3 Generic channel models 3.1 Physical propagation mechanisms The reflection,diffraction and scattering are usually considered the three basic gation mechanism.they occu an obstacle much larger than the w e.,reflection occurs.Diffraction happens when the obstacle's Diffraction arises because of the way in which wave propagates.This is described by the Huygens-Fresnel principle and the principle of waves.Th he propaga n of wth e can be Visualized by consi ing every poin the sum of these secondary waves.When waves are added together,their sum is determined by the relative phases and minima. agation considered,by using the high-r esolution parameter estimation algorithm plane waves can be val,of.delay,Doppler queny and complex especially for those with one-bouncen the pro tion.Figure 3.1 illustrates the estimated aths o photographs of th rajectories of the propagation paths are also d picted by o and with DoDs to the left of the sed by th a ade of thebuildings in the environment. Modeling the channel based on different mechanisms has where the uoofhave the constructed Paths Estimated Dire of Departure oD) Figure 3.1 The estimated propagation paths in an outdoor environment
3 Generic channel models 3.1 Physical propagation mechanisms The reflection, diffraction and scattering are usually considered the three basic propagation mechanism. They occur depending on the size L of the object compared with the wavelength λ. When a plane electromagnetic wave encounters an obstacle much larger than the wavelength, i.e. L ≫ λ, reflection occurs. Diffraction happens when the obstacle’s size is in the same order of the wavelength, i.e. L ≈ λ. Diffraction arises because of the way in which wave propagates. This is described by the Huygens-Fresnel principle and the principle of superposition of waves. The propagation of wave can be visualized by considering every point on a wave front as a point source for a secondary spherical wave. The wave displacement at any subsequent point is the sum of these secondary waves. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima. Scattering happens when a plane wave encounters an obstacle much smaller than the wavelength, i.e. L << λ. This obstacle becomes a new source emitting waves towards multiple directions. In the propagation considered, by using the high-resolution parameter estimation algorithm, plane waves can be estimated with their direction of arrival, direction of departure, delay, Doppler frequency and complex attenuation. With knowledge of the locations of the transmitter and the receiver, we are able to reconstruct the propagation paths especially for those with one-bounce in the propagation. Figure 3.1 illustrates the estimated paths overlapping with the photographs of the background. Reconstructed trajectories of the propagation paths are also depicted by overlaying on the map. It can be observed that some paths, for example, no. 1, to 6 paths are due to the diffraction at the edge of building “B3”. The paths no. 9 to 19, with DoAs to the right of upper “DoAs” plot, and with DoDs to the left of the “DoDs” plot, are more caused by the scattering due to the mixture of thin tree stems and also combined with the edges of the buildings. The paths no. 21, 22, 23, may be caused by the reflections of the sculptures and tree, as well as the facade of the buildings in the environment. Modeling the channel composition based on different propagation mechanisms has been performed for channel characterization in the elevation domain for outdoor environments in ?. It was found that the estimated paths are clustered in directions where the building walls, roof edges or trees have line-of-sight (LoS) conditions to both the This is a Book Title Name of the Author/Editor c XXXX John Wiley & Sons, Ltd E Estimated Directions of Arrival (DoAs) Reconstructed Paths Estimated Directions of Departure (DoDs) 25 24 23 22 21 20 19 17 18 16 15 14 13 12 11 10 9 8 5 7 6 4 3 2 1 Azimuth [°] Coelevation [°] 40 30 20 10 0 -10 -20 -30 -40 -50 85 90 95 100 105 110 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 98 674532 1 Azimuth [°] Coelevation [°] 150 100 50 0 -50 -100 -150 60 80 100 120 Relative delay [ns] 0 82 164 246 329 23 22 21 12 25 24 7 Tx 2 1 15 6 5 10 3 4 13 11 16 17 Rx 9 18 14 8 19 20 Figure 3.1 The estimated propagation paths in an outdoor environment
46 Generic channel models the latter paths. 3.2 Channel spread function the measurement equipment in these dimensions.At present,a channel can be estimated in the following dimensions tion or departure. R Figure 3.2 The diagram showing the dispersion of channel in multiple dimensions. The following descipion of hhas been published in Fleury's paper vity,ony the part relevant with e parame d.The reta methat the mponents may be specular path comp A numb or waves a Rx sites,we assume individual coordinate sy tems being specified at an arbitrary origin andO in the region transmits signals can be written as r红，)=oexpfj22r'(xT}ep02=6'(nLxu》epi2mtst-r小 (3.1) n the v()denoe the( the departu direct and the Doppler frequenc of the th impinging In(3.1),(.denotes the A direction is represe ented with a unit vector or direct ction h its nin d as depicted in Pigure 3.3Jakes(197)Watson(19).For direction of departure,the direction has uletely deteninsr rt d on a sphere S2 of uni cated at the origin ORx Srx =e(Tx,Orx)=[cos(orx)sin(0rx),sin(orx)sin(0Tx).cos(0rx)IT ES2. (3.2)
46 Generic channel models base station and the user equipment, i.e. those paths can be considered as one-bounce paths. Among those paths, the specular/reflected paths and the diffracted paths can be further identified. The former paths exhibit less path loss than the latter paths. 3.2 Channel spread function Figure 3.2 shows a diagram of a propagation channel which is considered to be dispersive in multiple dimensions. The dispersion dimensions that are necessary for channel parameter estimation depend on the intrinsic resolution ability of the measurement equipment in these dimensions. At present, a channel can be estimated in the following dimensions: delay, Doppler frequency, direction of arrival (i.e. the azimuth of arrival and the elevation of arrival), direction of departure (i.e. the azimuth of departure and the elevation of departure), polarization matrices, complex attenuation. Tx Moving Scatterer Path 1 Path D Nominal Direction of Departure Nominal Direction of Arrival Rx Scatterer Scatterer Spread in Direction of Departure Spread in Direction of Arrival Specular path Dispersive path Figure 3.2 The diagram showing the dispersion of channel in multiple dimensions. The following description of channel in terms of its spread function has been published in Fleury’s paper Fleury (2000). For brevity, only the part relevant with the parameter estimation is introduced. The readers who are interested at the channel characterization using the spread function may read Fluery’s paper for more details. As illustrated in Figure 3.2, a propagation channel may contain multiple separable components. For simplicity, we assume that these components may be specular path components. A number L of waves are departing from the location of the transmitter (Tx) and impinging in a region surrounding the location of receiver (Rx). For both the Tx and the Rx sites, we assume individual coordinate systems being specified at an arbitrary origin OTx and ORx in the region surrounding the Tx and the Rx respectively. The location of the Tx and the Rx antenna are determined by two unique vectors xTx ∈ R 3 and xRx ∈ R 3 respectively, where R denotes the real line. In the case where L components are all specular path components, the output signal of the Rx antenna located at xRx while the Tx antenna located at xTx transmits signals can be written as r(xTx, xRx;t) = X L ℓ=1 αℓ exp{j2πλ−1 0 (Ωℓ,Tx · xTx)} exp{j2πλ−1 0 (Ωℓ,Rx · xRx)} exp{j2πνℓt}s(t − τℓ). (3.1) In the above expression, s(t) denotes the modulating signal at the input of the transmitter (Tx) antenna and λ0 is the wavelength. The other parameters are, respectively, the complex amplitude αℓ, the delay τℓ, the incident direction Ωℓ,Rx, the departure direction Ωℓ,Tx, and the Doppler frequency νℓ of the ℓth impinging wave. In (3.1), ( · ) denotes the scalar product. A direction is represented with a unit vector Ω. For direction of arrival, the direction has its terminal point located at the origin O of the coordinate system, and its initial point located on a sphere S2 of unit radius centered at OTx as depicted in Figure 3.3 Jakes (1974) Watson (1983). For direction of departure, the direction has its terminal point located on a sphere S2 of unit radius centered at ORx, and its initial point located at the origin ORx. The direction of arrival is uniquely determined by its spherical coordinates (φTx, θTx) according to ΩTx = e(φTx, θTx) .= [cos(φTx) sin(θTx),sin(φTx) sin(θTx), cos(θTx)]T ∈ S2, (3.2) where φTx, θTx represent the azimuth and the co-elevation of the direction of departure respectively. Similarly, the direction of arrival ΩRx is uniquely determined by its spherical coordinate (φRx, θRx). Notice that the αℓ in (3.1) is
Generic channel models 47 ion of the complex clectrie field of the th wave as well as of the R antenna response such s,g,its fiel n(0)cos( Figure .3 Direction of incidence characterized by The summation of individual specular path components as represented in (3.1)can be generalized to the form of integral expression r(Tx,rx:t)=exp{j (Sx))exp(j ()]exp(j2rut)s(t-T)h(,Sax,T,v) (3.3) In the special case where the channel can be decomposed into multiple specular path components.the function h(Tx,SRs,Tv)takes the form h(s,m)=(s-)6(us-Su)6(r-)6(v-) (3.4 the chanel's bidirection-delay-Doppler spread function.Spread funcion has been paper,the delay-Dopplersp ead function is defined.Fleury(2000),the spread function is extended to the direction delay-Doppler scenario.Ir this book,we further extend the spread function inorder toinclude dispersion of a channel he preaboderbe the io ofiernm e部op0rcc se sca on objects with a large geometrical extent.The integ scribes can ementary faces of the scatt The expectation of the spread function is0,i.e. E[h(Tx,Rx,T,v】=0. (3.5) of the to fley 2000).this er consideration
Generic channel models 47 a function of the complex electric field of the ℓth wave as well as of the Rx antenna response such as, e.g., its field pattern. Direction S2 rh rv rz (0, 0, 1) (0, 1, 0) (1, 0, 0) sin(θ) cos(θ) sin(θ) sin(φ) sin(θ) cos(φ) θ φ O Ω Figure 3.3 Direction of incidence characterized by Ω The summation of individual specular path components as represented in (3.1) can be generalized to the form of integral expression r(xTx, xRx;t) = Z Z Z Z exp{j2πλ−1 0 (ΩTx · xTx)} exp{j2πλ−1 0 (ΩRx · xRx)} exp{j2πνt}s(t − τ)h(ΩTx, ΩRx, τ, ν)dΩTxdΩRxdτd (3.3) In the special case where the channel can be decomposed into multiple specular path components, the function h(ΩTx, ΩRx, τ, ν) takes the form h(ΩTx, ΩRx, τ, ν) = X L ℓ=1 αℓδ(ΩTx − ΩTx,ℓ)δ(ΩRx − ΩRx,ℓ)δ(τ − τℓ)δ(ν − νℓ). (3.4) Function h(ΩTx, ΩRx, τ, ν) is called the channel’s bidirection-delay-Doppler spread function. Spread function has been used to denote the dispersive behavior of a time-variant channel considered as a linear system in Bello (1963). In this paper, the delay-Doppler spread function is defined. In Fleury (2000), the spread function is extended to the directiondelay-Doppler scenario. In this book, we further extend the spread function in order to include dispersion of a channel in direction of arrival and direction of departure. The spread function can be used to describe the situation of diffuse scattering, i.e. a large amount of impinging waves arising from the diffuse scattering on objects with a large geometrical extent. The integral expression in (3.3) describes the input-output relationship of the linear system which comprises the Tx antenna, the propagation channel, and the Rx antenna. This linear system is called Radio Channel. The integral expression in (3.3) is only an approximation of the real input-output relationship of the radio channel when the range xTx, xRx and t are small so that the received signal can be considered to be the superposition of infinitesimal elementary plane waves originating from the elementary surfaces of the scattering objects. The expectation of the spread function is 0, i.e. E[h(ΩTx, ΩRx, τ, ν)] = 0. (3.5) Here, E[·] denotes the expectation of the random element given as an argument. According to Fleury (2000), this property is justified from a physical point of view as the phase of the impinging waves can be reasonably assumed to be uniformly distributed between 0 and 2π at the carrier frequencies under consideration
48 Generic channel models Figure 3.4 Power spectrum of direction of Bello (3),the following identity Elh(NTx:SRs:T:v)h(x:x'v)]P(STx:SRx.T.v)6(NTs-rx)6(SRxs -ix)6(T-)6(v-v)(3.6) is justified by the uncorrelated scattering assumption and thus,the zero-mean process)is an OSM.In (3.6), P（Tx,nRx,T,v）=Eh(LTx,Rx,T,w的 (3.7) 、中the,how the ege pgng power h(,S)=h(fs,Sne.T.vdrdv. 3.8) Similarly,the bidirection power spectrum is calculated as Pr,a=∫∫P(.)drdv 3.9 rture spectrum P(Tx;T=60 ns),P(Tx;T =90 ns), The delay-Doppler power spectrum P()can also be calculated
48 Generic channel models 0 10 20 30 40 50 60 70 −120 −100 −80 Delay in sample (sample interval: 5 ns ) Power in dB Figure 3.4 Power spectrum of direction of departure τ = 60 ns The radio channel is called uncorrelated scattering (US) when the bidirection-delay-Doppler spread function is an orthogonal stochastic measure (OSM) (Gihman and Skorohod 1974, Ch. IV). Similar with the notation introduced in Bello (1963), the following identity E[h(ΩTx, ΩRx, τ, ν)h(Ω′ Tx, Ω′ Rx, τ′ , ν′ )] = P(ΩTx, ΩRx, τ, ν)δ(ΩTx − Ω′ Tx)δ(ΩRx − Ω′ Rx)δ(τ − τ ′ )δ(ν − ν ′ ) (3.6) is justified by the uncorrelated scattering assumption and thus, the zero-mean process h(ΩTx, ΩRx, τ, ν) is an OSM. In (3.6), P(ΩTx, ΩRx, τ, ν) = E[|h(ΩTx, ΩRx, τ, ν)| 2 ] (3.7) is called the bidirection-delay-Doppler power spectrum, which describes how the average impinging power is distributed in the dimensions of the bidirection, delay and Doppler frequency. The spread function and the power spectrum of the channel in single dimensions can be calculated by computing the marginal of the multi-dimensional spread function and power spectrum respectively. For example, the spread function in bidirection h(ΩTx, ΩRx) can be computed as h(ΩTx, ΩRx) = Z Z h(ΩTx, ΩRx, τ, ν)dτdν. (3.8) Similarly, the bidirection power spectrum is calculated as P(ΩTx, ΩRx) = Z Z P(ΩTx, ΩRx, τ, ν)dτdν. (3.9) Figure 3.4 to 3.7 depict respectively, the power direction of departure spectrum P(ΩTx; τ = 60 ns), P(ΩTx; τ = 90 ns), P(ΩTx; τ = 145 ns), and P(ΩTx; τ = 300 ns) of a channel obtained from the measurement data obtained in the indoor environment. These power spectra are calculated by using the beamforming method. It can be observed that the spectrum of the noise components appears to be random. For the LoS component which appears at τ = 90 ns, the spectrum is more concentrated on a single direction. In the slope of the power delay profile, e.g. for τ = 145 ns, the channel is more dispersive and concentrated on four directions of arrival. For the second peak which appears at τ = 300 ns, the channel becomes concentrated again. The delay-Doppler power spectrum P(τ, ν) can also be calculated
Generic channel models 49 00 Figure 3.5 Power spectrum of direction of departure=115 ns 100 Figure 3.6 Power spectrum of direction of departure=145 ns 100 Figure 3.7 Power spectrum of direction of departure=300ns
Generic channel models 49 0 10 20 30 40 50 60 70 −120 −100 −80 Delay in sample (sample interval: 5 ns ) Power in dB Figure 3.5 Power spectrum of direction of departure τ = 115 ns 0 10 20 30 40 50 60 70 −120 −100 −80 Delay in sample (sample interval: 5 ns ) Power in dB Figure 3.6 Power spectrum of direction of departure τ = 145 ns 0 10 20 30 40 50 60 70 −120 −100 −80 Delay in sample (sample interval: 5 ns ) Power in dB Figure 3.7 Power spectrum of direction of departure τ = 300 ns
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
