2 Characterization of Propagation Channels a ysis of any neoosaadndomspeoadosp coya r() N Figure 2.1 A wireless communication system consisting of three parts
2 Characterization of Propagation Channels In general, any wireless communication systems include three parts, i.e., transmitter (Tx), receiver (Rx), and wireless channel in between to connect them, as shown in FIGURE 3.1. Unlike the Tx and Rx, which can be designed to make the system present better tradeoff between reliability and efficiency, the wireless channel cannot be engineered. However, reliable knowledge of the propagation channel serves as the enabling foundation for the design and analysis of any wireless communication system. Various concepts and definitions of the wireless channel usually make the beginner get lost. This chapter will manage to provide a unified and conceptually simple explanation of a morass of concepts for wireless channels. This is a Book Title Name of the Author/Editor c XXXX John Wiley & Sons, Ltd Figure 2.1 A wireless communication system consisting of three parts
22 Characterization of Propagation Channels 2.1 Three Phenomena in Wireless Channels is what wireless channels ding can path oss and Path loss,P,and shadowing,S,belong to large scale fading since they are dominant when the mobile station moves over distance al tens of wavelengths wavelengths.Shadowingis the slow variations of the received signal power over distances of several ten or hundred of the T and Rx.and are observed over distance of the order of the wavelength g=P.S.h (2.1) where P is path loss,s is shadowing,and h is multipath fading.It must be noted that throughout this book,we
22 Characterization of Propagation Channels 2.1 Three Phenomena in Wireless Channels Wireless channels are the real environments, in which the Tx and Rx are operating. Fading is what wireless channels bring to us. Fading refers to the time variation of the received signal power induced by changes in the transmission medium or path. Generally speaking, fading can be categorized as large-scale fading, consisting of path loss and shadowing, and small-scale fading. Therefore, in total we have three phenomena in wireless channels. Path loss, P, and shadowing, S, belong to large-scale fading since they are dominant when the mobile station moves over distances of several tens of wavelengths. As shown in FIGURE 3.2, path lose P means the attenuation in the transmitted signal while propagating from the Tx to Rx and is observed over distance of several hundred or thousand wavelengths. Shadowing S is the slow variations of the received signal power over distances of several ten or hundred wavelengths due to large terrain features such as buildings and hills. Large-scale fading is very important for the system design at the network level. For example, the cell coverage area, outage, and handoffs are influenced by these effects. On the other hand, small-scale fading appears due to the multipath propagation. As shown in FIGURE 3.2, multipath fading, h, refers to fast variations of the received signal power due to the constructive and destructive interference of the multiple signal paths between the Tx and Rx, and are observed over distance of the order of the wavelength. Small-scale fading plays an important role in determining the link level performance according to bit error rates, average fade durations, etc. Therefore, as shown in FIGURE 3.1, to completely characterize wireless channels, we can use the following expression g = P · S · h (2.1) where P is path loss, S is shadowing, and h is multipath fading. It must be noted that throughout this book, we constrain our interests in the investigation and modeling of small-scale fading for different types of channels, e.g., F2M channels, V2V channels, cooperative MIMO channels, etc
Characterization of Propagation Channels 23 Mutipath fading Path loss Shadowing Distance between the Tx and Rx Path Loss,PShedowisg,.合Mtipsth Fading,工+g=P.S.h Large Sehle Fading Small Schle Fading Figure 2.2 Three phenomena in wireless channels
Characterization of Propagation Channels 23 Figure 2.2 Three phenomena in wireless channels
24 Characterization of Propagation Channels 2.2 Path loss and shadowing Path loss is the attenuation in the transmitted sign al while pr ation ie cauced PR-PrG1GRD 入2 2.2) e the smit and re elv A is the A.Shorter the wavelength,higher the path loss. e p path loss in different propagation environments such as urban,rural,and indoor areas.Experiments show that the pootieaearohn8etmghieheatrgnuaioathanfie-pacepropagaioncondions.A ving can be modeled as a log-normal random variable,which is consistent with shown shadowing is due to the pow wer loss blocked by big objects,e.g high bu 1n8 e know tha the shad wing fulfills norm al distribution,ie.,Gaussian distribution,in the log domain,and thus the (2.3 206 can befod of the sh wing.Typic 2.3 Multipath fading agation mechanism manifested when the transmitted signal reaches the Rx by two ns and buildings,ofter path 0) obstructsa direct waves must pr ays.The muliple waves combine vectorially at the receiver antenna s mentioned above.the presence of local scatterers gives rise to NLoS scenarios,where Rayleigh distribution is the most popular distribution used to describe the Input delay-spread function (channel impulse response)) Output doppler-spread function)
24 Characterization of Propagation Channels 2.2 Path loss and shadowing Path loss is the attenuation in the transmitted signal while propagating from the Tx and Rx. This attenuation is caused by the effects such as free-space loss, refraction, diffraction, and reflection. Significant variations in the path loss are observed over distance of several hundred or thousand wavelengths. The simplest path loss model corresponds to a propagation in free space, i.e., line-of-sight (LoS) link between the Tx and Rx. In this case, the received signal power can be expressed as ? PR = PT GTGR λ 2 4πD2 (2.2) where PT is the transmitted power, GT and GR are the transmit and receive antenna gains, respectively, λ is the carrier wavelength, and D is the distance between the Tx and Rx. Note that the path loss exponents (i.e., the power of the distance dependence D) are 2 for free-space propagation. Therefore, the received power decreases with a factor of distance-squared under free-space propagation. (2.2) also shows the path loss dependency on the carrier wavelength λ. Shorter the wavelength, higher the path loss. However, in a real environment, the wireless signals seldom experience the free-space propagation. Therefore, several different models such as the Okumura-Hata ??, Lee ?, Walfish-Ikegami ?, etc., have been proposed to model path loss in different propagation environments such as urban, rural, and indoor areas. Experiments show that the actual path loss exponents are around 3–8, suggesting higher attenuation than free-space propagation conditions. A detailed description of different path loss models can be found in ?. The aforementioned path loss models assume that the path loss is constant at a given distance. However, the presence of obstacles, e.g., buildings and trees, leads to random variations of the received power at a given distance. This effects is termed shadowing (shadow fading). Experiments illustrate that shadowing can be modeled as a log-normal random variable, which is consistent with our intuition. As shown in FIGURE 3.2, shadowing is due to the power loss blocked by big objects, e.g., high buildings. Therefore, the total power loss, i.e., shadowing, can be calculated by multiplying every power loss caused by big objects. In log domain, the multiplication becomes the addition of every power loss. Based on the central limit theory ?, we know that the shadowing fulfills normal distribution, i.e., Gaussian distribution, in the log domain, and thus the shadowing can be modeled as a log-normal distribution. Therefore, the shadowing distribution is given by ? fΩp (x) = 10 xσΩp √ 2π ln 10 exp " − 10 log10 x − µΩp(dBm) 2 2σ 2 Ωp # (2.3) where Ωp denotes the mean squared envelope level, µΩp designates the area mean expressed in dBm, and σΩp is the standard deviation of the shadowing. Typical values of σΩp range from 5 to 10 dB. Detailed discussions of shadowing can be found in ?. 2.3 Multipath fading Multipath propagation is the propagation mechanism manifested when the transmitted signal reaches the Rx by two or more paths. The presence of local scatterers, e.g., mountains and buildings, often obstructs a direct wave path between the Tx and Rx (i.e., the LoS). Therefore, a non-LoS (NLoS) propagation path will appear between the Tx and Rx. Consequently, the waves must propagate through reflection, diffraction, and scattering. This results in the received waves from various directions with different delays. The multiple waves combine vectorially at the receiver antenna (a phenomenon called multipath fading) to produce a composite received signal. As mentioned above, the presence of local scatterers gives rise to NLoS scenarios, where Rayleigh distribution is the most popular distribution used to describe the fading envelope. Some types of scattering environments have a specular component, i.e., LoS or a strong reflected path. These scattering environments are called LoS scenarios, where Ricean distribution is used to describe the fading envelope. A non-directional channel can be characterised by one of the four system functions also termed as the first set of Bello’s functions ?. These four system functions of non-directional channel are the • Input delay-spread function (channel impulse response) h(t, τ′ ) • Output doppler-spread function H(fD, fc)
Characterization of Propagation Channels 25 h(t.r) G,) Figure 2.3 Fourier relationship between the system functions of non-directional channels. fodm(fp,,到 Fori Trasfomm G,到。t foog(f. Mt,,。 fH(p fe. Figure 2.4 Fourier relationship between the system functions of directional channels. Delay-Doppler-spread function(spread function)(fp.) .Time-variant transfer function G(t,f) Wheresfoeeight system functions?,which are extended from the traditional four system fnctionstbxineoporninganothertioimYdomansdirecion sPme.needConsndeinhthe system eaking.double-directional channel description)is ver ns here we gi dired onal channels and invite interested readers to refer to ??The eight system functions of
Characterization of Propagation Channels 25 Figure 2.3 Fourier relationship between the system functions of non-directional channels. Figure 2.4 Fourier relationship between the system functions of directional channels. • Delay-Doppler-spread function (spread function) g(fD, τ′ ) • Time-variant transfer function G(t, fc) where t denotes the time, τ ′ designates the time delay, fc is the carrier frequency, and fD is the Doppler shift. The Fourier relationship between the system functions is shown in FIGURE 3.3. Whereas for a directional channel, eight system functions ?, which are extended from the traditional four system functions by incorporating another two terms/domains (direction and space), can be used. Considering that the system functions of directional channels include those of non-directional channels as special cases and directional channel description (strictly speaking, double-directional channel description) is very useful for MIMO systems, here we give a brief overview of directional channels and invite interested readers to refer to ??. The eight system functions of directional channels are the
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
Characterization of Propagation Channels 29 2.4.1 Received envelope and phase distribution Received Envelop Distribution distribution.In this case,the complex envelope))has a Rayleigh distribution at any =ep-云}x20 (2.13) whereis the average envelope power. fading envelope can be expressed as ronments have a specular or Los component.In this case,the complex (2.14) =荒m20 (2.15) listic and flexible models are developed models,Nakagami and Weibull models can mimic Ricean fadir Rayleigh fading.and also worse than Rayleigh fading.We invite interested readers to refer to Parsons(2000);Stuber(2001) ●Phase Distribution The phase of the complex fading envelope()(t)+(t)can be also expressed as 0=an-l(h h(©1 (2.16) fee=2元-≤x≤ (2.17刀 2.4.2 Envelope level cross rate and average fade duration(3.4.2) envelope c Le(ra)=v2(K+1)fDpe-K-(K+I o(2pVK(K+1)) (2.18)
Characterization of Propagation Channels 29 2.4.1 Received envelope and phase distribution • Received Envelop Distribution In many wireless communication scenarios, the composite received signal consists of a large number of plane waves as expressed in (2.5). According to the central limit theory ?, the PDF of complex fading envelope hl(t) in (2.5) fulfills the complex Gaussian distribution. In this case, the complex envelope c(t) = |hl(t)| has a Rayleigh distribution at any time t, as shown in following fc(x) = 2x Ωp exp{− x 2 Ωp } x ≥ 0 (2.13) where Ωp = E[c 2 ] is the average envelope power. Some types of wireless communication environments have a specular or LoS component. In this case, the complex fading envelope can be expressed as hl(t) = r K K + 1 cLoSe −jφLoS + r 1 K + 1 X N n=1 cn(t)e −jφn(t) (2.14) where K is the Ricean factor and defined as the ratio of the specular power s 2 to scattered power Ωp, cLoS and φLoS are the amplitude and phase of LoS component, respectively. The complex envelope c(t) = |hl(t)| has a Ricean distribution at any time t, as shown in following fc(x) = 2x Ωp exp{−x 2 + s 2 Ωp }I0( 2xs Ωp ) x ≥ 0 (2.15) with s 2 = h I l (t) 2 + h Q l (t) 2 where h I l (t) and h Q l (t) are the inphase and quadrature components of hl(t), respectively, i.e., hl(t) = h I l (t) + h Q l (t). In the modeling history, Rayleigh model and Ricean model are the most popular models to characterize multipath fading due to its simplicity and acceptable accuracy. Nowadays, some more realistic and flexible models are developed, such as Nakagami fading Stüber (2001) and Weibull fading Parsons (2000). By adjusting the parameters of these models, Nakagami and Weibull models can mimic Ricean fading, Rayleigh fading, and also worse than Rayleigh fading. We invite interested readers to refer to Parsons (2000); Stüber (2001) • Phase Distribution The phase of the complex fading envelope hl(t) = h I l (t) + h Q l (t) can be also expressed as φ(t) = tan−1 ( h Q l (t) h I l (t) ). (2.16) The phase is normally non-uniform distribution and has a very complicated form, except for Rayleigh fading, where the phase fulfills uniform distribution over the interval [−π, π], i.e., fφ(t)(x) = 1 2π − π ≤ x ≤ π. (2.17) 2.4.2 Envelope level cross rate and average fade duration (3.4.2) Envelope LCR and AFD are two important second order statistics associated with envelope fading. The LCR, Lc(rg), is by definition the average number of crossings per second that the signal envelope, c(t) = |hl(t)|, crosses a specified level rg with positive/negative slope. Therefore, the LCR means how often the envelope crosses a specified level. Using the traditional PDF-based method ?, the closed form expression of LCR for Ricean fading can be derived as Stüber (2001) Lc(rg) = p 2π(K + 1)fDρe−K−(K+1)ρ 2 I0(2ρ p K(K + 1)) (2.18)
30 Characterization of Propagation Channels 5。几h首动。的 。R民6 L. RUaa:.)g Figure27 Fourier relationship between the rre functions of the non-drection system functions. or the case of Rayligh =0)and isotropic scattering,the above expression can be Le(rg)=v2ppe-. (2.19) Heg6o)anawRoeacatoeheghmteaeeCaneraageteaRoeroayafaionadayodiCke lark in are nor is the average time over which the signal envelope.().remains below a certain level 9 1-Q(V2,V2K+西rg Le(ra) (2.20) ative 2.4.3 Correlation functions erties will discover the important issue about how the multipath fading varies with time,frequency. 之 h ote that the rate of channe Ra,t,)=Eh(,)h(t红， (2.21) Rn(fe,fe2:fo.fp)=EH(fe.fD)H(fe2,fp2) (2.22) Rc(fa,.t)=E[G"(fa,t)G(,a] (2.23) Rg(fDi,fo)=Elg"(ri,fDi)g(r2,fpa)l. (2.24) fcionso in,there double ons as shown in FIGURE 3
30 Characterization of Propagation Channels Figure 2.7 Fourier relationship between the correlation functions of the non-directional system functions. where ρ = rg/ p Ωp. For the case of Rayleigh fading (K = 0) and isotropic scattering, the above expression can be simplified to Lc(rg) = √ 2πfDρe−ρ 2 . (2.19) Here, we would like to highlight that isotropic scattering environments are first mentioned by Clark in Clarke (1968) and refer to ideal scenarios where the scatterers are assumed to be uniformly distributed around the Tx/Rx. Real wireless communication environments are normally non-isotropic scattering scenarios. In this book, we will concentrate on the investigation and modeling of wireless channels for non-isotropic scattering scenarios. The envelope AFD, Tc−(rg), is the average time over which the signal envelope, c(t), remains below a certain level rg. Therefore, the envelope AFD means how long the envelope remains below a specified level. In general, the AFD Tc−(rg) for Ricean fading channels is defined by ? Tc−(rg) = Pc−(rg) Lc(rg) = 1 − Q √ 2K, p 2(K + 1)rg Lc(rg) (2.20) where Pc−(rg) indicates a cumulative distribution function of c(t) with Q(·, ·) denoting the generalized Marcum Q function. By setting K = 0 in (2.20), we can have the envelope AFD for Rayleigh fading case. 2.4.3 Correlation functions Correlation properties will discover the important issue about how the multipath fading varies with time, frequency, and distance. Note that the rate of channel variation has significant impact on several aspects of the communication problem. For the case of non-directional SISO channels, we have the following 4 correlation functions based on 4 system functions introduced in Chapter 3.3 Rh(t1, t2; τ ′ 1 , τ′ 2 ) = E[h ∗ (t1, τ′ 1 )h(t2, τ′ 2 )] (2.21) RH(fc1, fc2; fD1, fD2) = E[H ∗ (fc1, fD1)H(fc2, fD2)] (2.22) RG(fc1, fc2;t1, t2) = E[G ∗ (fc1, t1)G(fc2, t2)] (2.23) Rg(τ ′ 1 , τ′ 2 ; fD1, fD2) = E[g ∗ (τ ′ 1 , fD1)g(τ ′ 2 , fD2)]. (2.24) According to the Fourier relationship between Bello’s system functions as shown in FIGURE 3.3, there is a double Fourier transform relationship between the correlation functions as shown in FIGURE 3.7. In accordance to the aforementioned SISO case, we can define correlation functions of the MIMO system functions, i.e., the directional system functions introduced in Chapter 3.3. This leads again to 8 different correlation functions where the angle-resolved correlation functions are listed as following