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