=aanl (3.14) (3.15) The equation(6.13)can be recast as following s(t0）=exp(j2rut{[ae.1c2.1(2.)cf,(1.）+a1.2c2.1(2.)cf2(1.） +a4.21c22(2.cf(）+a2,2e2.2(2）ci2(1,ut-n} (3.16) (3.17刀 3.3.1 Model for time-division-multiplexing channel sounding Channel sounding can be conducted by structure depicted in Pigure 3.the signal mode The mth antenna element of Array 1 is active during the sounding windows 91,m(因=∑9m,(t-m4+Tg,m1=1,,M (3.18) whereidenotes the cycle index and t.m1=(位-1)Tw+(m1-1)T q1(④）三【m.1().,91.M( used,the soun ows()and thes )need merely
Generic channel models 51 1 Tt 2 M1-1 M1 1 2 M2-1 M2 Cycle 1 Cycle 2 1 2 1 2 1 2 Tg Tsc Tcy t Tr Array 1 Switch 1 Tx Array 2 Switch 2 Rx M2-1 M2 M2-1 M2 M2-1 M2 Figure 3.9 Timing structure of sounding and sensing windows Written in matrix form, (6.12) is reformulated as s(t; θℓ) = exp(j2πνℓt)C2(Ω2,ℓ)AℓC T 1 (Ω1,ℓ)u(t − τℓ), (3.11) with C2(Ω2,ℓ) = c2,1(Ω2,ℓ) c2,2(Ω2,ℓ) (3.12) C1(Ω1,ℓ) = c1,1(Ω1,ℓ) c1,2(Ω1,ℓ) (3.13) Aℓ = αℓ,1,1 αℓ,1,2 αℓ,2,1 αℓ,2,2 = [αℓ,p2,p1 ] (3.14) u(t) = [u1(t), . . . , uM(t)]T . (3.15) The equation (6.13) can be recast as following s(t; θℓ) = exp(j2πνℓt) · αℓ,1,1c2,1(Ω2,ℓ)c T 1,1 (Ω1,ℓ) + αℓ,1,2c2,1(Ω2,ℓ)c T 1,2 (Ω1,ℓ) +αℓ,2,1c2,2(Ω2,ℓ)c T 1,1 (Ω1,ℓ) + αℓ,2,2c2,2(Ω2,ℓ)c T 1,2 (Ω1,ℓ) u(t − τℓ) (3.16) = exp(j2πνℓt) · X 2 p2=1 X 2 p1=1 αℓ,p2,p1 c2,p2 (Ω2,ℓ)c T 1,p1 (Ω1,ℓ) u(t − τℓ). (3.17) 3.3.1 Model for time-division-multiplexing channel sounding Channel sounding by using multiple Tx and Rx antennas can be conducted by using RF-switch which connects single Tx antenna with the transmit front-end chain, or Rx antenna with the receiver front-end chain sequentially. We call this kind of sounding technique as time-division-multiplexing (TDM) sounding technique. Examples of the measurement equipments using the TDM sounding systems are the PROPsound, RUSK, and rBECS. We consider a widely used TDM structure depicted in Figure 3.9 to construct the signal model. The m1th antenna element of Array 1 is active during the sounding windows1 q1,m1 (t) = X I i=1 qTt (t − ti,m1 + Tg), m1 = 1, ..., M1, (3.18) where i denotes the cycle index and ti,m1 = (i − 1)Tcy + (m1 − 1)Tt. Here q1,m1 (t) is a real function, with value of 1 or 0 corresponding to the active or inactive moments of the m1th window. Let us define the sounding window vector q1(t) .= [q1,1(t), ..., q1,M1 (t)]T . 1Remarks: If another ordering of polarization sounding/sensing is used, the sounding windows q1(t) and the sensing windows q2(t) need merely to be appropriately redefined.
52 Generic channel models The so-called sensing window me(t-tm1.m,m2=1,,M,m1=1,M corresponds to the case where .The mth Tx antenna is active; The math Rx antenna is sensing. where t.mm=(-1Tg+(m-1T+(m2-1)T, The sensing window for the math Rx dual antenna is given by the real function q1.md-∑∑rt-t.m2m) (3.19 m= We can define the sensing window vector q2()=21(),2.2t)P. as well as 3.20) 3.3.2 Transmitted signal u()=q1(t)u(). (3.21) 3.3.3 Received signal The signal at the output of Switch 2 can be written as ((0w(. (3.22) with s(t:0c)=exp(j2vt)(t)C2()A.C(L)g(t-Te)u(t-Te). (3.23) Implementing(6.18),we can rewrite 8助=pU2-(o(ei(0 Pamlpim1 u(t-r). (3.24) li e ee the ne mm ension of the first equation in (7)in Fleury et al.(2002)to incorporate polarization. U(t:n)=q2(t)a(t)"u(t-r) (3.25) With this definition,(6.26)can be further written as (a.U(t:T)(S) (3.26)
52 Generic channel models The so-called sensing window qTsc (t − ti,m1,m2 ), m2 = 1, . . . , M2, m1 = 1, . . . , M1 corresponds to the case where • The m1th Tx antenna is active; • The m2th Rx antenna is sensing, where ti,m2,m1 = (i − 1)Tcy + (m1 − 1)Tt + (m2 − 1)Tr. The sensing window for the m2th Rx dual antenna is given by the real function q1,m2 (t) = X I i X M1 m1=1 qTsc (t − ti,m2,m1 ). (3.19) We can define the sensing window vector q2(t) .= [q2,1(t), ..., q2,M2 (t)]T . as well as q2(t) = X I i=1 X M2 m2=1 X M1 m1=1 qTsc (t − ti,m2,m1 ). (3.20) 3.3.2 Transmitted signal Making use of the sounding window vector q1(t), we have the explicit transmitted signal u(t) by concatenating the inputs of the M1 elements of Array 1 u(t) = q1(t)u(t). (3.21) 3.3.3 Received signal The signal at the output of Switch 2 can be written as Y (t) = X L ℓ=1 q T 2 (t)s(t; θℓ) + r No 2 q2(t)W(t), (3.22) with s(t; θℓ) = exp(j2πνℓt)q T 2 (t)C2(Ω2,ℓ)AℓC1(Ω1,ℓ) T q1(t − τℓ)u(t − τℓ). (3.23) Implementing (6.18), we can rewrite s(t; θℓ) = exp(j2πνℓt) · X 2 p2=1 X 2 p1=1 αℓ,p2,p1 q T 2 (t)c2,p2 (Ω2,ℓ)c T 1,p1 (Ω1,ℓ)q1(t) · u(t − τℓ). (3.24) Expression (6.26) is the extension of the first equation in (7) in Fleury et al. (2002) to incorporate polarization. Following the same approach as in this paper, we define the M2 × M1 sounding matrices U(t; τℓ) = q2(t)q1(t) Tu(t − τℓ). (3.25) With this definition, (6.26) can be further written as s(t; θℓ) = exp(j2πνℓt) X 2 p2=1 X 2 p1=1 αℓ,p2,p1 c T 2,p2 (Ω2,ℓ)U(t; τℓ)c1,p1 (Ω1,ℓ). (3.26)
Generic channel models 53 We can also express s()as (3.27 where smm(：化0)兰.pa.:exp(j2rvut)cn(L2t)U(tT)c1p(L1） 3.28) an expression similar to(7)in Fleury et al (2002). 3.4 Dispersive-path model 3.4.1 Motivation for proposing dispersive-path model along uncorrelated paths.The parameters of thes paths have differences larger than the intrinsicr tion.the obiective function which is maximized in the maximization p is derived based on a single wave signal m SAGE is not capable to separate them.Thus the mism n the non-correlated single-wave model and the h pathsisassumed to be s such as the first order el in Tan et al.(2003)and a d mode 100 nd oue 613nd Bengtsso fitting tec signal model is implemented in the SAGE agorithm.But the detail scheme of the extendedSAtandits performance was not covered by the article 3.4.2 Original model of slightly distributed sources C Le sn(8)=】 3.29 ath and The complex weight e. phase.O coul e thawhere constant in a stationary time-invariant environment. dassume tha the onary tim nvanant e ronment are independent random variables uniformly
Generic channel models 53 We can also express s(t; θℓ) as s(t; θℓ) = X 2 p2=1 X 2 p1=1 sp2,p1 (t; θℓ), (3.27) where sp2,p1 (t; θℓ) .= αℓ,p2,p1 exp(j2πνℓt)c T 2,p2 (Ω2,ℓ)U(t; τℓ)c1,p1 (Ω1,ℓ), (3.28) an expression similar to (7) in Fleury et al. (2002). 3.4 Dispersive-path model 3.4.1 Motivation for proposing dispersive-path model In the current SAGE algorithm, the received signal is assumed to be the superposition of multiple waves propagating along uncorrelated paths. The parameters of these paths have differences larger than the intrinsic resolutions of the measurement equipment, therefore they can be well separated by the SAGE algorithm. In another word, these paths are supposed to be uncorrelated. Based on this assumption, the objective function which is maximized in the maximization step is derived based on a single wave signal model. However in real situation, the propagation paths could be correlated, for instance, the multi-ray scenario introduced in the above topic is one scenario where the multiple paths are sufficiently close that the single-wave model based SAGE is not capable to separate them. Thus the mismatch between the non-correlated single-wave model and the reality of correlation results at the poor performance of the SAGE algorithm. The solution proposed in the former topic is specified for the multi-ray scenario, where the number of correlated paths is assumed to be small. When the number is large, which corresponds to the diffuse scattering scenario, the computational complexity becomes prohibitive for practical implementation. It is therefore necessary to find an appropriate solution for this special case. Diffuse scattering cluster estimation, which is also called slightly spatially distributed source estimation, has received attention recently. Different approaches have been published in recent publications, which can be generally categorized into two classes: 1), finding approximate models for the slightly distributed sources, such as the first order Taylor expansion approximation model in Tan et al. (2003) and a two-ray model proposed by Bengtsson and Ottersten (2000); 2) finding high-resolution estimators for estimating the slightly distributed sources, such as DSPE Valaee et al. (1995), DISPARE Meng et al. (1996) and Trump and Ottersten (1996), and spread root-MUSIC, ESPRIT, MODE Bengtsson and Ottersten (2000). These high-resolution estimators are derived more or less by employing subspace fitting techniques or covariance matrix fitting techniques. Extending the SAGE algorithm for estimating diffuse scattering cluster has been briefly mentioned in an electronic letter Tan et al. (2003). In this paper, an approximation signal model is implemented in the SAGE algorithm. But the detail scheme of the extended SAGE algorithm and its performance was not covered by the article. 3.4.2 Original model of slightly distributed sources The contribution of multiple slightly distributed sources to the received signal at the output of the mth Rx antenna array can be modelled as sm(θ) = X C c=1 X Lc ℓ=1 γc,ℓ · cm(θc,ℓ), (3.29) where C is the number of clusters, Lc is the number of multipaths in cth cluster, m is the data index in the frequency and spatial domain, θ is a parameter vector containing all the unknown parameters in the model, γc,ℓ is the path weight of the ℓth path in cth cluster, cm(θc,ℓ) denotes the response which has the expression as cm(θc,ℓ) .= e −j2π(m−1)(∆s/λ) sin(θc,ℓ) . Here ∆s is the array element spacing, θc,ℓ is the direction-of-arrival (DoA) of the ℓth path in cth cluster, λ is the carrier wavelength. The complex weight γc,ℓ = αc,ℓe jψc,ℓ , where αc,ℓ represents a real-valued amplitude, and ψc,ℓ denotes the initial phase. One could assume that the initial phase is fixed as a constant in a stationary time-invariant environment, then sm(θ) is a deterministic signal. However in some applications, it is difficult to ensure the same initial phase in all snapshots. Thus, it is reasonable to assume that the initial phases are independent random variables uniformly distributed on [−π, π]. Correspondingly γc,ℓ are random variables as well
54 Generic channel models The deviation of he direcion ofrival is calculated aswithdenoting the nominaldirectoof For simplicity reason,we focuson one cluster scenario.So thebo of thethuteris -w (3.30) eom密aeedsyaogeomesaud n=sn(e)+wn (3.31) d sm(。)=ecm(0e, (3.32) whereand are respectively the complex attenuation and the incident angle of the cth spectral wave. compare th performance of the estimatorusin the proposed approximate models with the one mate m 3.4.3 First-order Taylor expansion modelI The first order Taylor expansion with respect to the spread of the parameter is a6。+d≈ao)+daal where represents the nominal value and denotes the spread.Applying this principle to (3.30),we obtain an approximation model for slightly distributed sources as -低+i 侧+u2 (3.33) ndthe ne pdof theted
54 Generic channel models The deviation of the direction of arrival is calculated as ˜θc,ℓ = θc,ℓ − θc with θc denoting the nominal direction of arrival of all rays. For some applications, ˜θc,ℓ are assumed to be constant for multiple channel realizations. However they can be also assumed to be random variables, following approximately Gaussian distribution N ∼ (0, σ2 θ ). For simplicity reason, we focus on one cluster scenario. So the contribution of the cth cluster is sm(θc) = X Lc ℓ=1 γc,ℓ · cm(θc,ℓ), (3.30) Assuming that the transmitted signal u(t) is unitary one, the received signal originating from the slightly distributed source and additive white Gaussian noise can be described as xm = sm(θc) + wm = X Lc ℓ=1 γc,ℓ · cm(θc,ℓ) + wm, (3.31) where w is complex circularly symmetric additive white Gaussian noise with variance of σ 2 w. Traditionally the slightly distributed source is approximated with spectral wave. We call this approximation model as spectral wave model (SWM). The signal contribution of spectral wave at the output of the mth antenna can be written as sm(θc) = γc · cm(θc), (3.32) where γc and θc are respectively the complex attenuation and the incident angle of the cth spectral wave. Here we propose another two models that are used to approximate the slightly distributed source. In the simulation study section, we compare the performance of the estimators using the proposed approximate models with the one using the SWM model. 3.4.3 First-order Taylor expansion model I The first order Taylor expansion with respect to the spread of the parameter is a(φo + φ˜) ≈ a(φo) + φ˜ ∂a(φo) ∂φo , where φo represents the nominal value and φ˜ denotes the spread. Applying this principle to (3.30), we obtain an approximation model for slightly distributed sources as sm(θc) = X Lc ℓ=1 γc,ℓ · cm(θc,ℓ) = X Lc ℓ=1 γc,ℓ · cm(θc + ˜θc,ℓ) ≈ X Lc ℓ=1 γc,ℓ · cm(θc) +X Lc ℓ=1 γc,ℓ · ˜θc,ℓ · ∂cm(θc) ∂θc , (3.33) where θc is the nominal direction of arrival and ˜θc,ℓ is the angle spread of the ℓth wave in the cth distributed source. Introducing parameters γc = X Lc ℓ=1 γc,ℓ, ψc = X Lc ℓ=1 γc,ℓ · ˜θc,ℓ