26 Characterization of Propagation Channels h(t,r) Figure 2.5 A diagram of a signal system represented for a wireless system with multipath fading channels. .Time-variant direction-spread impulse response (channel impulse response)h(t,,) .Time-and space-variant impulse response s(t,,r) .Direction-Doppler-spread transfer function H(pf) .Doppler-spread space-variant transfer function T) .Doppler-direction-spread impulse response(spread function)(fp,,) .Doppler-spread space-variant impulse response m(f,) .Time-variant direction-spread transfer function M(f) .Time-space-variant transfer function G(t.f..r) where t denotes nctions is shown 3.4.The is upon differen requency-pace domain.Ho ever,since the channel impulse response ,t,到=∑-50-n)） (2.4) of resolvable multipat cmvelope)fmer ofths and an beas (2.5) where N is the number of multipaths,()denotes a time-variant amplitude,and()is the time-variant phase
26 Characterization of Propagation Channels Multipath Figure 2.5 A diagram of a signal system represented for a wireless system with multipath fading channels. • Time-variant direction-spread impulse response (channel impulse response) h(t, τ′ , Ω) • Time- and space-variant impulse response s(t, τ′ , x) • Direction-Doppler-spread transfer function H(fD, fc, Ω) • Doppler-spread space-variant transfer function T (fD, fc, x) • Doppler-direction-spread impulse response (spread function) g(fD, τ′ , Ω) • Doppler-spread space-variant impulse response m(fD, τ′ , x) • Time-variant direction-spread transfer function M(t, fc, Ω) • Time-space-variant transfer function G(t, fc, x) where t denotes the time, τ ′ designates the time delay, fc and fD are the carrier frequency and Doppler shift, respectively, x denotes the location of an antenna element in the antenna array in the Tx/Rx, and Ω is the direction of a antenna element in the antenna array in the Tx/Rx including both azimuth angle φ and elevation angle θ. The Fourier relationship between the system functions is shown in FIGURE 3.4. These system functions lay emphasis upon different aspects of the channels. For example, the channel impulse response h(t, τ′ , Ω) focuses on the description of channels in the time-direction domain, while the time-space-variant transfer function G(t, fc, x) describes the channels in the frequency-space domain. However, since the channel impulse response h(t, τ′ , Ω) can directly relate the multipath components, it is the most often-used system function and thus will be mainly used throughout this book. Based on the basic knowledge of signals and systems ? and the above introduced knowledge of multipath fading, a wireless system shown in FIGURE 3.1 can be represented as a general signal system as shown in FIGURE 3.5. In this case, the channel impulse response h(t, τ′ , Ω) can be expressed by h(t, τ′ , Ω) = X L l=1 hl(t)δ(τ ′ − τ ′ l )δ(Ω − Ωl) (2.4) where L is the total number of resolvable multipath components, hl(t) is the time-variant complex fading envelope associated with the lth resolvable multipath component arriving with an average time delay τ ′ l and an average direction Ωl . Each time-variant complex fading envelope hl(t) consists of a number of multipaths and can be expressed as hl(t) = X N n=1 cn(t)e −jφn(t) (2.5) where N is the number of multipaths, cn(t) denotes a time-variant amplitude, and φn(t) is the time-variant phase
Characterization of Propagation Channels 27 La. &-cos (a) (c) Figure2.Typical wireless communication scenarios,(a)fixed Tx and Rx;(b)movingRx;(c)multiple antennas
Characterization of Propagation Channels 27 ˄a˅ (b) (c) Figure 2.6 Typical wireless communication scenarios, (a) fixed Tx and Rx; (b) moving Rx; (c) multiple antennas
28 Characterization of Propagation Channels a=Geo。 (2.6) From FIGURE 3.6(a),it is clear that the phase is caused by the multipaths and can be shown as 4=2号 2.7 distance diffe the epet us now consider that the is moving and thus the complex fading envelope is time-varant with 2.8 In this case,as shown in FIGURE 3.6(b),the travel distance difference consists of two parts,i.e.,the travel distance bnd th金accd7 ehedhe phae c四 0=2红号=2a1+tem (2.9) 0-0e0 (2.10) e the nna in the y a in the ry multipaths,the motion of the Rx,and multiple antennas.In this case,the phase can be expressed as ()co(aa) (2.110 aaaa6.netae 9g=24-业+2=fnms0j+2=X4elo） (2.12) er f is clear that the 2.4 Stochastic characterization of multipath fading The feasible manner to characterize the multipath fading channel is to characterize its statistics. nd ofre is only given by the mu
28 Characterization of Propagation Channels Since multipath fading appears only over distance of the order of the wavelength, the fast variations of the received signal power due to the multipath fading are mainly because of the change of phase φn(t). We will show in what follows how the phase φn(t) is obtained. Let us start from the most simple scenario, where we consider a fixed Tx and Rx as shown in FIGURE 3.6(a). In this case, the complex fading envelope is time-invariant and can be expressed as hl = X N n=1 cne −jφn . (2.6) From FIGURE 3.6(a), it is clear that the phase φn is caused by the multipaths and can be shown as φn = 2πfc d ′ c (2.7) where fc is the carrier frequency, c is the speed of light, and d ′ = dn − dmin denotes the travel distance difference between the travel distance dn via the nth scatterer from the Tx to the Rx and the minimum travel distance dmin from the Tx to the Rx. Let us now consider that the Rx is moving and thus the complex fading envelope is time-variant with the expression as hl(t) = X N n=1 cn(t)e −jφn(t) . (2.8) In this case, as shown in FIGURE 3.6(b), the travel distance difference consists of two parts, i.e., the travel distance difference caused by multipaths and the travel distance difference caused by the motion of the Rx, and the phase can be expressed as φn(t) = 2πfc d ′′ c = 2πfc d ′ n + νt cos(θn) c (2.9) where ν denotes the moving velocity and θn is the angle of incidence. If we consider multiple antennas scenarios as shown in FIGURE 3.6(c), the time-variant complex fading envelope can be expressed as h pq l (t) = X N n=1 c pq n (t)e −jφpq n (t) (2.10) where the superscript (·) pq means the link between the pth antenna in the Tx and the qth antenna in the Rx. From FIGURE 3.6(c), the travel distance difference includes three parts, i.e., the travel distance difference caused by the multipaths, the motion of the Rx, and multiple antennas. In this case, the phase can be expressed as φ pq n (t) = 2πfc d ′′′ c = 2πfc d ′′ + ε cos(αn) c (2.11) where ε denotes the antenna element spacing and αn is the radiation angle as shown in FIGURE 3.6(c). Therefore, in total, the phase φ pq n (t) in (2.11) actually includes three parts and can be expressed completely as φ pq n (t) = 2πfc dn − dmin c + 2πfD cos(θn) + 2πλ−1 ε cos(αn) (2.12) where fD = fc ν c is the maximum Doppler frequency and λ = fc c is the wavelength. From (2.12), it is clear that the complete phase shift includes three terms, which can be named as the multipath-induced phase shift, the motioninduced phase shift, and the multiple antenna-induced phase shift, respectively. 2.4 Stochastic characterization of multipath fading The characterization of the multipath fading channel is essential for understanding and modelling it properly. Due to the huge number of factors that influence the channel, a deterministic characterization is not possible. The only feasible manner to characterize the multipath fading channel is to characterize its statistics. A full statistical description of the system functions is only given by the multidimensional PDFs of them, which is practically also not feasible. The most important and often-used approximation descriptions are the first-order received envelope and phase PDF of multipath fading and some second-order descriptions of multipath fading, e.g., LCR, AFD, and correlation properties