Lecture 2. Fading Channel
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1 1 Lecture 2. Fading Channel Characteristics of Fading Channels Modeling of Fading Channels Discrete-time Input/Output Model
2 2 Radio Propagation in Free Space Speed: c = 299,792,458 m/s Isotropic Received power at a particular location decays with distance: P ~ r -2 T Transmitted signal: s() t cos2 ft R Received signal: yt () cos 2 f ( t r/ c) r
3 Radio Propagation in Terrestrial Environment 3 T Distance between transmitter and receiver: r R Transmission Power: P t Pr P t Received Power: P r Fluctuation around the local mean: small-scale fading (fixed r) Fluctuation of the local mean: large-scale fading
4 4 Large-scale Fading Large-scale fading -- Log-normal shadowing Attributed to the random variation of propagation environment. Empirically modeled as a log-normal random variable with mean and variance 2. Mean is determined by the path loss: ~ r -, >2. log( P / P) r t is called path-loss factor, which depends on the propagation environment. The higher the density of obstacles, the larger. log(r)
5 Small-scale Fading 5 Small-scale fading Attributed to 1) multiple arriving paths; and 2) movements of the transmitter and/or receiver.
6 A Simple Two-path Model (1) 6 Reflecting wall, fixed antenna Transmitted signal: s() t cos2 ft T r d R wall Received signal: yt () cos 2 f ( t r/ c) r cos 2 f ( t (2 d r) / c) 2d r The phase difference between the two waves: 2 f (2 d r) 2 fr c c 4 f ( d r) c What happens if changes from to 2? =2k: Constructively add =(2k+1): Destructively add
7 Coherence Bandwidth vs. Delay Spread 7 An appreciable change of received signal will be observed if c the frequency of transmitted signal f changes by: 4( d r ) Coherence Bandwidth W c The difference between the propagation delays along the two signal paths is: 2d r r 2( d r) c c c Delay Spread T d 1 Td ~ W c
8 A Simple Two-path Model (2) 8 Reflecting wall, moving antenna T r rt () rvt 0 d R wall Transmitted signal: s() t cos2 ft Received signal: yt () cos2 f ( t r( t)/ c) cos2 f( t(2 d r( t))/ c) rt () 2 d rt () Doppler Shift: D 1 =-fv/c Doppler Shift: D 2 =fv/c cos 2 f ((1 v/ c) tr0 / c) cos 2 f((1 v/ c) t( r0 2 d) / c) r vt 2d r vt sin2 f ( vt / c ( r0 d)/ c)sin2 f ( t d / c) r vt 0
9 Coherence Time vs. Doppler Spread 9 An appreciable change of received signal will be observed if time t c changes by: 4 fv Coherence Time T c The difference between the Dopper shifts of the two paths is: Tc ~ 1 D s Ds D2 D1 2 fv/ c Doppler Spread D s
10 Modeling of Fading Channels 10 Transmitted signal s(t) Channel Received signal y(t) A fading channel can be modeled as a Linear Time-Varying system. yt () h(,) t s( t ) d Suppose that h(,t) is a deterministic function of time t and delay. Time-variant transfer function: Delay Doppler function: Doppler-variant transfer function: () denotes Fourier transform. H( f, t) ( h(, t)) S(, ) ( h(, t)) t D( f, ) ( S(, )) t ( H( f, t))
11 Modeling of Fading Channels 11 Transmitted signal s(t) Channel Received signal y(t) A fading channel can be modeled as a Linear Time-Varying system. yt () h(,) t s( t ) d Suppose that h(,t) is a WSS process with uncorrelated scatterers: The autocorrelation function * R ( h, 1 t, 1, 2 t ) 2 E[ h(, t ) h (, t )] R (, t) Time-variant transfer function: Delay Doppler function: R ( f, t) ( R (, t)) R (, ) ( R (, t)) S t h Doppler-variant transfer function: R ( f, ) ( R (, )) ( R ( f, t)) H D S h t h H
12 Coherence Bandwidth vs. Delay Spread 12 Let t 0: R (,0) H f 1 () f R (,0) h () Coherence Bandwidth W c T d ~ 1 W c Delay Spread T d Flat fading: Signal Bandwidth W << Coherence Bandwidth W c Symbol Time T >> Delay Spread T d Frequency-selective fading: Signal Bandwidth W >> Coherence Bandwidth W c Symbol Time T << Delay Spread T d
13 Coherence Time vs. Doppler Spread 13 Let 0: R R (0, ( t) ) h t () t R S( f ) S (0, ) 1 () Coherence Time ~ T T c c D 1s Doppler Spread D s Slow fading: Symbol Time T << Coherence Time T c Signal Bandwidth W >> Doppler Spread D s Fast fading: Symbol Time T >> Coherence Time T c Signal Bandwidth W << Doppler Spread D s
14 Discrete-time Input/Output Model 14 yt () h(,) t s( t ) d s[ n] ai( t) T ( t i( t) nt ) h(,) t a ()( t ()) t i i a i (t): attenuation at time t of path i i (t): propagation delay at time t of path i i n i 0 0 st () sn [ ] ( tnt) n T 0 0 () T t : modulation pulse 0 Sampled output at t=mt 0 : Let l=m-n. y[ m] s[ n] a ( mt ) (( mn) T ( mt )) n i i 0 T 0 i 0 0 h i 0 T0 0 i 0 l[ m ] y[ m] s[ ml] a ( mt ) ( lt ( mt )) l Discrete-time Input/Output Model: ym [ ] hl [ msm ] [ l] z[ m] i l
15 Discrete-time Input/Output Model L-1 ym [ ] hl [ msm ] [ l] z[ m] l=0 15 How many channel filter taps can be obtained (how to determine the value of L)? Delay spread: Sampling rate: W T max d i j i, j Flat fading: L=1 Frequency-selective fading: L>>1 L WT d How fast does the channel filter tap gain h l [m] vary with time in one symbol time T? Depends on the coherence time T c. Slow fading: h l [m] can be regarded as constant in T Fast fading: h l [m] varies with time in T
16 More about h l [m] 16 Without line-of-sight (LOS) paths: h l [m] can be modeled as a zero- 2 mean complex Gaussian random variable (0, ). The magnitude follows Rayleigh distribution. s With one LOS path: h l [m] can be modeled as h m k 1 k1 k1 j 2 l[ ] se (0, s) k: the ratio of the power in the LOS path and that in the scattered paths. The magnitude follows Ricean distribution.
17 More about h l [m] 17 Availability of Channel State Information (CSI) CSIR: CSI is available at the receiver side. available through channel measurement required for coherent detection CSIT: CSI is available at the transmitter side. available through channel measurement and feedback optional Ergodicity indispensable condition to achieve the Shannon capacity may not hold in delay-limited scenarios
18 Summary 18 Fading Channel Large-scale fading Path loss Shadowing Small-scale fading Delay spread (caused by multipath) Doppler spread (caused by mobility) Flat fading Frequency-selective fading Slow fading Fast fading Discrete-time input/output model L 1 y[ m] h[ m] s[ ml] z[ m] l0 l
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